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273577

Numerical Condition of Discrete Wavelet Transforms.

The recursive algorithm of a (fast) discrete wavelet transform, as well as its generalizations, can be described as repeated applications of block-Toeplitz operators or, in the case of periodized wavelets, multiplications by block circulant matrices. Singular values of a block circulant matrix are the singular values of some matrix trigonometric series evaluated at certain points. The norm of a block-Toeplitz operator is then the essential supremum of the largest singular value curve of this series. For all reasonable wavelets, the condition number of a block-Toeplitz operator thus is the lowest upper bound for the condition of corresponding block circulant matrices of all possible sizes. In the last section, these results are used to study conditioning of biorthogonal wavelets based on B-splines.

Introduction . Orthogonality is a very strong property. It might exclude other useful properties like, for example, symmetry in the case of compactly supported wavelets [6, 7]. Consequently, in many applications biorthogonal wavelets have been used rather than the orthogonal ones. Stability of such bases has been studied and conditions for Riesz bounds to be finite were established [2, 3, 4, 5]. However, when dealing with applications, one would like to have some quantitative information about sensitivity to such things like noise in the data or quantization. Some relevant estimates can be found in the engineering literature on multirate filter banks, where noise is modelled as a random process and its transmission through the system is studied; see, e.g., [12]. However, most of these results concern particular designs and implementations. Here we will use an alternative approach-we will look at discrete wavelet transforms from the point of view of linear algebra. For example, let us consider the process of image compression using wavelets (see, e.g., [1, 11, 14]). The algorithm has three steps. First, the discrete wavelet transform is applied to the image, then the resulting data is quantized and finally it is coded in some efficient way. The purpose of the transform is to increase the compressibility of the data and to restructure the data so as, after decompression, the error caused by quantizing is less disturbing for a human viewer then if the image was quantized directly without a transform. The encoded image can be manipulated in different ways (e.g., transmitted over network) which can cause further distortions. To decompress the image we just need to decode the data and to apply the inverse transform. Let us denote the error vector that is added to the transformed data y before the reconstruction by u and let us suppose that we know the magnitude of the denotes the original image, the relative error in the reconstructed image is This work was supported by the Flinders University of South Australia, the Cooperative Research Centre for Sensor Signal and Information Processing, Adelaide, and the Australian Government y National Institute of Standards and Technology, Gaithersburg, MD 20899-0001 A If no further assumptions are imposed on the image and type of the error, this estimate is the best possible. Also in other applications, the sensitivity to errors can be shown to be naturally related to the condition number of the transform matrix with respect to solving a system of linear equations, Condition number depends on the norm. For finite matrices we will use here matrix 2-norm, which is induced by the Euclidean vector norm. When necessary, we will use subscript 2 to emphasise that we deal with these norms. We will speak also about condition number of an operator l 2 ( Z Z ). We define it also by (1.1); the norm is the operator norm induced by the norm of l 2 ( Z Due to the translational character of wavelet bases, matrices and operators involved happen to have a characteristic structure-they are block circulant and block- Toeplitz, respectively. This structure can be employed when the condition numbers are computed; Fourier techniques can be used to transform them to a block diagonal form. This then leads to studying the (point-wise) singular values of certain trigonometric matrix series. In (Section 3 we study the finite case. The singular values of a block circulant matrix are shown to be the singular values of small matrices arising from the "block discrete Fourier transform" of the first block row of the block circulant matrix. In (Section 4 we generalize this result for block-Toeplitz operators Z ). Situation is rather more complicated there, because the Fourier transform maps the discrete space l 2 ( Z Z ) onto the functional space L 2 ([0; 2-)). As the main result we show there that ess sup oe where C(A) is the block-Toeplitz operator the infinite matrix of which is generated by strip Z , being square blocks) and The proof is based on the point-wise singular decomposition of A; some difficulties arising from the fact that we have to ensure that the singular vector we want to construct has square integrable components must be overcome on the way. For reasonable wavelets, the curves of singular values of A have some smoothness and essential supremum and infimum become supremum and infimum or even maximum and minimum. Condition number of C(A) is then lowest upper bound on condition of periodized wavelet transforms for all possible lengths of data. We also describe how some particular properties of the wavelets imply a certain structure of the singular values. These observations can be used to further improve the efficiency of computing the condition numbers. In the last section of this paper, we apply this technique to study conditioning of biorthogonal B-spline wavelets constructed by Cohen, Daubechies and Feauveau [5], nowadays probably the most often applied biorthogonal wavelets. We show there NUMERICAL CONDITION OF DISCRETE WAVELET TRANSFORMS 3 that the condition number increases exponentially with the order of the spline. Conditioning can be significantly improved by suitable scaling of the wavelet functions, but, even for the optimal scaling, the growth has exponential character. After finishing the first version of this paper, I became familiar with related works by Keinert [8] and Strang [9]. While Strang's work concerns mostly Riesz bounds for subspaces in a multiresolution analysis and wavelet decomposition, Keinert concentrates on conditioning of finitely sized transforms and asymptotic estimates for deep recursive transforms. He also presents a number of numerical experiments that show how these estimates are realistic when some specific types of introduced errors are considered (e.g., white noise). In this revised version I have tried to emphasise results that are complementary to those of Keinert and Strang. 2. Translational and wavelet bases and the operators of the change of a basis. Let us consider some translation-invariant subspace of L 2 ( R I ) with a translational Riesz basis fu k Zg generated by some r-tuple of functions u being the translation step. Let this subspace have another, similar, basis fv k Z g. Each of the functions v k , can be expressed in the terms of the first basis; there exist sequences Z ) such that r a (k;l) Let us form from these coefficients r \Theta r matrices An , n 2 Z n will be the element of An in kth row and lth column. We denote by A the infinite strip of concatenated matrices An , n 2 Z Z , and we define C(A) to be an infinite block-Toeplitz matrix We will denote by C(A) also an operator l 2 ( Z Z ) that can be represented by such a matrix. If r ff r for some l 2 ( Z to fff n gn2Z Z , that is, it is the operator of the change of a basis. Because of practical reasons (handling of finite data), periodized bases are often used in the wavelet context. If, for some integer N , we denote per A then fu per are bases for some subspace of L 2 ([0; Nh)) and the operator of the change of basis from the latter to the former can be represented by a block circulant matrix SN where A multiresolution analysis is a sequence of embedded subspaces of L 2 ( R I ) generated by the translates of an appropriately dilated scaling function. In particular, Z g: There are wavelet subspaces generated by a wavelet function, Z and these subspaces satisfy The scaling and wavelet function thus have to conform to the two-scale relations that are usually written as In the (fast) discrete wavelet transform, we perform recursively the change of basis from f2 j=2 '(2 Z g to f2 (j \Gamma1)=2 . We can consider both bases to be generated by two functions, former by u 1), the latter by x). The translation step h is here. The recursive inverse transform thus can be associated with repeated applications of C(A) , where that is, In fact, C ( A and the recursive transform itself can be seen as repetitive applications of a block-Toeplitz operator. As in the case of A, ~ NUMERICAL CONDITION OF DISCRETE WAVELET TRANSFORMS 5 sequences f ~ hn gn2Z Z and f~g n gn2Z Z determine the biorthogonal counterparts of the scaling and wavelet function, ~ ' and ~ / by relations analogous to (2.1). Although the conditioning of this basic step of recursive transform is crucial, we want to study also how the error cumulates in the recursive transform. Since all the bases involved have translational character, we can use the same approach as for one step for the transform of any finite depth; we can always find a common translation step. For example, let us consider two steps of recursion. We perform, in fact the change of basis in V j from f2 j=2 '(2 Z g to f2 (j \Gamma2)=2 Z Z g. All these bases can be considered to be translational bases with translation step generated by four functions. We have and The infinite strip A thus will have four rows; the entries can be easily found by recursive applications of (2.1). In particular, if we denote the sequences that form rows of A by fb 0 , we have b (1) b (2) An analogous approach can be used for generalizations of classical wavelet transforms like those based on more than one scaling and wavelet function and general integer dilation parameter m - 2 (multiwavelets, higher multiplicity wavelets) or non-stationary wavelets, where different block-Toeplitz operators applied in the recursive algorithm. Also wavelet packets transforms, where also wavelet spaces are further decomposed, can be described in a similar way. 3. Numerical condition of block circulant matrices. Any circulant matrix is unitarily similar to a diagonal matrix. This matrix has (up to scale) the discrete Fourier transform of the first row of the original matrix on the diagonal and the similarity matrix is the matrix of the discrete Fourier transform itself. This fact can be generalized for block circulant matrices as follows. Theorem 3.1. Each block circulant matrix is unitarily similar to a block diagonal matrix. In particular, CN (A) is similar to a matrix with diagonal blocks equal to Proof. Let !N is the primitive Nth root of unity, us first create the matrix of the "block discrete Fourier transform"; I I I (3. 6 R. TURCAJOV ' A I being the r \Theta r identity matrix. Such a matrix is unitary and the r \Theta r block in row and (n + 1)th block column of\Omega r;N CN (A)\Omega An\Gammal+kN being the Kronecker delta. Since the singular values are preserved by unitary transformations and the singular values of a block diagonal matrix are the singular values of the diagonal blocks, the theorem above has the following corollary. Corollary 3.2. A number oe is a singular value of CN (A) if and only if it is a singular value of A(2-in=N) for some Let us remind here, that the 2-norm of a matrix M equals to its largest singular value, which we will denote here oe max (M ). Similarly, oe min (M) will stand for the smallest singular value, the 2-norm of M \Gamma1 . Corollary 3.3. If N 1 is a divisor of N , cond 2 (CN1 (A)) - cond 2 (CN (A)), because all the singular values of CN1 (A) are simultaneously singular values of CN (A). This means that, for the recursive transform, we could estimate the condition in each step by the condition number of the largest block circulant matrix involved, applied in the first step of the recursion, since in each next step just m-times smaller matrix is used, m being the dilation factor. It would be useful to have some estimate completely independent of the size of the block circulant matrix. One such estimate is straightforward, Notice that if the curves of the largest and smallest singular values are continuous (which happens, for example, for compactly supported wavelets, when A contains only a finite number of non-zero entries) this is the lowest upper bound for cond(CN (A)) independent of N . We will show in the next section that for any reasonable wavelet the right hand side of (3.2) represents, in fact, the condition number of C(A). 4. Norm and condition number of block-Toeplitz operators. Similarly as in the previous section, we will apply here a "block Fourier transform". However, here the situation is a little more complicated than in the case of finite matrices. Let us denote l 2 Z ) the Hilbert space of (column) vectors of length r with all components in l 2 ( Z Z ). We can see this space also as a space of vector-valued sequences. The inner product is r r a NUMERICAL CONDITION OF DISCRETE WAVELET TRANSFORMS 7 subscripts determine entries of sequences, while superscripts entries of vectors. Sim- ilarly, r ([0; 2-)) is the Hilbert space of r-vectors of square integrable functions on [0; 2-) with the inner product r r Z 2-f Z 2-g(-) f(-) d-: To find the norm of the operator C(A) induced by the norm of l 2 ( Z Z ), we employ Hilbert space isomorphisms of these spaces. First, there is a trivial isomorphism between l 2 ( Z Z ) and l 2 Z Second, component-wise Fourier transform is a Hilbert space isomorphism l 2 r ([0; 2-)). For a sequence c 2 l 2 ( Z Z ) the Fourier transform bc 2 L 2 ([0; 2-)) is defined as where the sum converges in L 2 ([0; 2-)) sense. Since 1 e \Gammaik- , k 2 ZZ is an orthonormal basis for L 2 ([0; 2-)), the inverse mapping is given by Z 2-bc(-)e ik- dand the Fourier transform as defined above is a Hilbert space isomorphism l 2 ( Z 2-)). The extension to the vector case is obvious. Infinite Toeplitz matrices represent convolution operators. For sequences a; b 2 Z ), the convolution c = a b has entries Convolution operators are closely related to multipliers. The link is the Fourier transform Lemma 4.1. Let a; b 2 l 2 ( Z Z ) and let a b 2 l 2 ( Z Z ) or ba b b 2 L 2 ([0; 2-)). Then d a Proof. For any l 2 Z Z , d Because the Fourier transform is a Hilbert space isomorphism, A The last term represents the lth entry of the inverse Fourier transform of Theorem 4.2. The operator l 2 ( Z represented by C(A) is isomorphic with a matrix multiplier k2ZZ A k e ik- that maps L 2 r ([0; 2-)) \Gamma! L 2 r ([0; 2-)), u(-) \Gamma! A(-)u(-). Proof. By the former isomorphism, C(A) is isomorphic with the operator l 2 l 2 Z ), for which d, the image of c, is given by the formula A k\Gammal c k ; l 2 Z We will slightly abuse the notation and denote this operator also by C(A). Since we assume that C(A) represents the change from one Riesz basis to another, Z ) and the series k2ZZ A k e ik- converges component-wise in straightforward calculation shows that (4.1) can be extended to matrix/vector case (the Fourier transform being defined component-wise). Because fA \Gammak g k2ZZ A(\Gamma-), d A(\Gamma-)b c(-); Z ) or b r ([0; 2-)). A convolution-type operator thus becomes in the Fourier domain, indeed, the matrix multiplier A. The norm of C(A) induced by l 2 ( Z thus equals to the norm of the matrix multiplier A as an operator L 2 r ([0; 2-)) \Gamma! L 2 r ([0; 2-)). The following theorem gives formulae for the norm of a multiplier and its inverse. Theorem 4.3. Let A be an r \Theta r matrix multiplier with measurable components. Then sup kAuk ess sup r ([0;2-)) kAuk ess inf min Proof. Let us set ess sup and let x 2 L 2 r ([0; 2-)). Then r ([0;2-)) and hence sup r ([0;2-)) r ([0;2- We need to show that we have the lowest upper bound, in other words, that for each NUMERICAL CONDITION OF DISCRETE WAVELET TRANSFORMS 9 Let us take point-wise the singular value decomposition of A; where V(-) and U(-) are r \Theta r unitary matrices and \Sigma(-) is the diagonal matrix with the singular values of A(-) on the diagonal, in decreasing order. We will denote these singular values oe j (-), To construct x ffl we need a path of right singular vectors corresponding to the largest singular value, something like the first column of U , but we have to ensure that this path is square integrable. First, oe 1 is a measurable function, because A has measurable components and the matrix norm is a continuous function of the entries. Let us define Then C has measurable components and, for k ! +1, C(-) k \Gamma! P(-), where and D(-) is a diagonal matrix with the elements on the diagonal equal to either 1 or 0; if oe 1 (-) is of multiplicity m (m depending on -), then first m elements are 1 and all the others are 0. Notice that P(-) is the orthogonal projector onto the subspaces spanned by all right singular vectors corresponding to singular values oe 1 Because P is the limit of a sequence of matrices with measurable components, its components are measurable too. Now, for any ffl ? 0, the set is a measurable set and -(S ffl ) ? 0. Since P(-) 6= 0 for any -, there exist j and a set ~ that p(-), the jth column of P(-), is non-zero for - 2 ~ Let us set -( ~ Because x ffl has measurable components and j~x have r ([0; 2-)). A simple calculation shows that We have consequently, and Finally, -( ~ A which finishes the proof of the first part of the theorem. us concentrate on the second part of the statement. Let us denote ~ ess inf min Clearly, for any u, kuk kAuk Z 2-' min We now have to show that for every ffl ? 0 there exists x ffl , that In order to do that we first need to construct a square integrable path of right singular vectors corresponding to the path of the smallest singular values, oe r . Let us take, again, point-wise singular value decomposition of A, Now, for a positive integer k, let us consider a matrix A(-) A(-) k I . We have I therefore such a matrix is invertible and the norm of the inverse is (oe r If we set oe r (-) 2 +k I then C k has measurable components and C k (-) l \Gamma! P(-), k ! +1, l ! +1, where is, again, a diagonal matrix with the elements on the diagonal equal to either 1 or 0, but now, if oe r (-) is of multiplicity m, then the first r \Gamma m elements are 0 and all the others are 1. The components of the matrix P are measurable functions and the matrix P(-) is now the orthogonal projector onto the subspace spanned by right singular vectors corresponding to the singular values oe r\Gammam+1 for any vector x(-) of unit norm from its range, The rest of the proof would follow the lines of the proof of the first part, with S ffl being chosen as NUMERICAL CONDITION OF DISCRETE WAVELET TRANSFORMS 11 we would have oe r (-) 2 -( ~ The norm of A induced by the norm of L 2 r ([0; 2-)) thus is ess sup oe the mapping is invertible if and only if ess inf min and the norm of the inverse equals ess inf oe min (A(-)): Combining the results above we obtain the following theorem. Theorem 4.4. The condition of the operator C(A) (in the norm induced by the norm of l 2 ( Z Z ess sup -2[0;2-) oe max (A(-)) ess inf -2[0;2-) oe min For all wavelets of practical interest, A has only a finite number of non-zero entries or at least the sequences forming its rows decay very fast. This implies some smoothness of entries of A and, consequently, essential supremum of oe max and essential infimum of oe min coincide with the supremum and infimum, respectively. As we already pointed out, cond(C(A)) then represents sup N cond(CN (A)). Let us make a few comments about the structure of singular values of A(-) in relation with some special properties of A. First, when the underlying bases comprise of real functions, the entries of A are real and, consequently, means that the singular values in - \Gamma - and - are the same and we can restrict our attention onto interval [0; -], only. Another interesting effect is caused by all the scaling and wavelet functions and their biorthogonal counterparts being compactly supported. This corresponds to the fact that only a finite number of square blocks both in A and in ~ A that generates are non-zero. It is well known, particularly in the filter bank context (see, e. g., [12], [13]), that this happens if and only if there exist a non-zero constant ff and an integer p such that det( A k z \Gammak for any z 2 CI , z 6= 0. Because determinant is the product of singular values, the equation above implies that Y A for some positive constant fi independent of -. This is particularly useful when A has only two rows. The singular values of A(-) are then inversely proportional and the maximum and minimum over - then occur at the same point. That is, 5. Alternative expression. Let the sequences that form the rows of A be . Sometimes it is easier to deal with Fourier series than with A. We will see an example in (Section 6, when we will study conditioning of biorthogonal spline wavelets. In these cases it is better to use a different matrix function. Theorem 5.1. A number oe is a singular value of A(-) if and only if it is a singular value of B(\Gamma-=r), where b (2) (-) b (2) (- . Proof. Using the notation introduced in the proof of (Theorem 3.1, we have, for any 2-l r r =r r a (s;k+1) This is because r equals to r if divisible by r and it is 0 otherwise. Consequently, B(-)\Omega rA(\Gammar-)D r (-); where\Omega 1;r is the r \Theta r matrix of the discrete Fourier transform and D r (-) is the diagonal matrix with the diagonal entries equal to e \Gammaik- , particular order). 1;r is unitary and so is D r (-) (for any -), the statement of the theorem holds. Just let us point here that, instead of considering each row of A separately, we could use block rows, each of them comprising of, let say, p rows. We then would obtain similar result with some p \Theta p matrices B n instead of scalars b instead of \Omega 1;r we would use\Omega p;r=p and, similarly, D r (-) would be replaced by a matrix with diagonal blocks equal to e \Gammaik- I , This might be useful for the case of multiwavelets (more than one scaling function) when the two-scale equations analogous to (2.1) have matrix coefficients (see, e.g., [10]). NUMERICAL CONDITION OF DISCRETE WAVELET TRANSFORMS 13 6. Conditioning of biorthogonal wavelets based on B-splines. B-splines. Biorthogonal wavelets based on B-splines were introduced by Cohen, Daubechies and Feauveau in [5]. To the B-spline basis function of a particular order (which represents a scaling function) there exists a whole family of possible biorthogonal counterparts with different size of support and regularity. We will use here the notation of [2], where the sequences determining the scaling and wavelet functions through the two-scale equations of type (2.1) are given in terms of trigonometric polynomials ~ Since the scaling function ' equals to the B-spline of order n, For any integer K such that 2K - ~ determines a biorthogonal scaling function; PK is the solution of Bezout problem in particular, The corresponding to the wavelet filters are then defined as We have and, because we deal with compactly supported real classical wavelets with dilations by 2, we are interested in the maximum of the condition number of B(-) on [0; -=2]. Squares of the singular values of B(-) are the eigenvalues of the matrix B(-)B(-) and they satisfy a quadratic equation Fairly straightforward although somewhat tedious calculations show that the coefficients of this equation are det(B(-)B(-) 14 R. TURCAJOV ' A and where Theorem 6.1. The numerical condition of one level of the (fast) discrete wavelet transform based on B-spline biorthogonal wavelets of order n defined above is at least independently of the value of K. Proof. Since (cf. (6.1)), substituting -=2 into the formulae above we obtain Squares of the singular values of B(-=2) thus equal \Gamman , respectively, and the condition of this matrix is 2 n . The condition hence must be at least 2 n . Numerical experiments show that the condition number often equals 2 n . From the point of view of conditioning, it is better to choose K smaller for low order splines and larger for higher order splines; see Table 1 at the end of the paper. Once the scaling filters m 0 and ~ are given, (6.2) is not the only possibility for the corresponding wavelet filters. The entire freedom can be described as follows: Z , ff 6= 0. The choice of k is, from the point of view of numerical condition, irrelevant, but the scaling by ff can be used to improve the condition. In the case of the spline wavelets improvement can be significant. However, it turns out that whatever scaling we choose, we can't beat the exponential growth with the order of the spline. Theorem 6.2. For any scaling factor ff, the condition of one step of discrete wavelet transform with spline biorthogonal wavelet of order n is at least 2 n=2 . Proof. Instead of the condition of B(-) we need to study here the condition of before, in (6.2). For -=2 the singular values of B ff (-=2) are jffj 2 n=2 and 2 \Gamman=2 and its condition hence is jffj 2 n for \Gamman and 1=(jffj2 n ) for \Gamman . On the other hand, for and its condition is jffj for jffj - 1 and 1=jffj for jffj ! 1. Combining these results we see that the condition of the wavelet transform can not be better than jffj2 n if NUMERICAL CONDITION OF DISCRETE WAVELET TRANSFORMS 15 Consequently, whatever jffj we choose, the condition is at least 2 n=2 . The optimal scaling parameter is usually equal or close to 2 \Gamman=2 , see Table 2 and 3. Notice that this is true especially for the wavelets that have condition number equal to 2 n . The condition of the optimally scaled wavelet then equals 2 n=2 , in most cases. Figures 1-5 show some typical behaviour of the singular value curves in dependence on the order of the spline, parameter K, scaling of the wavelet and depth of the transform. There are some interesting details there like, for example, the presence of points where the plot looks almost like if two curves were intersecting each other, but, in fact, we have two different curves that have turning points and are well separated. Acknowledgements . The author was a postgraduate research scholar supported by the Australian government. She thanks Jaroslav Kautsky for suggesting the topic and many fruitful discussions. --R Image coding using wavelet transform in Wavelets: A Tutorial in Theory and Applications A stability criterion for biorthogonal wavelet bases and their related subband coding scheme Biorthogonal bases of compactly supported wavelets Orthonormal bases of compactly supported wavelets Numerical stability of biorthogonal wavelet transforms Inner products and condition numbers for wavelets and filter banks Short wavelets and matrix dilation equations Compact image coding using wavelets and wavelet packets based on non-stationary and inhomogeneous multiresolution analyses Multirate Systems and Filter Banks Perfect reconstruction FIR filter banks: Some properties and factorizations --TR --CTR Zhong-Yun Liu, Some properties of centrosymmetric matrices, Applied Mathematics and Computation, v.141 n.2-3, p.297-306, 5 September

block-Toeplitz operators;translational bases;biorthogonal wavelets;numerical condition;block circulant matrices

273582

Robust Solutions to Least-Squares Problems with Uncertain Data.

We consider least-squares problems where the coefficient matrices A,b are unknown but bounded. We minimize the worst-case residual error using (convex) second-order cone programming, yielding an algorithm with complexity similar to one singular value decomposition of A. The method can be interpreted as a Tikhonov regularization procedure, with the advantage that it provides an exact bound on the robustness of solution and a rigorous way to compute the regularization parameter. When the perturbation has a known (e.g., Toeplitz) structure, the same problem can be solved in polynomial-time using semidefinite programming (SDP). We also consider the case when A,b are rational functions of an unknown-but-bounded perturbation vector. We show how to minimize (via SDP) upper bounds on the optimal worst-case residual. We provide numerical examples, including one from robust identification and one from robust interpolation.

Introduction . Consider the problem of finding a solution x to an overdetermined set of equations Ax ' b, where the data matrices A 2 R n\Thetam , b 2 R n are given. The Least Squares (LS) fit minimizes the residual k\Deltabk subject to resulting in a consistent linear model of the form \Deltab) that is closest to the original one (in Euclidean norm sense). The Total Least Squares (TLS) solution described by Golub and Van Loan [17] finds the smallest error subject to the consistency equation \Deltab. The resulting closest consistent linear model \Deltab) is even more accurate than the LS one, since modifications of A are allowed. Accuracy is the primary aim of LS and TLS, so it is not surprising that both solutions may exhibit very sensitive behavior to perturbations in the data matrices b). Detailed sensitivity analyses for the LS and TLS problems may be found in [12, 18, 2, 44, 22, 14]. Many regularization methods have been proposed to de- To appear in SIAM Journal on Matrix Analysis and Applications, 1997. y Ecole Nationale Sup'erieure de Techniques Avanc'ees, 32, Bd. Victor, 75739 Paris, France. Internet: (elghaoui, lebret)@ensta.fr L. EL GHAOUI AND H. LEBRET crease sensitivity, and make LS and TLS applicable. Most regularization schemes for LS, including Tikhonov regularization [43], amount to solve a weighted LS problem for an augmented system. As pointed out in [18], the choice of weights (or regularization parameter) is usually not obvious, and application-dependent. Several criteria for optimizing the regularization parameter(s) have been proposed (see e.g. [23, 11, 15]). These criteria are chosen according to some additional a priori information, of deterministic or stochastic nature. The extensive surveys [31, 8, 21] discuss these problems and some applications. In contrast with the extensive work on sensitivity and regularization, relatively little has been done on the subject of deterministic robustness of LS problems, in which the perturbations are deterministic, and unknown-but-bounded (not necessarily small). Some work has been done on a qualitative analysis of the problem, where entries of are unspecified but by their sign [26, 39]. In many papers mentioning least-squares and robustness, the latter notion is understood in some stochastic sense, see e.g. [20, 47, 37]. A notable exception concerns the field of identification, where subject has been explored using a framework used in control system analysis [40, 9], or using regularization ideas combined with additional a priori information [34, 42]. In this paper, we assume that the data matrices are subject to (non necessarily small) deterministic perturbations. First, we assume that the given model is not a single pair (A; b), but a family of matrices is an unknown-but-bounded matrix, precisely, k\Deltak - ae, where ae - 0 is given. For x fixed, we define the worst-case residual as (1) We say that x is a Robust Least Squares (RLS) solution if x minimizes the worst-case residual r(A; b; ae; x). The RLS solution trades accuracy for robustness, at the expense of introducing bias. In our paper, we assume that the perturbation bound ae is known, but in x3.5, we also show that TLS can be used as a preliminary step to obtain a value of ae that is consistent with data matrices A; b. In many applications, the perturbation matrices \DeltaA, \Deltab have a known structure. For instance, \DeltaA might have a Toeplitz structure inherited from A. In this case, the worst-case residual (1) might be a very conservative estimate. We are led to consider the following Structured RLS (SRLS) problem. Given A (2) For ae - 0, and x we define the structured worst-case residual as kffik-ae We say that x is a Structured Robust Least Squares (SRLS) solution if x minimizes the worst-case residual r S (A; b; ae; x). ROBUST LEAST SQUARES 3 Our main contribution is to show that we can compute the exact value of the optimal worst-case residuals using convex, second-order cone or semidefinite programming (SOCP or SDP). The consequence is that the RLS and SRLS problems can be solved in polynomial-time, and great practical efficiency, using e.g. recent interior-point methods [33, 46]. Our exact results are to be contrasted with those of Doyle et. al [9], which also use SDP to compute upper bounds on the worst-case residual for identification problems. In the preliminary draft [5], sent to us shortly after submission of this paper, the authors provide a solution to an (unstructured) RLS problem, which is similar to that given in x3.2. Another contribution is to show that the RLS solution is continuous in the data matrices A; b. RLS can thus be interpreted as a (Tikhonov) regularization technique for ill-conditioned LS problems: the additonal a priori information is ae (the perturbation level), and the regularization parameter is optimal for robustness. Similar regularity results hold for the SRLS problem. We also consider a generalisation of the SRLS problem, referred to as the linear- fractional SRLS problem in the sequel, in which the matrix functions A(ffi), b(ffi) in (2) depend rationally on the parameter vector ffi. (We describe a robust interpolation problem that falls in this class in x7.6.) Using the framework of [9], we show that the problem is NP-complete in this case, but that we may compute, and optimize, upper bounds on the worst-case residual using SDP. In parallel with RLS, we interpret our solution as one of a weighted LS problem for an augmented system, the weights being computed via SDP. The paper's outline is as follows. Next section is devoted to some technical lemmas. Section 3 is devoted to the RLS problem. In section 4, we consider the SRLS problem. Section 5 studies the linear-fractional SRLS problem. Regularity results are given in Section 6. Section 7 shows numerical examples. 2. Preliminary results. 2.1. Semidefinite and second-order cone programs. We briefly recall some important results on semidefinite programs (SDPs) and second-order cone programs (SOCPs). These results can be found in e.g. [4, 33, 46]. A linear matrix inequality is a constraint on a vector x of the form where the symmetric matrices F are given. The minimization problem subject to F(x) - 0 called a semidefinite program (SDP). SDPs are convex optimization problems and can be solved in polynomial-time with e.g. primal-dual interior-point methods [33, 45]. 4 L. EL GHAOUI AND H. LEBRET The problem dual to problem (5) is maximize \GammaTrF 0 Z subject to Z - 0; TrF i where Z is a symmetric N \Theta N matrix and c i is the i-th coordinate of vector c. When both problems are strictly feasible (that is, when there exists x; Z which satisfy the constraints strictly), the existence of optimal points is guaranteed [33, thm.4.2.1], and both problems have equal optimal objectives. In this case, the optimal primal-dual pairs (x; Z) are those pairs (x; Z) such that x is feasible for the primal problem, Z is feasible for the dual one, and A second-order cone programming problem is one of the form subject to kC L. The dual problem of problem (7) is subject to are the dual variables. Optimality conditions similar to those for SDPs can be obtained for SOCPs. SOCPs can be expressed as SDPs, therefore they can be solved in polynomial-time using interior-point methods for SDPs. However the SDP formulation is not the most efficient numerically, as special interior-point methods can be devised for SOCPs [33, 28, 1]. complexity results on interior-point methods for SOCPs and SDPs are given by Nesterov and Nemirovsky [33, p.224,236]. In practice, it is observed that the number of iterations is almost constant, independent of problem size [46]. For the SOCP, each iteration has complexity O((n for the SDP, we refer the reader to [33]. 2.2. S-procedure. The following lemma can be found e.g. in [4, p.24]. It is widely used, e.g. in control theory, and in connection with trust region methods in optimization [41]. Lemma 2.1 (S-procedure). Let F be quadratic functions of the variable . The following condition on F ROBUST LEAST SQUARES 5 holds if there exist 0: When the converse holds, provided that there is some i 0 such that F 1 (i The next lemma is a corollary of the above result, in the case Lemma 2.2. Let T real matrices of appropriate size. We have for every \Delta, k\Deltak - 1, if and only if kT 4 k ! 1, and there exists a scalar - 0 such that 0: Proof. If T 2 or T 3 equal zero, the result is obvious. Now assume which in turn implies kT 4 k ! 1. Thus, for a given - , (10) holds if and only if kT 4 k ! 1 and for every (u; p), we have 4 p. Since T 2 6= 0, the constraint q T q - p T p is qualified, that is, satisfied strictly for some Using the S-procedure, we obtain that there exists - 2 R such that (10) holds if and only if 1, and for every (u; p) such that q T q - p T p, we have u our proof by noting that for every pair (p; q), only if p T p - q T q. Next lemma is a "structured" version of the above, which can be traced back to [13]. Lemma 2.3. Let T real matrices of appropriate size. Let D be a subspace of R N \ThetaN , and denote by S (resp. G) the set of symmetric (resp. skew- symmetric) matrices that commute with every element of D. We have and (9) for every \Delta 2 D, k\Deltak - 1, if there exist S 2 S, G 2 G such that If the condition is necessary and sufficient. Proof. The proof follows the scheme of that of lemma 2.2, except that p T p - q T q is replaced with p T G. Note that 0, the above result is a simple application of lemma 2.2 to the scaled matrices 6 L. EL GHAOUI AND H. LEBRET 2.3. Elimination lemma. The last lemma is proven in [4, 24]. Lemma 2.4 (Elimination). Given real matrices of appropriate size, there exists a real matrix X such that if and only if ~ U T W ~ where ~ U , ~ are orthogonal complements of U; V . If U; V are full column-rank, and (12) holds, a solution X to the inequality (11) is oe is any scalar such that Q ? 0 (the existence of which is guaranteed by (12)). 3. Unstructured Robust Least-Squares. In this section, we consider the RLS problem, which is to compute For we recover the standard LS problem. For every ae ? 0, OE(A; b; aeOE(A=ae; b=ae; 1), so we take in the sequel, unless otherwise stated. In the remainder of this paper, OE(A; b) (resp. r(A; b; x)) denotes OE(A; b; 1) (resp. r(A; b; 1; x)). In the definition above, the norm used for the perturbation bound is the Frobenius norm. As seen shortly, the worst-case residual is the same when the norm used is the largest singular value norm. 3.1. Optimizing the worst-case residual. The following results yield a numerically efficient algorithm for solving the RLS problem in the unstructured case. Theorem 3.1. When ae = 1, the worst-case residual (1) is given by The problem of minimizing r(A; b; x) over x has a unique solution x RLS , referred to as the RLS solution. This problem can be formulated as the second-order cone program subject to kAx \Gamma bk x# Proof. Fix x . Using the triangle inequality, we have ROBUST LEAST SQUARES 7 Now choose if Ax 6= b; any unit-norm vector otherwise. Since \Delta is rank-one, we have k\Deltak In addition, we have which implies that \Delta is a worst-case perturbation (for both the Frobenius and maximum singular value norms), and that equality always holds in (16). Finally, unicity of the x follows from the strict convexity of the worst-case residual. Using an interior-point primal-dual potential reduction method for solving the unstructured RLS problem (15), the number of iterations is almost constant [46]. Further- more, each iteration takes O((n+m)m 2 ) operations. A rough summary of this analysis is that the method has the same order of complexity as one SVD of A. 3.2. Analysis of the optimal solution. Using duality results for SOCPs, we have the following theorem. Theorem 3.2. When ae = 1, the (unique) solution x RLS to the RLS problem is given by A y b else, where (-) are the (unique) optimal points for problem (15). Proof. Using the results of x2.1, we obtain that the problem dual to (15) is subject to A T z Since both primal and dual problems are strictly feasible, there exist optimal points for both of them. If - at the optimum, then In this case, the optimal x is the (unique) minimum-norm solution to Now assume - . Again, both primal and dual problems are strictly feasible, therefore the primal and dual optimal objectives are equal: Using and 8 L. EL GHAOUI AND H. LEBRET Replace these values in A T z to obtain the expression of the optimal x: A T b; with Remark 3.1. When - , The RLS solution can be interpreted as the solution of a weighted LS problem for an augmented system: A I3 \Theta where -). The RLS method amounts to compute the weighting matrix \Theta that is optimal for robustness, via the SOCP (15). We shall encounter a generalization of the above formula for the linear-fractional SRLS problem of x5. Remark 3.2. It is possible to solve the problem when only A is perturbed In this case, the worst-case residual is kAx kxk, and the optimal x is determined by (17), where bk. (See the example in x7.2). 3.3. Reduction to a one-dimensional search. When the SVD of A is available, we can use it to reduce the problem to a one-dimensional convex differentiable problem. The following analysis will also be useful in x6. Introduce the SVD of A and a related decomposition for b: Assume that - at the optimum of problem (15). From (18), we have never feasible, we may define Multiplying by -, we obtain that From 1). Thus, the optimal worst-case residual is ROBUST LEAST SQUARES 9 where f is the following function The function f is convex and twice differentiable on [' min 1[. If b 62 Range(A), f is infinite at twice differentiable on the closed interval [' min 1]. Therefore, the minimization of f can be done using standard Newton methods for differentiable optimization. Theorem 3.3. When ae = 1, the solution of the unstructured RLS can be computed by solving the one-dimensional convex differentiable problem (19), or by computing the unique real root inside [' min 1] (if any) of the equation' 2 r The above theorem yields an alternative method for computing the RLS solution. This method is similar to the one given in [5]. A related approach was used for quadratically constrained LS problems in [19]. The above solution, which requires one SVD of A, has cost O(nm 2 +m 3 ). The SOCP method is only a few times more costly (see the end of x3.1), with the advantage that we can include all kinds of additional constraints on x (nonnegativity and/or quadratic constraints, etc) in the SOCP (15), with low additional cost. Also, the SVD solution does not extend to the structured case considered in x4. 3.4. Robustness of LS solution. It is instructive to know when the RLS and LS solutions coincide, in which case we can say the LS solution is robust. This happens if and only if the optimal ' in problem (19) is equal to 1. The latter implies b (that is, b 2 Range(A)). In this case, f is differentiable at its minimum over [' min 1] is at only if df d' We obtain a necessary and sufficient condition for the optimal ' to be equal to 1. This condition is If (21) holds, then the RLS and LS solutions coincide. Otherwise, the optimal ' ! 1, and x is given by (17). We may write the latter condition in the case when the norm-bound of the perturbation ae is different from 1 as: ae ? ae min , where ae min Thus, ae min can be interpreted as the perturbation level that the LS solution allows. We note that, when b 2 Range(A), the LS and TLS solution also coincide. Corollary 3.4. The LS, TLS and RLS solutions coincide whenever the norm- bound on the perturbation matrix ae satisfies ae - ae min (A; b), where ae min (A; b) is defined in (22). Thus, ae min (A; b) can be seen as a robustness measure of the LS (or TLS) solution. When A is full rank, the robustness measure aemin is non zero, and decreases as the condition number of A increases. Remark 3.3. We note that the TLS solution x TLS is the most accurate, in the sense it minimizes the distance function (see [18]), and the least robust, in the sense of the worst-case residual. The LS solution, x is intermediate (in the sense of accuracy and robustness). In fact, it can be shown that 3.5. Robust and Total Least-Squares. The RLS framework assumes that the data matrices (A; b) are the "nominal" values of the model, which are subject to unstructured perturbation, bounded in norm by ae. Now, if we think of (A; b) as "mea- sured" data, the assumption that (A; b) correspond to a nominal model may not be judicious. Also, in some applications, the norm-bound ae on the perturbation may be hard to estimate. The Total Least-Squares (TLS) solution, when it exists, can be used in conjunction with RLS to address this issue. Assume that the TLS problem has a solution. Let \DeltaA TLS , \Deltab TLS , x TLS be minimizers of the TLS problem minimize subject to and let ae TLS finds a consistent, linear system that is closest (in Frobenius norm sense) to the observed data (A; b). The underlying assumption is that the observed data (A; b) is the result of a consistent, linear system which, under the measurement process, has been subjected to unstructured perturbations, unknown but bounded in norm by ae TLS . With this assumption, any point of the ball ROBUST LEAST SQUARES 11 can be observed, just as well as (A; b). Thus, TLS computes an "uncertain linear system" representation of the observed phenomenon: is the nominal model, and ae TLS is the perturbation level. Once this uncertain system representation choosing x TLS as a "solution" to Ax ' b amounts to finding the exact solution to the nominal system. Doing so, we compute a very accurate solution (with zero residual), which does not take into account the perturbation level ae TLS . A more robust solution is given by the solution to the following RLS problem The solution to the above problem coincides with the TLS one (that is, in our case, with x TLS ) when ae TLS - ae min is stricly positive, except when A With standard LS, the perturbations that account for measurement errors are structured (with To be consistent with LS, one should consider the following RLS problem instead of (23): k\Deltabk-ae LS It turns out that the above problem yields the same solution as LS itself. To summarize, RLS can be used in conjunction with TLS for "solving" a linear system Ax ' b. Solve the TLS problem to build an ``uncertain linear system'' re-presentation of the observed data. Then, take the solution x RLS to the RLS problem with the nominal matrices ae TLS . Note that computing the TLS solution (precisely, A TLS , b TLS and ae TLS ) only requires the computation of the smallest singular value and associated singular subspace [17]. 4. Structured Robust Least Squares. In this section, we consider the SRLS problem, which is to compute kffik-ae where A; b are defined in (2). As before, we assume with no loss of generality that by r S (A; b; x). Throughout the section, we use the following 4.1. Computing the worst-case residual. We first examine the problem of computing the worst-case residual r S (A; b; x) for a given x With the above notation, we have F Now let - 0. Using the S-procedure (lemma 2.1), we have F for every ffi, only if there exists a scalar - 0 such that \Gammag Using the fact that - 0 is implied by -I - F , we may rewrite the above condition as \Gammag 0: The consequence is that the worst-case residual is computed by solving a SDP with two scalar variables. A bit more analysis shows how to reduce the problem to a one- dimensional, convex differentiable problem, and obtain the corresponding worst-case perturbation. Theorem 4.1. For every x fixed, the squared worst-case residual (for can be computed by solving the SDP in two variables subject to (29); or, alternatively, by minimizing a one-dimensional convex differentiable function: where If - is optimal for problem (30), the equations in ffi have a solution, any of which is a worst-case perturbation. Proof. See Appendix A, where we also show how to compute a worst-case perturbation 4.2. Optimizing the worst-case residual. Using theorem 4.1, the expression of F; g; h given in (27), and Schur complements, we obtain following result. Theorem 4.2. When ae = 1, the Euclidean-norm SRLS can be solved by computing an optimal solution (-; x) of the SDP subject to6 4 ROBUST LEAST SQUARES 13 where M(x) is defined in (26). Remark 4.1. Straightforward manipulations show that the result are coherent with the unstructured case. Although the above SDP is not directly amenable to the more efficient SOCP formulation, we may devise special interior-point methods for solving the problem. These special-purpose methods will probably have much greater efficiency than general-purpose SDP solvers. This study is left for the future. Remark 4.2. The discussion of x3.5 extends to the case when the perturbations are structured. TLS problems with (affine) structure constraints on perturbation matrices are discussed in [7]. While the structured version of the TLS problem becomes very hard to solve, the SRLS problem retains polynomial-time complexity. 5. Linear-Fractional SRLS. In this section, we examine a generalization of the SRLS problem. Our framework encompasses the case when the functions A(ffi), b(ffi) are rational. We show that the computation of the worst-case residual is NP-complete, but that upper bounds can be computed (and optimized) using SDP. First, we need to motivate the problem, and develop a formalism for posing it. This formalism was introduced by Doyle and coauthors [9] in the context of robust identification. 5.1. Motivations. In some structured robust least-squares problems such as (3), it may not be convenient to measure the perturbation size with Euclidean norm. Indeed, the latter implies a correlated bound on the perturbation. One may instead consider a SRLS problem, in which the bounds are not correlated, that is, the perturbation size in (3) is measured by the maximum norm: kffik 1-1 Also, in some RLS problems, we may assume that some columns of [A b] are perfectly known, for instance the error [\DeltaA \Deltab] has the form and otherwise unknown. More generally, we may be interested in SRLS problems, where the perturbed data matrices write A(\Delta) b(\Delta) A b are given matrices, and \Delta is a (full) norm-bounded matrix. In such a problem, the perturbation is not structured, except via the matrices L; RA (Note that a special case of this problem is solved in [5].) Finally, we may be interested in SRLS problems in which the matrix functions A(ffi), b(ffi) in (3) are rational functions of the parameter vector ffi. One example is given in x7.6. It turns out that the above three extensions can be addressed using the same formalism, which we detail now. 5.2. Problem definition. Let D be a subspace of R N \ThetaN , A 2 R n\Thetam R n\ThetaN , RA 2 R N \Thetam , R b 2 R N , D 2 R N \ThetaN . For every \Delta 2 D such that det(I \GammaD\Delta) 6= 0, 14 L. EL GHAOUI AND H. LEBRET we define the matrix functions A(\Delta) b(\Delta) A b For a given x we define the worst-case residual by r D (A; b; ae; x) \Delta \Delta2D; k\Deltak-ae We say that x is a Structured Robust Least Squares (SRLS) solution if x minimizes the worst-case residual above. As before, we assume no loss of generality, and denote r D (A; b; 1; x) by r D (A; b; x). The above formulation encompasses the three situations referred to in x5.1. First, the maximum-norm SRLS problem (33) is readily transformed into problem (35), as follows. Let n\ThetaN be such that [A i b i R T Problem (33) can be formulated as the minimization of (35), with D defined as above. Also, we recover the case when the perturbed matrices write as in (34), when we allow \Delta to be any full matrix (that is, In particular, we recover the unstructured RLS problem of x3, as follows. Assume n ? m. We have \Deltab \Theta refers to dummy elements that are added to the perturbation matrix in order to make it a square, n \Theta n matrix.) In this case, the perturbation set D is R n\Thetan . Finally, the case when A(ffi) and b(ffi) are rational functions of a vector ffi (well- defined over the unit ball fffi j kffik 1 - 1g) can be converted (in polynomial time) into the above framework (see e.g. [48] for a conversion procedure). We give an example of such a conversion in x7.6. 5.3. Complexity analysis. In comparison with the SRLS problem of x4, the linear-fractional SRLS problem offers two levels of increased complexity. First, checking whether the worst-case residual is finite is NP-complete [6]. The linear-fractional dependence (that is, D 6= 0) is a first cause of increased complexity. The SRLS problem above remains hard even when matrices A(ffi), b(ffi) depend affinely on the perturbation elements 0). Consider for instance the SRLS problem, with and in which D is defined as in (36). In this case, the problem of computing the worst-case residual can be formulated as kffik 1-1 F ROBUST LEAST SQUARES 15 for appropriate F; g; h. The only difference with the wost-case residual defined in (28) is the norm used to measure perturbation. Computing the above quantity is NP-complete (it is equivalent to a MAX CUT problem [36, 38]). The following lemma, which we provide for the sake of completeness, is a simple corollary of a result by Nemirovskii [32]. Lemma 5.1. The problem P(A;b; D; x): Given a positive rational number -, matrices A; b; L; RA of appropriate size, and an m-vector x, all with rational entries, and a linear subset D, determine whether r D (A; b; x) - is NP-complete. Proof. See appendix B. 5.4. An upper bound on the worst-case residual. Although our problem is NP-complete, we can minimize upper bounds in polynomial-time, using SDP. Introduce the following linear subspaces: R. The inequality - ? r D (A; b; x) holds if and only if, for every \Delta 2 D, 0: Using Lemma 2.3, we obtain that - ? r D (A; b; x) holds if there exist S 2 S, G 2 G, such that G; x) =6 4 \Theta Ax \Gamma b where \Theta \Delta Minimizing - subject to the above semidefinite constraint yields an upper bound for r D (A; b; x). It turns out that the above estimate of the worst-case residual is actually exact, in some "generic" sense. Theorem 5.2. When ae = 1, an upper bound on the worst-case residual r D (A; b; x) can be obtained by solving the SDP - subject to S 2 G; (38): The upper bound is exact when D = R N \ThetaN . If \Theta ? 0 at the optimum, the upper bound is also exact. Proof. See appendix C. L. EL GHAOUI AND H. LEBRET 5.5. Optimizing the worst-case residual. Since x appears linearly in the constraint (38), we may optimize the worst-case residual's upper bound using SDP. We may reduce the number of variables appearing in the previous problem, using the elimination lemma 2.4. Inequality in (38) can be written as in (11), with \GammaR b A where \Theta is defined in (39). Denote by N the orthogonal complement of [A T R T . Using the elimination lemma 2.4, we obtain an equivalent condition for (38) to hold for some x namely G; \Theta ? 0; (N \GammaR b \Gammab \GammaR b -7 5 (N For every -; S; G that are stricly feasible for the above constraints, an x that satisfies (38) is given, when RA is full-rank, by A A \Theta \Gamma1 A RA A A \Theta \Gamma1 (To prove this, we applied formula (13), and took oe !1.) Theorem 5.3. When ae = 1, an upper bound on the optimal worst-case residual can be obtained by solving the SDP - subject to S 2 G; (38); or, alternatively, the SDP - subject to (41): The upper bound is always exact when D = R N \ThetaN . If \Theta ? 0 at the optimum, the upper bound is also exact. The optimal x is then unique, and given by (42) when RA is full-rank. Proof. See appendix C. Remark 5.1. In parallel to the unstructured case (see remark 3.1), the linear- fractional SRLS can be interpreted as a weighted LS for an augmented system. Precisely, when \Theta ? 0, the linear-fractional SRLS solution can be interpreted as the solution of a weighted LS problem: A RA The SRLS method amounts to compute the weighting matrix \Theta that is optimal for robustness. ROBUST LEAST SQUARES 17 Remark 5.2. Our results are coherent with the unstructured case: replace L by I, R by [I 0] T , variable S by -I, and set G = 0. The parameter - of theorem 3.2 can be interpreted as the Schur complement of -I \Gamma LSL T in the matrix \Theta. Remark 5.3. We emphasize that the above results are exact (non conservative) when the perturbation structure is full. In particular, we recover (and generalize) the results of [5] in the case when only some columns of A are affected by otherwise unstructured perturbations. Remark 5.4. When is possible to use the approximation method of [16] to obtain solutions (based on the SDP relaxations given in theorem 5.3) that have expected value within 14% of the true value. 6. Link with regularization. The standard LS solution x LS is very sensitive to errors in A; b when A is ill-conditioned. In fact, the LS solution might not be a continuous function of A; b when A is near-deficient. This has motivated many researchers for ways to regularize the LS problem, which is to make the solution x unique and continuous in the data matrices (A; b). In this section, we briefly examine the links of our RLS and SRLS solution with regularization methods for standard LS. Beforehand, we note that since all our problems are formulated as SDPs, we could invoke the quite complete sensitivity analysis results obtained by Bonnans, Cominetti and Shapiro [3]. The application of these general results to our SDPs is considered in [35]. 6.1. Regularization methods for LS. Most regularization methods for LS amount to impose an additional bound on the solution vector x. One way is to minimize where\Omega is some squared-norm (see [23, 43, 8]). Another way is to use constrained least-squares (see [18, p.561-571]). In a classical Tikhonov regularization method, \Omega\Gamma some "regularization" parameter. The modified value of x is obtained by solving an augmented LS problem and is given by (Note that for every - ? 0, the above x is continuous in (A; b).) The above expression also arises in the Levenberg-Marquardt method for optimiza- tion, or in the Ridge regression problem [17]. As mentioned in [18], the choice of an appropriate - is problem-dependent, and in many cases, not obvious. In more elaborate regularization schemes of the Tikhonov type, the identity matrix in (46) is replaced with a positive semidefinite weighting matrix (see for instance [31, 8]). Again, this can be interpreted as a (weighted) least-squares method for an augmented system. L. EL GHAOUI AND H. LEBRET 6.2. RLS and regularization. Noting the similarity between (17) and (46), we can interpret the (unstructured) RLS method as one of Tikhonov regularization. The following theorem yields an estimate of the "smoothing effect" of the RLS method. Note that improved regularity results are given in [35]. Theorem 6.1. The (unique) RLS solution x RLS and the optimal worst-case residual are continuous functions of the data matrices A; b. Furthermore, if K is a compact set of R n , and then for every uncertainty size ae ? 0, the function R n\Thetam \Theta K \Gamma! [1 dK is Lipschitzian, with Lipschitz constant 1 Theorem 6.1 shows that any level of robustness (that is, any norm-bound on perturbations ae ? regularization. We describe in x7 some numerical examples that illustrate our results. Remark 6.1. In the RLS method, the Tikhonov regularization parameter - is chosen by solving a second-order cone problem, in such a way that - is optimal for robustness. The cost of the RLS solution is equal to the cost of solving a small number of least-squares problems of the same size as the classical Tikhonov regularization problem (45). Remark 6.2. The equation that determines - in the RLS method is ae This choice has resemblance with Miller's choice [30], where - is determined recursively by the equations This formula arises in RLS when there is no perturbation in b (see remark 3.2). Thus, Miller's solution corresponds to a RLS problem in which the perturbation affects only the columns of A. We note that this solution is not necessarily regular (continuous). Total least-squares (TLS) deserves a special mention here. When the TLS problem has a solution, it is given by x oe is the smallest singular value of [A b]. This corresponds to (46). The negative value of - implies that the TLS is a "deregularized" LS, a fact noted in [17]. In view of our link between regularization and robustness, the above is consistent with the fact that RLS trades off the accuracy of TLS with robustness and regularity, at the expense of introducing bias in the solution. See also remark 3.3. 6.3. SRLS and regularization. Similarly, we may ask whether the solution to the SRLS problem of x4 is continuous in the data matrices A as was the case for unstructured RLS problems. We only discuss continuity of the optimal worst-case ROBUST LEAST SQUARES 19 residual with respect to problems, the coefficient matrices A are fixed). In view of Theorem 4.2, continuity holds if the feasible set of the SDP (32) is bounded. Obviously, the objective - is bounded above by Thus the variable - is also bounded, as (32) implies 0 -. With - bounded above, we see that (32) implies that x is bounded if bounded implies x bounded. The above property holds if and only if [A T Theorem 6.2. A sufficient condition for continuity of the optimal worst-case residual (as a function of 6.4. Linear-fractional SRLS and regularization. Precise conditions for continuity of the optimal upper bound on worst-case residual in the linear-fractional case are not known. We may however regularize this quantity using a method described in [29] for a related problem. For a given ffl ? 0, define the bounded set ae ffl I oe where S is defined in (37). It is easy to show that restricting the condition number of variable S also bounds the variable G in the SDP (44). This yields the following result. Theorem 6.3. An upper bound on the optimal worst-case residual can be obtained by computing the optimal value -(ffl) of the SDP min - subject to S 2 G; (41): The corresponding upper bound is a continuous function of [A b]. As ffl ! 0, the corresponding optimal value -(ffl) has a limit, equal to the optimal value of SDP (44). As noted in remark 5.1, the linear-fractional SRLS can be interpreted as a weighted LS, and so can the above regularization method. Thus, the above method belongs to the class of Tikhonov (or weighted LS) regularization methods referred to in 6.1, the weighting matrix being optimal for robustness. 7. Numerical examples. The following numerical examples were obtained using two different codes: for SDPs, we used the code SP [45], and a matlab interface to SP called [10]. For the (unstructured) RLS problems, we used the second-order cone program described in [28]. L. EL GHAOUI AND H. LEBRET #iter Vertical bars indicate deviation for 20 trials, with mean min #iter Vertical bars indicate deviation for 20 trials, with mean min Fig. 1. Average, minimum and maximum number of iterations for various RLS problems using the SOCP formulation. In the left figure, we show these numbers for values of n ranging from 100 to 1000. For each value of n, the vertical bar indicates the minimum and maximum values obtained with 20 trials of A; b, with In the right figure, we show these numbers for values of m ranging from 11 to 100. For each value of n, the vertical bar indicates the minimum and maximum values obtained with 20 trials of A; b, with 1000. For both plots, the plain curve is the mean value. 7.1. Complexity estimates of RLS. We first did "large-scale" experiments for the RLS problem of x3. As mentioned in x2.1, the number of iterations is almost independent of the size of the problem for SOCPs. We have solved problem (15) for uniformly generated random matrices A and vectors b with various sizes of n; m. Figure 1 shows the average number of iterarions as well as the minimum and maximum number of iterations for various values of n; m. The experiments confirm the fact the number of iterations is almost independent of problem size for the RLS problem. 7.2. LS, TLS and RLS. We now compare the LS, TLS and RLS solutions for On the left and right plots in Fig. 2, we show the four points signs and the corresponding linear fits for LS problems (solid line), TLS problems (dotted line) and RLS problems for (dashed lines). The left plot gives the RLS solution with perturations [A+\DeltaA; b+\Deltab] whereas the right plot considers perturbation in A only, [A In both plots, the worst-case points for the RLS solution are indicated by 0 2. As ae increases, the slope of the RLS solution decreases, and goes to zero when ae ! 1. The plot confirms remark 3.3: the TLS solution is the most accurate and the least robust, and LS is intermediate. In the case when we have perturbations in A only (right plot), we obtain an instance of a linear-fractional SRLS (with a full perturbation matrix), as mentioned in x5.1. (It is also possible to solve this problem directly, as in x3.) In this last case of course, the worst-case perturbation can only move along the A-axis. ROBUST LEAST SQUARES 21 A TLS RLS A TLS RLS Fig. 2. Least-squares (solid), total least-squares (dotted) and robust least-squares (dashed) solutions. The signs + correspond to the nominal [A b]. The left plot gives RLS solution with perturations the right plot considers perturbation in A only, [A b]. The worst-case perturbed points for the RLS solution are indicated by 0 2. 7.3. RLS and regularization. As mentioned in x6, we may use RLS to regularize an ill-conditioned LS problem. Consider the RLS problem for The matrix A is singular when Fig. 3 shows the regularizing effect of the RLS solution. The left (resp. right) figure shows the optimal worst-case residual (resp. norm of RLS solution) as a function of the parameter ff, for various values of ae. When ae = 0, we obtain the LS solution. The latter is not a continuous function of ff, and both the solution norm and residual exhibit a spike for becomes singular). For ae ? 0, the RLS solution is smooth. The spike is more and more flattened as ae grows, which illustrates theorem 6.1. For 1, the optimal worst-case residual becomes flat (independent of ff), and equal to 7.4. Robustness of LS solution. The next example illustrates that sometimes (precisely, if b 2 Range(A)), the LS solution is robust, up to the perturbation level ae min defined in (22). This "natural" robustness of the LS solution degradates as the condition number of A grows. For " A ? 0, consider the RLS problem for ":1 We have considered six values of " A (which equals the inverse of the condition number of A) from .05 to .55. Table 1 shows the values of ae min (as defined in (22)) for 22 L. EL GHAOUI AND H. LEBRET ff dashed dashed dotted ff Fig. 3. Optimal worst-case residual and norm of RLS solution vs. ff for various values of perturbation level ae. For the optimal residual and solution are discontinuous. The spike is smoothed as more robustness is asked for (that is, when ae increases). On the right plot the curves for are not visible. Table Values of ae min for various " A . curve ae min 0.06 0.34 0.78 1.12 1.28 1.35 the six values of " A . When the condition number of A grows, the robustness of the LS solution (measured by ae min ) decreases. The right plot of Fig. 4 gives the worst-case residual vs. the robustness parameter ae for the six values of " A . The plot illustrates that for ae ? ae min , the LS solution (in our case, A differs from the RLS one. Indeed, for each curve, the residual remains equal to zero as long as ae - ae min . For example, the curve labeled '1' (corresponding to quits the x-axis for ae - ae The left plot of Fig. 4 corresponds to the RLS problem with of " A . This plot shows the various functions f(') as defined in (20). For each value of " A , the optimal ' (hence the RLS solution) is obtained by minimizing the function f . The three smallest values of " A induce functions f (as defined in (20)) that are minimal 1. For the three others, the optimal ' is 1. This means that ae min is smaller than 1 in the first three cases and larger than 1 in the other cases. This is confirmed in Table 1. 7.5. Robust identification. Consider the following system identification prob- lem. We seek to estimate the impulse response h of a discrete-time system from its input u and output y. Assuming that the system is single-input and single-ouput, linear, and of order m, and that u is zero for negative time indices, y, u and h are related by the ROBUST LEAST SQUARES 23 6 531 Fig. 4. The left plot shows function f(') (as defined in (20)) for the six values of " A (for ae = 1). The right plot gives the optimal RLS residuals versus ae for the same values of " A . The labels correspond to values of " A given in Table 1. convolution equations y, where and U is a lower triangular Toeplitz matrix whose first column is u. Assuming are known exactly leads to a linear equation in h, which can be computed with standard LS. In practive however, both y and u are subject to errors. We may assume for instance that the actual value of y is y ffiy, and that of u is u are unknown-but-bounded perturbations. The perturbed matrices U; y write is the i-th column of the m \Theta m identity matrix, and U i are lower triangular Toeplitz matrices with first column equal to e i . We first assume that the sum of the input and output energies is bounded, that is We adress the following min kffik-ae As an example, we consider the following nominal values for In Fig. 5, we have shown the optimal worst-case residual and that corresponding to the LS solution, as given by solving problems (30) and (32), respectively. Since the LS L. EL GHAOUI AND H. LEBRET solution has zero residual (U is invertible), we can prove (and check on the figure) that the worst-case residual grows linearly with ae. In contrast, the RLS optimal worst-case residual has a finite limit as ae !1. RLS ae Fig. 5. Worst-case residuals of LS and euclidean-norm SRLS solutions for various values of perturbation level ae. The worst-case residual for LS has been computed by solving problem (30), with fixed. We now assume that the perturbation bounds on y; u are not correlated. For instance, we consider problem (48), with the bound kffik - ae replaced with Physically, the above bounds mean that the output energy and peak input are bounded. This problem can be formulated as minimizing the worst-case residual (35), with [A and \Delta has the following structure: Here, the symbols \Theta denote dummy elements of \Delta that were added in order to work with a square perturbation matrix. The above structure corresponds to the set D in (36), with In Fig.6, we show the worst-case residual vs. ae, the uncertainty size. We show the curves corresponding to the values predicted by solving the SDP (43), with x variable ROBUST LEAST SQUARES 25 upper bound (LS solution) lower bound (LS solution) upper bound (RLS solution) lower bound (RLS solution) ae Fig. 6. Upper and lower bounds on worst-case residuals for LS and RLS solutions. The upper bound for LS has been computed by solving the SDP (38), with fixed. The lower bounds corresponds to the largest residuals kU(ffi trial )x \Gamma y(ffi trial )k among 100 trial points ffi trial , with (RLS solution), and x fixed to the LS solution x LS . We also show lower bounds on the worst-case, obtained using 100 trial points. This plot shows that, for the LS solution, our estimate of the worst-case residual is not exact, and the discrepancy grows linearly with uncertainty size. In contrast, for the RLS solution the estimate appears to be exact for every value of ae. 7.6. Robust interpolation. The following example is a robust interpolation problem that can be formulated as a linear-fractional SRLS problem. For given integers a polynomial of degree interpolates given points (a that is If we assume that (a are known exactly, we obtain a linear equation in the unknown x, with a Vandermonde structure:6 6 4 which can be solved via standard LS. Now assume that the interpolation points are not known exactly. For instance, we may assume that the b i 's are known, while the a i 's are parameter-dependent: a where the ffi i 's are unknown-but-bounded: jffi We seek a robust interpolant, that is, a solution x that minimizes kffik 1-ae 26 L. EL GHAOUI AND H. LEBRET where The above problem is a linear-fractional SRLS problem. Indeed, it can be shown that where and, for each i, . a i . a i . 1 (Note that det(I \Gamma D\Delta) 6= 0, since D is stricly upper triangular.) In Fig. 7, we have shown the result a 1 =6 423 The LS solution is very accurate (zero nominal residual: every point is interpolated ex- actly), but has a (predicted) worst-case residual of 1:7977. The RLS solution trades off this accuracy (only one point interpolated, and nominal residual of 0:8233) for robustness (with a worst-case residual less than 1:1573). As ae ! 1, the RLS interpolation polynomial becomes more and more horizontal. (This is consistent with the fact that we allow perturbations on vector a only.) In the limit, the interpolation polynomial is the solid line ROBUST LEAST SQUARES 27 Fig. 7. Interpolation polynomials: LS and RLS solutions for 0:2. The LS solution interpolates the points exactly, while the RLS one guarantees a worst-case residual error less than 1:1573. For the RLS solution is the zero polynomial. 8. Conclusions. This paper shows that several robust least-squares (RLS) problems with unknown-but-bounded data matrices are amenable to (convex) second-order cone or semidefinite programming (SOCP or SDP). The implication is that these RLS problems can be solved in polynomial-time, and efficiently in practice. When the perturbation enters linearly in the data matrices, and its size is measured by Euclidean norm, or in a linear-fractional problem with full perturbation matrix \Delta, the method yields the exact value of the optimal worst-case residual. In the other cases we have examined (such as arbitrary rational dependence of data matrices on the perturbation parameters), computing the worst-case residual is NP-complete. We have shown how to compute, and optimize, using SDP, an upper bound on the worst-case residual, that takes into account structure information. In the unstructured case, we have shown that both the worst-case residual and the (unique) RLS solution are continuous. The unstructured RLS can be interpreted as a regularization method for ill-conditioned problems. A striking fact is that the cost of the RLS solution is equal to a small number of least-squares problems arising in classical Tikhonov regularization approaches. This method provides a rigorous way to compute the optimal parameter from the data and associated perturbation bounds. Similar (weighted) least-squares interpretations and continuity results were given for the structured case. In our examples, we have demonstrated the use of a SOCP code [27], and a general-purpose semidefinite programming code, SP [45]. Future work could be devoted to writing special code that exploits the structure of these problems, in order to further increase the efficiency of the method. For instance, it seems that, in many problems, the perturbation matrices are sparse, and/or have special (e.g., Toeplitz) structure. The method can be used for several related problems. Constrained RLS. We may consider problems where additional (convex) constraints are added on the vector x. (Such constraints arise naturally in e.g., image processing). For instance, we may consider problem (1) with an addi- 28 L. EL GHAOUI AND H. LEBRET tional linear (resp. quadratic convex) constraint (Cx) i - 0, To solve such a problem, it suffices to add the related constraint to corresponding SOCP or SDP formulation. (Note that the SVD approach of x3.3 fails in this case.) ffl RLS problems with other norms. We may consider RLS problems in which the worst-case residual error in measured in other norms, such as the maximum ffl Matrix RLS. We may of course, derive similar results when the constant term b is a matrix. The worst-case error can be evaluated in a variety of norms. ffl Error-in-Variables RLS. We may consider problems where the solution x is also subject to uncertainty (due to implementation and/or quantization errors). That is, we may consider a worst-case residual of the form are given. We may compute (and optimize) upper bounds on the above quantity using SDP. This subject is examined in [25]. Acknowledgments . The authors wish to thank the anonymous reviewers for their precious comments, which led to many improvements over the first version of this paper. We are particularly indebted to the reviewer who pointed out the SOCP formulation for the unstructured problem. We also thank G. Golub and R. Tempo for providing us with some related references, and A. Sayed for sending us the preliminary draft [5]. The paper has also benefited from many fruitful discussions with S. Boyd, F. Oustry, B. Rottembourg and L. Vandenberghe. A. Proof of Theorem 4.1. Introduce the eigendecomposition of F and a related decomposition for g: writes at the optimum, then there exists a nonzero vector u such that (-I \Gamma F From inequality (29), we conclude that g T In other words, - g)-controllable, and u is an eigenvector that proves this uncontrollability. Using in (49), we obtain the optimal value of - in this case: Thus, the worst-case residual can be computed as claimed in the theorem. ROBUST LEAST SQUARES 29 For every pair (-) that is optimal for problem (29), we can compute a worst-case perturbation as follows. Define We have - at the optimum if and only - (that is, g)-controllable and the function f defined in (31) satisfies df d- 0: In this case, the optimal - satisfies that is, kffi 1. Using this and (50), we obtain F This proves that ffi 0 is a worst-case perturbation. at the optimum, then df d- which implies that kffi 0 k - 1. Since - max there exists a vector u such that loss of generality, we may assume that the vector We have F This proves that ffi defined above is a worst-case perturbation. In both cases seen above (- equals - worst-case perturbation is any vector ffi such that (We have just shown that the above equations always have a solution ffi when - is optimal.) This ends our proof. B. Proof of Lemma 5.1. We use the following result, due to Nemirovsky [32]. Lemma B.1. Let \Gamma(p; a) be a scalar function of positive integer p and p-dimensional vector a, such that, first, \Gamma is well-defined and takes rational values from (0; kak \Gamma2 ) for all positive integers p and all p-dimensional vectors a with kak - 0:1, and second, the value of this function at a given pair (p; a) can be computed in time polynomial in p and the length of the standard representation of the (rational) vector a. Then the problem L. EL GHAOUI AND H. LEBRET Given an integer p - 0 and a 2 R p , kak - 0:1, with rational positive entries, determine whether kffik 1-1 is NP-complete. Besides this, either (51) holds, or kffik 1-1 where d(a) is the smallest common denominator of the entries of a. To prove our result, it suffices to show that for some appropriate function \Gamma satisfying to the conditions of lemma B.1, we can reduce, for any given p; a, problem to ours, in polynomial time. Set 2a T a (a T a This function satisfies all requirements of lemma B.1, so problem P \Gamma (p; a) is NP-hard. Given rational positive entries, set A, b, D and x as follows. First, set D to be the set of diagonal matrices of R p\Thetap . Set Finally, set A, b as in (34) and 1, the worst-case residual for this problem is r D (A; b; kffik 1-1 kffik 1-1 Our proof is now complete. C. Proof of Theorem 5.3. In this section, we only prove theorem 5.3. The proof of theorem 5.2 follows the same lines. We start from problem (43), the dual of which is the maximization of 2(b T w +R T b u) subject to and the linear constraints A G; TrG(Y ROBUST LEAST SQUARES 31 Since both primal and dual problems are strictly feasible, every primal and dual feasible points are optimal if and only if ZF(-; G; is defined in (38) (see [46]). One obtains, in particular, G. Using equation (58) and (55), we obtain which implies that from equality of the primal and dual objectives (the trivial case can be easily ruled out). Assume that the matrix \Theta defined in (39) is positive-definite at the optimum. From equations (57)-(59), we deduce that the dual variable Z is rank-one: w Using (57) and (59), we obtain \Theta From (55), it is easy to derive the expression (42) for the optimal x in the case when \Theta ? 0 at the optimum, and RA is full-rank. We now show that the upper bound is exact at the optimum in this case. If we use condition (54), and the expression for Z; V deduced from (53), we obtain This implies that there exists I, such that \Theta ? 0, a straightforward application of lemma 2.3 shows that det(I \Gamma D\Delta) 6= 0, so we obtain (from (61)) and (from (53)), we have 1=2. We can now compute (from (55) and (60)). L. EL GHAOUI AND H. LEBRET Therefore, (from We obtain which proves that the matrix \Delta is a worst-case perturbation. --R An efficient newton barrier method for minimizing a sum of euclidean norms Pertubed optimization under the second order regularity hypothesis Linear Matrix Inequalities in System and Control Theory A new linear least-squares type model for parameter estimation in the presence of data uncertainties Computing the real structured singular value is NP-hard Structured total least squares and L 2 approximation problems Image reconstruction and restoration: overview of common estimation problems Unifying robustness analysis and system ID LMITOOL: A front-end for LMI op- timization Algorithms for the regularization of ill conditioned least-squares problems Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics Collinearity and total least squares Optimization of weighting constant for regularization in least squares system identification An analysis of the total least squares problem Quadratically constrained least squares and quadratic prob- lems The robust generalized least-squares estimator Regularization methods for large-scale problems Backward error and condition of structured linear systems The application of constrained least-squares estimation to image restoration by digital computer All controllers for the general H1 control problem: LMI existence conditions and state space formulas Social Sciences Synth'ese de diagrammes de r'eseaux d'antennes par optimisation convexe On continuity/discontinuity in robustness indicators Least squares methods for ill-posed problems with a prescribed bound Several NP-hard problems arising in robust stability analysis Interior point polynomial methods in convex programming: Theory and applications and application of bounded parameter models Robust solutions to uncertain semidefinite pro- grams Checking robust nonsingularity is NP-hard Matrix Anal. a connection between robust control and identifica- tion Indefinite trust region subproblems and nonsymmetric eigenvalue perturbations Solutions of Ill-Posed Problems The total least squares problem: computational aspects and analysis Robust estimation techniques in regularized image restora- tion Robust and Optimal Control --TR --CTR Michele Covell , Sumit Roy , Beomjoo Seo, Predictive modeling of streaming servers, ACM SIGMETRICS Performance Evaluation Review, v.33 n.2, p.33-35, September 2005 Jos F. 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Trafalis, Robust multiclass kernel-based classifiers, Computational Optimization and Applications, v.38 n.2, p.261-279, November 2007 Dimitris Bertsimas , Dessislava Pachamanova, Robust multiperiod portfolio management in the presence of transaction costs, Computers and Operations Research, v.35 n.1, p.3-17, January, 2008 Juan Liu , Ying Zhang , Feng Zhao, Robust distributed node localization with error management, Proceedings of the seventh ACM international symposium on Mobile ad hoc networking and computing, May 22-25, 2006, Florence, Italy D. Goldfarb , G. Iyengar, Robust portfolio selection problems, Mathematics of Operations Research, v.28 n.1, p.1-38, February Pannagadatta K. Shivaswamy , Chiranjib Bhattacharyya , Alexander J. 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uncertainty;robustness;ill-conditioned problem;regularization;least-squares problems;second-order cone programming;robust identification;robust interpolation;semidefinite programming

273585

Locality of Reference in LU Decomposition with Partial Pivoting.

This paper presents a new partitioned algorithm for LU decomposition with partial pivoting. The new algorithm, called the recursively partitioned algorithm, is based on a recursive partitioning of the matrix. The paper analyzes the locality of reference in the new algorithm and the locality of reference in a known and widely used partitioned algorithm for LU decomposition called the right-looking algorithm. The analysis reveals that the new algorithm performs a factor of $\Theta(\sqrt{M/n})$ fewer I/O operations (or cache misses) than the right-looking algorithm, where $n$ is the order of the matrix and $M$ is the size of primary memory. The analysis also determines the optimal block size for the right-looking algorithm. Experimental comparisons between the new algorithm and the right-looking algorithm show that an implementation of the new algorithm outperforms a similarly coded right-looking algorithm on six different RISC architectures, that the new algorithm performs fewer cache misses than any other algorithm tested, and that it benefits more from Strassen's matrix-multiplication algorithm.

Introduction . Algorithms that partition dense matrices into blocks and operate on entire blocks as much as possible are key to obtaining high performance on computers with hierarchical memory systems. Partitioning a matrix into blocks creates temporal locality of reference in the algorithm and reduces the number of words that must be transferred between primary and secondary memories. This paper describes a new partitioned algorithm for LU-factorization with partial pivoting, called the recursively-partitioned algorithm. The paper also analyzes the number of data transfers in a popular partitioned LU-factorization algorithm, the so-called right-looking algorithm, which is used in LAPACK [1]. The performance characteristics of other popular partitioned LU-factorization algorithms, in particular Crout and the left-looking algorithm used in the NAG library [4], are similar to those of the right-looking algorithm so they are not analyzed. The analysis of the two algorithms leads to two interesting conclusions. First, there is a simple system-independent formula for choosing the block size for the right- looking algorithm which is almost always optimal. Second, the recursively-partitioned algorithm generates asymptotically less memory traffic between memories than the right-looking algorithm, even if the block size for the right looking algorithm is chosen optimally. Numerical experiments indicate that the recursively-partitioned algorithm generates fewer cache misses and runs faster than the right-looking algorithm. The recursively-partitioned algorithm computes the LU decomposition with partial pivoting of an n-by-m matrix while transferring only \Theta(nm words between primary and secondary memories, where M is the size of the primary memory. The right-looking algorithm, on the other hand, transfers at least words. The number of words actually transferred by conventional algorithms depends on a parameter r, which is not chosen optimally in Parts of this research were performed while the author was a postdoctoral fellow at the IBM T.J. Watson Research Center and a postdoctoral associate at the MIT Laboratory for Computer Science. The work at MIT was supported in part by ARPA under Grant N00014-94-1-0985. y Xerox Palo Alto Research Center, 3333 Coyote Hill Road, Palo Alto, CA 94304. S. TOLEDO LAPACK. The new algorithm is optimal in the sense that the number of words that it transfers is asymptotically the same as the number transferred by partitioned (or blocked) algorithms for matrix multiplication and solution of triangular systems (at least when the number of columns is not very small compared to the size of primary memory). The right looking algorithm achieves such performance only when the matrix is so large that a few rows fill the primary memory. The recursively-partitioned algorithm algorithm has other advantages over conventional algorithms. It has no block-size parameter that must be tuned in order to achieve high performance. Since it is recursive, it is likely to perform better when the memory system has more than two levels, for example, on computer systems with two levels of cache or with both cache and virtual memory. To understand the main idea behind the new algorithm, let us look first at the conventional right-looking LU factorization algorithm. The algorithm decomposes the input matrix into dn=re blocks of at most r columns. Starting from the leftmost block of columns, the algorithm iteratively factors a block of r columns using a column- oriented algorithm. After a block is factored, the algorithm updates the entire trailing submatrix. The parameter r must be carefully chosen to minimize the number of words transferred between memories. If r is larger than M=n, many words must be transferred when a block of columns is factored. If r is too small, many trailing submatrices must be updated, and most of the updates require the entire trailing submatrix to be read from secondary memory. The main insight behind the recursively-partitioned algorithm is that there is no need to update the entire trailing submatrix after a block of columns is factored. After factoring the first column of the matrix, the algorithm updates just the next column to the right, which enables it to proceed. Once the second column is factored, we must apply the updates from the first two columns before we can proceed. The algorithm updates two more columns and proceeds. Once four columns are factored, they are used to update four more, and so on. In other words, the algorithm does not look all the way to the right every time a few columns are factored. As we shall see below, this short-sighted approach pays off. From another point of view, the new algorithm is a recursive algorithm. We know that the larger r (the number of columns in a block), the smaller the number of data transfers required for updating trailing submatrices. The algorithm therefore chooses the largest possible size, m=2. If that many columns do not fit within primary memory, they are factored recursively using the same algorithm, rather than being factored using a naive column oriented algorithm. Once the left m=2 columns are factored, they are used to update the right m=2 columns which are subsequently The rest of the paper is organized as follows. Section 2 describes and analyzes the recursively-partitioned algorithm. Section 3 analyzes the block-column right-looking algorithm. The actual performance of LAPACK's right-looking algorithm and the performance of the recursively-partitioned algorithm are compared in Section 4 on several high-end workstations. Section 5 concludes the paper with a discussion of the results and of related research. 2. Recursively-Partitioned LU Factorization. The recursively-partitioned algorithm is not only more efficient than conventional partitioned algorithms, but it is also simpler to describe and analyze. This section first describes the algorithm, and then analyzes the complexity of the algorithm in terms of arithmetic operations and in terms of the amount of data transferred between memories during its execution. LOCALITY OF REFERENCE IN LU DECOMPOSITION 3 The algorithm. The algorithm factors an n-by-m matrix A into an n-by-n permutation matrix P , an n-by-m unit lower triangular matrix L (that is, L's upper triangle is all zeros), and an m-by-m upper triangular matrix U , such that . A is treated as a block matrix A 11 A 21 A 21 A 22 where A 11 is a square matrix of order m=2-by-m=2. 1. If (that is, perform pivoting and scaling) A 11 A 21 U 11 and return. 2. Else, recursively factor A 11 A 21 U 11 3. Permute A 0A 0- A 12 A 22 4. Solve the triangular system L 11 5. A 00/ A 0\Gamma L 21 6. Recursively factor P 2 A 00= L 22 U 22 7. Permute L 0/ P 2 8. Return A 11 A 12 A 21 A 22 U 11 U 12 Complexity Analysis. It is not hard to see that the algorithm is numerically equivalent to the conventional column-oriented algorithm. Therefore, the algorithm has the name numerical properties as the conventional algorithm, and it performs same number of floating point operations, about nm In fact, all the variants of the LU-factorization algorithm discussed in this paper are essentially different schedules for the same algorithm. That is, they all have the same data-flow graph. We now analyze the number of words that must be transferred between the primary and secondary memories for n - m. The size of primary memory is denoted by M . For ease of exposition, we assume that the number of columns is a power of two. We denote the number of words that the algorithm must transfer between memories by IORP (n; m). We denote the number of words that must be transferred to solve an n-by-n triangular linear system with m right hand sides where the solution overwrites the right hand side by IOTS (n; m). We denote the number of words that must be transferred to multiply multiply an n-by\Gammam matrix by an m-by-k matrix and add the result to an n-by-k matrix by IOMM (n; m; k). Since the factorization algorithm uses matrix multiplication and solution of triangular linear system as subroutines, the number of I/O's it performs depends on the 4 S. TOLEDO number of I/O's performed by these subroutines. A partitioned algorithm for solving triangular linear systems performs at most IOTS (m; m) - M=3 M=3 M=3 I/O's. The actual number of I/O's performed is smaller, since the real crossover point is M=2, not M=3. Incorporating the improved bound into the analysis complicates the analysis with little effect on the final outcome. The number of I/O's performed by a standard matrix-multiplication algorithm is at most IOMM (n; n; m) - M=3 M=3 The bound for matrix multiplication holds for all values of n - m. The analysis here assumes the use of a conventional triangular solver and matrix multiplication, rather than so-called "fast" or Strassen-like algorithms. The asymptotic bounds for fast matrix-multiplication algorithms are better [5]. We analyze the recursively-partitioned algorithm using induction. Initially, the analysis that does not take into account the permutation of rows that the algorithm performs. We shall return to these permutations later in this section. The recurrence that governs the total number of words that are transferred by the algorithm is We first prove by induction that if 1=2 - m=2 - M=3, then IORP (n; m) - 2nm(1+ lg m). The base case true. Assuming that the claim is true for m=2, for 2:5m 2+ 3nm We now prove by induction that IORP (n; m) - 2nm mp M=3 for m=2 - M=3. The claim is true for the base case M=3 since m=2 - M=3 and since m=(2 1. Assuming that the claim is true for m=2, we have IORP (n; m) - 2nm mp M=3 LOCALITY OF REFERENCE IN LU DECOMPOSITION 5 M=3 M=3 mp M=3 M=3 nm 2p M=3 M=3 mp M=3 nm 2p M=3 M=3 mp M=3 nm 2p M=3 mp M=3 To bound the number word transfers due to permutations we compute the number of permutations a column undergoes during the algorithm. Each column is permuted either in the factorization in Step 2 and in the permutation in Step 7, or in the permutation in Step 3 and in the factorization in Step 6. It follows that each column is permuted 1 times. If each word is brought from secondary memory, then the total number of I/O's required for permutations is at most 2n m). This bound can be achieved when reading entire columns to primary memory and permuting them in primary memory. The following theorem summarizes the main result of this section. Theorem 2.1. Given a matrix multiplication subroutine whose I/O performance satisfies Equation (2.2) and a subroutine for solving triangular linear systems whose I/O performance satisfies Equation (2.1), the recursively-partitioned LU decomposition algorithm running on a computer with M words of primary memory computes the LU decomposition with partial pivoting of an n-by-m matrix using at most IORP (n; m) - 2nm mp M=3 I/O's. 3. Analysis of The Right-Looking LU Factorization. To put the performance of the recursively-partitioned algorithm in perspective, we now analyze the performance of the column-block right-looking algorithm. We first describe the algorithm and then analyze the number of data transfers, or I/O's, it performs. While 6 S. TOLEDO the bounds we obtain are asymptotically tight, we focus on lower bounds in terms of the constants. The number of I/O's required during the solution of triangular linear systems is smaller than the number of I/O's required during the updates to the trailing submatrix (a rank r update to a matrix), so we ignore the triangular solves in the analysis. Right-Looking LU. The algorithm factors an n-by-m matrix A such that m. The algorithm factors r columns in every iteration. In the kth iteration we decompose A into PA =4 A 11 A 12 A 13 A 21 A 22 A 23 where A 11 is a square matrix of order is a square matrix of order r. In the kth iteration the algorithm performs the following steps: 1. Factor A 22 A 2. Permute A 23 A 33 A 23 A 33 3. Permute 4. Solve the triangular system L 22 U 5. Update A 33 U The number of I/O's required to factor an n-by-r matrix using the column-by- column algorithm is nr 2when M - nr=2, but only when M - nr. To simplify the analysis, we ignore the range of M in which more than half then matrix fits within primary memory but less then the entire matrix. (Using one level of recursion leads to \Theta(nr) I/O's in this range). We use the facts that for M=3 M=3 rs if r - M=3 and that for r - s - t 2ts rs M=3 trs M=3 2ts if r - M=3 (3. LOCALITY OF REFERENCE IN LU DECOMPOSITION 7 The bound 2ts rs is an underestimate when M ! rs. We ignore this small slack in the analysis. The number of I/O's the algorithm performs depends on the relation of r to the dimensions of the matrix and to the size of memory. If r is so small that M - nr, then the updates to the trailing submatrix dominate the number of I/O's the algorithm performs. The updates to the trailing submatrix require at least I/O's. In particular, the first m=2r updates require at least If r is larger, factoring the m=r blocks of r columns requires at least r nr 2= nmrI/O's. The number of I/O's required for the rank-r updates depends on the value of r. If M=n - r - M=3, then the total number of I/O's performed by the rank-r updates is at least Therefore, the number of I/O's performed by the algorithm is at least nmr which is minimized at For m, the optimal value of r lies between (The exact value might deviate slightly from this range, since the expression we derived for the number of I/O's is only a lower bound). Substituting the optimal value of r, we find that the algorithm performs at least nm 1:5 \Gamma4 nm 1:5 I/O's in this range. If then the value performance than m. If M=3, then the value M=3 yields better performance than m. If r is yet larger, r - M=3, then the rank-r updates require I/O's. In particular, the first m=2r updates require at least M=3 nm 2p M=3 M=3 nm 2p M=3 8 S. TOLEDO I/O's. The total number of I/O's in this range, including both the updates and the factoring of blocks of columns, is therefore at least nm 2p M=3 M=3;M=n. The number of I/O's is minimized by choosing the smallest possible M=3. If the matrix is not very large compared to the size of main memory, n 2 =3 - M , it is also possible to choose r such that M=n. In this case, the total number of I/O's is at least nm 2p M=3 The analysis can be summarized as follows. A value of r close to max(M=n; m) is optimal for almost all cases. The only exception is for truly huge matrices, where m. For such matrices, M=3 is better than m. Combining the results, we obtain the following theorem. Theorem 3.1. Given a matrix multiplication subroutine whose I/O performance satisfies Equation (3.1) and a subroutine for solving triangular linear systems whose I/O performance satisfies Equation (3.2), the right-looking LU decomposition algorithm running on a computer with M words of primary memory computes the LU decomposition with partial pivoting of an n-by-m matrix using at least IORL (n; m) -! M=3 I/O's. The first case, r = M=n, leads to better performance only when more than columns fit within primary memory. Although these are lower bounds, they are asymptotically tight. The value 1=4 is a lower bound on the actual constant, which is higher than that. 4. Experimental Results. We have implemented and tested the recursively- partitioned algorithm 1 . The goal of the experiments was to determine whether the recursively partitioned algorithm is more efficient than the right-looking algorithm in practice. The results of the experiments clearly show that the recursively-partitioned algorithm performs less I/O and is that it is faster, at least on the computer on which the experiments were conducted. The results of the experiments complement our analysis of the two algorithms. The analysis shows that the recursively-partitioned algorithm performs less I/O than the right looking algorithm for most values of n and M . The analysis stops short of demonstrating that one algorithm is faster than another in three respects. First, the bounds in the analysis are not exact. Second, the analysis counts the total number Our Fortran 90 implementation is available online by anonymous ftp from theory.lcs.mit.edu as /pub/people/sivan/dgetrf90.f. The code can be compiled by many Fortran 77 compilers, in- luding compilers from IBM, Silicon Graphics, and Digital, by removing the RECURSIVE keyword and using a compiler option that enables recursion (see [11] for details). LOCALITY OF REFERENCE IN LU DECOMPOSITION 9 of I/O's in the algorithm, but the distribution of the I/O within the algorithm is significant. Finally, the analysis uses a simplified model of a two-level hierarchical memory that does not capture all the subtleties of actual memory systems. The experiments show that even though our analysis is not exact in these respects, the recursively-partitioned algorithm is indeed faster. Three sets of experiments are presented in this section. The first set presents and analyzes in detail experiments on IBM RS/6000 workstations. The goal of this set of experiments is to establish that the recursively-partitioned algorithm is faster than the right-looking algorithm. The second set of experiments show, in less detail, that the recursively-partitioned algorithm outperforms LAPACK's right-looking algorithm on a wide range of architectures. The goal of the second set of experiments is to establish the robustness of the performance of the recursively-partitioned algorithm. The third set of experiments shows that using Strassen's matrix multiplication algorithm speeds up the recursively-partitioned algorithm, but does not seem to speed up the right- looking algorithm. Some of the technical details of the experiments, such as operating system ver- sions, compiler versions, and compiler options are omitted from this paper. These details are fully described in our technical report [11]. Detailed Experimental Analyzes. The first set of experiments was performed on an IBM RS/6000 workstation with a 66.5 MHz POWER2 processor [14], 128 Kbytes 4-way set associative level-1 data-cache, a 1 Mbytes direct mapped level-2 cache, and a 128-bit-wide main memory bus. The POWER2 processor is capable of issuing two double-precision multiply-add instructions per clock cycle. Both LAPACK's right looking LU-factorization subroutine DGETRF and the recursively partitioned algorithm were compiled by IBM's XLF compiler version 3.2. All the algorithms used the BLAS from IBM's Engineering and Scientific Subroutine Library (ESSL). On square matrices we have also measured the performance of the LU-factorization subroutine DGEF from ESSL. The interface of this subroutine only allows for the factorization of square matrices. The coding style and the data structures used in the recursively-partitioned algorithm are the same as the ones used by LAPACK. In par- ticular, permutations are represented in both algorithms as a sequence of exchanges. In all cases, the array that contains the matrix to be factored was allocated statically and aligned on a 16-byte boundary. The leading dimension of the matrix was equal to the number of rows (no padding). The performance of the algorithms was assessed using measurements of both running time and cache misses. Time was measured using the machines real-time clock, which has a resolution of one cycle. The number of cache misses was measured using the POWER2 performance monitor [13]. The performance monitor is a hardware sub-system in the processor capable of counting cache misses and other processor events. Both the real-time clock and the performance monitor are oblivious to time sharing. To minimize the risk that measurements are influenced by other processes, we ran the experiments when no other users used the machine (but it was connected to the network). We later verified that the measurements are valid by comparing the real- time-clock measurements with the user time reported by AIX's getrusage system call on an experiment by experiment basis. All measurements reported here are based on an average of 10 executions. We have coded two variants of the recursively-partitioned algorithm. The two versions differ in the way permutations are applied to submatrices. In one version, permutations are applied using LAPACK's auxiliary subroutine DLASWP. This sub- Table The performance in millions of operations per second (Mflops) and the number of cache misses per thousand floating point operations (CM/Kflop) of five LU-factorization algorithms on an IBM RS/6000 Workstation, on square matrices. The figures for LAPACK's DGETRF are those of the block size r with the best running time, in upright letters, and those of the block size with the smallest number of cache misses, in italics. The minimum number of cache misses does not generally coincide with the minimum running time. See the text for a full description of the experiments. Subroutine Mflops CM/Kflop Mflops CM/Kflop LAPACK's DGETRF, row exchanges 178, 176 5.81, 5.65 170, 168 5.45, 5.29 Recursively-partitioned, row exchanges 201 3.76 186 4.14 LAPACK's DGETRF, permuting by columns 201, 199 2.94, 2.81 198, 195 3.11, 3.02 Recursively-partitioned, permuting by columns 222 1.61 223 1.59 ESSL's DGEF 228 2.15 221 3.42 routine, which is also used by LAPACK's right-looking algorithm, permutes the rows of a submatrix by exchanging rows using the vector exchange subroutine DSWAP, a level-1 BLAS. The second version permutes the rows of the matrix by applying the entire sequence of exchanges to one column after another. The difference amounts to swapping the inner and outer loops. This change was suggested by Fred Gustavson. The first experiment, whose results are summarized in Table 4.1, was designed to determine the effects of a complex hierarchical memory system on the partitioned algorithms. Four facts emerge from the table. 1. The recursively partitioned algorithm performs less cache misses and delivers higher performance than the right-looking algorithm. ESSL's subroutine performs less cache misses than LAPACK but more than the recursively- partitioned algorithm, but it achieves best or close to best performance. 2. Permuting one column at a time leads to less cache misses and faster execution than exchanging rows. This is true for both the right-looking algorithm and the recursively-partitioned algorithm. This is probably a result of the advantage of the stride-1 access to the column in the column permuting over the large stride access to rows in the row exchanges. 3. The performance in term of both time and cache misses of all the algorithms except the recursively-partitioned with column permuting is worse when the leading dimension of the matrix is a power of 2 than when it is not. The performance of the recursively-partitioned algorithm with column permuting improves by less than half a percent. The degradation in performance on a power of 2 is probably caused by fact that the caches are not fully associative. 4. The running time depends on the measured number of cache misses, but not completely. This can be seen both from the fact that ESSL's DGEF performs more cache misses than the recursively partitioned algorithm, but it is faster, and from the fact that the block size that leads to the minimum number of cache misses in the DGETRF does not lead to the best running time. The discrepancy can be caused by several factors that are not mea- sured, including misses and conflicts in the level-2 cache, TLB misses, and instruction scheduling. In all four cases in the table the minimum running time is achieved with a value of r that is higher than the number that leads to a minimum number of cache misses. For example, on with row exchanges performed the least number of cache misses with but the fastest running time was achieved with may mean that LOCALITY OF REFERENCE IN LU DECOMPOSITION 11 500 1000 1500 2000160200240 Mflops Order of Matrix RL, optimal r and r=64 500 1000 1500 2000246Cache Reloads Per KFlop Order of Matrix RL, optimal r RL, r=64 Fig. 4.1. The performance in Mflops (on the left) and the number of cache misses per Kflop (on the right) of LU factorization algorithms on an IBM RS/6000 Workstation. These graphs depict the performance of the recursively-partitioned (PR) and right-looking (RL) algorithms on square matrices. The optimal value of r was selected experimentally from powers of 2 between 2 and 256. The dashed lines represent the performance of the recursively-partitioned algorithms with column permuting (CP). the cause of the discrepancy is misses in the level-2 cache, which is larger than the level-1 cache and therefore may favor a larger block size (since more columns fit in it). In summary, the experiment shows that although the implementation details of the memory system influence the performance of the algorithms, the recursively-parti- tioned algorithm still emerges as faster than the right-looking one when they are implemented in a similar way. The second set of experiments was designed to assess the performance of the algorithms over a wide range of input sizes. The performance and number of cache misses of the algorithms are presented in Figure 4.1 on square matrices ranging in order from 200 to 2000. The level-1 cache is large enough to store a matrix of order 128. The following points emerge from the experiment. 1. Beginning with matrices of order 300, the the recursively-partitioned algorithm with column permuting is faster than the same algorithm with row exchanges which is still faster than LAPACK's DGETRF with row exchanges (we did not measure the performance of DGETRF with column permuting in this experiment). 2. The performance of DGETRF with optimal block size r and with essentially the same except at although the optimal block size clearly leads to a smaller number of cache misses from 3. The recursively-partitioned algorithm performs less cache misses than ESSL's DGEF on all input sizes, but it is not faster. As in the first experiment, the experiment itself does not indicate what causes this phenomenon. We speculate that it is caused by better instruction scheduling or fewer misses in the level-2 cache. The next experiment was designed to determine the sensitivity of the performance of the right-looking algorithm to the block size r. We used the column permuting strategy which proved more efficient in the previous experiments. The experiment consists of running the algorithm on a range of block sizes on a square matrix of order 1007 and on a rectangular 62500-by-64 matrix. The factorization of a rectangular Mflops Block Size r Cache Reloads Per KFlop Block Size r Fig. 4.2. The performance in Mflops (on the left) and the number of cache misses per Kflop (on the right) of the right-looking algorithm with column permuting with as a function of the block size r. The order of the square matrix used is 1007. Note that the y-axes do not start from zero. Block Size r Reloads Per KFlop Block Size r Fig. 4.3. The performance in Mflops (on the left) and the number of cache misses per Kflop (on the right) of the right-looking algorithm with column permuting with as a function of the block size r. The dimensions of the matrix are 62500 by 64. For comparison, the performance of the recursively partitioned algorithm on this problem is 118 Mflops and 11:03 CM/Kflop. matrix with arises as a subproblem in out-of-core LU factorization algorithms that factor blocks of columns that fit within core. The specific dimensions of the matrices were chosen so as to minimize the effects of conflicts in the memory system on the results. The results for shown in Figures 4.2 show that the minimum number of cache misses occurs at which is higher than and that the best performance is achieved with an even higher value of r, 55. The performance is not very sensitive to the choice of r, however, and all values between about 50 and 70 yield essentially the same performance, 201 Mflops. The results for matrices, shown in Figures 4.3, show that the minimum number of cache misses occur at and the best performance occurs at happens to coincide exactly with m. The sensitivity to r is greater here than in the square case, especially below the optimal value. The last experiment in this set, presented in Figure 4.4, was designed to determine whether the discrepancy between the optimal block size in terms of level-1 cache misses LOCALITY OF REFERENCE IN LU DECOMPOSITION 13 Mflops Block Size r Cache Reloads Per KFlop Block Size r Fig. 4.4. The performance in Mflops (on the left) and the number of cache misses per Kflop (on the right) of the right-looking algorithm with column permuting with as a function of the block size r. The order of the square matrix used is 1007. The machine used here has a bigger level-1 cache and no level-2 cache than the machine used in all the other experiments. Compare to Figure 4.2. For comparison, the performance of the recursively-partitioned algorithm on this problem on this machine is 229 Mflops and 0.650 CM/Kflop. and the optimal block size in terms of running time was caused by the level-2 cache. The experiment repeats the last experiment for square matrices of order 1007, except that the experiment was conducted on a machine with a 256-bit-wide main memory bus, 256 Kbytes level-1 cache, and no level-2 cache. The two machines are identical in all other respects. There is a discrepancy in optimal block sizes in Figure 4.4, but it is smaller than the discrepancy in Figure 4.2. The experiment shows that the discrepancy is not caused solely by the level-2 cache. It is not possible to determine whether the smaller discrepancy in this experiment is due to the lack of level-2 cache or to the larger level-1 cache. Robustness Experiments. The second set of experiments show that the performance advantage of the recursively partitioned algorithm, which was demonstrated by the first set of experiments, is not limited to a single computer architecture. The experiments accomplish this goal by showing that the recursively partitioned algorithm outperforms the right looking algorithm on a wide range of architectures. All the experiments in this set compare the performance of the recursively partitioned algorithm with the performance of LAPACK's right-looking on two sizes of square matrices, when the larger matrices do not fit within main memory). These sizes were chosen so as to minimize the impact of cache associativity on the results. Each measurement reported represents the average of the best 5 out of 10 runs, to minimize the effect of other processes in the system. The block size for the right-looking algorithm was LAPACK's default We used the following machine configurations: ffl A 66.5 MHz IBM RS/6000 workstation with a POWER2 processor, 128 Kbytes 4-way set associative data-cache, a 1 Mbytes direct mapped level- cache. and a 128-bit-wide bus. We used the BLAS from IBM's ESSL. ffl A 25 MHz IBM RS/6000 workstation with a POWER processor, 64 Kbytes 4-way set associative data-cache, and a 128-bit-wide bus. We used the BLAS from IBM's ESSL. ffl A 100 MHz Silicon Graphics Indy workstation with a MIPS R4600/R4610 14 S. TOLEDO Table The running time in seconds of LU factorization algorithms on several machines. For each machine and each matrix order, the table shows the running times of the recursively-partitioned (RP) algorithm and the right-looking (RL) algorithm with row exchanges and column permutations. Some measurements are not available and marked as N/A because the amount of main memory is insufficient to factor the larger matrix in core. See the text for a full description of the experiments. Row Column Row Column Exchanges Pivoting Exchanges Pivoting Machine RL RP RL RP RL RP RL RP IBM POWER 22.81 19.07 17.87 16.86 146.1 143.4 135.8 129.7 CPU/FPU pair, a 16 Kbytes direct mapped data cache, and a 64-bit-wide bus. We used the SGI BLAS. This machine has only 32 Mbytes of main memory, so the experiment does not include matrices of order ffl A 250 MHz Silicon Graphics Onyx workstation with 4 MIPS R4400/R4010 CPU/FPU pairs, a 16 Kbytes direct mapped data cache per processor, a 4 Mbytes level-2 cache per processor, and a 2-way interleaved main memory system with a 256-bit-wide bus. The experiment used only one processor. We used the SGI BLAS. ffl A 150 MHz DEC 3000 Model 500 with an Alpha 21064 processor, 8 Kbytes direct mapped cache, and a 512 Kbytes level-2 cache. We used the BLAS from DEC's DXML for IEEE floating point. A limit on the amount of physical memory allocated to a process prevented us from running the experiment on matrices of order ffl A 300 MHz Digital AlphaServer with 4 Alpha 21164 processors, each with an 8 Kbytes level-1 data cache, a 96 Kbytes on-chip level-2 cache, and a 4 Mbytes level-2 cache. The experiment used only one processor. We used the BLAS from DEC's DXML for IEEE floating point. The results, which are reported in Table 4.2, show that the recursively partitioned algorithm consistently outperforms the right-looking algorithm. The results also show that permuting columns is almost always faster than exchanging rows. Experiments using Strassen's Algorithm. Performing the updates of the trailing submatrix using a variant of Strassen's algorithm [10] improved the performance of the recursively partitioned algorithm. We replaced the call to DGEMM, the level-3 BLA subroutine for matrix multiply-add by a call to DGEMMB, a public domain implementation 2 of a variant of Strassen algorithm [3]. (Replacing the calls to DGEMM by calls to a Strassen matrix-multiplication subroutine in IBM's ESSL gave similar results). DGEMMB uses Strassen's algorithm only when all the dimensions of the input matrices are greater than a machine-dependent constant. The authors of DGEMMB set this constant to 192 for IBM RS/6000 workstations. In the recursively-partitioned algorithm with column permuting, the replacement Available online from http://www.netlib.org/linalg/gemmw. LOCALITY OF REFERENCE IN LU DECOMPOSITION 15 of DGEMM by DGEMMB reduced the factorization time on the POWER2 machine to 2:99 seconds for 1007 and to 22:18 seconds for 2014. The factorization times with the conventional matrix multiplication algorithm, reported in the first line of Table 4.2, are 3:05 and 23:45 seconds. The running time was reduced from 182:7 to 166:8 seconds on a matrix of order 4028. The change would have no effect on the right-looking algorithm, since in all the matrices it multiplies at least one dimension is r which was smaller than 192 in all the experiments. A similar experiment carried out by Bailey, Lee, and Simon [2] showed that Strassen's algorithm can accelerate the LAPACK's right-looking LU factorization on a Cray Y-MP. The largest improvements in performance, however, occured when large values of r were used. The fastest factorization of a matrix of order example, was obtained with Such a value is likely to cause poor performance on machines with caches. (The Cray Y-MP has no cache.) On the IBM POWER2 ma- chine, which has caches, increasing r from 64 to 512 causes the factorization time with a conventional matrix multiplication algorithm to increase from 30:8 seconds 54 sec- onds. Replacing the matrix multiplication subroutine by DGEMMB with reduces the solution time, but by less than 2 seconds. 5. Conclusions. The recursively-partitioned algorithm should be used instead of the right-looking algorithm because it delivers similar or better performance without parameters that must be tuned. No parameter to choose means that there is no possibility of a poor choice, and hence the new algorithm is more robust. Section 4 shows that the performance of the right-looking algorithm can be sensitive to r, and that the best performance does not always coincide with the block size that causes the smallest number of cache misses. Choosing r can be especially difficult on machines with more than two levels of memory. A recursive algorithm, on the other hand, is a natural choice for hierarchical memory systems with more than two levels. The recursively partitioned algorithm provides a good opportunity to use a fast matrix multiplication algorithm, such as Strassen's algorithm. Since a significant fraction of the work performed by the recursively partitioned algorithm is used to multiply large matrices, the benefit of using Strassen's algorithm can be large. The right-looking algorithm performs the same work by several multiplications of smaller matrices, so the benefit of Strassen's algorithm should be smaller. The analysis of the right-looking algorithm in Section 3 shows how the block size r should be chosen. The value r - m is optimal with two exceptions. When a single row is too large to fit within primary memory, a value M=3 leads to better performance. When more than columns fit within primary memory, r should be set to M=n to minimize memory traffic. The extreme cases are the source of the difficulty in choosing a good value of r for hierarchical memory systems with more than two levels. In our experiments, the performance of the right-looking algorithm on matrices with more rows than columns was very sensitive to the choice of r, but it was not sensitive on large square matrices. In the typical cases, when at least one row fits within primary memory, the right-looking algorithm with an optimal choice of r performs a factor of \Theta( more data transfers than the recursively partitioned algorithm. In our experiments this factor led to a significant difference in both the number of cache misses and the running time. The conclusion that the value m is often close to optimal shows that there is a system-independent way to choose r. In comparison, the model implementation of ILAENV, LAPACK's block-size-selection subroutine, uses a fixed value, 64, S. TOLEDO and LAPACK's User's Guide advises that system-dependent tuning of r could improve performance. The viewpoint of the LAPACK designers seems to be that r is a system-dependent parameter whose role is to hide the low bandwidth of the secondary memory system during the updates of the trailing submatrices. Our analysis here shows that the true role of r is to balance the number of data transfers between the two components of the algorithm: the factorization of blocks of columns and the updates of the trailing submatrices. Designers of out-of-core LU decomposition codes often propose to use block- column (or row) algorithms. Many of them propose to choose r = M=n so that an entire block of columns fits within primary memory [4, 6, 7, 15]. This approach works well when the columns are short and a large number of them fits within primary memory, but the performance of such algorithms would be unacceptable when only few columns fit within primary memory. Some researchers [7, 8, 9] suggest that algorithms that use less primary memory than is necessary for storing a few columns might have difficulty implementing partial pivoting. The analysis in this paper shows that it is possible to achieve a low number of data transfers even when a single row or column does not fit within primary memory. Womble et al. [15] presented a recursively-partitioned LU decomposition algorithm without pivoting. They claimed, without a proof, that pivoting can be incorporated into the algorithm without asymptotically increasing the number of I/O's the algorithm performs. They suggested that a recursive algorithm would be difficult to implement, so they implemented instead a partitioned left-looking algorithm using Toledo and Gustavson [12] describe a recursively-partitioned algorithm for out- of-core LU decomposition with partial pivoting. Their algorithm uses recursion on large submatrices, but switches to a left-looking variant on smaller submatrices (that would still not fit within main memory). Depending on the size of main memory, their algorithm can factor a matrix in 2=3 the amount of time used by an out-of-core left-looking algorithm with a fixed block size. 6. Acknowledgments . Thanks to Rob Schreiber for reading several early versions of this paper and commenting on them. Thanks to Fred Gustavson and Ramesh Agarwal for helpful suggestions. Thanks to the anonymous referees for several helpful comments. --R Using Strassen's algorithm to accelerate the solution of linear systems GEMMW: A portable level 3 BLAS Winograd variant of Strassen's matrix-matrix multiply algorithm A note on matrix multiplication in a paging environment Solving systems of large dense linear equations Matrix computations with Fortran and paging Gaussian elimination is not optimal Locality of reference in LU decomposition with partial pivoting The design and implementation of SOLAR The POWER2 performance monitor POWER2: Next generation of the RISC System/6000 family Beyond core: Making parallel computer I/O practical --TR --CTR Bradley C. 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Gustavson, A recursive formulation of Cholesky factorization of a matrix in packed storage, ACM Transactions on Mathematical Software (TOMS), v.27 n.2, p.214-244, June 2001 Rezaul Alam Chowdhury , Vijaya Ramachandran, Cache-oblivious dynamic programming, Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm, p.591-600, January 22-26, 2006, Miami, Florida Matteo Frigo , Volker Strumpen, The memory behavior of cache oblivious stencil computations, The Journal of Supercomputing, v.39 n.2, p.93-112, February 2007 Vladimir Rotkin , Sivan Toledo, The design and implementation of a new out-of-core sparse cholesky factorization method, ACM Transactions on Mathematical Software (TOMS), v.30 n.1, p.19-46, March 2004 Alexander Tiskin, Communication-efficient parallel generic pairwise elimination, Future Generation Computer Systems, v.23 n.2, p.179-188, February 2007 Lars Arge , Michael A. Bender , Erik D. Demaine , Bryan Holland-Minkley , J. Ian Munro, Cache-oblivious priority queue and graph algorithm applications, Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, May 19-21, 2002, Montreal, Quebec, Canada Jack Dongarra , Victor Eijkhout , Piotr uszczek, Recursive approach in sparse matrix LU factorization, Scientific Programming, v.9 n.1, p.51-60, January 2001 Siddhartha Chatterjee , Alvin R. Lebeck , Praveen K. Patnala , Mithuna Thottethodi, Recursive Array Layouts and Fast Matrix Multiplication, IEEE Transactions on Parallel and Distributed Systems, v.13 n.11, p.1105-1123, November 2002 Michael A. Bender , Ziyang Duan , John Iacono , Jing Wu, A locality-preserving cache-oblivious dynamic dictionary, Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms, p.29-38, January 06-08, 2002, San Francisco, California Dror Irony , Gil Shklarski , Sivan Toledo, Parallel and fully recursive multifrontal sparse Cholesky, Future Generation Computer Systems, v.20 n.3, p.425-440, April 2004 Isak Jonsson , Bo Kgstrm, Recursive blocked algorithms for solving triangular systemsPart I: one-sided and coupled Sylvester-type matrix equations, ACM Transactions on Mathematical Software (TOMS), v.28 n.4, p.392-415, December 2002 Richard Vuduc , James W. Demmel , Jeff A. Bilmes, Statistical Models for Empirical Search-Based Performance Tuning, International Journal of High Performance Computing Applications, v.18 n.1, p.65-94, February 2004

LU factorization;gaussian elimination;cache misses;partial pivoting;locality of reference

273592

A Survey of Combinatorial Gray Codes.

The term combinatorial Gray code was introduced in 1980 to refer to any method for generating combinatorial objects so that successive objects differ in some prespecified, small way. This notion generalizes the classical binary reflected Gray code scheme for listing n-bit binary numbers so that successive numbers differ in exactly one bit position, as well as work in the 1960s and 1970s on minimal change listings for other combinatorial families, including permutations and combinations.The area of combinatorial Gray codes was popularized by Herbert Wilf in his invited address at the SIAM Conference on Discrete Mathematics in 1988 and his subsequent SIAM monograph [Combinatorial Algorithms: An Update, 1989] in which he posed some open problems and variations on the theme. This resulted in much recent activity in the area, and most of the problems posed by Wilf are now solved.In this paper, we survey the area of combinatorial Gray codes, describe recent results, variations, and trends, and highlight some open problems.

Introduction One of the earliest problems addressed in the area of combinatorial algorithms was that of efficiently generating items in a particular combinatorial class in such a way that each item is generated exactly once. Many practical problems require for their solution the sampling of a random object from a combinatorial class or, worse, an exhaustive search through all objects in the class. Whereas early work in combinatorics focused on counting, by 1960, it was clear that with the aid of a computer it would be feasible to list the objects in combinatorial classes [Leh64]. However, in order for such a listing to be possible, even for objects of moderate size, combinatorial generation methods must be extremely efficient. A common approach has been to try to generate the objects as a list in which successive elements differ only in a small way. The classic example is the binary reflected Gray code [Gil58, Gra53] which is a scheme for listing all n-bit binary numbers so that successive numbers differ in exactly one bit. The advantage anticipated by such an approach is two-fold. First, generation of successive objects might be faster. Although for many combinatorial families, a straightforward lexicographic listing algorithm requires only constant average time per element, for other families, such as linear extensions, such performance has only been achieved by a Gray code approach [PR94]. Secondly, for the application at hand, it is likely that combinatorial objects which differ in only a small way are associated with feasible solutions which differ by only a small computation. For example in [NW78], Nijenhuis and show how to use a binary Gray code to speed up computation of the permanent. Aside from computational considerations, open questions in several areas of mathematics can be posed as Gray code problems. Finally, and perhaps one of the main attractions of the area, Gray codes typically involve elegant recursive constructions which provide new insights into the structure of combinatorial families. The term combinatorial Gray code first appeared in [JWW80] and is now used to refer to any method for generating combinatorial objects so that successive objects differ in some pre-specified, usually small, way. However, the origins of minimal change listings can be found in the early work of Gray [Gra53], Wells [Wel61], Trotter [Tro62], Johnson [Joh63], Lehmer [Leh65], Chase [Cha70], Ehrlich [Ehr73], and Nijenhuis and Wilf [NW78], and in the work of campanologists [Whi83]. In his article on the origins of the binary Gray code, Heath describes a telegraph invented by Emile Baudot in 1878 which used the binary reflected Gray code [Hea72]. (According to Heath, Baudot received a gold medal for his telegraph at the Universal Exposition in Paris in 1978, as did Thomas Edison and Alexander Graham Bell.) Examples of combinatorial Gray codes include (1) listing all permutations of that consecutive permutations differ only by the swap of one pair of adjacent elements [Joh63, Tro62], (2) listing all k-element subsets of an n-element set in such a way that consecutive sets differ by exactly one element [BER76, BW84, EHR84, EM84, NW78, Rus88a], (3) listing all binary trees so that consecutive trees differ only by a rotation at a single node [Luc87, LRR93], (4) listing all spanning trees of a graph so that successive trees differ only by a single edge [HH72, Cum66] (5) listing all partitions of an integer n so that in successive partitions, one part has increased by one and one part has decreased by one [Sav89], (6) listing the linear extensions of certain posets so that successive elements differ only by a transposition [Rus92, PR91, Sta92, Wes93], and (7) listing the elements of a Coxeter group so that successive elements differ by a reflection [CSW89]. Gray codes have found applications in such diverse areas as circuit testing [RC81], signal encoding [Lud81], ordering of documents on shelves [Los92], data compression [Ric86], statistics [DH94], graphics and image processing [ASD90], processor allocation in the hyper-cube hashing [Fal88], computing the permanent [NW78], information storage and retrieval [CCC92], and puzzles, such as the Chinese Rings and Towers of Hanoi [Gar72]. In recent variations on combinatorial Gray codes, generation problems have been considered in which the difference between successive objects, although fixed, is not required to be small. An example is the problem of listing all permutations of so that consecutive permutations differ in every location [Wil89]. The problem of generating all objects in a combinatorial class, each exactly once, so that successive objects differ in a pre-specified way, can be formulated as a Hamilton path/cycle problem: the vertices of the graph are the objects themselves, two vertices being joined by an edge if they differ from each other in the pre-specified way. This graph has a Hamilton path if and only if the required listing of combinatorial objects exists. A Hamilton cycle corresponds to a cyclic listing in which the first and last items also differ in the pre-specified way. But since the problem of determining whether a given graph has a Hamilton path or cycle is NP-complete [GJ79], there is no efficient general algorithm for discovering combinatorial Gray codes. Frequently in Gray code problems, however, the associated graph possesses a great deal of symmetry. Specifically, it may belong to the class of vertex transitive graphs. A graph G is vertex transitive if for any pair of vertices u, v of G, there is an automorphism OE of G with v. For example, permutations differing by adjacent transpositions give rise to a vertex transitive graph, as do k-subsets of an n-set differing by one element. It is a well-known open problem, due to Lov'asz, whether every undirected, connected, vertex transitive graph has a Hamilton path [Lov70]. Thus, schemes for generating combinatorial Gray codes in many cases provide new examples of vertex transitive graphs with Hamilton paths or cycles, from which we hope to gain insight into the more general open questions. It is also unknown whether all connected Cayley graphs (a subclass of the vertex transitive graphs) are hamiltonian. For many Gray code problems, especially those involving permutations, the associated graph is a Cayley graph. Although many Gray code schemes seem to require strategies tailored to the problem at hand, a few general techniques and unifying structures have emerged. The paper [JWW80] considers families of combinatorial objects, whose size is defined by a recurrence of a particular form, and some general results are obtained about constructing Gray codes for these families. Ruskey shows in [Rus92] that certain Gray code listing problems can be viewed as special cases of the problem of listing the linear extensions of an associated poset so that successive extensions differ by a transposition. In the other direction, the discovery of a Gray code frequently gives new insight into the structure of the combinatorial class involved. So, the area of combinatorial Gray codes includes many questions of interest in com- binatorics, graph theory, group theory, and computing, including some well-known open problems. Although there has been steady progress in the area over the past fifteen years, the recent spurt of activity can be traced to the invited address of Herbert Wilf at the SIAM Conference on Discrete Mathematics in San Francisco in June 1988, Generalized Gray Codes, in which Wilf described some results and open problems. (These are also reported in his SIAM monograph [Wil89].) All of the open problems on Gray codes posed by Wilf in [Wil89] have now been solved, as well as several related problems, and it is our intention here to follow up on this work. In this paper, we give a brief survey of the area of combinatorial Gray codes, describe recent results, variations and trends, and highlight some (new and old) open problems. This paper is organized into sections as follows: 1. Introduction; 2. Binary Numbers and Variations; 3. Permutations; 4. Subsets, Combinations, and Compositions; 5. Integer Partitions; 6. Set Partitions and Restricted Growth Functions; 7. Catalan Families; 8. Necklaces and Variations; 9. Linear Extension of Posets; 10. Acyclic Orientations; 11. Cayley Graphs and Permutation Gray Codes; 12. Generalizations of de Bruijn Sequences; 13. Concluding Remarks. In the remainder of this section, we discuss some notation and terminology which will be used throughout the paper. A Gray code listing of a class of combinatorial objects will be called max-min if the first element on the list is the lexicographically largest in the class and the last element is the lexicographically smallest. The Gray code is cyclic if the first and last elements on the list differ in the same way prescribed for successive elements of the list by the adjacency criterion. In many situations, the graph associated with a particular adjacency criterion is bipar- tite. If the sizes of the two partite sets differ by more than one, the graph cannot have a Hamilton cycle and thus there is no Gray code listing of the objects corresponding to the vertices, at least for the given adjacency criterion. In this case, we say that a parity problem exists. An algorithm to exhaustively list elements of a class C is called loop-free if the worst case time delay between listing successive elements is constant; the algorithm is called if, after listing the first element, the total time required by the algorithm to list all elements is O(N ), where N is the total number of elements in the class C. The term CAT was coined a. Binary Reflected b. Balanced c. Maximum Gap d. Non-composite 00000 11000 00000 10111 00000 00101 00000 01001 01000 10000 00111 00100 00111 00010 01011 00010 Figure 1: Examples of 5-bit binary Gray codes. by Frank Ruskey to stand for constant amortized time per element. Finally, we note that a Gray code for a combinatorial class is intrinsically bound to the representation of objects in the class. If sets A and B are two alternative representations of a class C under the bijections ff B, the closeness of ff(x) and ff(y) need not imply closeness of fi(x) and fi(y). That is, Gray codes are not necessarily preserved under bijection. Examples of this will be seen for several families, including integer partitions, set partitions, and Catalan families. Binary Numbers and Variations A Gray code for binary numbers is a listing of all n-bit numbers so that successive numbers (including the first and last) differ in exactly one bit position. The best known example is the binary reflected Gray code [Gil58, Gra53] which can be described as follows. If L n denotes the listing for n-bit numbers, then L 1 is the list 0, 1; for n ? 1, L n is formed by taking the list for L n\Gamma1 and pre-pending a bit of '0' to every number, then following that list by the reverse of L n\Gamma1 with a bit of '1' prepended to every number. So, for example, shown in Figure 1(a). Since the first and last elements of L n also differ in one bit position, the code is in fact a cycle. It can be implemented efficiently as a loop-free algorithm [BER76]. Note that a binary Gray code can be viewed as a Hamilton cycle in the n-cube. In practice, Gray codes with certain additional properties may be desirable (see [GLN88] for a survey). For example, note that as the elements of L n are scanned, the lowest order times, whereas the highest order bit changes only twice, counting the return to the first element. In certain applications, it is necessary that the number of bit changes be more uniformly distributed among the bit positions, i.e., a balanced Gray code is required. (See Figure 1(b) for an example from [VS80].) Uniformly balanced Gray codes were shown to exist for n a power of two by Wagner and West [WW91]. For general were suggested, but not proved, in [LS81, VS80, RC81]. Recently we have shown, using the Robinson-Cohn construction [RC81], that balanced Gray codes exist for all n in the following sense: Let a = b2 n =nc or b2 n =nc \Gamma 1, so that a is even. For each n ? 1 there is a cyclic n-bit Gray code in which each bit position changes either a or a In other applications, the requirement is to maximize the gap in a Gray code, which is defined in [GLN88] to be the shortest maximal consecutive sequence of 0's (or 1's) among all bit positions. (See Figure 1(c) for an example from [GLN88] in which the gap is 4, which is best possible for report a construction in which GAP(n)/n goes to 1 as n goes to infinity [GG]. Another variation, non-composite n-bit Gray codes, requires that no contiguous subsequence correspond to a path in any k-cube for Non-composite Gray codes have been constructed for all n [Ram90]. (See Figure 1(d) for an example from [Ram90].) A new constraint is considered in [SW95]. Define the density of a binary string to be the number of 1's in the string. Clearly, no Gray code for binary strings can list them in non-decreasing order of density. However, suppose the requirement is relaxed somewhat. Call a Gray code monotone if it runs through the density levels two at a time, that is, consecutive pairs of strings of densities those of densities Figure 2: A monotone Gray code for n. It is shown in [SW95] that monotone Gray codes can be constructed for all n. An example for shown in Figure 2. be the Boolean lattice of subsets of the set inclusion, and let H n denote the Hasse diagram of B n . The correspondence ng is a bijection from n-bit binary numbers to subsets of [n] and, under this bijection, a binary Gray code corresponds to a Hamilton path in H n . The vertices of H n can be partitioned into level sets, contains all of the i-element subsets of [n]. Then, a monotone Gray code is a Hamilton path in H n in which edges between levels i and must precede edges between levels j and Figure 3.) Monotone Gray codes have applications to the theory of interconnection networks, providing an embedding of the hypercube into a linear array which minimizes dilation in both directions [SW95]. In Section 4 we discuss their relationship to the middle two levels problem Fix a binary string ff and let B(n; ff) be the set of clean words for ff, i.e., the n-bit strings which do not contain ff as a contiguous substring. Does the subgraph of the n-cube induced by B(n; ff) have a Hamilton path, i.e., is there a Gray code for B(n; ff)? Squire has shown that the answer is yes if ff can be written as is a string with the property that no nontrivial prefix of fi is also a suffix of fi; otherwise there are parity @ @ @ @ @ @ @\Phi \Phi \Phi \Phi \Phi \Phi \Phi \Phi \Gamma\Phi \Phi \Phi \Phi \Phi \Phi \Phi \Phi \Delta Figure 3: The Hamilton path in H 5 corresponding to the monotone Gray code in Figure 2. problems for infinitely many n [Squ96]. It is natural to consider an extension of binary Gray codes to m-ary Gray codes. It was shown in [JWW80], using a generalization of the binary reflected Gray code scheme, that it is always possible to list the Cartesian product of finite sets so that successive elements differ only in one coordinate. A similar result is obtained in [Ric86] where each coordinate i is allowed to assume values in some fixed range results on clean words to m-ary Gray codes [Squ96], but leaves open the case when m is odd. Another listing problem for binary numbers, posed by Doug West, involves a change in the underlying graph. View an n-bit string as a subset of ng under the natural bijection g. Call two sets adjacent if they differ only in that one element increases by 1, one element decreases by 1, or the element '1' is deleted. The problem is to determine whether there is a Hamilton path in the corresponding graph, called the augmentation graph. When n(n \Gamma 1)=2 is even, a parity argument shows there is no Hamilton path. Otherwise, the question is open for n ? 7. Consider another criterion: two binary strings are adjacent if they differ either by (1) a rotation one position left or right or (2) by a negation of the last bit. The underlying graph is the shuffle-exchange network and a Hamilton path would be a "Gray code" for binary strings respecting this adjacency criterion. The existence of a Hamilton path in the shuffle-exchange graph, a long-standing open problem, was recently established by Feldman and Mysliwietz [FM93]. In [Fre79], Fredman considers complexity issues involved in generating arbitrary subsets of the set of n-bit strings so that successive strings differ only in one bit. He calls these quasi-Gray codes and establishes bounds and trade-offs on the resources required to generate successors using a decision assignment tree model of computation. Permutations Algorithms for generating all permutations of surveyed by Sedgewick in [Sed77]. Efficiency considerations provided the motivation for several early attempts to generate permutations in such a way that successive permutations Figure 4: Johnson-Trotter scheme for generating permutations by adjacent transpositions. Figure 5: Generating permutations by derangements, due to Lynn Yarbrough. differ only by the exchange of two elements. Such a Gray code for permutations was shown to be possible in several papers, including [Boo65, Boo67, Hea63, Wel61], which are described in [Sed77]. One disadvantage of these algorithms is that the elements exchanged are not necessarily in adjacent positions. It was shown independently by Johnson [Joh63] and Trotter [Tro62] that it is possible to generate permutations by transpositions even if the two elements exchanged are required to be in adjacent positions. The recursive scheme, illustrated in Figure 4, inserts into each permutation on the list for the element 'n' in each of the n possible positions, moving alternately from right to left, then left to right. A contrary approach to the problem is to require that permutations be listed so that each one differs from its predecessor in every position, that is, by a derangement. This problem was posed independently in [Rab84, Wil89]. The existence of such a list when n 6= 3 was established in [Met85] using Jackson's theorem [Jac80] and a constructive solution was presented in [EW85]. A simpler construction, ascribed to Lynn Yarbrough, is discussed in [RS87]. Yarbrough's solution is illustrated in Figure 5 and works as follows. Take each permutation on the Johnson-Trotter list for append an 'n', and rotate the resulting permutation, one position at a time, through its n possible cyclic shifts. As a final twist, swap the last two cyclic shifts. It is straightforward to argue that successive permutations differ in every position, using the property of the Johnson-Trotter list that successive permutations differ by adjacent transpositions. To generalize the problems of generating permutations, at one extreme, by adjacent transpositions, and at the other extreme, by derangements, consider the following. Given n and k satisfying n - k - 2, is it possible to list all permutations so that successive permutations differ in exactly k positions? This is shown to be possible, unless [Put89] and in [Sav90], where the listing is cyclic. It was shown further in [RS94a] that the positions (in which successive permutations differ) could be required to be contiguous. Putnam claims in [Put90] that when k is even (odd) all permutations (even permutations) can be generated by k-cycles of elements in contiguous positions. (Putnam's k-cycles need not be of the form (i; An interesting question arose in connection with a problem on Hamilton cycles in Cayley graphs (see Section 11.) Is it possible to generate permutations by "doubly adjacent" transpositions, i.e., so that successive transpositions are of neighboring pairs? Pair is considered to be a neighbor of (i The Johnson-Trotter scheme satisfies this requirement for n - 3, but not for Such a listing was shown to be possible by Chris Compton in his Ph.D. thesis [Com90]. It might be hoped that this could result in a very efficient permutation generation algorithm: it would become unnecessary to decide which of the adjacent pairs to transpose, only whether the next transposition is to the left or right of the current one. However, in its current form, Compton's algorithm is not practical, and is quite complex, even with the simplifications in [CW93]. The problem of generating all permutations of a multiset by adjacent interchanges was introduced by Lehmer as the Motel Problem [Leh65]. He shows that, because of parity problems, this is not always possible. It becomes possible, however, if the interchanged elements are not required to be adjacent and Ko and Ruskey give a CAT algorithm to generate multiset permutations according to this criterion [KR92]. 4 Subsets, Combinations, and Compositions Since there is a bijection between the subsets of an n-element set (an n-set) and the n-bit binary numbers, any binary Gray code defines a Gray code for subsets: two binary numbers differing in one bit correspond to two subsets differing by the addition or deletion of one element. For the subclass of combinations ( k-subsets of an n-set for fixed k), several Gray codes have been surveyed in [Wil89]. As observed in [BER76], a Gray code for combinations can be extracted from the binary reflected Gray code for n-bit numbers: delete from the binary reflected Gray code list all those elements corresponding to subsets which do not have exactly k elements. That which remains is a list of all k-subsets in which successive sets differ in exactly one element (see Figure 6(a) and compare to Figure 1(a)). The same list is generated by the revolving door algorithm in [NW78] and it can be described by a a. Revolving Door b. Strong Minimal Change c. Adjacent Interchange Figure Examples of Gray codes for combinations. simple recursive expression. A more stringent requirement is to list all k-sets with the strong minimal change property [EM84]. That is, if a k-set is represented as a sorted k-tuple of its elements, successive k-sets differ in only one position (see Figure 6(b)). Eades and McKay have shown that such a listing is always possible. An earlier solution was reported by Chase in [Cha70]. Perhaps the most restrictive Gray code which has been proposed for combinations is to generate k-subsets of an n-set so that successive sets differ in exactly one element and this element has either increased or decreased by one. This is called the adjacent interchange property since if the sets are represented as binary n-tuples, successive n-tuples may differ only by the interchange of a 1 and a 0 in adjacent positions (see Figure 6(c)). However, this is not always possible: it was shown that k-subsets of an n-set can be generated by adjacent interchanges if (i) k=0, 1, n, or is even and k is odd. In all other cases, parity problems prevent adjacent interchange generation [BW84, EHR84, HR88, Rus88a]. It was shown by Chase [Cha89] and by a simpler construction in [Rus93] that combinations can be generated so that successive elements differ either by an adjacent transposition or by the transposition of two bits that have a single '0' bit between them. There are several open problems about paths between levels of the Hasse diagram of the Boolean lattice, B n . The most notorious is the middle two levels problem which is attributed in [KT88] to Dejter, Erd-os, and Trotter and by others to H'avel and Kelley. The middle two levels of B 2k+1 have the same number of elements and induce a bipartite, vertex transitive graph on the k- and k + 1- element subsets of [2k 1]. The question is whether there is a Hamilton cycle in the middle two levels of B 2k+1 . At first glance, it would appear that one could take a Gray code listing of the k-subsets, in which successive elements differ in one element, and, by taking unions of successive elements, create a list of 1-subsets. Alternating between the lists would give a walk in the middle two levels graph, but, unfortunately, not a Hamilton path, at least not for any known Gray code on k-subsets. The graph formed by the middle two levels is a connected, undirected, vertex-transitive graph. Thus, either it has a Hamilton path, or it provides a counterexample to the Lov'asz conjecture. One approach to this problem which has been considered is to try to form a Hamilton cycle as the union of two edge-disjoint matchings. In [DSW88], it was shown that a Hamilton cycle in the middle two levels cannot be the union of two lexicographic matchings. However, other matchings may work and new matchings in the middle two levels have been defined [KT88, DKS94]. The largest value of k for which a Hamilton cycle is known to exist is Figure 7 for an example when 3. This unpublished work was done by Moews and Reid using a computer search [MR]. To speed up the search, they used a necklace-based approach, gambling that there would be a Hamilton path through necklaces which could be lifted to a Hamilton cycle in the original graph. We feel that a focus on the middle two levels of the necklace poset, as described in Section 8, is a promising approach to the middle two levels problem. Is there at least a good lower bound on the length of a longest cycle in the middle two levels of the Boolean lattice? Since this graph is vertex-transitive, a result of Babai [Bab79] shows that there is a cycle of length at least (3N(k)) 1=2 , where N(k) is the total number of vertices in the middle two levels of B 2k+1 . A result of Dejter and Quintana gives a cycle of length This was improved in [Sav93] to Figure 7: A Hamilton cycle in the middle two levels of B 7 . In a welcome breakthrough, Felsner and Trotter showed the existence of cycles of length at least 0:25N(k) [FT95]. The monotone Gray code, described in Section 1, contains as a subpath, a path in the middle two levels of length at least 0:5N(k) [SW95]. In [SW95], this was strengthened to get 'nearly Hamilton' cycles in the following sense: for every ffl ? 0, there is an h - 1 so that if a Hamilton cycle exists in the middle two levels of B 2k+1 for h, then there is a cycle of length at least (1 \Gamma ffl)N (k) in the mid-levels of B 2k+1 for all k - 1. Since Hamilton cycles are known for 1 - k - 11, the construction guarantees a cycle of length at least 0:839N(k) in the middle two levels of B 2k+1 for all k - 1. A variation on this problem is the antipodal layers problem: for which values of k is there a Hamilton path among the k-sets and sets of ng for all n, where two sets are joined by an edge if and only if one is a subset of the other? Results for limited values of k and n are given in [Hur94] and [Sim]. A composition of n into k parts is a sequence nonnegative integers whose sum is n. This is traditionally viewed as a placement of n balls into k boxes. Nijenhuis and asked in the first edition of [NW78] (p. 292, problem 34) whether it was possible to a. P (7; Figure 8: Gray codes for various families of integer partitions generate the k-compositions of n so that each is obtained from its predecessor by moving one ball from its box to another. Knuth solved this in 1974 while reading the galleys of the book and in [Kli82], Klingsberg gives a CAT implementation of Knuth's Gray code. Combinations and compositions can be simultaneously generalized as follows. Let denote the set of all ordered t-tuples t. If m i - s for is the set of s-combinations of a t-element set. If X is the multiset consisting of m i copies of element is the collection of s-element submultisets, or s- combinations of X . In [Ehr73], Ehrlich provides a loopless algorithm to generate multiset combinations so that successive elements differ in only two positions, but not necessarily by just \Sigma1 in those positions. It is shown in [RS95] that a Gray code still exists when the two position can change by only \Sigma1, thereby generalizing Gray code results for both combinations and compositions. 5 Integer Partitions A partition of an integer n is a sequence of positive integers x 1 satisfying Algorithms for generating integer partitions in standard orders such as lexicographic and antilexicographic were presented in [FL80] and [NW78]. The performance of the algorithms in [FL80] is analyzed in [FL81]. An integer partition in standard representation, can also be written as a list of pairs (y are the distinct integers appearing in the sequence -, and m i is the number of times y i appears in -. Ruskey notes in [Rus95] that a lexicographic listing of partitions in this ordered pairs representation has the property that successive elements of the list differ at most in the last three ordered pairs. asked the following question regarding a Gray code for integer partitions in the standard representation there a way to list the partitions of an integer n in such a way that consecutive partitions on the list differ only in that one part has increased by 1 and one part has decreased by 1? may decrease to 0 or a 'part' of size 0 may increase to 1.) Yoshimura demonstrated that this was possible for integers In [Sav89], it is shown constructively to be possible for all n. The result is a bit more general: for all n - k - 1, there is a way to list the set P (n; k), of all partitions of n into integers of size at most k, in Gray code order. Unless (n; the Gray code is max-min. As a consequence, each of the following can also be listed in Gray code order for all n, partitions of n whose largest part is k, (2) all partitions of n into k or fewer parts, and (3) all partitions of n into exactly k parts. See Figure 8(a) for a Gray code listing of P(7; 6). Exponents in the figure indicate the number of multiple copies. The approach in [Sav89], was to decompose the partitions problem, P (n; k), into sub-problems of two forms, a 'P ' form, which was the same form as the original problem, and a new 'M ' form. It was then shown that the P and M forms could be recursively defined in terms of (smaller versions of) both forms, thereby yielding a doubly recursive construction of the partitions Gray code. The algorithm has been implemented [Bee90] and can be modified to run in time O(jP (n; k)j). This strategy has been applied to yield Gray codes for other families of integer partitions. be the set of all partitions of n into parts of size at most k, in which the parts are required to be congruent to 1 modulo ffi . When just P (n; k). When 2, the elements of P ffi (n; are the partitions of n into odd parts of size at most k. It is shown in [RSW95] that P ffi (n; can be listed so that between successive partitions, one part increases by ffi (or ffi ones may appear) and another part decreases by ffi (or ffi ones may disappear.) (See Figure 8(b).) The Gray code is max-min unless (n; where a max-min Gray code is impossible. For the case of D(n; k), the set of partitions of n into odd parts of size at most k, the same strategy can be applied, but the construction becomes more complex. Surprisingly, it is still possible to list D(n; in Gray code order: between successive partitions one part increases by two and one part decreases by two [RSW95]. (See Figure 8(c).) The Gray code is max-min unless (n; or (12; 6), in which cases a max-min Gray code is impossible. One observation that follows from this work is that although there are bijections between the sets of partitions of n into odd parts and partitions of n into distinct parts, no bijection can preserve such Gray codes. The same techniques can be used to investigate Gray codes in other families of integer partitions, but each family has its own quirks: a small number of cases must be handled specially and subsets needed for linking recursively listed pieces can become empty. Never- theless, we conjecture that each of the following families has a Gray code enumeration, for arbitrary values of the parameters n, partitions of n into (a) distinct odd parts, (b) distinct parts congruent to 1 modulo ffi , (c) at most t copies of each part, (d) parts congruent to 1 modulo ffi , at most t copies of each part, and (e) exactly d distinct parts. 6 Set Partitions and Restricted Growth Functions A set partition is a decomposition of ng into a disjoint union of nonempty subsets called blocks. Let S(n) denote the set of all partitions of ng. For example, S(4) is shown in Figure 9(a). The restricted growth functions (RG functions) of length n, denoted R(n), are those strings a of non-negative integers satisfying a There is a well-known bijection between S(n) and R(n). For - 2 S(n), order the blocks of - according to their smallest element, for example, the blocks of would be ordered f1; 2; 7g, f3; 5; 6; 8g, f4; 10; 11g, f9g. Label the blocks of - in order by 0; :. The bijection assigns to - the string a (a)S(4) (b)L(4) in (c) Knuth's (d) modified (e)Ehrlich's lexicographic Gray code Knuth algorithm order Figure 9: Listings of S(4) and R(4). is the label of the block containing i. The associated string for - above is 2. For 4, the bijection is illustrated in the first two columns of Figure 9. In [Kay76], Kaye gives a CAT implementation of a Gray code for S(n), attributed to Knuth in [Wil89]. This was another problem posed by Nijenhuis and Wilf in their book [NW78] (p. 292, problem 25) and solved by Knuth while reading the galleys. In this Gray code, successive set partitions differ only in that one element moved to an adjacent block Figure 9(c).) However, the associated RG functions may differ in many positions. Ruskey [Rus95] describes a modification of Knuth's algorithm in which one element moves to a block at most two away between successive partitions and the associated RG functions differ only in one position by at most two (Figure 9(d).) Call a Gray code for RG functions strict if successive elements differ in only one position and in that position by only \Sigma1. Strict Gray codes for R(n) were considered in an early paper of Ehrlich where it was shown that for infinitely many values of n, they do not exist [Ehr73]. Nevertheless, Ehrlich was able to find an efficient listing algorithm for R(n) (loop- free) which has the following interesting property: successive elements differ in one position and the element in that position can change by 1, or, if it is the largest element in the string, it can change to 0. Conversely, a 0 can change to a the the largest value v in the string or to v + 1. For example, conversely. In the associated list of set partitions, this change corresponds to moving one element to an adjacent block in the partition, where the first and last blocks are considered adjacent (Figure 9(e).) Ehrlich's results are generalized in [RS94b] to the set of restricted growth tails, T (n; k), which are strings of non-negative integers satisfying a 1 - k and a i - 1+maxfa 1g. (These are a variation of the T (n; m) used in [Wil85] for ranking and unranking set partitions.) Note that T (n; R(n). Because of parity problems, for all k there are infinitely many values of n for which T (n; has no strict Gray code, that is, one in which only one position changes by 1. However, Gray codes satisfying Ehrlich's relaxed criterion are constructed and they can be made cyclic or max-min, properties not possessed by the earlier Gray codes. Consider now set partitions into a fixed number of blocks. For (n) be the set of partitions of ng into exactly b blocks. The bijection between S(n) and R(n) restricts to a bijection between S b (n) and The Ehrlich paper presents a loop-free algorithm for generating S b (n) in which successive partitions differ only in that two elements have moved to different blocks [Ehr73]. Ruskey describes a Gray code for R b (n) (and a CAT implementation) in which successive elements differ in only one position, but possibly by more than 1 in that position [Rus93]. It is shown in [RS94b] that in general, R b (n) does not have a strict Gray code, even under the relaxed criterion of Ehlich. It remains open whether there are strict Gray codes for R(n) and T (n; when the parity difference is 0. 7 Catalan Families In several families of combinatorial objects, the size is counted by the Catalan numbers, defined for n - 0 by These include binary trees on n vertices [SW86], well-formed sequences of 2n parentheses [SW86], and triangulations of a labeled convex polygon with bijections are known between most members of the Catalan family, a Gray code for one member of the family gives implicitly a listing scheme for every other member of the family. However, the resulting lists may not look like Gray codes, since bijections need not preserve minimal changes between elements. The problem of generating all binary trees with a given number of nodes was considered in several early papers, including [RH77], [Zak80], and [Zer85]. However, Gray codes in the Catalan family were first considered in [PR85], where binary trees were represented as strings of balanced parentheses. It was shown in [PR85] that strings of balanced parentheses could be listed so that consecutive strings differ only by the interchange of one left and one right parenthesis. For example '(()())(())' could follow `(()(())())'. The same problem was considered in [RP90] with the additional restriction that only adjacent left and right parentheses could be interchanged. For example, now '(()())(())' could not follow `(()(())())', but could follow '((()))(())'. The result of [RP90] is that all balanced strings of n pairs of parentheses can be generated by adjacent interchanges if and only if n is even or n ! 5, and for these cases, a CAT algorithm is given. A different minimal change criterion, focusing on binary trees, was considered in [Luc87] and [LRR93]: list all binary trees on n nodes so that consecutive trees differ only by a left or right rotation at a single node. The rotation operation is common in data structures where it is used to restructure binary search trees, while preserving the ordering properties. It was shown that such a Gray code is always possible and it can be generated efficiently [LRR93]. With a more intricate construction, Lucas was able to show that the associated graph was hamiltonian [Luc87], giving a cyclic Gray code. It so happens that under a particular bijection between binary trees with n nodes and the set of all triangulations of a labeled convex polygon with vertices, rotation in a binary tree corresponds to the flip of a diagonal in the triangulation [STT88]. So, the results of [Luc87, LRR93] also give a listing of all triangulations of a polygon so that successive triangulations differ only by the flip of a single diagonal. 8 Necklaces and Variations An n-bead, k-color necklace is an equivalence class of k-ary n-tuples under rotation. Figure lists the lexicographically smallest representatives of the n-bead, k-color necklaces for (n; asked if it is possible to generate necklaces effi- ciently, possibly in constant time per necklace. A proposed solution, the FKM algorithm of Fredricksen, Kessler, and Maiorana, had no proven upper bound better than O(nk n ) [FK86, FM78]. In [WS90] a new algorithm was presented with time complexity O(nN n where N n k is the number of n-bead necklaces in k colors. Subsequently, a tight analysis of the original FKM algorithm showed that it could, in fact, be implemented to run in time O(N n giving an optimal solution [RSW92]. Neither of the algorithms above gives a Gray code for necklaces. Can representatives of all binary n-bead necklaces be listed so that successive strings differ only in one bit position (as in Figure 10)? A parity argument shows that this is impossible for even n, but for odd n the question remains open. However, in the case of necklaces with a fixed number of 1's, Wang showed, with a very intricate construction, how to construct a Gray code in which successive necklace representatives differ only by the swap of a 0 and a 1 [Wan94, WS94] Figure 11.) It remains open whether necklaces with a fixed number of 1's can be generated in constant amortized time, either by a modification the FKM algorithm, by a Gray code, or by any other method. The Gray code adjacency criterion can be generalized to necklaces with k - 2 beads by requiring that successive necklaces differ in exactly one position and in that position by only 1. We conjecture that this can be done if and only if nk is odd. (Parity problems a. 5-bead binary b. 7-bead binary c. 3-bead ternary Figure 10: Examples of Gray codes for necklaces. prevent a Gray code when nk is even.) For necklaces of fixed weight, when is it possible to list all n-bead k-color necklaces of weight w so that successive necklaces differ in exactly two positions, one of which has increased by one and the other, decreased by one? We know of no counterexamples. To construct a slightly different set of objects, call two k-ary strings equivalent if one is a rotation or a reversal of the other. The equivalence classes under this relation are called bracelets. Lisonek [Lis93] shows how to modify the necklace algorithm of [WS90] to generate bracelets. We know of no Gray code for bracelets and it is open whether it is possible to generate bracelets in constant amortized time. When beads have distinct colors, bracelets are the rosary permutations of [Har71, Rea72]. Define a new relation R on n-bead binary necklaces by xRy if some member of x becomes a member of y by changing a 0 to a 1 in one bit position. The reflexive transitive closure, R is a partial order and the resulting poset is the necklace poset. For k - 0 and the middle two levels of this poset, consisting of necklaces of density k and k have the same number of elements. Does the bipartite subgraph induced by the middle two levels have a Hamilton path? This graph is not vertex-transitive, but it may encapsulate the "hard part" of the middle two levels problem described in Section 4. A necktie of n bands in k colors is an equivalence class of k-ary n-tuples under reversal. 7 beads, 4 ones 9 beads, 3 ones 8 beads, 4 ones Figure 11: Examples of Gray codes for binary necklaces with a fixed number of ones. If a necktie is identified with the lexicographically smallest element in its equivalence class, Wang [Wan93] shows that for n - 3, a Gray code exists if and only if either n or k is odd. For this result, two neckties are adjacent if and only if they differ only in one position and in that position by \Sigma1 modulo k. Further results on neckties appear in [RW94]. 9 Linear Extension of Posets A partially ordered set (S; -) is a set S together with a binary relation - on S which is reflexive, transitive, and antisymmetric. A linear extension of a poset is a permutation of the elements of the poset which is consistent with the partial order, that is, if x i - x j in the partial order, then i - j. The problem of efficiently generating all the linear extensions of a poset, in any order, has been studied in [KV83, KS74, VR81]. The area of Gray codes for linear extensions of a poset was introduced by Frank Ruskey in [Rus88b, PR91] as a setting in which to generalize the study of Gray codes for combinatorial objects. For example, if the Hasse diagram of the poset consists of two disjoint chains, one of length m and the other of length n, then there is a one-to-one correspondence between the linear extensions of the poset and the combinations of m objects chosen from m If the poset consists of a collection of disjoint chains, the linear extensions correspond to multiset permutations. Other examples are described in [Rus92]. To study the existence of Gray codes, Ruskey constructs a transposition graph corresponding to a given poset. The vertices are the linear extensions of the poset, two vertices being joined by an edge if they differ by a transposition. The resulting graph is bipartite. In [Rus88b], Ruskey makes the conjecture that whenever the parity difference is at most one, the graph of the poset has a Hamilton path. The conjecture is shown to be true for some special cases in [Rus92], including posets whose Hasse diagram consists of disjoint chains and for series parallel posets in [PR93]. The techniques which have been successful so far involve cutting and linking together listings for various subposets in rather intricate ways. In many cases where it is known how to list linear extensions by transpositions, it is also possible to require adjacent transpositions, although possibly with a more complicated construction [PR91, RS93, Sta92, Wes93]. It has been shown that if the linear extensions of a poset Q, with jQj even, can be listed by adjacent transpositions, then so can the linear extensions of QjP , for any poset P [Sta92], where QjP represents the union of posets P and Q with the additional relations fp qg. However, most problems in this area remain open. For example, even if the Hasse diagram of the poset consists of a single tree, the parity difference may be greater than one. This makes an inductive approach difficult. If the Hasse diagram consists of two trees, each with an odd number of vertices, the parity difference will be at most one, but it is unknown whether the linear extensions can be listed in this case. The problem is also open for posets whose Hasse diagram is a grid or tableau tilted ninety degrees [Rus]. Calculating the parity difference itself can be difficult and Ruskey [Rus] has several examples of posets for which the parity difference is unknown. (Some parity differences are calculated in [KR88].) Recently, Stachowiak has shown that computing the parity difference is #P-complete [Sta]. Even counting the number of linear extensions of a poset is an open problem for some specific posets, for example, the Boolean lattice [SK87]. Brightwell and Winkler have recently shown that the problem of counting the number of linear extensions of a given poset is #-P complete [BW92]. On the brighter side, Pruesse and Ruskey [PR94] have found a CAT algorithm for listing linear extensions so that successive extensions differ by one or two adjacent transpositions and Canfield and Williamson [CW95] have shown how to make it loop-free. In [PR93], Pruesse and Ruskey consider antimatroids, of which posets are a special case. Analogous to the case of linear extensions of a poset, they show that the sets of an antimatroid can be listed so that successive sets differ by at most two elements. In particular, this gives a listing of the ideals of a poset so that successive ideals differ only in one or two elements. Orientations For an undirected graph G, an acyclic orientation of G is a function on the edges of G which assigns a direction (u; v) or (v; u) to each edge uv of G in such a way that the resulting digraph has no directed cycles. Consider the problem of listing the acyclic orientations of G so that successive list elements differ only by the orientation of a single edge. It is not hard to see that when G is a tree with n edges, such a listing corresponds to an n bit binary Gray code; when G is K n , an acyclic orientation corresponds to a permutation of the vertices and the Johnson-Trotter Gray code for permutations provides the required listing for acyclic orientations. Denote by AO(G) the graph whose vertices are the acyclic orientations of G, two vertices adjacent if and only if the corresponding orientations differ only in the orientation of a single edge. The graph AO(G) is bipartite and is connected as long as G is simple. Edelman asked, whenever the partite sets have the same size, whether AO(G) is hamiltonian. It is shown in [SSW93] that the answer is yes for several classes of graphs, including trees, odd length cycles, complete graphs, odd ladder graphs, and chordal graphs. On the other hand, the parity difference is shown to be more than one for several cases, including cycles of even length and the complete bipartite graphs K m;n with m; n ? 1 and m or n even. The problem appears to be difficult and it is even open whether AO(K m;n ) is hamiltonian when mn is odd. However, the square of AO(G) is hamiltonian for any G [PR95, SZ95, Squ94c], which means that acyclic orientations can be listed so that successive elements differ in the orientations of at most two edges. The problem of counting acyclic orientations is #P-complete [Lin86] and it is an open question whether there is a CAT algorithm to generate them. The fastest listing algorithm known, due to Squire [Squ94b], requires O(n) average time per orientation, where n is the number of vertices of the graph. The linear extensions and acyclic orientations problems can be simultaneously generalized as follows. For a simple undirected graph G and a subset R of the edges of G, fix an acyclic orientation oe R of the edges of R. Let AOR (G) be the subgraph of AO(G) induced by the acyclic orientations of G which agree with oe R on R. Is this bipartite graph hamiltonian whenever the parity difference allows? is the acyclic orientations graph of G. AOR (G) becomes the linear extensions adjacency graph of an n element poset P when when and oe R are defined by the covering relations in P . In contrast to the situation for linear extensions and acyclic orientations, the square of AOR (G) is not necessarily hamiltonian. Counterexamples appear in [Squ94c] and [PR95]. Cayley Graphs and Permutation Gray Codes Many Gray code problems for permutations are best discussed in the setting of Cayley graphs. Given a finite group G and a set X of elements of G, the Cayley graph of G, C[G; X ], is the undirected graph whose vertices are the elements of G and in which there is an edge joining u and v if and only if Equivalently, uv is an edge in G if and only if u \Gamma1 v or v \Gamma1 u is in X . C[G; X ] is always vertex transitive and is connected if and only if X [X \Gamma1 generates G. It is an open question whether every Cayley graph is hamiltonian. (There are generating sets for which the Cayley digraph is not hamiltonian [Ran48].) This is a special case of the more general conjecture of Lov'asz that every connected, undirected, vertex-transitive graph has a Hamilton path [Lov70]. Results on Hamilton cycles are surveyed in [Als81] for vertex transitive graphs and in [Gou91] for general graphs. A survey of Hamilton cycles in Cayley graphs can be found in [WG84] and in the recent update of Curran and Gallian [CG96]. We focus here on a few recent questions which arose in the context of Gray codes. Suppose the group G is S n , the symmetric group of permutations of n symbols, and let X be a generating set of S n . Then a Hamilton cycle in the Cayley graph C[G; X ] can be regarded as a Gray code for permutations in which successive permutations differ only by a generator in X . Even in the special case of G = S n , it is still open whether every Cayley graph of S n has a Hamilton cycle. One of the most general results on the hamiltonicity of Cayley graphs for permutations was discovered by Kompel'makher and Liskovets in 1975. First note that for S n generated by the basis n)g, the Johnson- Trotter algorithm from Section 3 for generating permutations by adjacent transpositions gives a Hamilton cycle in C[G; X ]. Kompel'makher and Liskovets generalized this result to show that if X is any set of transpositions generating S n , then C[S [KL75]. Independently, and with a much simpler argument, Slater showed that these graphs have Hamilton paths [Sla78]. Tchuente [Tch82] extended the results of [KL75, Sla78] to show that the Cayley graph of S n , on any generating set X of transpositions, is not only hamiltonian, but Hamilton-laceable, that is, for any two vertices u; v of different parity there is a Hamilton path which starts at u and ends at v. It is unknown whether any of these results generalize to the case when X is a generating set of involutions (elements of order 2) for S n . An involution need not be a transposition: for example the product of disjoint transpositions is an involution. In perhaps the simplest nontrivial case, when S n is generated by three involutions, it is easy to show that if any two of the generators commute, then the Cayley graph is hamiltonian. (Cayley graphs arising in change ringing frequently have this property [Ran48, Whi83].) However if no two of the three involutions commute, it is open whether the Cayley graph is hamiltonian. As a specific example, we have not been able to determine whether there is a Gray code for permutations in which successive permutations may differ only by one of the three operations: (i) exchange positions 1 and 2, (ii) reverse the sequence, and (iii) reverse positions 2 through n of the sequence. Conway, Sloane, and Wilks have a related result on Gray codes for reflection groups: if G is an irreducible Coxeter group (a group generated by geometric reflections) and if X is the canonical basis of reflections for the group, then C[G; X ] is hamiltonian [CSW89]. This result makes use of the fact that in any set of three or more generators from this basis, there is always some pair of generators which commute (the Coxeter diagram for the basis is a tree). It is straightforward to show for groups G and H , with generating sets X and Y , respectively, that if C[G; X ] and C[H; Y ] are hamiltonian, and at least one of G and H have even order, then C[G \Theta H; X \Theta Y ] is hamiltonian. As noted in [CSW89], since any reflection group R is a direct product of irreducible Coxeter groups G 1 \Theta G 2 \Theta canonical basis for G i , the Cayley graph of R with respect to the basis is hamiltonian. This result has an interesting geometric interpretation. We can associate with a finite reflection group a tessellation of a surface in n-space by a spherical (n \Gamma 1)-simplex. The spherical simplices of the tesselation correspond to group elements, and the boundary shared by two simplices corresponds to a reflection in the bounding hyperplane. Thus, a Hamilton cycle in the Cayley graph corresponds to a traversal of the surface, visiting each simplex exactly once. It seems likely that if there do exist non-hamiltonian Cayley graphs of S n , there will be examples in which the vertices have small degree, such as S n generated by three involutions as described above. As a candidate counterexample, Wilf suggested the group of permutations generated by the two cycles (1 2) and (1 and Williamson were able to find a Hamilton cycle in this graph using their Gray code for generating permutations by doubly adjacent transpositions, described in Section 3 [CW93]. The results of [KL75, Sla78] were generalized in a different way in [RS93]. It was shown there that when n - 5, for any generating set X of transpositions of S n and for any transposition permutations in S n can be listed so that successive permutations differ only by a transposition in X and every other transposition is - . That is, the perfect matching in C[S defined by - is contained in a Hamilton cycle. One application of this result is to Cayley graphs for the alternating group, A n , consisting of all even permutations n. For example, letting n), the result in [RS93] implies that the Cayley graph of A n with respect to the generating set is hamiltonian. This result was obtained earlier with a direct argument by Gould and Roth [GR87]. 12 Generalizations of de Bruijn Sequences A de Bruijn sequence of order n is a circular binary sequence of length 2 n in which every n-bit number appears as a contiguous subsequence. This provides a Gray code listing of binary sequences in which successive elements differ by a rotation one position left followed by a change of the last element. It is known that these sequences exist for all n and the standard proof shows that a de Bruijn sequence of order n corresponds to an Euler tour in the de Bruijn digraph whose vertices are the binary n-tuples, with one edge binary n-tuple x This result has been generalized for k-ary n-tuples [Fre82], for higher dimensions (de Bruijn tori) [Coc88, FFMS85], and for k-ary tori [HI95]. It is known also, for any m satisfying , that there is a cyclic binary sequence of length m in which no n-tuple appears more than once [Yoe62]. So, the de Bruijn graph contains cycles of all lengths n. Results on de Bruijn cycles have been applied to random number generation in information theory [Gol64] and in computer architecture, where the de Bruijn graph is recognized as a bounded degree derivative of the shuffle-exchange network [ABR90]. Chung, Diaconis, and Graham generalized the notion of a de Bruijn sequence for binary numbers to universal cycles for other families of combinatorial objects [CDG92]. Universal cycles for combinations were studied by Hurlbert in [Hur90] and some interesting problems remain open. A universal cycle of order n for permutations is a circular sequence x of length of symbols from Mg in which every permutation is order isomorphic to some contiguous subsequence x means that for 1 - As an example, the sequence 123415342154213541352435 is a universal cycle of order 4 with 5. In [CDG92], the goal is to choose M as small as possible to guarantee the existence of a universal cycle. It is clear that M must satisfy 2. It is conjectured in [CDG92] that although the best upper bound they were able to obtain was sufficient. As another approach, we can relax the constraint on the length of the sequence, while requiring for the shortest circular sequence of symbols from ng which contains every permutation as a contiguous subsequence at least once. Jacobson and West have a simple construction for such a sequence of length 2n! [JW]. Concluding Remarks This paper has included a sampling of Gray code results in several areas, particularly those which have appeared since the survey of Wilf [Wil89], in which many of these problems were posed. Good references for early work on Gray codes are [Ehr73] and [NW78]. For a comprehensive treatment of Gray codes and other topics in combinatorial generation, we look forward to the book in preparation by Ruskey [Rus95]. Additional information on Gray codes also appears in the survey of Squire [Squ94a]. In [Gol93], Goldberg considers generating combinatorial structures for which achieving even polynomial delay is hard. For surveys on related material, see [Als81] for long cycles in vertex transitive graphs, [Gou91] for hamiltonian cycles, [WG84] and the recent update [CG96] for Cayley graphs, and [Sed77] for permutations. Acknowledgements I am grateful to Herb Wilf for collecting and sharing such an intriguing array of 'Gray code' problems. 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Explicit matchings in the middle levels of the Boolean lattice. On the generation of all topological sortings. The machine tools of combinatorics. Permutation by adjacent interchanges. Hard enumeration problems in geometry and combinatorics. Generating bracelets A Gray code based ordering for documents on shelves: Classificaton for browsing and retrieval. Problem 11. A technique for generating Gray codes. The rotation graph of binary trees is hamiltonian. Gray code generation for MPSK signals. A problem in arrangements. Problem 1186. Electronic mail communication (via J. Combinatorial Algorithms for Computers and Calculators. Binary tree Gray codes. Generating the linear extensions of certain posets by adjacent transpositions. Gray codes from antimatroids. Generating linear extensions fast. The prism of the acyclic orientation graph is hamiltonian. A Gray code variant: sequencing permutations via fixed points. 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Generating neckties: algorithms Gray code sequences of partitions. Generating permutations with k-differences Long cycles in the middle two levels of the Boolean lattice. Permutation generation methods. Hamiltonian bipartite graphs. The number of linear extensions of subset ordering. Generating all permutations by graphical transpositions. Combinatorial Gray codes and efficient generation. Generating the acyclic orientations of a graph. Two new Gray codes for acyclic orientations. Gray codes for A-free strings Gray code results for acyclic orientations. Finding parity difference by involutions. Hamilton paths in graphs of linear extensions for unions of posets. Rotation distance Constructive Combinatorics. Monotone Gray codes and the middle two levels problem. A note on the connectivity of acyclic orientations graphs Generation of permutations by graphical exchanges. An algorithm to generate all topological sorting arrangements. A technique for generating specialized Gray codes. A note on Gray codes for neckties A Gray Code for Necklaces of Fixed Density. Generation of permutations by transposition. Generating linear extensions by adjacent transpositions. A survey - hamiltonian cycles in Cayley graphs Ringing the changes. Combinatorics for Computer Science. Generalized Gray codes. Combinatorial Algorithms: An Update. A new algorithm for generating necklaces. Gray codes for necklaces of fixed density Construction of uniform Gray codes. Binary ring sequences. Ranking and unranking algorithms for trees and other combinatorial ob- jects Lexicographic generation of ordered trees. Generating binary trees using rotations. --TR --CTR Elizabeth L. Wilmer , Michael D. Ernst, Graphs induced by Gray codes, Discrete Mathematics, v.257 n.2-3, p.585-598, 28 November Colin Murray , Carsten Friedrich, Visualisation of satisfiability using the logic engine, proceedings of the 2005 Asia-Pacific symposium on Information visualisation, p.147-152, January 01, 2005, Sydney, Australia Khaled A. S. Abdel-Ghaffar, Maximum number of edges joining vertices on a cube, Information Processing Letters, v.87 n.2, p.95-99, 31 July Tadao Takaoka , Stephen Violich, Combinatorial generation by fusing loopless algorithms, Proceedings of the 12th Computing: The Australasian Theroy Symposium, p.69-77, January 16-19, 2006, Hobart, Australia V. V. Kuliamin, Test Sequence Construction Using Minimum Information on the Tested System, Programming and Computing Software, v.31 n.6, p.301-309, November 2005 Vincent Vajnovszki, A loopless algorithm for generating the permutations of a multiset, Theoretical Computer Science, v.307 n.2, p.415-431, 7 October James Korsh , Paul Lafollette, A loopless Gray code for rooted trees, ACM Transactions on Algorithms (TALG), v.2 n.2, p.135-152, April 2006 Gerard J. Chang , Sen-Peng Eu , Chung-Heng Yeh, On the (n,t)-antipodal Gray codes, Theoretical Computer Science, v.374 n.1-3, p.82-90, April, 2007 Kenneth A. Ross, Selection conditions in main memory, ACM Transactions on Database Systems (TODS), v.29 n.1, p.132-161, March 2004 Jean Pallo, Generating binary trees by Glivenko classes on Tamari lattices, Information Processing Letters, v.85 n.5, p.235-238, March

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273702

Fully Discrete Finite Element Analysis of Multiphase Flow in Groundwater Hydrology.

This paper deals with the development and analysis of a fully discrete finite element method for a nonlinear differential system for describing an air-water system in groundwater hydrology. The nonlinear system is written in a fractional flow formulation, i.e., in terms of a saturation and a global pressure. The saturation equation is approximated by a finite element method, while the pressure equation is treated by a mixed finite element method. The analysis is carried out first for the case where the capillary diffusion coefficient is assumed to be uniformly positive, and is then extended to a degenerate case where the diffusion coefficient can be zero. It is shown that error estimates of optimal order in the $L^2$-norm and almost optimal order in the $L^\infty$-norm can be obtained in the nondegenerate case. In the degenerate case we consider a regularization of the saturation equation by perturbing the diffusion coefficient. The norm of error estimates depends on the severity of the degeneracy in diffusivity, with almost optimal order convergence for nonsevere degeneracy. Implementation of the fractional flow formulation with various nonhomogeneous boundary conditions is also discussed. Results of numerical experiments using the present approach for modeling groundwater flow in porous media are reported.

Introduction . In this paper we develop and analyze a fully-discrete finite element procedure for solving the flow equations for an air-water system in groundwater hydrology, @t kk rff porous medium, OE and k are the porosity and absolute permeability of the porous system, ae ff , s ff , p ff , u ff , and - ff are the density, saturation, pressure, volumetric velocity, and viscosity of the ff-phase, f ff is the source/sink term, k rff is the relative permeability of the ff-phase, and g is the gravitational, downward- pointing, constant vector. Flow simulation in groundwater reservoirs has been extensively studied in past years (see, e.g., [26], [28] and the bibliographies therein). However, in most previous works the air-phase equation is eliminated by the assumption that the air-phase remains essentially at atmospheric pressure. This assumption, as mentioned in [13], is reasonable in most cases because the mobility of air is much larger than that of water, due to the viscosity difference between the two fluids. When the air-phase pressure is assumed constant, the air-phase mass balance equation can be eliminated and thus only the water-phase equation remains. Namely, the Richards equation is used to model the movement of water in groundwater reservoirs. However, it provides * Department of Mathematics and the Institute for Scientific Computation, Texas A&M Uni- versity, College Station, Partly supported by the Department of Energy under contract DE-ACOS-840R21400. email: zchen@isc.tamu.edu, ewing@ewing.tamu.edu. no information on the motion of air. If contaminant transport is the main concern and the contaminant can be transported in the air-phase, the air-phase needs to be included to determine the advective component of air-phase contaminant transport [7]. Furthermore, the dynamic interaction between the air and water phases is also important in vapor extraction systems. Hence in these cases the coupled system of nonlinear equations for the air-water system must be solved. It is the purpose of this paper that is to develop and analyze a finite element procedure for approximating the solution of the coupled system of nonlinear equations for the air-water system in groundwater hydrology. In petroleum reservoir simulation the governing equations that describe fluid flow are usually written in a fractional flow formulation, i.e., in terms of a saturation and a global pressure [1], [8]. The main reason for this fractional flow approach is that efficient numerical methods can be devised to take advantage of many physical properties inherent in the flow equations. However, this pressure-saturation formulation has not yet achieved application in groundwater hydrology. In petroleum reservoirs total flux type boundary conditions are conveniently imposed and often used, but in groundwater reservoirs boundary conditions are very complicated. The most commonly encountered boundary conditions for a groundwater reservoir are of first-type (Dirichlet), second-type (Neumann), third-type (mixed), and "well" type [8]. The problem of incorporating these nonhomogeneous boundary conditions into the fractional flow formulation has been a challenge [12]. In particular, in using the fractional flow approach a difficulty arises when the Dirichlet boundary condition is imposed for one phase (e.g. air) and the Neumann type is used for another phase (e.g. water). This paper follows the fractional flow formulation. Based on this approach, we develop a fully-discrete finite element procedure for the saturation and pressure equa- tions. The saturation equation is approximated by a Galerkin finite element method, while the pressure equation is treated by a mixed finite element method. It is well known that the physical transport dominates the diffusive effects in incompressible flow in petroleum reservoirs. In the air-water system studied here, the transport again dominates the entire process. Hence it is important to obtain good approximate velocities. This motivates the use of the parabolic mixed method, as in [17], in the computation of the pressure and the velocity. Also, due to its convection-dominated feature, more efficient approximate procedures should be used to solve the saturation equation. However, since this is the first time to carry out an analysis for the present problem, it is of some importance to establish that the standard finite element method for this model converges at an asymptotically optimal rate for smooth problems. Characteristic Petrov-Galerkin methods based on operator splitting [20], transport diffusion methods [32], and other characteristic based methods will be considered in forthcoming papers. The main part of this paper deals with an asymptotical analysis for the fully discrete finite element method for the first-type and second-type boundary conditions where p ffD and d ff are given functions, being disjoint, and - is the outer unit normal to @ We point out that petroleum reservoir simulation is different from groundwater reservoir simulation. The flow of two incompressible fluids (e.g. water and oil) is usually considered in the former case, while the latter system consists of the air and water phases. Consequently, the finite element analyses for these two cases differ. As shown here, compressibility and combination of the boundary conditions (1.3) and (1.4) complicate error analyses. Indeed, if optimality is to be preserved for the finite element method, the standard error argument just fails unless we work with higher order time-differentiated forms of error equations, which require properly scaling initial conditions. Also, we mention that a slightly compressible miscible displacement problem was treated in [14], [18], [23], [33]; however, only the single phase was handled, gravitational terms were omitted, and total flux type boundary conditions were assumed. Furthermore, the so-called "quadratic" terms in velocity were neglected. The dropping of these quadratic terms may not be valid near wells, and so the miscible displacement model was oversimplified both physically and mathematically. The analysis of this paper includes these terms. Finally, only the Raviart-Thomas mixed finite element spaces [34] have been considered in these earlier papers. We are here able to discuss all existing mixed spaces. The error analysis is given first for the case where the capillary diffusion coefficient is assumed to be uniformly positive. In this case, we show error estimates of optimal order in the L 2 -norm and almost optimal order in the L 1 -norm. Then we treat a degenerate case where the diffusion coefficient vanishes for two values of saturation. In the degenerate case we consider a regularization of the saturation equation by perturbing the diffusion coefficient to obtain a nondegenerate problem with smooth solutions. It is shown that the regularized solutions converge to the original solution as the perturbation parameter goes to zero with specific convergence rates given. The norm of error estimates depends on the severity of the degeneracy in diffusivity, with almost optimal order convergence for the degeneracy under consideration. The rest of this paper is concerned with implementation of the fractional flow formulation with various nonhomogeneous boundary conditions. We show that all the commonly encountered boundary conditions can be incorporated in the fractional flow formulation. Normally the "global" boundary conditions are highly nonlinear functions of the physical boundary conditions for the original two flow phases. This means that we have to iterate on these global boundary conditions as part of the solution process. We here develop a general solution approach to handle these boundary conditions. Results of numerical experiments using the present approach for modeling groundwater flow are reported here. The paper is organized as follows. In x2, we define a fractional flow formulation for equations (1.1)-(1.4). Then, in x3 we introduce weak forms of the pressure-saturation equations, and in x4 a fully-discrete finite element procedure for solving these equa- tions. An asymptotical analysis is given in x5 and x6 for the nondegenerate case and the degenerate case, respectively. Finally, in x7 we discuss implementation of various nonhomogeneous boundary conditions and present the results of numerical experiments. 2. A pressure-saturation formulation. In addition to (1.1)-(1.4), we impose the customary property that the fluid fills the volume: and define the capillary pressure function p c by Introduce the phase mobilities and the total mobility To devise our numerical method, it is important to choose a reasonable set of dependent variables. Since equal to the water residual saturation [3], pw cannot generally be expected to lie in any Sobolev space. Air being a continuous phase implies that p a is well behaved. Hence, as mentioned in the introduction, we define the global pressure [1] with sc -w d- d- Z pc (s)\Gamma -w c (-) The integral in the right-hand side of (2.3) is well defined [1], [8]. As usual, assume that ae ff depends on p [8]. Then we define the total velocity where g: Now it can be easily seen that (2.5a) where q Consequently, Equations (1.1) and (1.2) can be manipulated using (2.1)-(2.6) to have the pressure equation @t a ff=wae ff OEs ff @ae ff @t and the saturation equation OE @sw @t \Gammas w @t ae w OEs w @t Terms of the form u ff \Delta rae ff , neglected in compressible miscible displacement problems [14], [18], [23], [33]. The dropping of these terms may not be valid near wells. Also, if they are neglected, the model may not be qualitatively equivalent to the usual formulation of two phase flow. Hence we keep them in this paper. However, the water phase is usually assumed to be incompressible. With the incompressibility of the water phase and the following notation: c(s; ae a dae a dp ds ae a dae a dp ~ ae w ae a dae a dp ae) f(s; p) =ae a dae a dp k-w q a (rp c \Gamma ~ f a ae a ae w @t equations (2.7) and (2.8) can be now written as c(s; p) @p @t OE @s @t @t The boundary conditions for the pressure-saturation equations become (D(s)rs where s D and pD are the transforms of pwD and paD by (2.2) and (2.3), and ~ The model is completed by specifying the initial conditions The later analysis for the nondegenerate case in x5 is given under a number of assumptions. First, the solution is assumed smooth; i.e., the external source terms are smoothly distributed, the coefficients are smooth, the boundary and initial data satisfy the compatibility condition, and the domain has at least the regularity required for a standard elliptic problem to have H 2(\Omega\Gamma671/64748/ y and more if error estimates of order bigger than one are required. Second, the coefficients a(s), OE, and c(s; p) are assumed bounded below positively: Finally, the capillary diffusion coefficient D(s) is assumed to satisfy While the phase mobilities can be zero, the total mobility is always positive [31]. The assumptions (2.18) and (2.19) are physically reasonable. Also, the present analysis obviously applies to the incompressible case where c(s; In this case, the analysis is simpler since we have an elliptic pressure equation instead of the parabolic equation (2.9). Thus we assume condition (2.20) for the compressible case under consideration. Next, although the reasonableness of the assumption (2.21) is discussed in [16], the diffusion coefficient D(s) can be zero in reality. It is for this reason that section six is devoted to consideration of the case where the solution is not required smooth and the assumption (2.21) is removed. As a final remark, we mention that for the case where point sources and sinks occur in a porous medium, an argument was given in [22] for the incompressible miscible displacement problem and can be extended to the present case. 3. Weak forms. To handle the difficulty associated with the inhomogeneous Neumann boundary condition (2.13) in the analysis of the mixed finite element method, let d be such that d \Delta d and introduce the change of variable equations (2.9)-(2.11). Then the homogeneous Neumann boundary condition holds for ~ u. Thus, without loss of generality, we assume that ~ To be compatible, we also require that this homogeneous condition holds when In the two-dimensional case, let while it is accordingly defined in the three-dimensional case as follows: Also, set The weak form of (2.9)-(2.11) on which the finite element procedure is based is given below. Let is the time interval of interest. The mixed formulation for the pressure is defined by seeking a pair of maps 2(\Omega\Gamma such that (3.1a) (c(s; p) @p @t the inner products (\Delta; \Delta) are to be interpreted to be in L or (L 2 (\Omega\Gamma4 d , as appropriate, and h\Delta; \Deltai denotes the duality between H 1=2 H \Gamma1=2 (\Gamma 1 ). The weak form for the saturation s : J OE @s @t where the boundary condition (2.15) is used. Finally, to treat the nonzero initial conditions imposed on s and p in (2.16) and (2.17), we introduce the following transformations 7in (3.1) and (3.2): where we have zero initial conditions for s, p, and u. Hence, without loss of generality again, we assume that The reason for introducing these transformations to have zero initial conditions is to validate equation (5.15) later. 4. Fully-discrete finite element procedures. Let\Omega be a polygonal domain. For partitions into ele- ments, say, simplexes, rectangular parallelepipeds, and/or prisms. In both partitions, we also need that adjacent elements completely share their common edge or face. Let be a standard C 0 -finite element space associated with T h such that where hK is the norm in the Sobolev space W k;q (K) (we omit K when K =\Omega and kvk (\Omega\Gamma be the Raviart-Thomas-Nedelec [34], [29], the Brezzi-Douglas-Fortin-Marini [5], the Brezzi-Douglas-Marini [6] (if 2), the Brezzi-Douglas-Dur'an-Fortin [4] (if or the Chen-Douglas [11] mixed finite element space associated with the partition T hp of index such that the approximation properties below are satisfied: kr where h p;K for the first two spaces, for the second two spaces, and both cases are included in the last space. Finally, let ft n g nT n=0 be a quasi-uniform partition of J with t set \Deltat We are now in a position to introduce our finite element procedure. The fully-discrete finite element method is given as follows. The approximation procedure for the pressure is defined by the mixed method for a pair of maps fu n (ff(s (4.5a) (c(s (4.5b) and the finite element method for the saturation is given for s n h )rs n @t The initial conditions satisfy After startup, for equations (4.5) and (4.6) are computed as follows. First, using s h , and (4.5), evaluate fu n g. Since it is linear, (4.5) has a unique solution for each n [10], [27]. Next, using s h g, and (4.6), calculate s n h . Again, (4.6) has a unique solution for \Deltat n sufficiently small for each n [39]. We end this section with a remark. While the backward Euler scheme is used in (4.5b) and (4.6), the Crank-Nicolson scheme and more accurate time stepping procedures (see, e.g., [21]) can be used. The present analysis applies to these schemes. 5. An error analysis for the fully-discrete scheme. In this section we give a convergence analysis for the finite element procedure (4.5) and (4.6) under assumption (2.21). As usual, it is convenient to use an elliptic projection of the solution of into the finite element space M h . Let ~ defined by Then it follows from standard results of the finite element method [15], [30], [37] that (5.3a) 1. The same result applies to the time- differentiated forms of (5.1) [40]: @t @t @s @t As for the analysis of the mixed finite element method, we use the the following two projections instead of the elliptic projections introduced in [14] and [18]. So the present analysis is different from and in fact simpler than those in [14] and [18]. Each of our mixed finite element spaces [4]-[6], [11], [29], [34] has the property that there are projection operators \Pi such that kr and (see, e.g., [9], [19]) (r Note that, by (3.3) and (4.7), Finally, we prove some bounds of the projections ~ s and ~ p. Let be the interpolant of s in M h . Then we see, by (4.1), (5.3b), the approximation property of s, and an inverse inequality in M h , that ks ks ks \Gamma sk 0;1 ks where fl is given as in (5.3b). This implies that k~sk 1;1 is bounded for sufficiently smooth solutions since k - 1. The same argument applies to k@~s=@tk 1;1 . Next, note that, by the approximation property of the projection P h [27], These bounds on ~ are used below. We are now ready to prove some results. Below " is a generic positive constant as small as we please. 5.1. Analysis of the mixed method. We first analyze the mixed method (4.5). We set The following error equation is obtained by subtracting (4.5) from (3.1) at applying (5.8) and (ff(s @t @t Below C i indicates a generic constant with the given dependencies. Lemma 5.1. Let (u; p) and solve (3.1) and (4.5), respectively. Then ks @t @t @t Proof. Set add the resulting equations at use (3.3), (4.7), and (5.12) to see that where @t @t Then (5.15) can be easily seen. Lemma 5.2. Let (u; p) and and (4.5), respectively. Then ae @t ks \Theta k@' @t \Psi \Deltat n oe @t @t @t Proof. Difference equations (5.13) and (5.14) with respect to n, set in the resulting equations, divide by \Deltat n , and add to obtain where \Deltat n \Deltat n \Deltat n \Deltat n @t @t @t @t \Deltat n \Deltat n \Deltat n \Deltat n \Deltat n Observe that the left-hand side of (5.16) is larger than the quantity (5.17)2\Deltat n where We estimate the new term T n 2 in detail. Other terms can be bounded by a simpler argument. To estimate T n 2 , we write \Gamma\Phi \Deltat n \Deltat n \Deltat n Note that [A(s @s (bs @A @s (bs @A where and similar inequalities hold for b s Consequently, with - see that @s (bs @p@s (p \Deltat (bp \Deltat so that ks n\Gamma2 where and an analogous inequality holds for b s Also, we see that [A(s \Phi @A @s (p which implies that ks Next, it can be easily seen that ks Finally, since we find that Hence T n 2 can be bounded in terms of T n Other terms are bounded as follows: ks n\Gamma2 ks n\Gamma2 ks ks @t @t ks ks n\Gamma2 can be bounded as in (5.19), e.g., kbs ks ks ks ks apply these inequalities and (5.17)-(5.20), multiply (5.16) by \Deltat n , sum n, and properly arrange terms to complete the proof of the lemma. The error equations (5.13) and (5.14) are usually exploited to derive error estimates in the parabolic mixed finite element method [18], [27]. To handle the difficulty arising from the combination of the Dirichlet boundary condition (1.3) and the non-linearity of the differential system (2.9)-(2.11), we must use their time-differentiated forms, as mentioned before. Also, the three terms T n care of the quadratic terms in the velocities, which require more regularity on u than those without these quadratic terms, as seen from Lemma 5.2. 5.2. Analysis of the saturation equation. We now turn to analyzing the finite element method (4.6). Lemma 5.3. Let s and s h solve (3.2) and (4.6), respectively. Then ae ks @t ks 0;1 \Deltat n oe @t @t Proof. Subtract (4.6) from (3.2) at use (5.1) at set the test function to see that where @t @t The left-hand side of (5.21) is bigger than the quantity \Gamma2\Deltat n (D(s n\Gamma2 defined by and is bounded by Next, it can be easily seen that @t To avoid an apparent loss of a factor h in B n use summation by parts on these items. We work on B n 3 in detail, and other quantities can be estimated similarly. Applying summation by parts in n and the fact that - we see that \Psi so that, using the same argument as for (5.18), 3 \Deltat n ks ks oe where kbs n can be estimated as in (5.20). The term 7 \Deltat n has the same bound as in (5.25). Also, we find that 4 \Deltat n oe and 5 \Deltat n ks ks oe multiply (5.21) by \Deltat n , sum n, and use (5.22)-(5.27) to complete the proof of the lemma. 5.3. estimates. We now prove the main result in this section. Define K2Thp @t K2Thp @t @s @t Theorem 5.4. Let (u; p; s) and respectively. Then, if the parameters \Deltat, h p , and h satisfy we have ks @t @t Proof. Take a 1)-multiple of the inequality in Lemma 5.3, add the resulting inequality and the inequality in Lemma 5.2, and use (5.3)-(5.7), (5.15), and the extension of the solution for t - 0 to obtain ae \Theta (k@- oe where In deriving (5.29), we required that the " appearing in Lemma 5.3 be sufficiently small that (C 1 increases C 2 , but not C 1 . Observe that, by (5.12), The same result holds for - fl and fi fl . Combine (5.29), (5.30), and an inverse inequality to see that ae \Theta (k@- oe We now make the induction hypothesis that . Note that, by (5.12), (5.32) holds trivially for (5.32), (5.31) becomes ae \Theta (k@- oe Using (5.28), we choose the discretization parameters so small that Then it follows from (5.33) that ae oe which, together with Gronwall's inequality, implies that where for \Deltat not too large. Consequently, the induction argument is completed and the theorem follows. We remark that, if h and h p are of the same order as they tend to zero, then since k - k + 1. Since k - 1, 3: Also, if k - 2, we see that 3: Thus, for (5.28) to be satisfied, we assume that k - 2. This excludes the mixed finite element spaces of lowest order, i.e., k 1. The lowest order case has to be treated using different techniques. If the nonlinear coefficients ff(s) and c(s; p) in (4.5) are projected into the finite element space W h , the technique developed in [10] can be used to handle the lowest order case. We shall not pursue this here. 5.4. estimates. The main objective of this paper is to establish the estimates given in Theorem 5.4. For completeness, we end this section with a statement of L 1 -estimates for the errors in the two-dimensional case. Theorem 5.5. Assume that (p; s) and (p h ; s h ) satisfy (3.1), (3.2) and (4.5), (4.6), respectively, and the parameters h p and h satisfy (5.28). Then (\Omega\Gamma/ ks Proof. First, it follows from the approximation property of the projection P h [27] that Also, from [27, Lemma 1.2] and (5.13), we see that so that, by Theorem 5.4, This, together with (5.37), implies (5.35). Finally, apply the embedding inequality (5.3b), and (5.34) to obtain (5.36). 6. Finite elements for a degenerate problem. In this section we consider a degenerate case where the diffusion coefficient D(s) can be zero. Since the pressure equation is the same as before, we here focus on the saturation equation. For simplicity we neglect gravity. Then the saturation equation (2.11) can be written as @s @t @t 2\Omega \Theta J: For technical reasons we only consider the Neumann boundary condition (2.15): @\Omega \Theta J; and the initial condition is given by We impose the following conditions on the degeneracy of D(s): where the fi i are positive constants and ff j and - satisfy the conditions: 2: Difficulties arise when trying to derive error estimates for the approximate solution of (6.1) and (6.2) with D(s) satisfying the condition (6.3). To get around this problem, we consider the perturbed diffusion coefficient D - (s) defined by [13], [24], [35], [38] g. Since the coefficient D - (s) is bounded away from zero, the previous error analysis applies to the perturbed problem: OE @t @t 2\Omega \Theta J; (6.4a) (D @\Omega \Theta J; (6.4c) We now state a result on the convergence of s - to s as - tends to zero. Its proof is given in [24] for the case where dw j 0 and the right-hand side of (6.1) is zero, and can be easily extended to the present case. Theorem 6.1. Assume that D(s) satisfies (6.3) and there is a constant C ? 0 such that where Z sD(-)d-: Then there is C independent of -, s, and - such that As shown in [24], the requirement (6.5) is reasonable. We now consider a fully- discrete finite element method for (6.4). Let M h be the standard C 0 piecewise linear polynomial space associated with T h ; due to the roughness of the solution to (6.1) and (6.2), no improvements in the asymptotic convergence rates result from taking higher order finite element spaces. Also, we extend the domain of D - and q w as follows: ae D - (1) if - and Now the finite element solution s n to (6.4) is given by (6.7a) where P h is the L 2 -projection onto M h . The following theorem states the convergence of s h to s. For (6.8) below to be satisfied, we see from (6.6) that the perturbation parameter - need to satisfy the relation Theorem 6.2. Let s and s h solve (6.1), (6.2) and (6.7), respectively, and let the hypotheses of Theorem 6.1 be satisfied. Then there is C independent of -, s, and - such that The proof can be carried out as in [25], [35], and [38]; we omit the details. 7. Simulation with various boundary conditions. Let @\Omega be a set of four disjoint regions As mentioned in the introduction, the most commonly encountered boundary conditions for the two-pressure equations are of first-type, second-type, third-type, and well type. Then we consider for (7.3a) (7.4a) are given functions, d j is an arbitrary scaling constant, and - is the outer unit normal to @ Note that \Gamma 1 is of the first type, \Gamma 2 is of the third type (it reduces to the second type as - ff j 0), \Gamma 3 is of the well type, and on \Gamma 4 we have the Dirichlet condition for the air phase and the Neumann condition for the water phase. Let \Gamma Then the global boundary conditions for the pressure-saturation equations (2.9)-(2.11) become (7.7a) where pD and s D are the transforms of pwD and paD by (2.2) and (2.3), and Z pc (s)q a c (-) Z pc (s)q a c (-) Z pc (s)q w c (-) We now incorporate the boundary conditions (7.5)-(7.10) in the finite element scheme given in (4.5) and (4.6). The constraint V h ae V says that the normal components of the members of V h are continuous across the interior boundaries in T hp . Following [2], [9], we relax this constraint on V h by introducing Lagrange multipliers over interior boundaries. Since the mixed space V h is finite dimensional and defined locally on each element K in T hp , let V h . Then we define ~ for each K 2 T hp ae e for each j; oe and W h and M h are given as before. The mixed finite element solution of the pressure equation is fu n (c(s (ff(s and the finite element method for the saturation is given for s n satisfying h )rs n @t . The computation of these equations can be carried out as in (4.5) and (4.6). Note that the last equation in the unconstrained mixed formulation above enforces the continuity requirement on u h , so in fact . It is well known [2], [9] that the linear system arising from this unconstrained mixed formulation leads to a symmetric, positive definite system for the Lagrange multipliers, which can be easily solved. Also, the introduction of the Lagrange multipliers makes it easier to incorporate the boundary conditions (7.5)-(7.10). We now present a numerical example. The relative permeability functions are taken as follows: where s rw and s ra are the irreducible saturations of the water and air phases, respec- tively. The capillary pressure function is of the form where fl and \Theta are functions of the irreducible saturations. The water and air viscosities and densities are set to be 1cP and 0:8cP , and 100kg=m 3 and 1:3kg=m 3 , respectively. The permeability rate is two-dimensional domain of 4m width by 1m depth is simulated. Finally, the boundary of the domain is divided into the following segments: A uniform partition of\Omega into rectangles with \Deltay is taken, and the time step \Deltat is required to satisfy (5.28). The Raviart-Thomas space of lowest-order over rectangles is chosen. Tables 1 and 2 describe the errors and convergence orders for the pressure and saturation at time respectively. Experiments at other times and on finer meshes are also carried out; similar results are observed and not reported here. Table 1. Convergence of p h at Table 2. Convergence of s h at From Table 1, we see that the scheme is first-order accurate both in L 2 and L 1 norms for the pressure, i.e., optimal order. Table 2 shows that the scheme is almost optimal order for the saturation. Thus the numerical experiments in the two tables are in agreement with our earlier analytic results. --R On the solvability of boundary value problems for degenerate two-phase porous flow equations Mixed and nonconforming finite element methods: implementation Dynamics of Fluids in Porous Media one dimensional simulation and air phase velocities Mathematical Models and Finite Elements for Reservoir Simula- tion Analysis of mixed methods using conforming and nonconforming finite element methods Multiphase flow simulation with various boundary conditions Mixed finite element methods for compressible miscible displacement in porous media The Finite Element Method for Elliptic Problems The approximation of the pressure by a mixed method in the simulation of miscible displacement Characteristic Petrov-Galerkin subdomain methods for two phase immiscible flow Efficient time-stepping methods for miscible displacement problems in porous media Galerkin methods for miscible displacement problems with point sources and sinks-unit mobility ratio case Timestepping along characteristics for a mixed finite element approximation for compressible flow of contamination from nuclear waste in porous media A priori estimates and regularization for a class of porous medium equations Fundamentals of Soil Physics estimates for some mixed finite element methods for parabolic type problems Mixed finite elements in Fundamentals of Numerical Reservoir Simulation On the transport-diffusion algorithm and its application to the Navier-Stokes equations An implicit diffusive numerical procedure for a slightly compressible miscible displacement problem in porous media A mixed finite element method for second order elliptic problems Numerical Methods for flow through porous media I Maximum norm stability and error estimates in parabolic finite element equations Optimal L 1 estimates for the finite element method on irregular meshes A near optimal order approximation to a class of two sided nonlinear degenerate parabolic partial differential equations Galerkin Finite Element Methods for Parabolic Problems A priori L 2 error estimates for Galerkin approximation to parabolic partial differential equations --TR --CTR M. Afif , B. Amaziane, On convergence of finite volume schemes for one-dimensional two-phase flow in porous media, Journal of Computational and Applied Mathematics, v.145 n.1, p.31-48, 1 August 2002 E. Abreu , J. Douglas, Jr. , F. Furtado , D. Marchesin , F. Pereira, Three-phase immiscible displacement in heterogeneous petroleum reservoirs, Mathematics and Computers in Simulation, v.73 n.1, p.2-20, 6 November 2006 Z. Chen , R. E. Ewing, Degenerate Two-Phase Incompressible Flow IV: Local Refinement and Domain Decomposition, Journal of Scientific Computing, v.18 n.3, p.329-360, June

error estimate;air-water system;finite element;porous media;numerical experiments;mixed method;compressible flow;time discretization

273705

Convergence of a Multigrid Method for Elliptic Equations with Highly Oscillatory Coefficients.

Standard multigrid methods are not so effective for equations with highly oscillatory coefficients. New coarse grid operators based on homogenized operators are introduced to restore the fast convergence rate of multigrid methods. Finite difference approximations are used for the discretization of the equations. Convergence analysis is based on the homogenization theory. Proofs are given for a two-level multigrid method with the homogenized coarse grid operator for two classes of two-dimensional elliptic equations with Dirichlet boundary conditions.

Introduction Consider the multigrid method arising from the finite difference approximations to elliptic equations with highly oscillatory coefficients of the following type @ a ffl where a ffl strictly positive, continuous and 1-periodic in each component of y. Also, the operator L ffl is uniformly elliptic. That is, there exist two positive constants q and Q independent of ffl, such that for any - i is assumed to be very small, representing the length of the oscillations. These equations have important practical applications, for example, in the study of elasticity and heat conduction for composite materials. One major mathematical technique to deal with these equations is homogenization theory. The theory associates the original equation with its microstructure to some macrostructure effective equation that does not have oscillatory coefficients [2]. By homogenization, as ffl gets small, the solution of (1.1) will converge to the solution u(x) of the following homogenized equation, are constants, given by the following expressions, )dy: Here - j (y) is 1-periodic in y and satisfies @ @ a ik (y): Also, the homogenized operator L - retains the ellipticity property of the operator L ffl . Multigrid methods are usually not so effective when applied to (1.1). Standard construction of coarse grid operators may produce operators with different properties than those of the fine grid operators [1, 3, 12]. In order to restore the high efficiency of the multigrid method, a new operator for the coarser grid operator is developed [5, 6]. This operator is called a homogenized coarse grid operator, based on the homogenized form of the equa- tion. For full multigrid or with more general coefficients, the homogenized operator can be numerically calculated from the finer grids based on the local solution of the so called cell problem [5]. For numerical examples on model problems and on the approximation of heat conduction in composite materials [6]. One difficulty for these problems is that the smaller eigenvalues do not correspond to very smooth eigenfunctions. It is thus not easy to represent these eigenfunctions on the coarser grids. After classical smoothing iterations on the fine grid, we know that the high frequency eigenmodes of the errors are reduced, and only the low frequency eigenmodes are significant. Partially following [8], one may realize that the low frequency eigenmodes can be approximated by the corresponding homogenized eigenmodes. This is the reason why effective or homogenized operators are useful when defining the coarse grid operator. In this paper, using homogenized coarse grid operators, the convergence of the two level method applied to two classes of (1.1) with Dirichlet boundary conditions is analyzed. In chapter 2, we consider the equation with coefficient oscillatory in x direction In chapter 3, we consider the equation with coefficient oscillatory diagonally. We show that as both ffl and h go to zeros, our two level multigrid method converges when the number of smoothing iteration fl is large enough as a function of h. We also require the ratio h=ffl not to belong to a small resonance set. More precisely the convergence is proved under the following conditions: ffl For the first case in chapter 2, ffl For the second case in chapter 3, if h belongs to the set S(ffl; h 0 ) of Diophantine number, kh In [4], Engquist called it the convergence essentially independent of ffl. The purpose of this paper is to present new analysis in order to give a theoretical explanation of the computational results presented in [5, 6]. The bounds on fl given above are overly pessimistic compared to the numerical experiments but the h dependence in fl exists also in the computations [5, 6]. The effect of not requiring h 2 S is also seen in the numerical tests [5, 9]. If the coarse grid operator is defined, i.e., by direct arithmetic averaging an eigenmode analysis of the type given in section 2 and 3 produces the estimate difference between the correctly homogenized coarse grid operators and other operators is qualitatively consistent with the computational results of [5, 6]. The O(h \Gamma2 ) estimate means that there is no multigird effect and the convergence is only produced by the smoothing iterations. The l 2 difference between the inverse of the analytic operator L ffl and that of the corresponding homogenized operator L - is of the order O(ffl) [2]. This indicates that an eigenmode analysis of the type used in this paper cannot give estimates better than This is close to the estimate in one space dimension fl - Ch \Gamma6=5 ln h in [9]. In special cases, it is possible to design prolongation, restriction and coarse grid operators under that the resulting method corresponds to a direct solver [7]. This type of algorithm and methods based on special discretizations with built in a priori knowledge of the oscillatory behavior is outside the scope of this paper. Since in the sequel of the paper the following lemma is often cited, we introduce it here first. Lemma 1.1 [4] Suppose g(x; y) 2 C 3 ([0; 1] \Theta [0; 1]) and is 1-periodic in y. Let x We always denote the domain [0; 1] \Theta [0; 1] by - (0; 1) by \Omega\Gamma and -\Omega =\Omega by @ We discretize the domain by the same number of grid points N with equal step size both in x- and y-directions. The step size h is chose to belong to the set of S(ffl; h 0 ). And, the ratio of h to the wavelength ffl is fixed to be an strictly irrational number. -\Omega h denotes the set of grid points (ih; jh) 2 - \Omega\Gamma\Omega h for (ih; jh) 2 \Omega\Gamma and @\Omega h for (ih; jh) 2 @ present some constants, independent of ffl and h. D i are standard forward and backward finite difference operators in x direction; D j are for y direction. k \Delta k h represents the discrete L 2 \Gammanorm, indexed from 1 to N \Gamma 1. Oscillation Along a Coordinate Direction 2.1 Model Equation Consider a special case of (1.1), a two-dimensional elliptic problem with coefficients oscillatory in x-direction only, @ @x a ffl (x) @OE ffl @ @y a ffl (x) @OE ffl @y where a ffl (x) is a strictly positive continuous function, and a ffl the operator satisfies the property of (1.2). From (1.3), the corresponding homogenized equation of (2.1) is: \Gamma- 0 a(x)dx are the harmonic and arithmetical averages respectively. As ffl goes to zero, we know that the solution OE ffl of (2.1) converges to the solution OE of (2.2). Now, consider a corresponding discretized equation of (2.1), where a Denote the discretization of the homogenized operator \Gamma- @ 2 a @ 2 in (2.2) by The operator of the two level method by using the homogenized coarse grid operator [5, 6] can be expressed as For simplicity, in the sequel of the paper, I H h and I h H always denote the weighting restriction and bilinear interpolation respectively, and the smoothing operator S, where ff is the inverse of the largest eigenvalue of L ffl;h , has order of h \Gamma2 . LH is taken to be the corresponding homogenized operator L -;H . 2.2 Convergence Analysis Instead of the operator M , we consider a simplified operator, denoted by M 1 , Theorem 2.1 If the ratio of h to ffl is fixed, h belongs to the set of Diophantine number defined in (1.6), then there exist two constants C and ae 0 such that whenever Let's introduce some lemmate first, which are used in the proof of Theorem 2.1. Lemma 2.1 Assume Then, Z i is bounded, and satisfies Proof. By Lemma 1.1, taking Z 11 a(y) Hence, Z 11 a(y) Z 11 a(y) Z i is bounded for satisfies the following equation, a Proof. Directly applying Lemma 1.1, we can establish the result. 2 Lemma 2.3 Assume U ij satisfies where ij ) is a normalized eigenpair of L ffl;h . That is, Then, (D i Proof. Multiplied by U ij , added by parts, (2.9) then follows. For (2.10), note first for any grid function U ij , vanishing on @\Omega h , we have (D i (D i (D i Multiply D i both sides of (2.8), and take summation, Then, we can establish for some constant C. Analogously, we can get (2.11), (2.12) and (2.13) complete the proof. 2 Lemma 2.4 Assume OE h are defined in Lemma 2.3. Then, Proof. Introduce the following discrete function where Z i is defined in Lemma 2.1. Such G ij vanishes at boundary, i.e., By calculation, we have where (i; 2\Omega h , and j i is defined in Lemma 2.2. Multiply G ij on both sides, and take summation, a a (D i (D j (D j \Theta (D i By Lemma 2.3, we have (D i Hence, v u u u t (D i By Poincare inequality, By Lemma 2.1 and Lemma 2.3, Hence, Meanwhile, the inequality tells us that The proof of lemma is completed. 2 We are now able to show Theorem 2.1. Proof of Theorem 2.1. Denote the eigenvalues of L ffl;h and \Delta h (Laplacian operator) by - ffl respectively, where by dividing the set of eigenvalues into two subsets, say f- we can split the complete eigenspace of L ffl;h into two orthogonal subspaces. Namely, the space of low frequency expanded by the eigenfunctions whose corresponding eigenvalues belong to the first set, and that of high frequency expanded by the eigenfunctions whose corresponding eigenvalues belong to the second set. By minimax principle of eigenvalues, it is easy to see that for some constants c and C. For any normalized vector - such that where Thus, In the following analysis, we consider (2.17) in two steps. Step 1: Low frequency subspace. I -;h L ffl;h )OE ffl By Lemma 2.4, -;h L ffl;h )OE ffl The corresponding eigenvalue of Laplacian operator \Delta h can be explicitly expressed as By Taylor expansion, it follows that Hence, I By the constraint \Sigma I Since the ratio of h to ffl is fixed to be an irrational number, h has the same order as ffl. We have I 1 - Chk 3 Therefore, in order to make I 1 ! 1, it's sufficient to have Step 2: High frequency subspace. I -;h L 1ffl;h )k h kL2 -;h L2 I r k 2), we have I For I 2 - 1, it is sufficient to have Combining (2.21) and (2.24), we have The proof is completed. 2 Now, we are ready to show the main result. Theorem 2.2 There exists constant C, the operator M defined by (2.5) satisfies whenever h belongs to the set S(ffl; h 0 ) of Diophantine number, and Before we carry out the proof, we need the following lemma. Let Lemma 2.5 For some constant C, Proof. Since L -;h and are the homogenized operators defined respectively on fine and coarser grid with constant coefficients, they are well behaved [11, 12]. Furthermore, Therefore, (2.26)Proof of Theorem 2.2. Note that Therefore, Since ae(M) - kMk h , by Theorem 2.1 and Lemma 2.5, the rest of the proof can easily Oscillation Along the Diagonal Direction 3.1 Model Equation Consider another special case of (1.1), a two-dimensional elliptic problem with coefficients oscillatory diagonally, @ @x a ffl @ @y a ffl @y where a ffl (x) is a strictly positive continuous function, and a ffl Also, the operator has the property of (1.2). From (1.3), the corresponding homogenized equation of (3.1) is: As ffl goes to zero, we know that the solution OE ffl of (3.1) converges to the solution OE of (3.2). Now, consider a corresponding discretized equation of (3.1), where Here, we assume the discretized coefficients have the following property, a Denote the discretization of the homogenized operator a in (3.2) by (D i (D i k=1a k0 k=1 a k0 . The operator of the two level method can be expressed as where ff is the inverse of the largest eigenvalue of L ffl;h , has order of h \Gamma2 . And 3.2 Convergence Analysis We still first consider the simplified operator M 1 defined in (2.6). Theorem 3.1 If the ratio of h to ffl is strictly irrational, h belongs to the set of Diophantine number defined in (1.6), then there exist two constants C and ae 0 such that whenever In order to prove Theorem 3.1, we introduce the following lemmate first. Lemma 3.1 Define two discrete functions on k=1a kj k=1a k0 are bounded, and satisfy for (i; 2\Omega h . Proof. Notice that by the assumption of the coefficients, a kj Applying the operator L ffl;h to Z 1 ij as follows, k=1a kj By Lemma 1.1, k=1a kj Hence, Z 1 ij is bounded for (i; -\Omega h , and satisfies The result can be deduced similarly for Z 2 ij . We can also show that Z 2 ij , and satisfies This proves the lemma. 2 Remark. The explicit forms of Z 1 ij and Z 2 ij depend on a ffl being a function of x \Gamma y. For the general angular dependences a ffl (ffx + fiy), these forms would not be possible. Lemma 3.2 Assume discrete function defined as a bounded, and has the following properties Proof. Using the symmetric properties of the coefficients, we can establish the results similarly as in Lemma 3.1. 2 Lemma 3.3 Assume U ij satisfies (3. where ij ) is a normalized eigenpair of L ffl;h . Then, we have and Proof. First, we observe that Applying U ij to the following equation and taking summation, we have then follows. For (3.12), since U ij vanishes at the boundary, it satisfies2 (D i (D i (D i Multiplying D i to (3.10), we get (D i (D i (D i (D i By the uniformly ellipticity property of the homogenized operator (3.4), we have An similar argument gives (3.13), (3.14) and (3.15) complete the proof of lemma. 2 Lemma 3.4 Assume that U ij defined in Lemma 3.3 satisfies the following boundary conditions Then, (D i (D i Proof. Since we have (D i (D i Then, (D i (D i (D i Combined (3.12) with the following relation, the rest of the proof follows. 2 Theorem 3.2 Assume OE h are defined in Lemma 3.3 and Lemma 3.4. Then, Proof. First, we introduce the following discrete functions for (i; \Omega h . By the assumption (3.16), G ij vanishes at boundary, i.e., For ij , we have (D i (D j Then, For ij , we establish similarly (D i (D j Then, a a (b ij+1 Z 2 For simplicity, we introduce another two operators, L 1 and L by (D i and a a Observe that from (3.19), (3.20) and (3.21), we get (b ij+1 Z 2 Meanwhile, by Lemma 3.2, taking summations by parts, we have (D j \Gammaffl \Gammaffl By the symmetric property of the coefficients, a Proceeding in the same way as before, we obtain Combining (3.23), (3.24), (3.25) and (3.26), we get (D j Further, [a (D i (D i (D i (D i The exact same order for the last three terms in (3.22) can be established similarly. Con- sequently, from (3.27) and (3.28), (D i By Poincare inequality, By Lemma 3.1 and Lemma 3.3, which implies, We hence complete the proof. 2 Remark. The result in (3.18) here is consistent with the result established in [8] for the continuous case. Proof of Theorem 3.1. The procedure is exactly the same as that in Theorem 2.1, except different inequalities estimated. By Theorem 3.2, we have I instead of (2.20). Therefore, in order to make I 1 ! 1, we set instead of (2.21). For I 2 - 1, it is sufficient to have Combining (3.31) and (3.32), we have what we have done in previous section, we consequently establish the following main Theorem. Theorem 3.3 There exists constant C, the operator M defined by (2.5) satisfies whenever the step size h belongs to the set S(ffl; h 0 ) of Diophantine numbers, and The analysis of the proof strongly indicates us the role of homogenization, which plays in the convergence process. If, for example, the coarse grid operator is replaced by its averaged operator in one dimensional problem [5], the direct estimate for multigrid convergence rate is not asymptotically better than just using the damped Jacobi smoothing operator. This follows from the effect of the oscillations on the low eigenmodes. The homogenized coarse grid operator reduces the number of smoothing operation from O(h \Gamma2 ) to O(h \Gamma6=5 ln h), when the step size h belongs to the set S(ffl; h 0 ) of Diophantine numbers. In [9], it has also been shown that the number of smoothing iteration needed for the convergence of the multigird method with the average coarse grid operator guarantees the one with the homogenized coarse grid operator. The theoretical results established in this paper seem a little bit disappointing. From a number of numerical experiments [5, 6], we can get much faster convergence rate in practice than that required in the theoretical results. However, numerical results do indicate that the convergent rate depends on the grid size h for these types of equations with oscillatory coefficients [5, 6]. There are some inequalities in the implementation of the proof, which are potential to be improved so that a sharper convergent rate is possible. One of them is to enlarge the space of low eigenmodes, which can be approximated by the corresponding homogenized eigenmodes. Such as to improve (3.18) to (2.14), which we think is the sharpest inequality one can establish. We established the same inequality for the one dimensional case as in the two dimensional case oscillatory along a coordinate direction [9]. However, the portion of the eigenmodes that can be approximated by the homogenized ones in later case is relatively much smaller than the previous one. That's why we obtain O(h \Gamma4=3 ln h) for the number of smoothing iterations instead of O(h \Gamma6=5 ln h) for the later case, although there have the same inequality (2.14) for the space of low eigenmodes. Nevertheless, from the analysis of homogenization, we understand that there always exists a boundary layer [2, 10], which makes it hard to get the first lower order correction of the eigenfunctions. The case we discussed in chapter 2, which is equivalently to one dimensional problem, doesn't have such a boundary layer. We hence get an estimate as in (2.14). For the case in chapter 3, all we can establish is (3.18), which consists of the result established in [8] for the continuous case. And, it also defines us a smaller low eigenspace. However, numerical examples tells us that there are also some difference between these two cases. That a complete understanding of the first lower order correction for the eigenfunctions is required to further improve the estimates. --R The Multi-Grid Method for the Diffusion Equation with Strongly Discontinuous Coefficients Asymptotic Analysis for Periodic Structure Computation of Oscillatory Solutions for Partial Differential Equations Multigrid Methods For Differential Equations With Highly Oscillatory Coefficients. New Coarse Grid Operators of Multigrid Methods for Highly Oscillatory Coefficient Elliptic Problems. Grid Transfer Operators for Highly Variable Coefficient Problems. Homogenization of Elliptic Eigenvalue Problems: Part 1 Multigrid Method for Elliptic Equation with Oscillatory Coefficients. First Order Corrections to the Homogenized Eigenvalues of a Periodic Composite Medium. New York: springer- Verlag Multigrid Methods --TR

convergence;oscillation;homogenization theory;elliptic equation;finite difference;multigrid method

273720

Multidimensional Interpolatory Subdivision Schemes.

This paper presents a general construction of multidimensional interpolatory subdivision schemes. In particular, we provide a concrete method for the construction of bivariate interpolatory subdivision schemes of increasing smoothness by finding an appropriate mask to convolve with the mask of a three-direction box spline Br,r,r of equal multiplicities. The resulting mask for the interpolatory subdivision exhibits all the symmetries of the three-direction box spline and with this increased symmetry comes increased smoothness. Several examples are computed (for in terms of the refinement mask are established and applied to the examples to estimate their smoothness.

iii. The function ' is in some H-older class C ff for suitable ff. The function ' is fundamental, if i holds, and it is refinable, if ii holds. The sequence h is called the refinement mask of the function '. In that sense the paper is a continuation of [25] where we considered compactly supported fundamental solutions given as linear combination of B-splines in the univariate setting and of box splines in the multivariate setting. While those fundamental solutions exhibit nice symmetry, regularity and approximation properties, they fail to satisfy a refinement relation, which precludes their use in subdivision schemes. In fact, it was proven in [19] for the univariate case that there are no compactly supported piecewise polynomial functions (splines) which are refinable and fundamental except for piecewise constant or piecewise linear with integer knots. On the otherhand, the univariate refinable and fundamental functions given in [7] and [8] are convolutions of B-splines with some distributions. Univariate refinable and fundamental functions can be derived also as the autocorrelations of refinable functions constructed in [5] in the context of wavelets; again these functions have the form of convolutions of B-splines with distributions. This indicates that multivariate functions satisfying i-iii may not necessarily be splines, but they could be convolutions of box splines with some distributions. It is our goal here to provide in the multivariate setting compactly supported and refinable fundamental solutions with the nice properties of symmetry, regularity and approximation. The functions given here are convolutions of box splines with distributions. Some compactly supported interpolatory subdivision schemes have already been given in the literature. In particular, we mention the work of [9], [12], [13] and [14]. The butterfly subdivision scheme given by [12] was the first example of bivariate C 1 refinable and fundamental function '. An improvement of the smoothness analysis of the scheme can be found in [14]. Several continuous bivariate refinable and fundamental functions are also given in [9]. Applications of interpolatory subdivision schemes to the generation of surfaces can be found in [9], [12], [13], [14] and [23], and applications to wavelets decompositions and image compression can be found in [10] and [11]. The use of fundamental solutions for cardinal interpolation to obtain fundamental solutions for Hermite interpolation on the lattice was discussed in [15]. Connections of fundamental and refinable functions to refinable functions having orthogonal shifts was discussed in [20] and [22]. We first establish some notation and some consequences of the refinement relation (1.1). For a finite sequence a, the symbol of a is the trigonometric polynomial ea on IR s with extension to a Laurent polynomial, e A, on C s as defined by the equations For a compactly supported continuous function ' on IR s , the symbol takes the form Introduction and Method the last equality by the Poisson summation formula. In other words, it is the symbol of the sequence restricted to ZZ s . Upon taking Fourier transforms, the refinement relation (1.1) is equivalent to When ZZ s represents the vertices of the unit cube in IR s , a standard argument yields -2ZZ se h( or, in terms of the symbols, se Before we apply relation (1.5) to the problem at hand, we introduce another set of polynomials e 2 , for a sequence a and relate them to the polynomials e A . We can decompose e A modulo ZZ s 2 as follows: se A - (z) := Then e A Hence, we see that the polynomials e A 2 , are obtained from the polynomials 2 g under the action of the unitary matrix U := f(\Gamma1) - 2 ZZ s A z - e s: Now, if the function ' is a fundamental solution for cardinal interpolation so that i. holds, then from (1.2) and (1.3), e se se Multidimensional Interpolatory Subdivision Schemes 3 Thus, the equation is necessary in order that the function ' be a compactly supported and refinable fundamental solution for cardinal interpolation. Our point of view will be to try to define an appropriate polynomial e H satisfying (1.8) and so that the subdivision mask derived from the coefficients of e h will converge to the desired fundamental function '. We hope to do this with good estimates on the smoothness of the resulting fundamental solution as well. To this end we take our cue from the construction of compactly supported refinable functions in the univariate case in [5] where the function e h takes the factored form The left hand factor is the symbol of the refinement mask for the cardinal B-spline of order N . It is this left hand factor that gives the smoothness to the resulting refinable functions while the contribution of the trigonometric polynomial factor G(y) takes away from that smoothness. An appropriately chosen G not only gives the basic orthogonality properties of the refinable functions, but also does not lessen the smoothness too much. For several variables, the appropriate generalization of the cardinal B-splines are the box splines defined for a given s \Theta n matrix \Xi of full rank with integer entries. The basic facts and much of the notation concerning box splines are taken from [2]; the reader is referred to [2] for the appropriate references. In the case of the univariate cardinal spline, the number N plays several roles: it is the order of the spline (one more than the degree of its polynomial pieces); the interval [0 : :N ] is the support of the spline; and, the spline belongs to C N \Gamma2 , or even finer, its 1st derivative is in For the multivariate box splines, these three things are encoded in the direction matrix \Xi, but in more complicated ways. The (total) degree of the polynomial pieces of the box spline does not exceed n \Gamma s. The support of the box spline is the polyhedron where the summation runs over the columns of the matrix \Xi and as the notation indicates, we shall apply set notation to \Xi as a set of its columns. Finally, the box spline belongs to C is the minimum number of columns that can be discarded from \Xi to obtain a matrix of rank ! s. A box spline B \Xi satisfies the refinement equation e Y In what follows, we shall drop the subscript indication of dependence on the direction set \Xi unless it is needed to resolve ambiguities. In the univariate case, the shifts of any cardinal B-spline form a Riesz basis, but this is no longer the case for box splines in higher dimensions. However, there is an easy to check criterion for when the shifts of a box spline do form a Riesz basis; namely, when the direction set \Xi is 4 1. Introduction and Method a unimodular matrix (all bases of columns from \Xi have determinant \Sigma1). The last condition is equivalent to there being no y 2 IR s at which all of the functions b vanish. The latter fact in turn implies that the symbol for the autocorrelation function B au := B B(\Gamma \Delta ) is positive for . In this case, (1.5) reads where e B, e M are the symbols associated with the box spline B, its autocorrelation and its refinement mask m respectively. That observation is the basic step in the proof of Proposition(1.11). If the direction matrix \Xi defining the box spline neither set of Laurent polynomials 2 g and have common zeros in (Cnf0g) s . Here f M is the symbol for the refinement mask m of the box spline. Proof. Since e B au is positive for implies that the first set of Laurent polynomials in the Proposition have no common zeros for The relation (1.7) then implies that the second set of Laurent polynomials in the Proposition have no common zeros for . But from (1.6) and (1.9) it follows that the second set of Laurent polynomials can only have common zeros where z = exp(\Gammaiy). Hence, again by (1.7), the first set of Laurent polynomials can only have common zeros where one of the components of z is zero. - We shall use this Proposition to find candidates for e H that satisfy (1.8). By Bezout's theorem, there exist Laurent polynomials e such that We set Q := -2ZZ sz \Gamma- e m(y); or e In that case, e se so that e se and (1.8) is satisfied. The proceeding paragraph outlines a general construction of the mask. There are many possibilities for the polynomials e and choices of ff 2 IN s in (1.12). Which ones give rise to the refinement mask of a regular fundamental solution for cardinal interpolation with the properties we desire? We present the two dimensional case in detail in the next section. Multidimensional Interpolatory Subdivision Schemes 5 Once a construction of the mask is carried out, the Fourier transform of the corresponding refinable function ' with mask h can be represented by If e the infinite product converges in the sense of distributions. In this case the function ' is a compactly supported distribution. The function ' can be constructed iteratively. Begin with one of the simplest continuous examples of a ', say the hat function ' 0 := B 1;1;1 (the piecewise linear box spline having value 1 at (1; 1)) and define recursively h(j)' This is called the cascade algorithm. If ' n converges to ' uniformly, the function ' can be obtained through this interative process and ' is a compactly supported L1 (IR s ) function. If ' is stable, converges to ' uniformly. Recall that a compactly supported continuous function ' is stable, if there is C ? 0, so that With the refinement mask h satisfying e we have a function ', at least in the distributional sense, which satisfies equation (1.1). It is still a matter to check whether the obtained refinement mask defines a continuous function ', whether the resulting refinable function is the fundamental solution for cardinal interpolation and whether the corresponding subdivision scheme converges uniformly. Since only the mask is at hand, all criteria used to test the above properties should be in terms of the refinement mask h and it should be possible to implement the criteria for reasonable examples. We first of all require that ' be continuous. Regularity or smoothness criteria that can be applied to our case are discussed in the x3. All the examples we shall construct in x4 will be better than C 1 . Once we know that the resulting function ' is continuous, then we must show that it is fundamental. As a corollary of the characterization of the stability and orthonormality of refinable functions in terms of refinement mask, the results in [20] provide necessary and sufficient conditions to test whether a mask that determines a continuous compactly supported ' is also fundamental: Theorem(1.16). ([20] Proposition 4.1) Suppose ' is a compactly supported continuous function satisfying the refinement equation (1.1) with the mask sequence h supported in [\Gamma(N \Gamma1) : :(N \Gamma1)] s . Then ' is fundamental if and only if the ffi sequence is the unique eigenvector (up to constant multiples) of the matrix IH := corresponding to a simple eigenvalue 1. For the examples of our construction given in Section 4, the task of determining the eigenvalues and eigenvectors can be carried out numerically; for example, using MATLAB. Once we know that ' is fundamental, then it is automatically stable, since there is C ? 0 so that 6 2. The Bivariate Construction (see [17]). Consequently ' n converges to ' uniformly (e.g. see [3]). Suppose that the mask h defines a fundamental and refinable continuous function '. An interpolatory subdivision scheme is defined as follows: Let c := fc ff : ff 2 ZZ s g be a sequence of control points. The subdivision operator S is defined by This gives us new control points, The new control point sequence c 1 is determined linearly from c by 2 s different convolution rules, and sequence c 1 consists of 2 s different copies of the original control point sequence c which are mixed together. Since the scaling factor is 2, the new control polygon is parameterized, so that the points c 1 ff correspond to the finer grid 2 c(fi), the new control point sequence interpolates the previous one. Continuing this process, we get the control point sequences c corresponding to the grids 2 \Gamman ZZ s . This process is called an interpolatory subdivision scheme. The subdivision scheme is said to converge for an arbitrary control point sequence c 2 ' 1 (ZZ s ), if there exists a continuous function f so that lim ck The limit function f is c ff '( In particular, if c then the limit function is '. It was shown in [3] that if ' is stable, the subdivision scheme converges. Therefore, the interpolatory subdivision schemes discussed in this paper converge. Interested readers should consult [3] for details. Finally, we remark that since the refinable functions constructed here possible to construct prewavelets from them. Interested readers should consult [17], [18] and [24] for details of the construction of prewavelets. 2. The Bivariate Construction Here we detail a construction in the two-dimensional case. The box splines that have stable shifts are those defined for direction matrices based on the three directions 1). We shall take these directions to be given with equal multiplicities r. These box splines will be denoted by B r;r;r . The symbol of the mask in this case takes the form The box spline B r;r;r belongs to C 2r\Gamma2 . In case r = 1, the box spline B 1;1;1 is itself interpolatory so there is no need to find a suitable multiplier e Q for the symbol. Hence, we assume that r - 2 in the sequel. Multidimensional Interpolatory Subdivision Schemes 7 It has been our experience [25] that out of the many possible fundamental solutions that one can obtain through the use of Bezout's theorem, most do not have practical (or aesthetic) value because they have large variation (and often large max norm) over their support. The method of construction here leads to examples of fundamental solutions of increasing smoothness with the classic shape: centrally symmetric with value equal 1 at the origin. In fact, from the cases we have computed, it appears that with greater symmetry comes greater smoothness. Our object in the end will be to preserve the well-known symmetry structure of the box spline [1]. The existence of a function e Q implies that there is some square that will contain the support of the mask h for e H. A smaller square means smaller support for the mask h and consequently, smaller support for the refinable function. Initially, we try to fit the support of the mask h for e H into a square of side length 4r \Gamma 1. It turns out that for our examples this is possible. Later, we will impose further conditions to make its support look like that of (twice) the support of the box spline within that square. We consider the even lattice in the somewhat smaller square S central point 2). Choosing this central point for the ff in (1.12), we find polynomials e such that The idea of the construction is simply this: The masks of f each occupy a smaller rectangle S - in the lower right hand corner of S. The mask rectangle S - (with its values) is permitted to shift in the three directions long as the shifted rectangles remain within S. Each of the distinct such shifts are assigned an unknown coefficient which multiplies each of the entries in the mask. In this way, the points in 2ZZ are assigned values which are a linear combination of the values of the masks for the f (the values in ZZ 2 n2ZZ 2 are automatically zero). This leads to a linear system of equations when we ask that the resulting mask be zero everywhere except at should be equal 1. Our first goal is to analyze this system, even when some additional symmetry conditions inherited from f are imposed. First, we observe from (2.1) and the definition of f M - that 2: Therefore, the upper right corner of the rectangle S 0;0 can be shifted to the even lattice points in the rectangle [2r . There are (r \Gamma 1) 2 such shifts. The new mask formed when these shifts are added together will be the mask for the polynomial@ X a 0;0 (j)z 2jA f and the candidate for e Q 0;0 is the first factor. Similar reasoning applies to the other rectangles S - 8 2. The Bivariate Construction and we find that the candidates for the e are: e a 0;0 (j)z 2j e a 1;0 (j)z 2j e a 0;1 (j)z 2j e a 1;1 (j)z 2j : There are (r \Gamma unknowns in the above equations which equals precisely the number of even lattice points in [0 . Thus, the resulting system of equations has a solution if it is consistent and then each such solution satisfies (2.2). In general, there may be infinitely many solutions because of the symmetries inherent in the masks of the f First, observe that f M is invariant under the interchange of z(1) and z(2). Thus, f f For any given set of solutions e to (2.2), we define e e Then the e - also provide solutions for (2.2) as does the combination e - )=2 and the latter also satisfy e e Next we observe that f also satisfies the reciprocal relation z M(z This implies that Again, given any set of solutions e Multidimensional Interpolatory Subdivision Schemes 9 Observe that if the e then so do the e - . Furthermore, -2ZZ 2z 4r\Gamma2 z \Gamma2-\Gamma2- f so that the e - also satisfy (2.2). Moreover, the monomials comprising e still correspond to shifts of S - that would remain in S. Finally, the combinations e with the property that Assume now that the e are solutions of (2.2) that satisfy both (2.7) and (2.10). We have shown that they can be obtained from any solution by the above procedure. Define Then the relations e e Q(z follow from (2.7) and (2.10). We are now in a position to define a suitable e With this definition, e H is symmetric in the components of the variable z = (z(1); z(2)) and is reciprocal H(z These two properties imply immediately that the associated mask of coefficients h is symmetric through the direction and is symmetric through the origin: Furthermore, e so that the necessary condition (1.8) is fulfilled. Moreover, since we find that 2. The Bivariate Construction The maximum support square for the mask of any f while that for the mask of any e . Thus, the support of the mask for f Q is in and We now restrict ourselves to consider the periodic symbol e h := The reciprocal relation (2.13) implies that e h is a real cosine polynomial. From (2.14) we have 2 nf0g, we also have This gives four linear equations for the , from which easily follows In particular, since e implies that The above construction does lead to solutions that provide fairly smooth convergent interpolatory subdivision schemes. But as it stands, the solutions are not uniquely determined since we have not taken into account the full symmetries of the box spline. Here we use the symmetries of the "centered" form of the box spline B r;r;r (see [1]), translated to our setting. The symmetries of the mask f M will be generated by the two matrices and G 2 := acting on the exponents of z through the two relations M(z); and z(2) r f For later purposes, we note that the complete set of symmetries are embodied in the matrices G := 'oe The symmetries for the box spline were derived by mapping the three directions into permutations of themselves. This has significance for the problem at hand because the three directions are in fact the non-zero elements of ZZ 2 2 n0; for all G 2 G: Multidimensional Interpolatory Subdivision Schemes 11 The first of relations (2.17) just combines the interchange of z(1) and z(2) and the reciprocal relation and so has already been taken into account in our construction above. The second relation is a much stronger requirement that will restrict further the shifts of the S - that will be permitted, thus reducing the number of unknowns. The restrictions will depend on the parity of r. We first check how the second relation in (2.17) translates to the f r even: r odd: The symmetry on e H imposed by the matrix G 2 acting on its exponents is This, together with the definition of e H , (2.12), and the second relation in (2.17), makes the following requirement of e Q: As was the case with f places restrictions on the e r even: re r odd: If we require that the e retain the form (2.4), then substituting that form into the right hand side of (2.22) places further restrictions on which monomials may have non-zero coefficients. Carrying this out, we find that the e should have the form: r even: e \Gamma( r a 0;0 (j)z 2j ; e \Gamma( r e a 0;1 (j)z 2j ; e \Gamma( r a 1;1 (j)z 2j ; r odd: e e e e 12 2. The Bivariate Construction First observe that in either case there are coefficients in (2.23). Next, we note that the relations (2.19) put similar restrictions on the possible nonzero coefficients of z(1) 2j(1) z(2) 2j(2) for the f f is even, or f r is even, or f is even, or f is even, or Thus, under these restrictions the monomials z(1) 2j(1) z(2) 2j(2) in coefficients have indices that satisfy Hence, there are equations when these coefficients are compared in (2.2) with the e Once again, a solution exists if the system is consistent. The two operations (2.6) and (2.9) both produce functions of the type (2.23) if the original functions are of that type. Hence as before, there exist solutions e of (2.2) that satisfy both (2.7) and (2.10). It is possible to make the solution satisfy (2.21) as well by making use of r even: e e e e r odd: e e e e If the e have the form (2.23) and already satisfy (2.7), (2.10), and (2.2), then the functions e e e e will be of the form (2.23) and satisfy (2.2), (2.7), (2.10) and (2.22). We summarize the findings so far in the following theorem: Multidimensional Interpolatory Subdivision Schemes 13 Theorem(2.25). Among the solution sets f e -2ZZ 2, of having the form (2.23), there is one that satisfies (2.7), (2.10) and (2.22). For that solution set the Laurent polynomial e Q(z) with e is invariant under the action of the group G acting on its exponents: e G: The mask sequence h corresponding to e H is supported in and satisfies G: The relation (2.27) together with (2.18) gives a very strong symmetry on the mask h. (The reader may find it very helpful to consult the masks of section 4 while reading this.) Since G; as a set, the numbers fh(2j are the same for each - 2 ZZ 2 2 nf0g. These numbers are arranged symmetrically in the six cones generated by the lines on hexagonal rings of lattice points about the origin: A consequence of this is that the mask h is symmetric on its support along any of the lines const. For example, if const - 0, then The relations (2.16) were a result of f f which implies e e Comparing coefficients in these equations and taking into account ffi(j), we arrive at further relations for the mask h on the lines j(2)=const j(2)=2const j(1)=const j(1)=2const j(1)\Gammaj(2)=const j(1)\Gammaj(2)=2const 14 3. Regularity of refinable functions In particular, the sum of entries in the interior of each of the 6 cones is 0. The final symmetry comes from the fact that f e 3. Regularity of refinable functions In this section we will provide some criteria for the regularity of refinable functions in terms of their masks. These criteria are useful for the estimation of the regularity of the refinable functions derived from interpolatory subdivision schemes constructed in the next section. Unvariate counterparts of our regularity theorems and more can be found in [6, x7.1.3], [16], [4] and references cited therein. The proofs of some of these results is carried to the multivariate case and the dilation matrix 2I in order to provide estimates for our examples. This can be done without encountering many difficulties; however, we include the proofs for the sake of completeness. The function for some constant independent of x. The number ff is related to weighted L p exponents - p defined as Z In this paper, we only use - 1 and - 2 . The regularity order ff is related to - 1 and - 2 by the inequality The idea here is the usual Littlewood-Paley technique. The domain of the Fourier transform of ' is broken into the pieces C n := 2 n TT s Z then Z Hence, log 2, or when To estimate R '(w)j p dw, we consider the operator on L 2 (TT s ) defined in terms of a eg with the symbol of a finite mask Wf := This is the Fourier transform of the transition operator W for the mask g as defined in [20]: Wa := Multidimensional Interpolatory Subdivision Schemes 15 As discussed in [20], if g is supported on [\Gamma(N the space of all sequences supported on [\Gamma(N is an invariant subspace of W . The restriction of W to ' N is called the restricted transition operator. The restricted transition operator W on ' N can be represented by the matrix G := (cf. Theorem 1.16). We only use the restricted operator in this section and assume that g is supported on . Similarly, the space b the space of Fourier transforms of all sequences in ' N , is an invariant subspace of c W . Proposition (3.2). Let b ' n be defined inductively by Then for any f 2 L 2 (TT s ), we have and Z Z \Gamman w)b' n Z \Gamman w)\Pi n with equality whenever both f - 0 and eg - 0. Proof. For For general n, using the fact that 3. Regularity of refinable functions we fiind that c -2ZZ se g(w=2 \Gamman (w Inequality, (3.4) follows directly by (3.3), since Z Z \Gamman (w Z \Gamman w)b' n (w)jdw: A similar result was proved in [21]. Suppose that eg satisfies Define the space of trigonometric polynomials Since D fi 2 nf0g, we have that ae and for all jfij - ae. Hence, the space V ae is an invariant subspace of the restricted transition operator c W defined by (3.1). Proposition(3.6). Suppose e satisfies conditions (3.5) and eg - 0. Let - be the spectral radius of c . For the function ' g defined by there is a constant C such that Z Multidimensional Interpolatory Subdivision Schemes 17 Proof. For any given ffi ? 0, there exists a norm k \Delta k on V ae so that for any f 2 V ae , Hence Z Since all the choices of the constants in this proof do not depend on n, for simplicity, we denote all the constants by C even though the value of C may change with each occurance. Cjwj, the function b ' g is bounded on TT s . Hence, Z Z ae, we have f 2 V ae . Note that Z Z \Gamman w)b' n (w)jdw Z \Gamman w)b' n (w)jdw Z It was proved in [20] that if ' g is fundamental, then - ! 1. Hence, the condition whenever the mask g is the mask for a convergent interpolatory subdivision scheme. Theorem(3.7). Suppose e g satisfies conditions (3.5) where e g is derived from an interpolatory subdivision scheme with mask h as follows: eg := fails to hold. Let - be the spectral radius of c W j V ae , then the fundamental solution ' for the interpolatory subdivision scheme is in C ff\Gammaffl for any ffl ? 0 and log -= log 2; if e h - 0 or; log -=2 log fails to hold. Proof. If e h - 0, then ' Proposition 3.6 and the result follows from the Littlewood- Paley technique mentioned at the beginning of this section with Similarly, if e h - 0 fails, then we use the autocorrelation function ' au := '( \Delta ) '(\Gamma \Delta ) of '. The function ' au is refinable with the refinement mask h au := 2 \Gammas h( \Delta ) h(\Gamma \Delta ). Hence e h nonnegative. Thus, ' Proposition 3.6 and the result follows from the Littlewood-Paley technique mentioned at the beginning of this section with 2. - The above regularity results only depended on the property (3.5) and the nonnegativity of the mask eg and did not require any factorization of the mask eg. However, factorization can be useful 3. Regularity of refinable functions both to establish (3.5) in some cases and to isolate a nonnegative part. This is the case for the bivariate construction discussed in the last section where we have the factorization and b ' is the product of the box spline B r;r;r 2 C 2r\Gamma2 with the Fourier transform of a distribution. Since D fi e 2 nf0g, we have that e h, and Hence, the space V 2r\Gamma1 is an invariant subspace of the restricted transition operator c W defined by (3.1) with e e h, while V 4r\Gamma1 is an invariant subspace of c W if the autocorrelation function is used. In applications, we use either the matrix IH defined in Theorem 1.16 or the matrix IH au := as an operator on the restricted sequence spaces to check the regularity of '. Since c W is the Fourier transform of the operator IH (respectively, IH au ) on the restricted sequence space, the eigenvalues of IH (respectively, IH au ) are the eigenvalues of c W and the Fourier transform of the eigenvectors of IH (respectively, IH au ) are the corresponding eigenvectors of c W . This observation together with Theorem 3.7 allows the following procedure to estimate the regularity of '. First find all eigenvalues and eigenvectors of IH (respectively, IH au ), then throw out all those eignvalues whose Fourier transform of the corresponding eigenvectors are not in V ae , where . Note that the Fourier transform of the eigenvector v := (v fi ) of IH (respectively, IH au ) is not in V ae if and only if 0-j-j-ae The maximum absolute value of the remaining eigenvalues is the - to be used in Theorem 3.7. For the examples of our construction in the next section, the task of determining which of the largest eigenvectors are definitely NOT in V ae can be carried out numerically using MATLAB; simply order the eigenvalues according to decreasing modulus and proceed down the list checking (3.8) until a numerically significant drop in value occurs. The problem with this procedure is that as the matrices get large, it becomes quite difficult to carry out the numerical procedures. The complexity of using IH au is substantially more than in using IH, yet we are forced to use it since e h will fail to be positive for odd r. The next observation will be useful for our particular examples. In the case e mjeq, where e m is the mask of a box spline B \Xi and e is a finitely supported mask of some distribution with e , we can define an operator, c W 2 \Gammas eq , as in (3.1) but using q. If the support of the mask q is in consider the operator c W 2 \Gammas eq restricted to b ' N . Multidimensional Interpolatory Subdivision Schemes 19 mjeq where e m is the refinement mask of a box spline B \Xi , and the trigonometric polynomial e q is nonnegative, e g satisfies (3.5). Let - be the spectral radius of the restricted transition operator c W 2 \Gammas eq , i.e. the spectral radius of the matrix restricted to V ae . Then ' 2 C m(\Xi)+s\Gamma1\Gammalog -= log 2\Gammaffl , where ffl ? 0 is arbitrary. Proof. We note that e q e q (cf. [2, p. 102]). Hence, just as in the proof of Proposition 3.6 and using Theorem 3.7, we have Z Z Y e dw Z (recall that the value of C may change with each occurance). Hence, that ' 2 C m(\Xi)+s\Gamma1\Gammalog -= log 2\Gammaffl follows from the Littlewood-Paley technique. - The factorization in Corollary 3.9 does not require that we take the largest box spline factor, just that the remaining factor be nonnegative. 4. Examples We present the bivariate examples corresponding to decreasing detail) and discuss their regularity. We note that we have chosen the three direction mesh used in box spline theory. It is an easy transformation to convert this to the hexagonal mesh used by several people in wavelet analysis. One should realize that the three-direction symmetry of our examples would transform to a beautiful hexagonal symmetry in the hexagonal mesh. 2. The box spline B 2;2;2 has continuous second partial derivatives and its third partial derivatives are in L1 (IR 2 ). If we apply our method to obtain e h, then the support of the mask h 2;2;2 is expected to be in the square [\Gamma3 : Indeed, we obtain a mask exhibiting the full symmetries of the box spline 4. Examples where the entry at the origin is distinguished by boldface type. Here we note that there are only nonzero entries and only the 6 nonzero entries in the cone are required since the rest follow from the symmetries. If we use Theorem 1.16 on the 49 \Theta 49 matrix IH 2;2;2 determined by h 2;2;2 we find that it has 1 as a simple eigenvalue with ffi as its eigenvector. Hence, the function defined via (1.14) is a fundamental solution for cardinal interpolation. That ' 2;2;2 enjoys the classic shape of a fundamental solution and the symmetries of the tree direction box spline is clearly seen in Figure 4.1. Figure (4.1). The function ' 2;2;2 on [\Gamma3 : :3] 2 and its level lines \Gamma:1 3. The box spline B 3;3;3 belongs to C 4 (IR 2 ). The support of the mask h 3;3;3 will be in the square Again we obtain a mask exhibiting the full symmetries of the box spline Multidimensional Interpolatory Subdivision Schemes 21 where the entry at the origin is distinguished by boldface type. Only the entries in the cone are needed to describe the matrix with the symmetries are taken into consideration. The eigenvalues and eigenvectors of the 121 \Theta 121 sparse matrix IH 3;3;3 are easily determined by MATLAB. Again, 1 is a simple eigenvalue with ffi as its eigenvector. Hence, the function ' 3;3;3 defined via (1.14) is a fundamental solution for cardinal interpolation by Theorem 1.16(see Figure 4.2). -5 -4 -3 -2 -5 Figure (4.2). The function ' 3;3;3 on [\Gamma5 : :5] 2 and its level curves \Gamma:1 4. The box spline B 4;4;4 belongs to C 6 (IR 2 ). The support of the mask h 4;4;4 will be in the square [\Gamma7 : . Again, the mask exhibits the full symmetries of the box spline 22 4. Examples6 6 \Gamma350 12950 \Gamma48650 36050 \Gamma48650 12950 \Gamma350 0 where the entry at the origin is distinguished by boldface type. Again, analysing the matrix IH 4;4;4 , we find that the resulting function ' 4;4;4 is fundamental (see, Figure 4.3). -226 -226 Figure (4.3). The function ' 4;4;4 on [\Gamma7 : :7] 2 and its level lines [\Gamma:1 Multidimensional Interpolatory Subdivision Schemes 23 8. As is already apparent in the case 4, the masks soon become too large to give fully in the text. We have given the three cases full so that the symmetries alluded to in Section 2 can be firmly established in the minds of the reader. We give the masks of the four remaining cases in a more compact form, using the cones of symmetry. We use the first quadrant with 1 in the lower left corner sitting at the origin, the rest of the diagonal is left blank and one mask is given in the cone and the other in 0 - k ! j. The two included tables give the masks for 8. While the mask for 8 is a 31 \Theta 31 matrix which is quite large for numerical purposes, it does compare favorably with using a fundamental box spline interpolant of comparable smoothness (for example, the fundamental solution corresponding to B 3;3;3 ) which has exponential decay and so must be truncated appropriately for use. The matrices IH grow more quickly and for is already an 941 \Theta 941 matrix. The computations still show that the eigenvalue 1 is simple with ffi as its eigenvector. The computations of the sums (3.8) do show the effects of the added complexity, but are none-the-less useful for the estimation of regularity. Regularity. The examples for sufficient variety to test various smoothness criteria. We use Theorem 3.7 and Corollary 3.9 to estimate the regularity of our examples. We have computed three separate quantities. The first quantity is the spectral radius of c W j V for e in all cases, even though by Theorem 3.7 it applies only in the cases when e h - 0. It turns out that e h - 0 holds if and only if r is even, since exp(i(r \Gamma 2) \Delta )eq - 4 in each case. We suspect that this quantity gives the best estimate for regularity even in the case of odd r when it has not been shown to apply. The second computation yields an estimate for the spectral radius of c for the size of the matrix IH au quickly impedes the computation and we were successful in carrying it out only for 4. In every case, the estimate for the smoothness is smaller than that obtained using the first quantity, although this estimate is valid in all cases. Finally, we use the factorization e in Corollary 3.9. In this case, is the spectral radius of c W j V 2r\Gamma3 for qj. The results of the computations are listed in the following table: Recall that ' r;r;r belongs to C ff\Gammaffl where ff is estimated by \Gamma log(-)= log(2) when r is even, by \Gamma log(-)=(2 and by 2 \Gamma log(-)= log(2) for r odd. e qj 3 2.8301e+00 2.1751e+00 2.6100e+00 6 4.7767e+00 ?? NA --R Bivariate cardinal interpolation by splines on a three-direction mesh splines A new technique to estimate the regularity of a refinable function Orthonormal bases of compactly supported wavelets Ten Lectures on Wavelets Interpolation through an iterative scheme Symmetric iterative interpolation processes A butterfly subdivision scheme for surface interpolation with tension control "Multivariate Approximation and Interpolation," Using parameters to increase smoothness of curves and surfaces generated by subdivision Hermite interpolation on the lattice hZZ d Subdivision schemes in L p spaces Using the refinement equation for the construction of pre- wavelets II: Powers of two Multiresolution and wavelets Complete characterization of refinable splines Stability and orthonormality of multivariate refinable functions Biorthogonal wavelet bases on IR d Interpolatory subdivision schemes and wavelets Smooth surface interpolation over arbitrary triangulations by subdivision algorithms Wavelets and pre-wavelets in low dimensions --TR --CTR Yun-Zhang Li, On the holes of a class of bidimensional nonseparable wavelets, Journal of Approximation Theory, v.125 n.2, p.151-168, December Guiqing Li , Weiyin Ma, Interpolatory ternary subdivision surfaces, Computer Aided Geometric Design, v.23 n.1, p.45-77, January 2005 Bin Han , Rong-Qing Jia, Quincunx fundamental refinable functions and Quincunx biorthogonal wavelets, Mathematics of Computation, v.71 n.237, p.165-196, January 2002 Bin Dong , Zuowei Shen, Construction of biorthogonal wavelets from pseudo-splines, Journal of Approximation Theory, v.138 n.2, p.211-231, February 2006

subdivision schemes;box splines;wavelets;interpolatory subdivision schemes;interpolation

273734

Quasi-Optimal Schwarz Methods for the Conforming Spectral Element Discretization.

The spectral element method is used to discretize self-adjoint elliptic equations in three-dimensional domains. The domain is decomposed into hexahedral elements, and in each of the elements the discretization space is the set of polynomials of degree N in each variable. A conforming Galerkin formulation is used, the corresponding integrals are computed approximately with Gauss--Lobatto--Legendre (GLL) quadrature rules of order N, and a Lagrange interpolation basis associated with the GLL nodes is used. Fast methods are developed for solving the resulting linear system by the preconditioned conjugate gradient method. The conforming finite element space on the GLL mesh, consisting of piecewise Q1 or P1 functions, produces a stiffness matrix Kh that is known to be spectrally equivalent to the spectral element stiffness matrix KN. Kh is replaced by a preconditioner $\tilde{K}_h$ which is well adapted to parallel computer architectures. The preconditioned operator is then $\tilde{K}_h^{-1} K_N$.Techniques for nonregular meshes are developed, which make it possible to estimate the condition number of $\tilde{K}_h^{-1} K_N$, where $\tilde{K}_h$ is a standard finite element preconditioner of Kh , based on the GLL mesh. Two finite element--based preconditioners: the wirebasket method of Smith and the overlapping Schwarz algorithm for the spectral element method are given as examples of the use of these tools. Numerical experiments performed by Pahl are briefly discussed to illustrate the efficiency of these methods in two dimensions.

Introduction . In the past decade, many preconditioners have been developed for the large systems of linear equations arising from the finite element discretization of elliptic self-adjoint partial differential equations; see e.g. [6], [14], [27]. A specially challenging problem is the design of preconditioners for three dimensional problems. More recently, spectral element discretizations of such equations have been proposed, and their efficiency has been demonstrated; see [15], [16], and references therein. In large scale problems, long range interactions of the basis elements produce quite dense and expensive factorizations of the stiffness matrix, and the use of direct methods is not economical due to the large memory requirements [12]. Early work on preconditioners for these equations was done by Pavarino [20], [21], [19]. His algorithms are numerically scalable (i.e., the number of iterations is independent of the number of substructures) and quasi-optimal (the number of iterations grows slowly with the degree of the polynomials.) However, each application of the preconditioner can be very expensive. Several iterative substructuring methods which preserve scalability and quasi- optimality was introduced by Pavarino and Widlund [22], [24]. These preconditioners can be viewed as block-Jacobi methods after transforming the matrix to a particular basis. The subspaces used are the analogues of those proposed by Smith [28] for piecewise linear finite element discretizations. The bounds for the condition number of the preconditioned operator grows only slowly with the polynomial degree, and are independent of the number of substructures. Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, N.Y. 10012. Electronic mail address: casarin@cims.nyu.edu. This work has been supported in part by a Brazilian graduate student fellowship from CNPq, and in part by the U. S. Department of Energy under contract DE-FG02-92ER25127. The tensorial character of the spectral element matrix can be exploited when evaluating its action in a vector [16], but does not help when evaluating the action of the inverse of blocks of this matrix, as in the case of the above preconditioners. Following Pahl [17], based on the work of Deville and Mund [7] and of Canuto [5], the above constructions give rise to different and spectrally equivalent preconditioners using the same block partitioning of the finite element matrix generated by Q 1 elements on the hexahedrals of the Gauss-Lobatto-Legendre (GLL) mesh. This observation and experiments for a model problem in two dimensions were made by Pahl [17], who demonstrated experimentally that this preconditioner is very efficient. Thus, high order accuracy can be combined with efficient and inexpensive low-order preconditioning. We remark that similar ideas also appear in [18] and [26]. The analysis of Schwarz preconditioners for piecewise linear finite elements for the h-method, has relied upon shape regularity of the mesh [10], [9], [3], which clearly does not hold for the GLL meshes. We extend the analysis to such meshes, deriving estimates for these finite element preconditioners of spectral element methods. We give polylogarithmic bounds on the condition number of the preconditioned operators for iterative substructuring methods, and a result analogous to the standard bound for overlapping Schwarz algorithms. Then, by applying Canuto's result, [5], we propose and analyze a new overlapping preconditioner that depends only on the spectral element matrix. We also give a new proof of one of the estimates in [23] by using the same equivalence. The remainder of this paper is organized as follows. The next section contains some notation and a precise definition of the discrete problem. Our motivation and strategy are presented in detail in Section 3. In Section 4 we give the statement and proofs of our technical results. In the two remaining sections we formulate and analyze our algorithms. 2. Differential and Discrete Model Problems. Let\Omega be a bounded polyhedral region in R 3 with diameter of order 1. We consider the following elliptic self-adjoint Find (1) where Z\Omega Z\Omega fv dx for f 2 L This problem is discretized by the spectral element method (SEM) as follows; see [16]. We triangulate\Omega into non-overlapping substructures of diameter on the order of H . Each\Omega i is the image of the reference cube a mapping F is an isotropic dilation and G i a C 1 mapping such that its derivative and inverse of the derivative are uniformly bounded by a constant close to one. Moreover, we suppose that the intersection between the closure of two substructures is either empty, a vertex, a whole edge or a whole face. Each substructure\Omega i is called a distorted cube. We notice that some additional properties of the mappings F i are required to guarantee an optimal convergence rate. We refer to [2], problem 2 and the references therein for further detail on this issue, but remark that affine mappings are covered by the available convergence theory for these methods. We assume for simplicity that k(x) has the constant value k i in each with possibly large jumps occurring only across substructure boundaries. This point is important only in the analysis of iterative substructuring algorithms in Section 5, where our estimates are independent of the jumps of k(x). We define the space P N ( -\Omega ) as the space of QN functions, i.e. polynomials of degree at most N in each of the variables separately. The space P is the space of functions v N such that v N ffi F i belongs to P N ( -\Omega ). The conforming space P N H 1(\Omega\Gamma is the space of continuous functions the restrictions of which belong to The discrete L 2 inner product is defined by (2) are, respectively, the Gauss-Lobatto-Legendre (GLL) quadrature points and weights in the interval [\Gamma1; +1]; see [2]. The discrete problem is: find uN 2 P N 0(\Omega\Gamma9 such that aQ The functions OE N of P N(\Omega\Gamma that are one at the GLL node j and zero at the other nodes form the nodal basis of this space which gives rise in the standard way to the linear system KN b. Note that the mass matrix of this nodal basis generated by the discrete L 2 inner product 2 is diagonal. The analysis of the SEM method just described and experimental evidence show that it achieves very good accuracy for reasonably small N for a wide range of problems; see [2], [16], [15] and references therein. The practical application of this approach for large scale problems, however, depends on fast and reliable solution methods for the system KN b. The condition number of KN is very large even for moderate values of N [2]. Our approach is to solve this system by a preconditioned conjugate gradient algorithm. The following low-order discretization is used to define several preconditioners in the next sections. The GLL points define a triangulation T - h of -\Omega into parallelepipeds, and on this triangulation we define the space P - h ( - \Omega\Gamma of continuous piecewise trilinear tions. The spaces P are defined analogously to P The finite element discrete problem associated with (1) is: Find u 0(\Omega\Gamma5 such that The standard nodal basis f - in P - h ( - \Omega\Gamma is mapped by the F j , 1 basis for P h(\Omega\Gamma4 This basis also gives rise to a system K h in the standard way. We use the following notations: x - y , z - u, and v i w to express that there are positive constants C and c such that Here and elsewhere c and C are moderate constants independent of H or N . Let - h be the distance between the first two GLL points in the interval [\Gamma1; +1]; - h is proportional to 1=N 2 [2], and the sides h of an element K belonging to T - h satisfy depending on the location of K inside -\Omega . The triangulation is therefore non-regular. In the case of a region of diameter H; such as a we use a norm with weights generated by dilation starting from a region of unit diameter, 3. General Setup and Simplifications. In this section, we give our plan of study. uN be a function belonging to P N ( - \Omega\Gamma3 and let - I h uN be the function of -\Omega ) for which uN for every GLL point xG in - and where a - Q is given by (2) and (3) with see [5] and [2]. We remark that the key point of these results is the stability of the interpolation operator at the GLL nodes for functions of proved by Bernardi and Maday [1], [2]. Consider now a function v defined in a substructure -\Omega with diameter of order H . Changing variables to the reference substructure by - using simple estimates on the Jacobian of F i , we obtain 2(\Omega where the dimension d is equal to 1, 2, or 3. These estimates can be interpreted as spectral equivalences of the stiffness and mass matrices generated by the norms introduced above. Indeed, the nodal basis f - is mapped by interpolation at the GLL nodes to a nodal basis of P N ( can be written as u is the vector of nodal values of both - uN or - u h , and - K h and - KN are the stiffness matrices corresponding to j and a - Therefore, if K (i) h and K (i) N are the stiffness matrices generated by the basis fOE h and fOE N respectively, for all nodes j in the closure and a (\Delta; \Delta), then where u is the vector of nodal values, by (9), (8), and (5). The stiffness matrices KN and K h are formed by subassembly [9], h for any nodal vector u, where u (i) are the sub-vectors of nodal values an analogous expression is true for KN . These last two relations imply for any vector u. All these matrix equivalences and their analogues in terms of norms are hereafter called FEM-SEM equivalence. The equivalence (11) shows that K h is an optimal preconditioner for KN in terms of number of iterations. However, the solution of systems K h expensive to be used as an efficient preconditioner for large scale problems, which typically involve many substructures. We next show that the same reasoning applies to the Schur complements S h and SN , i.e., the matrices obtained by eliminating the interior nodes of in a classical way; see [9]. Let uN be Q-discrete (piecewise) harmonic if aQ for all i and all v N belonging to P N The definition of h-discrete (piecewise) harmonic functions is analogous. It is easy to see that u T SN and uN are respectively Q and h-discrete harmonic and u is the vector of the nodal values on the interfaces of the substructures. The matrices S h and SN are spectrally equivalent. Indeed, by the subassembly equation (10), it is enough to verify the spectral equivalence for each substructure separately. For the where I h N is the interpolation at the nodes of T h , H h is the h-discrete harmonic extension of the interface values, and the subscript\Omega i indicates the restriction of the bilinear form to this substructure. Here, we have used FEM-SEM equivalence and the well known minimizing property of the discrete harmonic extension. The reverse inequality is obtained in an analogous way. This equivalence means that S h is also an optimal preconditioner for SN . As before, the action of the inverse of S h is too expensive to produce an efficient preconditioner for large problems. In his Master's thesis [17], Pahl proposed the use of easily invertible finite element preconditioners B h and S h;WB for K h and S h , respectively. If the condition number with a moderately increasing function C(N ), then a simple Rayleigh quotient argument shows that -(B \Gamma1 analogously for S h;WB and SN . Since the evaluation of the action of B h h;WB is much cheaper, these are very efficient preconditioners. Our goal is to establish (13) and its analogue for S h and S \Gamma1 h;WB . We note that the triangulation T h is non-regular, and that all the bounds of this form established in the literature require some kind of inverse condition, or regularity of the triangulation, which does not hold for the GLL mesh. 4. Technical Results. This section presents the technical lemmas needed to prove our results. As it is clear from the start, we draw heavily upon the results, techniques, and organization of Dryja, Smith, and Widlund [9]. 4.1. Some estimates for non-regular triangulations. We state here all the estimates necessary to extend the technical tools developed in [9] to the case of non-regular hexahedral triangulations. We let - 3 be the reference element, and K be its image under an affine mapping F . K ae -\Omega is an element of the triangulation sides proportional to h 1 , h 2 and h 3 . The function u is a piecewise trilinear defined in K. Notice that in this subsection we use hats to represent functions and points of - K. The first result concerns the expressions of the L 2 and H 1 norms in terms of the nodal values. Let - e i be one of the coordinate directions of - K, and let - a, - b, - c and - d be the nodes in one of the faces that is perpendicular to - a etc. be the corresponding points on the parallel face. The notation x i denotes a generic node of K, and a; b; are the images of -a and - b, etc. Lemma 1. x=a;b;c;d Proof. These expressions follow by changing variables, and by using the equivalence of norms in the finite dimensional space Q K). In the next lemma we give a bound on the gradient of a trilinear function in terms of bounds on the difference of the values at the nodes (vertices). Lemma 2. Let u be trilinear in the element K such that ju(a)\Gammau(b)j - Cdist(a; b)=r for some constants C and r, and for any two vertices a and b of the element K. Then r Proof. The functions u and u x can be written as fy The values of u x at the vertices belonging to the face are clearly bounded by C=r. This implies estimates for the coefficients of u x and then the desired estimate. The other derivatives of u are treated analogously. Lemma 3. Let u be a trilinear function defined in K, and let # be a C 1 function such that jr#j - C=r, and j#j - C for some constants C and r. Then Here C is independent of all the parameters, and I - h is the interpolation to a Q 1 function of the values in the vertices of K. Proof. By equation (15), and letting h 1 , h 2 , and h 3 be the sides of the element K: x=a;b;c;d Each term in the sum above can be bounded by The bound on r# implies that I - h (#u)jj 2 x=a;b;c;d x=a;b;c;d since # is bounded. 4.2. Technical tools. We introduce notations related to certain geometrical ob- jects, since the iterative substructuring algorithms are based on subspaces directly related to the interiors of the substructures, the faces, edges and vertices. be the union of two which share a common face, wirebasket of the which is the union of all the edges and vertices of this subdomain. We note that a face in the interior of the region\Omega is common to exactly two substructures, an interior edge is shared by more than two, and an interior vertex is common to still more substructures. All the substructures, faces, and edges are regarded as open sets. The following simple standard reductions greatly simplify our analysis in the next sections. The preconditioner S h;WB that we use is defined by subassembly of the matrices h;WB , see Section 5. Therefore we can restrict our analysis to one substructure. The results for the whole region follow by a standard Rayleigh quotient argument. It is also enough to estimate the preconditioning of - S h by - these results can be translated into results for each substructure by the equivalences (7), (8), and (5). The assumption that the fF i g M are arbitrary smooth mappings improves the flexibility of the triangulation, but does not make the situation essentially different from the case of affine mappings. This is seen from the estimates in Section 3, where we only used properties of the derivative of F i . Therefore, without loss of generality, we assume, from now on, that the F i are affine mappings. In some of the following results, we state the result for substructures of diameter proportional to H , but prove the theorem only for a reference substructure. The introduction of the scaling factors into the final formulas by the methods and results of Section 3 are routine. For a proof of Lemma 4 and a general discussion, see Bramble and Xu [4]. Lemma 4. Let Q H u h be the L 2 projection of the finite element function u h onto the coarse space and We remark that these bounds are not necessarily independent of the values K i of the coefficient. To guarantee that, one has to work with weighted norms, and insist that the coefficients k i satisfy the quasi-monotone condition [8], [25]. Lemma 5. Let - u h be the average value of u h on W j ; the wirebasket of the and Similar bounds also hold for an individual substructure edge. Proof. In the reference substructure, we know that P - h ae V - h , where V - h is a standard space defined on a shape regular triangulation that includes . This can be done by refining appropriately all the elements of T - h with sides bigger than, say, 3 - h=2. Now we apply the well-known result for shape regular triangulations, lemma 4.3 in [9], to get both estimates, recalling that in the reference substructure - h i 1=N 2 . In the abstract Schwarz convergence theory, the crucial point in the estimate of the rate of convergence of the algorithm is to demonstrate that all functions in the finite element space can be decomposed into components belonging to the subspaces such that the sum of the resulting energies are uniformly, or almost uniformly, bounded with respect to the parameters H and N . The main technique for deriving such a decomposition is the use of a suitable partition of unity. In the next two lemmas, we explicitly construct such a partition. Lemma 6. Let F k be the common face k be the function in P h (\Omega\Gamma that is equal to one at the interior nodes of F k , zero on the remainder of 1(\Omega The same bound also holds for the other Proof. We define the functions - and - in the reference cube; ' F k and #F k are obtained, as usual, by mapping, see subsection 3. We construct a function - having Fig. 1. One of the segments CCk the same boundary values as - ' F k , and then prove the bound for the former. The standard energy minimizing property of discrete harmonic extensions then implies the bound for - . The six functions which correspond to the six faces of the cube also form a partition of unity at all nodes at the closure of the substructure except those on the wirebasket; this property is used in the next lemma. We divide the substructure into twenty four tetrahedra by connecting its center C to all the vertices and to all the six centers C k of the faces, and by drawing the diagonals of the faces of Fig 1. The function - associated to the face F k is defined as being 1=6 at the point C. The values at the centers of the faces are defined by - is the Kronecker symbol. - is defined to be linear at the segments CC j for 6. The values inside each subtetrahedron formed by a segment CC j and one edge of the cube are defined to be constant on the intersection of any plane through that edge, and is given by the value, already known, at the segment CC j . The values at the edge of the cube belonging to this subtetrahedron are then modified to be equal to zero. Next, the whole function - is modified to be a piecewise Q 1 function by interpolating at the vertices of all the GLL nodes of the reference cube. We claim that jr - x is a point belonging to any element K that does not touch any edge of the cube, and r is the distance between the center of K and the closest edge of the cube. Let ab be a side of K. We analyze in detail the situation depicted in Fig 2, where ab is parallel to CC k . Let e be the intersection of the plane containing these two segments with the edge of the cube that is closest to ab. (a)j - D, by construction of - , where D is the size of the radial projection of ab on CC k . By similarity of triangles, we may write: where r 0 is the distance between e and the midpoint of ab. Here we have used that the distance between e and CC k is of order 1. If the segment ab is not parallel to CC k , the difference j - (a)j is even smaller, and (18) is still valid. Notice that r 0 is within a multiple of 2 of r. Therefore Lemma 2 implies that jr - In order to estimate the energy of - , we start with the elements K that touch one of the edges of the face F k . Let h 3 be the largest side of one of these elements. Since the nodal values of - at K are 0, 1, and 1=6, Fig. 2. Geometry underlying equation (18) a r by a simple use of equation (15). By summing over K, we conclude that the energy of - is bounded independently of N for the union of all elements that touch one of the edges of the face F k . To estimate the contribution to the energy from the rest of the substructure, we consider one subtetrahedron at a time and introduce cylindrical coordinates using the substructure edge, that belongs to the subtetrahedron, as the z-axis. The bound now follows from the bound on the gradient given above and elementary considerations. We refer to [9] for more details. The following lemma corresponds to Lemma 4.5 in [9]. This lemma and the previous one are the keys to avoiding H 1=2estimates and extension theorems. Lemma 7. Let #F k (x) be the function introduced in the proof of Lemma 6, let F k be a face of the I h denote the interpolation operator associated with the finite element space P h and the image of the GLL points under the mapping I h (# F k and Proof. The first part is trivial from the construction of - made in the previous lemma. For the second part, we first estimate the sum of the energy of all the elements K that touch the wirebasket. The nodal values of the interpolator I - h ( - in such an element are 0,0,0,0, - (c)-u(c) and - lies between 0 and 1. Moreover, we denote by h 3 the side of K that is larger than the other two sides Note that this larger side is parallel to the closest wirebasket edge. using equation (15), we obtain: Then, by using the expression of the L 2 norm in the two segments that are parallel to the edge, and lemma 5, we have: where the sum is taken over all elements K that touch the boundary of the face F k . We next bound the energy of the interpolant for the other elements. Since r - C=r where r is the distance between the element K and the nearest edge of - (see the proof of the previous lemma), Lemma 3 implies that Kae Kae where the sum is taken over all elements K that do not touch the edges of -\Omega . The bound of the first term in the sum is trivial, and to bound the second term we partition the elements of -\Omega into groups, in accordance to the closest edge of the exact rule for the assignment of the elements that are halfway between is of no importance. For each edge of the wirebasket, we use a local cylindrical coordinate system with the z axis coinciding with the edge, and the radial direction, normal to the edge. In cylindrical coordinates, we estimate the sum by an integral Kae -\Omega r \Gamma2 jj-ujj 2 Z C Z Z z drd'dz: The integral with respect to z can be bounded by using Lemma 5. We obtain Kae -\Omega r \Gamma2 jj-ujj 2 Z C and thus Kae This proof is an adaptation of an argument given in [9] for shape regular meshes. Note that equation (16) replaces the use of the inverse inequality, which cannot be used here because of the bad aspect ratios of the elements. Equation (16) is analogous to the L 2 bound of the derivative of a product in terms of L 2 norms of the functions and L 1 norms of the gradients, which cannot be applied directly to our case because we have the interpolation operator I h . Lemma 8. Let - W k be the averages of u h on @F k ; and W k , respectively. Then, The proofs are direct consequences of the Cauchy-Schwarz inequality. Lemma 9. Let u h be zero on the mesh points of the faces of\Omega j and discrete harmonic This result follows by estimating the energy norm of the zero extension of the boundary values by means of equation (15) and by noting that the harmonic extension has a smaller energy. 5. Iterative Substructuring Algorithms. The first algorithm we analyze is a wirebasket based method, based on Algorithm 6.4 in [9]. This is a block-diagonal preconditioner after transforming the original matrix to a convenient basis. According to the abstract framework of Schwarz methods [9], we only need to prescribe spaces whose union is the whole space, and the corresponding bilinear forms. Each internal face F k generates a local space VF k of all the h-discrete harmonic functions that are zero at all the interface nodes that do not belong to this face. Notice that the functions belonging to VF k have support in the union of the two substructures and\Omega j that share the face F k . The bilinear form used for this space is just a(\Delta; \Delta). We also define a wirebasket subspace that is the range of the following interpolation operator: I h Here, ' k is the discrete harmonic extension of the standard nodal basis functions OE k , W h is the set of nodes in the union of all the wirebaskets, and - u h @F k is the average of u h on @F k . The bilinear form for this coarse subspace is given by These subspaces and bilinear forms define, via the Schwarz framework, a preconditioner of S h that we call S h;WB . Theorem 1. For the preconditioner S h;WB , we have where the constant C is independent of the N , H, and the values k i of the coefficient. Proof. We apply word by word the proof of theorem 6.4 in [9] to the matrix S h , using now the tools developed in Section 4. This gives The harmonic FEM-SEM equivalence (12) and a Rayleigh quotient argument complete the proof, as explained in Section 3. We do not give the complete proof here because it would be a mere restatement of the proof in [9]. The next algorithm is obtained from the previous one by the discrete harmonic FEM-SEM equivalence, by which we find a preconditioner SN;WB from the preconditioner studied above. Each face subspace related to a face F k is composed of the set of all Q-discrete harmonic functions that are zero at all the interface nodes that do not belong to the interior of the face F k . The wirebasket subspaces are defined as before, by prescribing the values at the GLL nodes on a face to be equal to the average of the function on the boundary of the face. The bilinear forms used for the face and wirebasket subspaces are aQ and b 0 (\Delta; \Delta), respectively. Notice that this is the wirebasket method based on GLL quadrature given in [24]. The following lemma shows the equivalence of the two functions uN and u h with respect to the bilinear form b 0 (\Delta; \Delta). Lemma 10. Let u h be a Q 1 finite element function on the GLL mesh of the interval I = [\Gamma1; +1], and let uN be its polynomial interpolant. Then Proof. We prove only the - part. The inequality without the infimum is valid for the constant c r that realizes the inf in the right hand side by the FEM-SEM equivalence. By taking the inf in the left hand side we preserve the inequality. Theorem 2. For the preconditioner SN;WB , we have where the constant is independent of the parameters H, N and the the values k i of the coefficient. Proof. In this proof, the functions with indices h and N are all discrete harmonic functions with respect to the appropriate norms, related in the same way as uN and u h , i.e. According to Section 3, it is enough to analyze one substructure \Omega i at a time, and prove the following equivalence: 1(\Omega We prove only the - part, and the other inequality is analogous. Lemma 10 gives an upper bound of the first term in the left hand side by the corresponding term in the right hand side. Each term in the sum on the left hand side can be bounded by The first term of this expression can be bounded by the corresponding term on the right hand side by interpolation and the harmonic FEM-SEM equivalence. The second term is bounded by where c h;W i is the average of u h over W i . Here we used the estimate on the energy norm of ' h;F k which implies a similar estimate of ' N;F k . Applying the Cauchy-Schwarz inequality, as in lemma 8, and the FEM-SEM equivalence, we can bound this last expression in terms of the first term in the right hand side of equation (19). The polynomial analogues of the lemmas in Section 4 can be proved using the harmonic FEM-SEM equivalence. This provides a theory for polynomials, which is completely parallel to the one we have presented, that can be used to prove this theorem directly. A variation of this approach is taken in [22] and [24], but without the use of the FEM-SEM equivalence. 6. Overlapping Schwarz Algorithms. We now consider the additive overlapping Schwarz methods, which are presented for instance in [10]. We recall that an abstract framework, theorem 3.1 in [10], is available for the analysis of this type of algorithm. Here we only discuss the additive version, but the analysis also applies in a standard way to the multiplicative variant, which is more effective in many practical problems. In the abstract framework for the additive Schwarz methods, a preconditioner B h for K h can be defined by specifying a set of local spaces together with a coarse space. We can also provide approximate solvers for the elliptic problem restricted to each of the proposed subspaces. Here we only work with exact solvers, since the extension to inexact solvers is straightforward by using the abstract framework. The domain\Omega is covered by substructures\Omega i , which are the original spectral elements. We enlarge each of them to produce overlapping i , in such a way that the boundary i does not cut through any element of the triangulation generated by the GLL nodes. The overlap ffi is defined as the minimum distance between the boundaries i . When ffi is proportional to H the overlap is called generous, and when ffi is comparable to the size of the Q 1 elements it is called a small overlap. For the sake of simplicity, we again restrict our analysis to the case when all the mappings F j are affine mappings. The general situation is treated similarly. The local spaces are given by P h i ), the set of functions in P h that vanish at all the nodes on or outside . The coarse space is a Q 1 finite element space given by the mesh generated by the vertices and edges of the subregions\Omega i . Each subregion \Omega i is then one element of this coarse finite element space. We note that this coarse mesh is regular by assumption. This construction is completely parallel to that of Section 2.1 of [11] for this particular choice of subregions. This setting incorporates the small and the generous overlap preconditioners. We use the bilinear form a(\Delta; \Delta) for the coarse and local spaces. Theorem 3. For this additive Schwarz algorithm, the condition number of the preconditioned operator satisfies: The constant C is independent of the parameters H, N , and ffi . Proof. As before, we follow the proof of the analogous theorem, theorem 3 in [11]. The proof follows word by word, except for the estimate of aK where I h is the interpolation operator, f' i g is a partition of unity, w h is a finite element function, and aK is just the restriction of a(\Delta; \Delta) to one element. In this case it is known that j' i and the rest of the proof follows without any change. Remark 1. Even though the theory does not rule out the possibility of growth of the constant when the coefficient k has large jumps, such a growth is very moderate in numerical experiments; see e.g [13]. We note also that when the overlap is generous, the method is optimal in the sense that the condition number is uniformly bounded with respect to the parameters of the problem; see [19] for early work on this type of preconditioner. Our results and techniques allow a very flexible choice of subregions. We now apply FEM-SEM equivalence to the subspaces used to define B h;AS ; this is the same technique used to derive the preconditioner SN;WB from S h;WB . The coarse space is the same, and the local spaces are defined by N (v N where I h N (v N ) interpolates v N at the GLL points and belongs to P h . These subspaces and the use of the bilinear forms aQ (\Delta; \Delta) and a(\Delta; \Delta) for the local and coarse spaces, respectively, define our preconditioner BN;AS . Theorem 3 and a simple application of the FEM-SEM equivalence for each one of the local spaces immediately give: Theorem 4. 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Widlund A polylogarithmic bound for an iterative substructuring method for spectral elements in three dimensions. Preconditioned conjugate gradient methods for spectral elements in three dimensions. Finite element preconditioning for legendre spectral collocation approximations to elliptic equations and systems. editors. Domain Decomposition Methods in Science and Engineering: The Sixth International Conference on Domain Decomposition A domain decomposition algorithm for elliptic problems in three dimensions. --TR --CTR James W. Lottes , Paul F. Fischer, Hybrid Multigrid/Schwarz Algorithms for the Spectral Element Method, Journal of Scientific Computing, v.24 n.1, p.613-646, July 2005 Luca F. Pavarino , Elena Zampieri, Overlapping Schwarz and Spectral Element Methods for Linear Elasticity and Elastic Waves, Journal of Scientific Computing, v.27 n.1-3, p.51-73, June 2006 Dan Stefanica, FETI and FETI-DP Methods for Spectral and Mortar Spectral Elements: A Performance Comparison, Journal of Scientific Computing, v.17 n.1-4, p.629-638, December 2002 V. Korneev , J. E. Flaherty , J. T. Oden , J. Fish, Additive Schwarz algorithms for solving hp-version finite element systems on triangular meshes, Applied Numerical Mathematics, v.43 n.4, p.399-421, December 2002 Marcello Manna , Andrea Vacca , Michel O. Deville, Preconditioned spectral multi-domain discretization of the incompressible Navier-Stokes equations, Journal of Computational Physics, v.201 n.1, p.204-223, 20 November 2004

schwarz methods;preconditioned conjugate gradients;iterative substructuring;domain decomposition;spectral element method

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Theory of neuromata.

A finite automatonthe so-called neuromaton, realized by a finite discrete recurrent neural network, working in parallel computation mode, is considered. Both the size of neuromata (i.e., the number of neurons) and their descriptional complexity (i.e., the number of bits in the neuromaton representation) are studied. It is proved that a constraint time delay of the neuromaton output does not play a role within a polynomial descriptional complexity. It is shown that any regular language given by a regular expression of length n is recognized by a neuromaton with &THgr;(n) neurons. Further, it is proved that this network size is, in the worst case, optimal. On the other hand, generally there is not an equivalent polynomial length regular expression for a given neuromaton. Then, two specialized constructions of neural acceptors of the optimal descriptional complexity &THgr;(

Introduction Neural networks [7] are models of computation motivated by our ideas about brain functioning. Both their computational power and their efficiency have been traditionally investigated [4, 14, 15, 19, 21] within the framework of computer science. One less commonly studied task which we will be addressing is the comparison of the computational power of neural networks with the traditional finite models of computation, such as recognizers of regular languages. It appears that a finite This research was supported by GA - CR Grant No. 201/95/0976. discrete recurrent neural network can be used for language recognition in parallel mode: at each time step one bit of an input string is presented to the network via an input neuron and an output neuron signals, possibly with a constant time delay, whether the input string which has been read, so far, belongs to the relevant language. In this way, a language can be recognized by a neural acceptor (briefly, by a neuromaton). It is clear that the neuromata recognize just the regular languages [12]. A similar definition of a neural acceptor appeared in [2, 8], where the problem of language recognition by neural networks has been explored in the context of finite automata. It was shown in [2] that every m-state deterministic finite automaton can be realized as a discrete neural net with O(m 3 neurons and that at least neurons are necessary for such construction. This upper and lower bound was improved in [6, 8] by showing that \Theta(m 1 neurons suffice and that, in the worst case, this network size cannot be decreased assuming either at most O(logm)-time simulation delay [6] or polynomial weights [8]. Moreover, several experiments to train the (second-order) recurrent neural networks from examples to behave like deterministic finite automata either for a practical exploitation or for a further rule extraction have also been done [5, 13, 20, 22] using the standard neural learning heuristics back-propagation. In the present paper we relate the size of neural acceptors to the length of regular expressions that on one hand are known to possess the same expressive power as finite automata, but on the other hand they represent a tool whose descriptional efficiency can exceed that of deterministic finite automata. First, in section 2 and 3, respectively, we will introduce the basic formalism for dealing with regular languages and neuromata. We will prove that a constant time delay of the neuromaton output does not play a role within a linear neuromata size. Therefore, we will restrict ourselves to neuromata which respond only one computational time step after the input string is read. Then, in section 4 we will prove that any regular language described by a regular expression of a length n can be recognized by a neuromaton consisting of O(n) neurons. Subsequently, in section 5 we will show that, in general, this result cannot be improved because there is a regular language given by a regular expression of length n requiring neural acceptors of n). Therefore, the respective neuromaton construction from a regular expression is size-optimal. This can be used, for example, for constructive neural learning when a learning algorithm for regular expressions from example strings [3] is first employed. On the other hand, in section 6 this construction is proven not to be efficiently reversible because there exists a neuromaton for which every equivalent regular expression is of an exponential length. Next, in section 7 we will present two specialized constructions of neural acceptors for single n-bit string recognition that both require O(n 1 neurons and either O(n) connections with constant weights or O(n 1 weights of the size O(2 The number of bits required for the entire string acceptor description in both cases is proportional to the length of the string. This means that these automata constructions are optimal from the descriptional complexity point of view. They can be exploited as a part of a more complex neural network design, for exam- ple, for the construction of a cyclic neural network, with O(2 n neurons and edges, which computes any boolean function [9]. In section 8 we will introduce the concept of Hopfield languages as the languages that are recognized by the so-called Hopfield acceptors (Hopfield neuromata) which are based on symmetric neural networks (Hopfield networks). Hopfield networks have been studied widely outside of the framework of formal languages, because of their convergence properties. From the formal language theoretical point of view we will prove an interesting fact, namely that the class of Hopfield languages is strictly contained in the class of regular languages. Hence, they represent a natural proper subclass of regular languages. Furthermore, we will formulate the necessary and sufficient, so-called, Hopfield condition stating when a regular language is a Hopfield language. In section 9 we will show a construction of a Hopfield neuromaton with O(n) neurons for a regular language satisfying the Hopfield condition. Thus, we will obtain a complete characterization of the class of Hopfield languages. As far as, the closure properties of Hopfield languages are concerned, we will show that the class of Hopfield languages is closed under union, intersection, concatenation and complement and that it is not closed under iteration. Finally, in section 10 we investigate the complexity of the emptiness problem for regular languages given by neuromata or by Hopfield acceptors. We will prove that both problems are PSPACE-complete. This is a somewhat surprising because the identical problems for regular expressions, deterministic and non-deterministic finite automata are known to only be NL-complete [10, 11]. It confirms the fact from section 6 that neuromata can be stronger than regular expressions from the descriptional complexity point of view. As a next consequence we will obtain that the equivalence problem for neuromata is PSPACE-complete as well. All previous results jointly point to the fact that neuromata present quite an efficient tool not only for the recognition of regular languages and of their subclasses respectively, but also for their description. In addition, the above-mentioned constructions can be generalized for the analog neural networks [18]. A preliminary version of this paper concerning general neuromata and Hopfield languages, respectively, appeared in [16] and [17]. Regular Languages We recall some basic notions from language theory [1]. We introduce the definition of regular expressions which determine regular languages. The concept of a (deterministic) finite automaton is defined and Kleene's theorem about the correspondence between finite automata and regular languages is mentioned as well. An alphabet is a finite set of symbols. A string over an alphabet \Sigma is a finite-length sequence of symbols from \Sigma. The empty string, denoted by e, is the string with no symbols. If x and y are strings, then the concatenation of x and y is the string xy. The string xx ntimes is abbreviated to x n . The length of a string x, denoted by jxj, is the total number of symbols in x. language over an alphabet \Sigma is a set of strings over \Sigma. Let L 1 and L 2 be two languages. The language L 1 \Delta L 2 , called the concatenation of L 1 and L 2 , is g. Let L be a language. Then define for n - 1. The iteration of L, denoted L ? , is the language L n=0 L n . Similarly the positive iteration Definition 3 The set RE of regular expressions over an alphabet defined as the minimal language over an alphabet f0; the following conditions: 1. ;; 2. if ff; fi 2 RE then also (ff In writing a regular expression we can omit many parentheses if we assume that ? has higher precedence than concatenation and the latter has higher precedence than +. For example, ((0(1 ? may be written abbreviate the ntimes to ff n but jff n j remains n \Delta jffj. Definition 4 The set is the set of regular languages [ff] which are denoted by regular expressions ff as follows: 1. 2. if ff; fi 2 RE then [ff We also use a regular expression ff corresponding to the positive iteration [ff] Definition 5 A (deterministic) finite automaton is a 5-tuple where is a finite set of automaton states, \Sigma is an input alphabet (in our case \Gamma! Q is the transition function, q 0 2 Q is the initial state of the automaton, and F ' Q is a set of accepting states. Definition 6 The generalized transition function of the automaton is defined in the following way: 1. 2. Fg is the language recognized by the finite automaton A. Theorem 1 (Kleene). A language L is regular if f it is recognized by some finite automaton A (i.e., Neuromata In this section we formalize the concept of a neural acceptor - the so-called neu- romaton which is a discrete recurrent neural network (or neural network, for short) exploited for a language recognition in the following way: During the network com- putation, an input string is presented bit after bit to the network by means of a single predetermined input neuron. All neurons of the network work in paral- lel. Following this, with a possible constant time delay the output neuron shows whether the input string, that has been already read, is from the relevant language. A similar definition appeared in [2] and [8]. Definition 7 A neural acceptor (briefly, a neuromaton) is a 7-tuple out; is the set of n neurons including the input neuron inp 2 , and the output neuron out 2 V , is the set of edges, is the set of integers) is the weight function (we use the abbreviation Z is the threshold function (the abbreviation is the initial state of the network. The graph (V; E) is called the architecture of the neural network N and is the size of the neuromaton. The number of bits that are needed for the whole neuromaton representation (especially for the weight and threshold functions) is called the descriptional complexity of neuromaton. formally, due to the notational consistency, by arbitrary xm+l 2 f0; 1g; l - 1, be the input for the neuroma- ton Further, assume that all oriented paths from inp to out in the architecture (V; E) have length at least k 1. The state of the neural network at the discrete time t is a mapping s 1g. At the beginning of a neural network computation the state s 0 is set to s 0 . Then at each time step network computes its new state s t from the old state s t\Gamma1 as follows: otherwise. For the neural acceptor N and its input x 2 f0; 1g m we denote the state of the output neuron out 2 V in the time 1g is the language recognized by the neuromaton N with the time delay k. First, we show with respect to language recognition capabilities that a constant time delay of the neuromaton output does not play a role, within a linear neuromata size. More precisely, all languages recognized with the time delay k by a neuromaton of size n can be recognized with the time delay 1 by a neuromaton of size O(2 k n). be a neuromaton of size n such that all oriented paths from inp to out in the architecture (V; E) have length at least 1. Then there exists a neuromaton N init ) of the size 2 k (n Proof: The idea of the proof is to construct N ? in such a way that it checks ahead all possible computations of N for the next steps and cancels all wrong computations that do not match the actual input being read with the time delay For this purpose besides the input and output neurons inp ? , out ? , the neuroma- ton N ? consists of 2 k blocks N x which foresee the computation of N , each of them for one of the possible next k bits x of input. Denote neurons in these blocks in the same way as in N but each of them indexed by the relevant k bits. Then the state of the neuron v x of N ? is equal to the state of the neuron v 2 V of N after the computation over the next bits (i.e., over the first k \Gamma 1 bits of x 2 f0; 1g k ) is performed. This is achieved as follows. The block N yb where y 2 f0; 1g its new state from the old state of the block N ay where a 2 f0; 1g. Therefore neurons in N ay are connected to neurons of N yb and the corresponding edges are labeled with relevant weights with respect to the original weight function w of N . The block N yb has the constant input b which is taken into account by modifying all thresholds of neurons in this block, especially when The input bit c 2 f0; 1g of N ? indicates that the computations of N ay , where a 6= c, are not valid. Therefore the input neuron inp ? cancels these computations by setting all neuron states in N ay to zero. Thus, all neurons in 2 k\Gamma1 blocks N ay states when the previous input bit does not match a. This also enables the block N yb to execute the correct computation because the influence of one invalid block from either N 0y or N 1y which are connected to N yb , is suppressed, due to zero states. The neurons in the blocks N x which are connected to out x lead to the output neuron out ? as well. They are labeled with the same weights. We know that half of them do not have any influence on out ? . Moreover, among the remaining ones, the corresponding neurons v x (v 2 V ) from all blocks N x have the same state because the distance between inp and out in N is at least k and the last k input bits cannot influence the output. It is sufficient to multiply the threshold of out ? by 2 k\Gamma1 in order to preserve the function computed by out. In this way, the correct recognition is accomplished with lookahead. A formal definition of the neuromaton N follows: ay ay init (v ya where s k\Gamma1 (v)(y) is the state of neuron v in the neuromaton N for the input y 2 1g k\Gamma1 at the time step k \Gamma 1. The term bw(hinp; vi) in the threshold definition takes into account the weight of the constant input in the block N y1 (y 2 f0; 1g corresponding to the original weight associated with the edge leading from the input inp to the relevant neuron v in the neuromaton N . The definition of w ensures setting the states of all neurons in the block N 0y (y 2 f0; 1g k\Gamma1 ) to zero iff the input inp ? is 1. Similarly the definition of weights w together with the term aw ? (hinp in the threshold definition cause zero states of all neurons in the block N 1y (y 2 Clearly, the size of the neuromaton N ? is 2. 2 Following Lemma 1, we can restrict ourselves to neuromata which respond only one computational time step after the input string is read because their size is, up to a constant multiplication factor, as large as the size of the equivalent neuromata with a constant time delay. Therefore, in the rest of this paper we will assume that the time delay in the neuromaton recognition is 1 and that any neuromaton architecture does not contain an edge from the input to the output neuron. We will also denote L 1 (N ) by L(N ) for any neuromaton N . Next, we will prove that the neural acceptor can be viewed as a finite automaton [12] and therefore, neuromata recognize exactly regular languages due to Theorem 1. Theorem 2 Let language recognized by some neuromaton N . Then L is regular. Proof: Let neural acceptor. We define a deterministic finite automaton \Sigmag and q . The transition function is defined for s 2 Q and x 2 \Sigma as follows: Finally, \Sigmag. Then the proposition follows from Theorem 1. 2 4 Upper Bound We show that any regular language, given by a regular expression of length n, may be recognized by a neuromaton of size O(n). The idea of recognition by neuromaton is to compare an input string with all possible strings generated by the regular expression and to report via the output neuron whether some of these strings match the input. Therefore, the constructed architecture of the neural network corresponds to the structure of the regular expression. The neuromaton proceeds through all oriented network paths that correspond to all strings generated by this expression and that, at the same time, match the part of the input string that has been read so far. Theorem 3 For every regular language L 2 RL denoted by a regular expression there exists a neuromaton N of the size O (jffj) such that L is recognized by N (i.e., Proof: Let be a regular language denoted by a regular expression ff. We construct a neuromaton N O (jffj) so that We first build an architecture (V; E) of the neural network N ff recursively with respect to the structure of the regular expression ff. For that purpose we define the sequence of graphs corresponding to the whole expression ff which is recursively partitioned into shorter regular subexpressions, so that (V corresponding only to the elementary subexpressions 0 or 1 of ff (we say, vertices of the type 0 or 1). For the sake of notational simplicity we identify the subexpressions of ff with the vertices of these graphs. 1. 2. Assume that have already been constructed and a subexpression of ff different from 0 or 1. Hence, besides the empty language and the empty string, the regular expression fi can denote union, concatena- tion, or iteration of subexpressions of fi. With respect to the relevant regular operation the vertex fi is fractioned and possibly new vertices corresponding to the subexpressions of fi arise in the graph (V To be really rigorous we should first remove the vertex fi and then add the new vertices. However, due to the notational simplicity we do not insist on such rigor and therefore, we can identify one of the new vertices with the old fi. That is why we write rather inexactly, for example, 'fi has the form fi fl'. Moreover, we ffl fi is ;: V g. ffl fi is e: V fiigg. ffl fi has the form fi g. ffl fi has the form fi g. ffl fi has the form fiig. This construction is finished after contains only subexpressions 0 or 1. Then we define the network architecture in the following way: For Now we can define the weight function w and the threshold function is the neuron of type 1: is the neuron of type 0: The initial state is defined as s 0 The set V contains three special neurons inp; out; start, as well as, other neurons of the type 0 or 1 - one for each subexpression 0 or 1 in ff; hence, jV An example of the neuromaton for the regular language [(1(0 figure 1 (the types of neurons are depicted inside the circles representing neurons; thresholds are depicted as weights of edges with constant inputs \Gamma1). inp start 101 out Figure 1: Neuromaton for [(1(0 We prove that From the construction of (V above, it is easy to observe that this graph corresponds to the structure of the regular expression ff. This means that for every string there is an oriented path start out leading from to out 2 V p and containing the vertices of the relevant types (0 or 1). On the other hand, for any such path there is a corresponding string in L. The neural acceptor N ff passes through all possible paths that match the network input. In the beginning the only non-input neuron start 2 V is active (its state is 1). It sends a signal to all connected neurons and subsequently becomes passive (its state is 0) due to the dominant threshold. The connected neurons of the type 0 or 1 compare their types with the network input and become active only when they match, otherwise they remain passive. Due to the weight and threshold values, it follows that any neuron of the type 1 becomes active iff inp 2 V is active and at least one of j 6= inp, hj; ii 2 E, is active, and any neuron of the type 0 becomes active iff inp 2 V is passive and at least one of j 6= inp, hj; ii 2 E, is active. This way all relevant paths are being traversed, and their traverse ends in the neuron which realizes the logical disjunction, and is active iff the prefix of the input string, that has been read so far, belongs to L. This completes the proof that 5 Lower Bound In this section we show the lower n) for the number of neurons that, in the worst case, are necessary for the recognition of regular languages which are described by regular expressions of the length n. As a consequence, it follows that the construction of the neuromaton from section 4 is size-optimal. The standard technique is employed for this purpose. For a given length of regular expression we define a regular language and the corresponding set with an exponential number of prefixes for this language. We prove that these prefixes must bring any neuromaton to an exponential number of different states in order to provide a correct recognition. This will imply the desired lower bound. Definition 9 For we denote by Ln , \Pi k , and Pn , respectively, the following regular languages: hi It is clear that Pn , n - 1 is the set of prefixes for the language Ln . We prove several lemmas concerning properties of these regular languages. The regular expression which defines the language Ln in Definition 9 is in fact of O length because the abbreviation for a repeated concatenation is not included when determining its length. Therefore, we first show that there is a regular expression ff n , of the linear length only, denoting the same language Ln . The number of prefixes in Pn is shown to be exponential with respect to n. Proof: (i) In the regular expression which denotes the language Ln from Definition 9, we can subsequently factor out (n \Gamma times the subexpression 1(e + 0) to obtain the desired regular expression of the linear length jff which defines the same language (ii) It follows from Definition 9 that j\Pi k The following lemma shows how the prefixes in Pn can be completed to strings from Ln . Lemma 3 Proof: (i), (ii) follow from Definition 9. (iii) Assume . The language Ln is defined via iteration in Definition 9. Henceforth, we can write \Pi Now we prove that any two different prefixes from Pn can be completed by the same suffix, so that one of the resulting strings is in Ln while the other one is not. Lemma Proof: Assume x . Then there exist We will distinguish two cases: Without loss of generality, suppose n ? From (ii) Lemma 3 we obtain x 1 due to (ii) Lemma 3. 2. We can write x fe; 0g, for g. Without loss of generality a ffl Denote z \Pi From (i) Lemma 3 z 1 0 2 Ln and z 2 1 2 Ln from (ii) Lemma 3. This implies that x 1 because Ln is closed under concatenation . To the contrary suppose that x 2 exist. Hence, z from (ii) Lemma 3 it follows z which is a contradiction. Thus, x 2 y 62 Ln . 2 Now we are ready to prove the following theorem concerning the lower bound. Theorem 4 Any neuromaton N that recognizes the language at n) neurons. Proof: The neuromaton N , that recognizes the language must different states which are reached when taking input prefixes from Pn because any different x can be completed by y from Lemma 4, so that x 1 y 2 Ln and x 2 y 62 Ln . This implies that N needs\Omega\Gamma n) binary neurons. 2 6 Neuromata Are Stronger Although from the results of sections 4, 5 it seems that from the descriptional complexity point of view, neuromata and regular expressions are polynomially equiva- lent, in this section, we will show that there exists a neuromaton for which every equivalent regular expression is of an exponential length. This means that the neu- romata construction from regular expressions that has been described in section 4 is not efficiently reversible. Theorem 5 For every n - 1 there exists a regular language Ln recognized by a neuromaton Nn of size O(n) and of descriptional complexity O(n 2 ) such that any regular expression ff n which defines We define the finite language to be the set of all binary strings of length m. The neuromaton recognizes Ln is a binary n-bit counter which accepts the input string iff its length is m. The neuron corresponding to the ith bit of the counter should change its state iff 1. However, the corresponding Boolean function cannot be computed by only one neuron and, therefore, a small subnetwork of three neurons a introduced for this purpose. Then v i is a disjunction of a i and b i where a and at least one of v its state as required but it takes two computational steps. This means that the counter is two times slower and only are required to count till m. Moreover the neuron v 0 generates the binary sequence (0011) ? and a special neuron rst is introduced to suppress the firing of the output neuron v n\Gamma2 after m bits are read. A formal definition of the counter follows: Clearly, the size of Nn is of order O(n) and its descriptional complexity is O(n 2 ). Let ff n be a minimal length regular expression which defines the language We prove that jff n m). The expression ff does not contain iterations because it defines the finite language (the part containing iterations would denote the empty language and, thus, could be omitted). To generate strings from the expression ff n is to be read from the left to the right side without any returning since no iteration is allowed. But any string of Ln is of length m and that is why ff n must contain at least m symbols from f0; 1g. Hence jff Theorem 5 shows that, in some sense, there is a big gap between the descriptive power of regular expressions and that of neuromata. We will discuss this issue later in sections 10, 11. 7 Neural String Acceptors The previous results showed that neuromata have the descriptive capabilities of regular expressions. In this section we will study how powerful they are when we confine ourselves to a certain subclass of regular expressions. Here, we will deal with the simplest case only considering fixed binary strings from f0; 1g ? . We present two constructions of neural acceptors for a single string recognition. For n-bit strings they both require O(n 1 neurons and either O(n) connections with constant weights or O(n 1 weights of the O(2 The number of bits required for the entire string acceptor description is in both cases proportional to the length of the string. This means that these constructions are optimal from their descriptional complexity point of view. Studying this elementary case is useful because single string recognition is very often a part of more complicated tasks. The techniques developed for the architectural neural network design can, for example, sometimes improve the construction of neuromata from section 4, when a regular expression consists of long binary sub- strings. The other example of application is a constructive neural learning, where strings are viewed as training patterns, and the resulting network is composed of neural acceptors for these strings that work in parallel. Theorem 6 For any string a = a an 2 f0; 1g n there exists a neural acceptor of the size O(n 1 neurons with O(n) connections of constant weights. Thus, the descriptional complexity of the neuromaton is \Theta(n). Proof: For the sake of simplicity suppose first that positive integer p. The idea of the neural acceptor construction is to split the string a 2 f0; 1g n into p pieces of the length p and to encode these p substrings using p 2 (for each p) binary weights of edges leading to p comparator neurons c . The input string being gradually stored per p bits into p buffer neurons . When the buffer is full, the relevant comparator neuron c i compares the assigned part of the string a with the corresponding part of the input string x that is stored in the buffer and sends the result of this comparison to the next comparator neuron. The synchronization of the comparison is performed by 2p clock neurons, where neurons s tick at each time step of the network computation, and neurons tick once in a period of p such steps. The last comparator neuron c p represents the output neuron and reports at the end whether the input string x matches a. A formal definition of the neuromaton for the recognition of the string a follows (see also figure 2). Define Denote by d fi fi . Then put ae All remaining weights and thresholds are set to 1. Finally, the initial state is 'i 'i 'i 'i 'i 'i 'i 'i 'i 'i 'i 'i x 'i 'i Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ l l l l l l l l l l l l l l l l l l l l A A A A A A A A A A A A A A A A A A Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z ae ae ae ae ae ae ae ae ae ae ae ae ae ae ae ae ae ae ae ae ae ae ae ae \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega inp out sp start Figure 2: Architecture for a neural string acceptor with O(n) edges. Note that the weights w are not constant as required. It is due to the neuron c i which should keep its state 1 after it becomes active in order to transfer the possible positive result of the comparison to the next comparator neuron. Therefore, the relevant feedback must exceed the sum of all other inputs to achieve the threshold value. This can be avoided by inserting auxiliary neurons, that remember the result of preceding comparisons, between neighbor comparator neurons. All of these neurons have a constant feedback because they only have one input, from the previous comparator neuron, to be exceeded. The construction of the neural acceptor can also be easily adapted for p. The technique from the proof of theorem 3 is employed for the recognition of the last r bits of the string a. The resulting architecture of size O(r) is then connected to the neural string acceptor by identifying the neuron start with the above-mentioned neuron 'i 'i 'i 'i 'i 'i 'i 'i 'i x 'i 'i 'i 'i 'i 'i 'i 'i 'i 'i 'i 'i 'i Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z l l l l l l l l l l l l l l l l l l l l A A A A A A A A A A A A A A A A A A Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z ae ae ae ae ae ae ae ae ae ae ae ae ae ae ae ae ae ae ae ae ae ae ae ae \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega l l l l Z Z A A A A A A A A A A A A A A A A A A A A A A A A A A start out an a q+3 a q+2 a q+1 inp reset sp Figure 3: Architecture for a neural string acceptor with O(n 1 Theorem 7 For any string a 2 f0; 1g n there exists a neural acceptor of the size neurons with O(n 1 weights of the size O(2 Thus, the descriptional complexity of the neuromaton is \Theta(n). Proof: The design of the desired neural acceptor is very similar to the construction from the proof of Theorem 6, except that the number of comparator neurons is reduced to two neurons c - ; c - . In this case the substrings a are encoded by O(p) (each by 2) weights of the size O(2 p ) corresponding to the connection leading from the clock neurons m i to these comparison neurons. The contents of the input buffer, viewed as a binary number, are first converted into an integer. This integer is then compared with the relevant encoded part of the string a by the comparator neuron c - to see whether it is smaller or equal. At the same time, it is compared by the comparator neuron c - to see whether it is greater or equal. The neuron which realizes the logical conjunction of comparator outputs is added to indicate whether the part of the input string matches the corresponding part of the string a. However, this leads to one more computational step of the neural acceptor. The correct synchronization can be achieved by exploiting the above-mentioned additional architecture for the recognition of the last r bits a q+1 ; a of the string a. Details of the synchronization are omitted, as well as, a complete formal definition of the neural string acceptor. We only give the definition of the weights that are relevant for the comparisons: 2 p\Gammaj a (i\Gamma2)p+j for The weights w are defined as differences because all clock neurons are active just when the part of the input is in the buffer. The architecture of the neural string acceptor is depicted in figure 3 (instead of the above-mentioned conjunction, the neuron reset is added to realize the negation of comparator conjunction to possibly terminate the clock). 2 Piotr Indyk [9] pointed out that the latter string acceptor construction from Theorem 7 can also be exploited for building a cyclic neural network, with O(2 n neurons and edges, which computes any boolean function. In this case the binary vector of all function values is encoded into the string acceptor. The position of the relevant -bit part of this vector which includes the desired output is given by the first nbits of input. All possible corresponding 2 n -bit substrings are generated and presented to the acceptor to achieve the relevant part of the function value vector from which the relevant bit is extracted. Its position is determined from the last nbits of input. 8 Hopfield Languages In section 7 we have restricted ourselves to a special subclass of regular expres- sions. In this section we will concentrate on a special type of neural acceptors, the so-called Hopfield neuromata which are based on symmetric neural networks (Hopfield networks). In these networks the weights are symmetric and therefore, the architecture of such neuromata can be seen as an undirected graph. Hopfield networks have been traditionally studied [4, 21] and used, due to their favorite convergence properties. These networks are also of particular interest, because their natural physical realizations exist (e.g. Ising spin glasses, 'optical computers'). Using the concept of Hopfield neuromata, we will define a class of Hopfield languages that are recognized by these particular acceptors. The neuromaton that is based on symmetric neural network (Hopfield network) where hi; called a Hopfield acceptor (Hopfield neuromaton). The language recognized by Hopfield neuromaton N is called Hopfield language. First, we will show that the class of Hopfield languages is strictly contained within the class of regular languages. For this purpose we will formulate the necessary condition - the so-called Hopfield condition when a regular language is a Hopfield language. Intuitively, Hopfield languages cannot include words with those potentially infinite substrings which allow the Hopfield neuromaton to converge and to forget relevant information about the previous part of the input string which is recognized. The idea of the proof is to find a necessary condition to prevent a Hopfield neuromaton from converging. Definition 11 A regular language L is said to satisfy a Hopfield condition if f for every Theorem 8 Every Hopfield language satisfies the Hopfield condition. Proof: Let be a Hopfield language recognized by the Hopfield neuro- ng and define be the integer vectors of size n \Theta 1 and let \Gammafinpg be the integer matrix of size n \Theta n. Note that the matrix W is symmetric, since N is the Hopfield acceptor. Let us present an input string v 1 x to the acceptor N . For a sufficiently large m 0 , the network's computation must start cycling over this input because the network has only 2 n+1 of possible states. be the different states in this cycle and x be the corresponding input bits, so that the state s i is followed by the state s -(i) , is the index shifting permutation: be the inverse permutation of -, and let - r be the composed permutations, for any r - 1. Further, let ~c be the binary vectors of size n \Theta 1. Then ( ~ follows from Definition 8. For each state of the cycle we define an integer and the symbol T denotes the transposition of a vector. Obviously, p. Using the fact that the matrix W is symmetric we obtain Moreover, x 2. So we can write ~ We know that ( ~ Therefore, (~c T which implies p. But from E - p Then we cannot have both c - 2 n, at the same time because in this case the inequality of (~c T is strict. The complementary case of c - 2 simultaneously is impossible as well. Since then, the number of 1's in ~c - 2 (i) would be greater than the number of 1's in ~c i for Therefore, we can conclude that ~c - 2 consequently This implies that the cycle length p - 2. Hence, for every v 2 2 f0; 1g ? either This completes the proof that L satisfies the Hopfield condition. 2 For example, it follows from Theorem 8 that the regular languages [(000) ? are not Hopfield languages because they do not satisfy the Hopfield condition. 9 The Hopfield Condition Is Sufficient In this section we will prove that the necessary Hopfield condition from Definition 11, stating when a regular language is a Hopfield language, is sufficient as well. A construction of a Hopfield neuromaton is shown for a regular language satisfying the Hopfield condition. Theorem 9 For every regular language satisfying the Hopfield condition there exists a Hopfield neuromaton N of the size O (jffj) such that L is recognized by N . Hence, L is a Hopfield language. Proof: The architecture of the Hopfield neuromaton for the regular language [ff] satisfying the Hopfield condition is given by the general construction from the proof of Theorem 3. As a result we obtain an oriented network N 0 that corresponds to the structure of the regular expression ff where each neuron n s of N 0 (besides the special neurons inp, out, start) is associated with one symbol s 2 f0; 1g from ff (i.e., is of the type s). The task of n s is to check whether s agrees with the input bit. We will transform N 0 into an equivalent Hopfield network N . Supposing that ff contains iterations of binary substrings with at most two bits, the standard technique [15, 21] of the transformation of acyclic neural networks to Hopfield networks can be employed. The idea consists in adjusting weights to prevent propagating a signal backwards while preserving the original function of neurons. The transformation starts in the neuron out, it is carried out in the opposite direction to oriented edges and ends up in the neuron start. For a neuron whose outgoing weights have been already adjusted, its threshold and incoming weights are multiplied by a sufficiently large integer which exceeds the sum of absolute values of outgoing weights. This is sufficient to suppress the influence of outgoing edges on the neuron. After the transformation is accomplished all oriented paths leading from start to out are labeled by decreasing sequences in weights. The problem lies in realizing general iterations using only the symmetric weights. Consider a subnetwork I of N 0 corresponding to the iteration of some substring of ff. Let the subnetwork I have arisen from a subexpression fi + of ff in the proof of theorem 3. After the above-mentioned transformation is performed, any path leading from the incoming edges of I to the outgoing ones is labeled by a decreasing sequence of weights in order to avoid the backward signal from spreading. But the signal should be propagated from any output of the subnetwork I back to each sub-network input, as the iteration requires. On one hand, an integer weight associated with such a connection should be small enough in order to suppress the backward signal propagation. On the other hand, this weight should be sufficiently large enough to influence the subnetwork input neuron. Clearly, these two requirements are contradictory. Consider a simple cycle C in the subnetwork I consisting of an oriented path passing through I and of one backward edge leading from the end of this path (i.e., the output of I) to its beginning (i.e., the input of I). Let the types of neurons in the cycle C establish an iteration a + where a 2 f0; 1g ? and jaj ? 2. Moreover, suppose that x 2 f0; 1g 2 and a = x k for some k - 2. In the Hopfield condition set v 2 to be a postfix of L associated with a path leading from C to out. Similarly set v 1 to be a prefix of L associated with a path leading from start to C such that for every which contradicts the Hopfield condition. Such prefix v 1 exists because, otherwise, the cycle C could be realized as an iteration of two bits. Therefore a 6= x k . This implies that strings a i , contain a substring of the form by - b where b; b. Hence, the string a has the form either For notational simplicity we confine ourselves to the former case while the latter remains similar. Furthermore, we consider a = a 1 by - ba 2 with a minimal ja 2 j. We shift the decreasing sequence of weights in C to start and to end in the neuron n y while relevant weights in N 0 are modified by the above-mentioned procedure to ensure a consistent linkage of C within N 0 . For example, this means that all edges leading from the output neuron of I in C to the input neurons of I are evaluated by sufficiently large weights to realize the corresponding iterations. Now the problem lies in signal propagation from the neuron n b to the neuron n y . Assume To support the small weight in the connection between n b and n y a new neuron id that copies the state of the input neuron inp is connected to the neuron n y via a sufficiently large weight which strengthens the small weight in the connection from n b . Obviously, both the neuron n b (b = 1) and the new neuron id are active at the same time and both enable the required signal propagation from n b to n y together. On the other hand, when the neuron n- b ( - active, the neuron id is passive due to the fact that it is copying the input. This prevents the neuron n y from becoming active at that time. However, for some symbols b in ff there can be neurons n b 0 outside the cycle C (but within the subnetwork I) to which the edges from n y lead. This situation corresponds to y being concatenated with a union operation within fi. In this case the active neurons n b 0 , id would cause the neuron n y to fire. To avoid this, we add another new neuron n y 0 that behaves identically as the neuron n y for the symbol y 1g. Thus, the same neurons that are connected to n y are linked to n y 0 and the edges originally outgoing from n y to n b 0 for all corresponding b 0 are reconnected to lead only from n y 0 . A similar approach is used in the opposite case when above-described procedure is applied for each simple cycle C in the subnetwork I corresponding to the iteration fi + . These cycles are not necessarily disjoint, but the decomposition of a = a 1 by - ba 2 with minimal ja 2 j ensures their consistent synthesis. Similarly, the whole transformation process is performed for each iteration in ff. In this case some iteration can be a part of another iteration and the magnitude of weights in the inner iteration will need to be accommodated to embody this into the outer iteration. It is also possible that the neuron id has to support both iterations in the same point. Finally, the number of simple cycles in ff is O (jffj). Hence, the size of the resulting Hopfield neuromaton remains of order O (jffj). In figure 4 the preceding construction is illustrated by an example of the Hopfield neuromaton for the regular language [(1(0 A simple cycle consisting of neurons clarified here in details. Notice the decreasing sequence of weights (7,5,1) in this cycle, starting and ending in the neuron n y , as well as, the neuron id which enables the signal propagation from n b to n y . The neuron n y 0 , identical with n y , has also been created because the neuron originally connected to n y (see figure 1). 2 start inp id id out Figure 4: Hopfield neuromaton for [(1(0 Corollary 1 Let L be a regular language. Then L is a Hopfield language if f L satisfies the Hopfield condition. Finally, we will briefly investigate the closure properties of the class of Hopfield languages. Theorem 10 The class of Hopfield languages is closed under union, concatenation, intersection, and complement. It is not closed under iteration. Proof: The closeness of Hopfield languages under union and concatenation follows from Corollary 1. To obtain the Hopfield neuromaton for the complement, negate the function of the output neuron out by multiplying the associated weights (and the threshold) by -2 and by adding 1 to the threshold. Hence, the Hopfield languages are closed under intersection as well. Finally, due to Theorem 9, [1010] is a Hopfield language, whereas [(1010) ? ] is not a Hopfield language because it does not satisfy the Hopfield condition from Theorem 8. 2 Emptiness problem In order to further illustrate the descriptional power of neuromata we investigate the complexity of the emptiness problem for regular languages given by neuromata or by Hopfield acceptors. We will prove that both problems are PSPACE-complete. Definition 12 Given a (Hopfield) neuromaton N , the (Hopfield) neuromaton emptiness problem, which is denoted NEP (HNEP), is the issue of deciding whether the language recognized by the (Hopfield) neuromaton N is nonempty. Theorem 11 NEP, HNEP are PSPACE-complete. Proof: To show that both NEP; HNEP 2 PSPACE, an input string for the (Hopfield) neuromaton N is being guessed bit by bit and its acceptance is checked by simulating the network computation in a polynomial space to witness the non-emptiness of Next, we show that NEP is PSPACE-hard. Let A be an arbitrary language in PSPACE. For each x 2 f0; 1g ? we will, in a polynomial time, construct the corresponding neuromaton N such that x 2 A iff N 2 NEP . Further, let M be a polynomial space bounded Turing machine which recognizes A. First, a cyclic neural network N 0 which simulates M can be constructed in a polynomial time using the standard technique [4]. The idea of this construction is that for each tape cell of M there is a subnetwork which simulates a tape head when it is in this position during the computation (i.e., the local transition rule). The neighbor subnetworks are connected to enable the head moves. The input x for M is encoded in the initial state of N 0 and at the end of the neural network computation one neuron of N 0 , called result, signals whether x 2 A. The neural network N 0 is embodied into the neuromaton N as follows. The input neuron inp of N is not connected to any other neuron and the output neuron out of N is identified with the neuron result of N 0 . accepts x iff the neuron out is active at the end of the simulation iff L(N ) contains all words of the length which is equal to the length of the computation of M on x iff N 2 NEP . Thus, x 2 A iff N 2 NEP . This completes the proof that NEP is PSPACE-complete. For the Hopfield neuromata a similar simulation of M can be achieved using the symmetric neural network N 0 . It is because any convergent computation of an arbitrary asymmetric neural network can be simulated by a symmetric network of a polynomial size [4] and we can assume, without loss of generality, that M stops for every input. Hence, HNEP is PSPACE-complete as well. 2 Theorem 11 is a somewhat surprising because the identical problems for regular expressions, deterministic and non-deterministic finite automata are known to only be NL-complete [10, 11]. In some sense this shows that there is a big gap between the descriptive power of regular expressions and that of neuromata and confirms the result from section 6. We will discuss this issue later in section 11. The difference between the descriptive power of regular expressions and of neu- romata can also be illustrated by the complement operation. While the emptiness problem for the complement of the regular expression becomes PSPACE-complete [1] the emptiness problem complexity of the complement of a neuromaton does not change because the output neuron can be easily negated. As a next consequence we will show that the neuromaton equivalence problem is PSPACE-complete as well. Given two (Hopfield) neuromata N 1 , N 2 the (Hopfield) neuromaton equivalence problem which is denoted NEQP (HNEQP) is the issue of deciding whether the languages recognized by these (Hopfield) neuromata are the same, i.e. whether Corollary 2 NEQP, HNEQP are PSPACE-complete. Proof: To prove that NEQP 2 it is sufficient to show that its complement is in PSPACE. For this purpose an input string for the neuro- mata being guessed and its acceptance for one of these two neuromata and its rejection for the other one are checked by simulating both network computations in a polynomial space to witness the non-equivalence L(N 1 To show that NEQP is PSPACE-hard, the complement of NEP denoted co-NEP, which is PSPACE-complete as well, is in a polynomial time reduced to NEQP. Let the neuromaton N be an instance of co-NEP. We will in a polynomial time construct the corresponding instance N 1 , N 2 for NEQP so that L(N ) is empty iff The neuromaton N 1 is identified with N . Let N 2 be an arbitrary small neuromaton which recognizes the empty language. It is easy to see that this is a correct reduction. Hence, NEQP is PSPACE-complete. The PSPACE-completeness of HNEQP can be achieved similarly. 2 Conclusions In this paper we have explored an alternative formalism for the regular language representation, based on neural networks. Comparing the so-called neuromata with the classical regular expressions we have obtained the result that within a polynomial descriptive complexity, the non-determinism, which is captured in the regular expressions, can be simulated in a parallel mode by neuromata which have a deterministic nature. In our opinion, the descriptive power of the neuromata consists in an efficient encoding of the transition function. While the transition function of the deterministic (nondeterministic) finite automata is usually specified by the list of its values (old state, input symbol)/new state, the same function in the neuromata is given by the vector of formulae - each for one neuron which evaluates it. This encoding can be interpreted more like a general program. From this point of view it is easy to observe that the table of transition rules in the deterministic finite automata is a special case of such program. Moreover, the neuromata can encode the nondeterministic transition function efficiently using parallelism. In this way, the behavior of the neuromata in the exponential number of all possible states can be described in a polynomial size. However, this process is not reversible. The neural transition program cannot be generally rewritten as a polynomial size table or a polynomial length regular expression. Moreover, the number of neuromaton states (although exponential) is limited and that is why the matching lower bound on the neuromaton size can be achieved using the standard technique: there exists a regular language which requires the exponential number of neuromaton states to be recognized. Also, a more complex neural transition rule specification makes the emptiness problem for the neuromata harder than for the classical formalism (finite automata, regular expressions). On the other hand, while the complement operation in the nondeterministic case of this formalism can cause an exponential growth of the descriptional complexity, the complement of the neuromata can be easily achieved. We have also investigated the Hopfield neuromata which are studied widely due to their convergence properties. We have shown that Hopfield neuromata determine the proper subclass of regular languages - the so-called Hopfield languages. Via the so-called Hopfield condition, we have completely characterized the class of Hopfield languages. We can conclude that the neuromata present quite an efficient tool not only for the recognition of regular languages and of their subclasses respectively, but also for their description. Acknowledgement We are grateful to Markus Holzer, Piotr Indyk, and Petr Savick'y for the stimulating discussions related to the topics of this paper. Further we thank Tereza Beda-nov'a who realized the pictures in LaTex environment. --R The Design and Analysis of Computer Algorithms. Efficient Simulation of Finite Automata by Neural Nets. Learning of Regular Expressions by Pattern Matching. Complexity Issues in Discrete Hopfield Net- works Learning and Extracting Finite State Automata with Second-Order Recurrent Neural Networks Bounds on the Complexity of Recurrent Neural Network Implementations of Finite State Machines. Optimal Simulation of Automata by Neural Nets. Personal Communication. A Note on the Space Complexity of Some Decision Problems for Finite Automata. Representation of Events in Nerve Nets and Finite Au- tomata Computational Complexity of Neural Networks: A Survey. Circuit Complexity and Neural Networks. Discrete Neural Computation A Theoretical Foundation. Complexity Issues in Discrete Neurocomputing. Learning Finite State Machines with Self-Clustering Recurrent Networks --TR Efficient simulation of finite automata by neural nets Neurocomputing A note on the space complexity of some decision problems for finite automata Learning and extracting finite state automata with second-order recurrent neural networks Circuit complexity and neural networks Learning finite machines with self-clustering recurrent networks Discrete neural computation Learning and extracting initial mealy automata with a modular neural network model Bounds on the complexity of recurrent neural network implementations of finite state machines The Design and Analysis of Computer Algorithms Computational complexity of neural networks Hopfield Languages Learning of regular expressions by pattern matching

hopfield networks;finite neural networks;regular expressions;descriptional complexity;emptiness problem;string acceptors

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Online Learning versus Offline Learning.

We present an off-line variant of the mistake-bound model of learning. This is an intermediate model between the on-line learning model (Littlestone, 1988, Littlestone, 1989) and the self-directed learning model (Goldman, Rivest Schapire, 1993, Goldman & Sloan, 1994). Just like in the other two models, a learner in the off-line model has to learn an unknown concept from a sequence of elements of the instance space on which it makes guess and test trials. In all models, the aim of the learner is to make as few mistakes as possible. The difference between the models is that, while in the on-line model only the set of possible elements is known, in the off-line model the sequence of elements (i.e., the identity of the elements as well as the order in which they are to be presented) is known to the learner in advance. On the other hand, the learner is weaker than the self-directed learner, which is allowed to choose adaptively the sequence of elements presented to him.We study some of the fundamental properties of the off-line model. In particular, we compare the number of mistakes made by the off-line learner on certain concept classes to those made by the on-line and self-directed learners. We give bounds on the possible gaps between the various models and show examples that prove that our bounds are tight.Another contribution of this paper is the extension of the combinatorial tool of labeled trees to a unified approach that captures the various mistake bound measures of all the models discussed. We believe that this tool will prove to be useful for further study of models of incremental learning.

Introduction The mistake-bound model of learning, introduced by Littlestone [L88, L89], has attracted a considerable amount of attention (e.g., [L88, L89, LW89, B90a, B90b, M91, CM92, HLL92, GRS93, GS94]) and is recognized as one of the central models of computational learning theory. Basically it models a process of incremental learning, where the learner discovers the 'labels' of instances one by one. At any given stage of the learning process, the learner has to predict the label of the next instance based on his current knowledge, i.e. the labels of the previous instances that it has already seen. The quantity that the learner would like to minimize is the number of mistakes it makes along this process. Two variants of this model were considered, allowing the learner different degrees of freedom in choosing the instances presented to him: ffl The on-line model [L88, L89], in which the sequence of instances is chosen by an adversary and the instances are presented to the learner one-by-one. ffl The self-directed model [GRS93, GS94], in which the learner is the one who chooses the sequence of instances; moreover, it may make his choices adaptively; i.e., each instance is chosen only after seeing the labels of all previous instances. In the on-line model, the learner is faced with two kinds of uncertainties. The first is which function is the target function, out of all functions in the concept class which are consistent with the data. The second is what are the instances that it would be challenged on in the future. While the first uncertainty is common to almost any learning model, the second is particular to the on-line learning model. A central aim of this research is to focus on the uncertainty regarding the target function by trying to "neutralize" the uncertainty that is involved in not knowing the future elements, and to understand the effect that this uncertainty has on the mistake-bound learning model. One way of doing this, is to allow the learner a full control of the sequence of instances, as is done in the self-directed model. Our approach is different: we define the off-line learning model as a one in which the learner knows the sequence of elements in advance. Since the difference between the on-line learner and the off-line learner is the uncertainty regarding the order of the instances, this comparison gives insight into the "information" that is in knowing the sequence. Once we define the off-line cost of a sequence of elements, we can also define a best sequence (a sequence in which the optimal learner, knowing the sequence, makes the fewest mistakes) and a worst sequence (a sequence in which the optimal learner, knowing the sequence, makes the most mistakes). These are best compared to the on-line and self-directed models if we think of them in the following way: ffl The worst sequence (off-line) model is a one in which the sequence of instances is chosen by an adversary but the whole sequence (without the labels) is presented to the learner before the prediction process starts. ffl The best sequence (off-line) model is a one in which the whole sequence of instances is chosen by the learner before the prediction process starts. Denote by M on\Gammaline (C); Mworst (C); M best (C) and M sd (C) the number of mistakes made by the best learning algorithm in the online, worst sequence, best sequence and self-directed model (respec- tively) on the worst target concept in a concept class C. Obviously, for all C, Mworst best The main issue we consider is to what degree these measures can differ from one another. We emphasize that the only complexity measure is the number of mistakes and that other complexity measures, such as the computational complexity of the learning algorithms, are ignored. It is known that in certain cases M sd can be strictly smaller than M on\Gammaline . For example, consider the class of monotone monomials over n variables. It can be seen that this class has M In addition, we give examples that show, for certain concept classes, that M sd may be smaller than M best by a multiplicative factor of O(log n); hence, showing the power of adaptiveness. The following example shows that there are also gaps between M best and Mworst . Given n points in the interval [0; 1], consider the class of functions which are a suffix of this interval (i.e., the functions f a (x) that are 1 for x - a and 0 otherwise). As there are n points, there are only n+1 possible concepts, and therefore the Halving algorithm [L88, L89] is guaranteed to make at most O(log n) mistakes, i.e. M on\Gammaline = O(log n). For this class, a best sequence would be receiving the points in increasing order, in which case the learner makes at most one mistake (i.e., M best = 1). On the other hand, we show that the worst sequence forces Mworst = \Theta(log n) mistakes. An interesting question is what is the optimal strategy for a given sequence. We show a simple optimal strategy and prove that the number of mistakes is exactly the rank of a search tree (into which we insert the points in the order they appear in the sequence). We generalize this example, and show that for any concept class, the exact number of mistakes is the rank of a certain tree corresponding to the concept class and the particular sequence (this tree is based on the consistent extensions). This formalization of the number of mistakes, in the spirit of Littlestone's formalization for the on-line case [L88, L89], provides a powerful combinatorial characterization. All the above examples raise the question of how large can these gaps be. We prove that the gaps demonstrated by the above mentioned examples are essentially the largest possible; more precisely, we show that M on\Gammaline can be greater than M sd by at most a factor of log jX j, where X is the instance space (e.g., in the monomials example X is the set of 2 n boolean assignments). The above result implies, in particular, that the ratio between the number of mistakes for the best sequence and the worst sequence is O(log n), where n is the length of the sequence. 1 We also show that Mworst = \Omega\Gamma log M on\Gammaline ), which implies that either both are constant or both are non-constant. Finally we show examples in which M on\Gammaline = 3 Mworst , showing that Mworst 6= M on\Gammaline . In a few cases we are able to derive better bounds: for the cases that Mworst and Mworst = 2 we show simple on-line algorithms that have at most 1 and 3 mistakes, respectively. One way to view the relationships among the above models is through the model of "experts" [CFHHSW93, FMG92, MF93]. For each sequence oe there is an expert, E oe , that makes its predictions under the assumption that the sequence is oe. Let oe be the sequence chosen by the adversary, related result by [B90a] implies that if efficiency constraints are imposed on the model, then there are cases in which some orders are "easy" and others are computationally "hard". then the expert E oe makes at most Mworst mistakes. The on-line learner does not know the sequence oe in advance, so the question is how close can it get to the best expert, E oe . The problem is that the number of experts is overwhelming; initially there are n! experts (although the number of experts consistent with the elements of the sequence oe seen so far decreases with each element presented). Therefore, previous results about experts do not apply here. The rest of this paper is organized as follows: In Section 2, we give formal definitions of the model and the measures of performance that are discussed in this paper, followed by some simple properties of these definitions. In Section 3, we give the definition of the rank of a tree (as well as some other related definitions) and prove some properties of it. In Section 4, we present the various gaps. Then, we characterize the off-line complexity (for completeness, we present in Section 4.2.1 a characterization of the same nature, based on [L88, L89], for the on-line complexity and for the self-directed complexity [GS94]) and we use these characterizations to obtain some basic results. Finally, in Section 5, we use these characterizations to study the gap between the on-line complexity and off-line complexity. 2 The Model 2.1 Basic Definitions In this section we formally present our versions of the mistake bound learning model which is the subject of this work. The general framework is similar to the on-line learning model defined by Littlestone [L88, L89]. Let X be any set, and let C be a collection of boolean functions defined over the set X (i.e. We refer to X as the instance space and to C as the concept class. Let S be a finite subset of X . An on-line learning algorithm with respect to S (and a concept class C) is an algorithm A that is given (in advance) S as an input; Then, it works in steps as follows: In the i-th step the algorithm is presented with a new element s i 2 S. It then outputs its prediction p i 2 f0; 1g and in response it gets the true value c t (s i ), where c t 2 C denotes the target function. The prediction p i may depend on the set S, the values it has seen so far (and of course the concept class C). The process continues until all the elements of S have been presented. denote the order according to which the elements of S are presented to the learning algorithm. Denote by M(A[S]; oe; c t ) the number of mistakes made by the algorithm on a sequence oe as above and target function c t 2 C, when the algorithm is given S in advance (i.e., the number of elements for which p i the mistake bound of the algorithm, for a fixed S, as M(A[S]) 4 Finally, let The original definitions of [L88, L89] are obtained (at least for finite X ) by considering An off-line learning algorithm is an algorithm A that is given (in advance) not only the set S, but also the actual sequence oe as an input. The learning process remains unchanged (except that each prediction p i can now depend on the actual sequence, oe, and not only on the set of elements, S). Denote by M(A[oe]; c t ) the number of mistakes made by an off-line algorithm, A, on a sequence oe and a target c t . Define M(A[oe]) sequence oe, define A A For a given S, we are interested in the best and worst sequences. Denote by M best (S; C) the smallest value of M(oe; C) over all oe, an ordering of S, and let oe best be a sequence that achieves this minimum (if there are several such sequences pick one of them arbitrarily). Similarly, M worst (S; C) is the maximal value of M(oe; C) and oe worst is a sequence such that M(oe worst ; worst (S; C). A self-directed learning algorithm A is a one that chooses its sequence adaptively; hence the sequence may depend on the classifications of previous instances (i.e., on the target function). Denote by M sd (A[S]; c t ) the number of mistakes made by a self-directed algorithm A on a target function c t 2 C, when the algorithm is given in advance S (the set from which it is allowed to pick its queries). Define M sd (A[S]) 4 A A The following is a simple consequence of the definitions: Lemma 1: For any X ; C, and a finite S ' X , best (S; C) - M worst (S; C) - M on-line (S; C): 2.2 Relations to Equivalence Query Models It is well known that the on-line learning model is, basically, equivalent to the Equivalence Query model [L89]. It is not hard to realize that our versions of the on-line scenario give rise to corresponding variants of the EQ model. For this we need the following definitions: ffl An equivalence-query learning algorithm with respect to S (and a concept class C) is an algorithm A that is given in advance S as an input; Then, it works in steps as follows: In the i-th step the algorithm outputs its hypothesis, h i ' S, and in response it gets a counterexample; i.e., an element x denotes the target function. The process goes on until ffl Let F denote the function that chooses the counterexamples x i . We denote by EQ(A[S]; F; c t ) the number of counterexamples, x i , presented by F to the algorithm, A, in the course of a learning process on the target, c t , when A knows S in advance (but does not know F ). Finally, let EQ(S; C) Note that the original definitions of Angluin [A89] are obtained by considering . The following is a well known (and easy to prove) fact: 1: For every One aspect of the last definition above is that it considers the worst case performance of the learner over all possible choices, F , of counterexamples to its hypotheses. It turns out that by relaxing the definition so that the learner is only confronted with F 's of a certain type, one gets EQ models that match the various offline learning measures presented in the previous subsection. ffl Let oe denote an ordering of the set S. Let F oe be the following strategy for choosing coun- terexamples. Given a hypothesis h, the counterexample, F oe (h), is the minimal element of according to the ordering oe. ffl Let EQ(oe; C) 4 ffl Let EQ best (S; C) 4 worst (S; C) 4 This variant of the equivalence query model in which the minimal counterexample is provided to the algorithm is studied, e.g., in [PF88]. 2: For every X ; C and S ' X as above, for every ordering oe of S, EQ(oe; Proof: Given an EQ algorithm for (S; C; oe) construct an off-line algorithm by predicting, on each element s i , the value that the current hypothesis of the EQ algorithm, h k i , assigns to s i . Whenever the teacher's response indicates a prediction was wrong, present that element as a counterexample to the EQ algorithm (and replace the current hypothesis by its revised hypothesis). For the other direction, given an offline algorithm, define at each stage of its learning process a hypothesis by assigning to each unseen element, s 2 S, the value the algorithm would have guessed for s if it got responses indicating it made no mistakes along the sequence, oe, from the current stage up to that element. Whenever a counterexample is being presented, pass on to the offline algorithm the fact that it has erred on that element (and update the hypothesis according to its new guesses). Corollary 1: For every 1. EQ best (S; best (S; C). 2. EQ worst (S; worst (S; C). 3 Labeled Trees A central tool that we employ in the quantitative analysis in this work is the notion of ranks of trees. We shall consider certain classes of labeled trees, depending upon the classes to be learned and the type of learning we wish to analyze. The following section introduces these technical notions and their basic combinatorial properties. 3.1 Rank of Trees In this subsection we define the notion of the rank of a binary tree (see, e.g., [CLR90, EH89, B92]), which plays a central role in this paper. We then prove some simple properties of this definition. For a tree T , if T is empty then rank(T its left subtree and TR be its right subtree. Then, For example, the rank of a leaf is 0. Let -r . The following lemma is a standard fact about the rank: Lemma 2: A depth d rank r tree has at most -r leaves. Proof: By induction on d and r. If there is exactly one leaf (if there were two or more, then their least common ancestor is of rank 1). If there is one leaf (which is a special case of r = 0) or two leaves, in which case r must be 1. In all these cases the claim holds. For the induction step, let T be a depth d rank r tree. Each of and TR are of depth at most by the definition of rank, in the worst case one of them is of rank r and the other of Hence, by the induction hypothesis, the number of leaves is bounded by r d! r r d which completes the proof. If r is small relative to d then it may be convenient to use the weaker d r (- -r on the number of leaves. A subtree of a tree T is a subset of the nodes of T ordered by the order induced by T . Lemma 3: The rank of a binary tree T is at least k iff it has a subtree T 0 which is a complete binary tree of depth k. Proof: Mark in T the nodes where the rank increases. Those are the nodes of T 0 . For a marked node with rank i, each of its children in T has rank hence it has a marked descendant with binary tree. For the other direction, note that the rank of a tree is at least the rank of any of its subtrees, and that a complete binary tree of depth k has rank k. Lemma 4: Let T be a complete binary tree of depth k. Let L be a partition of the leaves of T into t disjoint subsets. For to be the subtree induced by the leaves in L i (that is, T i is the tree of all the nodes of T that have a member of L i below them). Then, there exists such that Proof: The proof is by induction on t. For hence the claim is obvious. For consider the nodes in depth bk=tc in T . There are two cases: (a) if all these nodes belong to all trees T i then each of these trees contains a complete subtree of depth bk=tc and by Lemma 3 each of them has rank of at least bk=tc. (b) if there exists a node v in depth bk=tc which does not belong to all the trees T i then we consider the subtree T 0 whose root is v and consists of all the nodes below v. By the definition of v, the leaves of the tree T 0 belong to at most t \Gamma 1 of the sets L i . In addition the depth of T 0 is at least (t\Gamma1)k . Hence, by induction hypothesis, one of the subtrees T 0 of T 0 is of rank at least Finally note that T 0 i is a subtree of T i hence T i has the desired rank. Let us just mention that the above lower bound, on the rank of the induced subtrees, is essentially the best possible. For example, take T to be a complete binary tree. Each leaf corresponds to a path from the root to this leaf. Call an edge of such a path a left (right) edge if it goes from a node to its left (right) son. Let L 0 (L 1 ) be the set of leaves with more left (right) edges. Then, it can be verified that rank(T 0 3.2 Labeled Trees Let X denote some domain set, S ' X and C ' f0; 1g X as above. -labeled tree is a pair, (T ; F ), where T is a binary tree and F a function mapping the internal nodes of T into X . Furthermore, we impose the following restriction on F : is an ancestor of t then F ffl A branch in a tree is a path from the root to a leaf. It follows that the above mapping F is one to one on branches of T . ffl A branch realizes a function if for all 1 son of t i if and only if h(F (t i 1. Note that a branch can realize more than one function. On the other hand, if then the branch realizes a single function. -labeled tree is an (S; C)-tree if the set of functions realized by its branches is exactly S denote the set of all (S; C)-trees. ffl For a sequence of elements of X , let T C oe denote the maximal tree in T C S for which every node v in the k-th level is labeled F Note, that using this notation, a class C shatters the set of elements of a sequence, oe, if and only if T C oe is a complete binary tree (recall that a class C shatters a set fs if for every there exists a function f 2 C that for all can therefore conclude that, for any class C, oe is a complete binary treeg: (We shall usually omit the superscript C when it is clear from the context.) 4 Gaps between the Complexity Measures Lemma 1 provides the basic inequalities concerning the learning complexity of the different models. In this section we turn to a quantitative analysis of the sizes of possible gaps between these measures. We begin by presenting, in Section 4.1, some examples of concept classes for which there exist large gaps between the learning complexity in different models. In Section 4.3, we prove upper bounds on the possible sizes of these gaps, bounds that show that the examples of Section 4.1 obtain the maximal possible gap sizes. A useful tool in our analysis is a characterization of the various complexity measures as the ranks of certain trees. This characterization is given in section 4.2. Let us begin our quantitative analysis by stating a basic upper bound on the learning complexity in the most demanding model (from the student's point of view), namely, M on-line (S; C). Given an instance space X , a concept class C and a set S we define C S to be the projection of the functions in C on the set S (note that several functions in C may collide into a single function in C S ). Using the Halving algorithm [L88, L89] we get, Theorem 2: For all X ; C and S as above M on-line (S; C) - log jC S j. 4.1 Some Examples The first example demonstrates that M best (S; C) may be much smaller than M worst (S; C) and on-line (S; C). This example was already mentioned in the introduction and appears here in more details. Example 1: Let X be the unit interval [0; 1]. For, 0 - a - 1 define the function f a (x) to be 0 if x - a and 1 if x ? a. Let C 4 1g. In other words, the concept class C consists of all intervals of the form [a; 1] for 0 - a - 1. Let g. By Theorem 2, it is easy to see that in this example M on-line (S; C) - log(n 1). We would like to understand how an off-line algorithm performs in this case. Clearly, for every sequence oe, an adversary can always force a mistake on the first element of the sequence. Hence, M best (S; C) - 1. To see that M best (S; this sequence the following strategy makes at most 1 mistake: predict "0" until a mistake is made. Then, all the other elements are "1"s of the function. For a worst sequence consider Figure 1: The concept class of Example 2 It may be seen that the adversary can force the learning algorithm to make one mistake on each of the sets , and hence a total of mistakes. This is the worst possible, as this performance is already granted for an on-line algorithm as discussed above (and, by Lemma 1 an off-line algorithm can always match the on-line performance). The next example shows that for all n there exist sets class C, such that M sd (S; C) - 2 and M best (S; C) n). loss of generality, assume that for some value d, dg. The concept class C (see Figure 1) consists of 2 \Delta 2 d functions d. Each function f i is defined as follows: f i all there is a single x i which is assigned 1 and hence can be viewed as an indicator for the corresponding function f i ). The elements are partitioned into 2 d =d "blocks" each of size d. In each of these blocks the 2 d functions f get all the 2 d possible combinations of values (as in Figure 1). The functions are defined similarly by switching the roles of x's and y's. More precisely, g i serves as an indicator for the corresponding function g i ). Again, the elements are partitioned into blocks each of size d. In each of these blocks the 2 d functions get all the 2 d possible combinations of values. To see that M sd (S; C) - 2 we describe a self-directed learner for this concept class. The learner first asks about z and predicts in an arbitrary way. Whether it is right or wrong the answer indicates whether the target functions is one of the f i 's or one of the g i 's. In each case the learner looks for the corresponding indicator. That is, if c t asks the x's one by one, predicting 0 on each. A mistake on some x i (i.e., c t immediately implies that the target function is f i and no mistakes are made anymore. Symmetrically, if c t the learner asks the y's one by one, predicting 0 on each. A mistake on some y i (i.e., c t (y implies that the target function is g i and again no mistakes are made anymore. In any case, the total number of mistakes is at most 2. We now prove that M best (S; C) n). 2 The idea is that a learner must choose its sequence in advance, but does not know whether it looks for one of the f i 's or one of the g i 's. Formally, let oe be the (best) sequence chosen by the learner. We describe a strategy for the adversary to choose a target function in a way that forces the learner at least d=4 =\Omega\Gamma363 n) mistakes. Let oe 0 be a prefix of oe of length 2 d . The adversary considers the number of x i 's queried in oe 0 versus the number of 's. Assume, without loss of generality, that the number of x i 's in oe 0 is smaller than the number of y i 's in oe 0 . The adversary then restricts itself to choosing one of the f i 's. Moreover, it eliminates all those functions f i whose corresponding element x i appears in oe 0 . Still, there are at least 2 d =2 possible functions to choose from. Now, consider the y's queried in oe 0 . By the construction of C we can partition the elements "groups" of size 2 d =d such that every function f j gives the same value for all elements in each group. There are at least 2 d =2 elements y's that are queried in oe 0 and they belong to ' groups. By simple counting, d- d. We estimate the number of possible behaviors on these ' groups as follows: originally all 2 d behaviors on the d groups were possible. Hence, to eliminate one of the behaviors on the ' elements one needs to eliminate 2 d\Gamma' functions. As we eliminated at most 2 d =2 functions, the number of eliminated behaviors is at most2 In other words, there are at least 12 ' behaviors on these ' elements. On the other hand, if we are guaranteed to make at most r mistakes it follows from Theorem 6 and Lemma 2 that the number of functions is at most -r must be at least '=2 - d=4 n). 4.2 Characterizing M(oe; C) Using the Rank The main goal of this section is to characterize the measure M(oe; C). As a by-product, we present an optimal offline prediction algorithm. I.e., an algorithm, A, such that for every sequence oe, The next theorem provides a characterization of M(oe; C) in terms of the rank of the tree T C oe (for any concept class C and any sequence oe). A similar characterization was proved by Littlestone [L88, L89] for the on-line case (see section 4.2.1 below). Theorem 3: For all Proof: To show that M(oe; C) - rank(T oe ) we present an appropriate algorithm. For predicting on s 1 , the algorithm considers the tree T oe , defined above, whose root is s 1 . Denote by its left subtree and by TR its right subtree. If rank(T predicts "0", if predicts "1", otherwise (rank(T L can predict arbitrarily. Again, recall that in the case that rank(T L by the definition of rank, both rank(T L ) and rank(T R ) are smaller than rank(T oe ). Therefore, at each step the algorithm uses for the prediction a subtree of T oe which is consistent with all the values it has seen so far. To conclude, at each step where the algorithm made a mistake, the rank decreased by (at least) 1, so no more than rank(T oe ) mistakes are made. To show that no algorithm can do better, we present a strategy for the adversary for choosing a target in C so as to guarantee that a given algorithm A makes at least rank(T oe ) mistakes. The adversary constructs for itself the tree T oe . At step i, it holds a subtree T whose root is a node Again, by the Halving algorithm, M on-line (S; C) and therefore also M best (S; C) are O(log n). marked s i which is consistent with the values it already gave to A as the classification of s After getting A's prediction on s i the adversary decides about the true values as follows: If one of the subtrees, either TR , has the same rank as the rank of T then it chooses the value according to this subtree. Note that, by definition of rank, at most one of the subtrees may have this property, so this is well defined. In this case, it is possible that A guessed the correct value (for example, the algorithm we described above does this) but the rank of the subtree that will be used by the adversary in the i 1-th step is not decreased. The second possible case, by the definition of rank, is that the rank of both and TR is smaller by 1 than the rank of T . In this case, the adversary chooses the negation of A's prediction; hence, in such a step A makes a mistake and the rank is decreased by 1. Therefore, the adversary can force a total of rank(T oe ) mistakes. The above theorem immediately implies: Corollary 4: For all worst (S; oe is an ordering of S best (S; oe is an ordering of S Remark 1: It is worth noting that, by Sauer's Lemma [S72], if the concept class C has V C dimension d then the size of T C oe is bounded by n d (where n, as usual, is the length of oe). It follows that, for C with small V C, the tree is small and therefore, if consistency can be checked efficiently then the construction of the tree is efficient. This, in turn, implies the efficiency of the generic (optimal) off-line algorithm of the above proof, for classes with "small" VC dimension. Example 3: Consider again the concept class of Example 1. Note that in this case, the tree T oe is exactly the binary search tree corresponding to the sequence Namely, T oe is the tree constructed by starting with an empty tree and performing the sequence of operations e.g. [CLR90]). Hence, M(oe; C) is exactly the rank of this search tree. 4.2.1 Characterizing the On-line and Self-Directed Learning To complete the picture one would certainly like to have a combinatorial characterization of on-line (S; C) as well. Such a characterization was given by Littlestone [L88, L89]. We reformulate this characterization in terms of ranks of trees. The proof remains similar to the one given by Littlestone [L88, L89] and we provide it here for completeness. Theorem 5: For all Proof: To show that M on-line (S; C) - max we use an adversary argument similar to the one used in the proof of Theorem 3. The adversary uses the tree that gives the maximum in the above expression to choose both the sequence and the classification of its elements, so that at each time that the rank is decreased by 1 the prediction algorithm makes a mistake. To show that M on-line (S; C) is at most we present an appropriate algorithm, which is again similar to the one presented in the proof of Theorem 3. For predicting on s 2 S, we first define C 0 S to be all the functions in C S consistent with S to be all the functions in C S consistent with 1. The algorithm compares max and and predicts according to the larger one. The crucial point is that at least one of these two values must be strictly smaller than m otherwise there is a tree in T C whose rank is more than m. The prediction continues in this way, so that the maximal rank is decreased with each mistake. Finally, the following characterization is implicit in [GS94]: Theorem Proof: Consider the tree T whose rank is the minimal one in T C S . We will show that M sd (S; C) is at most the rank of T . For this, we present an appropriate algorithm that makes use of this tree. At each point, the learner asks for the instance which is the current node in the tree. In addition, it predicts according to the subtree of the current node whose rank is higher (arbitrarily, if the ranks of the two subtrees are equal). The true classification determines the child of the current node from which the learner needs to proceed. It follows from the definition of rank that whenever the algorithm makes a mistake the remaining subtree has rank which is strictly smaller than the previous one. For the other direction, given a strategy for the learner that makes at most M sd (S; C) mistakes we can construct a tree T that describes this strategy. Namely, at each point the instances that the learner will ask at the next stage, given the possible classifications of the current instance, determine the two children of the current node. Now, if the rank of T was more than M sd (S; C) then this gives the adversary a strategy to fool the learner: at each node classify the current instance according to the subtree with higher rank. If the ranks of both subtrees are equal then on any answer by the algorithm the adversary says the opposite. By the definition of rank, this gives rank(T ) mistakes. Hence, rank(T ) is at most M sd (S; C) and certainly the minimum over all trees can only be smaller. 4.3 A Bound on the Size of the Gap A natural question is how large can be the gaps between the various complexity measures. For example, what is the maximum ratio between M worst (S; C) and M best (S; C). In Example 1 the best is 1 and the worst is log n, which can be easily generalized to k versus \Theta(k log n). The following theorem shows that the gap between the smallest measure, M sd (S; C), and the largest measure, the on-line cost, cannot exceed O(log n). This, in particular, implies a similar bound for the gap between oe best and oe worst . By Example 1, the bound is tight; i.e., there are cases which achieve this gap. Similarly, the gap between M sd (S; C) and M best (S; C) exhibited by Example 2 is also optimal. Theorem 7: For of size n as above, on-line (S; C) - M sd (S; C) \Delta log n: We shall present two quite simple but very different proofs for this theorem. The first proof employs the tool of labeled trees (but gives a slightly weaker result) while the second is by an information - theoretic argument. Proof: [using labeled trees] Consider the tree T that gives the minimum in Theorem 6. Its depth is n and its rank, by Theorem 6, is C). By Lemma 2, this tree contains at most -m leaves. That is, jC -m . By Theorem 2, M on-line (S; C) - log -m Proof: [information theoretic argument] Let C S be the projection of the functions in C on the set S (note that several functions in C may collide into a single function in C S ). Consider the number of bits required to specify a function in C S . On one hand, at least log jC S j bits are required. On the other hand, any self-directed learning algorithm that learns this class yields a natural coding scheme: answer the queries asked by the algorithm according to the function c 2 C S ; the coding consists of the list of names of elements of S on which the prediction of the algorithm is wrong. This information is enough to uniquely identify c. It follows that M sd (S; C) \Delta log n bits are enough. Hence, log Finally, by the Halving algorithm [L88, L89], it is known that on-line (S; C) - log jC S j: The theorem follows. Corollary 8: For worst (S; best (S; C) \Delta log n). Proof: Combine Theorem 7 with Lemma 1. worst (S; C) vs. M on-line (S; C) In this section we further discuss the question of how much can a learner benefit from knowing the learning sequence in advance. In the terminology of our model this is the issue of determining the possible values of the gap between M on-line (S; C) and M worst (S; C). We show (in Section 5.2) that if one of these two measures is non-constant then so is the other. Quantitatively, if the on-line algorithm makes k mistakes, then any off-line algorithm makes \Omega\Gamma log mistakes on oe. For the special cases where M worst (S; C) is either 1 or 2, we prove (in Section 5.1) that M on-line (S; C) is at most 1 or 3 (respectively). 5.1 Simple Algorithms In this section we present two simple on-line algorithms, E1 and E2, for the case that the off-line algorithm is bounded by one and two mistakes (respectively) for any sequence. Let S be a set of elements of X . If for every sequence oe, which is a permutation of S, the off-line learning algorithm makes at most one mistake, then we show that there is an on-line algorithm E1 that makes at most one mistake on S, without knowing the actual order in advance. The algorithm E1 uses the guaranteed off-line algorithm A and works as follows: ffl Given an element x 2 S, choose any sequence oe that starts with x, and predict according to A's prediction on oe, i.e. A[oe]. If a mistake is made on x, then A[oe] made a mistake and it will not make any more mistakes on this sequence oe. Hence, we can use A[oe](c t (x)) to get the true values for all the elements of the sequence (where by A[oe](c t (x)) we denote the predictions that A makes on the sequence oe after getting the value c t (x)). In other words, for any y 2 S there is a unique value that is consistent with the value c t (x) 6= A[oe] (otherwise A[oe] can make another mistake). Therefore, E1 will make at most one mistake. In the case that for any sequence the off-line learning algorithm makes at most two mistakes, we present an on-line algorithm E2 that makes at most three mistakes (which is optimal due to Call an element x bivalent with respect to y if there exist sequences oe 0 and oe 1 that both start with xy and for oe 0 the on-line algorithm predicts "c t the on-line algorithm predicts "c t 1). Otherwise x is univalent with respect to y (we say that x is 1-univalent with respect to y if the prediction is always 1 and 0-univalent if the prediction is always 0). Our on-line procedure E2, on input x, works as follows. ffl So far we made no mistakes: If there is no y such that x is 1-univalent with respect to y, predict "c t predict ffl So far we made one mistake on a positive w: If we made such a mistake then we predicted "c t which implies that there is no y such that w is 1-univalent with respect to y. In particular, with respect to x, w is either 0-univalent or bivalent. In both cases there is a sequence oe = wxoe 0 such that A[oe] predicts "c t and makes a mistake. Use as the prediction on x (where again, A[oe](1) denotes the prediction that A makes on sequence oe after getting the value c t In case of another mistake, we have a sequence on which we already made two mistakes so it will not make any more mistakes. Namely, we can use A[oe](1; - b) to get the value for all elements in S. ffl So far we made one mistake on a negative w: If w is either 1-univalent with respect to x or bivalent with respect to x then this is similar to the previous case. The difficulty is that this time there is also a possibility that w is 0-univalent with respect to x. However, in this case, if we made a mistake this means that we predicted which implies that there exists a y such that w is 1-univalent with respect to y. Consider a sequence . By the definition of y, A[oe] predicts "c t therefore makes its first mistake on w. Denote by the prediction on y. If this is wrong again then all the other elements of the sequence are uniquely determined. Namely, there is a unique function f 1 that is consistent with c t b. If, on the other hand, b is indeed the true value of y, we denote by its prediction on x. Again, if this is wrong, we have a unique function f 2 which is consistent with c t Therefore, we predict c on x. In case we made a mistake (this is our second mistake) we know for sure that the only possible functions are f 1 and f 2 (in fact, if we are lucky then f 1 and we are done). To know which of the two functions is the target we will need to make (at one more mistake (3 in total). 5.2 A General Bound In this section we further discuss the gap between the measures M on-line (S; C) and M worst (S; C). We show that if the on-line makes k mistakes, then any off-line algorithm makes \Omega\Gamma log on oe. The proof makes use of the properties proved in Section 3.1 and the characterizations of both the on-line and the off-line mistake bounds as ranks of trees, proved in Section 4. More precisely, we will take a tree in T C S with maximum rank (this rank by Theorem 5 exactly characterizes the number of mistakes made by the on-line algorithm) and use it to construct a tree with rank which is "not too small" and such that the nodes at each level are labeled by the same element of S. Such a tree is of the form T oe , for some sequence oe. Lemma 5: Given a complete labeled binary tree of depth k, (T ; F S , there is a sequence, oe, of elements of S, such that the tree T C oe has log k). Proof: We will construct an appropriate sequence oe in phases. At phase i (starting with we add (at most 2 i ) new elements to oe, so that the rank of T C oe is increased by at least one (hence, at the beginning of the ith phase the rank is at least i). At the beginning of the ith phase, we have a collection of 2 i subtrees of T , each is of rank at least k=2 O(i 2 (in particular, at the beginning of phase 0 there is a single subtree, T itself, whose rank is k). Each of these subtrees is consistent with one of the leaves of the tree T C oe we already built in previous phases (i.e., the subtree is consistent with some assignment to the elements included in oe in all previous phases). Moreover, the corresponding 2 i leaves induce a complete binary subtree of depth i. In the ith phase, we consider each of the 2 i subtrees in our collection. From each such subtree T 0 we add to the sequence oe, an element r such that the rank of the subtree of T 0 rooted at r is rank(T 0 ) and the rank in each of the subtrees TR corresponding to the sons of r is rank(T remark that the order in which we treat the subtrees within the ith phase is arbitrary and that if some of the elements r already appear in oe we do not need to add them again. After adding all the new elements of oe we examine again the trees TR corresponding to each subtree . For each of them the other partitions the leaves of the tree into 2 according to the possible values for the other Hence, by Lemma 4, there exists subtrees R of respectively which have rank at least and each of them is consistent with one of the leaves of the extended T C oe . The 2 i+1 subtrees that we get in this way form the collection of trees for phase i + 1. Finally note that by the choice of elements r added to oe, we now get in T C oe a complete binary subtree of depth i + 1. If before the ith phase the rank of the subtrees in our collection is at least k i then after the ith phase the rank of the subtrees in our collection is at least k i =2 i . Hence, a simple induction implies that Therefore, we can repeat this process log phases hence obtaining a tree T C oe of rank log k). Theorem 9: Let C be a concept class, X an instance space and S ' X the set of elements. Then worst (S; log M on-line (S; C)). Proof: Assume that M on-line (S; k. By Theorem 5, there is a rank k tree in T C S , and by Lemma 3 it contains a complete binary subtree T of depth k. By Lemma 5, there is a sequence oe for which the tree T C oe has log k). Hence, by Theorem 3, M(oe; C) - log k. A major open problem is what is the exact relationship between the on-line and the off-line mistake bounds. The largest gap we could show is a multiplicative factor of 3=2. worst (S; Proof: We first give an example for the case be a space of 4 elements, C 1 be the following 8 functions on the 4 elements: f0000; 0011; 0010; 0111; 1000; 1010; 1100; 1111g. It can be verified (by inspection) that M on-line (S; worst (S; 2. For a general k, we just take k independent copies of X 1 and C 1 . That is, let X k be a space of 4k elements partitioned into k sets of 4 elements. Let C k be the 8 k functions obtained by applying one of the 8 functions in C 1 to each of the k sets of elements. Let . Due to the independence of the k functions, it follows that M on-line (S; worst (S; 6 Discussion In this work we analyze the effect of having various degrees of knowledge on the order of elements in the mistake bound model of learning on the performance (i.e., the number of mistakes) of the learner. We remark that in our setting the learner is deterministic. The corresponding questions in the case of randomized learners remain for future research. We can also analyze quantitatively the advantage that an online algorithm may gain from knowing just the set of elements, S, in advance (without knowing the order, oe, of their presentation). That is, we wish to compare the situation where the online algorithm knows nothing a-priori about the sequence (other than that it consists of elements of X ) and the case that the algorithm knows the set S from which the elements of the sequence are taken (but has no additional information as for their order). The following example shows that the knowledge of S gives an advantage to the learning algorithm: Consider the intervals concept class of Example 1, with the instance space X restricted to f 1 g. As proven, M on-line (X 1). On the other hand, for every set S of size ', we showed that M on-line (S; 1). Therefore, if S is small compared to (i.e., ' is small compared to n) the number of mistakes can be significantly improved by the knowledge of S. Acknowledgment We wish to thank Moti Frances and Nati Linial for helpful discussions. --R "Equivalence Queries and Approximate Fingerprints" "Separating Distribution-Free and Mistake-Bound Learning Models over the Boolean Domain" "Learning Boolean Functions in an Infinite Attribute Space" "Rank-r Decision Trees are a Subclass of r-Decision Lists" "How to Use Expert Advice" "On-line Learning of Rectangles" "Learning Decision Trees from Random Examples" "Universal Prediction of Individual Sequences" "Learning Binary Relations and Total Orders" "The Power of Self-Directed Learning" "Apple Tasting and Nearly One-Sided Learning" "Learning when Irrelevant Attributes Abound: A New Linear-Threshold Algo- rithm" "Mistake Bounds and Logarithmic Linear-Threshold Learning Algorithms" "The Weighted Majority Algorithm" "On-line Learning with an Oblivious Environment and the Power of Randomization" "Universal Sequential Decision Schemes from Individual Sequences" "Learning Automata from Ordered Examples" "On the Density of Families of Sets" --TR Learning decision trees from random examples needed for learning Mistake bounds and logarithmic linear-threshold learning algorithms Introduction to algorithms Equivalence queries and approximate fingerprints Learning boolean functions in an infinite attribute space On-line learning with an oblivious environment and the power of randomization Learning Automata from Ordered Examples On-line learning of rectangles Rank-<italic>r</italic> decision trees are a subclass of <italic>r</italic>-decision lists Learning binary relations and total orders How to use expert advice The weighted majority algorithm The Power of Self-Directed Learning Learning Quickly When Irrelevant Attributes Abound --CTR Peter Damaschke, Adaptive Versus Nonadaptive Attribute-Efficient Learning, Machine Learning, v.41 n.2, p.197-215, November 2000 Paul Burke , Sue Nguyen , Pen-Fan Sun , Shelley Evenson , Jeong Kim , Laura Wright , Nabeel Ahmed , Arjun Patel, Writing the BoK: designing for the networked learning environment of college students, Proceedings of the 2005 conference on Designing for User eXperience, November 03-05, 2005, San Francisco, California

mistake-bound;On-Line Learning;rank of trees

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Factorial Hidden Markov Models.

Hidden Markov models (HMMs) have proven to be one of the most widely used tools for learning probabilistic models of time series data. In an HMM, information about the past is conveyed through a single discrete variablethe hidden state. We discuss a generalization of HMMs in which this state is factored into multiple state variables and is therefore represented in a distributed manner. We describe an exact algorithm for inferring the posterior probabilities of the hidden state variables given the observations, and relate it to the forwardbackward algorithm for HMMs and to algorithms for more general graphical models. Due to the combinatorial nature of the hidden state representation, this exact algorithm is intractable. As in other intractable systems, approximate inference can be carried out using Gibbs sampling or variational methods. Within the variational framework, we present a structured approximation in which the the state variables are decoupled, yielding a tractable algorithm for learning the parameters of the model. Empirical comparisons suggest that these approximations are efficient and provide accurate alternatives to the exact methods. Finally, we use the structured approximation to model Bachs chorales and show that factorial HMMs can capture statistical structure in this data set which an unconstrained HMM cannot.

Introduction Due to its flexibility and to the simplicity and efficiency of its parameter estimation algorithm, the hidden Markov model (HMM) has emerged as one of the basic statistical tools for modeling discrete time series, finding widespread application in the areas of speech recognition (Rabiner & Juang, 1986) and computational molecular biology (Krogh, Brown, Mian, Sj-olander, & Haussler, 1994). An HMM is essentially a mixture model, encoding information about the history of a time series in the value of a single multinomial variable-the hidden state-which can take on one of K discrete values. This multinomial assumption supports an efficient parameter estimation algorithm-the Baum-Welch algorithm-which considers each of the K settings of the hidden state at each time step. However, the multinomial assumption also severely limits the representational capacity of HMMs. For exam- ple, to represent bits of information about the history of a time sequence, an HMM would need distinct states. On the other hand, an HMM with a distributed state representation could achieve the same task with binary state Z. GHAHRAMANI AND M.I. JORDAN variables (Williams & Hinton, 1991). This paper addresses the problem of constructing efficient learning algorithms for hidden Markov models with distributed state representations. The need for distributed state representations in HMMs can be motivated in two ways. First, such representations let the model automatically decompose the state space into features that decouple the dynamics of the process that generated the data. Second, distributed state representations simplify the task of modeling time series that are known a priori to be generated from an interaction of multiple, loosely-coupled processes. For example, a speech signal generated by the superposition of multiple simultaneous speakers can be potentially modeled with such an architecture. Williams and Hinton (1991) first formulated the problem of learning in HMMs with distributed state representations and proposed a solution based on deterministic learning. 1 The approach presented in this paper is similar to Williams and Hinton's in that it can also be viewed from the framework of statistical mechanics and mean field theory. However, our learning algorithm is quite different in that it makes use of the special structure of HMMs with a distributed state representation, resulting in a significantly more efficient learning procedure. Anticipating the results in Section 3, this learning algorithm obviates the need for the two-phase procedure of Boltzmann machines, has an exact M step, and makes use of the forward-backward algorithm from classical HMMs as a subroutine. A different approach comes from Saul and Jordan (1995), who derived a set of rules for computing the gradients required for learning in HMMs with distributed state spaces. However, their methods can only be applied to a limited class of architectures Hidden Markov models with distributed state representations are a particular class of probabilistic graphical model (Pearl, 1988; Lauritzen & Spiegelhalter, 1988), which represent probability distributions as graphs in which the nodes correspond to random variables and the links represent conditional independence relations. The relation between hidden Markov models and graphical models has recently been reviewed in Smyth, Heckerman and Jordan (1997). Although exact probability propagation algorithms exist for general graphical models (Jensen, Lauritzen, & Olesen, 1990), these algorithms are intractable for densely-connected models such as the ones we consider in this paper. One approach to dealing with this issue is to utilize stochastic sampling methods (Kanazawa et al., 1995). Another approach, which provides the basis for algorithms described in the current paper, is to make use of variational methods (cf. Saul, Jaakkola, & Jordan, 1996). In the following section we define the probabilistic model for factorial HMMs and in Section 3 we present algorithms for inference and learning. In Section 4 we describe empirical results comparing exact and approximate algorithms for learning on the basis of time complexity and model quality. We also apply factorial HMMs to a real time series data set consisting of the melody lines from a collection of chorales by J. S. Bach. We discuss several generalizations of the probabilistic model FACTORIAL HIDDEN MARKOV MODELS 247 Y t+1 Y t-1 Y S (2) Y S (2) Y t+1 S (2) Y t-1 (a) (b) Figure 1. (a) A directed acyclic graph (DAG) specifying conditional independence relations for a hidden Markov model. Each node is conditionally independent from its non-descendants given its parents. (b) A DAG representing the conditional independence relations in a factorial HMM with underlying Markov chains. in Section 5, and we conclude in Section 6. Where necessary, details of derivations are provided in the appendixes. 2. The probabilistic model We begin by describing the hidden Markov model, in which a sequence of observations modeled by specifying a probabilistic relation between the observations and a sequence of hidden states fS t g, and a Markov transition structure linking the hidden states. The model assumes two sets of conditional independence relations: that Y t is independent of all other observations and states given S t , and that S t is independent of Markov property). Using these independence relations, the joint probability for the sequence of states and observations can be factored as Y The conditional independencies specified by equation (1) can be expressed graphically in the form of Figure 1 (a). The state is a single multinomial random variable that can take one of K discrete values, S t Kg. The state transition probabilities, are specified by a K \Theta K transition matrix. If the observations are discrete symbols taking on one of D values, the observation probabilities can be fully specified as a K \Theta D observation matrix. For a continuous observation vector, P (Y t jS t ) can be modeled in many different forms, such as a Gaussian, a mixture of Gaussians, or even a neural network. 2 In the present paper, we generalize the HMM state representation by letting the state be represented by a collection of state variables Z. GHAHRAMANI AND M.I. JORDAN each of which can take on K (m) values. We refer to these models as factorial hidden Markov models, as the state space consists of the cross product of these state variables. For simplicity, we will assume that K although all the results we present can be trivially generalized to the case of differing K (m) . Given that the state space of this factorial HMM consists of all K M combinations of the t variables, placing no constraints on the state transition structure would result in a K M \Theta K M transition matrix. Such an unconstrained system is uninteresting for several reasons: it is equivalent to an HMM with K M states; it is unlikely to discover any interesting structure in the K state variables, as all variables are allowed to interact arbitrarily; and both the time complexity and sample complexity of the estimation algorithm are exponential in M . We therefore focus on factorial HMMs in which the underlying state transitions are constrained. A natural structure to consider is one in which each state variable evolves according to its own dynamics, and is a priori uncoupled from the other state variables: Y A graphical representation for this model is presented in Figure 1 (b). The transition structure for this system can be represented as M distinct K \Theta K matrices. Generalizations that allow coupling between the state variables are briefly discussed in Section 5. As shown in Figure 1 (b), in a factorial HMM the observation at time step t can depend on all the state variables at that time step. For continuous observations, one simple form for this dependence is linear Gaussian; that is, the observation Y t is a Gaussian random vector whose mean is a linear function of the state variables. We represent the state variables as K \Theta 1 vectors, where each of the K discrete values corresponds to a 1 in one position and 0 elsewhere. The resulting probability density for a D \Theta 1 observation vector Y t is ae oe where Each W (m) matrix is a D \Theta K matrix whose columns are the contributions to the means for each of the settings of S (m) , C is the D \Theta D covariance matrix, 0 denotes matrix transpose, and j \Delta j is the matrix determinant operator. One way to understand the observation model in equations (4a) and (4b) is to consider the marginal distribution for Y t , obtained by summing over the possible states. There are K settings for each of the M state variables, and thus there FACTORIAL HIDDEN MARKOV MODELS 249 are K M possible mean vectors obtained by forming sums of M columns where one column is chosen from each of the W (m) matrices. The resulting marginal density of Y t is thus a Gaussian mixture model, with K M Gaussian mixture components each having a constant covariance matrix C. This static mixture model, without inclusion of the time index and the Markov dynamics, is a factorial parameterization of the standard mixture of Gaussians model that has interest in its own right (Zemel, 1993; Hinton & Zemel, 1994; Ghahramani, 1995). The model we are considering in the current paper extends this model by allowing Markov dynamics in the discrete state variables underlying the mixture. Unless otherwise stated, we will assume the Gaussian observation model throughout the paper. The hidden state variables at one time step, although marginally independent, become conditionally dependent given the observation sequence. This can be determined by applying the semantics of directed graphs, in particular the d-separation criterion (Pearl, 1988), to the graphical model in Figure 1 (b). Consider the Gaussian model in equations (4a)-(4b). Given an observation vector Y t , the posterior probability of each of the settings of the hidden state variables is proportional to the probability of Y t under a Gaussian with mean - t . Since - t is a function of all the state variables, the probability of a setting of one of the state variables will depend on the setting of the other state variables. 3 This dependency effectively couples all of the hidden state variables for the purposes of calculating posterior probabilities and makes exact inference intractable for the factorial HMM. 3. Inference and learning The inference problem in a probabilistic graphical model consists of computing the probabilities of the hidden variables given the observations. In the context of speech recognition, for example, the observations may be acoustic vectors and the goal of inference may be to compute the probability for a particular word or sequence of phonemes (the hidden state). This problem can be solved efficiently via the forward-backward algorithm (Rabiner & Juang, 1986), which can be shown to be a special case of the Jensen, Lauritzen, and Olesen (1990) algorithm for probability propagation in more general graphical models (Smyth et al., 1997). In some cases, rather than a probability distribution over hidden states it is desirable to infer the single most probable hidden state sequence. This can be achieved via the Viterbi (1967) algorithm, a form of dynamic programming that is very closely related to the forward-backward algorithm and also has analogues in the graphical model literature (Dawid, 1992). The learning problem for probabilistic models consists of two components: learning the structure of the model and learning its parameters. Structure learning is a topic of current research in both the graphical model and machine learning communities (e.g. Heckerman, 1995; Stolcke & Omohundro, 1993). In the current paper we deal exclusively with the problem of learning the parameters for a given structure. Z. GHAHRAMANI AND M.I. JORDAN 3.1. The EM algorithm The parameters of a factorial HMM can be estimated via the expectation maximization (EM) algorithm (Dempster, Laird, & Rubin, 1977), which in the case of classical HMMs is known as the Baum-Welch algorithm (Baum, Petrie, Soules, & Weiss, 1970). This procedure iterates between a step that fixes the current parameters and computes posterior probabilities over the hidden states (the E step) and a step that uses these probabilities to maximize the expected log likelihood of the observations as a function of the parameters (the M step). Since the E step of EM is exactly the inference problem as described above, we subsume the discussion of both inference and learning problems into our description of the EM algorithm for factorial HMMs. The EM algorithm follows from the definition of the expected log likelihood of the complete (observed and hidden) data: Q(OE new log where Q is a function of the parameters OE new , given the current parameter estimate OE and the observation sequence fY t g. For the factorial HMM the parameters of the model are consists of computing Q. By expanding (5) using equations (1)-(4b), we find that Q can be expressed as a function of three types of expectations over the hidden state variables: hS (m) t i, and t i, where h\Deltai has been used to abbreviate E f\DeltajOE; fY t gg. In the HMM notation of Rabiner and Juang (1986), hS (m) corresponds to fl t , the vector of state occupation probabilities, hS (m) corresponds to - t , the K \Theta K matrix of state occupation probabilities at two consecutive time steps, and hS (m) t i has no analogue when there is only a single underlying Markov model. The M step uses these expectations to maximize Q as a function of OE new . Using Jensen's inequality, Baum, Petrie, Soules & Weiss (1970) showed that each iteration of the E and M steps increases the likelihood, P (fY t gjOE), until convergence to a (local) optimum. As in hidden Markov models, the exact M step for factorial HMMs is simple and tractable. In particular, the M step for the parameters of the output model described in equations (4a)-(4b) can be found by solving a weighted linear regression problem. Similarly, the M steps for the priors, - (m) , and state transition matrices, are identical to the ones used in the Baum-Welch algorithm. The details of the M step are given in Appendix A. We now turn to the substantially more difficult problem of computing the expectations required for the E step. 3.2. Exact inference Unfortunately, the exact E step for factorial HMMs is computationally intractable. This fact can best be shown by making reference to standard algorithms for prob- FACTORIAL HIDDEN MARKOV MODELS 251 abilistic inference in graphical models (Lauritzen & Spiegelhalter, 1988), although it can also be derived readily from direct application of Bayes rule. Consider the computations that are required for calculating posterior probabilities for the factorial HMM shown in Figure 1 (b) within the framework of the Lauritzen and Spiegelhalter algorithm. Moralizing and triangulating the graphical structure for the factorial HMM results in a junction tree (in fact a chain) with cliques of size M+1. The resulting probability propagation algorithm has time complexity O(TMK M+1 ) for a single observation sequence of length T . We present a forward-backward type recursion that implements the exact E step in Appendix B. The naive exact algorithm which consists of translating the factorial HMM into an equivalent HMM with K M states and using the forward-backward algorithm, has time complexity O(TK 2M ). Like other models with multiple densely-connected hidden variables, this exponential time complexity makes exact learning and inference intractable. Thus, although the Markov property can be used to obtain forward-backward- like factorizations of the expectations across time steps, the sum over all possible configurations of the other hidden state variables within each time step is unavoid- able. This intractability is due inherently to the cooperative nature of the model: for the Gaussian output model, for example, the settings of all the state variables at one time step cooperate in determining the mean of the observation vector. 3.3. Inference using Gibbs sampling Rather than computing the exact posterior probabilities, one can approximate them using a Monte Carlo sampling procedure, and thereby avoid the sum over exponentially many state patterns at some cost in accuracy. Although there are many possible sampling schemes (for a review see Neal, 1993), here we present one of the simplest-Gibbs sampling (Geman & Geman, 1984). For a given observation sequence fY t g, this procedure starts with a random setting of the hidden states fS t g. At each step of the sampling process, each state vector is updated stochastically according to its probability distribution conditioned on the setting of all the other state vectors. The graphical model is again useful here, as each node is conditionally independent of all other nodes given its Markov blanket, defined as the set of children, parents, and parents of the children of a node. To sample from a typical state variable S (m) t we only need to examine the states of a few neighboring nodes: t sampled from P (S (m) Sampling once from each of the TM hidden variables in the model results in a new sample of the hidden state of the model and requires O(TMK) operations. The sequence of overall states resulting from each pass of Gibbs sampling defines a Markov chain over the state space of the model. Assuming that all probabilities are bounded away from zero, this Markov chain is guaranteed to converge to the Z. GHAHRAMANI AND M.I. JORDAN posterior probabilities of the states given the observations (Geman & Geman, 1984). Thus, after some suitable time, samples from the Markov chain can be taken as approximate samples from the posterior probabilities. The first and second-order statistics needed to estimate hS (m) are collected using the states visited and the probabilities estimated during this sampling process are used in the approximate E step of EM. 4 3.4. Completely factorized variational inference There also exists a second approximation of the posterior probability of the hidden states that is both tractable and deterministic. The basic idea is to approximate the posterior distribution over the hidden variables P (fS t gjfY t g) by a tractable distribution Q(fS t g). This approximation provides a lower bound on the log likelihood that can be used to obtain an efficient learning algorithm. The argument can be formalized following the reasoning of Saul, Jaakkola, and Jordan (1996). Any distribution over the hidden variables Q(fS t g) can be used to define a lower bound on the log likelihood log log Q(fS t g) log where we have made use of Jensen's inequality in the last step. The difference between the left-hand side and the right-hand side of this inequality is given by the Kullback-Leibler divergence (Cover & Thomas, 1991): Q(fS t g) log The complexity of exact inference in the approximation given by Q is determined by its conditional independence relations, not by its parameters. Thus, we can chose Q to have a tractable structure-a graphical representation that eliminates some of the dependencies in P . Given this structure, we are free to vary the parameters of Q so as to obtain the tightest possible bound by minimizing (6). We will refer to the general strategy of using a parameterized approximating distribution as a variational approximation and refer to the free parameters of the distribution as variational parameters. To illustrate, consider the simplest variational approximation, in which the state variables are assumed independent given the observations This distribution can be written as FACTORIAL HIDDEN MARKOV MODELS 253 S (2) S (2) S (2) S (2) S (2) S (2) (a) (b) Figure 2. (a) The completely factorized variational approximation assumes that all the state variables are independent (conditional on the observation sequence). (b) The structured variational approximation assumes that the state variables retain their Markov structure within each chain, but are independent across chains. Y Y Q(S (m) The variational parameters, t g, are the means of the state variables, where, as before, a state variable S (m) t is represented as a K-dimensional vector with a 1 in the k th position and 0 elsewhere, if the m th Markov chain is in state k at time t. The elements of the vector ' (m) therefore define the state occupation probabilities for the multinomial variable S (m) t under the distribution Q: Q(S (m) Y t;k t;k This completely factorized approximation is often used in statistical physics, where it provides the basis for simple yet powerful mean field approximations to statistical mechanical systems (Parisi, 1988). To make the bound as tight as possible we vary ' separately for each observation sequence so as to minimize the KL divergence. Taking the derivatives of (6) with respect to ' (m) t and setting them to zero, we obtain the set of fixed point equations (see Appendix C) defined by new ae Y (m) oe where ~ Y (m) t is the residual error in Y t given the predictions from all the state variables not including m: ~ Y (m) '6=m Z. GHAHRAMANI AND M.I. JORDAN \Delta (m) is the vector of diagonal elements of W (m) 0 C 'f\Deltag is the softmax operator, which maps a vector A into a vector B of the same size, with elements and log P (m) denotes the elementwise logarithm of the transition matrix P (m) . The first term of (9a) is the projection of the error in reconstructing the observation onto the weights of state vector m-the more a particular setting of a state vector can reduce this error, the larger its associated variational parameter. The second term arises from the fact that the second order correlation hS (m) evaluated under the variational distribution is a diagonal matrix composed of the elements of ' (m) t . The last two terms introduce dependencies forward and backward in time. 5 Therefore, although the posterior distribution over the hidden variables is approximated with a completely factorized distribution, the fixed point equations couple the parameters associated with each node with the parameters of its Markov blanket. In this sense, the fixed point equations propagate information along the same pathways as those defining the exact algorithms for probability propagation. The following may provide an intuitive interpretation of the approximation being made by this distribution. Given a particular observation sequence, the hidden state variables for the M Markov chains at time step t are stochastically coupled. This stochastic coupling is approximated by a system in which the hidden variables are uncorrelated but have coupled means. The variational or "mean-field" equations solve for the deterministic coupling of the means that best approximates the stochastically coupled system. Each hidden state vector is updated in turn using (9a), with a time complexity of O(TMK 2 ) per iteration. Convergence is determined by monitoring the KL divergence in the variational distribution between successive time steps; in practice convergence is very rapid (about 2 to 10 iterations of (9a)). Once the fixed point equations have converged, the expectations required for the E step can be obtained as a simple function of the parameters (equations (C.6)-(C.8) in Appendix C). 3.5. Structured variational inference The approximation presented in the previous section factors the posterior probability such that all the state variables are statistically independent. In contrast to this rather extreme factorization, there exists a third approximation that is both tractable and preserves much of the probabilistic structure of the original system. In this scheme, the factorial HMM is approximated by M uncoupled HMMs as shown in Figure (b). Within each HMM, efficient and exact inference is implemented via the forward-backward algorithm. The approach of exploiting such tractable substructures was first suggested in the machine learning literature by Saul and Jordan (1996). FACTORIAL HIDDEN MARKOV MODELS 255 Note that the arguments presented in the previous section did not hinge on the the form of the approximating distribution. Therefore, more structured variational approximations can be obtained by using more structured variational distributions Q. Each such Q provides a lower bound on the log likelihood and can be used to obtain a learning algorithm. We write the structured variational approximation as Y Q(S (m) Y Q(S (m) where ZQ is a normalization constant ensuring that Q integrates to one and Q(S (m) Y Q(S (m) Y t;k t;k Y t;k Y t;k where the last equality follows from the fact that S (m) is a vector with a 1 in one position and 0 elsewhere. The parameters of this distribution are t gthe original priors and state transition matrices of the factorial HMM and a time-varying bias for each state variable. Comparing equations (11a)-(11c) to equation (1), we can see that the K \Theta 1 vector h (m) t plays the role of the probability of an observation (P (Y t jS t ) in (1)) for each of the K settings of S (m) t . For example, Q(S (m) 1jOE) corresponds to having an observation at time that under state S (m) 1;j . Intuitively, this approximation uncouples the M Markov chains and attaches to each state variable a distinct fictitious observation. The probability of this fictitious observation can be varied so as to minimize the KL divergence between Q and P . Applying the same arguments as before, we obtain a set of fixed point equations for h (m) t that minimize KL(QkP ), as detailed in Appendix D: h (m) new ae Y (m) oe where \Delta (m) is defined as before, and where we redefine the residual error to be ~ Y (m) '6=m Z. GHAHRAMANI AND M.I. JORDAN The parameter h (m) t obtained from these fixed point equations is the observation probability associated with state variable S (m) t in hidden Markov model m. Using these probabilities, the forward-backward algorithm is used to compute a new set of expectations for hS (m) t i, which are fed back into (12a) and (12b). This algorithm is therefore used as a subroutine in the minimization of the KL divergence. Note the similarity between equations (12a)-(12b) and equations (9a)-(9b) for the completely factorized system. In the completely factorized system, since hS (m) t , the fixed point equations can be written explicitly in terms of the variational parameters. In the structured approximation, the dependence of hS (m) t i on h (m) is computed via the forward-backward algorithm. Note also that (12a) does not contain terms involving the prior, - (m) , or transition matrix, P (m) . These terms have cancelled by our choice of approximation. 3.6. Choice of approximation The theory of the EM algorithm as presented in Dempster et al. (1977) assumes the use of an exact E step. For models in which the exact E step is intractable, one must instead use an approximation like those we have just described. The choice among these approximations must take into account several theoretical and practical issues. Monte Carlo approximations based on Markov chains, such as Gibbs sampling, offer the theoretical assurance that the sampling procedure will converge to the correct posterior distribution in the limit. Although this means that one can come arbitrarily close to the exact E step, in practice convergence can be slow (especially for multimodal distributions) and it is often very difficult to determine how close one is to convergence. However, when sampling is used for the E step of EM, there is a time tradeoff between the number of samples used and the number of EM iterations. It seems wasteful to wait until convergence early on in learning, when the posterior distribution from which samples are drawn is far from the posterior given the optimal parameters. In practice we have found that even approximate steps using very few Gibbs samples (e.g. around ten samples of each hidden variable) tend to increase the true likelihood. Variational approximations offer the theoretical assurance that a lower bound on the likelihood is being maximized. Both the minimization of the KL divergence in the E step and the parameter update in the M step are guaranteed not to decrease this lower bound, and therefore convergence can be defined in terms of the bound. An alternative view given by Neal and Hinton (1993) describes EM in terms of the negative free energy, F , which is a function of the parameters, OE, the observations, Y , and a posterior probability distribution over the hidden variables, Q(S): where EQ denotes expectation over S using the distribution Q(S). The exact E step in EM maximizes F with respect to Q given OE. The variational E steps used FACTORIAL HIDDEN MARKOV MODELS 257 here maximize F with respect to Q given OE, subject to the constraint that Q is of a particular tractable form. Given this view, it seems clear that the structured approximation is preferable to the completely factorized approximation since it places fewer constraints on Q, at no cost in tractability. 4. Experimental results To investigate learning and inference in factorial HMMs we conducted two experi- ments. The first experiment compared the different approximate and exact methods of inference on the basis of computation time and the likelihood of the model obtained from synthetic data. The second experiment sought to determine whether the decomposition of the state space in factorial HMMs presents any advantage in modeling a real time series data set that we might assume to have complex internal structure-Bach's chorale melodies. 4.1. Experiment 1: Performance and timing benchmarks Using data generated from a factorial HMM structure with M underlying Markov models with K states each, we compared the time per EM iteration and the training and test set likelihoods of five models: ffl HMM trained using the Baum-Welch algorithm; ffl Factorial HMM trained with exact inference for the E step, using a straight-forward application of the forward-backward algorithm, rather than the more efficient algorithm outlined in Appendix B; ffl Factorial HMM trained using Gibbs sampling for the E step with the number of samples fixed at 10 samples per variable; 6 ffl Factorial HMM trained using the completely factorized variational approxima- tion; and ffl Factorial HMM trained using the structured variational approximation. All factorial HMMs consisted of M underlying Markov models with K states each, whereas the HMM had K M states. The data were generated from a factorial HMM structure with M state variables, each of which could take on K discrete values. All of the parameters of this model, except for the output covariance matrix, were sampled from a uniform [0; 1] distribution and appropriately normalized to satisfy the sum-to-one constraints of the transition matrices and priors. The covariance matrix was set to a multiple of the identity matrix The training and test sets consisted of 20 sequences of length 20, where the observable was a four-dimensional vector. For each randomly sampled set of parameters, a separate training set and test set were generated and each algorithm was run once. Z. GHAHRAMANI AND M.I. JORDAN Fifteen sets of parameters were generated for each of the four problem sizes. Algorithms were run for a maximumof 100 iterations of EM or until convergence, defined as the iteration k at which the log likelihood L(k), or approximate log likelihood if an approximate algorithm was used, satisfied At the end of learning, the log likelihoods on the training and test set were computed for all models using the exact algorithm. Also included in the comparison was the log likelihood of the training and test sets under the true model that generated the data. The test set log likelihood for N observation sequences is defined as log P (Y (n) obtained by maximizing the log likelihood over a training set that is disjoint from the test set. This provides a measure of how well the model generalizes to a novel observation sequence from the same distribution as the training data. Results averaged over 15 runs for each algorithm on each of the four problem sizes (a total of 300 runs) are presented in Table 1. Even for the smallest problem size the standard HMM with K M states suffers from overfitting: the test set log likelihood is significantly worse than the training set log likelihood. As expected, this overfitting problem becomes worse as the size of the state space increases; it is particularly serious for For the factorial HMMs, the log likelihoods for each of the three approximate EM algorithms were compared to the exact algorithm. Gibbs sampling appeared to have the poorest performance: for each of the three smaller size problems its log likelihood was significantly worse than that of the exact algorithm on both the training sets and test sets (p ! 0:05). This may be due to insufficient sampling. However, we will soon see that running the Gibbs sampler for more than 10 samples, while potentially improving performance, makes it substantially slower than the variational methods. Surprisingly, the Gibbs sampler appears to do quite well on the largest size problem, although the differences to the other methods are not statistically significant. The performance of the completely factorized variational approximation was not statistically significantly different from that of the exact algorithm on either the training set or the test set for any of the problem sizes. The performance of the structured variational approximation was not statistically different from that of the exact method on three of the four problem sizes, and appeared to be better on one of the problem sizes 0:05). Although this result may be a fluke arising from random variability, there is another more interesting (and speculative) explanation. The exact EM algorithm implements unconstrained maximization of F , as defined in section 3.6, while the variational methods maximize F subject to a constrained distribution. These constraints could presumably act as regularizers, reducing overfitting. There was a large amount of variability in the final log likelihoods for the models learned by all the algorithms. We subtracted the log likelihood of the true generative model from that of each trained model to eliminate the main effect of the randomly sampled generative model and to reduce the variability due to training and test sets. One important remaining source of variance was the random seed used in FACTORIAL HIDDEN MARKOV MODELS 259 Table 1. Comparison of the factorial HMM on four problems of varying size. The negative log likelihood for the training and test set, plus or minus one standard deviation, is shown for each problem size and algorithm, measured in bits per observation (log likelihood in bits divided by NT ) relative to the log likelihood under the true generative model for that data set. 7 True is the true generative model (the log likelihood per symbol is defined to be zero for this model by our measure); HMM is the hidden Markov model with K M states; Exact is the factorial HMM trained using an exact E step; Gibbs is the factorial HMM trained using Gibbs sampling; CFVA is the factorial HMM trained using the completely factorized variational approximation; SVA is the factorial HMM trained using the structured variational approximation. M K Algorithm Training Test HMM 1.19 \Sigma 0.67 2.29 \Sigma 1.02 Exact 0.88 \Sigma 0.80 1.05 \Sigma 0.72 Gibbs 1.67 \Sigma 1.23 1.78 \Sigma 1.22 CFVA 1.06 \Sigma 1.20 1.20 \Sigma 1.11 SVA 0.91 \Sigma 1.02 1.04 \Sigma 1.01 HMM 0.76 \Sigma 0.67 9.81 \Sigma 2.55 Exact 1.02 \Sigma 1.04 1.26 \Sigma 0.99 Gibbs 2.21 \Sigma 0.91 2.50 \Sigma 0.87 CFVA 1.24 \Sigma 1.50 1.50 \Sigma 1.53 Exact 2.29 \Sigma 1.19 2.51 \Sigma 1.21 Gibbs 3.25 \Sigma 1.17 3.35 \Sigma 1.14 CFVA 1.73 \Sigma 1.34 2.07 \Sigma 1.74 Exact 4.23 \Sigma 2.28 4.49 \Sigma 2.24 Gibbs 3.63 \Sigma 1.13 3.95 \Sigma 1.14 CFVA 4.85 \Sigma 0.68 5.14 \Sigma 0.69 260 Z. GHAHRAMANI AND M.I. JORDAN iterations of EM -log likelihood (bits) (a) iterations of EM -log likelihood (bits) (b) iterations of EM -log likelihood (bits) (c) iterations of EM -log likelihood (bits) (d) Figure 3. Learning curves for five runs of each of the four learning algorithms for factorial HMMs: (a) exact; (b) completely factorized variational approximation; (c) structured variational approx- imation; and (d) Gibbs sampling. A single training set sampled from the size was used for all these runs. The solid lines show the negative log likelihood per observation (in bits) relative to the true model that generated the data, calculated using the exact algorithm. The circles denote the point at which the convergence criterion was met and the run ended. For the three approximate algorithms, the dashed lines show an approximate negative log likelihood. 8 each training run, which determined the initial parameters and the samples used in the Gibbs algorithm. All algorithms appeared to be very sensitive to this random seed, suggesting that different runs on each training set found different local maxima or plateaus of the likelihood (Figure 3). Some of this variability could be eliminated by explicitly adding a regularization term, which can be viewed as a prior on the parameters in maximuma posteriori parameter estimation. Alternatively, Bayesian (or ensemble) methods could be used to average out this variability by integrating over the parameter space. The timing comparisons confirm the fact that both the standard HMM and the exact are extremely slow for models with large state spaces (Fig- FACTORIAL HIDDEN MARKOV MODELS 261 Time/iteration HMM Figure 4. Time per iteration of EM on a Silicon Graphics R4400 processor running Matlab. ure 4). Gibbs sampling was slower than the variational methods even when limited to ten samples of each hidden variable per iteration of EM. Since one pass of the variational fixed point equations has the same time complexity as one pass of Gibbs sampling, and since the variational fixed point equations were found to converge very quickly, these experiments suggest that Gibbs sampling is not as competitive time-wise as the variational methods. The time per iteration for the variational methods scaled well to large state spaces. 4.2. Experiment 2: Bach chorales Musical pieces naturally exhibit complex structure at many different time scales. Furthermore, one can imagine that to represent the "state" of the musical piece at any given time it would be necessary to specify a conjunction of many different features. For these reasons, we chose to test whether a factorial HMM would provide an advantage over a regular HMM in modeling a collection of musical pieces. The data set consisted of discrete event sequences encoding the melody lines of J. S. Bach's Chorales, obtained from the UCI Repository for Machine Learning originally discussed in Conklin and Witten (1995). Each event in the sequence was represented by six attributes, described in Table 2. Sixty-six chorales, with 40 or more events each, were divided into a training set (30 chorales) and a test set (36 chorales). Using the first set, hidden Markov models with state space ranging from 2 to 100 states were trained until convergence (30 \Sigma 12 steps of EM). Factorial HMMs of varying sizes (K ranging from 2 to 6; M ranging from 2 to were also trained on the same data. To 262 Z. GHAHRAMANI AND M.I. JORDAN Table 2. Attributes in the Bach chorale data set. The key signature and time signature attributes were constant over the duration of the chorale. All attributes were treated as real numbers and modeled as linear-Gaussian observations (4a). Attribute Description Representation pitch pitch of the event int [0; 127] fermata event under fermata? binary st start time of event int (1/16 notes) dur duration of event int (1/16 notes) approximate the E step for factorial HMMs we used the structured variational ap- proximation. This choice was motivated by three considerations. First, for the size of state space we wished to explore, the exact algorithms were prohibitively slow. Second, the Gibbs sampling algorithm did not appear to present any advantages in speed or performance and required some heuristic method for determining the number of samples. Third, theoretical arguments suggest that the structured approximation should in general be superior to the completely factorized variational approximation, since more of the dependencies of the original model are preserved. The test set log likelihoods for the HMMs, shown in Figure 5 (a), exhibited the typical U-shaped curve demonstrating a trade-off between bias and variance (Ge- man, Bienenstock, & Doursat, 1992). HMMs with fewer than 10 states did not predict well, while HMMs with more than 40 states overfit the training data and therefore provided a poor model of the test data. Out of the 75 runs, the highest test set log likelihood per observation was \Gamma9:0 bits, obtained by an HMM with hidden states. 9 The factorial HMM provides a more satisfactory model of the chorales from three points of view. First, the time complexity is such that it is possible to consider models with significantly larger state spaces; in particular, we fit models with up to 1000 states. Second, given the componential parametrization of the factorial HMM, these large state spaces do not require excessively large numbers of parameters relative to the number of data points. In particular, we saw no evidence of overfitting even for the largest factorial HMM as seen in Figures 5 (c) & (d). Finally, this approach resulted in significantly better predictors; the test set likelihood for the best factorial HMM was an order of magnitude larger than the test set likelihood for the best HMM, as Figure 5 (d) reveals. While the factorial HMM is clearly a better predictor than a single HMM, it should be acknowledged that neither approach produces models that are easily interpretable from a musicological point of view. The situation is reminiscent of that in speech recognition, where HMMs have proved their value as predictive models of the speech signal without necessarily being viewed as causal generative models of speech. A factorial HMM is clearly an impoverished representation of FACTORIAL HIDDEN MARKOV MODELS 263 log likelihood (bits) (a) (b) Size of state space (c) (d) Figure 5. Test set log likelihood per event of the Bach chorale data set as a function of number of states for (a) HMMs, and factorial HMMs with (b) dashed line) and line). Each symbol represents a single run; the lines indicate the mean performances. The thin dashed line in (b)-(d) indicates the log likelihood per observation of the best run in (a). The factorial HMMs were trained using the structured approximation. For both methods the true likelihood was computed using the exact algorithm. musical structure, but its promising performance as a predictor provides hope that it could serve as a step on the way toward increasingly structured statistical models for music and other complex multivariate time series. 5. Generalizations of the model In this section, we describe four variations and generalizations of the factorial HMM. 5.1. Discrete observables The probabilistic model presented in this paper has assumed real-valued Gaussian observations. One of the advantages arising from this assumption is that the conditional density of a D-dimensional observation, P (Y t jS (1) t ), can be compactly specified through M mean matrices of dimension D \Theta K, and one D \Theta D covariance matrix. Furthermore, the M step for such a model reduces to a set of weighted least squares equations. The model can be generalized to handle discrete observations in several ways. Consider a single D-valued discrete observation Y t . In analogy to traditional HMMs, the output probabilities could be modeled using a matrix. However, in the case of a factorial HMM, this matrix would have D \Theta K M entries for each combination of the state variables and observation. Thus the compactness of the representation would be entirely lost. Standard methods from graphical models suggest approximating such large matrices with "noisy-OR" (Pearl, 1988) or "sigmoid" (Neal, 1992) models of interaction. For example, in the softmax model, which generalizes the sigmoid model to D ? 2, P (Y t jS (1) t ) is multinomial with mean proportional to 264 Z. GHAHRAMANI AND M.I. JORDAN exp . Like the Gaussian model, this specification is again com- pact, using M matrices of size D \Theta K. (As in the linear-Gaussian model, the W (m) are overparametrized since they can each model the overall mean of Y t , as shown in Appendix A.) While the nonlinearity induced by the softmax function makes both the E step and M step of the algorithm more difficult, iterative numerical methods can be used in the M step whereas Gibbs sampling and variational methods can again be used in the E step (see Neal, 1992; Hinton et al., 1995; and Saul et al., 1996, for discussions of different approaches to learning in sigmoid networks). 5.2. Introducing couplings The architecture for factorial HMMs presented in Section 2 assumes that the underlying Markov chains interact only through the observations. This constraint can be relaxed by introducing couplings between the hidden state variables (cf. Saul & Jordan, 1997). For example, if S (m) t depends on S (m) equation (3) is replaced by the following factorization Y Similar exact, variational, and Gibbs sampling procedures can be defined for this architecture. However, note that these couplings must be introduced with caution, as they may result in an exponential growth in parameters. For example, the above factorization requires transition matrices of size K 2 \Theta K. Rather than specifying these higher-order couplings through probability transition matrices, one can introduce second-order interaction terms in the energy (log probability) function. Such terms effectively couple the chains without the number of parameters incurred by a full probability transition matrix. In the graphical model formalism these correspond to symmetric undirected links, making the model a chain graph. While the Jensen, Lauritzen and Olesen (1990) algorithm can still be used to propagate information exactly in chain graphs, such undirected links cause the normalization constant of the probability distribution-the partition function-to depend on the coupling parameters. As in Boltzmann machines (Hinton & Sejnowski, 1986), both a clamped and an unclamped phase are therefore required for learning, where the goal of the unclamped phase is to compute the derivative of the partition function with respect to the parameters (Neal, 1992). 5.3. Conditioning on inputs Like the hidden Markov model, the factorial HMM provides a model of the unconditional density of the observation sequences. In certain problem domains, some of the observations can be better thought of as inputs or explanatory variables, and FACTORIAL HIDDEN MARKOV MODELS 265 the others as outputs or response variables. The goal, in these cases, is to model the conditional density of the output sequence given the input sequence. In machine learning terminology, unconditional density estimation is unsupervised while conditional density estimation is supervised. Several algorithms for learning in hidden Markov models that are conditioned on inputs have been recently presented in the literature (Cacciatore & Nowlan, 1994; Bengio & Frasconi, 1995; Meila & Jordan, 1996). Given a sequence of input vectors g, the probabilistic model for an input-conditioned factorial HMM is Y \Theta Y Y The model depends on the specification of P (Y t jS (m) which are conditioned both on a discrete state variable and on a (possibly con- tinuous) input vector. The approach used in Bengio and Frasconi's Input Output HMMs (IOHMMs) suggests modeling P (S (m) separate neural networks, one for each setting of S (m) . This decomposition ensures that a valid probability transition matrix is defined at each point in input space if a sum-to-one constraint (e.g., softmax nonlinearity) is used in the output of these networks. Using the decomposition of each conditional probability into K networks, inference in input-conditioned factorial HMMs is a straightforward generalization of the algorithms we have presented for factorial HMMs. The exact forward-backward algorithm in Appendix B can be adapted by using the appropriate conditional probabilities. Similarly, the Gibbs sampling procedure is no more complex when conditioned on inputs. Finally, the completely factorized and structured approximations can also be generalized readily if the approximating distribution has a dependence on the input similar to the model's. If the probability transition structure not decomposed as above, but has a complex dependence on the previous state variable and input, inference may become considerably more complex. Depending on the form of the input conditioning, the Maximization step of learning may also change considerably. In general, if the output and transition probabilities are modeled as neural networks, the M step can no longer be solved exactly and a gradient-based generalized EM algorithm must be used. For log-linear models, the M step can be solved using an inner loop of iteratively reweighted least-squares (McCullagh & Nelder, 1989). 5.4. Hidden Markov decision trees An interesting generalization of factorial HMMs results if one conditions on an input X t and orders the M state variables such that S (m) t depends on S (n) t for 266 Z. GHAHRAMANI AND M.I. JORDAN S (2) S (2) S (2) Figure 6. The hidden Markov decision tree. Figure 6). The resulting architecture can be seen as a probabilistic decision tree with Markovian dynamics linking the decision variables. Consider how this probabilistic model would generate data at the first time step, Given the top node S (1) can take on K values. This stochastically partitions X space into K decision regions. The next node down the hierarchy, S (2), subdivides each of these regions into K subregions, and so on. The output Y 1 is generated from the input X 1 and the K-way decisions at each of the M hidden nodes. At the next time step, a similar procedure is used to generate data from the model, except that now each decision in the tree is dependent on the decision taken at that node in the previous time step. Thus, the "hierarchical mixture of experts" architecture (Jordan Jacobs, 1994) is generalized to include Markovian dynamics for the decisions. Hidden Markov decision trees provide a useful starting point for modeling time series with both temporal and spatial structure at multiple resolutions. We explore this generalization of factorial HMMs in Jordan, Ghahramani, and Saul (1997). 6. Conclusion In this paper we have examined the problem of learning for a class of generalized hidden Markov models with distributed state representations. This generalization provides both a richer modeling tool and a method for incorporating prior structural information about the state variables underlying the dynamics of the system generating the data. Although exact inference in this class of models is generally intractable, we provided a structured variational approximation that can be computed tractably. This approximation forms the basis of the Expectation step in an EM algorithm for learning the parameters of the model. Empirical comparisons to several other approximations and to the exact algorithm show that this approximation is both efficient to compute and accurate. Finally, we have shown that FACTORIAL HIDDEN MARKOV MODELS 267 the factorial HMM representation provides an advantage over traditional HMMs in predictive modeling of the complex temporal patterns in Bach's chorales. Appendix A The M step The M step equations for each parameter are obtained by setting the derivatives of Q with respect to that parameters to zero. We start by expanding Q using equations (1)-(4b): tr C tr (log P (m) )hS (m) log Z; (A.1) where tr is the trace operator for square matrices and Z is a normalization term independent of the states and observations ensuring that the probabilities sum to one. Setting the derivatives of Q with respect to the output weights to zero, we obtain a linear system of equations for the W 0: (A.2) Assuming Y t is a D\Theta1 vector, let S t be the MK \Theta1 vector obtained by concatenating the S (m) vectors, and W be the D \Theta MK matrix obtained by concatenating the W (m) matrices (of size D \Theta K). Then solving (A.2) results in where y is the Moore-Penrose pseudo-inverse. Note that the model is overparameterized since the D \Theta 1 means of each of the W (m) matrices add up to a single mean. Using the pseudo-inverse removes the need to explicitly subtract this overall mean from each W (m) and estimate it separately as another parameter. To estimate the priors, we solve @Q=@- subject to the constraint that they sum to one, obtaining 268 Z. GHAHRAMANI AND M.I. JORDAN Similarly, to estimate the transition matrices we solve @Q=@P subject to the constraint that the columns of P (m) sum to one. For element (i; new Finally, the re-estimation equations for the covariance matrix can be derived by taking derivatives with respect to C \Gamma1 The first term arises from the normalization for the Gaussian density function: Z is proportional to jCj T=2 and @jCj=@C Substituting (A.2) and re-organizing we get For equations reduce to the Baum-Welch re-estimation equations for HMMs with Gaussian observables. The above M step has been presented for the case of a single observation sequence. The extension to multiple sequences is straightforward. Appendix Exact forward-backward algorithm Here we specify an exact forward-backward recursion for computing the posterior probabilities of the hidden states in a factorial HMM. It differs from a straightforward application of the forward-backward algorithm on the equivalent K M state HMM, in that it does not depend on a K M \Theta K M transition matrix. Rather, it makes use of the independence of the underlying Markov chains to sum over M transition matrices of size K \Theta K. Using the notation fY - g r t to mean the observation sequence Y ff (1) ff (M) FACTORIAL HIDDEN MARKOV MODELS 269 Then we obtain the forward recursions and At the end of the forward recursions, the likelihood of the observation sequence is the sum of the K M elements in ff T . Similarly, to obtain the backward recursions we define from which we obtain The posterior probability of the state at time t is obtained by multiplying ff t and This algorithm can be shown to be equivalent to the Jensen, Lauritzen and Olesen algorithm for probability propagation in graphical models. The probabilities are defined over collections of state variables corresponding to the cliques in the equivalent junction tree. Information is passed forwards and backwards by summing over the sets separating each neighboring clique in the tree. This results in forward-backward-type recursions of order O(TMK M+1 ). Using the ff t , fi t , and fl t quantities, the statistics required for the E step are Z. GHAHRAMANI AND M.I. JORDAN Appendix Completely factorized variational approximation Using the definition of the probabilistic model given by equations (1)-(4b), the posterior probability of the states given an observation sequence can be written as Z where Z is a normalization constant ensuring that the probabilities sum to one and Similarly, the probability distribution given by the variational approximation (7)- (8) can be written as expf\GammaH where log ' (m) Using this notation, and denoting expectation with respect to the variational distribution using angular brackets h\Deltai, the KL divergence is Three facts can be verified from the definition of the variational approximation: diagf' (m) FACTORIAL HIDDEN MARKOV MODELS 271 where diag is an operator that takes a vector and returns a square matrix with the elements of the vector along its diagonal, and zeros everywhere else. The KL divergence can therefore be expanded to log ' (m) C tr C trf' (m) log P (m) Taking derivatives with respect to ' (m) t , we obtain log ' (m) C \Gamma(log where \Delta (m) is the vector of diagonal elements of W (m) 0 C c is a term arising from log ZQ , ensuring that the ' (m) t sum to one. Setting this derivative equal to 0 and solving for ' (m) t gives equation (9a). Appendix Structured approximation For the structured approximation, HQ is defined as log h (m) Using (C.2), we write the KL divergence as tr C tr C diag log Z: (D.2) Z. GHAHRAMANI AND M.I. JORDAN Since KL is independent of - (m) and P (m) , the first thing to note is that these parameters of the structured approximation remain equal to the equivalent parameters of the true system. Now, taking derivatives with respect to log h (n) - , we get @ log h (n) log h (m) '6=m C @ log h (n) The last term, which we obtained by making use of the fact that @ log ZQ @ log h (n) cancels out the first term. Setting the terms inside the brackets in (D.3) equal to zero yields equation (12a). Acknowledgments We thank Lawrence Saul for helpful discussions and Geoffrey Hinton for support. This project was supported in part by a grant from the McDonnell-Pew Foundation, by a grant from ATR Human Information Processing Research Laboratories, by a gift from Siemens Corporation, and by grant N00014-94-1-0777 from the Office of Naval Research. Zoubin Ghahramani was supported by a grant from the Ontario Information Technology Research Centre. Notes 1. For related work on inference in distributed state HMMs, see Dean and Kanazawa (1989). 2. In speech, neural networks are generally used to model P (S t jY t ); this probability is converted to the observation probabilities needed in the HMM via Bayes rule. 3. If the columns of W (m) and W (n) are orthogonal for every pair of state variables, m and n, and C is a diagonal covariance matrix, then the state variables will no longer be dependent given the observation. In this case there is no "explaining away": each state variable is modeling the variability in the observation along a different subspace. 4. A more Bayesian treatment of the learning problem, in which the parameters are also considered hidden random variables, can be handled by Gibbs sampling by replacing the "M step" with sampling from the conditional distribution of the parameters given the other hidden variables (for example, see Tanner and Wong, 1987). 5. The first term is replaced by log - (m) for the second term does not appear for 6. All samples were used for learning; that is, no samples were discarded at the beginning of the run. Although ten samples is too few to even approach convergence, it provides a run-time roughly comparable to the variational methods. The goal was to see whether this "impatient" Gibbs sampler would be able to compete with the other approximate methods. FACTORIAL HIDDEN MARKOV MODELS 273 7. Lower values suggest a better probabilistic model: a value of one, for example, means that it would take one bit more than the true generative model to code each observation vector. Standard deviations reflect the variation due to training set, test set, and the random seed of the algorithm. Standard errors on the mean are a factor of 3.8 smaller. 8. For the variational methods these dashed lines are equal to minus the lower bound on the log likelihood, except for a normalization term which is intractable to compute and can vary during learning, resulting in the apparent occasional increases in the bound. 9. Since the attributes were modeled as real numbers, the log likelihoods are only a measure of relative coding cost. 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Iwan Santoso, Efficient reasoning, ACM Computing Surveys (CSUR), v.33 n.1, p.1-30, March 2001

mean field theory;bayesian networks;hidden markov models;graphical models;time series;EM algorithm

274791

Approximate graph coloring by semidefinite programming.

We consider the problem of coloring k-colorable graphs with the fewest possible colors. We present a randomized polynomial time algorithm that colors a 3-colorable graph on n vertices with min{O(&Dgr;1/3 log1/2 &Dgr; log n), O(n1/4 log1/2 n)} colors where &Dgr; is the maximum degree of any vertex. Besides giving the best known approximation ratio in terms of n, this marks the first nontrivial approximation result as a function of the maximum degree &Dgr;. This result can be generalized to k-colorable graphs to obtain a coloring using min{O(&Dgr;1-2/k log1/2 &Dgr; log n), O(n13/(k+1) log1/2 n)} colors. Our results are inspired by the recent work of Goemans and Williamson who used an algorithm for semidefinite optimization problems, which generalize linear programs, to obtain improved approximations for the MAX CUT and MAX 2-SAT problems. An intriguing outcome of our work is a duality relationship established between the value of the optimum solution to our semidefinite program and the Lovsz &thgr;-function. We show lower bounds on the gap between the optimum solution of our semidefinite program and the actual chromatic by duality this also demonstrates interesting new facts about the &thgr;-function.

Introduction A legal vertex coloring of a graph G(V; E) is an assignment of colors to its vertices such that no two adjacent vertices receive the same color. Equivalently, a legal coloring of G by k colors is a partition of its vertices into k independent sets. The minimum number of colors needed for such a coloring is called the chromatic number of G, and is usually denoted by -(G). Determining the chromatic number of a graph is known to be NP-hard (cf. [20]). Besides its theoretical significance as a canonical NP-hard problem, graph coloring arises naturally in a variety of applications such as register allocation [11, 12, 13] and timetable/examination scheduling [8, 43]. In many applications which can be formulated as graph coloring problems, it suffices to find an approximately optimum graph coloring-a coloring of the graph with a small though non-optimum number of colors. This along with the apparent impossibility of an exact solution has led to some interest in the problem of approximate graph coloring. The analysis of approximation algorithms for graph coloring started with the work of Johnson [27] who shows that a version of the greedy algorithm gives an O(n= log n)-approximation algorithm for k-coloring. improved this bound by giving an elegant algorithm which uses O(n 1\Gamma1=(k\Gamma1) ) colors to legally color a k-colorable graph. Subsequently, other polynomial time algorithms were provided by Blum [9] which use O(n 3=8 log 8=5 n) colors to legally color an n-vertex 3-colorable graph. This result generalizes to coloring a k-colorable graph with O(n 1\Gamma1=(k\Gamma4=3) log 8=5 n) colors. The best known performance guarantee for general graphs is due to Halld'orsson [25] who provided a polynomial time algorithm using a number of colors which is within a factor of O(n(log log n) of the optimum. Recent results in the hardness of approximations indicate that it may be not possible to substantially improve the results described above. Lund and Yannakakis [34] used the results of Arora, Lund, Motwani, Sudan, and Szegedy [6] and Feige, Goldwasser, Lov'asz, Safra, and Szegedy [17] to show that there exists a (small) constant ffl ? 0 such that no polynomial time algorithm can approximate the chromatic number of a graph to within a ratio of n ffl unless NP. The current hardness result for the approximation of the chromatic number is due to Feige and Kilian [18] and H-astad [26], who show that approximating it to within n 1\Gammaffi , for any ffi ? 0, would imply NP=RP (RP is the class of probabilistic polynomial time algorithms making one-sided error). However, none of these hardness results apply to the special case of the problem where the input graph is guaranteed to be k-colorable for some small k. The best hardness result in this direction is due to Khanna, Linial, and Safra [28] who show that it is not possible to color a 3-colorable graph with 4 colors in polynomial time unless In this paper we present improvements on the result of Blum. In particular, we provide a randomized polynomial time algorithm which colors a 3-colorable graph of maximum degree \Delta with log n); O(n 1=4 log 1=2 n)g colors; moreover, this can be generalized to k- colorable graphs to obtain a coloring using O(\Delta 1\Gamma2=k log 1=2 \Delta log n) or O(n 1\Gamma3=(k+1) log 1=2 n) colors. Besides giving the best known approximations in terms of n, our results are the first non-trivial approximations given in terms of \Delta. Our results are based on the recent work of Goemans and used an algorithm for semidefinite optimization problems (cf. [23, 2]) to obtain improved approximations for the MAX CUT and MAX 2-SAT problems. We follow their basic paradigm of using algorithms for semidefinite programming to obtain an optimum solution to a relaxed version of the problem, and a randomized strategy for "rounding" this solution to a feasible but approximate solution to the original problem. Motwani and Naor [37] have shown that the approximate graph coloring problem is closely related to the problem of finding a CUT COVER of the edges of a graph. Our results can be viewed as generalizing the MAX CUT approximation algorithm of Goemans and Williamson to the problem of finding an approximate CUT COVER. In our techniques also lead to improved approximations for the MAX k-CUT problem [19]. We also establish a duality relationship between the value of the optimum solution to our semidefinite program and the Lov'asz #-function [23, 24, 33]. We show lower bounds on the gap between the optimum solution of our semidefinite program and the actual chromatic by duality this also demonstrates interesting new facts about the #-function. Alon and Kahale [4] use related techniques to devise a polynomial time algorithm for 3-coloring random graphs drawn from a "hard" distribution on the space of all 3-colorable graphs. Recently, Frieze and Jerrum [19] have used a semidefinite programming formulation and randomized rounding strategy essentially the same as ours to obtain improved approximations for the MAX k-CUT problem with large values of k. Their results required a more sophisticated version of our analysis, but for the coloring problem our results are tight up to poly-logarithmic factors and their analysis does not help to improve our bounds. Semidefinite programming relaxations are an extension of the linear programming relaxation approach to approximately solving NP-complete problems. We thus present our work in the style of the classical LP-relaxation approach. We begin in Section 2 by defining a relaxed version of the coloring problem. Since we use a more complex relaxation than standard linear programming, we must show that the relaxed problem can be solved; this is done in Section 3. We then show relationships between the relaxation and the original problem. In Section 4, we show that (in a sense to be defined later) the value of the relaxation bounds the value of the original problem. Then, in Sections 5, 6, and 7, we show how a solution to the relaxation can be "rounded" to make it a solution to the original problem. Combining the last two arguments shows that we can find a good approximation. Section 3, Section 4, and Sections 5-7 are in fact independent and can be read in any order after the definitions in Section 2. In Section 8, we investigate the relationship between our fractional relaxations and the Lov'asz #-function, showing that they are in fact dual to one another. We investigate the approximation error inherent in our formulation of the chromatic number via semi-definite programming in Section 9. Vector Relaxation of Coloring In this section, we describe the relaxed coloring problem whose solution is in turn used to approximate the solution to the coloring problem. Instead of assigning colors to the vertices of a graph, we consider assigning (n-dimensional) unit vectors to the vertices. To capture the property of a coloring, we aim for the vectors of adjacent vertices to be "different" in a natural way. The vector k-coloring that we define plays the role that a hypothetical "fractional k-coloring" would play in a classical linear-programming relaxation approach to the problem. Our relaxation is related to the concept of an orthonormal representation of a graph [33, 23]. Definition 2.1 Given a graph E) on n vertices, and a real number k - 1, a vector k-coloring of G is an assignment of unit vectors u i from the space ! n to each vertex , such that for any two adjacent vertices i and j the dot product of their vectors satisfies the inequality The definition of an orthonormal representation [33, 23] requires that the given dot products be equal to zero, a weaker requirement than the one above. 3 Solving the Vector Coloring Problem In this section we show how the vector coloring relaxation can be solved using semidefinite pro- gramming. The methods in this section closely mimic those of Goemans and Williamson [21]. To solve the problem, we need the following auxiliary definition. Definition 3.1 Given a graph E) on n vertices, a matrix k-coloring of the graph is an n \Theta n symmetric positive semidefinite matrix M , with m We now observe that matrix and vector k-colorings are in fact equivalent (cf. [21]). Thus, to solve the vector coloring relaxation it will suffice to find a matrix k-coloring. Fact 3.1 A graph has a vector k-coloring if and only if it has matrix k-coloring. Moreover, a vector ffl)-coloring can be constructed from a matrix k-coloring in time polynomial in n and log(1=ffl). Note that an exact solution cannot be found, as some of the values in it may be irrational. Proof: Given a vector k-coloring fv i g, the matrix k-coloring is defined For the other direction, it is well known that for every symmetric positive definite matrix M there exists a square matrix U such that UU is the transpose of U ). The rows of U are vectors that form a vector k-coloring of G. An ffi-close approximation to the matrix U can be found in time polynomial in n and log(1=ffi) can be found using the Incomplete Cholesky Decomposition [21, 22]. (Here by ffi-close we mean a matrix U 0 such that U 0 U has L1 norm less than ffi .) This in turn gives a vector coloring of the graph, provided ffi is chosen appropriately. Lemma 3.2 If a graph G has a vector k-coloring then a vector ffl)-coloring of the graph can be constructed in time polynomial in k, n, and log(1=ffl). Proof: Our proof is similar to those of Lov'asz [33] and Goemans-Williamson [21]. We construct a semidefinite optimization problem (SDP) whose optimum is \Gamma1=(k \Gamma 1) when k is the smallest real number such that a matrix k-coloring of G exists. The optimum solution also provides a matrix k-coloring of G. minimize ff is positive semidefinite subject to Consider a graph which has a vector (and matrix) k-coloring. This means there is a solution to the above semidefinite program with 1). The ellipsoid method or other interior point based methods [23, 2] can be employed to find a feasible solution where the value of the objective is at most \Gamma1=(k \Gamma 1)+ ffi in time polynomial in n and log 1=ffi. This implies that for all fi; jg 2 at most which is at most \Gamma1=(k Thus a matrix ffl)-coloring can be found in time polynomial in k, n and log(1=ffl). From the matrix coloring, the vector coloring can be found in polynomial time as was noted in the previous lemma Relating Original and Relaxed Solutions In this section, we show that our vector coloring problem is a useful relaxation because the solution to it is related to the solution of the original problem. In order to understand the quality of the relaxed solution, we need the following geometric lemma: Lemma 4.1 For all positive integers k and n such that k - n + 1, there exist k unit vectors in ! n such that the dot product of any distinct pair is \Gamma1=(k \Gamma 1). Proof: Clearly it suffices to prove the lemma for other values of n, we make the coordinates of the vectors 0 in all but the first k \Gamma 1 coordinates.) We begin by proving the claim for explicitly provide unit vectors v (k) for j. The vector v (k) k(k\Gamma1) in all coordinates except the ith coordinate. In the ith coordinate v (k) i is . It is easy to verify that the vectors are unit length and that their dot products are exactly \Gamma1 . As given, the vectors are in a k-dimensional space. Note, however, that the dot product of each vector with the all-1's vector is 0. This shows that all k of the vectors are actually in a (k-1)-dimensional hyperplane of the k-dimensional space. This proves the lemma. Corollary 4.2 Every k-colorable graph G has a vector k-coloring. Proof: Bijectively map the k colors to the k vectors defined in the previous lemma. Note that a graph is vector 2-colorable if and only if it is 2-colorable. Lemma 4.1 is tight in that it provides the best possible value for minimizing the maximum dot-product among k unit vectors. This can be seen from the following lemma. Lemma 4.3 Let G be vector k-colorable and let i be a vertex in G. The induced subgraph on the neighbors of i is vector be a vector k-coloring of G and assume without loss of generality that Associate with each neighbor j of i a vector v 0 obtained by projecting v j onto coordinates 2 through n and then scaling it up so that v 0 j has unit length. It suffices to show that for any two adjacent vertices j and j 0 in the neighborhood of i, hv 0 Observe first that the projection of v j onto the first coordinate is negative and has magnitude at least 1=(k \Gamma 1). This implies that the scaling factor for v 0 j is at least k\Gamma1 . Thus, A simple induction using the above lemma shows that any graph containing a 1)-clique is not k-vector colorable. Thus the "vector chromatic number" lies between between the clique and chromatic number. This also shows that the analysis of Lemma 4.1 is tight in that \Gamma1=(k \Gamma 1) is the minimum possible value of the maximum of the dot-products of k vectors. In the next few sections we prove the harder part, namely, if a graph has a vector k-coloring then it has an ~ and an ~ O(n )-coloring. Given the solution to the relaxed problem, our next step is to show how to "round" the solution to the relaxed problem in order to get a solution to the original problem. Both of the rounding techniques we present in the following sections produce the coloring by working through an almost legal semicoloring of the graph, as defined below. Definition 5.1 A k-semicoloring of a graph G is an assignment of k colors to the at least half it vertices such that no two adjacent vertices are assigned the same color. An algorithm for semicoloring leads naturally to a coloring algorithm as shown by the following lemma. The algorithm uses up at most a logarithmic factor more colors than the semicoloring algorithm. Furthermore, we do not even lose this logarithmic factor if the semicoloring algorithm uses a polynomial number of colors (which is what we will show we use). Lemma 5.1 If an algorithm A can k i -semicolor any i-vertex subgraph of graph G in randomized polynomial time, where k i increases with i, then A can be used to O(k n log n)-color G. Furthermore, if there exists ffl ? 0 such that for all i, k can be used to color G with O(k n ) colors. Proof: We show how to construct a coloring algorithm A 0 to color any subgraph H of G. A 0 starts by using A to semicolor H . Let S be the subset of vertices which have not been assigned a color by A. Observe that jSj - jV (H)j=2. A 0 fixes the colors of vertices not in S, and then recursively colors the induced subgraph on S using a new set of colors. Let c i be the maximum number of colors used by A 0 to color any i-vertex subgraph. Then c i satisfies the recurrence It is easy to see that this any c i satisfying this recurrence, must satisfy c i - k i log i. In particular this implies that c n - O(k n log n). Furthermore for the case where k the above recurrence is satisfied only when c Using the above lemma, we devote the next two sections to algorithms for transforming vector colorings into semicolorings. 6 Rounding via Hyperplane Partitions We now focus our attention on vector 3-colorable graphs, leaving the extension to general k for later. Let \Delta be the maximum degree in a graph G. In this section, we outline a randomized rounding scheme for transforming a vector 3-coloring of G into an O(\Delta log 3 2 )-semicoloring, and thus into an log 3 log n)-coloring of G. Combining this method with a technique of Wigderson [42] yields an O(n 0:386 )-coloring of G. The method is based on [21] and is weaker than the method we describe in the following section; however, it introduces several of the ideas we will use in the more powerful algorithm. Assume we are given a vector 3-coloring fv i g n . Recall that the unit vectors v i and v j associated with an adjacent pair of vertices i and j have a dot product of at most \Gamma1=2, implying that the angle between the two vectors is at least 2-=3 radians (120 degrees). Definition 6.1 Consider a hyperplane H. We say that H separates two vectors if they do not lie on the same side of the hyperplane. For any edge fi; jg 2 E, we say that the hyperplane H cuts the edge if it separates the vectors v i and v j . In the sequel, we use the term random hyperplane to denote the unique hyperplane containing the origin and having as its normal a random unit vector v uniformly distributed on the unit sphere . The following lemma is a restatement of Lemma 1.2 of Goemans-Williamson [21]. Lemma 6.1 (Goemans-Williamson [21]) Given two vectors at an angle of ', the probability that they are separated by a random hyperplane is exactly '=-. We conclude that give a vector 3-coloring, for any edge fi; jg 2 E, the probability that a random hyperplane cuts the edge is exactly 2=3. It follows that the expected fraction of the edges in G which are cut by a random hyperplane is exactly 2=3. Suppose that we pick r random hyperplanes independently. Then, the probability that an edge is not cut by one of these hyperplanes is (1=3) r , and the expected fraction of the edges not cut is also (1=3) r . We claim that this gives us a good semicoloring algorithm for the graph G. Notice that r hyperplanes can partition ! n into at most 2 r distinct regions. (For r - n this is tight since r hyperplanes create exactly 2 r regions.) An edge is cut by one of these r hyperplanes if and only if the vectors associated with its end-points lie in distinct regions. Thus, we can associate a distinct color with each of the 2 r regions and give each vertex the color of the region containing its vector. The expected number of edges whose end-points have the same color is (1=3) r m, where m is the number of edges in E. Theorem 6.2 If a graph has a vector 3-coloring, then it has an O(\Delta log 3 2 )-semicoloring which can be constructed from the vector 3-coloring in polynomial time with high probability. Proof: We use the random hyperplane method just described. Fix \Deltae, and note that As noted above, r hyperplanes chosen independently at random will cut an edge with probability 1=9\Delta. Thus the expected number of edges which are not cut is since the number of edges is at most n\Delta=2. By Markov's inequality (cf. [38], page 46), the probability that the number of uncut edges is more than twice the expected value is at most 1=2. Thus, with probability at least 1/2 we get a coloring with at most n=4 uncut edges. Delete one endpoint of each such edge leaves a set of 3n=4 colored vertices with no uncut edges-ie, a semicoloring. Repeating the entire process t times means that we will find a O(\Delta log 3 2 )-semicoloring with probability at least 1 \Gamma 1=2 t . Noting that log 3 2 ! 0:631 and that \Delta - n, this theorem and Lemma 5.1 implies a semicoloring using O(n 0:631 ) colors. By varying the number of hyperplanes, we can arrange for a tradeoff between the number of colors used and the number of edges that violate the resulting coloring. This may be useful in some applications where a nearly legal coloring is good enough. 6.1 Wigderson's Algorithm Our coloring can be improved using the following idea due to Wigderson [42]. Fix a threshold value ffi. If there exists a vertex of degree greater than ffi, pick any one such vertex and 2-color its neighbors (its neighborhood is vector 2-colorable and hence 2-colorable). The colored vertices are removed and their colors are not used again. Repeating this as often as possible (or until half the vertices are colored) brings the maximum degree below ffi at the cost of using at most 2n=ffi colors. Thus, we can obtain a semicoloring using O(n=ffi colors. The optimum choice of ffi is around 0:613 , which implies a semicoloring using O(n 0:387 ) colors. This semicoloring can be used to legally color G using O(n 0:387 ) colors by applying Lemma 5.1. Corollary 6.3 A 3-colorable graph with n vertices can be colored using O(n 0:387 ) colors by a polynomial time randomized algorithm. The bound just described is (marginally) weaker than the guarantee of a O(n 0:375 ) coloring due to Blum [9]. We now improve this result by constructing a semicoloring with fewer colors. 7 Rounding via Vector Projections In this section we start by proving the following more powerful version of Theorem 6.2. A simple application of Wigderson's technique to this algorithm yields our final coloring algorithm. Lemma 7.1 For every integer function vector k-colorable graph with maximum degree \Delta can be semi-colored with at most O(\Delta 1\Gamma2=k \Delta) colors in probabilistic polynomial time. As in the previous section, this has immediate consequences for approximate coloring. Given a vector k-coloring, we show that it is possible to extract an independent set of size \Delta)). If we assign one color to this set and recurse on the rest, we will end up using \Delta) colors in all to assign colors to half the vertices and the result follows. To find such a large independent set, we give a randomized procedure for selecting an induced subgraph with n 0 vertices and m 0 edges such that E[n \Delta)). It follows that with a polynomial number of repeated trials, we have a high probability of choosing a subgraph with \Delta)). Given such a graph, we can delete one endpoint of each edge, leaving an independent set of size n \Delta)), as desired. We now give the details of the construction. Suppose we have a vector k-coloring assigning unit vectors v i to the vertices. We fix a parameter to be specified later. We choose a random n-dimensional vector r according to a distribution to be specified soon. The subgraph consists of all vertices i with Intuitively, since endpoints of an edge have vectors pointing away from each other, if the vector associated with a vertex has a large dot product with r, then the vector corresponding to an adjacent vertex will not have such a large dot product with r and hence will not be selected. Thus, only a few edges are likely to be in the induced subgraph on the selected set of vertices. To complete the specification of this algorithm and to analyze it, we need some basic facts about some probability distributions in ! n . 7.1 Probability Distributions in ! n Recall that the standard normal distribution has the density function distribution function \Phi(x), mean 0, and variance 1. A random vector is said to have the n-dimensional standard normal distribution if the components r i are independent random variables, each component having the standard normal distribution. It is easy to verify that this distribution is spherically symmetric, in that the direction specified by the vector r is uniformly distributed. (Refer to Feller [14, v. II], Knuth [31, v. 2], and R'enyi [39] for further details about the higher dimensional normal distribution.) Subsequently, the phrase "random d-dimensional vector" will always denote a vector chosen from the d-dimensional standard normal distribution. A crucial property of the normal distribution which motivates its use in our algorithm is the following theorem paraphrased from R'enyi [39] (see also Section III.4 of Feller [14, v. II]). Theorem 7.2 (Theorem IV.16.3 [39]) Let n-dimensional vector. The projections of r onto two lines ' 1 and ' 2 are independent (and normally distributed) if and only if ' 1 and ' 2 are orthogonal. Alternatively, we can say that under any rotation of the coordinate axes, the projections of r along these axes are independent standard normal variables. In fact, it is known that the only distribution with this strong spherical symmetry property is the n-dimensional standard normal distribution. The latter fact is precisely the reason behind this choice of distribution 1 in our algorithm. In particular, we will make use of the following corollary to the preceding theorem. Corollary 7.3 Let u be any unit vector in ! n . Let be a random vector (of i.i.d. standard normal variables). The projection of r along u, given by dot product hu; ri, is distributed according to the standard (1-dimensional) normal distribution. It turns out that even if r is a random n-dimensional unit vector, the above corollary still holds in the limit: as n grows, the projections of r on orthogonal lines approach (scaled) independent normal distributions. Thus using a random unit vectors for our projection turns out to be equivalent to using random normal vectors in the limit, but is messier to analyze. Let N(x) denote the tail of the standard normal distribution. I.e., x OE(y) dy: We will need the following well-known bounds on the tail of the standard normal distribution. (See, for instance, Lemma VII.2 of Feller [14, v. I].) Lemma 7.4 For every x ? 0, x x Proof: The proof is immediate from inspection of the following equations relating the three quantities in the desired inequality to integrals involving OE(x), and the fact OE(x)=x is finite for every x ? 0. x x OE(y) \Gammay 4 x x OE(y) dy: Readers familiar with physics will see the connection to Maxwell's law on the distribution of velocities of molecules in ! 3 . Maxwell started with the assumption that in every Cartesian coordinate system in ! 3 , the three components of the velocity vector are mutually independent and had expectation zero. Applying this assumption to rotations of the axes, we conclude that the velocity components must be independent normal variables with identical variance. This immediately implies Maxwell's distribution on the velocities. 7.2 The Analysis We are now ready to complete the specification of the coloring algorithm. Recall that our goal is to repeatedly identify, color and delete large independent sets from the graph. We actually set an easier intermediate goal: find an induced subgraph with a large number n 0 of edges and a number vertices. Since each edge only covers 2 vertices, the induced subgraph has vertices with no incident edges. These vertices form an independent set that can be colored and removed. As discussed above, to find this sparse graph, we choose a random vector r and take all vertices whose dot product with r exceeds a certain value c. Let the induced subgraph on these vertices have edges. We show that for sufficiently larger we get an independent set of size roughly n 0 . Intuitively, this is true for the following reason. Any particular vertex has some particular probability landing near r and thus being "captured" into our set. However, if two vertices are adjacent, the probability that they both land near r is quite small because the vector coloring has placed them far apart. For example, in the case of 3-coloring, when the probability that a vertex is chosen is p, the probability that both endpoints of an edge are chosen is roughly p 4 . It follows that we end up capturing (in expectation) a set of pn vertices that contains (in expectation) only edges in a degree-\Delta graph. In such a set, at least pn \Gamma p 4 \Deltan of the vertices have no incident edges, and thus form an independent set. We would like this independent set to be large. Clearly, we need to make p small enough to ensure p 4 \Deltan - pn, meaning p - \Delta \Gamma1=3 . Taking p much smaller only decreases the size of the independent set, so it turns out that our best choice is to take yielding an indpendent set of Repeating this capture process many times therefore achieves an ~ We now formalize this intuitive argument. The vector r will be a random n-dimensional vector. We precisely compute the expectation of n 0 , the number of vertices captured, and the expectation of m 0 , the number of edges in the induced graph of the captured vertices. We first show that when r is a random normal vector and our projection threshold is c, the expectation of n for a certain constant a depending on the vector chromatic number. We also show that N(ac) grows roughly as N(c) a 2 . (For the case of 3-coloring we have a = 2, and thus if picking a sufficiently large c, we can find an independent set of size N(c)). (In the following lemma, n 0 and m 0 are functions of c: we do not make this dependence explicit.) Lemma 7.5 Let a = k\Gamma2 . Then for any c, Proof: We first bound E [n 0 ] from below. Consider a particular vertex i with assigned vector . The probability that it is in the selected set is just P normally distributed and thus this probability is N(c). By linearity of expectations, the expected number of selected vertices Now we bound E [m 0 ] from above. Consider an edge with endpoint vectors v 1 and v 2 . The probability that this edge is in the induced subgraph is the probability that both endpoints are selected, which is where the expression follows from Corollary 7.3 applied to the preceding probability expression. We now observe that It follows that the probability that both endpoints of an edge are selected is at most N(ac). If the graph has maximum degree \Delta, then the total number of edges is at most n\Delta=2. Thus the expected number of selected edges, E [m 0 ], is at most n\DeltaN (ac)=2. Combining the previous arguments, we deduce that We now determine the a c such that \DeltaN (ac) ! N(c). This will give us an expectation of at least N(c)=2 in the above lemma. Using the bounds on N(x) in Lemma 7.4, we find that N(c) N(ac) a p' (The last equation holds since a = 2.) Thus if we choose c so that 1 \Gamma 1=c 2 -pand e (a 2 \Gamma1)c 2 =2 - \Delta, then we get \DeltaN (ac) ! N(c). Both conditions are satisfied, for sufficiently large \Delta, if we set smaller values of \Delta we can use the greedy 1-coloring algorithm to get a color the graph with a bounded number of colors, where the bound is independent of n.) For this choice of c, we find that the independent set that is found has size at least ne \Gammac 2 =2 c \Gammac 3 -\Omega / as desired. This concludes the proof of Lemma 7.1. 7.3 Adding Wigderson's Technique To conclude, we now determine absolute approximation ratios independent of \Delta. This involves another application of Wigderson's technique. If the graph has any vertex of large degree, then we use the fact that its neighborhood is large and is vector 1)-chromatic, to find a large independent set in its neighborhood. If no such vertex exists, then the graph has small maximum degree, so we can use Lemma 7.1 to find a large independent set in the graph. After extracting such an independent set, we recurse on the rest of the graph. The following lemma describes the details, and the correct choice of the threshold degree. Lemma 7.6 For every integer function vector k-colorable graph on n vertices can be semicolored with O(n 1\Gamma3=(k+1) log 1=2 n) colors by a probabilistic polynomial time algorithm. Proof: Given a vector k-colorable graph G, we show how to find an independent set of size n) in the graph. Assume, by induction on k, that there exists a constant c ? 0 s.t. we can find an independent set of size ci 3=(k 0 +1) =(log 1=2 i) in any k 0 -vector chromatic graph on k. We now prove the inductive assertion for k. If G has a vertex of degree greater than \Delta k (n), then we find a large independent set in the neighborhood of G. By Lemma 4.3, the neighborhood is vector Hence we can find in this neighborhood, an independent set of size at least n). If G does not have a vertex of degree greater than then by Lemma 7.1, we can find an independent set of size at least in G. This completes the induction. By now assigning a new color to each such independent set, we find that we can color at least n=2 vertices, using up at most O(n 1\Gamma3=(k+1) log 1=2 n) colors. The semicolorings guaranteed by Lemmas 7.1 and 7.6 can be converted into colorings using Lemma 5.1, yielding the following theorem. Theorem 7.7 Any vector k-colorable graph on n nodes with maximum degree \Delta can be colored, in probabilistic polynomial time, using minfO(\Delta 1\Gamma2=k 8 Duality Theory The most intensively studied relaxation of a semidefinite programming formulation to date is the Lov'asz #-function [23, 24, 33]. This relaxation of the clique number of a graph led to the first polynomial-time algorithm for finding the clique and chromatic numbers of perfect graphs. We now investigate a connection between # and a close variant of the vector chromatic number. Intuitively, the clique and coloring problems have a certain "duality" since large cliques prevent a graph from being colored with few colors. Indeed, it is the equality of the clique and chromatic numbers in perfect graphs which lets us compute both in polynomial time. We proceed to formalize this intuition. The duality theory of linear programming has an extension to semidefinite programming. With the help of Eva Tardos and David Williamson, we have shown that in fact the #-function and a close variant of the vector chromatic number are semidefinite programming duals to one another and are therefore equal. We first define the variant. Definition 8.1 Given a graph E) on n vertices, a strict vector k-coloring of G is an assignment of unit vectors u i from the space ! n to each vertex , such that for any two adjacent vertices i and j the dot product of their vectors satisfies the equality As usual we say that a graph is strictly vector k-colorable if it has a strict vector k-coloring. The strict vector chromatic number of a graph is the smallest real number k for which it has a strict vector k-coloring. It follows from the definition that the strict vector chromatic number of any graph is lower bounded by the vector chromatic number. Theorem 8.1 The strict vector chromatic number of G is equal to #(G). Proof: The dual of our strict vector coloring semidefinite program is as follows (cf. [2]): is positive semidefinite subject to By duality, the value of this SDP is \Gamma1=(k \Gamma 1) where k is the strict vector chromatic number. Our goal is to prove As before, the fact that fp ij g is positive semidefinite means we can find vectors v i such that The last constraint says that the vectors v form an orthogonal labeling [24], i.e. that hv 2 E. We now claim that the above optimization problem can be reformulated as follows: over all orthogonal labelings fv i g. To see this, consider an orthogonal labeling and define this is the value of the first constraint in the first formulation of the dual (that is, the constraint is - 1) and of the denominator in the second formulation. Then in an optimum solution to the first formulation, we must have since otherwise we can divide each v i by - and get a feasible solution with a larger objective value. Thus the optimum of the second formulation is at least as large as that of the first. Similarly, given any optimum fv i g for the second feasible solution to the first formulation with the same value. Thus the optima are equal. We now manipulate the second formulation. min It follows from the last equation that the vector chromatic number is However, by the same argument as used to reformulate the dual, this is equal to problem of maximizing orthogonal labelings such that 1. This is simply Lov'asz's formulation of the #-function [24, page 287]. 9 The Gap between Vector Colorings and Chromatic Numbers The performance of our randomized rounding approach seems far from optimum. In this section we ask why, and show that the problem is not in the randomized rounding but in the gap between the original problem and its relaxation. We investigate the following question: given a vector k- colorable graph G, how large can its chromatic number be in terms of k and n? We will show that a graph with chromatic number n\Omega\Gamma23 can have bounded vector chromatic number. This implies that our technique is tight in that it is not possible to guarantee a coloring with n o(1) colors on all vector 3-colorable graphs. Definition 9.1 The Kneser graph K(m; defined as follows: the vertices are all possible r-sets from a universe of size m; and, the vertices v i and v j are adjacent if and only if the corresponding r-sets satisfy t. We will need following theorem of Milner [36] regarding intersecting hypergraphs. Recall that a collection of sets is called an antichain if no set in the collection contains another. Theorem 9.1 (Milner) Let S 1 ff be an antichain of sets from a universe of size m such that, for all i and j, Then, it must be the case that Notice that using all q-sets, for example for this theorem. The following theorem establishes that the Kneser graphs have a large gap between their vector chromatic number and chromatic numbers. Theorem 9.2 Let r \Delta denote the number of vertices of the graph K(m; m=8, the graph K(m; colorable but has chromatic number at least n 0:0113 . Proof: We prove a lower bound on the Kneser graph's chromatic number - by establishing an upper bound on its independence number ff. It is easy to verify that the ff in Milner's theorem is exactly the independence number of the Kneser graph. To bound - observe that ff r large enough m. In the above sequence, the fourth line uses the approximation for every fi 2 (0; 1), where c fi is a constant depending only on fi. Using the inequality r we obtain m - lg n and thus Finally, it remains to show that the vector chromatic number of this graph is 3. This follows by associating with each vertex v i an m-dimensional vector obtained from the characteristic vector of the set S i . In the characteristic vector, +1 represents an element present in S i and \Gamma1 represents elements absent from S i . The vector associated with a vertex is the characteristic vector of S i scaled down by a factor of m to obtain a unit vector. Given vectors corresponding to sets S i and S j , the dot product gets a contribution of \Gamma1=m for coordinates in S i \DeltaS j and +1=m for the others. (Here A\DeltaB represents the symmetric difference of the two sets, i.e., the set of elements which occur in exactly one of A or B.) Thus the dot product of two adjacent vertices, or sets with intersection at most t, is given by This implies that the vector chromatic number is 3. More refined calculations can be used to improve this bound somewhat. Theorem 9.3 There exists a Kneser graph K(m; which is 3-vector colorable but has chromatic number exceeding n 0:016101 , where r denotes the number of vertices in the graph. Further, for large k, there exists a Kneser graph K(m; which is k-vector colorable but has chromatic number exceeding n 0:0717845 . Proof: The basic idea is to improve the bound on the vector chromatic number of the Kneser graph using an appropriately weighted version of the characteristic vectors. We use weights a and to represent presence and absence, respectively, of an element in the set corresponding to a vertex in the Kneser graph, with appropriate scaling to obtain a unit vector. The value of a which minimizes the vector chromatic number can be found by differentiation and is mt Setting proves that the vector chromatic number is at most At the same time, using Milner's Theorem proves that the exponent of the chromatic number is at least r By plotting these functions, we have shown that there is a set of values with vector chromatic number 3 and chromatic number at least n 0:016101 . For large constant vector chromatic numbers, the limiting value of the exponent of the chromatic number is roughly 0:0717845. Conclusions The Lov'asz number of a graph has been a subject of active study due to the close connections between this parameter and the clique and chromatic numbers. In particular, the following "sandwich theorem" was proved by Lov'asz [33] (see Knuth [32] for a survey). This led to the hope that the following question may have an affirmative answer. Do there exist ffl, graph G on n vertices Our work in this paper proves a weak but non-trivial upper bound on the the chromatic number of G in terms of #(G). However, this is far from achieving the bound conjectured above and subsequent to our work, two results have ended up answering this question negatively. Feige [16] has shown that for every ffl ? 0, there exist families of graphs for which -(G) ? #(G)n 1\Gammaffl . Interestingly, families of graphs exhibited in Feige's work use the construction of Section 9 as a starting point. Even more conclusively, the results of H-astad [26] and Feige and Kilian [18] have shown that no polynomial time computable function approximates the clique number or chromatic number to within factors of n 1\Gammaffl , unless NP=RP. Thus no simple modification of the # function is likely to provide a much better approximation guarantee. In related results, Alon and Kahale [5] have also been able to use the semidefinite programming technique in conjunction with our techniques to obtain algorithms for computing bounds on the clique number of a graph with linear-sized cliques, improving upon some results due to Boppana and Halldorsson [10]. Independent of our results, Szegedy [41] has also shown that a similar construction yields graphs with vector chromatic number at most 3 but which are not colorable using n 0:05 colors. Notice that the exponent obtained from his result is better than the one in Section 9. Alon [3] has obtained a slight improvement over Szegedy's bound by using an interesting variant of the Kneser graph construction. Finally, the main algorithm presented here has been derandomized in a recent work of Mahajan and Ramesh [35]. Acknowledgments Thanks to David Williamson for giving us a preview of the MAX-CUT result [21] during a visit to Stanford. We are indebted to John Tukey and Jan Pedersen for their help in understanding multi-dimensional probability distributions. Thanks to David Williamson and Eva Tardos for discussions of the duality theory of SDP. We thank Noga Alon, Don Coppersmith, Jon Kleinberg, Laci Lov'asz and Mario Szegedy for useful discussions, and the anonymous referees for the careful comments. --R Probability Approximations via the Poisson Clumping Heuristic. Interior point methods in semidefinite programming with applications to combinatorial optimization. Personal Communication A spectral technique for coloring random 3-colorable graphs Approximating the independence number via the Theta function. Proof Verification and Hardness of Approximation Problems. Improved Non-approximability Results New approximation algorithms for graph coloring. Approximating maximum independent sets by excluding subgraphs. Coloring heuristics for register alloca- tion Register allocation and spilling via graph coloring. Register allocation via coloring. An Introduction to Probability Theory and Its Applications. Forbidden Intersections. Randomized graph products Interactive proofs and the hardness of approximating cliques. Zero knowledge and chromatic number. Improved approximation algorithms for MAX k-CUT and MAX BISECTION Computers and Intractability: A Guide to the Theory of NP-Completeness Improved approximation algorithms for maximum cut and satisfiability problems. Matrix Computations. The ellipsoid method and its consequences in combinatorial optimization. Geometric Algorithms and Combinatorial Optimization. A still better performance guarantee for approximate graph coloring. Clique is hard to approximate within n 1 Worst case behavior of graph coloring algorithms. On the Hardness of Approximating the Chromatic Number. On Syntactic versus Computational Views of Approximability. Aufgabe 300. The Art of Computer Programming. The Sandwich Theorem. On the Shannon capacity of a graph. On the hardness of approximating minimization problems. Derandomizing semidefinite programming based approximation algorithms. A combinatorial theorem on systems of sets. On Exact and Approximate Cut Covers of Graphs. Randomized Algorithms. Probability Theory. A note on the ' number of Lov'asz and the generalized Delsarte bound. Personal Communication. Improving the Performance Guarantee for Approximate Graph Coloring. A Technique for Coloring a Graph Applicable to Large-Scale Optimization Prob- lems --TR Improving the performance guarantee for approximate graph coloring Coloring heuristics for register allocation A still better performance guarantee for approximate graph coloring On the hardness of approximating minimization problems Approximating maximum independent sets by excluding subgraphs New approximation algorithms for graph coloring Improved non-approximability results A spectral technique for coloring random 3-colorable graphs (preliminary version) Randomized algorithms Randomized graph products, chromatic numbers, and Lovasz j-function Interactive proofs and the hardness of approximating cliques Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming An MYAMPERSANDOtilde;(<italic>n</italic><supscrpt>3/14</supscrpt>)-coloring algorithm for 3-colorable graphs Computers and Intractability The Art of Computer Programming, 2nd Ed. (Addison-Wesley Series in Computer Science and Information Zero Knowledge and the Chromatic Number Derandomizing semidefinite programming based approximation algorithms Register allocation MYAMPERSANDamp; spilling via graph coloring Clique is hard to approximate within n1- --CTR Yonatan Bilu, Tales of Hoffman: three extensions of Hoffman's bound on the graph chromatic number, Journal of Combinatorial Theory Series B, v.96 n.4, p.608-613, July 2006 Eran Halperin , Ram Nathaniel , Uri Zwick, Coloring k-colorable graphs using relatively small palettes, Journal of Algorithms, v.45 n.1, p.72-90, October 2002 Robert A. Stubbs , Sanjay Mehrotra, Generating Convex Polynomial Inequalities for Mixed 01 Programs, Journal of Global Optimization, v.24 n.3, p.311-332, November 2002 Eran Halperin , Ram Nathaniel , Uri Zwick, Coloring Amin Coja-Oghlan , Lars Kuhtz, An improved algorithm for approximating the chromatic number of G Michael Krivelevich , Ram Nathaniel , Benny Sudakov, Approximating coloring and maximum independent sets in 3-uniform hypergraphs, Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms, p.327-328, January 07-09, 2001, Washington, D.C., United States Uriel Feige , Michael Langberg, The RPR2 rounding technique for semidefinite programs, Journal of Algorithms, v.60 n.1, p.1-23, July 2006 Sanjeev Arora , Eden Chlamtac, New approximation guarantee for chromatic number, Proceedings of the thirty-eighth annual ACM symposium on Theory of computing, May 21-23, 2006, Seattle, WA, USA Michel X. Goemans , David Williamson, Approximation algorithms for MAX-3-CUT and other problems via complex semidefinite programming, Proceedings of the thirty-third annual ACM symposium on Theory of computing, p.443-452, July 2001, Hersonissos, Greece Irit Dinur , Elchanan Mossel , Oded Regev, Conditional hardness for approximate coloring, Proceedings of the thirty-eighth annual ACM symposium on Theory of computing, May 21-23, 2006, Seattle, WA, USA Miroslav Chlebk , Janka Chlebkov, Complexity of approximating bounded variants of optimization problems, Theoretical Computer Science, v.354 n.3, p.320-338, 4 April 2006 Amin Coja-Oghlan, Solving NP-hard semirandom graph problems in polynomial expected time, Journal of Algorithms, v.62 n.1, p.19-46, January, 2007 Michel X. Goemans , David P. Williamson, Approximation algorithms for MAX-3-CUT and other problems via complex semidefinite programming, Journal of Computer and System Sciences, v.68 n.2, p.442-470, March 2004 Moses Charikar, On semidefinite programming relaxations for graph coloring and vertex cover, Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms, p.616-620, January 06-08, 2002, San Francisco, California D. Sivakumar, Algorithmic derandomization via complexity theory, Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, May 19-21, 2002, Montreal, Quebec, Canada Alon , Konstantin Makarychev , Yury Makarychev , Assaf Naor, Quadratic forms on graphs, Proceedings of the thirty-seventh annual ACM symposium on Theory of computing, May 22-24, 2005, Baltimore, MD, USA Bernard Chazelle , Carl Kingsford , Mona Singh, The side-chain positioning problem: a semidefinite programming formulation with new rounding schemes, Proceedings of the Paris C. Kanellakis memorial workshop on Principles of computing & knowledge: Paris C. Kanellakis memorial workshop on the occasion of his 50th birthday, p.86-94, June 08-08, 2003, San Diego, California, USA Arash Behzad , Izhak Rubin, Multiple Access Protocol for Power-Controlled Wireless Access Nets, IEEE Transactions on Mobile Computing, v.3 n.4, p.307-316, October 2004 Amin Coja-Oghlan, Finding Large Independent Sets in Polynomial Expected Time, Combinatorics, Probability and Computing, v.15 n.5, p.731-751, September 2006 Luca Trevisan, Non-approximability results for optimization problems on bounded degree instances, Proceedings of the thirty-third annual ACM symposium on Theory of computing, p.453-461, July 2001, Hersonissos, Greece Sanjeev Arora , Satish Rao , Umesh Vazirani, Expander flows, geometric embeddings and graph partitioning, Proceedings of the thirty-sixth annual ACM symposium on Theory of computing, p.222-231, June 13-16, 2004, Chicago, IL, USA Per Austrin, Balanced max 2-sat might not be the hardest, Proceedings of the thirty-ninth annual ACM symposium on Theory of computing, June 11-13, 2007, San Diego, California, USA Ccile Murat , Vangelis Th. Paschos, On the probabilistic minimum coloring and minimum k-coloring, Discrete Applied Mathematics, v.154 n.3, p.564-586, 1 March 2006 Lars Engebretsen , Piotr Indyk , Ryan O'Donnell, Derandomized dimensionality reduction with applications, Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms, p.705-712, January 06-08, 2002, San Francisco, California Martin Skutella, Convex quadratic and semidefinite programming relaxations in scheduling, Journal of the ACM (JACM), v.48 n.2, p.206-242, March 2001 Bernard Chazelle , Carl Kingsford , Mona Singh, A Semidefinite Programming Approach to Side Chain Positioning with New Rounding Strategies, INFORMS Journal on Computing, v.16 n.4, p.380-392, Fall 2004 Amin Coja-oghlan, The Lovsz Number of Random Graphs, Combinatorics, Probability and Computing, v.14 n.4, p.439-465, July 2005 V. Th. Paschos, Polynomial approximation and graph-coloring, Computing, v.70 n.1, p.41-86, March

approximation algorithms;randomized algorithms;NP-completeness;graph coloring;chromatic number

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Compiler blockability of dense matrix factorizations.

The goal of the LAPACK project is to provide efficient and portable software for dense numerical linear algebra computations. By recasting many of the fundamental dense matrix computations in terms of calls to an efficient implementation of the BLAS (Basic Linear Algebra Subprograms), the LAPACK project has, in large part, achieved its goal. Unfortunately, the efficient implementation of the BLAS results often in machine-specific code that is not portable across multiple architectures without a significant loss in performance or a significant effort to reoptimize them. This article examines wheter most of the hand optimizations performed on matrix factorization codes are unnecessary because they can (and should) be performed by the compiler. We believe that it is better for the programmer to express algorithms in a machine-independent form and allow the compiler to handle the machine-dependent details. This gives the algorithms portability across architectures and removes the error-prone, expensive and tedious process of hand optimization. Although there currently exist no production compilers that can perform all the loop transformations discussed in this article, a description of current research in compiler technology is provided that will prove beneficial to the numerical linear algebra community. We show that the Cholesky and optimized automaticlaly by a compiler to be as efficient as the same hand-optimized version found in LAPACK. We also show that the QR factorization may be optimized by the compiler to perform comparably with the hand-optimized LAPACK version on modest matrix sizes. Our approach allows us to conclude that with the advent of the compiler optimizations dicussed in this article, matrix factorizations may be efficiently implemented in a BLAS-less form

Introduction The processing power of microprocessors and supercomputers has increased dramatically and continues to do so. At the same time, the demand on the memory system of a computer is to increase dramatically in size. Due to cost restrictions, typical workstations cannot use memory chips that have the latency and bandwidth required by today's processors. Instead, main memory is constructed of cheaper and slower technology and the resulting delays may be up to hundreds of cycles for a single memory access. To alleviate the memory speed problem, machine architects construct a hierarchy of memory where the highest level (registers) is the smallest and fastest and each lower level is larger but Research supported by NSF Grant CCR-9120008 and by NSF grant CCR-9409341. The second author was also supported by the U.S. Department of Energy Contracts DE-FG0f-91ER25103 and W-31-109-Eng-38. y Department of Computer Science, Michigan Technological University, Houghton MI 49931, carr@cs.mtu.edu. z Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL 60439 lehoucq@mcs.anl.gov, http://www.mcs.anl.gov/home/lehoucq/index.html. slower. The bottom of the hierarchy for our purposes is main memory. Typically, one or two levels of cache memory fall between registers and main memory. The cache memory is faster than main memory, but is often a fraction of the size. The cache memory serves as a buffer for the most recently accessed data of a program (the working set). The cache becomes ineffective when the working set of a program is larger than its size. The three factorizations considered in this paper, the LU, Cholesky, and QR, are among the most frequently used by numerical linear algebra and its applications. The first two are used for solving linear systems of equations while the last is typically used in linear least squares problems. For square matrices of order n, all three factorizations involve on the order of n 3 floating point operations for data that needs n 2 memory locations. With the advent of vector and parallel supercomputers, the efficiency of the factorizations were seen to depend dramatically upon the algorithmic form chosen for the implementation [16, 18, 32]. These studies concluded that managing the memory hierarchy is the single most important factor governing the efficiency of the software implementation computing the factorization. The motivation of the LAPACK [2] project was to recast the algorithms in the EISPACK [35] and LINPACK [14] software libraries with block ones. A block form of an algorithm restructures the algorithm in terms of matrix operations that attempt to minimize the amount of data moved within the memory hierarchy while keeping the arithmetic units of the machine occupied. LAPACK blocks many dense matrix algorithms by restructuring them to use the level 2 and 3 BLAS [11, 12]. The motivation for the Basic Linear Algebra Subprograms, BLAS [29], was to provide a set of commonly used vector operations so that the programmer could invoke the subprograms instead of writing the code directly. The level 2 and 3 BLAS followed with matrix-vector and matrix-matrix operations, respectively, that are often necessary for high efficiency across a broad range of high performance computers. The higher level BLAS better utilize the underlying memory hierarchy. As with the level 1 BLAS, responsibility for optimizing the higher level BLAS was left to the machine vendor or another interested party. This study investigates whether a compiler has the ability to block matrix factorizations. Although the compiler transformation techniques may be applied directly to the BLAS, it is interesting to draw a comparison with applying them directly to the factorizations. The benefit is the possibility of a BLAS-less linear algebra package that is nearly as efficient as LAPACK. For example, in [30], it was demonstrated that on some computers, the best LU factorization was an inlined approach even when a highly optimized set of BLAS were available. We deem an algorithm blockable if a compiler can automatically derive the most efficient block algorithm (for our study, the one found in LAPACK) from its corresponding machine-independent point algorithm. In particular, we show that LU and Cholesky factorizations are blockable algo- rithms. Unfortunately, QR factorization with Householder transformations is not blockable. How- ever, we show an alternative block algorithm for QR that can be derived using the same compiler methods as those used for LU and Cholesky factorizations. This study has yielded two major results. The first, which is detailed in another paper [9], reveals that the hand loop unrolling performed when optimizing the level 2 and 3 BLAS [11, 12] is often unnecessary. While the BLAS are useful, the hand optimization that is required to obtain good performance on a particular architecture may be left to the compiler. Experiments show that, in most cases, the compiler can automatically unroll loops as effectively as hand optimization. The second result, which we discuss in this paper, reveals that it is possible to block matrix factorizations automatically. Our results show that the block algorithms derived by the compiler are competitive with those of LAPACK [2]. For modest sized matrices (on the order of 200 or less), the compiler-derived variants are often superior. We begin our presentation with a review of background material related to compiler optimiza- tion. Then, we describe a study of the application of compiler analysis to derive the three block algorithms in LAPACK considered above from their corresponding point algorithms. We present an experiment comparing the performance of hand-optimized LAPACK algorithms with the compiler- derived algorithms attained using our techniques. We also briefly discuss other related approaches. Finally, we summarize our results and provide and draw some general conclusions. Background The transformations that we use to create the block versions of matrix factorizations from their corresponding point versions are well known in the mathematical software community [15]. This section introduces the fundamental tools that the compiler needs to perform the same transformations automatically. The compiler optimizes point versions of matrix factorizations through analysis of array access patterns rather than through linear algebra. 2.1 Dependence As in vectorizing and parallelizing compilers, dependence is a critical compiler tool for performing transformations to improve the memory performance of loops. Dependence is necessary for determining the legality of compiler transformations to create blocked versions of matrix factorizations by giving a partial order on the statements within a loop nest. A dependence exists between two statements if there exists a control flow path from the first statement to the second, and both statements reference the same memory location [26]. ffl If the first statement writes to the location and the second reads from it, there is a true dependence, also called a flow dependence. ffl If the first statement reads from the location and the second writes to it, there is an antide- pendence. ffl If both statements write to the location, there is an output dependence. ffl If both statements read from the location, there is an input dependence. A dependence is carried by a loop if the references at the source and sink (beginning and end) of the dependence are on different iterations of the loop and the dependence is not carried by an outer loop [1]. In the loop below, there is a true dependence from A(I,J) to A(I-1,J) carried by the I-loop, a true dependence from A(I,J) to A(I,J-1) carried by the J-loop and an input dependence from A(I,J-1) to A(I-1,J) carried by the I-loop. enhance the dependence information, section analysis can be used to describe the portion of an array that is accessed by a particular reference or set of references [5, 21]. Sections describe common substructures of arrays such as elements, rows, columns and diagonals. As an example of section analysis consider the following loop. If A were declared to be 100 \Theta 100, the section of A accessed in the loop would be that shown in the shaded portion of Figure 1. Matrix factorization codes require us to enhance basic dependence information because only a portion of the matrix is involved in the block update. The compiler uses section analysis to reveal that portion of the matrix that can be block updated. Section 3.1.1 discusses this in detail.10100 Figure 1 Section of A 3 Automatic Blocking of Dense Matrix Factorizations In this section, we show how to derive the block algorithms for the LU and the Cholesky factorizations using current compiler technology and section analysis to enhance dependence information. We also show that the QR factorization with Householder transformations is not blockable. How- ever, we present a performance-competitive version of the QR factorization that is derivable by the compiler. 3.1 LU Factorization The LU decomposition factors a non-singular matrix A into the product of two matrices, L and U , such that A = LU [20]. L is a unit lower triangular matrix and U is an upper triangular matrix. This factorization can be obtained by multiplying the matrix A by a series of elementary lower and A. The pivot matrices are used to make the LU factorization a numerically stable process. We first examine the blockability of LU factorization. Since pivoting creates its own difficulties, we first show how to block LU factorization without pivoting. We then show how to handle pivoting. 3.1.1 No Pivoting Consider the following algorithm for LU factorization. This point algorithm is referred to as an unblocked right-looking [13] algorithm. It exhibits poor cache performance on large matrices. To transform the point algorithm to the block algorithm, the compiler must perform strip-mine-and-interchange on the K-loop [38, 36]. This transformation is used to create the block update of A. To apply this transformation, we first strip the K-loop into fixed size sections (this size is dependent upon the target architecture's cache characteristics and is beyond the scope of this paper [28, 10]) as shown below. Here KS is the machine-dependent strip size that is related to the cache size. To complete the transformation, the KK-loop must be distributed around the loop that surrounds statement 20 and around the loop nest that surrounds statement 10 before being interchanged to the innermost position of the loop surrounding statement 10 [37]. This distribution yields: Unfortunately, the loop is no longer correct. This loop scales a number of values before it updates them. Dependence analysis allows the compiler to detect and avoid this change in semantics by recognizing the dependence cycle between A(I,KK) in statement 20 and A(I,J) in statement 10 carried by the KK-loop. Using basic dependence analysis only, it appears that the compiler would be prevented from blocking LU factorization due to the cycle. However, enhancing dependence analysis with section information reveals that the cycle only exists for a portion of the data accessed in both statements. Figure 2 shows the sections of the array A accessed for the entire execution of the KK-loop. The section accessed by A(I,KK) in statement 20 is a subset of the section accessed by A(I,J) in statement 10. Since the recurrence exists for only a portion of the iteration space of the loop surrounding statement 10, we can split the J-loop into two loops - one loop iterating over the portion of A where the dependence cycle exists, and one loop iterating over the portion of A where the cycle does not exist - using a transformation called index-set splitting [38]. J can be split at the point to create the two loops as shown below. Figure 2 Sections of A in LU Factorization DO DO Now the dependence cycle exists between statements 20 and 30, and statement 10 is no longer in the cycle. Strip-mine-and-interchange can be continued by distributing the KK-loop around the two new loops as shown below. DO DO DO To finish strip-mine-and-interchange, we need to move the KK-loop to the innermost position in the nest surrounding statement 10. However, the lower bound of the I-loop contains a reference to KK. This creates a triangular iteration space as shown in Figure 3. To interchange the KK and I loops, the intersection of the line I=KK+1 with the iteration space at the point (K,K+1) must be handled. Therefore, interchanging the loops requires the KK-loop to iterate over a trapezoidal region with an upper bound of I-1 until I-1 ? K+KS-1 (see Wolfe, and Carr and Kennedy for more details on transforming non-rectangular loop nests [38, 8]). This gives the following loop nest. DO DO DO I KK Figure Iterations Space of LU Factorization At this point, a right-looking [13] block algorithm has been obtained. Therefore, LU factorization is blockable. The loop nest surrounding statement 10 is a matrix-matrix multiply that can be further optimized depending upon the architecture. For superscalar architectures whose performance is bound by cache, outer loop unrolling on non-rectangular loops can be applied to the J- and I-loops to further improve performance [8, 9]. For vector architectures, a different loop optimization strategy may be more beneficial [1]. Many of the transformations that we have used to obtain the block version of LU factorization are well known in the compiler community and exist in many commercial compilers (e.g., HP, DEC and SGI). One of the contributions of this study to compiler research is to show how the addition of section analysis allows a compiler to block matrix factorizations. Note that none of the aforementioned compilers uses section analysis for this purpose. 3.1.2 Adding Partial Pivoting Although the compiler can discover the potential for blocking in LU decomposition without pivoting using index-set splitting and section analysis, the same cannot be said when partial pivoting is added (see Figure 4 for LU decomposition with partial pivoting). In the partial pivoting algorithm, a new recurrence exists that does not fit the form handled by index-set splitting. Consider the following sections of code after applying index-set splitting to the algorithm in Figure 4. DO The reference to A(IMAX,J) in statement 25 and the reference to A(I,J) in statement 10 access the same sections. Distributing the KK-loop around both J-loops would convert the true dependence from A(I,J) to A(IMAX,J) into an antidependence in the reverse direction. The rules for the preservation of data dependence prohibit the reversing of a dependence direction. This would seem to preclude the existence of a block analogue similar to the non-pivoting case. However, a block algorithm that ignores the preventing recurrence and distributes the KK-loop can still be mathematically derived [15]. Consider the following. If \Gammam 1 I then !/ This result shows that we can postpone the application of the eliminator M 1 until after the application of the permutation matrix P 2 if we also permute the rows of the eliminator. Extending Equation 1 to the entire formulation we have In the implementation of the block algorithm, P i cannot be computed until step i of the point algorithm. P i only depends upon the first i columns of A, allowing the computation of k P i 's and is the blocking factor, and then the block application of the - C . pick pivot - IMAX DO Figure 4 LU Decomposition with Partial Pivoting To install the above result into the compiler, we examine its implications from a data dependence viewpoint. In the point version, each row interchange is followed by a whole-column update in which each row element is updated independently. In the block version, multiple row interchanges may occur before a particular column is updated. The same computations (column updates) are performed in both the point and block versions, but these computations may occur in different locations (rows) of the array. The key concept for the compiler to understand is that row interchanges and whole-column updates are commutative operations. Data dependence alone is not sufficient to understand this. A data dependence relation maps values to memory locations. It reveals the sequence of values that pass through a particular location. In the block version of LU decomposition, the sequence of values that pass through a location is different from the point version, although the final values are identical. Unless the compiler understands that row interchanges and column updates commute, LU decomposition with partial pivoting is not blockable. Fortunately, a compiler can be equipped to understand that operations on whole columns are commutable with row permutations. To upgrade the compiler, one would have to install pattern matching to recognize both the row permutations and whole-column updates to prove that the recurrence involving statements 10 and 25 of the index-set split code could be ignored. Forms of pattern matching are already done in commercially available compilers. Vectorizing compilers pattern match for specialized computations such as searching vectors for particular conditions [31]. Other preprocessors pattern match to recognize matrix multiplication and, in turn, output a predetermined solution that is optimal for a particular machine. So, it is reasonable to believe that pivoting can be recognized and implemented in commercial compilers if its importance is emphasized. 3.2 Cholesky Factorization When the matrix A is symmetric and positive definite, the LU factorization may be written as and D is the diagonal matrix consisting of the main diagonal of U . The decomposition of A into the product of a triangular matrix and its transpose is called the Cholesky factorization. Thus we need only work with the lower triangular half of A and essentially the same dependence analysis that applies to the LU factorization without pivoting may be used. Note that with respect to floating point computation, the Cholesky factorization only differs from LU in two regards. The first is that there are n square roots for Cholesky and the second is that only the lower half of the matrix needs to be updated. The strip mined version of the Cholesky factorization is shown below. As is the case with LU factorization, there is a recurrence between A(I,J) in statement 10 and A(I,KK) in statement 20 carried by the KK-loop. The data access patterns in Cholesky factorization are identical to LU factorization (see Figure 2), index-set splitting can be applied to the J-loop at K+KS-1 to allow the KK-loop to be distributed, achieving the LAPACK block algorithm. 3.3 QR Factorization In this section, we examine the blockability of the QR factorization. First, we show that the algorithm from LAPACK is not blockable. Then, we give an alternate algorithm that is blockable. 3.3.1 LAPACK Version The LAPACK point algorithm for computing the QR factorization consists of forming the sequence 1. The initial matrix A rows and n columns, where for this study we assume m - n. The elementary reflectors update A k in order that the first k columns of A k+1 form an upper triangular matrix. The update is accomplished by performing the matrix vector multiplication w followed by the rank one update A Efficiency of the implementation of the level 2 BLAS subroutines determines the rate at which the factorization is computed. For a more detailed discussion of the QR factorization see the book by Golub and Van Loan [20]. The LAPACK block QR factorization is an attempt to recast the algorithm in terms of calls to level 3 BLAS [15]. If the level 3 BLAS are hand-tuned for a particular architecture, the block QR algorithm may perform significantly better than the point version on large matrix sizes (those that cause the working set to be much larger than the cache size). Unfortunately, the block QR algorithm in LAPACK is not automatically derivable by a compiler. The block application of a number of elementary reflectors involves both computation and storage that does not exist in the original point algorithm [15]. To block a number of eliminators together, the following is computed The compiler cannot derive I \Gamma V TV T from the original point algorithm using dependence infor- mation. To illustrate, consider a block of two elementary reflectors !/ The computation of the matrix is not part of the original algorithm. Hence, the LAPACK version of block QR factorization is a different algorithm from the point version, rather than just a reshaping of the point algorithm for better performance. The compiler can reshape algorithms, but, it cannot derive new algorithms with data dependence information. In this case, the compiler would need to understand linear algebra to derive the block algorithm. In the next section, a compiler-derivable block algorithm for QR factorization is presented. This algorithm gives comparable performance to the LAPACK version on small matrices while retaining machine independence. 3.3.2 Compiler-Derivable QR Factorization Consider the application of j matrices V k to A k , The compiler derivable algorithm, henceforth called cd-QR, only forms columns k through k of A k+j and then updates the remainder of matrix with the j elementary reflectors. The final update of the trailing columns is "rich" in floating point operations that the compiler organizes to best suit the underlying hardware. Code optimization techniques such as strip-mine- and-interchange and unroll-and-jam are left to the compiler. The derived algorithm depends upon the compiler for efficiency in contrast to the LAPACK algorithm that depends on hand optimization of the BLAS. Cd-QR can be obtained from the point algorithm for QR decomposition using array section analysis. For reference, segments of the code for the point algorithm after strip mining of the outer loop are shown in Figure 5. To complete the transformation of the code in Figure 5 to obtain cd-QR, the I-loop must be distributed around the loop that surrounds the computation of V i and around the update before being interchanged with the J-loop. However, there is a recurrence between the definition and use of A(K,J) within the update section and the definition and use of A(J,I) in computation of The recurrence is carried by the I-loop and appears to prevent distribution. * Generate elementary reflector V-i. ENDDO * Update A(i:m,i+1:n) with V-i. ENDDO ENDDO ENDDO ENDDO ENDDO Figure 5 Strip-Mined Point QR Decomposition II II Figure 6 Regions of A Accessed by QR Decomposition Figure 6 shows the sections of the array A(:,:) accessed for the entire execution of the I-loop. If the sections accessed by A(J,I) and A(K,J) are examined, a legal partial distribution of the I-loop is revealed (note the similarity to LU and Cholesky factorization). The section accessed by A(J,I) (the black region) is a subset of the section accessed by A(K,J) (both the black and gray regions) and the index-set of J can be split at the point to create a new loop that executes over the iteration space where the memory locations accessed by A(K,J) are disjoint from those accessed by A(J,I). The new loop that iterates over the disjoint region can be further optimized by the compiler depending upon the target architecture. 3.3.3 A Comparison of the Two QR Factorizations The algorithm cd-QR does not exhibit as much cache reuse as the LAPACK version on large matrices. The reason is that the LAPACK algorithm is able to take advantage of the level 3 BLAS routine DGEMM, which can be highly optimized. Cd-QR uses operations that are closer to the level 2 BLAS and that have worse cache reuse characteristics. Therefore, we would expect the LAPACK algorithm to perform better on larger matrices as it could possibly take advantage of a highly tuned matrix-matrix multiply kernel. 3.4 Summary of Transformations In summary, Table 1 lists the analyses and transformations that must be used by a compiler to block matrix factorizations. Items 1 and 2 are discussed in Section 2. Items 3 through 7 are discussed in Section 3.1. Item 8 is discussed in the compiler literature [28, 10]. Item 9 is discussed in Section 3.1.2. Many commercial compilers (e.g. IBM[34], HP, DEC, and SGI) contain items 1, 3, 4, 5, 6, 7 and 8. However, it should be noted that items 2 and 9 are not likely to be found in any of today's commercial compilers. Table 1 Summary of the compiler transformations necessary to block matrix factorizations. Dependence Analysis (Section 2.1 [26, 19]) Array Section Analysis (Section 2.1 [5, 21]) 3 Strip-Mine-and-Interchange (Section 3.1 [38, 36]) 4 Unroll-and-Jam (Section 3.1 [9]) We measured the performance of each block factorization algorithm on four different architectures: the IBM POWER2 model 590, the HP model 712/80, the DEC Alpha 21164 and the SGI model Indigo2 with a MIPS R4400. Table 2 summarizes the characteristics of each machine. These architectures were chosen because they are representative of the typical high-performance workstation. Table Machine Characteristics Machine Clock Speed Peak Mflops Cache Size Associativity Line Size DEC Alpha 250MHz 500 8KB 1 On all the machines, we used the vendor's optimized BLAS. For example, on the IBM POWER2 and SGI Indigo2, we linked with the libraries -lessl (Engineering and Scientific Subroutine Library [22]) and -lblas, respectively. Our compiler-optimized versions were obtained by hand using the algorithms in the literature. The reason that this process could not be fully automated is because of a current deficiency in the dependence analyzer of our tool [4, 6]. Table 3 lists the FORTRAN compiler and the flags used to compile our factorizations. Table 3 FORTRAN compiler and switches. Machine Compiler Flags HP 712 f77 v9.16 -O DEC Alpha f77 v3.8 -O5 SGI Indigo2 f77 v5.3 -O3 -mips2 In Tables 4-6, performance is reported in double precision megaflops. The number of floating point operations for the LU, QR and Cholesky factorizations are 2=3n 3 respectively, where m and n are the number of rows and columns, respectively. We used the LAPACK subroutines dgetrf,dgeqrf and dpotrf for the LU, QR and Cholesky factorizations, respectively. Each factorization routine is run with block sizes of 1, 2, 4, 8, 16, 24, 32, 48, and 64. 1 In each table, the columns should be interpreted as follows: LABlk: The best blocking factor for the LAPACK algorithm. LAMf: The best megaflop rate for the LAPACK algorithm (corresponding to LABlk). CBlk: The best blocking factor for the compiler-derived algorithm. CMf: The best megaflop rate for the compiler-derived algorithm (corresponding to CBlk). In order to explicitly set the block size for the LAPACK factorizations, we have modified the LAPACK integer function ILAENV to include a common block. All the benchmarks were run when the computer systems were free of other computationally intensive jobs. All the benchmarks were typically run two or more times. The differences in time were within 5 %. 4.1 LU Factorization Table 4 shows the performance of the compiler-derived version of LU factorization with pivoting versus the LAPACK version. Table 4 LU Performance on IBM, HP, DEC and SGI Size LABlk LAMf CBlk CMf Speedup LABk LAMf CBlk CMf Speedup 100x100 200x200 300x300 DEC Alpha SGI Indigo2 Size LABlk LAMf CBlk CMf Speedup LABk LAMf CBlk CMf Speedup 100x100 200x200 300x300 500x500 1 Although the compiler can effectively choose blocking factors automatically, we do not have an implementation of the available algorithms [28, 10]. The IBM POWER2 results show that as the size of the matrix increases to 100, the compiler derived algorithm's edge over LAPACK diminishes. And for the remaining matrix sizes, the compiler derived algorithm stays within 7 % of the LAPACK one. Clearly, the FORTRAN compiler on the IBM POWER2 is able to nearly achieve the performance of the hand optimized BLAS available in the ESSL library for the block matrix factorizations. For the HP 712, Table 4 indicates an unexpected trend. The compiler-derived version performs better on all matrix sizes except 50x50, with dramatic improvements as the matrix size increases. This indicates that the hand-optimized DGEMM does not efficiently use the cache. We have optimized for cache performance in our compiler derived algorithm. This is evident when the size of the matrices exceeds the size of the cache. The significant performance degradation for the 50x50 case is interesting. For a matrix this small, cache performance is not a factor. We believe the performance difference comes from the way code is generated. For superscalar architectures like the HP, a code generation scheme called software pipelining is used to generate highly parallel code [27, 33]. However, software pipelining requires a lot of registers to be successful. In our code, we performed unroll-and-jam to improve cache performance. However, unroll-and-jam can significantly increase register pressure and cause software pipelining to fail [7]. On our version of LU decomposition, the HP compiler diagnostics reveal that software pipelining failed on the main computational loop due to high register pressure. Given that the hand-optimized version is highly software pipelined, the result would be a highly parallel hand-optimized loop and a not-as-parallel compiler-derived loop. At matrix size 25x25, there are not enough loop iterations to expose the difference. At matrix size 50x50, the difference is significant. At matrix sizes 75x75 and greater, cache performance becomes a factor. At this time, there are no known compiler algorithms that deal with the trade-offs between unroll-and-jam and software pipelining. This is an important area of future research. For the DEC Alpha, Table 4 shows that our algorithm performs as well as or better than the LAPACK version on matrices of order 100 or less. After size 100x100, the second-level cache on the Alpha, which is 96K, begins to overflow. Our compiler-derived version is not blocked for multiple levels of cache, while the LAPACK version is blocked for 2 levels of cache [25]. Thus, the compiler- derived algorithm suffers many more cache misses in the level-2 cache than the LAPACK version. It is possible for the compiler to perform the extra blocking for multiple levels of cache, but we know of no compiler that currently does this. Additionally, the BLAS algorithm utilized the following architectural features that we do not [25]: ffl The use of temporary arrays to eliminate conflicts in the level-1 direct-mapped cache and the translation lookaside buffer [28, 10]. ffl The use of the memory-prefetch feature on the Alpha to hide latency between cache and memory. Although each of these optimizations could be done in the DEC product compiler, they are not. Each optimization would give additional performance to our algorithm. Using a temporary buffer may provide a small improvement, but prefetching can provide a significant performance improvement because the latency to main memory is on the order of 50 cycles. Prefetches cannot be issued in the source code, so we were unable to try this optimization. The results on the SGI are roughly similar to those for the DEC Alpha. It is difficult for us to determine exactly why our performance is lower on smaller matrices because we have no diagnostic tools. It could again be software pipelining or some architectural feature of which we are not aware. We do note that the code generated by the SGI compiler is worse than expected. Additionally, the 2-level cache comes into play on the larger matrices. Comparing the results on the IBM POWER2 and the multi-level cache hierarchy systems (DEC and SGI), shows that our compiler-derived versions are very effective for a single-level cache. It is evident that more work needs to be done in optimizing the update portion of the factorizations to obtain the same relative performance as a single-level cache system on a multi-level cache system. 4.2 Cholesky Factorization Table 5 shows the performance of the compiler-derived version of Cholesky factorization versus the version. The IBM POWER2 results show that as the size of the matrix increases to 200, the compiler derived algorithm's edge over the LAPACK diminishes. And for the remaining matrix sizes, the compiler derived algorithm stays within 8% of the LAPACK one. As was the case for the LU factorization, the compiler version performs very well. Only for the large matrix sizes does the highly tuned BLAS used by the LAPACK factorization cause LAPACK to be faster. Table 5 shows a slightly irregular pattern for the block size used by the compiler derived algorithm. We remark that for matrix sizes 50 through 200, the MFLOP rate for the two block sizes 8 and 16 were nearly equivalent. On the HP, we observe the same pattern as we did for LU factorization. When cache performance is critical, we outperform the LAPACK version. When cache performance is not critical, the LAPACK version gives better results, except when the matrix is small. Our algorithm performed much better on the 25x25 matrix most likely due to the high overhead associated with software pipelining on short loops. Since Cholesky factorization has fewer operations than LU factorization in the update portion of the code, we would expect a high overhead associated with small matrices. Also, the effects of cache are not seen until larger matrix sizes (compared to LU factorization). This is again due to the smaller update portion of the factorization. Table 5 Cholesky Performance on IBM, HP, DEC and SGI Size LABlk LAMf CBlk CMf Speedup LABlk LAMf CBlk CMf Speedup 50x50 100x100 200x200 300x300 On the DEC, we outperform the LAPACK version up until the 500x500 matrix. This is the same pattern as seen in LU factorization except that it takes longer to appear. This is due to the smaller size of the update portion of the factorization. The results on the SGI show that the compiler derived version performs better than the LAPACK for matrix sizes up to 100. As the matrix size increases to 500 from 150, the compiler derived algorithm's performance decreases by 15% compared to that of the LAPACK factorization. We believe that this has to do with the 2-level cache hierarchy. We finally remark that although Table 5 shows a similar pattern as Table 4, there are differences. Recall, that as explained in x 3.2, the Cholesky factorization only has approximately half of the the floating point operations of LU since it neglects the strict (above the diagonal) upper triangular portion of the matrix during the update phase. Moreover, there is the computation of the square root of the diagonal element during each of the n iterations. 4.3 QR Factorization Table 6 shows the performance of the compiler-derived version of QR factorization versus the LAPACK version. Since the compiler-derived algorithm for block QR factorization has worse cache performance than the LAPACK algorithm, but O(n 2 ) less computation, we expect worse performance when the cache performance becomes critical. In plain words, the LAPACK algorithm uses the level 3 BLAS matrix multiply kernel DGEMM but the compiler derived algorithm can only utilize operations similar to the level 2 BLAS. On the HP, we see the same pattern as before. However, since the cache performance of our algorithm is not as good as the LAPACK version, we see a much smaller improvement when our algorithm has superior performance. Again, we also see that when the matrix sizes stay within the limits of the cache, LAPACK outperforms our algorithm. Table 6 QR Performance on IBM and HP Size LABlk LAMf CBlk CMf Speedup LABlk LAMf CBlk CMf Speedup 28 0.75 300x300 For the other three machines, we see the same pattern as on the previous factorizations except that our degradations are much larger for large matrices. Again, this is due to the inferior cache performance of cd-QR. An interesting trend revealed by Table 6 is that the IBM POWER2 has a slightly irregular block size pattern as the matrix size increases. We remark that only for matrix sizes less than or equal to 75, is there interesting behavior. For the first two matrix sizes, the optimal block size is larger than the dimension of the matrix. This implies that no blocking was performed; the level 3 BLAS was not used by the LAPACK algorithm. For the matrix size 75, the rate achieved by the LAPACK algorithm with block size 8 was within 4-6 % of the unblocked factorization. 5 Related Work We briefly review and summarize other investigations parallel to ours. It is evident that there is an active amount of work to remove the substantial hand coding associated with efficient dense linear algebra computations. 5.1 Blocking with a GEMM based Approach Since LAPACK depends upon a set of highly tuned BLAS for efficiency, there remains the practical question of how they should be optimized. As discussed in the introduction, an efficient set of BLAS requires a non-trivial effort in software engineering. See [23] for a discussion on software efforts to provide optimal implementations of the level 3 BLAS. An approach that is both efficient and practical is the GEMM-based one proposed by K-agstr-om, Ling and Van Loan [23] in a recent study. Their approach advocates optimizing the general matrix multiply and add kernel GEMM and then rewriting the remainder of the level 3 BLAS in terms of calls to this kernel. The benefit of their approach is that only this kernel needs to be optimized- whether by hand or the compiler. Their thorough analysis highlights the many issues that must be considered when attempting to construct a set of highly tuned BLAS. Moreover, they provide high quality implementations of the BLAS for general use as well as a performance evaluation benchmark [24]. We emphasize that our study examines only whether the necessary optimizations may be left to the compiler, and, also whether they should be applied directly to the matrix factorizations themselves. What is beyond the ability of the compiler is that of recasting the level 3 BLAS in terms of calls to GEMM. 5.2 PHiPAC Another recent approach is the methodology expressed for developing a Portable High-Performance matrix-vector libaries in ANSI C (PHiPAC) [3]. The project is motivated by many of the reasons as outlined in our introduction. There is a major difference in approaches which does not make it a parallel study. As in the GEMM based approach, they seek to support the BLAS and aim to be more efficient than the vendor supplied BLAS. However, unlike our study or the GEMM one, PHiPAC assumes that ANSI C is the programming language. Because of various C semantics PHiPAC instead seeks to provide parameterized generators that produce the optimized code. See the report [3] for a discussion on the inhibitors in C that prevent an optimizing compiler from generating efficient code. 5.3 Auto-blocking Matrix Multiplication Frens and Wise present an alternative algorithm for matrix-matrix multiply that is based upon a quadtree representation of matrices [17]. Their solution is recursive and suffers from the lack of interprocedural optimization in most commercial compilers. Their results show that when paging becomes a problem on SGI multiprocessor systems, the quadtree algorithm has superior performance to the BLAS 3. On smaller problems, the quadtree algorithm has inferior performance. In relation to our work, we could not expect the compiler to replace the BLAS 3 with the quadtree approach when appropriate as it is a change in algorithm rather than a reshaping. In addition, the specialized storage layout used by Frens and Wise calls into question the effect on an entire program. 6 Summary We have set out to determine whether a compiler can automatically restructure matrix factorizations well enough to avoid the need for hand optimization. To that end, we have examined a collection of implementations from LAPACK. For each of these programs, we determined whether a plausible compiler technology could succeed in obtaining the block version from the point algorithm. The results of this study are encouraging: we have demonstrated that there exist implementable compiler methods that can automatically block matrix factorization codes to achieve algorithms that are competitive with those of LAPACK. Our results show that for modest-sized matrices on advanced microprocessors, the compiler-derived variants are often superior. These matrix sizes are typical on workstations. Given that future machine designs are certain to have increasingly complex memory hierarchies, compilers will need to adopt increasingly sophisticated memory-management strategies so that programmers can remain free to concentrate on program logic. Given the potential for performance attainable with automatic techniques, we believe that it is possible for the user to express machine-independent point matrix factorization algorithms without the BLAS and still get good performance if compilers adopt our enhancements to already existing methods. Acknowledgments Ken Kennedy and Richard Hanson provided the original motivation for this work. Ken Kennedy, Keith Cooper and Danny Sorensen provided financial support for this research when it was begun at Rice University. We also wish to thank Tomas Lofgren and John Pieper of DEC for their help with obtaining the DXML libraries and diagnosing the compiler's performance, respectively. We also thank Per Ling of the University of Ume-a, Ken Stanley of the University of California Berkeley for their help with the benchmarks and discussions. --R Automatic translation of Fortran programs to vector form. A portable A parallel programming environment. Analysis of interprocedural side effects in a parallel programming environment. Improving software pipelining with unroll-and-jam Compiler blockability of numerical algorithms. Improving the ratio of memory operations to floating-point operations in loops Tile size selection using cache organization. A set of level 3 Basic Linear Algebra Subprograms. An extended set of Fortran Basic Linear Algebra Subprograms. Solving Linear systems on Vector and shared memory computers. Solving Linear Systems on Vector and Shared-Memory Computers Implementing linear algebra algorithms for dense matrices on a vector pipeline machine. Parallel algorithms for dense linear algebra computations. Practical dependence testing. Matrix Computations. An implementation of interprocedural bounded regular section analysis. The Structure of Computers and Computations Volume Software pipelining: An effective scheduling technique for vliw machines. The cache performance and optimizations of blocked algorithms. Basic linear algebra subprograms for fortran usage. Implementing efficient and portable dense matrix factorizations. A comparative study of automatic vectorizing compilers. Introduction to Parallel and Vector Solutions of Linear Systems. Register allocation for software pipelined loops. Automatic Selection of High Order Transformations in the IBM XL Fortran Compilers. A data locality optimizing algorithm. Advanced loop interchange. Iteration space tiling for memory hierarchies. --TR Automatic translation of FORTRAN programs to vector form An extended set of FORTRAN basic linear algebra subprograms Software pipelining: an effective scheduling technique for VLIW machines Introduction to Parallel MYAMPERSANDamp; Vector Solution of Linear Systems Analysis of interprocedural side effects in a parallel programming environment Parallel algorithms for dense linear algebra computations A set of level 3 basic linear algebra subprograms The cache performance and optimizations of blocked algorithms Practical dependence testing A data locality optimizing algorithm Register allocation for software pipelined loops Compiler blockability of numerical algorithms Memory-hierarchy management Improving the ratio of memory operations to floating-point operations in loops Tile size selection using cache organization and data layout Matrix computations (3rd ed.) Auto-blocking matrix-multiplication or tracking BLAS3 performance from source code Automatic selection of high-order transformations in the IBM XL FORTRAN compilers Basic Linear Algebra Subprograms for Fortran Usage Solving Linear Systems on Vector and Shared Memory Computers Structure of Computers and Computations An Implementation of Interprocedural Bounded Regular Section Analysis Iteration Space Tiling for Memory Hierarchies Implementing Efficient and Portable Dense Matrix Factorizations Improving Software Pipelining With Unroll-and-Jam --CTR Mahmut Kandemir , J. Ramanujam , Alok Choudhary, Improving Cache Locality by a Combination of Loop and Data Transformations, IEEE Transactions on Computers, v.48 n.2, p.159-167, February 1999 Nikolay Mateev , Vijay Menon , Keshav Pingali, Fractal symbolic analysis, Proceedings of the 15th international conference on Supercomputing, p.38-49, June 2001, Sorrento, Italy Kgstrm , Per Ling , Charles van Loan, GEMM-based level 3 BLAS: high-performance model implementations and performance evaluation benchmark, ACM Transactions on Mathematical Software (TOMS), v.24 n.3, p.268-302, Sept. 1998 Steve Carr , Soner nder, A case for a working-set-based memory hierarchy, Proceedings of the 2nd conference on Computing frontiers, May 04-06, 2005, Ischia, Italy Jeremy D. Frens , David S. Wise, Auto-blocking matrix-multiplication or tracking BLAS3 performance from source code, ACM SIGPLAN Notices, v.32 n.7, p.206-216, July 1997 Qing Yi , Vikram Adve , Ken Kennedy, Transforming loops to recursion for multi-level memory hierarchies, ACM SIGPLAN Notices, v.35 n.5, p.169-181, May 2000 Qing Yi , Ken Kennedy , Haihang You , Keith Seymour , Jack Dongarra, Automatic blocking of QR and LU factorizations for locality, Proceedings of the 2004 workshop on Memory system performance, June 08-08, 2004, Washington, D.C. Vijay Menon , Keshav Pingali , Nikolay Mateev, Fractal symbolic analysis, ACM Transactions on Programming Languages and Systems (TOPLAS), v.25 n.6, p.776-813, November Qing Yi , Ken Kennedy , Vikram Adve, Transforming Complex Loop Nests for Locality, The Journal of Supercomputing, v.27 n.3, p.219-264, March 2004

LAPACK;LU decomposition;QR decomposition;BLAS;cholesky decomposition;cache optimization

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Component Based Design of Multitolerant Systems.

AbstractThe concept of multitolerance abstracts problems in system dependability and provides a basis for improved design of dependable systems. In the abstraction, each source of undependability in the system is represented as a class of faults, and the corresponding ability of the system to deal with that undependability source is represented as a type of tolerance. Multitolerance thus refers to the ability of the system to tolerate multiple fault-classes, each in a possibly different way. In this paper, we present a component based method for designing multitolerance. Two types of components are employed by the method, namely detectors and correctors. A theory of detectors, correctors, and their interference-free composition with intolerant programs is developed, that enables stepwise addition of components to provide tolerance to a new fault-class while preserving the tolerances to the previously added fault-classes. We illustrate the method by designing a fully distributed multitolerant program for a token ring.

Introduction Dependability is an increasingly relevant system-level requirement that encompasses the ability of a system to deliver its service in a desirable manner, in spite of the occurrence of faults, security intrusions, safety hazards, configuration changes, load variations, etc. Achieving this ability is difficult, essentially because engineering a system for the sake of one dependability property, say availability in the presence of faults, often interferes with another desired dependability property, say security in presence of intrusions. In this paper, to effectively reason about multiple dependability properties, we introduce the concept of multitolerance. Each source of undependability is treated as a class of "faults" and each dependability property is treated as a type of "tolerance". Thus, multitolerance refers to the ability of a system to tolerate multiple classes of faults, each in a possibly different way. Although there are many examples of multitolerant systems in practice [1 \Gamma 3] and there exists a growing body of research that presents instances of multitolerant systems [4 \Gamma 9], we are not aware of previous work that has considered the systematic design of multitolerance. Towards redressing this deficiency, we present in this paper a formal method for the design of multitolerant systems. To deal with the difficulty of interference between multiple types of tolerances, our method is based on the use of components. More specifically, a multitolerant system designed using the method consists of an intolerant system and a set of components, one for each desired type of tolerance. Thus, the method reduces the complexity of design to that of designing the components and that of correctly adding them to the intolerant system. Moreover, it enables reasoning about each type of tolerance and the interferences between different types of tolerance at the level of the components themselves, as opposed to involving the whole system. The method further reduces the complexity of adding multiple components to an intolerant system by adding each component in a stepwise fashion. In other words, the method considers the fault- classes that an intolerant system is subject to in some fixed total order, say F1 :: Fn. A component is added to the intolerant system so that it tolerates F1 in a desirable manner. The resulting system is then augmented with another component so that it tolerates F2 in a desirable manner and its tolerance to F1 is preserved. This process of adding a new tolerance and preserving all old tolerances is repeated until all n fault-classes are accounted for. It follows that the final system is multitolerant with respect to F1 :: Fn. Components used in our method are built out of two building blocks, namely detectors and correctors, that occur -albeit implicitly- in fault-tolerant systems. Intuitively, a detector "detects" whether some predicate is satisfied by the system state; and a corrector detects whether some predicate is satisfied by the system state and also "corrects" the system state in order to satisfy that predicate whenever the predicate is not satisfied. Detectors can be used to ensure that each step of the system is "safe" with respect to its "problem specification", while correctors can be used to ensure that the system eventually reaches a state from where its problem specification is (re)satisfied. Thus, in this paper, we are also able to show that components built out of detectors are sufficient for designing "fail-safe" tolerance in programs, that components built out of correctors are sufficient for designing "nonmasking" tolerance in programs, and that components built out of both detectors and correctors are sufficient for designing "masking" tolerance in programs. (We will formally define each of these terms shortly.) The rest of this paper is organized as follows. In Section 2, we give formal definitions of programs, their problem specifications, their faults, and their fault-tolerances. In Sections 3 and 4, we define detectors and correctors, discuss their role in the design of fault-tolerant systems, and illustrate how they can be designed hierarchically and efficiently. In Section 5, we present a theory of non-interference for composing detectors and correctors with intolerant programs. In Section 6, we define multitolerance and present our formal method for designing multitolerance. In Section 7, we illustrate our method by designing a multitolerant token ring program. Finally, we discuss some issues raised by our methodology in Section 8 and make concluding remarks in Section 9. Preliminaries In this section, we give formal definitions of programs, problem specifications, faults, and fault- tolerances. The formalization of programs is a standard one, that of specifications is adapted from Alpern and Schneider [9], and that of faults and fault-tolerances is adapted from earlier work of the first author with Mohamed Gouda [10] (with the exception of the portion on fail-safe tolerance which is new). Programs. A program is a set of variables and a finite set of actions. Each variable has a predefined nonempty domain. Each action has a unique name, and is of the form: The guard of each action is a boolean expression over the program variables. The statement of each action is such that its execution atomically and instantaneously updates zero or more program variables. Notation. To conveniently write an action as a restriction of another action, we use the notation hname to define an action hname 0 i whose guard is obtained by restricting the guard of action hnamei with hguard 0 i, and whose statement is identical to the statement of action hnamei. Operationally speaking, hname 0 i is executed only if the guard of hnamei and the guard hguard 0 i are both true. Likewise, to conveniently write a program as a restriction of another program, we use the notation to define a program consisting of the set of actions hguardi - ac for each action ac of hprogrami. Let p be a program. Definition (State). A state of p is defined by a value for each variable of p, chosen from the predefined domain of the variable. Definition (State predicate). A state predicate of p is a boolean expression over the variables of p. Note that a state predicate may be characterized by the set of all states in which the boolean expression (of the state predicate) is true. Definition (Enabled). An action of p is enabled in a state iff its guard (which is a state predicate) is true in that state. Definition (Computation). A computation of p is a fair, maximal sequences of states s that for each j, j ? 0, s j is obtained from state s j \Gamma1 by executing an action of p that is enabled in the state s j \Gamma1 . Fairness of the sequence means that each action in p that is continuously enabled along the states in the sequence is eventually chosen for execution. Maximality of the sequence means that if the sequence is finite then the guard of each action in p is false in the final state, i.e., the sequence cannot be further extended by executing an enabled action in its final state. Problem specification. problem specification is a set of sequences of states that is suffix closed and fusion closed. Suffix closure of the set means that if a state sequence oe is in that set then so are all the suffixes of oe. Fusion closure of the set means that if state sequences hff; x; fli and hfi; x; ffii are in that set then so are the state sequences hff; x; ffii and hfi; x; fli, where ff and fi are finite prefixes of state sequences, fl and ffi are suffixes of state sequences, and x is a program state. Note that the state sequences in a problem specification may be finite or infinite. Following Alpern and Schneider [9], it can be shown that any problem specification is the intersection of some "safety" specification that is suffix closed and fusion closed and some "liveness" specification, defined next. Definition (Safety). A safety specification is a set of state sequences that meets the following condition: for each state sequence oe not in that set, there exists a prefix ff of oe, such that for all state sequences fi, fffi is not in that set (where fffi denotes the concatenation of ff and fi). Definition (Liveness). A liveness specification is a set of state sequences that meets the following condition: for each finite state sequence ff, there exists a state sequence fi such that fffi is in that set. Defined below are some examples of problem specifications, namely, generalized pairs, closures, and converges to. Let S and R be state predicates. Definition (Generalized Pairs). A generalized pair (fSg; fRg) is a set of all state sequences, s such that for each j; j - 0, if S is true at s j then R is true at s j+1 . Definition (Closure). The closure of S, S , is the set of all state sequences s is true at s j then for all k, k - j, S is true at s k . Definition (Converges to). S converges to R is the set of all state sequences s in the intersection of S and R such that for all is true at s i then there exists k, k- i, where R is true at s k . Note that (fSg; converges to S. Program correctness with respect to a problem specification. Let SPEC be a problem specification. Definition (Satisfies). p satisfies SPEC for S iff each computation of p that starts at a state where S is true is in SPEC. Definition (Violates). p violates SPEC for S iff it is not the case that p satisfies SPEC for there exists a computation of p that starts at a state where S is true and is not in SPEC. For convenience in reasoning about programs that satisfy special cases of problem specifications, we introduce the following notational abbreviations. Definition (Generalized Hoare-triples). fSg p fRg iff p satisfies the generalized pair (fSg; fRg) for true. Definition (Closed in p). S is closed in p iff p satisfies S for true. Note that it is trivially true that the state predicates true and false are closed in p. Definition (Converges to in p). S converges to R in p iff p satisfies S converges to R for true. Informally speaking, proving the correctness of p with respect to SPEC involves showing that p satisfies SPEC for some state predicate S. (Of course, to be useful, the predicate S should not be false.) Now, since problem specifications are suffix closed, we may without loss of generality restrict our attention to proving that p satisfies the problem specification for some closed state predicate S. We call such a state predicate S an invariant of p. Invariants enable proofs of program correctness that eschew operational arguments about long (sub)sequences of states, and are thus methodologically advantageous. Definition (Invariant). S is an invariant of p for SPEC iff S is closed in p and p satisfies SPEC for S. Notational remark. Henceforth, whenever the problem specification is clear from the context, we will omit it; thus, "S is an invariant of p" abbreviates "S is an invariant of p for SPEC ". One way to calculate an invariant of p is to characterize the set of states that are reachable under execution of p starting from some designated "initial" states. Experience shows, however, that for ease of proofs of program correctness one may prefer to use invariants of p that properly include such a reachable set of states. This is a key reason why we have not included initial states in the definition of programs. Techniques for the design of invariants have been articulated by Dijkstra [11], using the notion of auxiliary variables, and by Gries [12], using the heuristics of state predicate ballooning and shrinking. Techniques for the mechanical calculation of invariants have been discussed by Alpern and Schneider [13]. Faults. The faults that a program is subject to are systematically represented by actions whose execution perturbs the program state. We emphasize that such representation is possible notwithstanding the type of the faults (be they stuck-at, crash, fail-stop, omission, timing, performance, or Byzantine), the nature of the faults (be they permanent, transient, or intermittent), or the ability of the program to observe the effects of the faults (be they detectable or undetectable). Definition (Fault-class). A fault-class for p is a set of actions over the variables of p. Let T be a state predicate, S an invariant of p, and F a fault-class for p. Definition (Preserves). An action ac preserves a state predicate T iff execution of ac in any state where T is true results in a state where T is true. is an F -span of p for S iff S ) T , T is closed in p, and each action of F preserves T . Thus, at each state where an invariant S of p is true, an F -span T of p for S is also true. Also, like S, T is also closed in p. Moreover, if any action in F is executed in a state where T is true, the resulting state is also one where T is true. It follows that for all computations of p that start at states where S is true, T is a boundary in the state space of p up to which (but not beyond which) the state of p may be perturbed by the occurrence of the actions in F . Notational remark. Henceforth, we will ambiguously abbreviate the phrase "each action in F preserves T " by "T is closed in F ". And, whenever the program p is clear from the context, we will omit it; thus, "S is an invariant" abbreviates "S is an invariant of p" and "F is a fault-class" abbreviates "F is a fault-class for p". Fault-tolerances. We are now ready to define what it means for a program p with an invariant S to tolerate a fault-class F . Definition -tolerant for S). p is F -tolerant for S iff there exists a state predicate T that satisfies the following three conditions: ffl At any state where S is true, T is also true. (In other words, S ) T .) ffl Starting from any state where T is true, if any action in p or F is executed, the resulting state is also one where T is true. (In other words, T is closed in p and each action in F preserves .) ffl Starting from any state where T is true, every computation of p alone eventually reaches a state where S is true. (In other words, since S and T are closed in p, T converges to S in p.) This definition may be understood as follows. The state predicate T is an F -span of p for S- a boundary in the state space of p up to which (but not beyond which) the state of p may be perturbed by the occurrence of faults in F . If faults in F continue to occur, the state of p remains within this boundary. When faults in F stop occurring, p converges from this boundary to the stricter boundary in the state space where the invariant S is true. It is important to note that there may be multiple such state predicates T for which p meets the above three requirements. Each of these multiple T state predicates captures a (potentially different) type of fault-tolerance of p. We will exploit this multiplicity in Section 6 in order to define multitolerance. Types of fault-tolerances. We now classify three types of fault-tolerances that a program can exhibit, namely masking, nonmasking, and fail-safe tolerance, using the above definition of F -tolerance. Informally speaking, this classification is based upon the extent to which the program satisfies its problem specification in the presence of faults. Of the three, masking is the strictest type of tolerance: in the presence of faults, the program always satisfies its safety specification, and the execution of p after execution of actions in F yields a computation that is in both the safety and liveness specification of p, i.e., the computation is in the problem specification of p. Nonmasking is less strict than masking: in the presence of faults, the program need not satisfy its safety specification but, when faults stop occurring, the program eventually resumes satisfying both its safety and liveness specification; i.e., the computation has a suffix that is in the problem specification. Fail-safe is also less strict than masking: in the presence of faults, the program always satisfies its safety specification but, when faults stop occurring, the program need not resume satisfying its liveness specification; i.e., the computation is in the safety specification -but not necessarily in the liveness specification- of p. Formally, these three types of tolerance may be expressed in terms of the definition of F -tolerance, as follows: Definition (masking tolerant). p is masking tolerant to F for S iff p is F -tolerant for S and S is closed in F . (In other words, if a fault in F occurs in a state where S is true, p continues to be in a state where S is true.) Definition (nonmasking tolerant). p is nonmasking tolerant to F for S iff p is F -tolerant for S and S is not closed in F . (In other words, if a fault in F occurs in a state where S is true, p may be perturbed to a state where S is violated. However, p then recovers to a state where S is true.) Definition (fail-safe tolerant). p is fail-safe tolerant to F for S iff there exists a state predicate R such that p is F -tolerant for S - R, S - R is closed in p and in F , and p satisfies the safety specification (of the problem specification) for S - R. (In other words, if a fault in F occurs in a state where S is true, p may be perturbed to a state where S or R is true. In the latter case, the subsequent execution of p yields a computation that is in the safety specification of p but not necessarily in the liveness specification.) Notation. In the sequel, whenever the fault-class F and invariant S are clear from the context, we omit them; thus, "masking tolerant" abbreviates "masking tolerant to F for S", and so on. Detectors In this section, we define the first of two building blocks which are sufficient for the design of fault-tolerant programs, namely detectors. We also present the properties of detectors, show how to construct them in a hierarchical and efficient manner, and discuss their role in the design of fault-tolerance. As mentioned in the introduction, intuitively, a detector is a program that "detects" whether a given state predicate is true in the current system state. Implementations of detectors abound in practice: Wellknown examples include comparators, error detection codes, consistency checkers, watchdog programs, snoopers, alarms, snapshot procedures, acceptance tests, and exception conditions. Z be state predicates of a program d and U be a state predicate that is closed in d. We say that "Z detects X in d for U " iff the following three conditions hold: ffl (Safeness) At any state where U is true, if Z is true then X is also true. (In other words, U ffl (Progress) Starting from any state where U -X is true, every computation of d either reaches a state where Z is true or a state where X is false. ffl (Stability) Starting from any state where U - Z is true, d falsifies Z only if it also falsifies X . (In other words, fU - Zg d fZ - :Xg.) The Safeness condition implies that a detector d never lets the predicate Z "witness" the detection predicate X incorrectly when executed in states where U is true. The Progress condition implies that in any computation of d starting from a state where U is true, if X is true continuously then d eventually detects X by truthifying Z. The Stability condition implies that once Z is truthified, it continues to be true unless X is falsified. Remark. If the detection predicate X is closed in d, our definition of the detects relation reduces to one given by Chandy and Misra [14]. We have considered this more general definition to accommodate the case -which occurs for instance in nonmasking tolerance- where X denotes that "something bad has happened"; in this case, X is not supposed to be closed since it has to be subsequently corrected. (End of Remark.) In the sequel, we will implicitly assume that the specification of a detector d (dn) is "Z detects X in d for U " (respectively, "Zn detects Xn in dn for Un"). Also, we will implicitly assume that U (Un) is closed in d (respectively, dn). Properties. The detects relation is reflexive, antisymmetric, and transitive in its first two arguments Lemma 3.0 Let X , Y , and Z be state predicates of d and U be a state predicate that is closed in d. The following statements hold. detects X in d for U ffl If Z detects X in d for U , and X detects Z in d for U then U ffl If Z detects Y in d for U , and Y detects X in d for U then Z detects X in d for U Lemma 3.1 Let V be a state predicate such that U - V is closed in d. The following statements hold. ffl If Z detects X in d for U then Z detects X in d for U-V ffl If Z detects X in d for U , and V ) X then Z-V detects X in d for U ffl If Z detects X in d for U , and Z then Z detects X-V in d for U Compositions. Regarding the construction of detectors, there are cases where a detection predicate X cannot be witnessed atomically, i.e., by executing at most one action of a detector program. To detect such predicates, we give compositions of "small" detectors that yield "large" detectors in a hierarchical and efficient manner. In particular, given two detectors, d1 and d2, we compose them in two ways: (i) in parallel and (ii) in sequence. Parallel composition of detectors. In the parallel composition of d1 and d2, denoted by d1[]d2, both d1 and d2 execute in an interleaved fashion. Formally, the parallel composition of d1 and d2 is the union of the (variables and actions of) programs d1 and d2. Observe that '[]' is commutative: d1[](d2[]d3), and that '-' distributes over `[]': g - Theorem 3.2 Let Z1 detect X1 in d1 for U and Z2 detect X2 in d2 for U . If the variables of d1 and d2 are mutually exclusive then Z1-Z2 detects X1-X2 in d1[]d2 for U Proof. The Safeness condition of d1[]d2 follows trivially from the Safeness of d1 and of d2. For the Progress condition, we consider two cases, (i) a computation of d1[]d2 falsifies X1 - X2 and (ii) a computation of d1[]d2 never falsifies X1 - X2: In the first case, Progress is satisfied trivially. In the second case, eventually Z1 is truthified, and by Stability of d1, Z1 continues to be true in the execution of d1. moreover, since the variables of d1 and d2 are disjoint, Z1 continues to be true in d2. Likewise, Z2 is eventually truthified and continues to be true. Thus, Progress is satisfied. Finally, the Stability condition is satisfied since Z1-Z2 can be falsified only if X1 or X2 are violated. In d1[]d2, since d1 and d2 perform their detection concurrently, the time required for detection of X1-X2 is the maximum time taken to detect X1 and to detect X2. (We are assuming that the unit for measuring time allows both d1 and d2 to attempt execution of an action each.) Also, the space complexity of d1[]d2 is the sum of the space complexity of d1 and d2, since the state space of d1[]d2 is the union of the state space of d1 and of d2. Sequential composition of detectors. In the sequential composition of d1 and d2, denoted by d1; d2, d2 executes only after d1 has completed its detection, i.e., after the witness predicate Z1 is true. Formally, the sequential composition of d1 and d2 is the program whose set of variables is the union of the variables of d1 and d2 and whose set of actions is the union of the actions of d1 and of Z1-d2. We postulate the axiom that ';' is left-associative: d1; d2; Observe that ';' is not commutative, that `;' distributes over '[]': d1; and that '-' distributes over `;': g - (d1; Suppose, again, that the variables of d1 and d2 are mutually exclusive. In this case, starting from any state where X1-X2 is true continuously, d1 eventually truthifies Z1. Only after Z1 is truthified are the actions of d2 executed; these actions eventually truthify Z2. Since Z2 is truthified only when Z1 (and, hence, X1) and X2 are true, it also follows that U assume U Theorem 3.3 Let Z1 detect X1 in d1 for U and Z2 detect X2 in d2 for U-X1. If the variables of d1 and d2 are mutually exclusive, and U then Z2 detects X1-X2 in d1; d2 for U In d1; d2, the time (respectively, space) taken to detect X1-X2 is the sum of the time (respectively, space) taken to detect X1 and to detect X2. The extra time taken by d1; d2 as compared to d1[]d2 is warranted in cases where the witness predicate Z2 can be witnessed atomically but Z1-Z2 cannot. Example: Memory access. Let us consider a simple memory access program that obtains the value stored at a given address (cf. Figure 1). The program is subject to two fault-classes: The first consists of protection faults which cause the given address to be corrupted so that it falls outside the valid address space, and the second consists of page faults which remove the address and its value from the memory. For tolerance to the first fault-class, there is a detector d1 that uses the page table TBL to detect whether the address addr is valid (X1). For tolerance to the second fault-class, there is another detector d2 that uses the memory MEM to detect whether the given address is in memory (X2). d2 MEM addr Figure 1: Memory access Formally, these detectors are as follows (for simplicity, we assume that TBL is a set of valid addresses and MEM is a set of objects of the form haddr; vali): Thus, we may observe: ffl Z1 detects X1 in d1 for U1 (1) ffl Z2 detects X2 in d2 for U1 - X1 (2) Note that an appropriate choice of initial state in U1 would be one where both Z1 and Z2 are false. Note also that, in U1, Z1 is truthified only when X1 is true and that Z2 is truthified only when X1 and X2 are both true. To detect X1-X2, we may compose d1 and d2 sequentially: d1 would first detect X1, and then d2 would detect X2. From Theorem 3.3, (1) and (2) we get: ffl Z2 detects X1 - X2 in d1; d2 for U1 (3) Application to design of fault-tolerance. Detectors suffice to ensure that a program satisfies its safety specification. To see this, recall that a safety specification essentially rules out certain finite prefixes of program computation. Now, consider any prefix of a computation that is not ruled out by the safety specification. Execution of a program action starting from this prefix does not violate the safety specification iff the elongated prefix is not ruled out by the safety specification. In other words, for each program action ac there exists a set of computation prefixes from which execution of ac does not violate the safety specification. It follows that there exists a "detection" state predicate such that execution of ac in any state where that state predicate is true does not violate the safety specification. (From the fusion closure of the safety specification, it suffices that this state predicate characterize a set of states, each state st of which yields upon executing ac a successor state st 0 such that there is some state sequence in the safety specification in which st and st 0 occur consecutively in that order.) Now, if detectors can be added to the program so that for each program action ac a detection predicate of ac is witnessed, and each program action can be restricted to execute only if its corresponding witness predicate is true, the resulting program satisfies the safety specification. To design fail-safe tolerance to F for S, we need to ensure that upon starting from a state where S is true, the execution of p in the presence of F always yields a computation that is in the safety specification of p. It follows that detectors suffice for the design of fail-safe tolerance. Likewise, to design masking tolerance to F for S, we need to ensure that upon starting from a state where S is true, the execution of p in the presence of F never violates the safety specification and the execution of p after execution of actions in F always yields a computation that is in both the safety and the liveness specification of p, i.e., that computation is in the problem specification of p. (From the fusion closure of the problem specification, it follows that a computation of p that is in the safety specification and that has a suffix in the problem specification is itself in the problem specification.) Now, regarding safety, it suffices that detectors be added to p. (Regarding liveness, it suffices that correctors be added to p, as discussed in the next section.) Detectors can also play a role in the design of nonmasking tolerance: They may be used to detect whether the program is perturbed to a state where its invariant is false. As discussed in the next section, such detectors can be systematically composed with correctors that restore the program to a state where its invariant is true. In this section, we discuss the second set of building blocks, namely correctors, in a manner analogous to our discussion of detectors. As mentioned in the introduction, intuitively, a corrector is a detector that also "corrects" the program state whenever it detects that its detection predicate is false in the current system state. Implementations of correctors also abound in practice: Wellknown examples include voters, error correction codes, reset procedures, rollback recovery, rollforward recovery, constraint (re)satisfaction, exception handlers, and alternate procedures in recovery blocks. Z be state predicates of a program c and U be closed in c. We say that "Z corrects X in c for U " iff the following four conditions holds: ffl (Safeness) At any state where U is true, if Z is true then X is also true. (In other words, U ffl (Convergence) Starting from any state where U is true, every computation of c eventually reaches a state where X is true, and subsequently, X continues to be true thereafter. (In other words, U converges to X in c.) Starting from any state where U - X are true, every computation of c either reaches a state where Z is true or a state where X is false. ffl (Stability) Starting from any state where U -Z is true, c falsifies Z only if it also falsifies X . (In other words, fU - Zg c fZ - :Xg.) From the above definition, it follows that if Z corrects X in c for U , then Z detects X in c for U . It also follows, that U - X is closed in c (cf. Convergence). Consequently, starting from any state where U -X is true, every computation of c reaches a state where Z is true (cf. Progress). moreover, U - Z is closed in c (cf. Stability). Remark. If the witness predicate Z is identical to the correction predicate X , our definition of the corrects relation reduces to one given by Arora and Gouda [10]. We have considered this more general definition to accommodate the case -which occurs for instance in masking tolerance- where the witness predicate Z can be checked atomically but the correction predicate X cannot. (End of Remark.) Properties. The corrects relation is antisymmetric and transitive in its first two arguments: Lemma 4.0 Let X , Y , and Z be state predicates of c and U be a state predicate that is closed in c. The following statements hold. ffl If Z corrects X in c for U , and X corrects Z in c for U then U ffl If Z corrects Y in c for U , and Y corrects X in c for U then Z corrects X in c for U Lemma 4.1 Let V be a state predicate such that U - V is closed in c. Then the following statements hold. ffl If Z corrects X in c for U then Z corrects X in c for U - V ffl If Z corrects X in c for U and V ) X then Z-V corrects X in c for U Compositions. Analogous to detection predicates that cannot be witnessed atomically, there exist cases where a correction predicate X cannot be corrected atomically, i.e., by executing at most one step (action) of a corrector. To correct such predicates, we construct "large" correctors from "small" correctors just as we did for detectors. Parallel composition of correctors. The parallel composition of two correctors c1 and c2, denoted by c1[]c2, is the union of the (variables and actions of) programs c1 and c2. Theorem 4.2 Let Z1 correct X1 in c1 for U and Z2 correct X2 in c2 for U . If the variables of c1 and c2 are mutually exclusive then Z1-Z2 corrects X1-X2 in c1[]c2 for U The time taken by c1[]c2 to correct X1-X2 is the maximum of the time taken to correct X1 and to correct X2. The space taken is the corresponding sum. Sequential composition of correctors. The sequential composition of correctors c1 and c2, denoted by c1; c2, is the program whose set of variables is the union of the variables of c1 and c2 and whose set of actions is the union of the actions of c1 and of Z1 - c2. Theorem 4.3 Let Z1 correct X1 in c1 for U and Z2 correct X2 in c2 for (U -X1). If the variables of c1 and c2 are mutually exclusive, and U then Z2 corrects X1-X2 in c1; c2 for U The time (respectively, space) taken by c1; c2 to correct X1-X2 is the sum of the time (respectively, space) taken to correct X1 and to correct X2. As mentioned in the previous section, one way to design a corrector for X is by sequential composition of a detector and a corrector: the detector first detects whether :X is true and, using this witness, the corrector then establishes X . Theorem 4.4 Let Z detect :X in d for U , Z , and X correct X in c for U - Z 0 If X is closed in d, and fU - Zg c fZ - Xg then X corrects X in (:Z - d); c for U If c is atomic, i.e., c satisfies Progress and Convergence in at most one step, the following corollary holds. Corollary 4.5 Let Z detect :X in d for U , Z , and X correct X in c for U - Z 0 If X is closed in d, and c is atomic then X corrects X in d; c for U Another way to design a corrector for X is by sequential composition of a corrector and a detector: the corrector first satisfies its correction predicate X and then the detector satisfies the desired witness predicate Z. Theorem 4.6 Let X correct X in c for U , and Z detect X in d for U . If X is closed in d then Z corrects X in (:X - c); d for U Again, if d is atomic, the following corollary holds. Corollary 4.7 Let X correct X in c for U , and Z detect X in d for U . If X is closed in d and c is atomic then Z corrects X in c; d for U Example: Memory access (continued). If the given address is valid but is not in memory, an object of the form haddr; \Gammai has to be added to the memory. (We omit the details of how this object is obtained, e.g., from a disk, a remote memory, or a network.) Thus, there is a corrector, c, which is formally specified as follows: addr d2 c Figure 2: Memory access Thus, we may observe: ffl X2 corrects X2 in c for true (4) ffl X2 corrects X2 in c for U1 (5) ffl X2 corrects X2 in c for U1 - X1 (6) Before detector d2 can witness that the value of the address is in memory, corrector c should execute. Hence, we compose c and d2 sequentially. From Corollary 4.7, (2) and (6) we have: ffl Z2 corrects X2 in c; d2 for U1 - X1 (7) moreover, detector d1 should witness that the address is valid, before either corrector c or detector d2 execute. Hence, we compose d1 and c; d2 sequentially. Recall from Section 3 that Z1 detects X1 in d1 for U1. Also, X1 is closed in d1. Hence, if d1 is started in a state satisfying U1 - X1, it will eventually satisfy Z1. It follows that Z1 corrects X1 in d1 for U1 - X1. Therefore, from Theorem 4.3 and (9), we have: ffl Z2 corrects X1 - X2 in d1; (c; d2) for U1 - X1 (8) Application to design of fault-tolerance. Correctors suffice to ensure that computations of a program have a suffix in the problem specification. To see this, observe that if the correction predicate of a corrector is chosen to be an invariant of the program, the corrector ensures the program will eventually reach a state where that invariant is true and henceforth the program computation is in the problem specification. To design nonmasking tolerance to F for an invariant S, we need to ensure that upon starting from a state where S is true, execution of p will, after execution of actions in F , always yield a computation that has a suffix in the problem specification. It follows that correctors whose correction predicate is the invariant S suffice for the design of nonmasking tolerance. Likewise, to design masking tolerance to F for S, we need to ensure that upon starting from a state where S is true, execution of p in the presence of F never violates its safety specification and execution of p after execution of actions in F always yields a computation that is in the problem specification of p. For the latter guarantee, it suffices that correctors be added to p (and, for the former, it suffices that detectors be added to p, as discussed in the previous section). 5 Composition of Detector/Corrector Components and Programs In this section, we discuss how a detector/corrector component is correctly added to a program so that the resulting program satisfies the specification of the component. As far as possible, the proof of preservation should be simpler than explicitly proving all over again that the specification is satisfied in the resulting program. This is achieved by a compositional proof that shows that the program does not "interfere" with the component, i.e., the program and the component when executed concurrently do not violate the specification of the component. Compositional proofs of interference-freedom have received substantial attention in the formal methods community [15\Gamma19] in the last two decades. Drawing from these efforts, we identify several simple sufficient conditions to ensure that when a program p is composed with a detector (respectively a corrector) q, the safety specification of q, viz Safeness and Stability, and liveness specification, viz Progress and Convergence, are not violated. Sufficient conditions for satisfying the safety specification of a detector. To demonstrate that p does not interfere with Safeness and Stability, a straightforward sufficient condition is that the actions of p be a subset of the actions of q; this occurs, for instance, when program itself acts as a detector. Another straightforward condition is that the variables of p and q be disjoint. A more general condition is that p only reads (but not writes) the variables of q; in this case, p is said to be "superposed" on q. Sufficient conditions for satisfying the liveness specification of a detector. The three conditions given above also suffice to demonstrate that p does not interfere with Progress of q, provided that the actions of p and q are executed fairly. Yet another condition for satisfying Progress of q is to require that q be "atomic", i.e., that q achieves its Progress in at most one step. It follows that even if p and q execute concurrently, Progress of q is satisfied. Alternatively, require that p executes only after Progress of q is achieved. It follows that p cannot interfere with Progress of q. Likewise, require that p terminates eventually. It follows that after p has terminated, execution of q in isolation satisfies its Progress. More generally, require that there exists a variant function f (whose range is over a well-founded set) such that execution of any action in p or q reduces the value of f until Progress of q is achieved. It follows that even if q is executed concurrently with p, Progress of q is satisfied. The sufficient conditions outlined above are formally stated in Table 1. Sufficient conditions for the case of a corrector are similar. In the following theorems, let Z detect X in q for U , and let U be closed in p. Theorem 5.0 (Superposition) If q does not read or write any variable written by p, and only reads the variables written by q then Z detects X in q[]p for U Theorem 5.1 (Containment) If actions of p are a subset of q then Z detects X in q[]p for U Theorem 5.2 (Atomicity) If fU - Zg p fZ - :Xg, and q is atomic then Z detects X in q[]p for U Theorem 5.3 (Order of execution) If fU - Zg p fZ - :Xg then Z detects X in q; p for U Theorem 5.4 (Termination) If fU - Zg p fZ - :Xg, and U converges to V in p[]q then Z detects X in (:V - p)[]q for U Theorem 5.5 (Variant function) If fU - (0!f =K)g q f(0!f then Z detects X in q[]p for U Table Sufficient conditions for interference-freedom The discussion above has addressed how to prove that a program does not interfere with a component, but not how a component does not interfere with a program. Standard compositional techniques suffice for this purpose. In practice, detectors such as snapshot procedures, watchdog programs, and snooper programs typically read but not write the state of the program to which they are added. Thus, these detectors do not interfere with the program. Likewise, correctors such as reset, rollback recovery, and forward recovery procedures are typically restricted to execute only in states where the invariant at hand is false. Thus, these correctors do not interfere with the program. Example: Memory access (continued). Consider an intolerant program p for memory access that assumes that the address is valid and is currently present in the memory. For ease of exposition, we let p access only one memory location instead of multiple locations. Thus, p is as follows: For p to not interfere with the specification of the corrector d1; (c; d2), it suffices that p execute only after Z2, the witness predicate of d1; (c; d2), is satisfied. Hence, d1; (c; d2) and p should be composed in sequence. From the analogue of Theorem 5.3 for the case of correctors, we have that p does not interfere with d1; (c; d2): ffl Z2 corrects X1 - X2 in d1; (c; d2); p for U1 - X1 6 Designing Multitolerance In this section, we first define multitolerance and then present our method for compositional, stepwise design of multitolerant programs. Let p be a program with invariant S, F1::Fn be n fault-classes, and l1; l2; ::; ln be types of tolerance (i.e., masking, nonmasking or fail-safe). We say that p is multitolerant to F1::Fn for S iff for each fault-class F j; 1-j- n, p is lj-tolerant to F j for S. The definition may be understood as follows: In the presence of faults in class F j, p is perturbed only to states where some F j-span predicate for S, T j, is true. (Note that there exists a potentially different fault-span for each fault-class.) After faults in F j stop occurring, subsequent execution of p always yields a computation that is in the problem specification as prescribed by the type of tolerance lj. For example, if lj is fail-safe, each computation of p starting from a state where T j is true is in the safety specification. Example: Memory access (continued). Observe that the memory access program, d1; (c; d2); p, discussed in Section 5, is multitolerant to the classes of protection faults and page faults: it is fail-safe tolerant to the former and masking tolerant to latter. In particular, in the presence of a page fault, it always obtains the correct data from the memory. And in the presence of a protection fault, it obtains no data value. Compositional and stepwise design method. As outlined in the introduction, our method starts with a fault-intolerant program and, in a stepwise manner, considers the fault-classes in some fixed total order, say F1::Fn. In the first step, the intolerant program is augmented with detector and/or corrector components so that it is l1-tolerant to F 1. The resulting program is then augmented with other detector/corrector components, in the second step, so that it is l2-tolerant to F2 and its l1-tolerance to F1 is preserved. And so on until, in the n-th step, the ln-tolerance to Fn is added while preserving the l1::ln\Gamma1tolerances to F1::Fn\Gamma1. The multitolerant program designed thus has the structure shown in Figure 3. detectors and/or correctors for Fn Fault-intolerant program detectors and/or correctors for F1 detectors and/or correctors for F2 Figure 3: Structure of a multitolerant program designed using our method First step. Let p be the intolerant program with invariant S. By calculating an F 1-span of p for S, detector and corrector components can be designed for satisfying l1-tolerance to F 1. As discussed in Section 3 and 4, it suffices to add detectors to design fail-safe tolerance to F 1, correctors to design nonmasking tolerance to F 1, and both detectors and correctors to design masking tolerance to F 1. tolerant program Nonmasking program Failsafe correctors detectors and correctors Intolerant program Masking tolerant program detectors tolerant Figure 4: Components that suffice for design of various tolerances Note that the detectors and correctors added to p are also subject to F . Hence, they themselves have to be tolerant to F . But it is not necessary that they be masking tolerant to F . More specifically, it suffices that detectors added to design fail-safe tolerance be themselves fail-safe tolerant to F ; this is because if the detectors do not satisfy their liveness specification in the presence of F , the resulting program can be made to not satisfy the liveness specification of p in the presence of F . Likewise, it suffices that correctors added to design nonmasking tolerance be themselves nonmasking tolerant to F ; this is because as long as the computations of the correctors have suffixes that are in their safety and liveness specification, the computations of the resulting program can be made to have suffixes in the safety and liveness specification of p. Lastly, as can be expected, it suffices that the detectors and correctors added to design masking tolerance be themselves masking tolerant to F . (See Figure 5.) In practice, the detectors and correctors added to p often possess the desired tolerance to F trivially. But if they do not, one way to design them to be tolerant to F is by the analogous addition of more detectors and correctors. Another way is to design them to be self tolerant, without using any more detector and corrector components, as is exemplified by self-checking, self-stabilizing, and inherently fault-tolerant designs. tolerant program nonmasking components tolerant program Intolerant program Failsafe Masking tolerant program Nonmasking masking components Figure 5: Tolerance requirements for the components With the addition of detector and/or corrector components to p, it remains to show that, in the resulting program p1, the components do not interfere with p and that p does not interfere with the components. Note that p1 may contain variables and actions that were not in p and, hence, invariants and fault-spans of p1 may differ from those of p. Therefore, letting S1 be an invariant of p1 and T1 be an F 1-span of p1 for S1, we show the following. 1. In the absence of F 1, i.e., in states where S1 is true, the components do not interfere with p, i.e., each computation of p is in the problem specification even if it executes concurrently with the new components. 2. In the presence of F 1, i.e., in states where T1 is true, p does not interfere with the components, i.e., each computation of the components is in the components' specification (in the sense prescribed by its type of tolerance) even if they execute concurrently with p. The addition of the detectors and correctors may itself be simplified by using a stepwise For instance, to design masking tolerance, we may first augment the program with detectors, and then augment the resulting fail-safe tolerant program with correctors. Alternatively, we may first augment the program with correctors, and then augment the resulting nonmasking tolerant program with detectors. (See Figure 6.) For reasons of space, we refer the interested reader to [20] for the formal details of this two-stage approach for designing masking tolerance.000000000000000000000000000000000000000000000000000000111111111111111111111111111111111111111111111111111111111111111000000000000000000000000000000000000000000000000000000000000000111111111111111111111111111111111111111111111111111111tolerant Masking tolerant program Intolerant program correctors detectors program correctors program detectors Failsafe Nonmasking tolerant Figure Two approaches for stepwise design of masking tolerance Second step. This step adds l2-tolerance to F2 and preserves the l1-tolerance to F 1. To add l2- tolerance to F 2, just as in the first step, we add new detector and corrector components to p1. Then, we account for the possible interference between the executions of these added components and of p1. More specifically, letting S2 be an invariant of the resulting program p2, T21 be an F 1-span of p2 for S2, and T22 denote an F 2-span of p2 for S2, we show the following. 1. In the absence of F1 and F 2, i.e., in states where S2 is true, the newly added components do not interfere with p1, i.e., each computation of p1 is in the problem specification even if it executes concurrently with the new components. 2. In the presence of F 2, i.e., in states where T22 is true, p1 does not interfere with the new com- ponents, i.e., each computation of the new components is in the new components' specification (in the sense prescribed by its type of tolerance) even if they execute concurrently with p1. 3. In the presence of F 1, i.e., in states where T21 is true, the newly added components do not interfere with the l1-tolerance of p1 for F 1, i.e., each computation of p1 is in the specification, l1-tolerant to F 1, even if p1 executes concurrently with the new components. Remaining steps. For the remaining steps of the design, where we add tolerance to F3::Fn, the procedure of the second step is generalized accordingly. 7 Case Study in Multitolerance Design : Token Ring Recall the mutual exclusion problem: Multiple processes may each access their critical section provided that at any time at most one process is accessing its critical section. moreover, no process should wait forever to access its critical section, assuming that each process leaves its critical section in finite time. Mutual exclusion is readily achieved by circulating a token among processes and letting each process enter its critical section only if it has the token. In a token ring program, in particular, the processes are organized in a ring and the token is circulated along the ring in a fixed direction. In this case study, we design a multitolerant token ring program. The program is masking tolerant to any number, K, of faults that each corrupt the state of some process detectably. Its tolerance is continuous in the sense that if K state corruptions occur, it corrects its state within \Theta(K) time. Thus, a quantitatively unique measure of tolerance is provided to each FK, where FK is the fault- class that causes at most K state corruptions of processes. By detectable corruption of the state of a process, we mean that the corrupted state is detected by that process before any action inadvertently accesses that state. The state immediately before the corruption may, however, be lost. (For our purposes, it is irrelevant as to what caused the corruption; i.e., whether it was due to the loss of a message, the duplication of a message, timing faults, the crash and subsequent restart of a process, etc.) We proceed as follows: First, we describe a simple token ring program that is intolerant to detectable state corruptions. Then, we add detectors and correctors so as to achieve masking tolerance to the fault that corrupts the state of one process. Progressively, we add more detectors and correctors so as to achieve masking tolerance to the fault-class that corrupts process states at most K, K ? 1, times. 7.1 Fault-Intolerant Binary Token Ring Processes 0::N are organized in a ring. The token is circulated along the ring such that process j, the token to its successor j +1. (In this section, + and \Gamma are in modulo N+1 Each process j maintains a binary variable x:j. Process j; j 6= N , has the token iff x:j differs from its successor x:(j +1) and process N has the token iff x:N is the same as its successor x:0. The program, TR, consists of two actions for each process j. Formally, these actions are as follows (where Invariant. Consider a state where process j has the token. In this state, since no other process has a token, the x value of all processes 0::j is identical and the x value of all processes (j+1)::N is identical. Letting X denote the string of binary values x:0; x:1; :::; x:N , we have that X satisfies the regular expression (0 l 1 (N+1\Gammal) [ 1 l 0 (N+1\Gammal) ), which denotes a sequence of length N+1 consisting of zeros followed by ones or ones followed by zeros. Thus, an invariant of the program TR is 7.2 Adding Tolerance to 1 State Corruption Based on our assumption that state corruption is detectable, we introduce a special value ?, such that when any process j detects that its state (i.e., the value of x:j) is corrupted, it resets x:j to ?. We can now readily design masking tolerance to a single corruption of state at any process j by ensuring that (i) the value of x:j is eventually corrected so that it is no longer ? and (ii) in the interim, no process (in particular, j+1) inadvertently gets the token as a result of the corruption of x:j. For (i), we add a corrector at each process j: it corrects x:j from ? to a value that is either 0 or 1. The corrector at j, j 6= 0, copies x:(j \Gamma 1); the corrector at the corrector action at j has the same statement as the action of TR at j, and we can merge the corrector and TR actions. For (ii), we add a detector at each process Its detection predicate is and it has no actions. The witness predicate of this detector (which, in this case, is the detection predicate itself) is used to restrict the actions of program TR at j. Hence, the actions of TR at j execute only when As a result, the execution of actions of TR is always safe (i.e., these actions cannot inadvertently generate a token). The augmented program, PTR, is Fault Actions. When the state of x:j is corrupted, x:j is set to ?. Hence, the fault action is Proof of interference-freedom. Starting from a state where S TR is true, in the presence of faults that set the x value of a process to ?, string X always satisfies the regular expression (0 [ ?) l (1 [ ?) (N+1\Gammal) or (1 [ ?) l (0 [ ?) (N+1\Gammal) . Thus, an invariant of PTR is SPTR , where Consider the detector at j: Both its detection and witness predicates are x:(j \Gamma 1) 6= ?. Since the detects relation is trivially reflexive in its first two arguments, it follows that in PTR. In other words, the detector is not interfered by any other actions. Consider the corrector at j: Both its correction and witness predicates are x:j 6= ?. Since the program actions are identical to the corrector actions, by Theorem 5.1, the corrector actions are not interfered by the actions of TR. Also, since the detectors have no actions, the detectors at processes other than j do not interfere with the corrector at j; moreover, since at most one x value is set to ?, when x:j =? and thus the corrector at j is enabled, the witness predicate of the detector at j is true and hence the corrector at j is not interfered by the detector at j. Consider the program actions of TR: Their safety follows from the safety of the detectors, described above. And, their progress follows from the progress of the correctors, which ensure that starting from a state where SPTR is true and a process state is corrupted every computation of PTR reaches a state where S TR is true, and the progress of the detectors, which ensures that no action of TR is indefinitely blocked from executing. Observe that our proof of mutual interference-freedom illustrates that we do not have to re-prove the correctness of TR for the new invariant. Observe, also, that if the state of process j is corrupted then within \Theta(1) time the corrector at j corrects the state of j. 7.3 Adding Tolerance to 2::N State Corruptions The proof of non-interference of program PTR can be generalized to show that PTR is also masking tolerant to the fault-class that twice corrupts process state. The generalization is self-evident for the case where the state corruptions are separated in time so that the first one is corrected before the second one occurs. For the case where both state corruptions occur concurrently, say at processes j and k, we need to show that the correctors at j and k truthify interference by each other and the other actions of the program. Let us consider two subcases: (i) j and k are non-neighboring, and (ii) j and k are neighboring. For the first subcase, j and k correct x:j and x:k from their predecessors j\Gamma1 and k\Gamma1, respectively. This execution is equivalent to the parallel composition of the correctors at j and k. By Theorem 4.2, PTR reaches a state where x:j and x:k are not ?. For the second subcase (letting j be the predessor of k), j corrects x:j from its predecessor truthifies x:j 6=? and then terminates. Since the corrector at j does not read any variables written by the corrector at k. Thus, from the analogue of Theorem 5.0 for the case of correctors, the corrector at j is not interfered by the corrector at k. After x:j 6= ? is truthified, the corrector at k corrects x:k from its predecessor j. By Theorem 4.4, the corrector at k is not interfered by the corrector at j. Since the correctors at j and k do not interfere with each other, it follows that the program reaches a state where x:j and x:k are not ?. In fact, as long as the number of faults is at most N , there exists at least one process j with x:j 6=?. PTR ensures that the state of such a j eventually causes j+1 to correct its state to x:(j Such corrections will continue until no process has its x value set to ?. Hence, PTR tolerates up to N faults and the time required to converge to S TR is \Theta(K), where K is the number of faults. 7.4 Adding Tolerance to More Than N State Corruptions Unfortunately, if more than N faults occur, program PTR deadlocks iff it reaches a state where the x value of all processes is ?. To be masking tolerant to the fault-classes that corrupt the state of processes more than N times, a corrector is needed that detects whether the state of all processes is ? and, if so, corrects the program to a state where the x value of some process (say 0) to be equal to 0 or 1. Since the x values of all processes cannot be accessed simultaneously, the corrector detects in a sequential manner whether the x values of all processes are ?. Let the detector added for this purpose at process j be denoted as dj and the (sequentially composed) detector that detects whether the x values of all processes is corrupted be dN To design dj, we add a value ? to the domain of x:j. When dN detects that x:N is equal to ?, it sets x:N to ?. Likewise, when dj, detects that x:j is equal to ?, it sets x:j to ?. Note that since dj is part of the sequential composition, it is restricted to execute only after j+1 has completed its detection, i.e., when x:(j+1) is equal to ?. It follows that when j completes its detection, the x values of processes j::N are corrupted. In particular, when d0 completes its detection, the x values of all processes are corrupted. Hence, when x:0 is set to ?, it suffices for the corrector to reset x:0 to 0. To ensure that while the corrector is executing, no process inadvertently gets the token as a result of the corruption of x:j, we add detectors that restrict the actions of PTR at j+1 to execute only in states where x:j 6=? is true. Actions. Program FTR consists of five actions at each process j. Like PTR, the first two actions, FTR1 and FTR2, pass the token from j to j+1 and are restricted by the trivial detectors to execute only when x:(j \Gamma1) is neither ? nor ?. Action FTR3 is dN ; it lets process N change x:N from ? to ?. Action FTR4 is dj for j ! N . Action FTR5 is the corrector action at process 0: it lets process correct x:0 from ? to 0. Formally, these actions are as follows: Invariant. Starting from a state where SPTR is true, the detector can change the trailing ? values in X to ?. Thus, FTR may reach a state where X satisfies the regular expression (1 [ ?) l (0 [ Subsequent state corruptions may perturb X to the form (1 [?) l (0 [?) m (? [?) (N+1\Gammal\Gammam) [ (0 [?) l (1 [?) m (?[?) (N+1\Gammal\Gammam) . Since all actions preserve this last predicate, an invariant of FTR is (0 Proof of interference-freedom. To design FTR, we have added a corrector (actions FTR3 \Gamma 5) to program PTR to ensure that for some j, x:j is not corrupted, i.e., the correction predicate of this corrector is V , where This corrector is of the form dN ; d(N\Gamma1); :::; d0; c0, where each dj is an atomic detector at process j and c0 is an atomic corrector at process 0. The detection predicate of dN is :V and its witness predicate is x:0 =?. To show that this detector in isolation satisfies its specification, observe that 1. x:N =? detects in dN for SFTR . 2. for (SFTR From (1) and (2), by Theorem 3.3, x:(N \Gamma1) =? detects . Using the same argument, x:0=? detects in dN Now, observe that SFTR converges to V in dN violated execution of eventually truthify x:0 =?, and execution of c0 will truthify V . Thus, V corrects V in dN The corrector is not interfered by the actions FTR1 and FTR2. This follows from the fact that FTR1 and FTR2 do not interfere with each dj and c0 (by using Theorem 5.2). In program FTR, we have also added a detector at process j that detects x:(j\Gamma1) 6=?. As described above (for the 1 fault case), this detector does not interfere with other actions, and it is not interfered by other actions. Finally, consider actions of program PTR: their safety follows from the safety of the detector described above. Also, starting from any state in SFTR , the program reaches a state where x value of some process is not corrupted. Starting from such a state, as in program PTR, eventually the program reaches a state where S TR is truthified, i.e., no action of PTR is permanently blocked. Thus, the progress of these actions follows. Theorem 7.0 Program FTR is masking tolerant for invariant SFTR to the fault-classes FK, K- 1, where FK detectably corrupts process states at most K times. moreover, SFTR converges to S TR in FTR within \Theta(K) time. Remark. We emphasize that the program FTR is masking tolerant to the fault-classes FK for the invariant SFTR and not for S TR . Thus, in the presence of faults in FK, SFTR continues to be true although S TR may be violated. Process j, j !N , has a token iff x:j differs from x:(j+1) and neither x:j nor x:(j+1) is corrupted, and process N has a token iff x:N is the same as x:0 and neither x:N nor x:0 is corrupted. Thus, in a state where SFTR is true at most one process has a token. Also starting from such a state eventually the program reaches a state where S TR is true. Starting from such a state, each process can get then token. Thus, starting from any state in SFTR , computations of FTR are in the problem specification of the token ring. In this section, we address some of the issues that our method for design of multitolerance has raised. We also discuss the motivation for the design decisions made in this work. Our formalization of the concept of multitolerance uses the abstractions of closure and convergence. Can other abstractions be used to formalize multitolerance? What are the advantages of using closure and convergence? In principle, one can formulate the concept of multitolerance using abstractions other than closure and convergence. As pointed out by John Rushby [21], the approaches to formulate fault-tolerance can be classified into two: specification approaches and calculational approaches. In specification approaches, a system is regarded as a composition of several subsystems, each with a standard specification and one or more failure specifications. A system is fault-tolerant if it satisfies its standard specification when all components do, and one of its failure specifications if some of its components depart from their standard specification. One example of this approach is due to Herlihy and Wing [22] who thus formulate graceful degradation, which is a special case of multitolerance. In calculational approaches, the set of computations permissible in the presence of faults is calculated. A system is said to be fault-tolerant if this set satisfies the specification of the system (or an acceptably degraded version of it). Our approach is calculational since we compute the set of states that are potentially reachable in the presence of faults (fault-span). While other approaches may be used to formulate the design of multitolerance, we are not aware of any formal methods for design of multitolerance using them. moreover, in our experience, the structure imposed by abstractions of closure and convergence has proven to be beneficial in several ways: (1) it has enabled us to discover the role of detectors and correctors in the design of all tolerance properties (cf. Sections 3 and 4); (2) it has yielded simple theorems for composing tolerance actions and underlying actions in an interference-free manner (cf. Sections 5 and 6); (3) it has facilitated our design of novel and complex distributed programs whose tolerances exceed those of comparable programs designed otherwise [5, 10, 20, 23, 24, 25]. We have represented faults as state perturbations. This representation readily handles transient faults, but does it also handle permanent faults? intermittent faults? detectable faults? undetectable All these faults can indeed be represented as state perturbations. The token ring case study illustrates the use of state perturbations for various classes of transient faults. In an extended version of this paper [24], we present a case study of tree-based mutual exclusion which illustrates the analogous representation for permanent faults and for detectable or undetectable faults. It is worth pointing out that representing permanent and intermittent faults, such as Byzantine faults and fail-stop and repair faults, may require the introduction of auxiliary variables [5, 10]. For example, to represent Byzantine faults that affects a process j, we may introduce an auxiliary boolean variable byz:j that is true iff j is Byzantine. If j is not Byzantine, it executes its "normal" actions. Otherwise, it executes some "abnormal" actions. When the Byzantine fault occurs, byz:j is truthified, thus, permitting j to execute its abnormal actions. Similarly, to represent fail-stop and repair faults that affects a process j, we may introduce an auxiliary boolean variable down:j that is true iff j has fail-stopped. All actions of j are restricted to be executed only when down:j is false. When a fail-stop fault occurs, down:j is truthified, thus preventing j from executing its actions. When a repair occurs, down:j is falsified. We have assumed that problem specifications are suffix closed and fusion closed. Where are these assumptions exploited in the design method? Do these assumptions restrict the applicability of the method? We have used these assumptions in three places: (1) Suffix closure of problem specifications implies the existence of invariant state predicates. (2) Fusion closure of problem specifications implies the existence of correction state predicates. (3) Suffix closure and fusion closure of problem specifications imply that the corresponding safety specifications are fusion closed, which, in turn, implies the existence of detection state predicates. These assumptions are not restrictive in the following sense: Let L be a set of state sequences that is not suffix closed and/or not fusion closed and let p be a program. Then, it can be shown that by adding history variables to the variables of p, there exists a problem specification L 0 such that the following condition holds: all computations of p that start at states where some "initial" state predicate is true are in L iff p satisfies L 0 for some state predicate. Thus, the language of problem specifications is not restrictive. How would our method of considering the fault-classes one-at-a-time compare with a method that considers them altogether? There is a sense in which the one-at-a-time and the altogether methods are equivalent: programs designed by the one method can also be designed by the other method. To justify this informally, let us consider a program p designed by using the altogether method to tolerate fault-classes F 1, F 2, . , Fn. Program p can also be designed using the one-at-a-time method as follows: Let p1 be a subprogram of p that tolerates F 1. This is the program designed in the first stage of the one-at-a-time method. Likewise, let p2 be a subprogram of p that tolerates F1 and F 2. This is the program designed in the second stage of the one-at-a-time method. And so on, until p is designed. To complete the argument of equivalence, it remains to observe that a program designed by the one-at-a-time n-stage method can trivially be designed by the altogether method. In terms of software engineering practice, however, the two methods would exhibit differences. Towards identifying these differences, we address three issues: (i) the structure of the programs designed using the two methods, (ii) the complexity of using them, and (iii) the complexity of the programs designed using them. On the first issue, the stepwise method may yield programs that are better structured. This is exemplified by our hierarchical token ring program which consists of three layers: the basic program that transmits the token, a corrector for the case when at least one process is not corrupted, and a corrector for the case when all processes are corrupted On the second issue, since we consider one fault-class at a time, the complexity of each step is less than the complexity of the altogether program. For example, in the token ring program, we first handled the case where the state of some process is not corrupted. Then, we handled the only case where the state of all processes is corrupted. Thus, each step was simpler than the case where we would need to consider both these cases simultaneously. On the third issue, it is possible that considering all fault-classes at a time may yield a program whose complexity is (in some sense) optimal with respect to each fault-class, whereas the one-at- a-time approach may yield a program that is optimal for some, but not all, fault-classes. This suggests two considerations for the use of our method. One, the order in which the fault-classes are considered should be chosen with care. (Again, in principle, programs designed with one order can be designed by any other order. But, in practice, different orders may yield different programs, and the complexity of these programs may be different.) And, two, in choosing how to design the tolerance for a particular fault-class, a "lookahead" may be warranted into the impact of this design choice on the design of the tolerances to the remaining fault-classes. How does our compositional method affect the trade-offs between dependability properties? Our method makes it possible to reason about the trade-offs locally, i.e., focusing attention only on the components corresponding to those dependability properties, as opposed to globally, i.e., by considering the entire program. Thus, our method facilitates reasoning about trade-offs between dependability properties. moreover, as can be expected, if the desired dependability properties are impossibility to cosatisfy, it will follow that there do not exist components that can be added to the program while complying with the interference-freedom requirements of our method. How does our compositional design method compare with the existing methods for designing fault-tolerant programs? Our compositional design method is rich in the sense that it subsumes various existing fault-tolerance design methods such as replication, checkpointing and recovery, Schneider's state machine approach, exception handling, and Randell's recovery blocks. (The interested reader is referred to [20, 24] for a detailed discussion of how properties such as replication, agreement, and order are designed by interference-free composition within our method.) How are fault-classes derived? Can our method be used if it is difficult to characterize the faults the system is subject to? Derivation of fault-classes is application specific. It begins with the identification of the faults that the program may be subject to. Each of these faults is then formally characterized using state perturbations. (As mentioned above, auxiliary variables may be introduced in this formalization.) The desired type of tolerance for each fault is then specified. Finally, the faults are grouped into (possibly overlapping) fault-classes, based on the characteristics of the faults or their corresponding types of tolerance. If it is difficult to characterize the faults in an application, a user of our method is obliged to guess some large enough fault-class that would accommodate all possible faults. It is often for this reason that designers choose weak models such as self-stabilization (where the state may be perturbed arbitrarily) or Byzantine failure (where the program may behave arbitrarily). 9 Concluding Remarks and Future Work In this paper, we formalized the notion of multitolerance to abstract a variety of problems in de- pendability. It is worthwhile to point out that multitolerance has other related applications as well. One is to reason about graceful degradation with respect to progressively increasing fault-classes. Another is to guarantee different qualities of service (QoS) with respect to different user requirements and traffics. A third one is to reason about adaptivity of systems with respect to different modes of environment behavior. We also presented a simple, compositional method for designing multitolerant programs, that added detector and corrector components for providing each desired type of tolerance. The addition of multiple components to an intolerant program was made tractable by adding tolerances to fault- classes one at a time. To avoid re-proving the correctness of the program in every step, we provided a theory for ensuring mutual interference-freedom in compositions of detectors and correctors with the intolerant program. To our knowledge, this is the first formal method for the design of multitolerant programs. Our method is effective for the design of quantitative as well as qualitative tolerances. As an example of quantitative tolerance, we presented a token ring protocol that recovers from upto K faults in time. For examples of qualitative tolerances, we refer the interested reader to our designs of multitolerant programs for barrier computations, repetitive Byzantine agreement, mutual exclusion, tree maintenance, leader election, bounded-space distributed reset, and termination detection [23, To apply our design method in practice, we are currently developing SIEFAST, a simulation and implementation environment that enables stepwise implementation and validation of multitolerant distributed programs. We are also studying the mechanical synthesis of multitolerant concurrent programs. Acknowledgments . We are indebted to the anonymous referees for their detailed and constructive comments on earlier versions of this paper, which significantly improved the presentation. Thanks also to Laurie Dillon for all her help during the review process. --R Reliable Computer Systems: Design and Evaluation. The AT&T Case The Galileo Case A foundation of fault-tolerant computing Superstabilzing protocols for dynamic distributed systems. Maximal flow routing. A highly safe self-stabilizing mutual exclusion algorithm Defining liveness. Closure and convergence: A foundation of fault-tolerant computing A Discipline of Programming. The Science of Programming. Proving boolean combinations of deterministic properties. Parallel Program Design: A Foundation. The existence of refinement mappings. A proof technique for communicating sequential processes. Stepwise refinement of parallel programs. Proofs of networks of processes. An axiomatic proof technique for parallel programs. Designing masking fault-tolerance via nonmasking fault-tolerance Critical system properties: Survey and taxonomy. Specifying graceful degradation. Multitolerance in distributed reset. Multitolerance and its design. Constraint satisfaction as a basis for designing nonmasking fault-tolerance Multitolerant barrier synchronization. Compositional design of multitolerant repetitive byzantine agreement. --TR --CTR Anil Hanumantharaya , Purnendu Sinha , Anjali Agarwal, A component-based design and compositional verification of a fault-tolerant multimedia communication protocol, Real-Time Imaging, v.9 n.6, p.401-422, December Orna Raz , Mary Shaw, An Approach to Preserving Sufficient Correctness in Open Resource Coalitions, Proceedings of the 10th International Workshop on Software Specification and Design, p.159, November 05-07, 2000 Anish Arora , Sandeep Kulkarni , Murat Demirbas, Resettable vector clocks, Proceedings of the nineteenth annual ACM symposium on Principles of distributed computing, p.269-278, July 16-19, 2000, Portland, Oregon, United States Anish Arora , Sandeep S. Kulkarni , Murat Demirbas, Resettable vector clocks, Journal of Parallel and Distributed Computing, v.66 n.2, p.221-237, February 2006 Paul C. Attie , Anish Arora , E. Allen Emerson, Synthesis of fault-tolerant concurrent programs, ACM Transactions on Programming Languages and Systems (TOPLAS), v.26 n.1, p.125-185, January 2004 Robyn R. Lutz, Software engineering for safety: a roadmap, Proceedings of the Conference on The Future of Software Engineering, p.213-226, June 04-11, 2000, Limerick, Ireland Anish Arora , Marvin Theimer, On modeling and tolerating incorrect software, Journal of High Speed Networks, v.14 n.2, p.109-134, April 2005 I-Ling Yen , Farokh B. Bastani , David J. Taylor, Design of Multi-Invariant Data Structures for Robust Shared Accesses in Multiprocessor Systems, IEEE Transactions on Software Engineering, v.27 n.3, p.193-207, March 2001 Anish Arora , Paul C. Attie , E. Allen Emerson, Synthesis of fault-tolerant concurrent programs, Proceedings of the seventeenth annual ACM symposium on Principles of distributed computing, p.173-182, June 28-July 02, 1998, Puerto Vallarta, Mexico Axel van Lamsweerde , Emmanuel Letier, Handling Obstacles in Goal-Oriented Requirements Engineering, IEEE Transactions on Software Engineering, v.26 n.10, p.978-1005, October 2000 Vina Ermagan , Jun-ichi Mizutani , Kentaro Oguchi , David Weir, Towards Model-Based Failure-Management for Automotive Software, Proceedings of the 4th International Workshop on Software Engineering for Automotive Systems, p.8, May 20-26, 2007 Felix C. Grtner, Fundamentals of fault-tolerant distributed computing in asynchronous environments, ACM Computing Surveys (CSUR), v.31 n.1, p.1-26, March 1999 Anish Arora , Sandeep S. Kulkarni, Designing Masking Fault-Tolerance via Nonmasking Fault-Tolerance, IEEE Transactions on Software Engineering, v.24 n.6, p.435-450, June 1998

formal methods;stepwise design;correctors;graceful degradation;dependability;compositional design;fault-tolerance;detectors;interference-freedom

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Generalized Queries on Probabilistic Context-Free Grammars.

AbstractProbabilistic context-free grammars (PCFGs) provide a simple way to represent a particular class of distributions over sentences in a context-free language. Efficient parsing algorithms for answering particular queries about a PCFG (i.e., calculating the probability of a given sentence, or finding the most likely parse) have been developed and applied to a variety of pattern-recognition problems. We extend the class of queries that can be answered in several ways: (1) allowing missing tokens in a sentence or sentence fragment, (2) supporting queries about intermediate structure, such as the presence of particular nonterminals, and (3) flexible conditioning on a variety of types of evidence. Our method works by constructing a Bayesian network to represent the distribution of parse trees induced by a given PCFG. The network structure mirrors that of the chart in a standard parser, and is generated using a similar dynamic-programming approach. We present an algorithm for constructing Bayesian networks from PCFGs, and show how queries or patterns of queries on the network correspond to interesting queries on PCFGs. The network formalism also supports extensions to encode various context sensitivities within the probabilistic dependency structure.

Introduction pattern-recognition problems start from observations generated by some structured stochastic process. Probabilistic context-free grammars (PCFGs) [1], [2] have provided a useful method for modeling uncertainty in a wide range of structures, including natural languages [2], programming languages [3], images [4], speech signals [5], and RNA sequences [6]. Domains like plan recognition, where non-probabilistic grammars have provided useful models [7], may also benefit from an explicit stochastic model. Once we have created a PCFG model of a process, we can apply existing PCFG parsing algorithms to answer a variety of queries. For instance, standard techniques can efficiently compute the probability of a particular observation sequence or find the most probable parse tree for that sequence. Section II provides a brief description of PCFGs and their associated algorithms. However, these techniques are limited in the types of evidence they can exploit and the types of queries they can answer. In particular, the existing PCFG techniques generally require specification of a complete observation sequence. In many contexts, we may have only a partial sequence available. It is also possible that we may have evidence beyond simple observations. For example, in natural language processing, we may be able to exploit contextual information about a sentence in determining our beliefs about certain unobservable variables in its parse tree. In addition, we may be interested in computing the probabilities of alternate types of events (e.g., future observations or abstract features of the parse) that the extant techniques do not directly support. The restricted query classes addressed by the existing algorithms limit the applicability of the PCFG model in domains where we may require the answers to more complex queries. A flexible and expressive representation for the distribution of structures generated by the grammar would support broader forms of evidence and queries than supported by the more specialized algorithms that currently exist. We adopt Bayesian networks for this purpose, and define an algorithm to generate a network representing the distribution of possible parse trees (up to a specified string length) generated from a flies (0.4) like (0.4) flies (0.45) verb pp (0.2) noun ! ants (0.5) Fig. 1. A probabilistic context-free grammar (from Charniak [2]). given PCFG. Section III describes this algorithm, as well as our algorithms for extending the class of queries to include the conditional probability of a symbol appearing anywhere within any region of the parse tree, conditioned on any evidence about symbols appearing in the parse tree. The restrictive independence assumptions of the PCFG model also limit its applica- bility, especially in domains like plan recognition and natural language with complex dependency structures. The flexible framework of our Bayesian-network representation supports further extensions to context-sensitive probabilities, as in the probabilistic parse tables of Briscoe & Carroll [8]. Section IV explores several possible ways to relax the independence assumptions of the PCFG model within our approach. Modified versions of our PCFG algorithms can support the same class of queries supported in the context-free case. II. Probabilistic Context-Free Grammars A probabilistic context-free grammar is a tuple h\Sigma; the disjoint sets \Sigma and N specify the terminal and nonterminal symbols, respectively, with S 2 N being the start symbol. P is the set of productions, which take the form the probability that E will be expanded into the string -. The sum of probabilities p over all expansions of a given nonterminal E must be one. The examples in this paper will use the sample grammar (from Charniak [2]) shown in Fig. 1. This definition of the PCFG model prohibits rules of the the empty string. However, we can rewrite any PCFG to eliminate such rules and still represent the original distribution [2], as long as we note the probability Pr(S ! "). For clarity, the algorithm descriptions in this paper assume Pr(S ! negligible amount of additional bookkeeping can correct for any nonzero probability. The probability of applying a particular production to an intermediate string is conditionally independent of what productions generated this string, or what productions will be applied to the other symbols in the string, given the presence of E. Therefore, the probability of a given derivation is simply the product of the probabilities of the individual productions involved. We define the parse tree representation of each such derivation as for non-probabilistic context-free grammars [9]. The probability of a string in the language is the sum taken over all its possible derivations. vp!verb np pp: 0.0014 vp!verb np: 0.00216 np!noun pp: 0.036 np!noun: verb!flies: 0.4 prep!like: 1.0 noun!ants: 0.5 noun!swat: noun!flies: 0.45 verb!like: 0.4 Fig. 2. Chart for Swat flies like ants. A. Standard PCFG Algorithms Since the number of possible derivations grows exponentially with the string's length, direct enumeration would not be computationally viable. Instead, the standard dynamic programming approach used for both probabilistic and non-probabilistic CFGs [10] exploits the common production sequences shared across derivations. The central structure is a table, or chart, storing previous results for each subsequence in the input sentence. Each entry in the chart corresponds to a subsequence x of the observation string . For each symbol E, an entry contains the probability that the corresponding subsequence is derived from that symbol, Pr(x jE). The index i refers to the position of the subsequence within the entire terminal string, with indicating the start of the sequence. The index j refers to the length of the subsequence. The bottom row of the table holds the results for subsequences of length one, and the top entry holds the overall result, which is the probability of the observed string. We can compute these probabilities bottom-up, since we know that Pr(x i if E is the observed symbol x i . We can define all other probabilities recursively as the sum, over all productions of the product p \Delta Altering this procedure to take the maximum rather than the sum yields the most probable parse tree for the observed string. Both algorithms require time O(L 3 ) for a string of length L, ignoring the dependency on the size of the grammar. To compute the probability of the sentence Swat flies like ants, we would use the algorithm to generate the table shown in Fig. 2, after eliminating any unused intermediate entries. There are also separate entries for each production, though this is not necessary if we are interested only in the final sentence probability. In the top entry, there are two listings for the production S !np vp, with different subsequence lengths for the right-hand side symbols. The sum of all probabilities for productions with S on the left-hand side in this entry yields the total sentence probability of 0.001011. This algorithm is capable of computing any inside probability, the probability of a particular string appearing inside the subtree rooted by a particular symbol. We can work top-down in an analogous manner to compute any outside probability [2], the probability of a subtree rooted by a particular symbol appearing amid a particular string. Given these probabilities we can compute the probability of any particular nonterminal appearing in the parse tree as the root of a subtree covering some subsequence. For example, in the sentence Swat flies like ants, we can compute the probability that like ants is a prepositional phrase, using a combination of inside and outside probabilities. swat (1,1,1) verb (1,1,2) flies (2,1,1) prep (3,1,2) noun (4,1,2) ants (4,1,1) like (3,1,1) noun (2,1,2) Fig. 3. Parse tree for Swat flies like ants, with (i; j; labeled. The Left-to-Right Inside (LRI) algorithm [10] specifies how we can use inside probabilities to obtain the probability of a given initial subsequence, such as the probability of a sentence (of any length) beginning with the words Swat flies. Furthermore, we can use such initial subsequence probabilities to compute the conditional probability of the next terminal symbol given a prefix string. B. Indexing Parse Trees Yet other conceivable queries are not covered by existing algorithms, or answerable via straightforward manipulations of inside and outside probabilities. For example, given observations of arbitrary partial strings, it is unclear how to exploit the standard chart directly. Similarly, we are unaware of methods to handle observation of nonterminals only (e.g., that the last two words form a prepositional phrase). We seek, therefore, a mechanism that would admit observational evidence of any form as part of a query about a PCFG, without requiring us to enumerate all consistent parse trees. We first require a scheme to specify such events as the appearance of symbols at designated points in the parse tree. We can use the indices i and j to delimit the leaf nodes of the subtree, as in the standard chart parsing algorithms. For example, the pp node in the parse tree of Fig. 3 is the root of the subtree whose leaf nodes are like and ants, so 2. However, we cannot always uniquely specify a node with these two indices alone. In the branch of the parse tree passing through np, n, and flies, all three nodes have To differentiate them, we introduce the k index, defined recursively. If a node has no child with the same i and j indices, then it has index is one more than the k index of its child. Thus, the flies node has the noun node above it has its parent np has 3. We have labeled each node in the parse tree of Fig. 3 with its (i; j; indices. We can think of the k index of a node as its level of abstraction, with higher values indicating more abstract symbols. For instance, the flies symbol is a specialization of the noun concept, which, in turn, is a specialization of the np concept. Each possible specialization corresponds to an abstraction production of the form only one symbol on the right-hand side. In a parse tree involving such a production, the nodes for E and E 0 have identical i and j values, but the k value for E is one more than that of E 0 . We denote the set of abstraction productions as PA ' P . All other productions are decomposition productions, in the set two or more symbols on the right-hand side. If a node E is expanded by a decomposition production, the sum of the j values for its children will equal its own j value, since the length of the original subsequence derived from E must equal the total lengths of the subsequences of its children. In addition, since each child must derive a string of nonzero length, no child has the same j index as E, which must then have Therefore, abstraction productions connect nodes whose indices match in the i and j components, while decomposition productions connect nodes whose indices differ. III. Bayesian Networks for PCFGs A Bayesian network [11], [12], [13] is a directed acyclic graph where nodes represent random variables, and associated with each node is a specification of the distribution of its variable conditioned on its predecessors in the graph. Such a network defines a joint probability distribution-the probability of an assignment to the random variables is given by the product of the probabilities of each node conditioned on the values of its predecessors according to the assignment. Edges not included in the graph indicate conditional independence; specifically, each node is conditionally independent of its nondescendants given its immediate predecessors. Algorithms for inference in Bayesian networks exploit this independence to simplify the calculation of arbitrary conditional probability expressions involving the random variables. By expressing a PCFG in terms of suitable random variables structured as a Bayesian network, we could in principle support a broader class of inferences than the standard PCFG algorithms. As we demonstrate below, by expressing the distribution of parse trees for a given probabilistic grammar, we can incorporate partial observations of a sentence as well as other forms of evidence, and determine the resulting probabilities of various features of the parse trees. A. PCFG Random Variables We base our Bayesian-network encoding of PCFGs on the scheme for indexing parse trees presented in Section II-B. The random variable N ijk denotes the symbol in the parse tree at the position indicated by the (i; j; indices. Looking back at the example parse tree of Fig. 3, a symbol E labeled (i; j; indicates that N combinations not appearing in the tree correspond to N variables taking on the null value nil. Assignments to the variables N ijk are sufficient to describe a parse tree. However, if we construct a Bayesian network using only these variables, the dependency structure would be quite complicated. For example, in the example PCFG, the fact that N 213 has the value np would influence whether N 321 takes on the value pp, even given that parent in the parse tree) is vp. Thus, we would need an additional link between N 213 and N 321 , and, in fact, between all possible sibling nodes whose parents have multiple expansions. To simplify the dependency structure, we introduce random variables P ijk to represent the productions that expand the corresponding symbols N ijk . For instance, we add the node P 141 , which would take on the value vp!verb np pp in the example. N 213 and N 321 are conditionally independent given P 141 , so no link between siblings is necessary in this case. However, even if we know the production P ijk , the corresponding children in the parse tree may not be conditionally independent. For instance, in the chart of Fig. 2, entry (1,4) has two separate probability values for the production S !np vp, each corresponding to different subsequence lengths for the symbols on the right-hand side. Given only the production used, there are again multiple possibilities for the connected N variables: N four of these sibling nodes are conditionally dependent since knowing any one determines the values of the other three. Therefore, we dictate that each variable P ijk take on different values for each breakdown of the right-hand symbols' subsequence lengths. The domain of each P ijk variable therefore consists of productions, augmented with the j and k indices of each of the symbols on the right-hand side. In the previous example, the domain of P 141 would require two possible values, S ! np[1; 3]vp[3; 1] and where the numbers in brackets correspond to the j and k values, respectively, of the associated symbol. If we know that P 141 is the former, then N and N probability one. This deterministic relationship renders the child N variables conditionally independent of each other given P ijk . We describe the exact nature of this relationship in Section III-C.2. Having identified the random variables and their domains, we complete the definition of the Bayesian network by specifying the conditional probability tables representing their interdependencies. The tables for the N variables represent their deterministic relationship with the parent P variables. However, we also need the conditional probability of each P variable given the value of the corresponding N variable, that is, E). The PCFG specifies the relative probabilities of different productions for each nonterminal, but we must compute the probability, fi(E; j; (analogous to the inside probability [2]), that each symbol E t on the right-hand side is the root node of a subtree, at abstraction level k t , with a terminal subsequence length j t . B. Calculating fi B.1 Algorithm We can calculate the values for fi with a modified version of the dynamic programming algorithm sketched in Section II-A. As in the standard chart-based PCFG algorithms, we can define this function recursively and use dynamic programming to compute its values. Since terminal symbols always appear as leaves of the parse tree, we have, for any terminal symbol x 2 \Sigma, fi(x; For any nonterminal symbol since nonterminals can never be leaf nodes. For is the sum, over all productions expanding E, of the probability of that production expanding E and producing a subtree constrained by the parameters j and k. abstraction productions are possible. For an abstraction production need the probabilities that E is expanded into E 0 and that E 0 derives a string of length j from the abstraction level immediately below E. The former is given by the probability associated with the production, while the latter is simply According to the independence assumptions of the PCFG model, the expansion of E 0 is independent of its derivation, so the joint probability is simply the product. We can compute these probabilities for every abstraction production expanding E. Since the different expansions are mutually exclusive events, the value for fi(E; j; is merely the sum of all the separate probabilities. We assume that there are no abstraction cycles in the grammar. That is, there is no sequence of productions since if such a cycle existed, the above recursive calculation would never halt. The same assumption is necessary for termination of the standard parsing algorithm. The assumption does restrict the classes of grammars for which such algorithms are applicable, but it will not be restrictive in domains where we interpret productions as specializations, since cycles would render an abstraction hierarchy impossible. For productions are possible. For a decomposition production we need the probability that E is thus expanded and that each E t derives a subsequence of appropriate length. Again, the former is given by p, and the latter can be computed from values of the fi function. We must consider every possible subsequence length j t for each E t such that In addition, the could appear at any level of abstraction k t , so we must consider all possible values for a given subsequence length. We can obtain the joint probability of any combination of t=1 values by computing since the derivation from each is independent of the others. The sum of these joint probabilities over all possible t=1 yields the probability of the expansion specified by the production's right-hand side. The product of the resulting probability and p yields the probability of that particular expansion, since the two events are independent. Again, we can sum over all relevant decomposition productions to find the value of fi(E; j; 1). The algorithm in Fig. 4 takes advantage of the division between abstraction and decomposition productions to compute the values fi(E; j; strings bounded by length. The array kmax keeps track of the depth of the abstraction hierarchy for each subsequence length. B.2 Example Calculations To illustrate the computation of fi values, consider the result of using Charniak's grammar from Fig. 1 as its input. We initialize the entries for have probability one for each terminal symbol, as in Fig. 5. To fill in the entries for we look at all of the abstraction productions. The symbols noun, verb, and prep can all be expanded into one or more terminal symbols, which have nonzero fi values at 1. We enter these three nonterminals at values equal to the sum, over all relevant abstraction productions, of the product of the probability of the given production and the value for the right-hand symbol at 1. For instance, we compute the value for noun by adding the product of the probability of noun!swat and the value for swat, that of noun!flies and flies, and that of noun!ants and ants. This yields the value one, since a noun will always derive a string of length one, at a single level abstraction above the terminal string, given this grammar. The abstraction phase continues until we find S at 4, for which there are no further abstractions, so we go Compute-Beta(grammar,length) for each symbol x 2Terminals(grammar) for each symbol E 2Nonterminals(grammar) then /* Decomposition phase */ for each production for each sequence fj t g m t=1 such that for each sequence fk t g m t=1 such that 1 - k t -kmax[j t ] result/ p for t / 1 to m result Abstraction Phase */ while for each production then fi[E; return fi, kmax Fig. 4. Algorithm for computing fi values. on to begin the decomposition phase. To illustrate the decomposition phase, consider the value for fi(S; 3; 1). There is only one possible decomposition production, s!np vp. However, we must consider two separate cases: when the noun phrase covers two symbols and the verb phrase one, and when the noun phrase covers one and the verb phrase two. At a subsequence length of two, both np and vp have nonzero probability only at the bottom level of abstraction, while at a length of one, only at the third. So to compute the probability of the first subsequence length combination, we multiply the probability of the production by fi(np; 2; 1) and fi(vp; 1; 3). The probability of the second combination is a similar product, and the sum of the two values provides the value to enter for S. The other abstractions and decompositions proceed along similar lines, with additional summation required when multiple productions or multiple levels of abstraction are possible. The final table is shown in Fig. 5, which lists only the nonzero values. np 0:0672 np 0:176 np 0:08 vp 0:3 vp 0:1008 vp 0:104 vp 0:12 2 prep 1:0 pp 0:176 pp 0:08 pp 0:4 verb 1:0 noun 1:0 like 1:0 swat 1:0 flies 1:0 ants 1:0 Fig. 5. Final table for sample grammar. B.3 Complexity For analysis of the complexity of computing the fi values for a given PCFG, it is useful to define d to be the maximum length of possible chains of abstraction productions (i.e., the maximum k value), and m to be the maximum production length (number of symbols on the right-hand side). A single run through the abstraction phase requires time O(jPA j), and for each subsequence length, there are O(d) runs. For a specific value of j, the decomposition phase requires time O(jPD jj each decomposition production, we must consider all possible combinations of subsequence lengths and levels of abstractions for each symbol on the right-hand side. Therefore, the whole algorithm would take time O(n[djP A j C. Network Generation Phase We can use the fi function calculated as described above to compute the domains of random variables N ijk and P ijk and the required conditional probabilities. C.1 Specification of Random Variables The procedure Create-Network, described in Fig. 6, begins at the top of the abstraction hierarchy for strings of length n starting at position 1. The root symbol variable, N 1n(kmax[n]) , can be either the start symbol, indicating the parse tree begins here, or nil , indicating that the parse tree begins below. We must allow the parse tree to start at any j and k where fi(S; because these can all possibly derive strings (of any length bounded by n) within the language. Create-Network then proceeds downward through the N ijk random variables and specifies the domain of their corresponding production variables, P ijk . Each such production variable takes on values from the set of possible expansions for the possible nonterminal symbols in the domain of N ijk . If k ? 1, only abstraction productions are possible, so the procedure Abstraction-Phase, described in Fig. 7, inserts all possible expansions and draws links from P ijk to the random variable N ij(k\Gamma1) , which takes on the value of the right-hand side symbol. If the procedure Decomposition-Phase, described in Fig. 8, performs the analogous task for decomposition productions, except that it must also consider all possible length breakdowns and abstraction levels for the symbols on the right-hand side. if fi[S; length; then Insert-State(N 1(length)kmax[length] ,S) if Start-Prob(fi,kmax,length,kmax[length]\Gamma1)? 0:0 then Insert-State(N 1(length)kmax[length] ,nil ) for k / kmax[j] down-to 1 for for each symbol E 2Domain(N ijk ) then then else else Fig. 6. Procedure for generating the network. for each production Insert-State(N Fig. 7. Procedure for finding all possible abstraction productions. for each production for each sequence fj t g m t=1 such that for each sequence fk t g m t=1 such that 1 - k t -kmax[j t ] then Insert-State(P ijk for t / 1 to m Fig. 8. Procedure for finding all possible decomposition productions. then if fi[S; then Insert-State(P ijk ,nil ! S[j; Insert-State(N ij(k\Gamma1) ,S) else if then Insert-State(P ijk ,nil Insert-State(N i(j \Gamma1)kmax[j \Gamma1] ,S) if Start-Prob(fi,kmax,j,k)? 0:0 then Insert-State(P ijk ,nil !nil ) then Add-Parent(N ij(k\Gamma1) ,P ijk ) else Add-Parent(N i(j \Gamma1)kmax[j \Gamma1] ,P ijk ) Insert-State(N i(j \Gamma1)kmax[j \Gamma1] ,nil ) Fig. 9. Procedure for handling start of parse tree at next level. if j=0 then return 0.0 else if k=0 then return else return fi[S; Fig. 10. Procedure for computing the probability of the start of the tree occurring for a particular string length and abstraction level. Create-Network calls the procedure Start-Tree, described in Fig. 9, to handle the possible expansions of nil : either nil ! S, indicating that the tree starts immediately below, or nil ! nil , indicating that the tree starts further below. Start-Tree uses the procedure Start-Prob, described in Fig. 10, to determine the probability of the parse tree starting anywhere below the current point of expansion. When we insert a possible value into the domain of a production node, we add it as a parent of each of the nodes corresponding to a symbol on the right-hand side. We also insert each symbol from the right-hand side into the domain of the corresponding symbol variable. The algorithm descriptions assume the existence of procedures Insert-State and Add-Parent. The procedure Insert-State(node,label) inserts a new state with name label into the domain of variable node. The procedure Add-Parent(child,parent) draws a link from node parent to node child. C.2 Specification of Conditional Probability Tables After Create-Network has specified the domains of all of the random variables, we can specify the conditional probability tables. We introduce the lexicographic order OE over the set f(j; k)j1 - (j For the purposes of simplicity, we do not specify an exact value for each probability specify a weight, We compute the exact probabilities through normalization, where we divide each weight by the sum . The prior probability table for the top node, which has no parents, can be defined as follows: (j;k)OE(n;kmax[n]) For a given state ae in the domain of any P ijk node, where ae represents a production and corresponding assignment of j and k values to the symbols on the right-hand side, of the form (p), we can define the conditional probability of that state as: E) / p Y For any symbol E in the domain of N ijk , Pr(P For the productions for starting or delaying the tree, the probabilities are: (j The probability tables for the N ijk nodes are much simpler, since once the productions are specified, the symbols are completely determined. Therefore, the entries are either one or zero. For example, consider the nodes N with the parent node P (among others). For the rule ae representing . For all symbols other than E ' in the domain of N i ' , this conditional probability is zero. We can fill in this entry for all configurations of the other parent nodes (represented by the ellipsis in the condition part of the probability), though we know that any conflicting configurations (i.e., two productions both trying to specify the symbol N are impossible. Any configuration of the parent nodes that does not specify a certain symbol indicates that the node takes on the value nil with probability one. C.3 Network Generation Example As an illustration, consider the execution of this algorithm using the fi values from Fig. 5. We start with the root variable N 142 . The start symbol S has a fi value greater than zero here, as well as at points below, so the domain must include both S and nil . To obtain Pr(N simply divide fi(S; 4; 2) by the sum of all fi values for S, yielding 0.055728. The domain of P 142 is partially specified by the abstraction phase for the symbol S in the domain of N 142 . There is only one relevant production, S !vp, which is a possible expansion since fi(vp; 4; 1) ? 0. Therefore, we insert the production into the domain of P 142 , with conditional probability one given that N since there are no other possible expansions. We also draw a link from P 142 to N 141 , whose domain now includes vp with conditional probability one given that P To complete the specification of P 142 , we must consider the possible start of the tree, since the domain of N 142 includes nil . The conditional probability of P is 0.24356, the ratio of fi(S; 4; 1) and the sum of fi(S; j; 1). The link from P 142 to N 141 has already been made during the abstraction phase, but we must also insert S and nil into the domain of N 141 , each with conditional probability one given the appropriate value of P 142 . We then proceed to N 141 , which is at the bottom level of abstraction, so we must perform a decomposition phase. For the production S ! np vp, there are three possible combinations of subsequence lengths which add to the total length of four. If np derives a string of length one and vp a string of length three, then the only possible levels of abstraction for each are three and one, respectively, since all others will have zero values. Therefore, we insert the production s!np[1,3] vp[3,1] into the domain of where the numbers in brackets correspond to the subsequence length and level of abstraction, respectively. The conditional probability of this value, given that N is the product of the probability of the production, fi(np; 1; 3), and fi(vp; 3; 1), normalized over the probabilities of all possible expansions. We then draw links from P 141 to N 113 and N 231 , into whose domains we insert np and vp, respectively. The i values are obtained by noting that the subsequence for np begins at the same point as the original string while that for vp begins at a point shifted by the length of the subsequence for np. Each occurs with probability one, given that the value of P 141 is the appropriate production. Similar actions are taken for the other possible subsequence length combinations. The operations for the other random variables are performed in a similar fashion, leading to the network structure shown in Fig. 11. C.4 Complexity of Network Generation The resulting network has O(n 2 d) nodes. The domain of each N i11 variable has O(j\Sigmaj) states to represent the possible terminal symbols, while all other N ijk variables have O(jN possible states. There are n variables of the former, and O(n 2 d) of the latter. For k ? 1, the P ijk variables (of which there are O(n 2 d)) have a domain of O(jPA states. For P ij1 variables, there are states for each possible decomposition production, for each possible combination of subsequence lengths, and for each possible level of abstraction of the symbols on the right-hand side. Therefore, the P ij1 variables (of which there are have a domain of O(jPD jj states, where we have again defined d to be the maximum value of k, and m to be the maximum production length. Unfortunately, even though each particular P variable has only the corresponding N variable as its parent, a given N variable could have potentially O(n) P variables as Fig. 11. Network from example grammar at maximum length 4. parents. The size of the conditional probability table for a node is exponential in the number of parents, although given that each N can be determined by at most one P (i.e., no interactions are possible), we can specify the table in a linear number of parameters. If we define T to be the maximum number of entries of any conditional probability table in the network, then the abstraction phase of the algorithm requires time O(jPA jT ), while the decomposition phase requires time O(jPD jn Handling the start of the parse tree and the potential space holders requires time O(T ). The total time complexity of the algorithm is then O(n 2 jP D jn O(jP jn m+1 d m T m ), which dwarfs the time complexity of the dynamic programming algorithm for the fi function. However, this network is created only once for a particular grammar and length bound. D. PCFG Queries We can use the Bayesian network to compute any joint probability that we can express in terms of the N and P random variables included in the network. The standard Bayesian network algorithms [11], [12], [14] can return joint probabilities of the form conditional probabilities of the form Pr(X each X is either N or P . Obviously, if we are interested only in whether a symbol E appeared at a particular location in the parse tree, we need only examine the marginal probability distribution of the corresponding variable. Thus, a single network query will yield the probability Pr(N E). The results of the network query are implicitly conditional on the event that the length of the terminal string does not exceed n. We can obtain the joint probability by multiplying the result by the probability that a string in the language has a length not exceeding n. For any j, the probability that we expand the start symbol S into a terminal string of length j is which we can then sum for To obtain the appropriate unconditional probability for any query, all network queries reported in this section must be multiplied by D.1 Probability of Conjunctive Events The Bayesian network also supports the computation of joint probabilities analogous to those computed by the standard PCFG algorithms. For instance, the probability of a particular terminal string such as Swat flies like ants corresponds to the probability ants). The probability of an initial subsequence like Swat flies. , as computed by the LRI algorithm [10], corresponds to the probability Pr(N like). Since the Bayesian network represents the distribution over strings of bounded length, we can find initial subsequence probabilities only over completions of length bounded by However, although in this case our Bayesian network approach requires some modification to answer the same query as the standard PCFG algorithm, it needs no modification to handle more complex types of evidence. The chart parsing and LRI algorithms require complete sequences as input, so any gaps or other uncertainty about particular symbols would require direct modification of the dynamic programming algorithms to compute the desired probabilities. The Bayesian network, on the other hand, supports the computation of the probability of any evidence, regardless of its structure. For in- stance, if we have a sentence Swat flies . ants where we do not know the third word, a single network query will provide the conditional probability of possible completions well as the probability of the specified evidence Pr(N This approach can handle multiple gaps, as well as partial information. For example, if we again do not know the exact identity of the third word in the sentence Swat flies . ants, but we do know that it is either swat or like, we can use the Bayesian network to fully exploit this partial information by augmenting our query to specify that any domain values for N 311 other than swat or like have zero probability. Although these types of queries are rare in natural language, domains like speech recognition often require this ability to reason when presented with noisy observations. We can answer queries about nonterminal symbols as well. For instance, if we have the sentence Swat flies like ants, we can query the network to obtain the conditional probability that like ants is a prepositional phrase, Pr(N like; ants). We can answer queries where we specify evidence about nonterminals within the parse tree. For instance, if we know that like ants is a prepositional phrase, the input to the network query will specify that N as well as specifying the terminal symbols. Alternate network algorithms can compute the most probable state of the random variables given the evidence, instead of a conditional probability [11], [15], [14]. For example, consider the case of possible four-word sentences beginning with the phrase Swat flies. The probability maximization network algorithms can determine that the most probable state of terminal symbol variables N 311 and N 411 is like flies, given that N 111 =swat, N 211 =flies, and N 511 =nil. D.2 Probability of Disjunctive Events We can also compute the probability of disjunctive events through multiple network queries. If we can express an event as the union of mutually exclusive events, each of the form X then we can query the network to compute the probability of each, and sum the results to obtain the probability of the union. For instance, if we want to compute the probability that the sentence Swat flies like ants contains any prepositions, we would query the network for the probabilities In a domain like plan recognition, such a query could correspond to the probability that an agent performed some complex action within a specified time span. In this example, the individual events are already mutually exclusive, so we can sum the results to produce the overall probability. In general, we ensure mutual exclusivity of the individual events by computing the conditional probability of the conjunction of the original query event and the negation of those events summed previously. For our example, the overall probability would be Pr(N prepjE)+Pr(N prep; N 312 6= prepjE), where E corresponds to the event that the sentence is Swat flies like ants. The Bayesian network provides a unified framework that supports the computation of all of the probabilities described here. We can compute the probability of any event is a set of mutually exclusive events fX i t1 j t1 k t1 t=1 with each X being either N or P . We can also compute probabilities of events where we specify relative likelihoods instead of strict subset restrictions. In addition, given any such event, we can determine the most probable configuration of the uninstantiated random variables. Instead of designing a new algorithm for each such query, we have only to express the query in terms of the network's random variables, and use any Bayesian network algorithm to compute the desired result. D.3 Complexity of Network Queries Unfortunately, the time required by the standard network algorithms in answering these queries is potentially exponential in the maximum string length n, though the exact complexity will depend on the connectedness of the network and the particular network algorithm chosen. The algorithm in our current implementation uses a great deal of preprocessing in compiling the networks, in the hope of reducing the complexity of answering queries. Such an algorithm can exploit the regularities of our networks (e.g., the conditional probability tables of each N ijk consist of only zeroes and ones) to provide reasonable response time in answering queries. Unfortunately, such compilation can itself be prohibitive and will often produce networks of exponential size. There exist Bayesian network algorithms [16], [17] that offer greater flexibility in compilation, possibly allowing us to to limit the size of the resulting networks, while still providing acceptable query response times. Determining the optimal tradeoff will require future research, as will determining the class of domains where our Bayesian network approach is preferable to existing PCFG algorithms. It is clear that the standard dynamic programming algorithms are more efficient for the PCFG queries they address. For domains requiring more general queries of the types described here, the flexibility of the Bayesian network approach may justify the greater complexity. IV. Context Sensitivity For many domains, the independence assumptions of the PCFG model are overly restrictive. By definition, the probability of applying a particular PCFG production to expand a given nonterminal is independent of what symbols have come before and of what expansions are to occur after. Even this paper's simplified example illustrates some of the weaknesses of this assumption. Consider the intermediate string Swat ants like noun. It is implausible that the probability that we expand noun into flies instead of ants is independent of the choice of swat as the verb or the choice of ants as the object. Of course, we may be able to correct the model by expanding the set of nonterminals to encode contextual information, adding productions for each such expansion, and thus preserving the structure of the PCFG model. However, this can obviously lead to an unsatisfactory increase in complexity for both the design and use of the model. Instead, we could use an alternate model which relaxes the PCFG independence assumptions. Such a model would need a more complex production and/or probability structure to allow complete specification of the distribution, as well as modified inference algorithms for manipulating this distribution. A. Direct Extensions to Network Structure The Bayesian network representation of the probability distribution provides a basis for exploring such context sensitivities. The networks generated by the algorithms of this paper implicitly encode the PCFG assumptions through assignment of a single nonterminal node as the parent of each production node. This single link indicates that the expansion is conditionally independent of all other nondescendant nodes, once we know the value of this nonterminal. We could extend the context-sensitivity of these expansions within our network formalism by altering the links associated with these production nodes. We can introduce some context sensitivity even without adding any links. Since each production node has its own conditional probability table, we can define the production probabilities to be a function of the (i; j; values. For instance, the number of words in a group strongly influences the likelihood of that group forming a noun phrase. We could model such a belief by varying the probability of a np appearing over different string lengths, as encoded by the j index. In such cases, we can modify the standard PCFG representation so that the probability information associated with each production is a function of i, j, and k, instead of a constant. The dynamic programming algorithm of Fig. 4 can be easily modified to handle production probabilities that depend on j and k. However, a dependency on the i index as well would require adding it as a parameter of fi and introducing an additional loop over its possible values. Then, we would have to replace any reference to the production probability, in either the dynamic programming or network generation algorithm, with the appropriate function of i, j, and k. Alternatively, we may introduce additional dependencies on other nodes in the net- work. A PCFG extension that conditions the production probabilities on the parent of the left-hand side symbol has already proved useful in modeling natural language [18]. In this case, each production has a set of associated probabilities, one for each non-terminal symbol that is a possible parent of the symbol on the left-hand side. This new probability structure requires modifications to both the dynamic programming and the network generation algorithms. We must first extend the probability information of the fi function to include the parent nonterminal as an additional parameter. It is then straightforward to alter the dynamic programming algorithm of Fig. 4 to correctly compute the probabilities in a bottom-up fashion. The modifications for the network generation algorithm are more complicated. Whenever we add P ijk as a parent for some symbol node N - k , we also have to add N ijk as a parent of P - k . For example, the dotted arrow in the subnetwork of Fig. 12 represents the additional dependency of P 112 on N 113 . We must add this link because N 112 is a possible child nonterminal, as indicated by the link from P 113 . The conditional probability tables for each P node must now specify probabilities given the current nonterminal and the parent nonterminal symbols. We can compute these by combining the modified fi values with the conditional production probabilities. Returning to the example from the beginning of this section, we may want to condition the production probabilities on the terminal string expanded so far. As a first approximation to such context sensitivity, we can imagine a model where each production has an associated set of probabilities, one for each terminal symbol in the language. Each represents the conditional probability of the particular expansion given that the corresponding terminal symbol occurs immediately previous to the subsequence derived from the nonterminal symbol on the left-hand side. Again, our fi function requires an additional parameter, and we need a modified version of the dynamic programming algorithm to compute its values. However, the network generation algorithm needs to introduce only one additional link, from N i11 for each P (i+1)jk node. The dashed arrows Fig. 12. Subnetwork incorporating parent symbol dependency. Fig. 13. Subnetwork capturing dependency on previous terminal symbol. in the subnetwork of Fig. 13 reflect the additional dependencies introduced by this context sensitivity, using the network example from Fig. 11. The P 1jk nodes are a special case, with no preceding terminal, so the steps from the original algorithm are sufficient. We can extend this conditioning to cover preceding terminal sequences rather than individual symbols. Each production could have an associated set of probabilities, one for each possible terminal sequence of length bounded by some parameter h. The fi function now requires an additional parameter specifying the preceding sequence. The network generation algorithms must then add links to P ijk from nodes N (i\Gammah)11 ,. , h, or from N 111 ,. , N (i\Gamma1)11 , if i ! h. The conditional probability tables then specify the probability of a particular expansion given the symbol on the left-hand side and the preceding terminal sequence. In many cases, we may wish to account for external influences, such as explicit context representation in natural language problems or influences of the current world state in planning, as required by many plan recognition problems [19]. For instance, if we are processing multiple sentences, we may want to draw links from the symbol nodes of one sentence to the production nodes of another, to reflect thematic connections. As long as our network can include random variables to represent the external context, then we can represent the dependency by adding links from the corresponding nodes to the appropriate production nodes and altering the conditional probability tables to reflect the effect of the context. In general, the Bayesian networks currently generated contain a set of random variables sufficient for expressing arbitrary parse tree events, so we can introduce context sensitivity by adding the appropriate links to the production nodes from the events on which we wish to condition expansion probabilities. Once we have the correct network, we can use any of the query algorithms from Section III-D to produce the corresponding conditional probability. B. Extensions to the Grammar Model Context sensitivities expressed as incremental changes to the network dependency structure represent only a minor relaxation of the conditional independence assumptions of the PCFG model. More global models of context sensitivity will likely require a radically different grammatical form and probabilistic interpretation framework. The History-Based Grammar (HBG) [20] provides a rich model of context sensitivity by conditioning the production probabilities on (potentially) the entire parse tree available at the current expansion point. Since our Bayesian networks represent all positions of the parse tree, it is theoretically possible to represent these conditional probabilities by introducing the appropriate links. However, since the HBG model uses decision tree methods to identify equivalence classes of the partial trees and thus produce simple event structures to condition on, it is unclear exactly how to replicate this behavior in a systematic generation algorithm. If we restrict the types of context sensitivity, then we are more likely to find such a network generation algorithm. In the non-stochastic case, context-sensitive grammars [9] provide a more structured model than the general unrestricted grammar by allowing only productions of the form ff 1 the ffs are arbitrary sequences of terminal and/or nonterminal symbols. This restriction eliminates productions where the right-hand side is shorter then the left-hand side. Such a production indicates that A can be expanded into B only when it appears in the surrounding context of ff 1 immediately precedent and ff 2 immediately subsequent. Therefore, perhaps an extension to a probabilistic context-sensitive grammar (PCSG), similar to that for PCFGs, could provide an even richer model for the types of conditional probabilities briefly explored here. The intuitive extension involves associating a likelihood weighting with each context-sensitive production and computing the probability of a particular derivation based on these weights. These weights cannot correspond to probabilities, because we do not know, a priori, which expansions may be applicable at a given point in the parse (due to the different possible contexts). Therefore, a set of fixed production values may not produce weights that sum to one in a particular context. We can instead use these weights to determine probabilities after we know which productions are applicable. The probability of a particular derivation sequence is then uniquely determined, though it could be sensitive to the order in which we apply the productions. We could then define a probability distribution over all strings in the context-sensitive language so that the probability of a particular string is the sum of the probabilities over all possible derivation sequences for that string. This definition appears theoretically sound, though it is unclear whether any real-world domains exist for which such a model would be useful. If we create such a model, we should be able to generate a Bayesian network with the proper conditional dependency structure to represent the distribution. We would have to draw links to each production node from its potential context nodes, and the conditional probability tables would reflect the production weights in each particular context possibility. It is an open question whether we could create a systematic generation algorithm similar to that defined for PCFGs. Although the proposed PCSG model cannot account for dependence on position or parent symbol, described earlier in this section, we could make similar extensions to account for these types of dependencies. The result would be similar to the context-sensitive probabilities of Pearl [21]. However, Pearl conditions the probabilities on a part-of-speech trigram, as well as on the sibling and parent nonterminal symbols. If we allow our model to specify conjunctions of contexts, then it may be able to represent these same types of probabilities, as well as more general contexts beyond siblings and trigrams. It is clearly difficult to select a model powerful enough to encompass a significant set of useful dependencies, but restricted enough to allow easy specification of the productions and probabilities for a particular language. Once we have chosen a grammatical formalism capable of representing the context sensitivities we wish to model, we must define a network generation algorithm to correctly specify the conditional probabilities for each production node. However, once we have the network, we can again use any of the query algorithms from Section III-D. Thus, we have a unified framework for performing inference, regardless of the form of the language model used to generate the networks. Probabilistic parse tables [8] and stochastic programs [22] provide alternate frameworks for introducing context sensitivity. The former approach uses the finite-state machine of the chart parser as the underlying structure and introduces context sensitivity into the transition probabilities. Stochastic programs can represent very general stochastic processes, including PCFGs, and their ability to maintain arbitrary state information could support general context sensitivity as well. It is unclear whether any of these approaches have advantages of generality or efficiency over the others. V. Conclusion The algorithms presented here automatically generate a Bayesian network representing the distribution over all parses of strings (bounded in length by some parameter) in the language of a PCFG. The first stage uses a dynamic programming approach similar to that of standard parsing algorithms, while the second stage generates the network, using the results of the first stage to specify the probabilities. This network is generated only once for a particular PCFG and length bound. Once created, we can use this network to answer a variety of queries about possible strings and parse trees. Using the standard Bayesian network inference algorithms, we can compute the conditional probability or most probable configuration of any collection of our basic random variables, given any other event which can be expressed in terms of these variables. These algorithms have been implemented and tested on several grammars, with the results verified against those of existing dynamic programming algorithms when applicable, and against enumeration algorithms when given nonstandard queries. When answering standard queries, the time requirements for network inference were comparable to those for the dynamic programming techniques. Our network inference methods achieved similar response times for some other types of queries, providing a vast improvement over the much slower brute force algorithms. However, in our current implementation, the memory requirements of network compilation limit the complexity of the grammars and queries, so it is unclear whether these results will hold for larger grammars and string lengths. Preliminary investigation has also demonstrated the usefulness of the network formalism in exploring various forms of context-sensitive extensions to the PCFG model. Relatively minor modifications to the PCFG algorithms can generate networks capable of representing the more general dependency structures required for certain context sen- sitivities, without sacrificing the class of queries that we can answer. Future research will need to provide a more general model of context sensitivity with sufficient structure to support a corresponding network generation algorithm. Although answering queries in Bayesian networks is exponential in the worst case, our method incurs this cost in the service of greatly increased generality. Our hope is that the enhanced scope will make PCFGs a useful model for plan recognition and other domains that require more flexibility in query forms and in probabilistic structure. In addition, these algorithms may extend the usefulness of PCFGs in natural language processing and other pattern recognition domains where they have already been successful. Acknowledgments We are grateful to the anonymous reviewers for careful reading and helpful suggestions. This work was supported in part by Grant F49620-94-1-0027 from the Air Force Office of Scientific Research. --R An Introduction "Probabilistic languages: A review and some open questions," "Recognition of equations using a two-dimensional stochastic context-free grammar," "Stochastic grammars and pattern recognition," "Stochastic context-free grammars for modeling RNA," "Getting serious about parsing plans: A grammatical analysis of plan recognition," "Generalized probabilistic LR parsing of natural language (corpora) with unification-based grammars," Introduction to Automata Theory "Basic methods of probabilistic context free grammars," Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference Probabilistic Reasoning in Expert Systems: Theory and Algorithms An Introduction to Bayesian Networks "Bucket elimination: A unifying framework for probabilistic inference," "Cost-based abduction and MAP explanation," "Topological parameters for time-space tradeoff," "Query DAGs: A practical paradigm for implementing belief-network infer- ence," "Context-sensitive statistics for improved grammatical language models," "Accounting for context in plan recognition, with application to traffic monitoring," "Towards history-based grammars: Using richer models for probabilistic parsing," "Pearl: A probabilistic chart parser," "Effective Bayesian inference for stochastic programs," --TR --CTR Jorge R. Ramos , Vernon Rego, Feature-based generators for time series data, Proceedings of the 37th conference on Winter simulation, December 04-07, 2005, Orlando, Florida

probabilistic context-free grammars;bayesian networks

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Efficient Sparse LU Factorization with Partial Pivoting on Distributed Memory Architectures.

AbstractA sparse LU factorization based on Gaussian elimination with partial pivoting (GEPP) is important to many scientific applications, but it is still an open problem to develop a high performance GEPP code on distributed memory machines. The main difficulty is that partial pivoting operations dynamically change computation and nonzero fill-in structures during the elimination process. This paper presents an approach called S* for parallelizing this problem on distributed memory machines. The S* approach adopts static symbolic factorization to avoid run-time control overhead, incorporates 2D L/U supernode partitioning and amalgamation strategies to improve caching performance, and exploits irregular task parallelism embedded in sparse LU using asynchronous computation scheduling. The paper discusses and compares the algorithms using 1D and 2D data mapping schemes, and presents experimental studies on Cray-T3D and T3E. The performance results for a set of nonsymmetric benchmark matrices are very encouraging, and S* has achieved up to 6.878 GFLOPS on 128 T3E nodes. To the best of our knowledge, this is the highest performance ever achieved for this challenging problem and the previous record was 2.583 GFLOPS on shared memory machines [8].

Currently with the Computer Science Department, University of Illinois at Urbana-Champaign. pivoting operations interchange rows based on the numerical values of matrix elements during the elimination process, it is impossible to predict the precise structures of L and U factors without actually performing the numerical factorization. The adaptive and irregular nature of sparse LU data structures makes an efficient implementation of this algorithm very hard even on a modern sequential machine with memory hierarchies. There are several approaches that can be used for solving nonsymmetric systems. One approach is the unsymmetric-pattern multi-frontal method [5, 25] that uses elimination graphs to model irregular parallelism and guide the parallel computation. Another approach [19] is to restructure a sparse matrix into a bordered block upper triangular form and use a special pivoting technique which preserves the structure and maintains numerical stability at acceptable levels. This method has been implemented on Illinois Cedar multi-processors based on Aliant shared memory clusters. This paper focuses on parallelization issues for a given column ordering with row interchanges to maintain numerical stability. Parallelization of sparse LU with partial pivoting is also studied in [21] on a shared memory machine by using static symbolic LU factorization to overestimate nonzero fill-ins and avoid dynamic variation of LU data structures. This approach leads to good speedups for up to 6 processors on a Sequent machine and further work is needed to assess the performance of the sequential code. As far as we know, there are no published results for parallel sparse LU on popular commercial distributed memory machines such as Cray-T3D/T3E, Intel Paragon, IBM SP/2, TMC CM-5 and Meiko CS-2. One difficulty in the parallelization of sparse LU on these machines is how to utilize a sophisticated uniprocessor architecture. The design of a sequential algorithm must take advantage of caching, which makes some previously proposed techniques less effective. On the other hand, a parallel implementation must utilize the fast communication mechanisms available on these ma- chines. It is easy to get speedups by comparing a parallel code to a sequential code which does not fully exploit the uniprocessor capability, but it is not as easy to parallelize a highly optimized sequential code. One such sequential code is SuperLU [7] which uses a supernode approach to conduct sequential sparse LU with partial pivoting. The supernode partitioning makes it possible to perform most of the numerical updates using BLAS-2 level dense matrix-vector multiplications, and therefore to better exploit memory hierarchies. SuperLU performs symbolic factorization and generates supernodes on the fly as the factorization proceeds. UMFPACK is another competitive sequential code for this problem and neither SuperLU nor UMFPACK is always better than the other [3, 4, 7]. MA41 is a code for sparse matrices with symmetric patterns. All of them are regarded as of high quality and deliver excellent megaflop performance. In this paper we focus on the performance analysis and comparison with SuperLU code since the structure of our code is closer to that of SuperLU. In this paper, we present an approach called S that considers the following key strategies together in parallelizing the sparse LU algorithm: 1. Adopt a static symbolic factorization scheme to eliminate the data structure variation caused by dynamic pivoting. 2. data regularity from the sparse structure obtained by the symbolic factorization scheme so that efficient dense operations can be used to perform most of computation and the impact of nonzero fill-in overestimation on overall elimination time is minimized. 3. Develop scheduling techniques for exploiting maximum irregular parallelism and reducing memory requirements for solving large problems. We observe that on most current commodity processors with memory hierarchies, a highly optimized BLAS-3 subroutine usually outperforms a BLAS-2 subroutine in implementing the same numerical operations [6, 9]. We can afford to introduce some extra BLAS-3 operations in re-designing the LU algorithm so that the new algorithm is easy to be parallelized but the sequential performance of this code is still competitive to the current best sequential code. We use the static symbolic factorization technique first proposed in [20, 21] to predict the worst possible structures of L and U factors without knowing the actual numerical values, then we develop a 2-D L/U supernode partitioning technique to identify dense structures in both L and U factors, and maximize the use of BLAS-3 level subroutines for these dense structures. We also incorporate a supernode amalgamation [1, 10] technique to increase the granularity of the computation. In exploiting irregular parallelism in the re-designed sparse LU algorithm, we have experimented with two mapping methods, one of which uses 1-D data mapping and the other uses 2-D data mapping. One advantage of using 1-D data mapping is that the corresponding LU algorithm can be easily modeled by directed acyclic task graphs (DAGs). Graph scheduling techniques and efficient run-time support are available to schedule and execute DAG parallelism [15, 16]. Scheduling and executing DAG parallelism is a difficult job because parallelism in sparse problems is irregular and execution must be asynchronous. The important optimizations are overlapping computation with communication, balancing processor loads and eliminating unnecessary communication overhead. Graph scheduling can do an excellent job in exploiting irregular parallelism but it leads to extra memory space per node to achieve the best performance. Also the 1-D data mapping can only expose limited parallelism. Due to these restrictions, we have also examined a 2-D data mapping method and an asynchronous execution scheme which exploits parallelism under memory constraints. We have implemented our sparse LU algorithms and conducted experiments with a set of nonsymmetric benchmark matrices on Cray-T3D and T3E. Our experiments show that our approach is quite effective in delivering good performance in terms of high megaflop numbers. In particular, the 1-D code outperforms the current 2-D code when processors have sufficient memory. But the 2-D code has more potential to solve larger problems and produces higher megaflop numbers. The rest of the paper is organized as follows. Section 2 gives the problem definition. Section 3 describes structure prediction and 2-D L/U supernode partitioning for sparse LU factorization. Section 4 describes program partitioning and data mapping schemes. Section 5 addresses the asynchronous computation scheduling and execution. Section 6 presents the experimental results. Section 7 concludes the paper. Find m such that ja mk (03) if a then A is singular, stop; row k with row m; with a ik 6= 0 with a kj 6= 0 with a ik 6= 0 Figure 1: Sparse Gaussian elimination with partial pivoting for LU factorization. Preliminaries Figure shows how a nonsingular matrix A can be factored into two matrices L and U using GEPP. The elimination steps are controlled by loop index k. For elements manipulated at step k, we use i for row indexing and j for column indexing. This convention will be used through the rest of this paper. During each step of the elimination process, a row interchange may be needed to maintain numerical stability. The result of LU factorization process can be expressed by: L is a unit lower triangular matrix, U is a upper triangular matrix, and P is a permutation matrix which contains the row interchange information. The solution of a linear system be solved by two triangular solvers: y. The triangular solvers are much less time consuming than the Gaussian elimination process. Caching behavior plays an important role in achieving good performance for scientific computations. To better exploit memory hierarchy in modern architectures, supernode partitioning is an important technique to exploit the regularity of sparse matrix computations and utilize BLAS routines to speed up the computation. It has been successfully applied to Cholesky factorization [26, 30, 31]. The difficulty for the nonsymmetric factorization is that supernode structure depends on pivoting choices during the factorization thus cannot be determined in advance. SuperLU performs symbolic factorization and identifies supernodes on the fly. It also maximizes the use of BLAS- level operations to improve the caching performance of sparse LU. However, it is challenging to parallelize SuperLU on distributed memory machines. Using the precise pivoting information at each elimination step can certainly optimize data space usage, reduce communication and improve load balance, but such benefits could be offset by high run-time control and communication overhead. The strategy of static data structure prediction in [20] is valuable in avoiding dynamic symbolic factorization, identifying the maximum data dependence patterns and minimizing dynamic control overhead. We will use this static strategy in our S approach. But the overestimation does introduce extra fill-ins and lead to a substantial amount of unnecessary operations in the numerical factorization. We observe that in SuperLU [7] the DGEMV routine (the BLAS-2 level dense matrix vector multiplication) accounts for 78% to 98% of the floating point operations (excluding the symbolic factorization part). It is also a fact that BLAS-3 routine DGEMM (matrix-matrix multiplication) is usually much faster than BLAS-1 and BLAS-2 routines [6]. On Cray-T3D with a matrix of size \Theta 25, DGEMM can achieve 103 MFLOPS while DGEMV only reaches 85 MFLOPS. Thus the key idea of our approach is that if we could find a way to maximize the use of DGEMM after using static symbolic factorization, even with overestimated nonzeros and extra numerical operations, the overall code performance could still be competitive to SuperLU which mainly uses DGEMV. 3 Storage prediction and dense structure identification 3.1 Storage prediction The purpose of symbolic factorization is to obtain structures of L and U factors. Since pivoting sequences are not known until the numerical factorization, the only way to allocate enough storage space for the fill-ins generated in the numerical factorization phase is to overestimate. Given a sparse matrix A with a zero-free diagonal, a simple solution is to use the Cholesky factor L c of A T A. It has been shown that the structure of L c can be used as an upper bound for the structures of L and U factors regardless of the choice of the pivot row at each step [20]. But it turns out that this bound is not very tight. It often substantially overestimates the structures of the L and U factors (refer to Table 1). Instead we consider another method from [20]. The basic idea is to statically consider all possible pivoting choices at each step. The space is allocated for all the possible nonzeros that would be introduced by any pivoting sequence that could occur during the numerical factorization. We summarize the symbolic factorization method briefly as follows. The nonzero structure of a row is defined as a set of column indices at which nonzeros or fill-ins are present in the given n \Theta n matrix A. Since the nonzero pattern of each row will change as the factorization proceeds, we use R k i to denote the structure of row i after step k of the factorization and A k to denote the structure of the matrix A after step k. And a k ij denotes the element a ij in A k . Notice that the structures of each row or the whole matrix cover the structures of both L and U factors. In addition, during the process of symbolic factorization we assume that no exact numerical cancelation occurs. Thus, we have ij is structurally nonzerog: We also define the set of candidate pivot rows at step k as follows: ik is structurally nonzerog: We assume that a kk is always a nonzero. For any nonsingular matrix which does not have a zero-free diagonal, it is always possible to permute the rows of the matrix so that the permuted matrix has a zero-free diagonal [11]. Though the symbolic factorization does work on a matrix that contains zero entries in the diagonal, it is not preferable because it makes the overestimation too generous. The symbolic factorization process will iterate n steps and at step k, for each row its structure will be updated as: R Essentially the structure of each candidate pivot row at step k will be replaced by the union of the structures of all the candidate pivot rows except those column indices less than k. In this way it is guaranteed that the resulting structure A n will be able to accommodate the fill-ins introduced by any possible pivot sequence. A simple example in Figure 2 demonstrates the whole process. Nonzero Fill-in A Figure 2: The first 3 steps of the symbolic factorization on a sample 5 \Theta 5 sparse matrix. The structure remains unchanged at steps 4 and 5. This symbolic factorization is applied after an ordering is performed on the matrix A to reduce fill-ins. The ordering we are currently using is the multiple minimum degree ordering for A T A. We also permute the rows of the matrix using a transversal obtained from Duff's algorithm [11] to make A have a zero-free diagonal. The transversal can often help reduce fill-ins [12]. We have tested the storage impact of overestimation for a number of nonsymmetric testing matrices from various sources. The results are listed in Table 1. The fourth column in the table is original number of nonzeros, and the fifth column measures the symmetry of the structure of the original matrix. The bigger the symmetry number is, the more nonsymmetric the original matrix is. A unit symmetry number indicates a matrix is symmetric, but all matrices have nonsymmetric numerical values. We have compared the number of nonzeros obtained by the static approach and the number of nonzeros obtained by SuperLU, as well as that of the Cholesky factor of A T A, for these matrices. The results in Table 1 show that the overestimation usually leads to less than 50% extra nonzeros than SuperLU scheme does. Extra nonzeros do imply additional computational cost. For example, one has to either check if a symbolic nonzero is an actual nonzero during a numerical factorization, or directly perform arithmetic operations which could be unnecessary. If we can aggregate these floating point operations and maximize the use of BLAS-3 subroutines, the sequential code performance will still be competitive. Even the fifth column of Table 1 shows that the floating operations from the overestimating approach can be as high as 5 times, the results in Section 6 will show that actual ratios of running times are much less. Thus it is necessary and beneficial to identify dense structures in a sparse matrix after the static symbolic factorization. It should be noted that there are some cases that static symbolic factorization leads to excessive overestimation. For example, memplus matrix [7] is such a case. The static scheme produces 119 times as many nonzeros as SuperLU does. In fact, for this case, the ordering for SuperLU is applied based on A T + A instead of A T A. Otherwise the overestimation ratio is 2.34 if using A T A for SuperLU also. For another matrix wang3 [7], the static scheme produces 4 times as many nonzeros as SuperLU does. But our code can still produce 1 GFLOPS for it on 128 nodes of T3E. This paper focuses on the development of a high performance parallel code when overestimation ratios are not too high. Future work is to study ordering strategies that minimize overestimation ratios. factor entries/jAj S =SuperLU Matrix 9 e40r0100 17281 553562 1.000 14.76 17.32 26.48 1.17 3.11 Table 1: Testing matrices and their statistics. 3.2 2-D L/U supernode partitioning and dense structure identification Supernode partitioning is a commonly used technique to improve the caching performance of sparse code [2]. For a symmetric sparse matrix, a supernode is defined as a group of consecutive columns that have nested structure in the L factor of the matrix. Excellent performance has been achieved in [26, 30, 31] using supernode partitioning for Cholesky factorization. However, the above definition is not directly applicable to sparse LU with nonsymmetric matrices. A good analysis for defining unsymmetric supernodes in an L factor is available in [7]. Notice that supernodes may need to be further broken into smaller ones to fit into cache and to expose more parallelism. For the SuperLU approach, after L supernode partitioning, there are no regular dense structures in a U factor that could make it possible to use BLAS-3 routines (see Figure 3(a)). However in the S approach, there are dense columns (or subcolumns) in a U factor that we can identify after the static symbolic factorization (see Figure 3(b)). The U partitioning strategy is explained as follows. After an L supernode partition has been obtained on a sparse matrix A, i.e., a set of column blocks with possible different block sizes, the same partition is applied to the rows of the matrix to further break each supernode panel into submatrices. Now each off-diagonal submatrix in the L part is either a dense block or contains dense blocks. Furthermore, the following theorem identifies dense structure patterns in U factors. This is the key to maximizing the use of BLAS-3 subroutines in our algorithm. (a) (b) Figure 3: (a) An illustration of dense structures in a U factor in the SuperLU approach; (b) Dense structures in a U factor in the S approach. In the following theorem, we show that the 2-D L/U partitioning strategy is successful and there is a rich set of dense structures to exploit. The following notations will be used through the rest of the paper. ffl The L and U partitioning divides the columns of A into N column blocks and the rows of A into N row blocks so that the whole matrix is divided into N \Theta N submatrices. For submatrices in the U factor, we denote them as U ij for 1 . For submatrices in the L factor, we denote them as L ij for 1 denotes the diagonal submatrix. We use A ij to denote a submatrix when it is not necessary to distinguish between L and U factors. ffl Define S(i) as the starting column (or row) number of the i-th column (or row) block. For convenience, we define S(N ffl A subcolumn (or subrow) is a column (or row) in a submatrix. For simplicity, we use a global column (or row) index to denote a subcolumn (or subrow) in a submatrix. For example, by subcolumn k in the submatrix block U ij , it means the subcolumn in this submatrix with the global column index k where 1). Similarly we use a ij to indicate an individual nonzero element based on global indices. A compound structure in L or U is a submatrix, a subcolumn, or a subrow. ffl A compound structure is nonzero if it contains at least one nonzero element or fill-in. We use A ij 6= 0 to indicate that block A ij is nonzero. Notice that an algorithm only needs to operate on nonzero compound structures. A compound structure is structurally dense if all of its elements are nonzeros or fill-ins. In the following we will not differentiate between nonzero and fill-in entries. They are all considered as nonzero elements. Theorem 1 Given a sparse matrix A with a zero-free diagonal, after the above static symbolic factorization and 2-D L/U supernode partitioning are performed on A, each nonzero submatrix in the U factor of A contains only structurally dense subcolumns. Proof: Recall that P k is the set of candidate pivot rows at symbolic factorization step k. Given a supernode spanning from column k to k + s, from its definition and the fact that after step k the static symbolic factorization will only affect the nonzero patterns in submatrix a k+1:n;k+1:n , and A has a zero-free diagonal, we have Notice at each step k, the final structures of row i (i 2 P k ) are updated by the symbolic factorization procedure as R For the structure of a row i where k - i - k +s, we are only interested in nonzero patterns of the U part (excluding the part belonging to L kk ). We call this partial structure as UR i . Thus for UR It can be seen that after the k-th step updating, UR k Knowing that the structure of row k is unchanged after step k, we only need to prove that UR k k+s as shown below. Then we can infer that the nonzero structures of rows from k to k + s are same and subcolumns at the U part are either structurally dense or zero. Now since P k oe P k+1 , and it is clear that: Similarly we can show that UR k+s k . The above theorem shows that the L/U partitioning can generate a rich set of structurally dense subcolumns or even structurally dense submatrices in a U factor. We also further incorporate this result with supernode amalgamation in Section 3.3 and our experiments indicate that more than 64% of numerical updates is performed by the BLAS-3 routine DGEMM in S , which shows the effectiveness of the L/U partitioning method. Figure 4 demonstrates the result of a supernode partitioning on a 7 \Theta 7 sample sparse matrix. One can see that all the submatrices in the upper triangular part of the matrix only contain structurally dense subcolumns. Based on the above theorem, we can further show a structural relationship between two submatrices in the same supernode column block, which will be useful in implementing our algorithm to detect nonzero structures efficiently for numerical updating. Corollary 1 Given two nonzero submatrices U ij , U k in U ij is structurally dense, then subcolumn k in U i 0 j is also structurally dense. Nonzero Figure 4: An example of L/U supernode partitioning. Proof: The corollary is illustrated in Figure 5. Since L i 0 i is nonzero, there must be a structurally dense subrow in L i 0 i . This will lead to a nonzero element in the subcolumn k in U the subcolumn k of U ij is structurally dense. According to Theorem 1, subcolumn k in U i 0 j is structurally dense. U Figure 5: An illustration for Corollary 1. Corollary 2 Given two nonzero submatrices U ij , U is structurally dense, U must be structurally dense. Proof: That is straightforward using Corollary 1. 3.3 Supernode amalgamation For most tested sparse matrices, the average size of a supernode after L/U partitioning is very small, about 1:5 to 2 columns. This results in very fine grained tasks. Amalgamating small supernodes can lead to great performance improvement for both parallel and sequential sparse codes because it can improve caching performance and reduce interprocessor communication overhead. There could be many ways to amalgamate supernodes [7, 30]. The basic idea is to relax the restriction that all the columns in a supernode must have exactly the same nonzero structure below diagonal. The amalgamation is usually guided by a supernode elimination tree. A parent could be merged with its children if the merging does not introduce too many extra zero entries into a supernode. Row and column permutations are needed if the parent is not consecutive with its children. However, a column permutation introduced by the above amalgamation method could undermine the correctness of the static symbolic factorization. We have used a simpler approach that does not require any permutation. This approach only amalgamates consecutive supernodes if their nonzero structures only differ by a small number of entries and it can be performed in a very efficient manner which only has a time complexity of O(n) [27]. We can control the maximum allowed differences by an amalgamation factor r. Our experiments show that when r is in the range of gives the best performance for the tested matrices and leads to improvement on the execution times of the sequential code. The reason is that by getting bigger supernodes, we are getting larger dense structures, although there may be a few zero entries in them, and we are taking more advantage of BLAS-3 kernels. Notice that after applying the supernode amalgamation, the dense structures identified in the Theorem 1 are not strictly dense any more. We call them almost-dense structures and can still use the result of Theorem 1 with a minor revision. That is summarized in the following corollary. All the results presented in Section 6 are obtained using this amalgamation strategy. Corollary 3 Given a sparse matrix A, if supernode amalgamation is applied to A after the static symbolic factorization and 2-D L/U supernode partitioning are performed on A, each nonzero sub-matrix in the U factor of A contains only almost-structurally-dense subcolumns. 4 Program partitioning, task dependence and processor mapping After dividing a sparse matrix A into submatrices using the L/U supernode partitioning, we need to partition the LU code accordingly and define coarse grained tasks that manipulate on partitioned dense data structures. Program partitioning. Column block partitioning follows supernode structures. Typically there are two types of tasks. One is F actor(k), which is to factorize all the columns in the k-th column block, including finding the pivoting sequence associated with those columns. The other is Update(k; j), which is to apply the pivoting sequence derived from F actor(k) to the j-th column block, and modify the j-th column block using the k-th column block, where Instead of performing the row interchange to the right part of the matrix right after each pivoting search, a technique called "delayed-pivoting" is used [6]. In this technique, the pivoting sequence is held until the factorization of the k-th column block is completed. Then the pivoting sequence is applied to the rest of the matrix, i.e., interchange rows. Delayed-pivoting is important, especially to the parallel algorithm, because it is equivalent to aggregating multiple small messages into a larger one. Here the owner of the k-th column block sends the column block packed together with the pivoting information to other processors. An outline of the partitioned sparse LU factorization algorithm with partial pivoting is described in Figure 6. The code of F actor(k) is summarized in Figure 7. It uses BLAS-1 and BLAS- subroutines. The computational cost of the numerical factorization is mainly dominated by tasks. The function of task Update(k; j) is presented in Figure 8. The lines (05) and are using dense matrix multiplications. (2) Perform task F actor(k); Perform task Update(k; j); Figure partitioned sparse LU factorization with partial pivoting. (3) Find the pivoting row t in column m; row t and row m of the column block k; (5) Scale column m and update rest of columns in this column block; Figure 7: The description of task F actor(k). We use directed acyclic task graphs (DAGs) to model irregular parallelism arising in this partitioned sparse LU program. The DAGs are constructed statically before numerical factorization. Previous work on exploiting task parallelism for sparse Cholesky factorization has used elimination trees (e.g. [28, 30]), which is a good way to expose the available parallelism because pivoting is not required. For sparse LU, an elimination tree of A T A does not directly reflect the available paral- lelism. Dynamically created DAGs have been used for modeling parallelism and guiding run-time execution in a nonsymmetric multi-frontal method [5, 25]. Given the task definitions in Figures 6, 7 and 8 we can define the structure of a sparse LU task graph in the following. These four properties are necessary. ffl There are N tasks F actor(k), where 1 - k - N . ffl There is a task Update(k; . For a dense matrix, there will be a total of N(N \Gamma 1)=2 updating tasks. ffl There is a dependence edge from F actor(k) to task Update(k; j), where (02) Interchange rows according to the pivoting sequence; be the lower triangular part of L kk ; (04) if the submatrix U kj is dense else for each dense subcolumn c u of U kj for each nonzero submatrix A ij if the submatrix U kj is dense else for each dense subcolumn c u of U kj b be the corresponding dense subcolumn of A ij ; Figure 8: A description of task Update(k; j). ffl There is a dependence from Update(k; k 0 ) to F actor(k 0 ), where exists no task Update(t; k 0 ) such that We add one more property, that while not necessary, simplifies implementation. This property essentially does not allow exploiting commutativity among Update() tasks. However, according to our experience with Cholesky factorization [16], the performance loss due to this property is not substantial, about 6% in average when graph scheduling is used. ffl There is a dependence from Update(k; j) to Update(k there exists no task Update(t; j) such that Figure 9(a) shows the nonzero pattern of the partitioned matrix shown in Figure 4. Figure 9(b) is the corresponding task dependence graph. 1-D data mapping. In the 1-D data mapping, all submatrices, from both L and U part, of the same column block will reside in the same processor. Column blocks are mapped to processors in a cyclic manner or based on other scheduling techniques such as graph scheduling. Tasks are assigned based on owner-compute rule, i.e., tasks that modify the same column block are assigned to the same processor that owns the column block. One disadvantage of this mapping is that it serializes the computation in a single F actor(k) or In other words, a single F actor(k) or Update(k; task will be performed by (b)3 4 5125 Figure 9: (a) The nonzero pattern for the example matrix in Figure 4. (b) The dependence graph derived from the partitioning result. For convenience, F () is used to denote F actor(), U() is used to denote Update(). one processor. But this mapping strategy has an advantage that both pivot searching and subrow interchange can be done locally without any communication. Another advantage is that parallelism modeled by the above dependence structure can be effectively exploited using graph scheduling techniques. data mapping. In the literature 2-D mapping has been shown more scalable than 1-D for sparse Cholesky [30, 31]. However there are several difficulties to apply the 2-D block-oriented mapping to the case of sparse LU factorization even the static structure is predicted. Firstly, pivoting operations and row interchanges require frequent and well-synchronized interprocessor communication when submatrices in the same column block are assigned to different processors. Effective exploitation of limited irregular parallelism in the 2-D case requires a highly efficient asynchronous execution mechanism and a delicate message buffer management. Secondly, it is difficult to utilize and schedule all possible irregular parallelism from sparse LU. Lastly, how to manage a low space complexity is another issue since exploiting irregular parallelism to a maximum degree may need more buffer space. Our 2-D algorithm uses a simple standard mapping function. In this scheme, p available processors are viewed as a two dimensional grid: c . A nonzero submatrix block A ij (could be an L block or a U block) is assigned to processor P i mod pr ; j mod pc . The 2-D data mapping is considered more scalable than 1-D data mapping because it enables parallelization of a single F actor(k) or Update(k; j) task on p r processors. We will discuss how 2-D parallelism is exploited using asynchronous schedule execution. 5 Parallelism exploitation 5.1 Scheduling and run-time support for 1-D methods We discuss how 1-D sparse LU tasks are scheduled and executed so that parallel time can be minimized. George and Ng [21] used a dynamic load balancing algorithm on a shared memory machine. For distributed memory machines, dynamic and adaptive load balancing works well for problems with very coarse grained computations, but it is still an open problem to balance the benefits of dynamic scheduling with the run-time control overhead since task and data migration cost is too expensive for sparse problems with mixed granularities. We use task dependence graphs to guide scheduling and have investigated two types of scheduling schemes. ffl Compute-ahead scheduling (CA). This is to use block-cyclic mapping of tasks with a compute-ahead execution strategy, which is demonstrated in Figure 10. This idea has been used to speed up parallel dense factorizations [23]. It executes the numerical factorization layer by layer based on the current submatrix index. The parallelism is exploited for concurrent updating. In order to overlap computation with communication, the F actor(k executed as soon as F actor(k) and Update(k; k so that the pivoting sequence and column block k for the next layer can be communicated as early as possible. ffl Graph scheduling. We order task execution within each processor using the graph scheduling algorithms in [36]. The basic optimizations are balancing processor loads and overlapping computation with communication to hide communication latency. These are done by utilizing global dependence structures and critical path information. (01) if column block 1 is local (02) Perform task F actor(1); Broadcast column block 1 and the pivoting sequence; local Receive column block k and the pivoting choices; rows according to the pivoting sequence; Perform task F actor(k Broadcast column block k and the pivoting sequence; local if column block k has not been received Receive column block k and the pivoting choices; rows according to the pivoting sequence; Perform task Update(k; j); Figure 10: The 1-D code using compute-ahead schedule. Graph scheduling has been shown effective in exploiting irregular parallelism for other applications (e.g. [15, 16]). Graph scheduling should outperform the CA scheduling for sparse LU because it does not have a constraint in ordering F actor() tasks. We demonstrate this point using the LU task graph in Figure 9. For this example, the Gantt charts of the CA schedule and the schedule derived by our graph scheduling algorithm are listed in Figure 11. It is assumed that each task has a computation weight 2 and each edge has communication weight 1. It is easy to see that our scheduling approach produces a better result than the CA schedule. If we look at the CA schedule carefully, we can see that the reason is that CA can look ahead only one step so that the execution of task F actor(3) is placed after Update(1; 5). On the other hand, the graph scheduling algorithm detects that F actor(3) can be executed before Update(1; 5) which leads to better overlap of communication with computation. P1(a) Figure 11: (a) A schedule derived by our graph scheduling algorithm. (b) A compute-ahead schedule. For convenience F () is used to denote F actor(), U() is used to denote Update(). However the implementation of the CA algorithm is much easier since the efficient execution of a sparse task graph schedule requires a sophisticated run-time system to support asynchronous communication protocols. We have used the RAPID run-time system [16] for the parallelization of sparse LU using graph scheduling. The key optimization is to use Remote Memory Access(RMA) to communicate a data object between two processors. It does not incur any copying/buffering during a data transfer since low communication overhead is critical for sparse code with mixed granularities. RMA is available in modern multi-processor architectures such as Cray-T3D [34], T3E [32] and Meiko CS-2 [15]. Since the RMA directly writes data to a remote address, it is possible that the content at the remote address is still being used by other tasks and then the execution at the remote processor could be incorrect. Thus for a general computation, a permission to write the remote address needs to be obtained before issuing a remote write. However in the RAPID system, this hand-shaking process is avoided by a carefully designed task communication protocol [16]. This property greatly reduces task synchronization cost. As shown in [17], the RAPID sparse code can deliver more than 70% of the speedup predicted by the scheduler on Cray-T3D. In addition, using RAPID system greatly reduces the amount of implementation work to parallelize sparse LU. (01) Let (my rno; my cno) be the 2-D coordinates of this processor; Perform ScaleSwap(k); Perform Update Perform Update 2D(k; j); Figure 12: The SPMD code of 2-D asynchronous code. 5.2 Asynchronous execution for the 2-D code As we discussed previously, 1-D data mapping can not expose parallelism to a maximum extent. Another issue is that a time-efficient schedule may not be space-efficient. Specifically, to support concurrency among multiple updating stages in both RAPID and CA code, multiple buffers are needed to keep pivoting column blocks of different stages on each processor. Therefore for a given problem, the per processor space complexity of the 1-D codes could be as high as O(S 1 ), where S 1 is the space complexity for a sequential algorithm. For sparse LU, each processor in the worst case may need a space for holding the entire matrix. The RAPID system [16] also needs extra memory space to hold dependence structures. Based on the above observation, our goal for the 2-D code is to reduce memory space requirement while exploiting a reasonable amount of parallelism so that it can solve large problem instances in an efficient way. In this section, we present an asynchronous 2-D algorithm which can substantially overlap multi-stages of updating but its memory requirement is much smaller than that of 1-D methods Figures 12 shows the main control of the algorithm in an SPMD coding style. Figure 13 shows the SPMD code for F actor(k) which is executed by processors of column k mod p c . Recall that the algorithm uses 2-D block-cyclic data mapping and the coordinates for the processor that owns are (i mod Also we divide the function of Update() (in Figure 8) into two parts: ScaleSwap() which does scaling and delayed row interchange for submatrix A k:N; k+1:N as shown in Figure 14; Update 2D() which does submatrix updating as shown in Figure 15. In all figures, the statements which involve interprocessor communication are marked with . It can be seen that the computation flow of this 2-D code is still controlled by the pivoting tasks F actor(k). The order of execution for F actor(k), is sequential, but Update 2D() tasks, where most of the computation comes from, can execute in parallel among all processors. The asynchronous parallelism comes from two levels. First a single stage of tasks Update 2D(k; Find out local maximum element of column m; (05)* Send the subrow within column block k containing the local maximum to processor P k mod pr ; k mod pc ; (06) if this processor owns L kk (07)* Collect all local maxima and find the pivot row t; (08)* Broadcast the subrow t within column block k along this processor column and interchange subrow t and subrow m if necessary. Scale local entries of column m; Update local subcolumns from column (12)* Multicast the pivot sequence along this processor row; (13)* if this processor owns L kk then Multicast L kk along this processor row; (14)* Multicast the part of nonzero blocks in L k+1:N; k owned by this processor along this processor row; Figure 13: Parallel execution of F actor(k) for the 2-D asynchronous code. can be executed concurrently on all processors. In addition, different stages of Update 2D() tasks from Update 2D(k; can also be overlapped. The idea of compute-ahead scheduling is also incorporated, i.e., F actor(k + 1) is executed as soon as Update finishes. Some detailed explanation for pivoting, scaling and swapping is given below. In line (5) of Figure 13, the whole subrow is communicated when each processor reports its local maximum to the processor that owns the L kk block. Let m be the current global column number on which the pivoting is conducted, then without further synchronization, processor locally swap subrow m with subrow t which contains the selected pivoting element. This shortens the waiting time to conduct further updating with a little more communication volume. However in line (08), processor P k mod pr ; k mod pc must send the original subrow m to the owner of subrow t for swapping, and the selected subrow t to other processors as well for updat- ing. In F actor() tasks, synchronizations take place at lines (05), (07) and (08) when each processor reports its local maximum to P k mod pr ; k mod pc , and P k mod pr ; k mod pc broadcasts the subrow containing global maximum along the processor column. For task ScaleSwap(), the main role is to scale U k; k+1:N and perform delayed row interchanges for remaining submatrices A k+1:N; k+1:N . We examine the degree of parallelism exploited in this algorithm by determining number of updating stages that can be overlapped. Using this information we can also determine the extra buffer space needed per processor to execute this algorithm correctly. We define the stage overlapping degree then receive the pivot sequence from P my rno; k mod pc ; (04) if This processor own a part of row m or the pivot row t for column m (05)* Interchange nonzero parts of row t and row m owned by this processor; (08)* if my cno 6= k mod p c then receive L kk from P my rno; k mod pc ; Scale nonzero blocks in U k; k:N owned by this processor; (10)* Multicast the scaling results along this processor column; (11)* if my cno 6= k mod p c then receive L k:N; k from P my rno; k mod pc ; (12)* if my rno 6= k mod p r then receive U k; k:N from P k mod pr ; my cno ; Figure 14: Task ScaleSwap(k) for the 2-D asynchronous code. (1) Update 2D(k; using L ik and U kj ; Figure 15: Update 2D(k; j) for the 2-D asynchronous code. for updating tasks as There exist tasks Update 2D(k; ) and Update 2D(k executed concurrently.g Here Update 2D(k; ) denotes a set of Update 2D(k; tasks where Theorem 2 For the asynchronous 2-D algorithm on p processors where p ? 1 and the reachable upper bound of overlapping degree is p c among all processors; and the reachable upper bound of overlapping degree within a processor column is min(p r \Gamma Proof: We will use the following facts in proving the theorem: ffl Fact 1. F actor(k) is executed at processors with column number k mod p c . Processors on this column are synchronized. When a processor completes F actor(k), this processor can still do Update shown in Figure 13, but all Update tasks belonging to this processor where t ? 1 must have been completed on this processor. ffl Fact 2. ScaleSwap(k) is executed at processors with row number k mod p r . When a processor completes ScaleSwap(k), all Update tasks belonging to this processor where t ? 0 must have been completed on this processor. Part 1. First we show that Update 2D() tasks can be overlapped to a degree of p c among all processors. When trivial based on Fact 1. When p c ? 1, we can imagine a scenario in which all processors in column 0 have just finished task F actor(k), and some of them are still working on Update processors in column 1 could go ahead and execute Update 2D(k; ) tasks. After processors in column 1 finish Update 2D(k; k+1) task, they will execute F actor(k+1). Then after finishing Update 2D(k; ) tasks, processors in column 2 could execute Update 2D(k Finally, processors in column p c \Gamma 1 could execute F actor(k moment, processors in column 0 may be still working on Update Thus the overlapping degree is p c . Now we will show by contradiction that the maximum overlapping degree is p c . Assume that at some moment, there exist two updating stages being executed concurrently: Update 2D(k; ) and Update must have been completed. Without loss of generality, assuming that processors in column 0 execute F actor(k 0 ), then according to Fact 1 all Update should be completed before this moment. Since block cyclic mapping is used, it is easy to see each processor column has performed one of the F actor(j) tasks should be completed on all processors. Then for any concurrent stage Update 2D(k; ), k must satisfy which is a contradiction. Part 2. First, we show that overlapping degree min(p r \Gamma can be achieved within a processor column. For the convenience of illustration, we consider a scenario in which all delayed row interchanges in ScaleSwap() take place locally without any communication within a processor column. Therefore there is no interprocessor synchronization going on within a processor column except in F actor() tasks. Assuming , we can imagine at some moment, processors in column 0 have completed F actor(s), and P 0;0 has just finished ScaleSwap(s), and starts executing Update 2D(s; ), where s mod processors in column 1 will execute Update 1), after which P 1;0 can start ScaleSwap(s then Update 2D(s Following this reasoning, after Update 2D(s been finished on processors of column could complete previous Update 2D() tasks and ScaleSwap(s+p r \Gamma 1), and start Update 2D(s+p r \Gamma 1; ). Now P 0;0 may be still working on Update 2D(s; ). Thus the overlapping degree is obviously the above reasoning will stop when processors of column and F actor(s 1). In that case when P pc \Gamma1;0 is to start Update 2D(s+ pr \Gamma1;0 could be still working on Update 2D(s \Gamma because of the compute ahead scheduling. Hence the overlapping degree is p c . Now we need to show that the upper bound of overlapping degree within a processor column is We have already shown in the proof of Part 1 that the overall overlapping degree is less than p c , so is the overlapping degree within a processor column. To prove it is also less than 1, we can use the similar proof as that for part 1, except using ScaleSwap(k) to replace F actor(k), and using Fact 2 instead of Fact 1. Knowing degree of overlapping is important in determining the amount of memory space needed to accommodate those communication buffers on each processor for supporting asynchronous execu- tion. Buffer space is additional to data space needed to distribute the original matrix. There are four types of communication that needs buffering: 1. Pivoting along a processor column (lines (05), (07), and (08) in Figure 13), which includes communicating pivot positions and multicasting pivot rows. We call the buffer for this purpose Pbuffer. 2. Multicasting along a processor row (line (12), (13) and (14) in Figure 13). The communicated data includes L kk , local nonzero blocks in L k+1:N; k , and pivoting sequences. We call the buffer for this purpose Cbuffer. 3. Row interchange within a processor column (line (05) in Figure 14). We call this buffer Ibuffer. 4. Multicasting along a processor column (line (10) in Figure 14). The data includes local nonzero blocks of a row panel. We call the buffer Rbuffer. Here we assume that p r - because based on our experimental results, setting p r - always leads to better performance. Thus the overlapping degree of Update 2D() tasks within a processor row is at most p c , and the overlapping degree within a processor column is at most p r \Gamma 1. Then we need p c separate Cbuffer's for overlapping among different columns and Rbuffer's for overlapping among different rows. We estimate the size of each Cbuffer and Rbuffer as follows. Assuming that the sparsity ratio of a given matrix is s after fill-in and the maximum block size is BSIZE, each Cbuffer is of size: maxfspace for local nonzero blocks of L k:N;k Similarly each Rbuffer is of size: local nonzero blocks of U We ignore the buffer size for Pbuffer and Ibuffer because they are very small (the size of Pbuffer is only about BSIZE \Delta BSIZE and the size of Ibuffer is about s \Delta n=p c ). Thus the total buffer space needed for the asynchronous execution is: C Notice that the sequential space complexity In practice, we set p c =p 2. Therefore the buffer space complexity for each processor is 2:5 which is very small for a large matrix. For all the benchmark matrices we have tested, the buffer space is less than 100 K words. Given a sparse matrix, if the matrix data is evenly distributed onto p processors, the total memory requirement per processor is S 1 =p +O(1) considering n AE p and n AE BSIZE. This leads us to conclude that the 2-D asynchronous algorithm is very space scalable. 6 Experimental studies Our experiments were originally conducted on a Cray-T3D distributed memory machine at San Supercomputing Center. Each node of the T3D includes a DEC Alpha EV4(21064) processor with 64 Mbytes of memory. The size of the internal cache is 8 Kbytes per processor. The BLAS-3 matrix-matrix multiplication routine DGEMM can achieve 103 MFLOPS, and the BLAS-2 matrix-vector multiplication routine DGEMV can reach 85 MFLOPS. These numbers are obtained assuming all the data is in cache and using cache read-ahead optimization on T3D, and the matrix block size is chosen as 25. The communication network of the T3D is a 3-D torus. Cray provides a shared memory access library called shmem which can achieve 126 Mbytes/s bandwidth and 2:7-s communication overhead using shmem put() primitive [34]. We have used shmem put() for the communications in all the implementations. We have also conducted experiments on a newly acquired Cray-T3E at San Diego Supercomputing Center. Each T3E node has a clock rate of 300 MHZ, an 8Kbytes internal cache, 96Kbytes second level cache, and 128 Mbytes main memory. The peak bandwidth between nodes is reported as 500 Mbytes/s and the peak round trip communication latency is about 0.5 to 2 -s [33]. We have observed that when block size is 25, DGEMM achieves 388 MFLOPS while DGEMV reaches 255 MFLOPS. We have used block size 25 in our experiments since if the block size is too large, the available parallelism will be reduced. In this section we mainly report results on T3E. In some occasions that the absolute performance is concerned, we also list the results on T3D to see how our approach scales when the underline architecture is upgraded. All the results are obtained on T3E unless explicitly stated. In calculating the MFLOPS achieved by our parallel algorithms, we do not include extra floating point operations introduced by the overestimation. We use the following formula: Achieved Operation count obtained from SuperLU Parallel time of our algorithm on T3D or T3E : The operation count for a matrix is reported by running SuperLU code on a SUN workstation with large memory since SuperLU code cannot run for some large matrices on a single T3D or T3E node due to memory constraint. We also compare the S sequential code with SuperLU to make sure that the code using static symbolic factorization is not too slow and will not prevent the parallel version from delivering high megaflops. 6.1 Impact of static symbolic factorization on sequential performance We study if the introduction of extra nonzero elements by the static factorization substantially affects the time complexity of numerical factorization. We compare the performance of the S sequential code with SuperLU code performance in Table 2 1 for those matrices from Table 1 that 1 The times for S in this table do not include symbolic preprocessing cost while the times for SuperLU include symbolic factorization because SuperLU does it on the fly. Our implementation for static symbolic preprocessing is can be executed on a single T3D or T3E node. We also introduce two other matrices to show how well the method works for larger matrices and denser matrices. One of the two matrices is b33 5600 which is truncated from BCSSTK33 because of the current sequential implementation is not able to handle the entire matrix due to memory constraint, and the other one is dense1000. Matrix S Approach SuperLU Exec. Time Ratio Seconds Mflops Seconds Mflops S /SuperLU sherman5 2.87 0.94 8.81 26.9 2.39 0.78 10.57 32.4 1.21 1.22 sherman3 6.06 2.03 10.18 30.4 4.27 1.68 14.46 36.7 1.56 1.21 jpwh991 2.11 0.69 8.24 25.2 1.62 0.56 10.66 31.0 1.34 1.23 goodwin 43.72 17.0 15.3 39.4 - dense1000 10.48 4.04 63.6 165.0 19.6 8.39 34.0 79.4 0.53 0.48 Table 2: Sequential performance: S versus SuperLU. A "-" implies the data is not available due to insufficient memory. Though the static symbolic factorization introduces a lot of extra computation as shown in Table 1, the performance of S after 2-D L/U partitioning is consistently competitive to that of highly optimized SuperLU. The absolute single node performance that has been achieved by the S approach on both T3D and T3E is consistently in the range of 5 \Gamma 10% of the highest DGEMM performance for those matrices of small or medium sizes. Considering the fact that sparse codes usually suffer poor cache reuse, this performance is reasonable. In addition, the amount of computation for the testing matrices in Table 2 is small, ranging from to 107 million double precision floating operations. Since the characteristic of the S approach is to explore more dense structures and utilize BLAS-3 kernels, better performance is expected on larger or denser matrices. This is verified on a matrix b33 5600. For even larger matrices such as vavasis3, we cannot run S on one node, but as shown later, the 2-D code can achieve 32.8 MFLOPS per node on 16 T3D processors. Notice that the megaflops performance per node for sparse Choleksy reported in [24] on 16 T3D nodes is around 40 MFLOPS, which is also a good indication that S single-node performance is satisfactory. We present a quantitative analysis to explain why S can be competitive to SuperLU. Assume the speed of BLAS-2 kernel is ! 2 second=f lop and the speed of BLAS-3 kernel is ! 3 second=f lop. The total amount of numerical updates is C f lops for SuperLU and C 0 f lops for the S . Apparently simplicity, we ignore the computation from the scaling part within each column because it contributes very little to the total execution time. Hence we have: very efficient. For example, the preprocessing time is only about 2.76 seconds on a single node of T3E for the largest matrix we tested (vavasis3). where T symbolic is the time spent on dynamic symbolic factorization in the SuperLU approach, ae is the percentage of the numerical updates that are performed by DGEMM in S . Let j be the ratio of symbolic factorization time to numerical factorization time in SuperLU, then we simplify Equation (1) to the following: We estimate that j - 0:82 for the tested matrices based on the results in [7]. In [17], we have also measured ae as approximately ae - 0:67. The ratios of the number of floating point operations performed in S and SuperLU for the tested matrices are available in Table 1. In average, the value of C 0 is 3:98. We plug in these typical parameters in Equation 2 and 3, and we have: For lop. Then we can get T S - 1:93. For T3E, lop. And we get T S - 1:68. These estimations are close to the ratios obtained in Table 2. The discrepancy is caused by the fact that the submatrix sizes of supernodes are non-uniform, which leads to different caching performance. If submatrices are of uniform sizes, we expect our prediction is more accurate. For instance, in the dense case, C 0 is exactly 1. The ratio T S is calculated as 0:48 for T3D and 0:42 for T3E, which are almost the same as the ratios listed in Table 2. The above analysis shows that using BLAS-3 as much as possible makes S competitive to SuperLU. Suppose in a machine that DGEMM outperforms DGEMV substantially and the ratio of the computation that is performed by DGEMM is high enough, S could be faster than SuperLU for some matrices. The last two entries in Table 2 have already shown this. 6.2 Parallel performance of 1-D codes In this subsection, we report a set of experiments conducted to examine the overall parallel performance of 1-D codes, the effectiveness of scheduling and supernode amalgamation. Performance: We list the MFLOPS numbers of the 1-D RAPID code obtained on various number of processors for several testing matrices in Table 3 entry implies the data is not available due memory constraint, same below). We know that the megaflops of DGEMM on T3E is about 3.7 times as large as that on T3D, and the RAPID code after using a upgraded machine is speeded up about 3 times in average, which is satisfactory. For the same machine, the performance of the RAPID code increases when the number of processors increases and speedups compared to the pure S sequential code (if applicable) can reach up to 17.7 on 64 T3D nodes and 24.1 on 64 T3E nodes. From 32 to 64 nodes, the performance gain is small except for matrices goodwin, e40r0100 and b33 5600, which are much larger problems than the rest. The reason is that those small tested matrices do not have enough amount of computation and parallelism to saturate a large number of processors when the elimination process proceeds toward the end. It is our belief that better and more scalable performance can be obtained on larger matrices. But currently the available memory on each node of T3D or T3E limits the problem size that can be solved with the current version of the RAPID code. Matrix P=2 P=4 P=8 P=16 P=32 P=64 sherman5 14.7 44.4 25.8 79.0 40.8 133.1 53.8 168.6 64.9 210.7 68.4 229.9 sherman3 16.4 51.4 30.0 90.7 45.7 143.5 61.1 192.8 64.3 199.0 66.3 212.7 jpwh991 13.3 41.4 23.2 75.6 40.5 124.2 51.2 173.9 58.0 193.2 60.0 217.3 orsreg1 17.4 53.4 30.6 90.6 51.2 160.3 68.7 215.6 75.3 223.3 75.3 231.6 goodwin 29.6 73.6 54.0 135.7 87.9 238.0 136.4 373.7 182.0 522.6 218.1 655.8 Table 3: Absolute performance (MFLOPS) of the 1-D RAPID code. Effectiveness of Graph Scheduling: We compare the performance of 1-D CA code with 1-D RAPID code in Figure 16. The Y axis is stands for parallel time. For 2 and 4 processors, in certain cases, the compute-ahead code is slightly faster than the RAPID code. But for the number of processors more than 4, the RAPID code runs faster. The more processors involved, the bigger the performance gap tends to be. The reason is that for a small number of processors, there are sufficient tasks making all processors busy and the compute- ahead schedule performs well while the RAPID code suffers a certain degree of system overhead. For a larger number of processors, schedule optimization becomes important since there is limited parallelism to exploit. Effectiveness of supernode amalgamation: We have examined how effective our supernode amalgamation strategy is using the 1-D RAPID code. Let PT a and PT be the parallel time with and without supernode amalgamation respectively. The parallel time improvement ratios on T3E for several testing matrices are listed in Table 4 and similar results on T3D are in [17]. Apparently the supernode amalgamation has brought significant improvement due to the increase of supernode size which implies an increase of the task granularities. This is important to obtaining good parallel performance [22]. Comparison of the RAPID Code with the 1-D CA Code #proc *: sherman5 +: sherman3 -0.10.10.30.50.7Comparison of the RAPID Code with the 1-D CA Code #proc *: jpwh991 x: goodwin Figure Impact of different scheduling strategies on 1-D code approach. Matrix P=1 P=2 P=4 P=8 P=16 P=32 sherman5 47% 47% 46% 50% 40% 43% sherman3 20% 25% 23% 28% 22% 14% jpwh991 48% 48% 48% 50% 47% 40% Table 4: Parallel time improvement obtained by supernode amalgamation. 6.3 2-D code performance As mentioned before, our 2-D code exploits more parallelism but maintains a lower space complexity, and has much more potential to solve large problems. We show the absolute performance obtained for some large matrices on T3D in Table 5. Since some matrices cannot fit for a small number of processors, we only list results on 16 or more processors. The maximum absolute performance achieved on 64 nodes of T3D is 1.48 GFLOPS, which is translated to 23.1 MFLOPS per node. For nodes, the per-node performance is 32.8 MFLOPS. Table 6 shows the performance numbers on T3E for the 2-D code. We have achieved up to 6.878 GFLOPS on 128 nodes. For 64 nodes, megaflops on T3E are from 3.1 to 3.4 times as large as that on T3D. Again considering that DGEMM megaflops on T3E is about 3.7 times as large as that on T3D, our code performance after using a upgraded machine is good. Notice that 1-D codes cannot solve the last six matrices of Table 6. For those matrices solvable using both 1-D RAPID and 2-D codes, we compare the average parallel time differences by computing the and the result is in Figure 17. The 1-D RAPID code achieves Matrix Time(Sec) Mflops Time(Sec) Mflops Time(Sec) Mflops goodwin 6.0 110.7 4.6 145.2 3.6 184.8 ex11 87.9 305.0 53.4 501.8 33.4 802.6 raefsky4 129.8 242.9 76.0 413.8 43.2 719.2 Table 5: Performance results of the 2-D code for large matrices on T3D. Matrix P=8 P=16 P=32 P=64 P=128 Time Mflops Time Mflops Time Mflops Time Mflops Time Mflops goodwin 3.1 215.2 1.9 344.6 1.3 496.3 1.1 599.2 0.9 715.2 ex11 50.7 528.8 28.3 946.2 16.2 1654.2 9.9 2703.1 6.4 4182.2 raefsky4 79.4 391.2 43.2 718.9 24.1 1290.7 13.9 2233.3 8.6 3592.9 inaccura 16.8 244.6 9.9 415.2 6.3 655.8 3.9 1048.0 3.0 1391.4 af23560 22.3 285.4 12.9 492.9 8.12 784.3 5.7 1123.2 4.2 1512.7 Table Performance results of 2-D asynchronous algorithm on T3E. All times are in seconds. better performance because it uses sophisticated graph scheduling technique to guide the mapping of column blocks and ordering of tasks, which results in better overlapping of communication with computation. The performance difference is larger for the matrices listed in the left of Figure 17 compared to the right of Figure 17. We partially explain the reason by analyzing load balance factors of the 1-D RAPID code and the 2-D code in Figure 18. The load balance factor is defined as work total =(P \Delta work max ) [31]. Here we only count the work from the updating part because it is the major part of the computation. The 2-D code has better load balance, which can make up for the impact of lacking of efficient task scheduling. This is verified by Figure 17 and Figure 18. One can see that when the load balance factor of the 2-D code is close to that of the RAPID code (e.g., lnsp3937), the performance of the RAPID code is much better than the 2-D code; when the load balance factor of the 2-D code is significantly better than that of the RAPID code (e.g., jpwh991 and orserg1), the performance differences are smaller. Synchronous versus asynchronous 2-D code. Using a global barrier in the 2-D code at each elimination step can simplify the implementation, but it cannot overlap computations among different updating stages. We have compared parallel time reductions between the asynchronous code and the synchronous code for some testing matrices in Table 7. It shows that asynchronous design improves performance significantly, especially on large number of processors on T3E. It demonstrates the importance of exploiting parallelism using asynchronous execution. The experiment 0.20.4Comparison of the RAPID Code with the 2-D Code #proc *: sherman5 +: sherman3 Comparison of the RAPID Code with the 2-D Code #proc *: jpwh991 x: goodwin Figure 17: Performance improvement of 1-D RAPID over 2-D code: balance comparison of RAPID v.s. 2-D #proc load balance factor x sherman3 balance comparison of RAPID v.s. 2-D #proc load balance factor x jpwh991 Figure 18: Comparison of load balance factors of 1-D RAPID code and 2-D code. data on T3D is in [14]. 7 Concluding remarks In this paper we present an approach for parallelizing sparse LU factorization with partial pivoting on distributed memory machines. The major contribution of this paper is that we integrate several techniques together such as static symbolic factorization, scheduling for asynchronous parallelism, 2-D L/U supernode partitioning techniques to effectively identify dense structures, and maximize the use of BLAS-3 subroutines in the algorithm design. Using these ideas, we are able to exploit more data regularity for this open irregular problem and achieve up to 6.878 GFLOPS on 128 T3E nodes. This is the highest performance known for this challenging problem and the previous record was 2.583 GFLOPS on shared memory machines [8]. Matrix P=2 P=4 P=8 P=16 P=32 P=64 sherman5 7.7% 6.4% 19.4% 28.1% 25.9% 24.1% sherman3 10.2% 12.4% 20.3% 22.7% 26.0% 25.0% jpwh991 8.7% 10.0% 23.8% 33.3% 35.7% 28.6% orsreg1 6.1% 7.7% 17.5% 28.0% 20.5% 28.2% goodwin 5.4% 14.1% 14.2% 24.6% 26.0% 30.2% Table 7: Performance improvement of 2-D asynchronous code over 2-D synchronous code. The comparison results show that the 2-D code has a better scalability than 1-D codes because 2-D mapping exposes more parallelism with a carefully designed buffering scheme. But the 1-D RAPID code still outperforms the 2-D code if there is sufficient memory since the scheduling and execution techniques for the 2-D code are simple, and are not competitive to graph scheduling. Recently we have conducted research on developing space efficient scheduling algorithms while retaining good time efficiency [18]. It is still an open problem to develop advanced scheduling techniques that better exploit parallelism for 2-D sparse LU factorization with partial pivoting. There are other issues which are related to this work and need to be further studied, for example, alternative for parallel sparse LU based on Schur complements [13] and static estimation and parallelism exploitation for sparse QR [29, 35]. It should be noted that the static symbolic factorization could fail to be practical if the input matrix has a nearly dense row because it will lead to an almost complete fill-in of the whole matrix. It might be possible to use different matrix reordering to avoid that. Fortunately, this is not the case in most of matrices we have tested. Therefore our approach is applicable to a wide range of problems using a simple ordering strategy. It is interesting in the future to study ordering strategies that minimize overestimation ratios so that S can consistently deliver good performance for various classes of sparse matrices. Acknowledgment This work is supported by NSF RIA CCR-9409695, NSF CDA-9529418, the UC MICRO grant with a matching from SUN, NSF CAREER CCR-9702640, and ARPA DABT-63-93-C-0064 through the Rutgers HPCD project. We would like to thank Kai Shen for the efficient implementation of the static symbolic factorization algorithm, Xiaoye Li and Jim Demmel for helpful discussions and providing us their testing matrices and SuperLU code, Cleve Ashcraft, Tim Davis, Apostolos Gerasoulis, Esmond Ng, Ed Rothberg, Rob Schreiber, Horst Simon, Chunguang Sun, Kathy Yelick and anonymous referees for their valuable comments. --R The Influence of Relaxed Supernode Partitions on the Multifrontal Method. Progress in Sparse Matrix Methods for Large Sparse Linear Systems on Vector Supercomputers. User's guide for the Unsymmetric-pattern Multifrontal Package (UMFPACK) Personal Communication An Unsymmetric-pattern Multifrontal Method for Sparse LU factor- ization Numerical Linear Algebra on Parallel Processors. A Supernodal Approach to Sparse Partial Pivoting. An Asynchronous Parallel Supernodal Algorithm for Sparse Gaussian Elimination. An Extended Set of Basic Linear Algebra Subroutines. The Multifrontal Solution of Indefinite Sparse Symmetric Systems of Equations. On Algorithms for Obtaining a Maximum Transversal. Personal Communication Structural Representations of Schur Complements in Sparse Matrices A Comparison of 1-D and 2-D Data Mapping for Sparse LU Factorization with Partial Pivoting Efficient Run-time Support for Irregular Task Computations with Mixed Granularities Sparse LU Factorization with Partial Pivoting on Distributed Memory Machines. Space and Time Efficient Execution of Parallel Irregular Computations. The Parallel Solution of Nonsymmetric Sparse Linear Systems Using H Symbolic Factorization for Sparse Gaussian Elimination with Partial Pivoting. Parallel Sparse Gaussian Elimination with Partial Pivoting. On the Granularity and Clustering of Directed Acyclic Task Graphs Scientific Computing: An Introduction with Parallel Computing Compilers Highly Scalable Parallel Algorithms for Sparse Matrix Factorization. A Parallel Unsymmetric-pattern Multifrontal Method Parallel Algorithms for Sparse Linear Systems Parallel sparse gaussian elimination with partial pivoting and 2-d data mapping Computational Models and Task Scheduling for Parallel Sparse Cholesky Factorization. Distributed Sparse Gaussian Elimination and Orthogonal Factorization. Exploiting the Memory Hierarchy in Sequential and Parallel Sparse Cholesky Factorization. Improved Load Distribution in Parallel Sparse Cholesky Fac- torization Synchronization and Communication in the T3E Multiprocess. The Cray T3E Network: Adaptive Routing in a High Performance 3D Torus. Decoupling Synchronization and Data Transfer in Message Passing Systems of Parallel Computers. Parallel Sparse Orthogonal Factorization on Distributed-memory Multiprocessors PYRROS: Static Task Scheduling and Code Generation for Message-Passing Multiprocessors --TR --CTR Kai Shen , Xiangmin Jiao , Tao Yang, Elimination forest guided 2D sparse LU factorization, Proceedings of the tenth annual ACM symposium on Parallel algorithms and architectures, p.5-15, June 28-July 02, 1998, Puerto Vallarta, Mexico Xiaoye S. Li , James W. Demmel, Making sparse Gaussian elimination scalable by static pivoting, Proceedings of the 1998 ACM/IEEE conference on Supercomputing (CDROM), p.1-17, November 07-13, 1998, San Jose, CA Patrick R. Amestoy , Iain S. Duff , Jean-Yves L'excellent , Xiaoye S. Li, Analysis and comparison of two general sparse solvers for distributed memory computers, ACM Transactions on Mathematical Software (TOMS), v.27 n.4, p.388-421, December 2001 Xiaoye S. Li , James W. Demmel, SuperLU_DIST: A scalable distributed-memory sparse direct solver for unsymmetric linear systems, ACM Transactions on Mathematical Software (TOMS), v.29 n.2, p.110-140, June

dense structures;Sparse LU factorization;gaussian elimination with partial pivoting;irregular parallelism;asynchronous computation scheduling

275845

Strong Interaction Fairness Via Randomization.

AbstractWe present MULTI, a symmetric, distributed, randomized algorithm that, with probability one, schedules multiparty interactions in a strongly fair manner. To our knowledge, MULTI is the first algorithm for strong interaction fairness to appear in the literature. Moreover, the expected time taken by MULTI to establish an interaction is a constant not depending on the total number of processes in the system. In this sense, MULTI guarantees real-time response. MULTI makes no assumptions (other than boundedness) about the time it takes processes to communicate. It, thus, offers an appealing tonic to the impossibility results of Tsay and Bagrodia, and Joung concerning strong interaction fairness in an environment, shared-memory, or message-passing, in which processes are deterministic and the communication time is nonnegligible. Because strong interaction fairness is as strong a fairness condition that one might actually want to impose in practice, our results indicate that randomization may also prove fruitful for other notions of fairness lacking deterministic realizations and requiring real-time response.

Introduction A multiparty interaction is a set of I/O actions executed jointly by a number of processes, each of which must be ready to execute its own action for any of the actions in the set to occur. An attempt to participate in an interaction delays a process until all other participants are available. After the actions are executed, the participating processes continue their local computation. Although a relatively new concept, the multiparty interaction has found its way into various distributed programming languages and algebraic models of concurrency. See [11] for a taxonomy of programming languages offering linguistic support for multiparty interaction. Although multiparty interactions are executed synchronously, the underlying model of communication is usually asynchronous and bipartied. The multiparty interaction scheduling problem then is concerned with synchronizing asynchronous processes to satisfy the following requirements: (1) an interaction can be established only if it is enabled (i.e., all of its participants are ready), and (2) a process can participate in only one interaction at a time. Moreover, some notion of fairness is typically associated with the implementation to prevent "unfair" computations that favor a particular process or interaction. Several important fairness notions have been proposed in the literature [1, 2, 3], including: weak interaction fairness, where if an interaction is continually enabled, then some of its participants will eventually engage in an interaction; and strong interaction fairness, where an interaction that is infinitely often enabled will be established infinitely often. A distinguishing characteristic of interaction fairness is that it is much weaker than most known fairness notions, while strong interaction fairness is much stronger. In general, stronger fairness notions induce more liveness properties, but are also more difficult to implement. Therefore, it is not surprising to see that only weak interaction fairness has been widely implemented (e.g., [18, 15, 4, 14, 12, 20, 10]). It is also not surprising to see that all of these algorithms are asymmetric and deterministic, as weak interaction fairness (and thus strong interaction fairness) has been proven impossible by any symmetric, deterministic, and distributed algorithm [8, 13]. Given that a process decides autonomously when it will attempt an interaction, and at a time that cannot be predicted in advance, strong interaction fairness is still not possible even if the symmetry requirement is dropped [19, 9]. Note that these impossibility results do not depend on the type of communication primitives (e.g., message-passing or shared-memory) provided by the underlying execution model. They hold as long as one process's readiness for multiparty interaction can be known by another only through communications, and the time it takes two processes to communicate is nonnegligible (but can be finitely bounded). In the case of CSP communication guard scheduling, a special case of the multiparty interaction scheduling problem where each interaction involves exactly two processes, randomization has proven to be an effective technique for coping with the aforementioned impossibility phenomena. For ex- ample, the randomized algorithm of Reif and Spirakis [17] is symmetric, weak interaction fair with probability 1, and guarantees real time response: if two processes are continuously willing to interact with each other within a time interval \Delta, then they establish an interaction within \Delta time with high likelihood, and the expected time for establishment of interaction is constant. The randomized algorithm of Francez and Rodeh [8] is simpler: a process p i expresses its willingness to establish an interaction with a process p j by setting a Boolean variable shared by the two may then need to wait a certain amount of time ffi before reaccessing the variable to determine if p j is likewise interested in the interaction. The authors show that, under the proviso that the time to access a shared variable is negligible compared to ffi, the algorithm is weak interaction fair with probability 1. Note, however, that this assumption, combined with the fact that no lower bound on ffi is provided, may significantly limit the algorithm's practicality. Furthermore, no strong interaction fairness is claimed by either algorithm. In this paper, we present Multi, an extension of Francez and Rodeh's randomized algorithm to the multiparty case. We prove that Multi is weak interaction fair with probability 1. We also show that if the transition of a process to a state in which it is ready for interaction is independent of the random draws of the other processes, then, with probability 1, Multi is strong interaction fair. To our knowledge, Multi is the first algorithm for strong interaction fairness to appear in the literature. We also present a detailed timing analysis of Multi and establish a lower bound on how long a process must wait before reaccessing a shared variable. Consequently, our algorithm can be fine-tuned for optimal performance. Moreover, we show that the expected time to establish an interaction is a constant not depending on the total number of processes in the system. Thus, Multi also guarantees real-time response. Because strong interaction fairness is as strong a fairness condition that one might actually want to impose in practice, our results indicate that randomization may also prove fruitful for other notions of fairness lacking deterministic realizations and requiring real-time response. The rest of the paper is organized as follows. Section 2 describes the multiparty interaction scheduling problem in a more anthropomorphic setting known as Committee Coordination. Our randomized algorithm is presented in Section 3 and analyzed in Section 4. Section 5 concludes. 2 The Committee Coordination Problem The problem of scheduling multiparty interactions in asynchronous systems has been elegantly characterized by Chandy and Misra as one of Committee Coordination [5]: Professors (cf. processes) in a certain university have organized themselves into committees (cf. interactions) and each committee has a fixed membership roster of one or more professors. From time to time, a professor may decide to attend a committee meeting; it starts waiting and continues to wait until a meeting of a committee of which it is a member is established. To state the Committee Coordination problem formally, we need the following two assumptions: (1) a professor attending a committee meeting will not leave the meeting until all other members of the committee have attended the meeting; and (2) given that all members have attended a committee meeting, each committee member leaves the meeting in finite time. The problem then is to devise an algorithm satisfying the following three requirements: Synchronization: When a professor attends a committee meeting, all other members of the committee will eventually attend the meeting as well. Exclusion: No two committees meet simultaneously if they have a common member. Weak Interaction Fairness: If all professors of a committee are waiting, then eventually some professor will attend a committee meeting. We shall also consider strong interaction fairness, i.e., a committee that is infinitely often enabled will be established infinitely often. A committee is enabled if every member of the committee is waiting, and is disabled otherwise. The overall behavior of a professor can be described by the state transition diagram of Figure 1, where state T corresponds to thinking, W corresponds to waiting for a committee meeting, and E means that the professor is actively engaged in a meeting. Note that any algorithm for the problem should only control the transition of a professor from state W to state E, but not the other two transitions. That is, the transitions from T to W and from to T are autonomous to each professor. Moreover, we do not assume any upper bound on the time 'i 'i 'i meeting start meeting ready for finish meeting Figure 1: State transition diagram of a professor. a professor can spend in thinking. Otherwise, an algorithm for the problem could simply wait long enough until all professors become waiting, and then schedule a committee meeting of its choosing. All three requirements for the problem and strong interaction fairness would then be easily satisfied. 3 The Algorithm In this section, we present Multi, our randomized algorithm for committee coordination. In the algorithm, we associate with each committee M a counter CM whose value ranges over [0 \Delta \Delta jprof.Mj \Gamma1], where prof :M is the set of professors involved in M . CM can be accessed only by the professors in prof :M and only through the TEST&OP instruction as follows: result := TEST&OP(CM , zero-op, nonzero-op) The effect of this instruction is to apply to CM the operation zero-op if its value is zero and the operation nonzero-op otherwise, and to assign to the variable result the old value (i.e., the value before the operation) of CM . The operations used here are no-op, inc, and dec, where For example, if returns 2. If TEST&OP(CM , no-op, dec) leaves CM unchanged and returns 0. To simplify the presentation of the algorithm, we assume that the execution of the TEST&OP instruction is "atomic." This assumption is removed in Section 4.4, where a more concurrent implementation of TEST&OP is considered. Algorithm Multi can be informally described as follows. Initially, all the shared counters are set to zero. When a professor p i decides to attend a committee meeting, it randomly chooses a committee M of which it is a member. It then attempts to start a meeting of M by increasing the value of CM by 1 (all increments and decrements are to be interpreted modulo jprof.Mj). If the new value of CM is 0 (i.e., before the increment), then professor p i concludes that each of the other members of M has increased CM by one, and is waiting for p i to convene M . So goes to state E to start the meeting. If the new value of CM is not zero, then at least one of the professors in prof.M is not yet ready. waits for some period of time (hoping that its partners will become ready) and then reaccesses CM . If CM has now been set to 0, then all the professors which were not ready for M are now ready, and so p i can attend the meeting. If CM is still not zero, then some professor is still not ready for M . So p i withdraws its attempt to start M by decreasing the value of CM by 1 and tries another committee. The algorithm to be executed by each professor p i is presented in Figure 2, where waiting (line 1) is a Boolean flag indicating whether or not p i is waiting for a committee meeting. The constant ffi (line is the amount of time a professor waits before reaccessing a counter. We will later require (see Section 4) that ffi be greater than is the maximum amount of time a professor can spend in executing lines 2 and 3. 1 Note that the algorithm is symmetric in the sense that all professors execute the same code and make no use of their process ids. 4 Analysis of the Algorithm In this section we prove that Multi satisfies the synchronization and exclusion requirements of the Committee Coordination problem, and, with probability 1, is weak and strong interaction fair. We also analyze the expected time Multi takes to schedule a committee meeting. 4.1 Definitions We assume a discrete global time axis where, to an external observer, the events of the system are totally ordered. Internally, however, processors may execute instructions simultaneously at the same time instance. Simultaneous access to a shared counter will be arbitrated in the implementation of precisely, jmax should also include the time it takes to execute line 1. To simplify the analysis, we assume that the Boolean flag waiting only serves to indicate the state of the executing professor, and so no explicit test of the flag is needed. Moreover, we assume that an action is executed instantaneously at some time instance. The time it takes to execute an action is the difference between the time the action is executed and the time the previous action (of the same professor) was executed. 1. while waiting do f 2. randomly choose a committee M from fM j 3. if TEST&OP(CM , inc, 4. then /* a committee meeting is established */ 5. attend the meeting of M 6. else f wait 7. if TEST&OP(CM , no-op, 8. then /* a committee meeting is established */ 9. attend the meeting of M ; 10. /* else try another committee */ g 11. g Figure 2: Algorithm Multi for professor p i . the TEST&OP instruction, which we assume is executed atomically. Since the time axis is discrete, it is meaningless to say that "there are infinitely many time instances in some finite time interval such that ." Therefore, throughout this paper, the phrase "there are infinitely many time instances" refers to the interval [0; 1). For analysis purposes, we present in Figure 3 a refinement of the state transition diagram of Figure 1, where state W is refined into three sub-states . The actions taken by the professors from these sub-states are: randomly choose a new committee. execute the instruction TEST&OP(CM ; inc; inc). executing the instruction TEST&OP(CM ; no-op; dec). We say that a professor accesses counter CM when it executes the TEST&OP instruction of state W 1 , reaccesses CM when it executes the TEST&OP instruction of state W 2 , and monitors committee M while it is in state W 2 waiting for reaccess to CM . According to the algorithm, if at time t a professor p accesses a counter CM by TEST&OP(CM ; inc; inc) in state W 1 , then it will be in state W 2 or E right after 2 t, depending on the value of CM . Further- 2 If an action, which transits a professor p from state s1 to state s2 , occurs at time t, then we say that p is in state s1 just before t, and is in state s2 right after t. For p's state to be defined at every time instance, we stipulate that p's 'i 'i 'i 'i 'i ready for meeting reaccess CM (=0) access CM draw finish meeting access CM reaccess CM (6= random Figure 3: State transition diagram of a professor executing the algorithm. ffistart waitingrandom drawrandom drawrandom draw Figure 4: Timing constraints on the actions executed by a professor. more, if p enters state W 2 at time t to monitor a committee M , then at time t reaccess CM by TEST&OP(CM ; no-op; dec). Depending on the value of CM , after time the professor will either return to state W 0 to choose another committee, or enter state E to attend the meeting of M . In executing algorithm Multi, a professor starts waiting for a committee meeting in state W 0 and then repeatedly cycles through states W 0 through W 2 until entering a committee meeting via a transition from state W 1 or W 2 . The actions it performs along this cycle are subject to the timing constraints depicted in Figure 4. In particular, the interval between consecutive access and reaccess actions must be of length ffi, while the interval between consecutive reaccess and access actions must have length no greater than j max . We shall sometimes refer to the former as a "ffi-interval." As will be made explicit in Section 4.3, the duration of a ffi-interval may vary from iteration to iteration of the algorithm; we will require only that ffi is greater than a lower bound determined by j max and the number of professors in the committee currently under consideration. Figure 5 illustrates a possible scenario for four professors executing the algo- state at time t is s2 if p executes the action at time t. For example, if p accesses CM at time t and then reaccesses CM at time t is in state W2 at any time instance in [t; t ffi). Note that the interval is open at t + ffi. So if we say that p is in state W2 at time t, then p must have accessed CM at some time in prof prof Echoose M 14choose M 123choose M 123T Echoose M 234choose M 234choose M 123T Echoose M 123choose M 234choose M 123T Echoose M 14choose M 234choose M 14 Figure 5: A partial computation of four professors. rithm, where p 1 and p 4 are involved in committee M 14 , are involved in M 123 , and p 2 , are involved in M 234 . For each professor, we explicitly depict its state (on the Y-axis) at each global time instance (on the X-axis). For example, at time 1 professor p 1 starts waiting for a committee meeting and so it enters state W 0 from state T . At time 2, it randomly chooses M 14 and then accesses CM 14 at time 3. Since CM before the access, p 1 enters state W 2 to monitor M 14 for units and then reaccesses CM 14 at time 6. Since p 4 will not access CM 14 returns to state W 0 to try another committee. Later at time 12, p 1 chooses committee M 123 and then accesses CM 123 at time 13. When p 1 reaccesses CM 123 at time 16, it finds that both p 2 and p 3 are willing to start the meeting of M 123 . So p 1 enters state E to attend the meeting. The meeting of 123 ends at time 19, after which the committee members can return to state T at a time of their own choosing. The shaded area between time 17 and 19 represents a synchronization interval for the three professors. 4.2 Properties of the Algorithm That Hold with Certainty We now analyze the correctness of the algorithm. We begin with an invariant about the value of a shared counter CM , which we will use in proving that Multi satisfies the synchronization condition of the Committee Coordination Problem. Lemma 1 If at time t there are k professors in state W 2 monitoring committee M and no professor, since last entering state W 0 , has entered state E to attend a meeting of M , then the value of CM at time t is k and k ! jprof :M j. If, however, at time t some professor has entered state E to attend a meeting of M , then professors in prof :M will have entered state E to attend the meeting of M . Proof: We prove the lemma by induction on t i , the time at which the i th system event occurs. The lemma holds at time t 0 because initially every professor in prof :M is in state T and the induction hypothesis, assume the following at time t prof :M is the set of professors in state W 2 monitoring committee M , no professor, since last entering state W 0 , has entered state E to attend a meeting of M , and Consider now the nearest time t j , j - i, at which some professor p accesses or reaccesses CM . Since no professor accesses or reaccesses CM in [t the induction hypothesis holds as well in this interval. Suppose first that p accesses CM through the instruction TEST&OP(CM ; inc; inc). If before the access CM ! jprof after the access (which is less than jprof and p enters state W 2 . That is, after time t j there are jQj professors in state W 2 monitoring committee M , and Conversely, if before the access after the access enters state E. Since just before t j the other professors in Q are all monitoring M , by time reaccessed CM by TEST&OP(CM ; no-op; dec). Moreover, when they reaccess CM they will find that CM = 0 and so they will all enter state E to start M . To see why this last statement is true, recall that by the first of the two assumptions we put forth in defining the Committee Coordination problem (Section 2), no professor attending M will leave M before all of M 's members have entered state E to attend M . Consequently, no professor attending M can leave M to attempt another instance of M (by accessing CM ), from which the desired result follows. Suppose now that p reaccesses CM through the instruction TEST&OP(CM ; no-op; dec). Then p 2 Q. Since CM 6= 0, right after p's reaccess, returns to state W 0 . So right after t j , professors are in state W 2 monitoring M . 2 Theorem 1 (Synchronization) If a professor in prof :M enters state E at time t to attend a meeting of M , then within ffi time all professors in prof :M will have entered state E to attend the meeting of M . Proof: The theorem follows immediately from Lemma 1. 2 Theorem simultaneously if they have a common member Proof: The result follows from the fact that a professor monitors one committee at a time. 2 4.3 Properties of the Algorithm That Hold with Probability 1 We move on to prove that Multi is weak and strong interaction fair, and analyze its time complexity. For this we need the the following two assumptions: A1: The ffi-interval a professor chooses to wait for committee M satisfies the condition is the maximum amount of time a professor can spend in executing lines 2 and 3 of Multi. A2: A professor's transition from thinking to waiting (see Figure 1) does not depend on the random draws performed by other professors. Note that A2 is required only for strong interaction fairness. We also require some definitions about the "random draw" a professor performs in state W 0 when deciding which meeting to attempt. Recall that we say that a professor accesses a counter CM when it executes the instruction TEST&OP(CM ; inc, inc) (line 3 of the algorithm) and reaccesses CM when it executes TEST&OP(CM , no-op, dec) (line 7). Now suppose that professor p accesses some counter in the time interval there is more than one such access, choose the most recent one. Then the choice of counter must be the result of the random draw performed before the access (line 2). Let D t;p denote this random draw; D t;p is not defined if p does not access any counter in Furthermore, let D t;prof prof :M and D t;p is defined g. For example, if p is in state W 2 at time t, then p must have accessed some counter in the interval must be defined. As we shall see in Lemma 4, the definition of D t;p guarantees that if D t;p is defined for all p 2 prof :M and these random draws yield the same outcome M , then M will be established. Henceforth we shall use / p;M to denote the fixed non-zero probability that professor p 2 prof :M chooses committee M in a random draw. Thus, Y p2prof :M is the probability that a set of mutually independent random draws, one by each professor in prof :M , yields the same outcome M . The following three lemmas are used in the fairness proofs. The first one says that D t;prof :M and must refer to mutually disjoint sets of random draws if t and t 0 are at least ffi apart. Proof: Directly from the definition of D t;p . 2 Lemma 3 Assume A1 and that committee M is continuously enabled in the interval is, M is enabled at any time instance in [t there exists a time instance t, t Since M is continuously enabled in [t every professor of M is, by definition, in a -state throughout this interval. Clearly, either one of the following holds: such that p is in state W 2 throughout such that p is in state W 0 or W 1 throughout In case (i), let t g. Then D t;p is defined for every p 2 prof :M and every Given that t and that ' - j max , we thus have that there exists some t, j. So the lemma is proven. For case (ii), suppose that some professor p 1 2 prof :M stays in W 0 or W 1 in [t then accesses some counter at t i +m 1 , where by the assumed lower bound on ', t If D t i +m 1 ;p is defined for all p 2 prof :M , then we are done. Otherwise, there must exist another professor 2. By the lower bound on ', still enabled at time t i +m 1 . So p 2 is in a W-state at t i +m 1 . cannot be in state W 2 , for otherwise D t i +m 1 ;p 2 would be defined. So p 2 is in state W 0 or must access some counter within j max time. Assume that p 2 accesses some counter at We argue that D t i +m 1 +m 2 ;p 1 is also defined. To see this, recall that By the lower bound on ' and the fact that jprof :M j - 2, we have still in a W -state at any time instance in some counter at t i +m 1 , p 1 must have entered state W 2 after the access, and stays in W 2 throughout must be defined for all t in [t i and the fact that Note that t defined for every other professor in prof :M , then we are done. Otherwise, similar to the above reasoning, there must exist a third professor prof :M such that D t i +m 1 +m 2 ;p 3 is not defined. Using the same argument, we can show that there exists m 3 where is defined for all Continuing in this fashion, we can show that there exists k professors in prof :M , and and D t i +m 1 +m 2 +:::+m k ;p l is defined for all 1 - l - k. Given that there are only a finite number of professors in prof :M , eventually we will establish that there exists some t, t D t;p is defined for each p 2 prof :M . The lemma is then proven. 2 The proof of Lemma 3 is illustrated in Figure 6 for a committee of size 4 with t the smallest ffi allowed by A1. As a consequence of Lemma 3, different professors can choose different values for ffi; these values need only satisfy the lower bound established by A1. 3 Therefore, the clocks used by the professors to implement time-outs need not be adjusted to the same accuracy. Lemma 3 says that, under assumption A1, if a committee M is continuously enabled sufficiently long, then there exists an interval of length ' within which every professor in prof :M performs a random draw. The following lemma ensures that if their random draws yield the same outcome, then they must establish M . Lemma 4 If jD t;prof :M all the random draws in D t;prof :M yield the same outcome 3 As such, the ffi referred to in the definition of D t;prof :M and in the statement of Lemma 1 should be understood as the minimum and the maximum of the relevant ffi values, respectively. M is continuously enabled \Gammareadyaccessreaccess \Gammareaccessaccess \Gammareaccessaccess access Figure Illustration of the proof of Lemma 3. is the maximum possible interval throughout which M is enabled but jD t;prof:M j 6= jprof:M j. Here jprof:M must be defined for all . Note that if ffi min would equal would not be defined. M , then by time t some professor must have already entered state E to start M , and by time all professors in prof :M will enter state E to start M . Proof: Assume the hypotheses described in the lemma. Let p i 2 prof :M be the first professor which, after performing its random draw in D t;prof :M , accesses CM by TEST&OP(CM ; inc; inc), and prof :M be the last professor to do so. Let t i and t j be the time at which p i and p j , respectively, access CM . Then, t. CM at time t i , it will not reaccess CM until professors that access CM in [t remain in state W 2 before p j accesses CM . By Lemma 1, just before access. So when p j accesses CM at t j , it will set CM to zero and enter state E to start M . Moreover, by time ffi, every other professor of M will learn that M has been started when it reaccesses CM by TEST&OP(CM ; no-op; dec), and so will also enter state E to start M . Since ffi, the lemma is thus established. 2 Note that Lemma 4 relies on the fact that the access involved in the definition of D t;p occurs in the interval that is open at and closed at t. If we were to relax the definition to the closed interval then the correctness of Lemma 4 would depend on how an access/reaccess conflict to the same counter is resolved in the implementation. To see this, suppose that p 1 accesses a counter at a counter at t. So by the new definition both D t;p 1 and D t;p 2 are defined. Suppose further that both random draws yield the same outcome M , which involves only access the same counter CM at t, respectively. Assume that before By the algorithm, p 1 will wait ffi time and then reaccess CM at t, causing a conflict with p 2 's access at the same time. Clearly, M will be established only if the conflict is resolved in favor of the access; i.e., p 2 gets to go first. Theorem 3 (Weak Interaction Fairness) Assume A1 and that all members of a committee M are waiting for committee meetings. Then the probability is 1 that eventually a meeting involving some member of M will be started. Proof: Assume A1, and suppose that M is enabled at t. Let jprof :M j. Consider the probability that M is continuously enabled in [t; t M is continuously enabled in [t; t exists a time instance t 1 , that jD t 1 ;prof :M j. If the random draws in D t 1 ;prof :M yield the same outcome M , then, by Lemma 4, M must be disabled at t 1 . (Even if the random draws do not yield the same outcome, some professor of M may still establish another committee meeting M 0 if its random draw has the outcome M 0 and at the same time all other professors of M 0 are also interested in M 0 .) So the probability that the random draws in D t 1 ;prof :M do not cause any committee involving a member of M to be started is no greater than and so the probability that M is continuously enabled in [t; t Similarly, if M is still enabled after t 1 , then by Lemmas 2 and 3 there must exist another time instance t 2 such that D t 2 ;prof :M contains a completely new set of random draws of size jprof :M j. Again, the probability that M remains enabled after these random draws is no greater than given that the random draws in D t 1 ;prof :M do not cause any member of M to attend a meeting. So the probability that M is continuously enabled up to time t 2 is no greater than if M is still enabled at t 2 then there will be another new set of random draws D t 3 ;prof :M of size jprof :M j. In general, the probability that M remains continuously enabled after i mutually disjoint sets of random draws D t 1 ;prof As i tends to infinity, tends to zero. So the probability is zero that M remains enabled forever. 2 Intuitively, A1 requires that the ffi parameter used in the algorithm is large enough so that a professor will not reaccess a counter before the other professors get a chance to access the counter. reaccess 6 access 6 reaccess 6 access 6 reaccess reaccess 6 access 6 reaccess 6 access 6 reaccess Figure 7: Two professors wait forever without establishing a meeting due to a bad choice of ffi. If this assumption is removed from Theorem 3, then a set of professors could access and reaccess a counter forever without ever establishing a committee meeting. To illustrate, consider Figure 7. Each professor reaccesses a counter before the other professor could access the same counter. So no matter what committees they choose in their random draws, at no time can a professor see the result of a counter set by the other professor. The strong interaction fairness property of the algorithm additionally requires assumption A2 and a lemma on the probabilistic behavior of a large number of random draws. Lemma 5 Assume A2. If there are infinitely many t's such that jD t;prof :M then the probability is 1 that all the random draws in D t;prof :M produce the same outcome for infinitely many t's. be an infinite sequence of increasing time instances at which jD t i ;prof :M jprof :M j. W.l.o.g. assume that 8i; t 2, the sets D t i ;prof :M are pairwise disjoint. Consider the random draws in set D t i ;prof :M . Let EM denote the event that the random draws in produce the same outcome M . By A2, the probability of EM 's occurrence is independent of the time these random draws are made and is given by /M . Define random variable A i to be 1 if EM occurs at t i , and 0 otherwise. Then A By the Law of Large Numbers (see, for example, [6]), for any ffl we have lim That is, when n tends to infinity, the probability is 1 that n tends to /M . Therefore, with probability 1, the set fi j A Hence, with probability 1, there are infinitely many i's such that the random draws in D t i ;prof :M produce the same outcome M . 2 Theorem 4 (Strong Interaction Fairness) Assume A1 and A2. Then if a committee is enabled infinitely often, with probability 1 the committee will be convened infinitely often. Proof: Since the algorithm satisfies weak interaction fairness, we assume that there are infinitely many i's such that M becomes enabled at time instance t i . Let either (1) there are infinitely many i's such that M is continuously enabled in each interval [t or (2) starting from some point t i 0 onward, whenever M becomes enabled at t i , some professor in prof :M attends a committee meeting in the interval Consider Case (1). By Lemma 3 and A1, there are infinitely many i's such that each interval contains some time instance t such that jD t;prof :M j. Then by Lemma 5 and A2, with probability 1 there are infinitely many t's such that all the random draws in D t;prof :M produce the same outcome. So by Lemma 4, with probability 1 M is convened infinitely often. Consider Case (2). W.l.o.g. assume each interval (t contains no time instance t such that by the previous argument we can also show that M will be convened infinitely often with probability 1. Let Q prof :M be the set of professors that have accessed a counter between the time t i at which they are waiting for M until the time at which they attend a committee meeting. For each q 2 Q i , let a be q's first such access, and let D 1 denote q's random draw performed right before a. For each p's latest random draw performed before time t i . Note that since p is in a W -state at t i , D 2 defined and its outcome must cause p to attend a committee meeting at some time after t i . Let g. By A2, the random draws in D 0 are mutually independent and have a nonzero probability /M to yield the same outcome M . Therefore, by the Law of Large Numbers (see the proof of Lemma 5), the probability is 1 that there are infinitely many i's such that all the random draws in D 0 yield the same outcome M . If all the random draws in D 0 yield the same outcome M , then either a meeting of M will be established, or each professor in prof :M will still be waiting for M and will perform another random draw to access a new counter. By a technique similar to Lemma 3, it can be seen that in the later case we would be able to find a time instance t in (t j. By the assumption of the case, we conclude that with probability 1 M is convened infinitely often. 2 Note that if Assumption A2 is dropped from Theorem 4, then a conspiracy against strong inter-action fairness can be devised. To illustrate, consider a system of two professors p 1 and p 2 , and three committees involving only involving only p 2 , and M 12 , which involves both p 1 and p 2 . Suppose that p 1 becomes waiting, and then tosses a coin to choose either M 1 or M 12 . The malicious could remain thinking until p 1 has selected M 1 ; then p 2 becomes waiting just before random draw is performed only if p 1 's latest random draw yields outcome once it selects M 1 , M 12 will not be started if p 1 remains in its meeting while p 2 is waiting. However, M 12 is enabled as soon as p 2 becomes waiting. Similarly, could also remain thinking until p 2 has selected M 2 . So if this scenario is repeated ad infinitum, then the resulting computation would not be strong interaction fair. The time complexity of the algorithm is analyzed in the following theorem, which assume a worst case scenario that a professor spend j max time in executing lines 2 and 3 of Multi. Theorem 5 (Time Complexity) Assume that each professor spends j max time in executing lines 2 and 3 of Multi, and assume A1, i.e., the amount of time ffi a professor spends in monitoring an interaction is greater than 1). Then the expected time it takes any member of a committee M to start a meeting from the time that M becomes enabled is no greater than Proof: Suppose that M becomes enabled at time t. Consider first that there exists a time instance Assume first that while M is enabled, no conflicting committee is enabled simultaneously. (Two committee conflict if they share a common member.) So when M is enabled, each professor in prof :M can only attend a meeting of M . By Lemma 4, if the random draws in jD t 1 ;prof :M j yield the same outcome M (an event that occurs with probability /M ), then some professor in prof :M will start a meeting of M by time t 1 . Otherwise, each professor in prof :M must perform another random draw and access the selected counter within j (from the time it reaccesses the previous selected counter). So there must exist another time instance t 2 , contains a completely new set of random draws, one by each professor in prof :M . Once again, if the new random draws yield the same outcome M , then some professor will start a committee meeting. So the probability that M will be disabled by time t 2 is In general, let D t i ;prof :M be the i th set of random draws performed by the professors, where ffi). Then the probability that M will be disabled by time t i is ffi). So the expected time starting from t until some member of M enters state E to start a meeting is no greater than We assumed above that no committee conflicting with M can be enabled while M is enabled; this implies that for each set of random draws D t i ;prof :M , the random draws must yield the same outcome M for the members of M to start a meeting. If a conflicting committee is enabled simultaneously, then some of the random draws in D t i ;prof :M may still lead to a committee meeting even if they do not yield outcome M . Hence, the expected time starting from t until some member of M enters state E to start a meeting will actually be less than (j conflicting committee is enabled simultaneously. Assume next that there exists no time instance t 1 , must be disabled before t 's disabledness must be the result of some professor's random draw leading to the establishment of some committee meeting involving that professor. So the disabling of M at some time before must also be a probabilistic event. Therefore, in this case the expected time starting from t until some member of M enters state E to start a meeting is no greater than j. Given that that the expected time is no greater than Therefore, in either case, the expected time starting from t until some member of M enters state E to start a meeting is no greater than (j Intuitively, Theorem 5 states that the expected time for any member of prof :M to start a committee meeting when they are all waiting is no greater than the amount of time to execute one round of the while-loop of Multi (i.e., divided by the probability that these professors in their random draws all choose the same committee M (i.e., /M ). Note that j max is a constant determined by the size (number of members) of the largest committee. Call this value S max and let j . The probability /M is a constant determined by the size of M , and the maximum number of committees of which a professor can be a member (call this value C p2prof :M / Finally, ffi is a constant determined by Therefore, the time complexity of the algorithm is bounded by the following constant: While in the worst case S max could be equal to the total number of professors, and C max could be equal to the total number of committees in the system (which, in turn, could be dependent on the total number of professors), in practice, it is generally known that both parameters must be kept small and independent of the total number of professors in the system [7]. 4 In contrast, deterministic algorithms for Committee Coordination such as [15, 12, 10] have time complexity where c 0 is a constant and N is the total number of professors in the system. 5 The time complexity of these algorithms depends explicitly on N because they use priority to beak the symmetry among professors. As such, a lower priority professor may have to wait for a higher priority professor if they attempt to establish a conflicting committee, and the higher priority professor in turn may have to wait for another higher priority professor, and so on. (Recall, Section 1, that there is no symmetric, deterministic, and distributed solution for Committee Coordination.) If C max and S max are kept small and independent of N , then Multi, in addition to guaranteeing strong interaction fairness, out-performs deterministic algorithms while providing real-time response. 4.4 A Non-Atomic Implementation of TEST&OP As promised in Section 3, we now present a non-atomic, and hence more concurrent, implementation of the TEST&OP instruction. Recall that the execution of the statement TEST&OP(CM , zero-op, nonzero-op) actually involves two 6 actions: read CM , then apply to CM the operation zero-op if and the operation nonzero-op otherwise. More precisely, the actions are read followed by inc when a professor executes TEST&OP(CM ; inc; inc) to access a counter, and read followed by dec/no-op when it executes TEST&OP(CM ; no-op; dec) to reaccess the counter. Clearly, we can apply a mutual exclusion algorithm (see [16] for a survey) to ensure that each access and reaccess to a counter proceeds atomically. This, however, is an overkill. For example, 4 A scheme of synchrony loosening is therefore proposed in [7] for reducing the size of an interaction in practical applications. 5 These algorithms and Multi all allow professors to distributedly establish a committee meeting on their own. Other deterministic algorithms such as [5, 4, 14] employ "managers" to coordinate committee meetings; the time complexity of these algorithms then depends on the number of managers they use. 6 Three, if you count the Boolean test. accesses to a counter can be executed concurrently. 7 To see this, consider three possible interleaved executions of two TEST&OP(CM ; inc; inc)'s: read 1 , read 2 , inc 1 , inc 2 read 1 , read 2 , inc 2 , inc 1 read 1 , inc 1 , read 2 , inc 2 Observe that the first two executions have the same effect: both will cause the executing professors to enter state W 2 to monitor M because the value of CM returned by both reads is less than value before the two accesses is less than jprof then the third execution, in which the two accesses proceed atomically, will also have the same effect. If CM 's value before the two accesses is jprof :M \Gamma 2j, then in the third execution the professor executing the first access will enter W 2 to monitor M , while the other professor will enter state E to start a meeting of M . When the first professor's ffi-interval expires, it will learn that a meeting of M has been established when it reaccesses CM , and so will also enter state E to start a meeting of M . The situation is similar in the first two executions: both professors will enter state E to start a meeting of M when they reaccess the counter. So all three interleaved executions preserve the synchronization property of the algorithm. (Breaking the atomicity of TEST&OP clearly has no effect to the algorithm's exclusion and fairness properties.) Note that the system performance may be increased if we reverse the order of execution of the read and inc actions in the implementation of TEST&OP(CM ; inc; inc). To see this, consider again the case where two professors attempt to access CM simultaneously. The following are two possible interleaved executions: inc 1 , inc 2 , read 1 , read 2 inc 1 , inc 2 , read 2 , read 1 Suppose that CM 's value before the access is jprof after the two increments, So each professor, upon reading the value of CM , learns that all professors of M are now interested in M and can enter state E to start a meeting of M . Moreover, the new implementation of 7 We assume that basic machine-level instructions such as inc, dec, load , and store are executed atomically. Thus, if two such instructions are executed concurrently, the result is equivalent to their sequential execution in some unknown order. 8 When concurrent accesses to the same counter are allowed, more than one professor may access CM simultaneously and then all enter state W2 to monitor M because CM ! jprof before the accesses. Likewise, Lemma 1, which assumes that access to a counter is atomic, needs to be slightly changed to reflect the possibility that professors in prof :M are in state W2 at the same time. inc) still ensures Multi's synchronization property regardless of how the actions of overlapping TEST&OP(CM ; inc; inc) instructions are interleaved. 9 Similarly, interleaving read and dec/no-op of different professors' reaccesses to the same counter cannot invalidate the algorithm's synchronization property. Only simultaneous access and reaccess to the same counter may conflict. To illustrate, suppose that p 1 wishes to access CM while p 2 wishes to reaccess. Suppose further that the value of CM before the attempt is jprof access proceeds atomically before p 2 's reaccess, both professors will enter state E to start the meeting. However, if the four constituent actions are interleaved as follows: read 2 , inc 1 , read 1 , dec 2 decrement CM by one and go to state W 0 to select a new committee. On the other hand, since before p 2 's decrement, it will discover that thus enter state E to start M . Hence, the synchronization requirement is violated. To ensure that access and reaccess to the same counter are mutually exclusive while at the same time allowing concurrent accesses and concurrent reaccesses, we can implement TEST&OP(CM ; inc; inc) and TEST&OP(CM , no-op; dec) using the algorithm shown in Figures 8 and 9. The algorithm is based on Dekker's algorithm for the bi-process critical section problem [16] and, as discussed above, now returns the new value of CM . CM access count is a counter recording the number of professors that are attempting to access CM , while CM reaccess count records the number of professors attempting to reaccess CM . Both counters are initialized to zero. Furthermore, variable CM turn, initialized to access, is used for resolving conflicts between accesses and reaccesses. It can be seen that a professor enters the critical section to access CM only if CM access count only professors attempting to access CM may modify CM access count , and they test CM reaccess count only when CM access count ? 0, it follows that when some professor p enters the critical section to access CM , no other professor can simultaneously enter the critical section to reaccess CM . Moreover, no professor can enter the critical section to 9 If this new implementation is adopted, then line 3 of Figure 2 needs to be changed to "if TEST&OP(CM ; inc; inc) = 0" because TEST&OP(CM ; inc; inc) now returns the value of CM after the access. Note, however, that TEST&OP(CM , no- op, dec) must still return the value of CM before the access. This is because if it returns the value of CM after the access, then when the returned value is zero, the executing professor p i would not be able to tell if (1) only p i itself is interested in M (so that p i should decrease CM by one and then return to state W0 to retry another committee), or (2) all members in prof :M are interested in M (so that p i should leave CM unchanged and then enter state E to start a meeting of M .) 1. inc (CM access count) ; 2. while CM reaccess count ? 0 do 3. if CM turn = reaccess then f 4. dec (CM access count) ; 5. while CM turn = reaccess do no-op ; 6. inc (CM access count) ; g 7. /* beginning of critical section */ 8. inc(C M 9. return read(C M 10. /* end of critical section */ 11. dec (CM access count) ; 12. if CM access count = 0 then CM turn := reaccess ; Figure 8: Implementation of TEST&OP(CM ; inc; inc). 1. inc (CM reaccess count) ; 2. while CM access count ? 0 do 3. if CM turn = access then f 4. dec (CM reaccess count) ; 5. while CM turn = access do no-op ; 6. inc (CM reaccess count) ; g 7. /* beginning of critical section */ 8. if read(C M 9. else f return read(C M 10. /* end of critical section */ 11. dec (CM reaccess count) ; 12. if CM reaccess count Figure 9: Implementation of TEST&OP(CM ; no-op; dec). reaccess CM while p is already in the critical section but has not yet left the critical section. Similarly, if a professor is in the critical section to reaccess CM , then no other professor can enter the critical section to access CM . The mutual exclusion property therefore holds. Note that it is possible that while a professor is in the critical section (say, to access CM ), some professor has already "flipped" CM turn to reaccess (line 12 of Figures 8). However, the premature flipping of CM turn cannot invalidate the algorithm's mutual exclusion property because the entering of the critical section to reaccess CM does not depend on the value of CM turn, but rather on the value of CM access count : as long as some professor is in the critical section to access CM , CM access count remains greater than 0, and so no professor can exit the while-loop of Figures 9 (lines 2-6) to reaccess CM . The algorithm is also deadlock-free. To see this, consider an arbitrary time instance at which A ' prof :M is the set of professors wishing to access CM and R ' prof :M is the set of professors wishing to reaccess CM definition). Consider now the plight of some p 2 A (similar reasoning applies in the case of reaccess). If obviously, p will succeed. Otherwise what happens next depends on the value of CM turn: If CM turn = access then each professor in R must undo its increment of CM reaccess count , and wait in line 5 of Figure 9 until CM turn is flipped to reaccess . When CM reaccess count has been reset to zero, p can then enter the critical section. Conversely, if CM turn = reaccess , then p and the other professors in A must undo their increments of CM access count , collectively resetting the value of this variable to zero, and wait in line 5 of Figure 8 until the professors in R enter the critical section and then flip CM turn to access . Moreover, the algorithm permits concurrent access (and concurrent reaccess, too), meaning that if a professor p 1 attempts to access CM while some other professor p 2 is accessing the counter, then may succeed even if there is already a third professor waiting for reaccess to CM . This is because is in the critical section while CM turn = access , then all professors waiting for reaccess to CM are blocked at line 5 of Figure 9, and CM reaccess count = 0. So p 1 can immediately enter the critical section. Note that allowing subsequent professors to concurrently access a counter cannot indefinitely delay a professor waiting for reaccess to the same counter because (1) the number of professors in a committee is finite, and (2) a professor p's access to CM must be followed by a reaccess to the same counter (unless p's access leads to a committee meeting; and if this is the case, then the professor must enter state E to wait for the other members to finish their reaccesses so that they can start a meeting). By assumption A1 the time between the access and reaccess, i.e., the ffi-interval, must be long enough for other professors to finish their accesses. Note further that permitting concurrent accesses is highly desirable because it increases the likelihood of establishing committee meetings. For example, suppose that two sets of professors are waiting for access and reaccess to CM , respectively, while some professor is already accessing the counter. Deferring the reaccesses until all accesses have proceeded can only help the members of M reach consensus, while scheduling the accesses and reaccesses in a fair manner (e.g., alternatively) adds no help at all to the establishment of M 's meeting. Conclusions We have presented Multi, a new randomized algorithm for scheduling multiparty interactions. We have shown that by properly setting the value of ffi (the amount of time a process is willing to wait for an interaction to be established), our algorithm is both weak and strong interaction fair with probability 1. Our results hold even if the time it takes to access a shared variable (the communication delay) is nonnegligible. To our knowledge, this makes Multi the first algorithm for strong interaction fairness to appear in the literature. Strong interaction fairness has been proven impossible by any deterministic algorithm. Our results therefore indicate that randomization is a feasible and efficient countermeasure to such impossibility phenomena. Furthermore, since most known fairness notions are weaker than strong interaction fairness, they too can be implemented via randomization. For example, strong process fairness [1], where a process infinitely often ready for an enabled interaction shall participate in an interaction infinitely often, is also realized by our randomized algorithm in spite of the fact that it cannot be implemented by any deterministic multiparty interaction scheduling [19, 9]. Multi is an extension of Francez and Rodeh's randomized algorithm for CSP-like biparty inter- actions. Francez and Rodeh were able to claim only weak interaction fairness for their algorithm, and then only under the limiting assumption that the communication time is negligible compared to ffi. In this case, strong interaction fairness would be possible even in a deterministic setting. We have also analyzed the time complexity of our algorithm. Like Reif and Spirakis's real-time algorithm [17], the expected time taken by Multi to establish an interaction is a constant not depending on the total number of processes in the system. Although Multi is presented in a shared-memory model, it can be easily converted to a message-passing algorithm by letting some processes maintain the shared variables, and other processes communicate with them by message passing to obtain the values of these variables. The time to read/write a shared variable then accounts for the time it takes to deliver a message. The ffi parameter in Assumption A1 can be properly adjusted to reflect the new communication delay so that both weak and strong interaction fairness notions can still be guaranteed with probability 1. Multi, as originally described in Section 3, uses an operation TEST&OP for processes to access a shared counter atomically. This operation is rather complex and will not be generally available. Moreover, it unnecessarily eliminates potential concurrency. So, in Section 4.4, we have proposed an implementation of TEST&OP that uses the more basic atomic instructions inc, dec, load , and store. Implementing Multi on a machine that does not support the atomic execution of these instructions, as could well be the case for inc and dec, is an interesting open problem. As discussed in Section 4.4, the implementation of TEST&OP would be much simpler if a general-purpose mutual exclusion algorithm was used instead. However, we know of no mutual exclusion algorithm that allows concurrent accesses to the critical section if the accesses themselves do not conflict with one another. Therefore we had to design our own solution. Finally, unlike deterministic algorithms, randomized algorithms such as Multi only "guarantee" average-case behavior, not a worst-case bound. It would therefore be interesting to conduct simulation studies on Multi to measure its response time in practical settings. Experiments in which the size of S max and C max (see Section 4.3) vary from small constants to large values approaching the number of professors in the system would be especially insightful. Acknowledgments . We would like to thank the anonymous referees for their careful reading of the manuscript and their valuable comments. --R Appraising fairness in languages for distributed program- ming On fairness as an abstraction for the design of distributed systems. Fairness and hyperfairness in multi-party interactions Process synchronization: Design and performance evaluation of distributed algo- rithms A Foundation of Parallel Program Design. A Course in Probability Theory. Interacting Processes: A Multiparty Approach to Coordinated Distributed Programming. A distributed abstract data type implemented by a probabilistic communication scheme. Characterizing fairness implementability for multiparty interaction. Coordinating first-order multiparty interactions A comprehensive study of the complexity of multiparty interaction. An implementation of N-party synchronization using tokens On the advantage of free choice: A symmetric and fully distributed solution to the dining philosophers problem (extended abstract). A distributed synchronization scheme for fair multi-process handshakes A new and efficient implementation of multiprocess synchronization. Algorithms for Mutual Exclusion. Real time synchronization of interprocess communications. Distributed algorithms for ensuring fair interprocess communications. Some impossibility results in interprocess synchronization. --TR --CTR Catuscia Palamidessi , Oltea Mihaela Herescu, A randomized encoding of the -calculus with mixed choice, Theoretical Computer Science, v.335 n.2-3, p.373-404, 23 May 2005 Rafael Corchuelo , Jos A. Prez , Antonio Ruiz-Corts, Aspect-oriented interaction in multi-organisational web-based systems, Computer Networks: The International Journal of Computer and Telecommunications Networking, v.41 n.4, p.385-406, 15 March

multiparty interaction;weak interaction fairness;randomized algorithms;committee coordination;distributed algorithms;strong interaction fairness

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Scheduling Block-Cyclic Array Redistribution.

AbstractThis article is devoted to the run-time redistribution of one-dimensional arrays that are distributed in a block-cyclic fashion over a processor grid. While previous studies have concentrated on efficiently generating the communication messages to be exchanged by the processors involved in the redistribution, we focus on the scheduling of those messages: how to organize the message exchanges into "structured" communication steps that minimize contention. We build upon results of Walker and Otto, who solved a particular instance of the problem, and we derive an optimal scheduling for the most general case, namely, moving from a CYCLIC(r) distribution on a P-processor grid to a CYCLIC(s) distribution on a Q-processor grid, for arbitrary values of the redistribution parameters P, Q, r, and s.

Introduction Run-time redistribution of arrays that are distributed in a block-cyclic fashion over a multidimensional processor grid is a difficult problem that has recently received considerable attention. This interest is motivated largely by the HPF [13] programming style, in which scientific applications are decomposed into phases. At each phase, there is an optimal distribution of the data arrays onto the processor grid. Typically, arrays are distributed according to a CYCLIC(r) pattern along one or several dimensions of the grid. The best value of the distribution parameter r depends on the characteristics of the algorithmic kernel as well as on the communication-to-computation ratio of the target machine [5]. Because the optimal value of r changes from phase to phase and from one machine to another (think of a heterogeneous environment), run-time redistribution turns out to be a critical operation, as stated in [10, 21, 22] (among others). Basically, we can decompose the redistribution problem into the following two subproblems: Message generation The array to be redistributed should be efficiently scanned or processed in order to build up all the messages that are to be exchanged between processors. Communication scheduling All the messages must be efficiently scheduled so as to minimize communication overhead. A given processor typically has several messages to send, to all other processors or to a subset of these. In terms of MPI collective operations [16], we must schedule something similar to an MPI ALLTOALL communication, except that each processor may send messages only to a particular subset of receivers (the subset depending on the sender). Previous work has concentrated mainly on the first subproblem, message generation. Message generation makes it possible to build a different message for each pair of processors that must communicate, thereby guaranteeing a volume-minimal communication phase (each processor sends or receives no more data than needed). However, the question of how to efficiently schedule the messages has received little attention. One exception is an interesting paper by Walker and Otto [21] on how to schedule messages in order to change the array distribution from CYCLIC(r) on a P - processor linear grid to CYCLIC(Kr) on the same grid. Our aim here is to extend Walker and Otto's work in order to solve the general redistribution problem, that is, moving from a CYCLIC(r) distribution on a P -processor grid to a CYCLIC(s) distribution on a Q-processor grid. The general instance of the redistribution problem turns out to be much more complicated than the particular case considered by Walker and Otto. However, we provide efficient algorithms and heuristics to optimize the scheduling of the communications induced by the redistribution operation. Our main result is the following: For any values of the redistribution parameters P , Q, r and s, we construct an optimal schedule, that is, a schedule whose number of communication steps is minimal. A communication step is defined so that each processor sends/receives at most one message, thereby optimizing the amount of buffering and minimizing contention on communication ports. The construction of such an optimal schedule relies on graph-theoretic techniques such as the edge coloring number of bipartite graphs. We delay the precise (mathematical) formulation of our results until Section 4 because we need several definitions beforehand. Without loss of generality, we focus on one-dimensional redistribution problems in this article. Although we usually deal with multidimensional arrays in high-performance computing, the problem reduces to the "tensor product" of the individual dimensions. This is because HPF does not allow more than one loop variable in an ALIGN directive. Therefore, multidimensional assignments and redistributions are treated as several independent one-dimensional problem instances. The rest of this article is organized as follows. In Section 2 we provide some examples of redistribution operations to expose the difficulties in scheduling the communications. In Section 3 we briefly survey the literature on the redistribution problem, with particular emphasis given to the Walker and Otto paper [21]. In Section 4 we present our main results. In Section 5 we report on some MPI experiments that demonstrate the usefulness of our results. Finally, in Section 6, we state some conclusions and future work directions. Motivating Examples Consider an array X[0:::M \Gamma 1] of size M that is distributed according to a block cyclic distribution CYCLIC(r) onto a linear grid of P processors (numbered from Our goal is to redistribute X using a CYCLIC(s) distribution on Q processors (numbered from For simplicity, assume that the size M of X is a multiple of Qs), the least common multiple of P r and Qs: this is because the redistribution pattern repeats after each slice of L elements. Therefore, assuming an even number of slices in X will enable us (without loss of generality) to avoid discussing side effects. Let L be the number of slices. Example 1 Consider a first example with 5. Note that the new grid of Q processors can be identical to, or disjoint of, the original grid of P processors. The actual total number of processors in use is an unknown value between 16 and 32. All communications are summarized in Table 1, which we refer to as a communication grid. Note that we view the source and target processor grids as disjoint in Table 1 (even if it may not actually be the case). We see that each source processor messages and that each processor receives 7 messages, too. Hence there is no need to use a full all-to-all communication scheme that would require 16 steps, with a total of 16 messages to be sent per processor (or more precisely, 15 messages and a local copy). Rather, we should try to schedule the communication more efficiently. Ideally, we could think of organizing the redistribution in 7 steps, or communication phases. At each step, 16 messages would be exchanged, involving disjoint pairs of processors. This would be perfect for one-port communication machines, where each processor can send and/or receive at most one message at a time. Note that we may ask something more: we can try to organize the steps in such a way that at each step, the 8 involved pairs of processors exchange a message of the same length. This approach is of interest because the cost of a step is likely to be dictated by the length of the longest message exchanged during the step. Note that message lengths may or may not vary significantly. The numbers in Table 1 vary from 1 to 3, but they are for a single slice vector. For a vector X of length lengths vary from 1000 to 3000 (times the number of bytes needed to represent one data-type element). A schedule that meets all these requirements, namely, 7 steps of 16 disjoint processor pairs exchanging messages of the same length, will be provided in Section 4.3.2. We report the solution schedule in Table 2. Entry in position (p; q) in this table denotes the step (numbered from a to g for clarity) at which processor p sends its message to processor q. In Table 3, we compute the cost of each communication step as (being proportional to) the length of the longest message involved in this step. The total cost of the redistribution is then the sum of the cost of all the steps. We further elaborate on how to model communication costs in Section 4.3.1. Table 1: Communication grid for 5. Message lengths are indicated for a vector X of size Communication grid for of msg. Nbr of msg. 7 7 Example 2 The second example, with shows the usefulness of an efficient schedule even when each processor communicates with every other processor. As illustrated in Table 4, message lengths vary with a ratio from 2 to 7, and we need to organize the all-to-all exchange steps in such a way that messages of the same length are communicated at each step. Again, we are able to achieve such a goal (see Section 4.3.2). The solution schedule is given in Table 5 (where steps are numbered from a to p), and its cost is given in Table 6. (We do check that each of the 16 steps is composed of messages of the same length.) Example 3 Our third motivating example is with As shown in Table 7, the communication scheme is severely unbalanced, in that processors may have a different number of messages to send and/or to receive. Our technique is able to handle such complicated situations. We provide in Section 4.4 a schedule composed of 10 steps. It is no longer possible to have messages of the same length at each step (for instance, processor messages only of length 3 to send, while processor messages only of length 1 or 2), but we do achieve a redistribution in communication steps, where each processor sends/receives at most one message per step. The number of communication steps in Table 8 is clearly optimal, as processor to send. The cost of the schedule is given in Table 9. Table 2: Communication steps for Communication steps for 9 - a - b - d f - g e - c Table 3: Communication costs for Communication costs for Step a b c d e f g Total Cost Example 4 Our final example is with P 6= Q, just to show that the size of the two processor grids need not be the same. See Table 10 for the communication grid, which is unbalanced. The solution schedule (see Section 4.4) is composed of 4 communication steps, and this number is optimal, since processor messages to receive. Note that the total cost is equal to the sum of the message lengths that processor must receive; hence, it too is optimal. 3 Literature overview We briefly survey the literature on the redistribution problem, with particular emphasis given to the work of Walker and Otto [21]. Table 4: Communication grid for are indicated for a vector X of size of msg. Nbr of msg. 3.1 Message Generation Several papers have dealt with the problem of efficient code generation for an HPF array assignment statement like where both arrays A and B are distributed in a block-cyclic fashion on a linear processor grid. Some researchers (see Stichnoth et al.[17], van Reeuwijk et al.[19], and Wakatani and Wolfe [20]) have dealt principally with arrays distributed by using either a purely scattered or cyclic distribution (CYCLIC(1) in HPF) or a full block distribution (CYCLIC(d n is the array size and p the number of processors). Recently, however, several algorithms have been published that handle general block-cyclic distributions. Sophisticated techniques involve finite-state machines (see Chatterjee et al. [3]), set-theoretic methods (see Gupta et al. [8]), Diophantine equations (see Kennedy et al. [11, 12]), Hermite forms and lattices (see Thirumalai and Ramanujam [18]), or linear programming (see Ancourt et al. [1]). A comparative survey of these algorithms can be found in Wang et al. [22], where it is reported that the most powerful algorithms can handle block-cyclic distributions as efficiently as the simpler case of pure cyclic or full-block mapping. At the end of the message generation phase, each processor has computed several different messages (usually stored in temporary buffers). These messages must be sent to a set of receiving processors, as the examples of Section 2 illustrate. Symmetrically, each processor computes the number and length of the messages it has to receive and therefore can allocate the corresponding memory space. To summarize, when the message generation phase is completed, each processor Table 5: Communication steps for Communication steps for 9 g a Table Communication costs for Communication costs for Step a b c d e f Cost has prepared a message for all those processors to which it must send data, and each processor possesses all the information regarding the messages it will receive (number, length, and origin). 3.2 Communication Scheduling Little attention has been paid to the scheduling of the communications induced by the redistribution operation. Simple strategies have been advocated. For instance, Kalns and Ni [10] view the communications as a total exchange between all processors and do not further specify the operation. In their comparative survey, Wang et al. [22] use the following template for executing an array assignment statement: 1. Generate message tables, and post all receives in advance to minimize operating systems overhead 2. Pack all communication buffers 3. Carry out barrier synchronization Table 7: Communication grid for 5. Message lengths are indicated for a vector X of size 14 Nbr of msg. Nbr of msg. 6 9 6 6 9 6 6 9 6 6 9 6 6 9 9 4. Send all buffers 5. Wait for all messages to arrive 6. Unpack all buffers Although the communication phase is described more precisely, note that there is no explicit scheduling: all messages are sent simultaneously by using an asynchronous communication pro- tocol. This approach induces a tremendous requirement in terms of buffering space, and deadlock may well happen when redistributing large arrays. The ScaLAPACK library [4] provides a set of routines to perform array redistribution. As described by Prylli and Tourancheau [15], a total exchange is organized between processors, which are arranged as a (virtual) caterpillar. The total exchange is implemented as a succession of steps. At each step, processors are arranged into pairs that perform a send/receive operation. Then the caterpillar is shifted so that new exchange pairs are formed. Again, even though special care is taken in implementing the total exchange, no attempt is made to exploit the fact that some processor pairs may not need to communicate. The first paper devoted to scheduling the communications induced by a redistribution is that of Walker and Otto [21]. They review two main possibilities for implementing the communications induced by a redistribution operation: Wildcarded nonblocking receives Similar to the strategy of Wang et al. described above, this asynchronous strategy is simple to implement but requires buffering for all the messages to be received (hence, the total amount of buffering is as high as the total volume of data to be redistributed). Table 8: Communication steps for Communication steps for Table 9: Communication costs for Communication costs for Step a b c d e f g h i j Total Cost Synchronous schedules A synchronized algorithm involves communication phases or steps. At each step, each participating processor posts a receive, sends data, and then waits for the completion of the receive. But several factors can lead to performance degradation. For instance, some processors may have to wait for others before they can receive any data. Or hot spots can arise if several processors attempt to send messages to the same processor at the same step. To avoid these drawbacks, Walker and Otto propose to schedule messages so that, at each step, each processor sends no more than one message and receives no more than one message. This strategy leads to a synchronized algorithm that is as efficient as the asynchronous version, as demonstrated by experiments (written in MPI [16]) on the IBM SP-1 and Intel Paragon, while requiring much less buffering space. Walker and Otto [21] provide synchronous schedules only for some special instances of the redistribution problem, namely, to change the array distribution from CYCLIC(r) on a P -processor linear grid to CYCLIC(Kr) on a grid of same size. Their main result is to provide a schedule composed of K steps. At each step, all processors send and receive exactly one message. If K is smaller than P , the size of the grid, there is a dramatic improvement over a traditional all-to-all implementation. Table 10: Communication grid for Message lengths are indicated for a vector X of size msg. Nbr of msg. 2 4 Our aim in this article is to extend Walker and Otto's work in order to solve the general re-distribution problem, that is, moving from a CYCLIC(r) distribution on a P -processor grid to a CYCLIC(s) distribution on a Q-processor grid. We retain their original idea: schedule the communications into steps. At each step, each participating processor neither sends nor receives more than one message, to avoid hot spots and resource contentions. As explained in [21], this strategy is well suited to current parallel architectures. In Section 4.3.1, we give a precise framework to model the cost of a redistribution. 4 Main Results 4.1 Problem Formulation Consider an array X[0:::M \Gamma 1] of size M that is distributed according to a block-cyclic distribution CYCLIC(r) onto a linear grid of P processors (numbered from Our goal is to redistribute X by using a CYCLIC(s) distribution on Q processors (numbered from 1). Equivalently, we perform the HPF assignment is CYCLIC(r) on a -processor grid, while Y is CYCLIC(s) on a Q-processor grid 1 . The block-cyclic data distribution maps the global index i of vector X (i.e., element X[i]) onto a processor index p, a block index l, and an item index x, local to the block (with all indices starting at 0). The mapping i \Gamma! (p; l; x) may be written as bi=rc We derive the relation 1 The more general assignment Y [a can be dealt with similarly. Table 11: Communication steps for Communication steps for Table 12: Communication costs for Communication costs for Step a b c d Total Cost Similarly, since Y is distributed CYCLIC(s) on a Q-processor grid, its global index j is mapped as y. We then get the redistribution equation Qs) be the least common multiple of P r and Qs. Elements i and L+i of X are initially distributed onto the same processor (because L is a multiple of P r, hence r divides L, and P divides L \Xi r). For a similar reason, these two elements will be redistributed onto the same processor In other words, the redistribution pattern repeats after each slice of L elements. Therefore, we restrict the discussion to a vector X of length L in the following. Let rQs). The bounds in equation (3) become s: Given the distribution parameters r and s, and the grid parameters P and Q, the redistribution problem is to determine all the messages to be exchanged, that is, to find all values of p and q such that the redistribution equation (3) has a solution in the unknowns l, m, x, and y, subject to the bounds in Equation (4). Computing the number of solutions for a given processor pair (p; q) will give the length of the message. We start with a simple lemma that leads to a handy simplification: Lemma 1 We can assume that r and s are relatively prime, that is, Proof The redistribution equation (3) can be expressed as Equation (3) can be expressed as If it has a solution for a given processor pair (p; q), then \Delta divides z, z = \Deltaz 0 , and we deduce a solution for the redistribution problem with r 0 , s 0 , P , and Q. Let us illustrate this simplification on one of our motivating examples: Back to Example 3 Note that we need to scale message lengths to move from a redistribution operation where r and s are relatively prime to one where they are not. Let us return to Example 3 and assume for a while that we know how to build the communication grid in Table 7. To deduce the communication grid for say, we keep the same messages, but we scale all lengths by This process makes sense because the new size of a vector slice is \DeltaL rather than L. See Table 13 for the resulting communication grid. Of course, the scheduling of the communications will remain the same as with while the cost in Table 9 will be multiplied by \Delta. 4.2 Communication Pattern Consider a redistribution with parameters r, s, P , and Q, and assume that Qs). The communication pattern induced by the redistribution operation is a complete all-to-all operation if and only if Proof We rewrite Equation (5) as ps \Gamma because P r:l \Gamma Qs:m is an arbitrary multiple of g. Since z lies in the interval [1 \Gamma whose length is r guaranteed that a multiple of g can be found within this interval if Conversely, assume that g - r s: we will exhibit a processor pair (p; q) exchanging no message. Indeed, is the desired processor pair. To see this, note that pr \Gamma (because g divides P r); hence, no multiple of g can be added to pr \Gamma qs so that it lies in the interval [1 \Gamma Therefore, no message will be sent from p to q during the redistribution. 2 In the following, our aim is to characterize the pairs of processors that need to communicate during the redistribution operation (in the case s). Consider the following function 2 For another proof, see Petitet [14]. Table 13: Communications for are indicated for a vector X of size 14 Nbr of msg. Nbr of msg. 6 9 6 6 9 6 6 9 6 6 9 6 6 9 9 Function f maps each processor pair (p; q) onto the congruence class of pr \Gamma qs modulo g. According to the proof of Lemma 2, p sends a message to q if and only if f(p; (modg). Let us illustrate this process by using one of our motivating examples. Back to Example 4 In this example, We have (as in the proof of Lemma 2). If receives no message from p. But if does receive a message (see Table 10 to check this). To characterize classes, we introduce integers u and v such that r \Theta (the extended Euclid algorithm provides such numbers for relatively prime r and s). We have the following result. Proposition 1 Assume that r ' u mod g: Proof First, to see that PQ indeed is an integer, note that Since g divides both P r and Qs, it divides PQ. Two different classes are disjoint (by definition). It turns out that all classes have the same number of elements. To see this, note that for all k 2 [0; integer d 0 , and Since there are g classes, we deduce that the number of elements in each class is PQ . Next, we see that (p - ; q -s mod (because Finally, (p both P r and Qs divide divides )rs. We deduce that PQ divides hence all the processors pairs (p are distinct. We have thus enumerated class(0). Definition 3 Consider a redistribution with parameters r, s, P , and Q, and assume that 1. Let length(p; q) be the length of the message sent by processor p to processor q to redistribute a single slice vector X of size As we said earlier, the communication pattern repeats for each slice, and the value reported in the communication grid tables of Section 2 are for a single slice; that is, they are equal to length(p; q). are interesting because they represent homogeneous communications: all processor pairs in a given class exchange a message of same length. Proposition 2 Assume that Qs) be the length of the vector X to be redistributed. Let vol(k) be the piecewise function given by Figure 1 for k 2 [1 \Gamma (recall that if (p; q) 2 class(k) where sends no message to q). vol(k s volr Figure 1: The piecewise linear function vol. Proof We simply count the number of solutions to the redistribution equation pr easily derive the piecewise linear vol function represented in Figure 1. We now know how to build the communication tables in Section 2. We still have to derive a schedule, that is, a way to organize the communications as efficiently as possible. 4.3 Communication Schedule 4.3.1 Communication Model According to the previous discussion, we concentrate on schedules that are composed of several successive steps. At each step, each sender should send no more than one message; symmetrically, each receiver should receive no more than one message. We give a formal definition of a schedule as follows. Definition 4 Consider a redistribution with parameters r, s, P , and Q. ffl The communication grid is a P \Theta Q table with a nonzero entry length(p; q) in position (p; q) if and only if p has to send a message to q. ffl A communication step is a collection of pairs t, and length(p t. A communication step is complete if senders or all receivers are active) and is incomplete otherwise. The cost of a communication step is the maximum value of its entries, in other words, maxflength(p ffl A schedule is a succession of communication steps such that each nonzero entry in the communication grid appears in one and only one of the steps. The cost of a schedule may be evaluated in two ways: 1. the number of steps NS, which is simply the number of communication steps in the schedule; or 2. the total cost TC, which is the sum of the cost of each communication step (as defined above). The communication grid, as illustrated in the tables of Section 2, summarizes the length of the required communications for a single slice vector, that is, a vector of size Qs). The motivation for evaluating schedules via their number of steps or via their total cost is as follows: ffl The number of steps NS is the number of synchronizations required to implement the sched- ule. If we roughly estimate each communication step involving all processors (a permutation) as a measure unit, the number of steps is the good evaluation of the cost of the redistribution. ffl We may try to be more precise. At each step, several messages of different lengths are exchanged. The duration of a step is likely to be related to the longest length of these messages. A simple model would state that the cost of a step is ff where ff is a start-up time and - the inverse of the bandwidth on a physical communication link. Although this expression does not take hot spots and link contentions into account, it has proven useful on a variety of machines [4, 6]. The cost of a redistribution, according to this formula, is the affine expression ff \Theta NS with motivates our interest in both the number of steps and the total cost. 4.3.2 A Simple Case There is a very simple characterization of processor pairs in each class, in the special case where r and Q, as well as s and P , are relatively prime. Proposition 3 Assume that respectively denote the inverses of s and r modulo g). Proof relatively prime with Qs, hence with g. Therefore the inverse of r modulo g is well defined (and can be computed by using the extended Euclid algorithm applied to r and g). Similarly, the inverse of s modulo g is well defined, too. The condition easily translates into the conditions of the proposition. In this simple case, we have a very nice solution to our scheduling problem. Assume first that 1. Then we simply schedule communications class by class. Each class is composed of PQ processor pairs that are equally distributed on each row and column of the communication grid: in each class, there are exactly Q sending processors per row, and P receiving processors per column. This is a direct consequence of Proposition 3. Note that g does divide P and Q: under the hypothesis gcd(r; To schedule a class, we want each processor g, to send a message to each processor (or equivalently, if we look at the receiving side). In other words, the processor in position p 0 within each block of g elements must send a message to the processor in position q 0 within each block of g elements. This can be done in max(P;Q) complete steps of messages. For instance, if there are five blocks of senders three blocks of receivers blocks of senders send messages to 3 blocks of receivers. We can use any algorithm for generating the block permutation; the ordering of the communications between blocks is irrelevant. If we have an all-to-all communication scheme, as illustrated in Example 2, but our scheduling by classes leads to an algorithm where all messages have the same length at a given step. If 1. In this case we simply regroup classes that are equivalent modulo g and proceed as before. We summarize the discussion by the following result Proposition 4 Assume that scheduling each class successively leads to an optimal communication scheme, in terms of both the number of steps and the total cost. Proof Assume without loss of generality that P - Q. According to the previous discussion, if (the number of classes) times P (the number of steps for each class) communication steps. At each step we schedule messages of the same class k, hence of same length vol(k). If times P communication steps, each composed of messages of the same length (namely, processing a given class k 2 [0; Remark 1 Walker and Otto [21] deal with a redistribution with We have shown that going from r to Kr can be simplified to going from the technique described in this section enables us to retrieve the results of [21]. 4.4 The General Case When gcd(s; P entries of the communication grid may not be evenly distributed on the rows (senders). Similarly, when entries of the communication grid may not be evenly distributed on the columns (receivers). Back to Example 3 We have 5. We see in Table 7 that some rows of the communication grid have 5 nonzero entries (messages), while other rows have 10. Similarly, hence r 3. Some columns of the communication grid have 6 nonzero entries, while other columns have 10. Our first goal is to determine the maximum number of nonzero entries in a row or a column of the communication grid. We start by analyzing the distribution of each class. , and in any class class(k), k 2 [0; 1], the processors pairs are distributed as follows: ffl There are P 0 entries per column in Q 0 columns of the grid, and none in the remaining columns. ffl There are Q 0 entries per row in P 0 rows of the grid, and none in the remaining rows. Proof First let us check that Since r" is relatively prime with Q 0 (by definition of r 0 ) and with s" (because have There are PQ elements per class. Since all classes are obtained by a translation of class(0), we can restrict ourselves to discussing the distribution of elements in this class. The formula in Lemma 1 states that r mod . But -s mod P can take only those values that are multiple of s 0 and -r mod Q can take only those values that are multiple of r 0 , hence the result. To check the total number of elements, note that Let us illustrate Lemma 3 with one of our motivating examples. Back to Example 3 Elements of each class should be located on P 0 columns of the processor grid. Let us check class(1) for instance. Indeed we have the following. Lemma 3 shows that we cannot use a schedule based on classes: considering each class separately would lead to incomplete communication steps. Rather, we should build up communication steps by mixing elements of several classes, in order to use all available processors. The maximum number of elements in a row or column of the communication grid is an obvious lower bound for the number of steps of any schedule, because each processor cannot send (or receive) more than one message at any communication step. Proposition 5 Assume that (otherwise the communication grid is full). If we use the notation of Lemma 3, 1. the maximum number mR of elements in a row of the communication grid is d and 2. the maximum number mC of elements in a column of the communication grid is d e: Proof According to Lemma 1, two elements of class(k) and class(k are on the same row of the communication grid if -s in the interval [0; PQ Necessarily, s 0 , which divides P and is relatively prime with u. A fortiori s 0 is relatively prime with u. Therefore s 0 divides share the same rows of the processor grid if they are congruent modulo s 0 . This induces a partition on classes. Since there are exactly Q 0 elements per row in each class, and since the number of classes congruent to the same value modulo s 0 is either b r+s\Gamma1 c or d r+s\Gamma1 e, we deduce the value of mR . The value of mC is obtained similarly. It turns out that the lower bound for the number of steps given by Lemma 5 can indeed be achieved. Theorem 1 Assume that (otherwise the communication grid is full), and use the notation of Lemma 3 and Lemma 5. The optimal number of steps NS opt for any schedule is Proof We already know that the number of steps NS of any schedule is greater than or equal to g. We give a constructive proof that this bound is tight: we derive a schedule whose number of steps is maxfmR ; mC g. To do so, we borrow some material from graph theory. We view the communication grid as a graph is the set of sending processors, and is the set of receiving processors; and only if the entry (p; q) in the communication grid is nonzero. G is a bipartite graph (all edges link a vertex in P to a vertex in Q). The degree of G, defined as the maximum degree of its vertices, is g. According to K-onig's edge coloring theorem, the edge coloring number of a bipartite graph is equal to its degree (see [7, vol. 2, p.1666] or Berge [2, p. 238]). This means that the edges of a bipartite graph can be partitioned in d G disjoint edge matchings. A constructive proof is as follows: repeatedly extract from E a maximum matching that saturates all maximum degree nodes. At each iteration, the existence of such a maximum matching is guaranteed (see Berge [2, p. 130]). To define the schedule, we simply let the matchings at each iteration represent the communication steps. Remark 2 The proof of Theorem 1 gives a bound for the complexity of determining the optimal number of steps. The best known maximum matching algorithm for bipartite graphs is due to Hopcroft and Karp [9] and has cost O(jV j 5 there are at most max(P; Q) iterations to construct the schedule, we have a procedure in O((jP j 2 to construct a schedule whose number of steps is minimal. 4.5 Schedule Implementation Our goal is twofold when designing a schedule: ffl minimize the number of steps of the schedule, and ffl minimize the total cost of the schedule. We have already explained how to view the communication grid as a bipartite graph E). More accurately, we view it as an edge-weighted bipartite graph: the edge of each edge (p; q) is the length length(p; q) of the message sent by processor p to processor q. We adopt the following two strategies: stepwise If we specify the number of steps, we have to choose at each iteration a maximum matching that saturates all nodes of maximum degree. Since we are free to select any of such matchings, a natural idea is to select among all such matchings one of maximum weight (the weight of a matching is defined as the sum of the weight of its edges). greedy If we specify the total cost, we can adopt a greedy heuristic that selects a maximum weighted matching at each step. We might end up with a schedule having more than NS opt steps but whose total cost is less. To implement both approaches, we rely on a linear programming framework (see [7, chapter 30]). Let A be the jV j \Theta jEj incidence matrix of G, where ae 1 if edge j is incident to vertex i Since G is bipartite, A is totally unimodular (each square submatrix of A has determinant 0, 1 or \Gamma1). The matching polytope of G is the set of all vectors x 2 Q jEj such that ae (intuitively, is selected in the matching). Because the polyhedron determined by Equation 7 is integral, we can rewrite it as the set of all vectors x 2 Q jEj such that To find a maximum weighted matching, we look for x such that where c 2 N jEj is the weight vector. If we choose the greedy strategy, we simply repeat the search for a maximum weighted matching until all communications are done. If we choose the stepwise strategy, we have to ensure that, at each iteration, all vertices of maximum degree are saturated. This task is not difficult: for each vertex v of maximum degree in position i, we replace the constraint (Ax) translates into Y t is the number of maximum degree vertices and Y 2 f0; 1g jV j whose entry in position i is 1 iff the ith vertex is of maximum degree. We note that in either case we have a polynomial method. Because the matching polyhedron is integral, we solve a rational linear problem but are guaranteed to find integer solutions. To see the fact that the greedy strategy can be better than the stepwise strategy in terms of total cost, consider the following example. Example 5 Consider a redistribution problem with 3. The communication grid is given in Table 14. The stepwise strategy is illustrated in Table 15: the number of steps is equal to 10, which is optimal, but the total cost is 20 (see Table 16). The greedy strategy requires more steps, namely, 12 (see Table 17), but its total cost is only (see Table 18). Table 14: Communication grid for Message lengths are indicated for a vector X of size of msg. Nbr of msg. 4.5.1 Comparison with Walker and Otto's Strategy Walker and Otto [21] deal with a redistribution where We know that going from r to Kr can be simplified to going from we apply the results of Section 4.3.2 (see Remark 1). In the general case (s are evenly distributed among the columns of the communication grid (because r 1), but not necessarily among the rows. However, all rows have the same total number of nonzero elements because s 0 divides In other words, the bipartite graph is regular. And since maximum matching is a perfect matching. Because messages have the same length: length(p; (p; q) in the communication grid. As a consequence, the stepwise strategy will lead to an optimal schedule, in terms of both the number of steps and the total cost. Note that NS opt = K under the hypotheses of Walker and Otto: using the notation of Lemma 5, we have We have d s Note that the same result applies when Because the graph is regular and all entries in the communication grid are equal, we have the following theorem, which extends Walker and Otto main result [21]. Table 15: Communication steps (stepwise strategy) for Stepwise strategy for Table Communication costs (stepwise strategy) for Stepwise strategy for Step a b c d e f g h i j Total Cost Proposition 6 Consider a redistribution problem with arbitrary P , Q and s). The schedule generated by the stepwise strategy is optimal, in terms of both the number of steps and the total cost. The strategy presented in this article makes it possible to directly handle a redistribution from an arbitrary CYCLIC(r) to an arbitrary CYCLIC(s). In contrast, the strategy advocated by Walker and Otto requires two redistributions: one from CYCLIC(r) to CYCLIC(lcm(r,s)) and a second one from CYCLIC(lcm(r,s)) to CYCLIC(s). 5 MPI Experiments This section presents results for runs on the Intel Paragon for the redistribution algorithm described in Section 4. Table 17: Communication steps (greedy strategy) for Greedy strategy for Table Communication costs (greedy strategy) for Greedy strategy for Step a b c d e f g h i j k l Total Cost 5.1 Description Experiments have been executed on the Intel Paragon XP/S 5 computer with a C program calling routines from the MPI library. MPI is chosen for portability and reusability reasons. Schedules are composed of steps, and each step generates at most one send and/or one receive per processor. Hence we used only one-to-one communication primitives from MPI. Our main objective was a comparison of our new scheduling strategy against the current re-distribution algorithm of ScaLAPACK [15], namely, the "caterpillar" algorithm that was briefly summarized in Section 3.2. To run our scheduling algorithm, we proceed as follows: 1. Compute schedule steps using the results of Section 4. 2. Pack all the communication buffers. 3. Carry out barrier synchronization. 4. Start the timer. 5. Execute communications using our redistribution algorithm (resp. the caterpillar algorithm). 6. Stop the timer. 7. Unpack all buffers. The maximum of the timers is taken over all processors. We emphasize that we do not take the cost of message generation into account: we compare communication costs only. Instead of the caterpillar algorithm, we could have used the MPI ALLTOALLV communication primitive. It turns out that the caterpillar algorithm leads to better performance than the MPI ALLTOALLV for all our experiments (the difference is roughly 20% for short vectors and 5% for long vectors). We use the same physical processors for the input and the output processor grid. Results are not very sensitive to having the same grid or disjoint grids for senders and receivers. 5.2 Results Three experiments are presented below. The first two experiments use the schedule presented in Section 4.3.2, which is optimal in terms of both the number of steps NS and the total cost TC. The third experiment uses the schedule presented in Section 4.4, which is optimal only in terms of NS. Back to Example 1 The first experiment corresponds to Example 1, with 5. The redistribution schedule requires 7 steps (see Table 3). Since all messages have same length, the theoretical improvement over the caterpillar algorithm, which as 16 steps, is 7=16 - 0:44. Figure 2 shows that there is a significant difference between the two execution times. The theoretical ratio is obtained for very small vectors (e.g., of size 1200 double-precision reals). This result is not surprising because start-up times dominate the cost for small vectors. For larger vectors the ratio varies between 0:56 and 0:64. This is due to contention problems: our scheduler needs only 7 step, but each step generates 16 communications, whereas each of the 16 steps of the caterpillar algorithm generates fewer communications (between 6 and 8 per step), thereby generating less contention. Back to Example 2 The second experiment corresponds to Example 2, with Our redistribution schedule requires 16 steps, and its total cost is 6). The caterpillar algorithm requires 16 steps, too, but at each step at least one processor sends a message of length (proportional to) 7, hence a total cost of 112. The theoretical gain 77=112 - 0:69 is to be expected for very long vectors only (because of start-up times). We do not obtain anything better than 0:86, because of contentions. Experiments on an IBM SP2 or on a Network of Workstations would most likely lead to more favorable ratios. Back to Example 4 The third experiment corresponds to Example 4, with experiment is similar to the first one in that our redistribution schedule requires much fewer steps than does the caterpillar (12). There are two differences, however: P 6= Q, and our algorithm is not guaranteed to be optimal in terms of total cost. Instead of obtaining the theoretical ratio of 4=12 - 0:33, we obtain results close to 0:6. To explain this, we need to take a closer look at the caterpillar algorithm. As shown in Table 19, 6 of the 12 steps of the caterpillar algorithm are indeed empty steps, and the theoretical ratio rather is 4=6 - 0:66. Global size of redistributed vector (64-bit double precision)500015000 Microseconds caterpillar optimal scheduling Figure 2: Comparing redistribution times on the Intel Paragon for Table 19: Communication costs for with the caterpillar schedule. Caterpillar for Step a b c d e f g h i j k l Total Cost 6 Conclusion In this article, we have extended Walker and Otto's work in order to solve the general redistribution problem, that is, moving from a CYCLIC(r) distribution on a P -processor grid to a CYCLIC(s) distribution on a Q-processor grid. For any values of the redistribution parameters P , Q, r, and s, we have constructed a schedule whose number of steps is optimal. Such a schedule has been shown optimal in terms of total cost for some particular instances of the redistribution problem (that include Walker and Otto's work). Future work will be devoted to finding a schedule that is optimal in terms of both the number of steps and the total cost for arbitrary values of the redistribution problem. Since this problem seems very difficult (it may prove NP-complete), another perspective is to further explore the use of heuristics like the greedy algorithm that we have introduced, and to assess their performances. We have run a few experiments, and these generated optimistic results. One of the next releases of the ScaLAPACK library may well include the redistribution algorithm presented in this article. Global size of redistributed vector (64-bit double precision)40008000Microseconds caterpillar optimal scheduling Figure 3: Time measurement for caterpillar and greedy schedule for different vector sizes, redistributed from --R A linear algebra framework for static HPF code distribution. Graphes et hypergraphes. Generating local addresses and communication sets for data-parallel programs A portable linear algebra library for distributed memory computers - design issues and performance Software libraries for linear algebra computations on high performance computers. Matrix computations. Handbook of combinatorics. Compiling array expressions for efficient execution on distributed-memory machines Processor mapping techniques towards efficient data redistribution. Efficient address generation for block-cyclic distributions A linear-time algorithm for computing the memory access sequence in data-parallel programs Steele Jr. Algorithmic redistribution methods for block cyclic decompositions. Efficient block-cyclic data redistribution MPI the complete reference. Generating communication for array state- ments: design Fast address sequence generation for data-parallel programs using integer lattices An implementation framework for HPF distributed arrays on message-passing parallel computer systems Redistribution of block-cyclic data distributions using MPI Redistribution of block-cyclic data distributions using MPI Runtime performance of parallel array assignment: an empirical study. --TR --CTR Prashanth B. Bhat , Viktor K. Prasanna , C. S. Raghavendra, Block-cyclic redistribution over heterogeneous networks, Cluster Computing, v.3 n.1, p.25-34, 2000 Stavros Souravlas , Manos Roumeliotis, A pipeline technique for dynamic data transfer on a multiprocessor grid, International Journal of Parallel Programming, v.32 n.5, p.361-388, October 2004 Ching-Hsien Hsu , Shih-Chang Chen , Chao-Yang Lan, Scheduling contention-free irregular redistributions in parallelizing compilers, The Journal of Supercomputing, v.40 n.3, p.229-247, June 2007 Hyun-Gyoo Yook , Myong-Soon Park, Scheduling GEN_BLOCK Array Redistribution, The Journal of Supercomputing, v.22 n.3, p.251-267, July 2002 Ching-Hsien Hsu, Sparse Matrix Block-Cyclic Realignment on Distributed Memory Machines, The Journal of Supercomputing, v.33 n.3, p.175-196, September 2005 Minyi Guo , Yi Pan, Improving communication scheduling for array redistribution, Journal of Parallel and Distributed Computing, v.65 n.5, p.553-563, May 2005 Minyi Guo , Ikuo Nakata, A Framework for Efficient Data Redistribution on Distributed Memory Multicomputers, The Journal of Supercomputing, v.20 n.3, p.243-265, November 2001 Neungsoo Park , Viktor K. Prasanna , Cauligi S. Raghavendra, Efficient Algorithms for Block-Cyclic Array Redistribution Between Processor Sets, IEEE Transactions on Parallel and Distributed Systems, v.10 n.12, p.1217-1240, December 1999 Ching-Hsien Hsu , Yeh-Ching Chung , Don-Lin Yang , Chyi-Ren Dow, A Generalized Processor Mapping Technique for Array Redistribution, IEEE Transactions on Parallel and Distributed Systems, v.12 n.7, p.743-757, July 2001 Ching-Hsien Hsu , Yeh-Ching Chung , Chyi-Ren Dow, Efficient Methods for Multi-Dimensional Array Redistribution, The Journal of Supercomputing, v.17 n.1, p.23-46, Aug. 2000 Saeri Lee , Hyun-Gyoo Yook , Mi-Soo Koo , Myong-Soon Park, Processor reordering algorithms toward efficient GEN_BLOCK redistribution, Proceedings of the 2001 ACM symposium on Applied computing, p.539-543, March 2001, Las Vegas, Nevada, United States Ching-Hsien Hsu , Kun-Ming Yu, A Compressed Diagonals Remapping Technique for Dynamic Data Redistribution on Banded Sparse Matrix, The Journal of Supercomputing, v.29 n.2, p.125-143, August 2004 Emmanuel Jeannot , Frdric Wagner, Scheduling Messages For Data Redistribution: An Experimental Study, International Journal of High Performance Computing Applications, v.20 n.4, p.443-454, November 2006 PeiZong Lee , Wen-Yao Chen, Generating communication sets of array assignment statements for block-cyclic distribution on distributed memory parallel computers, Parallel Computing, v.28 n.9, p.1329-1368, September 2002 Antoine P. Petitet , Jack J. Dongarra, Algorithmic Redistribution Methods for Block-Cyclic Decompositions, IEEE Transactions on Parallel and Distributed Systems, v.10 n.12, p.1201-1216, December 1999 Jih-Woei Huang , Chih-Ping Chu, An Efficient Communication Scheduling Method for the Processor Mapping Technique Applied Data Redistribution, The Journal of Supercomputing, v.37 n.3, p.297-318, September 2006

block-cyclic distribution;MPI;distributed arrays;scheduling;HPF;redistribution

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Timestep Acceleration of Waveform Relaxation.

Dynamic iteration methods for treating certain classes of linear systems of differential equations are considered. It is shown that the discretized Picard--Lindelf (waveform relaxation) iteration can be accelerated by solving the defect equations with a larger timestep, or by using a recursive procedure based on a succession of increasing timesteps. A discussion of convergence is presented, including analysis of a discrete smoothing property maintained by symmetric multistep methods applied to linear wave equations. Numerical experiments indicate that the method can speed convergence.

Introduction . Much of modern chemical and physical research relies on the numerical solution of various wave equations. Since these problems are extremely demanding of both storage and cpu-time, new numerical methods and fast algorithms are needed to make optimal use of advanced computers. The dynamic iteration or waveform relaxation (WR) method [9, 11] is an iterative decoupling scheme for ordinary differential equations which can facilitate concurrent processing of large ODE systems for applications such as VLSI circuit simulation [6, 15] and partial differential equations [1, 4]. In this article, accelerated dynamic iteration schemes are used to solve systems of linear differential equations, with emphasis on the ordinary differential equations arising from discretization of linear wave equations. Although our experiments use finite differences for the spatial derivatives, other spatial discretizations could be used. For time discretization, we use symmetric multistep methods, although other choices may also be appropriate. As is the case for stationary iterative methods applied to spatially discretized elliptic PDEs, it is found that finer fixed step (time) discretizations slow the convergence of the WR iteration, while large timesteps can be used to resolve the slow modes. The idea that is explored here is to use a coarse timestep on the defect equations to speed up convergence of the fine grid iteration. Nevanlinna already pointed out [14, 13] that for general applications of WR it makes sense from Department of Mathematics, University of Kansas, Lawrence, KS 66045, leimkuhl@math.ukans.edu. This work was supported by NSF grant DMS-9303223 an efficiency standpoint to use coarser discretization in the early sweeps (when the iteration error is large), and then incrementally to refine the time discretization near convergence. Our point of view is rather to vary the timestep to resolve different modes present in the solution, using two time stepsizes (or multiple stepsizes). The current article is related to recent work of Horton and Vandewalle [4] and Horton, Vandewalle and Worley [5] which considered space-time multigrid methods for solving parabolic equations. The new scheme will be referred to as timestep acceleration since it relies on adjustment of the integration timestep to accelerate the dynamic iteration. This approach shares some features of multigrid methods. For the convenience of the reader generally familiar with multigrid methods, we outline the algorithm in the abstract setting of solving an unspecified dynamical system as follows: Accelerated Waveform Relaxation. Given: fine timestep h and an approximation u 0 to the solution with fixed stepsize h. 1. Smoothing Starting from u 0 , perform a fixed number of iterations of a smoothing waveform relaxation iteration with timestep h. 2. Correction Compute the defect (residual) in this solution on the fine time mesh. If the timestep sufficiently large, solve the discrete defect equation restricted to the coarse time mesh directly (i.e. without relaxation). Else recursively apply some number of iterations of the algorithm using stepsize H to the defect equation restricted to the coarse time mesh. Next, correct the solution after prolonging onto the fine time mesh. 3. Smoothing Apply a fixed number of iterations of the fine stepsize smoothing iteration. (In x6, we present and analyze a more precisely defined version of this algorithm, TAWR.) A major barrier to efficient solution of large scale wave equations is the need for small timesteps. Due to the sequential character of standard ODE methods, this effectively reduces the potential for parallel speedups. Compared to standard timestepping schemes, the method discussed here directly addresses this problem by enabling the use of larger timesteps to recover at least a portion of the dynamics. Another important obstacle to computation-particularly in the case of high dimensional problems-is the necessary storage. The new method actually exacerbates this problem since solution information at many points must be stored. However, in waveform relaxation based on a block splitting, the storage is naturally segmented according to the decoupling, so the scheme may be appropriate for a parallel computer based on a distributed memory architecture. Although standard analytical results for multigrid methods or coarse-grid acceleration are typically developed for finite dimensional Hermitian positive definite problems, these can be relaxed to give at least partial convergence results. In fact proving theoretical convergence for timestep acceleration is easier than for standard multigrid due to the strong smoothing properties of the Picard-Lindel-of operator (it is a contraction on small intervals). Analysis of the behavior of the iteration on special linear model problems is also possible and is briefly discussed here. The scheme is found to work well in simple numerical experiments with linear wave equa- tions. Although our experiments are conducted in one space dimension, nothing in principle prevents application in higher dimensions (although many practical issues will need to be dealt with). 2. Waveform Relaxation. Consider a second order linear system of differential equation (1) where the eigenvalues of the matrix A are assumed to lie in the left half plane. A special case that we will frequently refer to is the 1-D wave equation U discretized with finite differences on the unit square with periodic boundary conditions: . 0 A is called the discrete Laplacian. Here is a vector of approximations at nodes x i , Another potential application is to the Schr-odinger equation. Discretizing, for example, with finite differences leads to d dt (2) where A is the discrete Laplacian. If v(x; t) is the potential energy function of the corresponding classical system, we have V In simplified settings v(x; t) is time-independent, hence so is V . The waveform relaxation method for (1) is based on a splitting results in ODE IVPs For example, we might choose A+ to be the diagonal of A (Jacobi splitting), a block-diagonal part of A (block-Jacobi splitting), the lower triangular part of A (Gauss-Seidel splitting), etc. Much work involving the discrete Laplacian in elliptic PDEs is based on Gauss-Seidel splitting in red-black ordering. Another useful splitting is the damped Jacobi splitting where with D the diagonal of A. The extreme case is called the Picard splitting. When referring to (1) and (3) we will generally limit discussion to the case where A and are symmetric negative semidefinite matrices. The WR iteration proceeds as follows: starting from a given initial waveform u (which may be constant), we solve (3) with as a forced linear system for u 1 over some time interval, say [0; T ]. (This interval is referred to as the window). The function u 1 then yields a forcing for the next iteration or sweep, and the process repeats. In practice, the systems are solved numerically over the entire interval, and the storage of the resulting discrete approximation is an important drawback of the method which may place severe limitations on the size of the time window. On the other hand, we gain in two ways: first, the systems we solve at each iteration can be decoupled into problems of reduced dimension, and second, the decoupled problems can often be solved on separate processors of a parallel computer. An alternative approach would be based on solving the linear equations that result at each step of a standard discretization using a parallel algorithm, however, depending on the computer architecture employed, the flexibility in the choice of window size may reduce the overall communication cost, e.g. by eliminating some of the time spent in initializing the transfer of data between processors. Preliminary convergence results for WR appear in the paper by Lelarasmee et al [9]. Miekkala and Nevanlinna [11, 12] and Nevanlinna [13] have developed an extensive theory for studying waveform relaxation for linear systems. Lubich and Osterman proposed to combine the WR method with spatial multigrid schemes [10]. Recent work by Horton and Vandewalle [4] and by Horton, Vandewalle and Worley [5] has shown that a careful implementation of (spa- tial) multigrid-WR methods for parabolic PDEs can provide excellent parallel speedups. The use of waveform relaxation for solving hyperbolic partial differential equations and relations to domain decomposition were explored by Bj-rhus [1]. 3. Mathematical Background. In this section we state some elementary results concerning the iteration (3). The reader is directed to the papers of Nevanlinna and Miekkala for basic theory. The waveform relaxation method for (1) can be viewed as an iteration u As shown in [11], implying superlinear convergence. On the other hand, for stiff dissipative linear systems, it makes sense to allow T ! 1 in which case meaningful spectral information is obtained [11]. Since the solution to the equations (1) and (2) does not in general lie in L 2 ([0; 1)), this approach must be modified. A reasonable practical approach is that taken in [13] where an exponential weighting function e \Gammafft is inserted into the usual L 2 norm. For ff ? 0, the space L 2 ff is normed by kuk ff := -Z 1je \Gammafft u(t)j 2 dt and ae ff refers to spectral radius in that space. If we take the Laplace transform of (3), we obtain: u(0). The following results are proved by Nevanlinna and Miekkala [11]: ae ff Rez-ff which follows from the Paley-Wiener theorem (the second expression follows from a maximum principle after a suitable remapping of the domain) and which follows from Parseval's identity. We now provide some simple estimates for the response of the iteration operator in weighted 2-norm. First, consider the behavior of the solution operator of (1) in the weighted space, where d=dt. Examining the spectral radius of the normal matrix L(z) along the line Rez = ff, we find the eigenvalues are: are the eigenvalues of A. Hence By maximizing these functions over y, we can compute the moduli of the eigenvalues of solution operator in the weighted space. Theorem 3.1. Define Then ae ff (L In particular eigenvalues near zero have the strongest influence. When A is the discrete Laplacian, or any symmetric negative semidefinite matrix with an eigenvalue at zero, we have ae We can use this to estimate the norm of the iteration matrix, since where the - are determined from Theorem 3.1 with - i the eigenvalues of A+ rather than A. Asymptotic estimates for the relation between spectral radius of S(z) and ff are given in [8]. Let us consider the wave equations with periodic boundary conditions on the square as a model problem. We will use damped Jacobi iteration with parameter ! 2 (0; 1]. In this case, A, are all diagonalized by the discrete Fourier transform, so we arrive readily at the eigenvalues e and In this case the spectral radius can be readily computed. We have \Deltax pp pp and ae ff We are interested in moderate weights ff which we define to mean ff ! ae(A). (Intuitively, this corresponds to looking in the time domain on intervals greater than the smallest period of the motion.) In the standard theory, one uses the value of ! in the damped Jacobi splitting to enhance a smoothing property: a damping in the iteration of the modes corresponding to larger eigenvalues. However, the important consideration for timestep acceleration is not the way in which the smoother acts on fast "spatial" modes but rather the response of the smoother to high frequency forcings. In fact, the real smoothing property we are interested in has to do with the shape of the graph of ae(S(ff + iy)) as a function of y. For example, when a damped Jacobi splitting is applied to solve the semidiscrete wave equation, we find that the spectral radius of S achieves its maximum on Rez = ff at the point (if ff ! (or at ). The maximum is typically achieved well away from For the Picard splitting, it is easy to see rather that the maximum occurs at In this case, we say that the iteration has a smoothing property with respect to high frequency forcings. It is not necessary to use a slowly converging splitting such as the Picard splitting to obtain a good smoothing property. A typical feature of a good splitting for this purpose is that A+ would have an eigenvalue at or near the origin. Thus the smoothing property of a block-Jacobi splitting of the discrete Laplacian would improve with the block size. To illustrate this smoothing concept, consider the time-dependent Schr-odinger equation (2). The iteration becomes d dt We could again use a Jacobi or damped Jacobi splitting, but in practice, a more useful choice might be or, more simply After time discretization, these choices will lead to equations at each timestep which can be efficiently solved by, for example, using a parallel implementation of the fast Fourier transform Still another possibility is to work directly in the Fourier coefficients. Let QAQ where and so that the equations become d dt \Psi: We can then apply a Jacobi splitting to this problem. One finds that the diagonal of QV (t)Q H is dI where The computation can be implemented efficiently using the FFT. The symbol of the WR iteration operator R for the Schr-odinger equation with V constant is and, trivially, ff Using the Laplacian+potential splitting, or one of its cousins can be expected to yield a good smoothing property with respect to high frequency forcings. 4. Discretization. In this section, we focus on (1) and apply a discrete transform as in [12] to analyze the symmetric multistep methods commonly used for integrating oscillatory problems. Multistep methods construct an approximating sequence fu n g to fu(t n )g at successive time points use fu k n g to refer to the numerical solution generated at the kth sweep of waveform relaxation. Symmetric multistep methods for - take the form: with sequences, for which ff These methods are used for integration of second order oscillatory problems. An important feature of this class of methods is their time-reversibility. Note that the multistep methods require k starting values. In discretizing dissipative problems it is sensible to replace the space L 2 by l 2 h with norm . For our investigations, we use the weighted space with norm je \Gammanhff u which can be viewed as a discretization of the L 2 ff norm. Following the usual practice we define operators a and b on sequences by We also use the symbols a and b to refer to the corresponding characteristic polynomials: To preserve the intuitive correspondence of results from the continuous-time to discrete worlds, define a discrete transform which takes fu n g to - u(z) by - tially a discretization of the Laplace transform, and equivalent to the discrete Laplace or i transform). Applying (5) to the linear problem (3) and computing the discrete transform, we find where and OE includes the effects due to the k starting values. We are going to assume that these starting values are exact (for the unsplit discrete problem) so they do not effect the convergence of the iteration. For example, if where In the general case, discrete versions of the Paley-Wiener theorem and Parseval's identity give (after modifying results in [12] to take into account the exponential weight): ae h;ff (S h Rez-ff ae(S h (z)) and In order for the discretized operator to be bounded, we evidently need to require for any - 2 We now consider an example. Ignoring rounding error, the popular leapfrog method for second order systems is equivalent to St-ormer's rule (also known as the Verlet method), a symmetric 2-step method with ff 1. Applying this scheme to the WR iteration for the linear problem and taking the discrete transform gives The function is an an O(h 2 z 4 ) approximation to z 2 . The poles of the transformed discrete iteration operator z =h cosh with - an eigenvalue of A+ . Explicit multistep scheme are always conditionally stable meaning that the stability of the schemes will depend on the stepsize being restricted roughly in inverse relation to the square root of the spectral radius of A+ . For the St-ormer method, the stability condition is that - ! 0 and \Gammah 2 - != 2, which is also the condition that the poles of S h remain on the imaginary axis. The function Im cosh(-) is monotone in the real variable - on [\Gamma1; 1], hence the ordering of the poles is preserved along the imaginary axis. Another popular second order method is the (implicit) trapezoidal rule which has transform 4.1. Decay of the discrete symbol. Theory due to Miekkala and Nevanlinna [12] compares the convergence of the discrete iteration in l 2 h to that of the continuous time iteration for dissipative problems and for methods that are not weakly stable. We need to modify this mechanism to cover convergence for stable methods for second order differential equations in the weighted spaces rather than L 2 and l 2 h . In what follows, it is assumed that the k starting values are held fixed as we iterate. These could also be obtained by some convergent process, but this does not seem a meaningful generalization. In the case of the discretized iteration we need to examine the images of segments ff . The situation for representative and in Figure 1 we see the images of this line for the trapezoidal and St-ormer discretizations, for various values of the stepsize. Putting real ff into each of the functions P t:r: h and z 2 one can show that for sufficiently small hff which means, somewhat surprisingly, that in the neighborhood of y = 0, the St-ormer discretization actually leads to a slightly more stable overall iteration than that generated by the trapezoidal rule. h=.2 h=.3 h=.4 h=.1 h=.2 Fig. 1. Approximation of z 2 by St-ormer (dashed lines) and trapezoidal rule (dotted lines) The situation for large h is more dramatic. For nonstiff problems with eigenvalues - very near the origin in the complex plane, large steps should be possible and one might suppose that the St-ormer and trapezoidal rule discretizations would behave similarly with respect to WR convergence. In fact, this is not the case and it turns out that the St-ormer method yields a much more stable WR iteration than the trapezoidal rule over comparable time intervals. Figure 2 shows the image of 1 h and z 2 . Figure also indicates that results such as Proposition 9 of [10] and Theorem 3.4 of [12] which bound the spectral radius of the discrete iteration in terms of that of the continuous iteration for A-stable multistep methods will not typically hold in our setting. The problem with generalizing the results of [12] is that they were based on the strengthened stability assumption that the stability region includes a disk on the negative real axis touching the origin, whereas many of the symmetric methods we consider (e.g. St-ormer's rule) do not satisfy this condition. We will use the exponential weight to correct for the weak stability of the method. For define the fl-stability region\Omega fl of the method as the set of all - 2 - C such that the roots lie in the disk jij - e fl and are simple on the boundary. The iteration operator S h is bounded in l 2 h;ff if 2.5 -2 -1.5 -113s.r. h=2 t.r. h=2 Fig. 2. Large time-step comparison of images of 1 Now observe that We can directly relate the spectral radius of the discrete iteration to that of the continuous time iteration. In fact, ae ff;h (S h Rez-ff ae Rez-ff ae s Let the notation bdyW be used to indicate the boundary of the set W . Since f r bdy\Omega hff , we have, analogous to a result in [12]. Theorem 4.1. Suppose int\Omega ffh , then ae ff;h (S h int\Omega ffh and @\Omega ffh g: Theorem 4.2. If the dynamic iteration converges in L 2 ff and the symmetric multistep method is irreducible and convergent then the discretized iteration converges in l 2 ff;h for sufficiently small h and ae ff;h (S h We will outline a proof of this result since the reasoning is somewhat different than that used in [12]. be the k zeros of a, with being the principle root counted with multiplicity two. For simplicity, assume that these zeros all lie on the unit circle S 1 and that they are ordered counterclockwise about the unit circle, thus fi (It would not be difficult to treat the case where some zeros lie inside the unit circle.) From consistency, we must have that fi double root, while all of the other roots are simple. We can view ja(e hff w)j 2 as a function of w on S 1 . For at the zeros of a; for h sufficiently small and ff ? 0, it has local minima located near the points . We can expand a in Taylor's series about the fi i to obtain Only multiple root of a, hence a 0 1. This means that, for e hff w in the vicinity of fi i , must have a(e hff must also hold at the local minimum. Using this, we can prove a small lemma which shows that the spectral radius is determined for small h by the approximation property of the principle root. Lemma 4.3. For h sufficiently small, ae(S ae(S and a similar result holds for kS h k. Proof: By symmetry, ' sufficiently small, the global minimum of ja(e hff e ihy )j 2 on I h must occur at one of its local minima over that interval or at the endpoints. Since b(e hz ) can be uniformly bounded in any bounded region, it is straightforward to see that the quantity - h (y) defined by s satisfies min and hence that ae(S Due to symmetry, the behavior of ae is the same on the intervals [' other words, we need only look in the latter subinterval for the maximizing value. 2. Given - consistency implies that 0-y-y Choose a large enough rectangular neighborhood N of the origin so that e.g. ae(S(z)) ! ae ff (S)=2 for z outside N . Now j- h (' 2 =h)j - h \Gamma1=2 , thus for h sufficiently small, the curve \Gamma h := f- h (y) : leaves N . After leaving N , it cannot reenter N (or j- h (y) 2 j would have another local minimum). Within N , the curve \Gamma h will approximate the line segment ff + iy to O(h). Thus for h sufficiently small, the maximum value of ae(S h (z)) will occur when - h (y) lies within N , and since this point lies within O(h) of ff iy we can see that asymptotically, the spectral radius of S h in the weighted space can differ by only O(h) from that of S. This concludes the proof of the theorem. 5. Aliasing effects. Consider the transformed discrete iterator S h on the vertical line ff R. The degree to which an eigenvalue \Gamma! 2 of A+ has an impact on the solution at frequency - depends inversely on the separation between P h (ff + i-) and \Gamma! 2 . Those frequencies - for which P h (ff lies far from the spectrum of A+ will be only weakly propagated by the iteration. For any multistep method, the function P h (ff + i-) is actually periodic in - with period 2-=h. This aliasing effect means that high frequencies can be excited with large stepsizes. Frequencies give the same response. Actually, the situation is even somewhat worse due to the symmetry about the real axis: the response to \Gamma- will be the same as the response to - 0 . Of course, if there are no frequencies present in the forcing function above say -=h then these anomalous excitations do no harm. We will illustrate with the wave equation example. We first look at the response of the discrete solution operator for the (unsplit) spatially discretized wave equation along the line 1 iy. The curves shown in Figure 3 show the spectral radii (hence also the norm) of versus y for :25. The maximum value is achieved near expected from Theorems 3.1 and 4.2. An increase in the stepsize h provides accuracy for small y while introducing some extraneous excitation at 2-k=h, k 2 Z. solid h=1 -. h=.5 . h=.25 Fig. 3. Spectral radius of L(1+iy), wave equation, N=32. We next examine the spectral radius of the transformed discrete iteration operator S(z) for the the Jacobi splitting of the spatially discretized wave equation along the line iy. The curves shown in Figure 4 show the spectral radii versus y for h in the progression As h decreases, the spectral radius has increasing maxima achieved at increasing values of y. solid h=1 -. h=.5 . h=.25 Fig. 4. Spectral radius of S(1+iy), wave equation, N=32. Using a large stepsize to resolve the small y response will not apparently improve the convergence of the small step Jacobi iteration, since the large stepsize solution operator does not even act on the high frequencies (where the spectral radius is large). The exception to this will be in the case that h is so small that the "coarse" grid is not coarse at all (in which case, little is gained through iteration). Moreover, unless ae(S h (z)) is small outside of an interval about the origin of length roughly 2-=H , the artifacts introduced at the high frequencies on the coarse grid will not be damped out. To see a substantial improvement, our iteration operator should be designed to achieve its maximum at or near As mentioned previously, for the wave equation, a natural (if slow) choice is standard Picard iteration. If we turn to the Schr-odinger equation and consider for example the splitting (4) for V constant. In case V is not too large, we would expect here that the maximum of ae(R(z)) is achieved near has an eigenvalue near the origin) and that substantial improvement may be possible by exploiting a coarse time step solution. 6. Timestep Acceleration of WR. The examples of the previous section suggest that an approach in which different time meshes are used at each sweep could be successful. The goal, as for standard multigrid is to iterate on successively coarser grids, thus resolving those components of the residual that are most difficult to obtain on the fine grid. We envision ultimately combining spatial multigrid with this timestep acceleration scheme. For the formulation and analysis of standard multigrid methods in the context of elliptic PDEs, the reader is referred to [2]. We will now define the steps of the algorithm described in the introduction. Let b and a represent the operators which define the discretization. We use h to represent the fine timestep, and H to represent the coarse timestep. Normally, in solving elliptic PDEs, we use In our case, this choice may or may not be appropriate; for the purposes of discussing an algorithm, we assume that the stepsize changes by a common factor at each iteration, but this is perhaps not essential in practice. Let I H h and I h prolongation operators, respectively, which act between the fine and coarse time-meshes, thus I H H;ff and I h h;ff . Note that whether we wish to solve problems with or without forcing, description for a forced problem permits an easy recursive definition. Algorithm TAWR(h). Given: fine timestep h, a sequence ff n g 2 l 2 h;ff , an approximation u 0 to the solution with fixed stepsize h, and a splitting , the following algorithm solves subject to k prescribed starting values ' i , 1. Small-timestep Pre-Smoothing. Starting from u 0 , perform - sweeps of WR iteration with timestep where u l+1 2. Large-timestep Correction. Compute the defect fd n g satisfying d and from the formula If the timestep sufficiently large, solve directly (i.e. without relaxation) using zeros for starting values. Else apply - iterations of TAWR(H) to (9), using zeros for starting values. Next, correct: 3. Small-timestep Post-Smoothing Apply - iterations of the fine mesh smoothing operation (8). Notes: ffl For this is the V-cycle, for - 2, it is called the W-cycle. ffl Different numbers of smoothing steps could be used in the pre- and post-smoothings. ffl To solve the problem using timestep acceleration, we first compute ff n g := )g. 7. Convergence Analysis. In this section we present an elementary general convergence result regarding two-grid acceleration. This result could be easily extended to the full timestep acceleration iteration. The iteration operator in the two stepsize case can be written as S - h where S h represents the smoothing sweep and C h;H represents the coarse-grid correction. In general it is enough to understand h . The operator C h;H can be where we have denoted the prolongation and restriction operators by p and r, respectively. On the other hand, It is enough to show that is O(h \Gamma2 ) in l 2 ff;h , while the norm of is O(h 2 OE(-)), where OE tends to zero as - ! 0. In fact we anticipate that the situation is often rather better than this result would indicate, but this approach allows us to state a quite general convergence result. Based on the relation R \Gamma1 h , the fact that b is a bounded operator, and the theorems of the last section, we have: Lemma 7.1. Suppose consistent, stable linear multistep is used and the restriction and prolongation operators are bounded operators. Then kM sufficiently small. The proof follows since (i) kL \Gamma1 O(h), (ii) the same thing holds for h replaced by H and qh, and (iii) the restriction and prolongation operators are bounded. 2. For the smoothing, we have We therefore have This converges to zero provided ae ff;h (S h Thus we can state: Theorem 7.2. If the undiscretized smoothing iteration is convergent (ae ff (S) ! 1), a consis- tent, stable linear multistep is used, the restriction and prolongation are bounded operators between l 2 h;ff and l 2 H;ff and enough smoothing iterations are performed, then the timestep-accelerated waveform relaxation algorithm converges. Because of the strong contractivity of the Picard operator on small time intervals, it would be straightforward to extend this result to the full multiple mesh recursive acceleration scheme. On the other hand, besides proving asymptotic convergence, this simplified approach provides no practical estimates of convergence. 7.1. Treatment of Model Problems. A key observation is that two modes are coupled via restriction. It is possible to write a formula for the "symbol" of the iteration operator as a matrix operating on the pair of coupled modes e hnz and e hn(z+i-) . As an example, taking the operators \Theta 111 (full weighting restriction) and linear interpolation), and assuming any symmetric multistep method (a; b), then we find the action of M on the pair of modes is given by where and (\Omega is the Kronecker product). Now for Jacobi or Picard iteration on the wave equation, for example, the matrix - M is easily reduced to a diagonal matrix of 2 \Theta 2 blocks, so the asymptotic convergence behavior can be determined relatively easily. For red-black Gauss-Seidel iteration on the square, we get a further pairing of the spatial modes, so - M actually is reduced to 4 \Theta 4 blocks. By studying the spectra of these blocks, various ODE discretizations could be compared for their effect on asymptotic rate of convergence, as could other choices of restriction/prolongation. Note that besides the restriction and prolongation having an adjoint relationship if we choose the second order, two-step discretization then we find that also rR h This appears to be the only consistent, stable two-step scheme for which this property holds with the given r and p. These resemble the conditions for "variational form", however the operator R and its discretization are not self-adjoint in our setting, so we do not have the space decomposition l 2 and the standard theoretical results cannot be directly applied. 8. Numerical Experiments. We performed experiments using the two-grid iteration on linear wave equations. We found that the performance improvements were sensitive to many factors, including timestep, time window length, and splitting. Unfortunately, we cannot expect to have complete flexibility in the choice of the time interval or "window" as this may be determined from a storage or communication limitation. Similarly, the timestep is typically chosen for accuracy reasons. Consider the standard 1D wave equation (1), N=16, using Jacobi iteration for the smooth- ing. We used the discretization (10) together with full weighting restriction and piecewise linear interpolation. We did not anticipate very good behavior since the smoothing property is relatively weak for this splitting (fast modes are not very strongly damped). Indeed, this is what we observed. For most values of the stepsize, the 2-grid acceleration improved convergence, but not by very much. (In some cases performance was even slightly degraded.) In each of the Figures the 2-norm of the error is graphed as a function of the sweep number s and the timestep n. In Figure 5 the error in Jacobi WR is indicated for stepsize shows the mild improvement in the error when a coarse grid correction is applied at each Jacobi WR sweep. We next examined a modified wave equation of the form where A is the discrete Laplacian and - is a scalar parameter. We used "Laplacian splitting" into A and -I . It is easy to see that this splitting possesses a strong "smoothing property". We first chose means that we had a substantial perturbation of timestep sweep error Fig. 5. Errors in Jacobi WR, h=.025, 40 steps without correction.103050515255001500timestep sweep error Fig. 6. Errors with coarse grid correction, h=.025, 40 steps, showing poor acceleration. The benefit of coarse grid correction is diminished by the poor smoothing property of the Jacobi smoother. the discrete Laplacian. Initial data excited the first two eigenfunctions of the Laplacian (slow modes), although this choice was not critical to the results we obtained. Twenty timesteps of size were used. In this case, the coarse grid corrections offer substantial improvement, as shown in Figure 7. The left figure shows the log 10 error versus timestep and sweep number; on the right we have shown the log 10 ratio of the errors with and without the two-grid acceleration. The improvement evidenced here is as much as a factor of 10 5 in 20 sweeps, or a little under a factor of 2 per sweep on average. The improvement is evident until the error reaches the level of roundoff. At the weaker perturbation the effect somewhat diminished (Figure 8). If we instead increased the strength of the applied field 100), the coarse grid correction continued to offer substantial acceleration; this is indicated in Figure 9. At larger or smaller timesteps, the improvement slightly diminished. A linear acceleration effect was observed on longer time intervals (Figure timestep iteration sweep log -5 timestep iteration sweep log error ratio Fig. 7. (a) log error and (b) log error ratio, steps . The splitting provides a strong smoothing property, and a substantial improvement is possible with the two-grid iteration.515020 timestep iteration sweep log error ratio515020 timestep iteration sweep log error Fig. 8. (a) log error and (b) log error ratio, Although these experiments suggest that time-mesh coarsening accelerations hold promise for improving the parallel waveform relaxation algorithm, they certainly do not settle all the issues. In particular, we do not have an easy and robust mechanism for determining what timestep iteration sweep log -5 -1timestep iteration sweep log error ratio Fig. 9. (a) log error and (b) log error ratio, -5 timestep iteration sweep log error ratio1030020 -55timestep iteration sweep log error Fig. 10. (a) log error and (b) log error ratio, steps . On longer time intervals, the convergence acceleration factor (ratio of errors with and without acceleration) becomes approximately linear in the sweep. splittings will benefit from acceleration, or for determining various parameters such as number of smoothing sweeps, optimal coarsening, etc. We also have not yet experimented with the use of more than two levels of time-mesh acceleration. Acknowledgements : The author is indebted to Pawel Szeptycki for several helpful discussions during the early stages of this work. Stefan Vandewalle read a preliminary version of the manuscript and contributed several very useful comments. The computers of the Kansas Institute for Theoretical and Computational Science were used for the numerical experiments. --R New York Solving Ordinary Differential Equations A spacetime multigrid method for parabolic PDEs An algorithm with polylog parallel complexity for solving parabolic partial differential equations Zukowski D. Estimating waveform relaxation convergence The waveform relaxation method for time-domain analysis of large scale integrated circuits Multigrid dynamic iteration for parabolic equations Convergence of dynamic iteration methods for initial value problems Sets of convergence and stability regions Remarks on Picard Lindel-of iteration Power bounded prolongations and Picard-Lindel-of iteration Partitioning algorithms and parallel implementation of waveform relaxation algorithms for circuit simulation --TR --CTR D. Guibert , D. Tromeur-Dervout, Parallel adaptive time domain decomposition for stiff systems of ODEs/DAEs, Computers and Structures, v.85 n.9, p.553-562, May, 2007

multigrid methods;waveform relaxation;wave equation

275958

A Fast Iterative Algorithm for Elliptic Interface Problems.

A fast, second-order accurate iterative method is proposed for the elliptic equation \[ \grad\cdot(\beta(x,y) \grad u) =f(x,y) \] in a rectangular region $\Omega$ in two-space dimensions. We assume that there is an irregular interface across which the coefficient $\beta$, the solution u and its derivatives, and/or the source term f may have jumps. We are especially interested in the cases where the coefficients $\beta$ are piecewise constant and the jump in $\beta$ is large. The interface may or may not align with an underlying Cartesian grid. The idea in our approach is to precondition the differential equation before applying the immersed interface method proposed by LeVeque and Li [ SIAM J. Numer. Anal., 4 (1994), pp. 1019--1044]. In order to take advantage of fast Poisson solvers on a rectangular region, an intermediate unknown function, the jump in the normal derivative across the interface, is introduced. Our discretization is equivalent to using a second-order difference scheme for a corresponding Poisson equation in the region, and a second-order discretization for a Neumann-like interface condition. Thus second-order accuracy is guaranteed. A GMRES iteration is employed to solve the Schur complement system derived from the discretization. A new weighted least squares method is also proposed to approximate interface quantities from a grid function. Numerical experiments are provided and analyzed. The number of iterations in solving the Schur complement system appears to be independent of both the jump in the coefficient and the mesh size.

Introduction . Consider the elliptic equation r (fi(x; 2\Omega Given BC on in a rectangular domain\Omega in two space dimensions. Within the region, suppose there is an irregular interface \Gamma across which the coefficient fi is discontinuous. Referring to Fig 1, we assume that fi(x; y) has a constant value in each sub-domain, The interface \Gamma may or may not align with a underline Cartesian grid. Depending on the properties of the source term f(x; y), we usually have jump conditions across the interface \Gamma: This work was supported by URI grant #N00014092-J-1890 from ARPA, NSF Grant DMS- 9303404, and DOE Grant DE-FG06-93ER25181. y Department of Mathematics, University of California at Los Angeles, Los Angeles, CA 90095. (zhilin@math.ucla.edu). Z. LI (a) -0.4 Interface (b) Fig. 1. Two typical computational domains and interfaces with uniform Cartesian grids. where (X(s); Y (s)) is the arc-length parameterization of \Gamma, the superscripts \Gamma or denotes the limiting values of a function from one side or the other of the interface. These two jump conditions can be either obtained by physical reasoning or derived from the differential equation, see [2, 9, 14] etc. Note in potential theory, v(s) 6j 0 corresponds to a single layer source along the interface \Gamma, while w(s) 6j 0 corresponds to a double layer source. The normal derivative u n usually has a kink across the interface due to the discontinuity in the coefficient fi. If w(s) 6j 0; then the solution would be discontinuous across the interface. There are many applications in solving elliptic equations with discontinuous coef- ficients, for example, steady state heat diffusion or electrostatic problems, multi-phase and porous flow, solidification problems, and bubble computations etc. There are two main concerns in solving (1.1)-(1.4) numerically: ffl How to discretize (1.1)-(1.4) to certain accuracy. It is difficult to study the consistency and the stability of a numerical scheme because of the discontinuities across the interface. ffl How to solve the resulting linear system efficiently. Usually if the jump in the coefficient is large, then the resulting linear system is ill-conditioned, and the number of iterations in solving such a linear system is large and proportional to the jump in the coefficient. There are a few numerical methods designed to solve elliptic equations with discontinuous coefficients, for example, harmonic averaging, smoothing method, and finite element approach etc., see [2] for a brief review of different methods. Most of these methods can be second order accurate in the l-1 or the l-2 norm, but not in the l-1 norm, since they may smooth out the solution near the interface. A. Mayo and A. Greenbaum [14, 15] have derived an integral equation for elliptic interface problems with piecewise coefficients. By solving the integral equation, they can solve such interface problems to second order accuracy in the l-1 norm using the techniques developed by A. Mayo in [13, 14] for solving Poisson and biharmonic equations on irregular regions. The total cost includes solving the integral equation and a regular Poisson equation using a fast solver, so this gives a fast algorithm. The possibility of extension to variable coefficients is mentioned in [14]. R.J. LeVeque and Z. Li have recently developed a different approach for discretizing A FAST ALGORITHM FOR INTERFACE PROBLEMS 3 elliptic problems with irregular interfaces called the immersed interface method (IIM) [2, 9], which can handle both discontinuous coefficients and singular sources. This approach has also been applied to three dimensional elliptic equations [7], parabolic wave equations with discontinuous coefficients [4, 5], and the incompressible Stokes flow problems with moving interfaces [3, 6]. L. Adams [1] has successfully implemented a multi-grid algorithm for the immersed interface method. However, there are some numerical examples with large jumps in the coefficients in which the immersed interface method may fail to give accurate answers or converge very slowly. In this paper, we propose a fast algorithm for elliptic equations with large jumps in the coefficients. The idea is to precondition the elliptic equation before using the immersed interface method. In order to take advantage of fast Poisson solvers on rectangular regions, we introduce an intermediate unknown function [u n ](s) which is defined only on the interface. Then we discretize a corresponding Poisson equation, which has different sources from the original one, using the standard five-point stencil with some modification in the right hand side. Our discretization is equivalent to using a second order difference scheme to approximate the Poisson equation in the interior a second order discretization for the Neumann-like interface condition Thus from the error analysis for elliptic equations with Neumann boundary conditions, for example, see [17], we would have second order accurate solution at all grid points including those near or on the interface. A GMRES method is employed to solve the Schur complement system derived from the discretization. A new weighted least squares method is proposed to approximate interface quantities such as u \Sigma n from a grid function defined on the entire domain. This new technique has been successfully applied in the multi-grid method for interpolating the grid function between different levels [1] with remarkable improvement in the computed solution. These ideas will be discussed in detail in the following sections. The method described in this paper seems to be very promising not only because it is second order accurate, but also because the number of iterations in solving the Schur complement system is almost independent of both the jumps in the coefficients and the mesh size. This has been observed from our numerous numerical experiments, though we have not been able to prove this theoretically. Our new method has been used successfully for the computation of some inverse problems [20]. This paper is organized as follows. In Section 2, we precondition (1.1)-(1.4) to get an equivalent problem. In Section 3, we use the IIM idea to discretize the equivalent problem and derive the Schur complement system. The weighted least squares approach to approximate u \Sigma n from the grid function u ij is discussed in Section 4. Some implementation details are addressed in Section 5. Brief convergence analysis is given in Section 6. An efficient preconditioner for the Schur complement system is proposed in Section 7. Numerical experiments and analysis can be found in Section 8. Some new approaches in the error analysis involving interfaces are also introduced there. 2. Preconditioning the PDE to an equivalent problem. The problem we intend to solve is the following: 4 Z. LI Problem (I). r (fi(x; (2.6a) Given BC on with specified jump conditions along the interface \Gamma (2.7a) Consider the solution set u g (x; y) of the following problem as a functional of g(s). Problem (II). f if x f if x (2.8a) Given BC on with specified jump conditions 1 (2.9a) Let the solution of Problem (I) be u (x; y), and define along the interface \Gamma. Then u (x; y) satisfies the elliptic equation (2.8a)-(2.8b) and jump conditions (2.9a)-(2.9b) with g(s) j g . In other words, u g (x; y) j u (x; y), and @n is satisfied. Therefore, solving Problem (I) is equivalent to finding the corresponding and then u g (x; y) in Problem (II). Notice that g is only defined along the interface, so it is one dimension lower than u(x; y): Problem (II) is an elliptic interface problem which is much easier to solve because the jump condition [u n ] is given instead of [fiu n ]. With the immersed interface method, it is very easier to construct a second order scheme which also satisfies the conditions of the maximum principle. In this paper, we suppose fi is piecewise constant as in (1.2), so Problem (II) is a Poisson equation with a discontinuous source term and given jump conditions. We can then use the standard five-point stencil to discretize the left hand side of (2.8a), but modify the right hand side to get a second order scheme, see [2, 9] for the detail. Thus we can take advantage of fast Poisson solvers for the discrete system. The cost in solving 1 The jump conditions (2.9a) and (2.9b) depend on the singularities of the source term f(x; y) along the interface. However, in the expression of (2.8a), we do not need information of f(x; y) on the interface \Gamma, so there is no need to write f(x; y) differently. A FAST ALGORITHM FOR INTERFACE PROBLEMS 5 Problem (II) is just a little more than that in solving a regular Poisson equation on the rectangle with a smooth solution. For more general variable coefficient, the discussions in this paper are still valid except we can not use a fast Poisson solver because of the convection term (rfi \Delta ru)=fi in (2.8a). However a multi-grid approach developed by L. Adams [1] perhaps can be used to solve Problem (II). We wish to find numerical methods with which we can compute u g (x; y) to second order accuracy. We also hope that the total cost in computing g and u g is less than that in computing u g through the original Problem (I). The key to success is computing g efficiently. Below we begin to describe our method to solve g . Once is found, we just need one more fast Poisson solver to get the solution u (x; y). 3. Discretization. We use a uniform grid on the rectangle [a; b] \Theta [c; d] where the Problem (I) is defined: We assume that simplicity. We use a cubic spline ~ passing through a number of control points (X express the immersed interface, where s is the arc-length of the interface and (X the position of the k-th point on the interface \Gamma. Other representations of the interface are possible. A level set formulation is currently under investigation. Any other quantity q(s) defined on the interface such as w(s) and g(s) can also be expressed in terms of a cubic spline with the same parameter s. Since cubic splines are twice differentiable we can gain access to the value of q(s) and its first or second derivatives at any point on the interface in a continuous manner. We use upper case letters to indicates the solution of the discrete problem and lower case letters for the continuous solutions. Given W k and G k , the discrete form of jump conditions (2.9a) and (2.9b), with the immersed interface method, the discrete form of (2.8a) can be written as where is the discrete Laplacian operator using the standard five-point stencil. Note that if happens to be on the interface, then f ij =fi ij is defined as the limiting value from a pre-chosen side of the interface. C ij is zero except at those irregular grid points where the interface cuts through the five-point stencil. A fast Poisson solver, for example, FFT, ADI, Cyclic reduction, or Multi-grid, can be applied to solve (3.12). The solution U ij depends on G k , W k , continuously. In matrix and vector form we have is the discrete linear system for the Poisson equation when W k , G k are all zero. The solution is smooth for such a system. B(W;G) is a mapping from (3.12). From [2, 9] we 6 Z. LI know that B(W;G) depends on the first and second derivatives of w(s), and the first derivative of g(s), where the differentiation is carried out along the interface. At this stage we do not know whether such a mapping is linear or not. However in the discrete case, all the derivatives are obtained by differentiating the corresponding splines which are linear combination of the values on those control points. Therefore B(W;G) is indeed linear function of W and G and can be written as are two matrices with real entries. Thus (3.13) becomes The solution U of the equation above certainly depends on G and we are interested in finding G which satisfies the discrete form of (2.7b) where the components of the vectors U are discrete approximation of the normal derivative at control points from each side of the interface. In the next section, we will discuss how to use the known jump G, and sometimes also V , to interpolate U ij to get U n in detail. As we will see in the next section, U depend on U , G and V linearly where E, D, and - are some matrices and P . Combine (3.14) and (3.16) to obtain the linear system of equations for U and G: G F The solution U and G are the discrete forms of u g (x; y) and g , the solution of Problem (II) which satisfies (2.11). The next question is how to solve (3.17) efficiently. The GMRES method applied to (3.17) directly or the multi-grid approach [1] are two attractive choices. However, in order to take advantage of fast Poisson solvers, we have decided to solve G in (3.17) first, and then to find the solution U by using one more fast Poisson solver. Eliminating U from (3.17) gives a linear system for G (D F This is an n b \Theta n b system for G, a much smaller linear system compared to the one for U . The coefficient matrix is the Schur complement of D in (3.17). In practice, the matrices A, B, E, D, P , and the vectors - F are never formed. The matrix and vector form are merely for theoretical purposes. Thus an iterative method, such as the GMRES iteration [18], is preferred. The way we compute (3.16) will dramatically change the condition number of (3.18). A FAST ALGORITHM FOR INTERFACE PROBLEMS 7 4. A weighted least squares approach for computing interface quantities from a grid function. When we apply the GMRES method to solve the Schur complement system of (3.18) for G , we need to compute the matrix-vector multiplica- tion, which is equivalent to computing U \Gamma n with the knowledge of U ij and the jump condition [U n ]. This turns out to be a crucial step in solving the linear system (3.18) for G . Our approach is based on a weighted least squares formulation. The idea described here can also be, and has been, applied to the case, where we want to approximate some quantities on the interface from a grid function. For example, interpolating U ij to the interface to get U \Gamma (X; Y ) or U on the interface. This new approach has also been successfully applied to the multi-grid method for interpolating the grid function between different levels by L. Adams[1] with remarkable improvement in the computed solution. We start from the continuous situation, the discrete version can be obtained ac- cordingly. Let u(x; y) be a piecewise smooth function, with discontinuities only along the interface. We want to interpolate u(x get approximations to the normal derivatives are only defined on the interface, to second order. Our approach is inspired by Peskin's method in interpolating a velocity field u(x; y) to get the velocity of the interface using a discrete delta function. The continuous and discrete forms are the following: ZZ\Omega where ~ is a discrete delta function. A commonly used one is Notice that ffi h (x) is a smooth function of x. Peskin's approach is very robust and only a few neighboring grid points near ~ are involved. However this approach is only first order accurate and may smear out the solution near the interface. Our interpolation formula for n , for example, can be written in the following form (j ~ where d ff (r) is a function of the distance measured from the point ~ X, d ff Q is a correction term which can be determined once fl ij are known. Although we are trying to approximate the normal derivatives here, the same principle also applies to the function values U as well with different choices of fl ij and Q. Note no extra effort is needed to decide which grid points should be involved. Therefore, expression (4.21) is robust and depends on the the grid function u ij continuously, two very attractive 8 Z. LI properties of Peskin's formula (4.20). In addition to the advantages of Peskin's ap- proach, we also have flexibility in choosing the coefficients fl ij and the correction term Q to achieve second order accuracy. The parameter ff in (4.21) can be fixed or chosen according to problems, see Section 8. Below we discuss how to use the immersed interface method to determine the coefficients and the correction term Q. They are different from point to point on the interface. So they should really be labeled as fl ij; ~ etc. But for simplicity of notation we will concentrate on a single point ~ drop the subscript ~ X. Since the jump condition is given in the normal direction, we introduce local coordinates at (X; Y ), where ' is the angle between the x-axis and the normal direction. Under such new coordinates, the interface can be parameterized by -(j); j. Note that and, provided the interface is smooth at (X; Y well. The solution of the Poisson equation Problem (II) will satisfies the following interface relations, see [2, 9] for the derivation, be the -j coordinates of where the sign depends on whether (- lies on the side of \Gamma. Using Taylor expansion of (4.25) about (X; Y ) in the new coordinates, after collecting terms we have (j ~ a 9 where the a j are given by A FAST ALGORITHM FOR INTERFACE PROBLEMS 9 (j ~ a (j ~ a (j ~ a (j ~ a (j ~ a (j ~ (j ~ a (j ~ a (j ~ a (j ~ (j ~ a (j ~ From the interface relations (4.24) we know that all the jumps in the expression above can be expressed in terms of the known information. Since - , we obtain the linear system of equations for the coefficients a 9 Note that we would have the exact same equation if we want to interpolate a smooth function to get an approximation u n at ~ X to second order accuracy. The discontinuities across the interface only contribute to the correction term Q. This agrees with our analysis in [2, 9] for Poisson equations with discontinuous and/or singular sources, where we can still use the classical five-point scheme but add a correction term to the right hand side at irregular grid points. If the linear system (4.26) has a solution, then we can obtain a second order approximation to the normal derivative n by choosing an appropriate correction term Q. Therefore we want to choose ff big enough, say ff - 1:6h, such that at least six grid points are involved. Usually we have an under-determined linear system which has infinitely many solutions. We should then choose the one fl ij with the minimal 2-norm subject to (4:26): For such a solution, each fl ij will have roughly the same magnitude O(1=h); so ij d ff (j ~ is roughly a decreasing function of the distance measured from ~ X. This is one of desired properties of our interpolation. In practice, only a hand full of grid points, controlled by the parameter ff, are involved. Those grid points which are closer to (X; Y ) have more influence than others which are further away. Z. LI When we know the coefficients we also know the a k 's. From the a k 's and the interface relations (4.24), we can determine the correction term Q easily, Thus we are able to compute n to second order accuracy. We can derive a formula for n in exactly the same way. However, with the relation u g, we can write down a second order interpolation scheme for immediately (j ~ is the solution we computed for n . In the next section, we will explain an important modification of either (4.21) or (4.28) depends on the magnitude of fi \Gamma or We should mention another intuitive approach, one-sided interpolation, in which we only use grid points on the proper side of the interface in computing a limiting value at the interface: This approach does not make use of the interface relations (4.24), so we have to have at least six points from each side in order to have a second order scheme. Note that we can also use the least squares technique described in this section for one-sided interpolation. This approach has been tested already. The weighted least squares approach using the interface relations (4.24) appears superior in practice. It has the following advantages: ffl Fewer grid points are involved. When we make use of the interface relation, compared to the one-sided interpolation, the number of grid points which are involved is reduced roughly by half. ffl Second order accuracy with smaller error constant. The grid points involved in our approach are clustered around the point (X; Y ) on the interface, and those which are closer to (X; Y ) have more influence than those which are further away in our weighted least squares approach. We have smaller error constant in the Taylor expansions compared to the one-sided interpolation. The error constant can be as much as 8 - 27 times smaller as the one-sided interpolation. In two dimensional computation, we can not take m and n to be very large, to have a smaller error constant sometimes is as important as to have a high order accurate method. ffl Robust and smoother error distribution. We have a robust way in choosing the grid points which are involved. The interpolation formulas (4.21) and depend continuously on the location (X; Y ) and the grid points and so does the truncation error for these two interpolation schemes. In other words, we will have a smooth error distribution. This is very important in moving interface problems where we do no want to introduce any non-physical oscillations. A FAST ALGORITHM FOR INTERFACE PROBLEMS 11 downs. In one-sided interpolation, sometimes we can not find enough grid points in one particular side of the interface, then the one-sided interpolation will break down. In our approach, every grid point on one side is connected to the other by the interface relations (4.24). So no break down will occur. ffl Trade off or disadvantages. The only trade off of our weighted least squares approach is that we have to solve a under-determined 6 by p linear system of equation (where p - instead of solving one that is 6 by 6. The larger ff is, the more computational cost in solving (4.26). Fortunately, the linear system has full row rank and can be solved by the LR-RU method [8] or other efficient least squares solvers. 5. Some details in implementation. The main process of our algorithm is to solve the Schur complement system (3.18) using the GMRES method with an initial guess G (0) We need to derive the right hand side, and explain how to compute the matrix-vector multiplication of the system without explicitly forming the coefficient matrix. The right hand side needs to be computed just once which is described below. 5.1. Computing the right hand side of the Schur complement system. If we take apply one step of the immersed interface method to solve Problem (II) to get U(0), then With the knowledge of U(0) and G = 0; we can compute the normal derivatives on each side of the interface to get U \Sigma using the approach described in the previous section. Thus the right hand side of the Schur complement system is F The last two equalities are obtained from (3.16) and (3.18) with Now we are able to compute the right hand side of the Schur complement system. 5.2. Computing the matrix-vector multiplication of the Schur comple- ment. Now consider the matrix-vector multiplication (D of the Schur complement, where Q is an arbitrary vector of dimension n b . This involves essentially two steps. 1. A fast Poisson solver for computing which is the solution of Problem (II) with Z. LI 2. The weighted least squares interpolation to compute U The residual vector in the flux jump condition is which is the same residual vector of the second equation in (3.17) from our definition. In other words, see also (3.16) The matrix-vector multiplication (5.29) then can be computed from the last equality of the following derivation: F from (5.30) V from (5.32): Note that from the second line to the third line we have used the following which is defined in (3.18). It worth to point out that once our algorithm is successfully terminated, which means that the residual vector is close to the zero vector, we not only have an approximation Q to the solution G , an approximation U(Q) to the solution U , bult also approximations U \Sigma n (Q) to the normal derivatives from each side of the interface. The normal derivative information is very useful for some moving interface problems where the velocity of the interface depends on the normal derivative of the pressure. 6. Convergence Analysis. As to this point, we have had a complete algorithm for solving the original elliptic equations of the form Problem (I). We have transformed the original elliptic equation to a corresponding Poisson equation with different source term and jump conditions, or internal boundary conditions, (2.9b) and (2.11). The jump condition (2.11) is Neumann-like boundary condition which involves the normal derivatives from each side of the interface. In our algorithm, the classical five-point difference scheme at regular grid points is used. This discetization is second order accurate. As discussed in Section 4, the Neumann-like internal boundary condition (2.11) is also discretized to second order. So from the analysis in Chapter 6 of [17] on Neumann conditions, we should be able to conclude second order convergence globally for our computed solution, provide that we can solve the Poisson Problem (II) to second order accuracy. This is confirmed in our early work [2, 9]. Numerical experiments have confirmed second order accuracy of the computed solution for numerous test problems, see Section 8. 7. An efficient preconditioner for the Schur complement system. With the algorithm described in previous sections, we are able to solve Problem I to second order accuracy. In each iteration we need to solve a Poisson equation with a modified right hand side. A fast Poisson solver such as a fast Fourier transformation method etc. [19], can be used. The number of iterations of A FAST ALGORITHM FOR INTERFACE PROBLEMS 13 the GMRES method depends on the condition number of the Schur complement. If we make use of both (4.21) and (4.28) to compute U \Sigma n , the condition number seems to be proportional to 1=h. Therefore the number of iterations will grow linearly as we increase the number of grid points. Below we propose a modification in the way of computing U \Sigma which seems to improve the condition number of the Schur complement system dramatically. If n and n are the exact solutions, that is then we can solve n or u n in terms of v, It is easy to get or The idea is simple and intuitive. We use one of the formulas (4.21) or (4.28) obtained from the weighted least squares interpolation to approximate n or u n , and then use or (7.33) to approximate u n or n to force the solution to satisfy the flux jump condition. This is actually an acceleration process, or a preconditioner for the Schur complement system (3.18). With this modification, the number of iterations for solving the Schur complement system seems to be independent of the mesh size h, and almost independent of the jump [fi] in the coefficient as well, see the next section for more details. Although we have not been able to prove this claim, the algorithm seems to be extraordinary successful. Whether we use the pair (4.21) and (7.34) or the other (4.28) and (7.33) have only a little affect on the accuracy of the computed solutions and the number of iterations. The algorithm otherwise behaves the same and the analysis in the next section seems to be true no mater what pair we choose. We have been using the following criteria to choose the desired pair n is determined by (4.28) n is determined by (4.21) which seems always better than the choice of the other way around. 8. Numerical Experiments. We have done many numerical experiments with different interfaces and various test functions. Since our scheme can handle jumps in the solution, we have great flexibility in choosing test problems. From the numerical tests we intend to determine: ffl The accuracy of computed solutions. Are they second order accurate? 14 Z. LI ffl The numbers of iterations as we change the mesh size h and the ratio of the discontinuous coefficient, ffl The ability of the algorithm to deal with complicated interfaces and large jumps in the coefficient. All the experiments are computed with double precision. The computational parameters include: Computational rectangle [a; b] \Theta [c; d]. ffl The number of grid points m and n in the x- and y- directions respectively, we assume that is the mesh size. ffl The number of control points n b . The interface is expressed in terms of cubic splines passing through the control points. ffl The parameter ff in the weighted least squares interpolation. We take specified differently. The maximum norm is used to measure the errors in the computed solution U ij , and the normal derivatives U \Gamma n p from each side of the interface at the p-th control point. The relative errors are defined as follows where ~ is one of control points on the interface. Each grid point is labeled as either in the or the outside\Omega + of the interface and the exact solution is determined accordingly. In other words, the exact solution is not determined from the exact interface but the discrete one. In Table 1, r i , 3 is the ratio of successive errors. A ratio of 4 corresponds to second-order accuracy. In Table 1, k is the number of iterations required in solving the Schur complement system (3.18). The ratio of coefficients is defined . In the figures, we use S to express the slopes of least squares line of experimental data (log(h i ); log(E i )). Example 1. Consider the following interface where within the computational domain Fig 1 shows some interfaces with different parameters r 0 , . Dirichlet boundary conditions, as well as the jump conditions [u] and [fiu n ] along the interface, are determined from the exact A FAST ALGORITHM FOR INTERFACE PROBLEMS 15 solution if (x; y) r 4 if (x; y) . The source term can be determined accordingly: if (x; y) if (x; y) We provide numerical results for three typical cases below. Case A. The interface parameters are chosen as r the interface is a circle centered at the origin, see Fig 2(a). With C the solution is continuous everywhere, but u n and fiu n are discontinuous across the circle. It is easy to verify that [fiu n when we take C Fig 3(a) is the plot of the solution \Gammau with Case B. The interface parameters are chosen as r 20, Fig 2(a). We shift the center of the interface a little bit to have a non-symmetric solution relative to the grid. We want our test problems to be as general as possible. The interface is irregular but the curvature has modest magnitude. So with a reasonable number of points on the interface, we can express it well. Now it is almost impossible to find an exact solution which is continuous but not smooth across the interface, so we simply set C Fig 3(b) is the plot of the solution \Gammau with Case C. The interface parameters are chosen as r 20, Fig 2(b). Now the magnitude of the curvature is very large at some points on the interface and we have to have enough control points to resolve it. The solution parameters are set to be the same as in Case B. Fig 4-6 and Table 1 are some plots and data from the computed solutions which we will analyze below. 8.1. Accuracy. Table 1 shows the results of grid refinement analysis for Case A with two very different ratio 0:5, the ratio r i are very close to 4 indicating second order convergence. With the error in the solution drops much more rapidly. This is because the solution in approaches a constant as fi becomes large, and it is quadratic order accurate method would give high accurate solution in both regions. So it is not surprising to see the ratio r 1 is much larger than 4. For the normal derivatives, we expect second order accuracy again since fi n is not quadratic and has magnitude of O(1). This agrees with the results r 2 and r 3 in Table 1. In Fig 4 we consider the opposite case when fi . In this case the solution is not quadratic so we see the expected second order accuracy. Fig 4(a) Z. LI (a) -0.4 A (b) -0.4 Fig. 2. Different interfaces of Example 1. (a) Case A and B. (b) Case C. (a) -0.4 -0.3 -0.2 -0.4 -0.3 -0.2 -0.1Fig. 3. The solutions \Gammau of Example 1 with 1. (a) Case A, a circular interface where the solution is continuous but [fiun Case B, an irregular interface where both the solution and the flux [fiun ] are discontinuous. Table Numerical results and convergence analysis for Case A with A FAST ALGORITHM FOR INTERFACE PROBLEMS 17 (a) (b) -5 log(h) log(E Fig. 4. (a): Error distribution for Case A. (b): Errors E i vs the mesh size h in log-log scale for Case A with (a) log(h) log(E (b) -6.4 -6.2 -6 -5.8 -5.6 -5.4 -5.2 -5 -4.8 -5 log(h) log(E) Fig. 5. Errors E i vs the mesh size h in log-log scale for Case B with The solid line: n . The dotted line: n (b) The solid line and dotted line are the same as in (a) but on a different scale. The dash-dotted line is the result obtained with log(h) log(E 2:71 2:07 Fig. 6. Errors E i vs the mesh size h in log-log scale for Case C with 1. The solid line: fixed n b (n 520). The dotted line: n Z. LI plots the error distribution over the region. The error seems to change continuously even though the maximum error occurs on or near the interface. Usually if the curvature is very big in some part of an interface, for example, near a corner, then we would observe large errors over the neighborhood of that part of the interface. For interface problems, the errors usually do not decrease monotonously as we refine the grid unless the interface is aligned with one of the axes. We need to study the asymptotic convergence rate which is usually defined as the slope of the least squares line of the experimental data (log(h i ); log(E i )). Fig 4(b) plots the errors versus the mesh size h in log-log scale for the case n. The asymptotic convergence rate is about 2:62 compared to 2 for a second order method. As h gets smaller we can see the curves for the errors become flatter indicating the asymptotic convergence rate will approach 2. The dotted curves in Fig 5 and Fig 6 are the results for case B and C, where the interfaces are more complicated compared to case A. Again we take The asymptotic convergence rates are far more than two. Such behavior can also be observed from Example 4 in [16]. Does it mean that our method is better than second order? Certainly this is not true from our discretization. Below we explain what is happening. For interface problems, the errors depend on the solution u(x; y), the mesh size h, the jump in the coefficient [fi], the interface \Gamma and its relative position to the grid, and the maximal distance between control points on the interface, h b . We can write the error in the solution, for example, as follows The first term in the right hand side of (8.4) is the error from the discretization of the differential equation. The term C (u; h; h b ; [fi]; \Gamma) has magnitude O(1) but does not necessarily approach to a constant. The second term in the right hand side of is the error from the discretization of the interface \Gamma. If we use a cubic spline interpolation, then q ? 2. For Case A, the interface is well expressed and the first term in (8.4) is dominant, we have clearly second order convergence. For Case B and C, the interfaces are more complicated and the second term in (8.4) is dominant. That is why we have higher than second order convergence. Eventually, the error in the first term will dominate and we will then observe second order convergence. To further verify the arguments above, we did some tests with fixed number of control points n b . For example, we take n the solid line in Fig 6. Presumably the interface is expressed well enough and the second term in (8.4) is negligible. We see the slopes of the least squares line of the errors E 1 and E 2 are 2:15 and 2:07 respectively indicating second order convergence. Usually the error in the normal derivatives n and u behaves the same, so we only need to study one of them. If we let n b change with the same speed as the number of grids m and n, then the second term in (8.4) is dominant and the slopes of the least squares line of the errors E 1 and E 2 are 2:71 and 2:69 respectively. Once n b is large enough, the first term will dominate in (8.4) and the error will decrease quadratically. This can also be seen roughly from Fig 6. Note that the errors oscillate as n gets large whether we fix n b or not. But the fluctuation becomes smaller as we refine the grid. The upper envelop of E 1 behaves the same as the least squares line of the experimental data (log(h i ); log(E i )). So it is reasonable to use the asymptotic convergence rate to discuss the accuracy when errors do not behave monotonously. A FAST ALGORITHM FOR INTERFACE PROBLEMS 19 As another test, we let n b change slower than m and n. The solid lines in Fig 5(a) are obtained with Now we have roughly and the errors decrease quadratically with the mesh size h, but not h b . The slopes of the least squares line of the errors E 1 and E 2 are 2:23 and 2:22 respectively. We now discuss the effect of the different choice of the parameter ff in the least squares approximation described in Section 4 on the solution. Most of the computations are done with Fig 5(b), the dash-dotted line where As we can expect, the smaller ff is, the higher accuracy in the computed solution because the points involved are clustered together and the error in the Taylor expansion will be smaller. However, the smaller ff is, the more oscillatory in the error as we refine the grid. For larger ff, the computation cost increases quickly, but the error behaves much smoother with the mesh size h. Usually we can take small ff for smooth interfaces, and larger ff if we want a smoother error distribution for more complicated interfaces. 8.2. The number of iterations versus the mesh size h. Fig 7(a), also see Fig 10(a) for Example 2, shows the number of iterations versus the number of grid points m and n for case A, B, and C. It is not surprising to see that the number of iterations depends on the shape of the interface. The number of iterations required for Case C is larger than that for Case A and B. But it is wonderful to see that the number of iterations is almost independent of the mesh size h. For Case A, where the interface is a circle, we only need about iterations for all choices of the mesh size h for two extreme cases We will see in the next paragraph that this is also true for different choices of the ratio . Note that the number of iterations is about two or three fewer than the numbers of calls of the fast Poisson solver. We need two or three of them for initial set up of the Schur complement system. In Fig 7(a), the lowest curve corresponds to case A with the lowest but the second curve corresponds to 1. For case B, the number of iterations required is about 17 - 21 for respectively. For case C, the most complicated interface, the number of iterations is about 46 with reasonable number of control points on the interface for 8.3. The number of iterations versus the jump ratio Fig 7(b), also see Fig 10(b) for Example 2 , plots the number of iterations versus the jump ratio ae in log-log scale with fixed number of grids goes away from the unit we have larger jump relatively in the coefficient. The number of iterations increases proportional to jlog(ae)j when ae is small but soon reaches a point after which the number of iterations will remain as a constant. Such points depends on the shape of the interface. For Case A, it requires only about 5 - 6 iterations at the most for iterations for ae ? 1 in solving the Schur complement system using the GMRES method. For Case B, the numbers are about 17 - 22. For Case C, the most complicated interface in our examples, the numbers are about 47 - 69. As we mentioned in the previous paragraph, also see Fig 7(a), for Case C, with only 160 control points we can not express the complicated interface Fig 2(b) very well. If we take more control points on the interface, then the number of iterations will be about Z. LI (a) A (b) A 6 91769 Fig. 7. The number of iterations for Example 1 with vs the number of grids n. Case A: lower curve, Case B: lower curve, the ratio of jumps in log-log scale with when Example 2. This geometry of this example is adapted from Problem 3 of [1]. The solution domain is the [\Gamma1; 1] \Theta [0; 3] rectangle and the interface is determined by Fig 8(a) show the solution domain and the interface \Gamma with x Again Dirichlet boundary condition, as well as the jump conditions [u] and [fiu n ] are determined from the exact solution The source term can be determined accordingly: Fig 8(b) is a plot of the computed solution. This example is different from Example 1 in several ways. The solution is independent of the coefficient fi. But the magnitude of the jump [fiu n ] and the source increase with the magnitude of the jump [fi]. However we have observed similar behaviors in the numerical results as we discussed in Example 1. Example 1 and Example 2 are two extreme samples of elliptic interface problems. So we should be able to get some insights about the method proposed in this paper. Fig 9 shows errors E i versus mesh size h in log-log scale with different choice of b . In Fig 9(a), . The solid lines correspond to a fixed discretization of the interface, n As we expected, the asymptotic convergence rate for are 2:1272. They are all close to 2 indicating A FAST ALGORITHM FOR INTERFACE PROBLEMS 21 (a) (b)0.51.52.5-11 y x Fig. 8. (a) The interface of Example 2. (b) The solution of Example 2. second order accuracy. The dotted line in Fig 9(a) correspond to a variable n b which changes in the same rate as the number of grid point m in x-direction. The asymptotic convergence rate of E i for are S 3:3473. They are Fig 9(b). These numbers are all larger than 2 similar to the cases we saw in Fig 5, and Fig 6. We have explained such phenomena already. Fig 10(a) plots the number of iteration versus the number of grids n with Again we consider two extreme cases, with Once the interface is well expressed somewhere after n ? 180, the number of iteration will slightly decrease to a constant which is about 28 for and 34 for Fig 10(b) plots the number of iteration versus the ratio ae with fixed grid ae ? 1. We observe the same behavior as in Fig 7(b). Initially the number of iterations increases proportional to jlog(ae)j as ae goes away from the unit, but it soon approaches a constant which is about 28 for ae ! 1 and 34 for ae ? 1. (a) -6.2 -6 -5.8 -5.6 -5.4 -5.2 -5 -4.8 -4.6 log(h) 3:35 2:14 (b) -6 -5.9 -5.8 -5.7 -5.6 -5.5 -5.4 -5.3 -5.2 -5.1 log(E Fig. 9. Errors E i vs the mesh size h in log-log scale for Example 2 with 1: The solid line: fixed n b , dotted line: n Summary of the numerical experiments. In our computations, the largest error usually occurs at those points which are close to the part of the interface which 22 Z. LI (a) (b) )28Fig. 10. The number of iterations for Example 2 with vs the number of grids n. Lower curve: vs the ratio of jumps in log-log scale with fixed grid has large curvature. Depending on the shape of the interface, we should take enough control points on the interface so the error in expressing the interface does not dominate the global error. However, once such a critical number is decided, we do not need to double it as we double the number of grid points, which saves some computational cost. We should still be able to maintain second order accuracy. The number of iteration for solving the Schur complement system using a GMRES method is almost independent of both the mesh size h as well as the jump in the coefficient. 9. Conclusions. We have developed a second-order accurate fast algorithm for a type of elliptic interface problems with large jumps in the coefficient across some irregular interface. We precondition the original partial differential equation to obtain an equivalent Poisson problem with different source terms and a Neumann-like interface condition. The fast Poisson solver proposed in [2, 9] can be employed to solve the Schur complement system for the intermediate unknown, the jump in the normal derivative along the interface. Then we proposed a preconditioning technique for the Schur complement system which seems to be very successful. Numerical tests revealed that the number of iterations in solving the Schur complement system is independent of both the mesh size h and the jump in the coefficient, though we have not proved this strictly in theory. The idea introduced in this paper might be applicable to other related problem, for example, to domain decomposition techniques. A new least squares approach to approximate interface quantities from a grid function is also proposed. By analyzing the numerical experiments, we have discussed some issues in error analysis involving interfaces. There is still a lot of room for improving the method described in this paper. For example, we have used cubic spline interpolations for closed interfaces. There are some advantages of this approach. But large errors can occur at the connection of the first and the last control points when we try to make the curve closed. That might also be one of reasons why the error does not decrease monotonously. As an alternative, a level set formulation is under investigation. The next project following this paper is to study the case with variable coefficients. A FAST ALGORITHM FOR INTERFACE PROBLEMS 23 We can rewrite (1.1) either as (2.8a) or r if x r f if x are the averages of the coefficients fi from each side of the interface. Whether (2.8a) or (9.5) is used, we shall still introduce an intermediate unknown, the jump in the normal derivative across the interface if the jump condition is given in the form of [fiu n ]. In this way, the coefficients of the difference scheme would be very close to those obtained form the classical five-point stencil. We can not take advantage of the fast Poisson solvers for variable coefficient anymore, but we can make use of the multi-grid method developed by L. Adams in [1]. 10. Acknowledgments . It is my pleasure to acknowledge the encouragements and advice from various people including Prof. Randy LeVeque, Stanley Osher, Tony Chan, Loyce Adams, Jun Zou and Barry Merriman. Thanks also to Prof. Yousef Saad and Dr. Victor Eijkhout for helping me to implement and understand the GMRES method. --R A multigrid algorithm for immersed interface problems. The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. Simulation of bubbles in creeping flow using the immersed interface method. Immersed interface methods for wave equations with discontinuous coefficients. Finite difference methods for wave equations with discontinuous coefficients. Immersed interface method for Stokes flow with elastic boundaries or surface tension. A note on immersed interface methods for three dimensional elliptic equations. Uniform treatment of linear systems - algorithm and numerical stability The Immersed Interface Method - A Numerical Approach for Partial Differential Equations with Interfaces Immersed interface method for moving interface problems. ADI methods for heat equations with discontinuties along an arbitrary interface. On the rapid evaluation of heat potentials on general regions. The fast solution of Poisson's and the biharmonic equations on irregular regions. The rapid evaluation of Volume Fast parallel iterative solution of Poisson's and the biharmonic equations on irregular regions. A fast poisson solver for complex geometries. Numerical Solution of Partial Differential Equations. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. Fast Poisson solver. Computing some inverse problems. --TR --CTR Kazufumi Ito , Zhilin Li, Solving a Nonlinear Problem in Magneto-Rheological Fluids Using the Immersed Interface Method, Journal of Scientific Computing, v.19 n.1-3, p.253-266, December Songming Hou , Xu-Dong Liu, A numerical method for solving variable coefficient elliptic equation with interfaces, Journal of Computational Physics, v.202 n.2, p.411-445, 20 January 2005 M. Oevermann , R. Klein, A Cartesian grid finite volume method for elliptic equations with variable coefficients and embedded interfaces, Journal of Computational Physics, v.219 n.2, p.749-769, December, 2006 Petter Andreas Berthelsen, A decomposed immersed interface method for variable coefficient elliptic equations with non-smooth and discontinuous solutions, Journal of Computational Physics, v.197 n.1, p.364-386, 10 June 2004 Shaozhong Deng , Kazufumi Ito , Zhilin Li, Three-dimensional elliptic solvers for interface problems and applications, Journal of Computational Physics, v.184 n.1, p.215-243, Peter Schwartz , Michael Barad , Phillip Colella , Terry Ligocki, A Cartesian grid embedded boundary method for the heat equation and Poisson's equation in three dimensions, Journal of Computational Physics, v.211 n.2, p.531-550, 20 January 2006 Xu-Dong Liu , Thomas C. Sideris, Convergence of the ghost fluid method for elliptic equations with interfaces, Mathematics of Computation, v.72 n.244, p.1731-1746, October Shi Jin , Xuelei Wang, Robust numerical simulation of porosity evolution in chemical vapor infiltration: II. Two-dimensional anisotropic fronts, Journal of Computational Physics, v.179 n.2, p.557-577, July 2002 Ming-Chih Lai , Zhilin Li , Xiaobiao Lin, Fast solvers for 3D Poisson equations involving interfaces in a finite or the infinite domain, Journal of Computational and Applied Mathematics, v.191 n.1, p.106-125, 15 June 2006 Do Wan Kim , Young-Cheol Yoon , Wing Kam Liu , Ted Belytschko, Extrinsic meshfree approximation using asymptotic expansion for interfacial discontinuity of derivative, Journal of Computational Physics, v.221 n.1, p.370-394, January, 2007 I. Klapper , T. Shaw, A large jump asymptotic framework for solving elliptic and parabolic equations with interfaces and strong coefficient discontinuities, Applied Numerical Mathematics, v.57 n.5-7, p.657-671, May, 2007 Carlos J. Garca-Cervera , Zydrunas Gimbutas , Weinan E., Accurate numerical methods for micromagnetics simulations with general geometries, Journal of Computational Physics, v.184 n.1, p.37-52, B. P. Lamichhane , B. I. Wohlmuth, Mortar finite elements for interface problems, Computing, v.72 n.3-4, p.333-348, May 2004 Chohong Min , Frdric Gibou , Hector D. Ceniceros, A supra-convergent finite difference scheme for the variable coefficient Poisson equation on non-graded grids, Journal of Computational Physics, v.218 n.1, p.123-140, 10 October 2006 Frederic Gibou , Ronald P. Fedkiw , Li-Tien Cheng , Myungjoo Kang, A second-order-accurate symmetric discretization of the Poisson equation on irregular domains, Journal of Computational Physics, v.176 n.1, p.205-227, February 10, 2002 John K. Hunter , Zhilin Li , Hongkai Zhao, Reactive autophobic spreading of drops, Journal of Computational Physics, v.183 n.2, p.335-366, December 10 Y. C. Zhou , G. W. Wei, On the fictitious-domain and interpolation formulations of the matched interface and boundary (MIB) method, Journal of Computational Physics, v.219 n.1, p.228-246, 20 November 2006 Y. C. Zhou , Shan Zhao , Michael Feig , G. W. Wei, High order matched interface and boundary method for elliptic equations with discotinuous coefficients and singular sources, Journal of Computational Physics, v.213 n.1, p.1-30, 20 March 2006 Xiaolin Zhong, A new high-order immersed interface method for solving elliptic equations with imbedded interface of discontinuity, Journal of Computational Physics, v.225 n.1, p.1066-1099, July, 2007 George Biros , Lexing Ying , Denis Zorin, A fast solver for the Stokes equations with distributed forces in complex geometries, Journal of Computational Physics, v.193 n.1, p.317-348, January 2004 Sining Yu , Yongcheng Zhou , G. W. Wei, Matched interface and boundary (MIB) method for elliptic problems with sharp-edged interfaces, Journal of Computational Physics, v.224 n.2, p.729-756, June, 2007

GMRES method;immersed interface method;discontinuous coefficients;elliptic equation;schur complement;preconditioning;cartesian grid

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Inner and Outer Iterations for the Chebyshev Algorithm.

We analyze the preconditioned Chebyshev iteration in which at each step the linear system involving the preconditioner is solved inexactly by an inner iteration. We allow the tolerance used in the inner iteration to decrease from one outer iteration to the next. When the tolerance converges to zero, the asymptotic convergence rate is the same as for the exact method. Motivated by this result, we seek the sequence of tolerance values that yields the lowest cost to achieve a specified accuracy. We find that among all sequences of slowly varying tolerances, a constant one is optimal. Numerical calculations that verify our results are presented. Asymptotic methods, such as the W.K. B. method for linear recurrence equations, are used with an estimate of the accuracy of the asymptotic result.

Introduction The Chebyshev iterative algorithm [1] for solving linear systems of equations often requires at each step the solution of a subproblem i.e. the solution of another linear system. We assume that the subproblem is also solved iteratively by an "inner iteration". The term "outer iteration" refers to a step of the basic algorithm. The cost of performing an outer iteration is dominated by the cost of solving the subproblem, and it can be measured by the number of inner iterations. A good measure of the total amount of work needed to solve the original problem to some accuracy ffl is then, the total number of inner iterations. To reduce the amount of work, one can consider solving the subproblems "inexactly" i.e. not to full accuracy. Although this diminishes the cost of solving each subproblem, it usually slows down the convergence of the outer iteration. It is therefore interesting to study the effect of solving each subproblem inexactly on the performance of the algorithm. We consider two measures of performance: the asymptotic convergence rate and the total amount of work required to achieve a given accuracy ffl. The accuracy to which the inner problem is solved may change from one outer iteration to the next. First, we evaluate the asymptotic convergence rate when the tolerance values converge to 0. Then, we seek the "optimal strategy", that is, the sequence of tolerance values that yields the lowest possible cost for a given ffl. The present results, contained in Giladi [2], extend those of Giladi [3]. The asymptotic convergence rate of the inexact Chebyshev iteration, with a fixed tolerance for the inner iteration, was derived in Golub and Overton [4] (see also [5], [6], [7], [8], [9], [10]). Previous work has mainly concentrated on the convergence rate, whereas we emphasize the cost of the algorithm. In section 2, we review the Chebyshev method and present the basic error bound for the inexact algorithm. Then, in section 3 we evaluate the asymptotic convergence rate when the sequence of tolerance values gradually decreases to j - 0. In section 4 we seek the "best strategy" i.e the one that yields the lowest possible cost. In section 5, we obtain an asymptotic approximation for the error bound when the sequence of tolerance values is slowly varying. In section 6 we analyze the error in this asymptotic approximation and present a few numerical calculations that demonstrate it's accuracy. In section 7 we use the analysis of section 5, to show that for the Chebyshev iteration, the optimal strategy is constant tolerance. We also estimate the optimal constant. Then, in section 8 we present a few numerical calculations that demonstrate the accuracy of the analysis of section 7. In Section 9, we generalize this result to other iterative schemes. iteration Chebyshev iteration (see Manteuffel [11]) to solve the real n \Theta n system of linear equations uses the splitting It requires that the spectrum of M \Gamma1 A be contained in an ellipse, symmetric about the real axis, in the open right half of the complex plane. We denote the foci of such an ellipse by l and u. Furthermore, we assume that M \Gamma1 A is diagonalizable. The exact Chebyshev method is defined by where c k+1 (-) In (3), the initial iterate x 0 is given, and in (7), c k denotes the Chebyshev polynomial of degree k. The inexact Chebyshev method is obtained by solving (5) iteratively for z k . This results in replacing (5) by In the variable strategy scheme the tolerance ffi k tends to j - 0 as k increases, while in the constant strategy scheme is constant. We denote the error at step k by We also define K, V , \Sigma and oe j by We use the same derivation as in [4] to show that when -oe where ffi represents a sequence of tolerance values fffi k g 1 . In equation (11), ae is defined by The function -(k; ffi) satisfies the recurrence equation with initial conditions The constant \Delta in (13) is given by ae The bound (11) is the product of two terms: ae k and -(k; ffi). The former is the bound for the exact algorithm and it is exponentially decaying. The latter is a monotonically increasing term which accounts for the accumulation of errors introduced by solving the inner problem inexactly. We shall obtain asymptotic approximations to -(k; ffi) under various assumptions on the sequence ffi k in order to analyze the performance of the inexact algorithm. 3 Asymptotic convergence rate We shall now estimate the asymptotic convergence rate of the inexact Chebyshev algorithm when the sequence of tolerance values for the inner iteration gradually decreases to 0. Our goal is to show that then, the asymptotic convergence rate of the inexact algorithm is the same as that of the exact scheme. This is in contrast to the case of constant tolerance for which the asymptotic convergence rate of the inexact algorithm is lower than that of the exact algorithm [4]. We base our analysis on the bound (11). Therefore, we wish to compute ae k -(k; ffi) In order to do so, we need to estimate the asymptotic behavior for large k of -(k; ffi). By making mild assumptions on the rate at which ffi k ! 0, we will show that lim Upon using (17) in (16), we find that the asymptotic convergence rate of the algorithm is lim ae where ae e is the asymptotic convergence rate of the exact algorithm. Equation (17) holds for many sequences ffi k of tolerance values. In order to obtain a general result, we shall assume only that The positive constant C in (19) is arbitrary. Hence, if C AE 1 the sequence of tolerance values can decay quite slowly. We show that (17) holds under assumption (19) in two steps. First we show that -(k; ffi) in (13), is bounded by the function oe(k; - ffi), where oe(k; - replaced by . Then, we show that lim k!1 oe(k; - ffi) 1=k = 1. As a first step, we prove the following proposition Proposition 1 Let -(k; ffi) be a solution to (13) and (14) and let oe(k; - ffi) be the solution to the same equation with replaced by - . Assume that - and that oe(0; - Then, for all k. We prove this proposition by induction. For we obtain from (14) that Then, we assume that assertions (20) and (21) are true for all In view of and By the induction hypothesis, oe(N; - -(N; ffi). Furthermore, - so the right side of (23) is greater than or equal to the right side of (24). We conclude that and that oe(N We shall now obtain the asymptotic behavior of oe(k; ffi) for large k from (13) with . We use the method of [12]. We first replace -(k; ffi) by oe(k; - ffi) in (13) and set - C=k. Then we introduce the stretched variables to obtain We seek for R(x) an asymptotic approximation valid for ffl - 1 of the form The functions /(x), K are to be determined so that R(x) satisfies equation (27). The constant c(ffl) is to be determined so that R(x) is independent of ffl. After substituting (28) into (27), we express each side of the resulting expression in power series in ffl 1=2 assuming that /(x 2ffl). can be expanded in Taylor series in powers of ffl. Then, we equate the coefficients of each power of ffl 1=2 on the left side of the resulting expression, to the same power of ffl 1=2 on the right side. The coefficients of ffl and of ffl 3=2 , yield the following equations for /(x; ffi) and K 0 (x; ffi) respectively: x x Upon solving (29) for / we find 2C \Deltax: (31) Introducing the right side of (31) into (30) and solving the resulting equation for K 0 we obtain To find the constant D in (32) we could match (28) to another expansion which satisfies the initial conditions (14). However, the value of D is unimportant for our purposes since We substitute (31) for / and (32) for K 0 into (28) for R. Then, we use the change of variables (26) to obtain To make the right side of (33) independent of ffl, we require that and we obtain Therefore, lim In a realistic numerical computation ffi k is bounded below by the machine precision j 0 . Moreover, the analysis of the iteration with sufficiently small, the performance of the inexact algorithm is for all practical purposes indistinguishable from that of the exact algorithm. Indeed, solving (13) with where It follows from (16), (18) and (36) that the asymptotic convergence rate is ae e e OE( - The number N(ffl; - ffi) of outer iterations required to achieve an accuracy ffl with tolerance - ffi is approximately log ffl log 'e: Hence, if the inexact scheme requires no more than one more iteration per thousand than the exact scheme. The difference is undetectable when N(ffl; This leads us to evaluate the asymptotic convergence rate when To obtain the behavior of oe(k; ffi) in (13) for large k, when into (13) to obtain (27) and seek an expansion for R(x) of the form We introduce (40) into (27) to obtain, after some manipulation, equations for / and K x Then, we solve (41) and (42) and substitute the results into (40) to obtain, with \Phi(j) defined in (37), and D a constant Hence, lim In view of (38) and (44) the asymptotic convergence rate is the same as that with The results (34) and (43) of this formal analysis can be made rigorous. We summarize the above analysis in the following theorem: Theorem 1 Assume that a linear system of equations is solved to accuracy ffl, using the Chebyshev iteration, with a variable strategy fffi k g. Assume that that positive constant C. Then, the asymptotic convergence rate of the Chebyshev iteration with the variable tolerance is the same as the asymptotic convergence rate of the scheme with the fixed tolerance j. 4 The optimal strategy problem Motivated by the result of section 3, we now wish to find the "best" sequence of tolerance values for the inner iterations. More precisely, we seek the sequence of tolerances that yields the lowest possible cost for the algorithm. To formalize this problem, we let , be a sequence of tolerance values. The jth component of ffi, is the tolerance, required in the solution of the subproblem at outer iteration j. Therefore and the number of inner iterations at step j is d \Gamma log e. In this estimate, ae is the convergence factor of the method which is used in the solution of the subproblem. Then, we define N(ffl;ffi) to be the number of outer iterations needed to reduce the initial error by a factor ffl when the problem is solved with strategy ffi: It follows that the total number of inner iterations required to achieve this accuracy ffl is proportional to log Our objective is to minimize C(ffl; ffi ) with respect to ffi. We consider the set S of slowly varying strategies In (46), the function ffi(x) is assumed to be twice continuously differentiable and ffi 0 denotes it's derivative. The condition ensures that ffi(fik) varies slowly as a function of k if fi - 1. In order to simplify the analysis, we use the fact that log Z N(ffl;ffi)log ffi(fit)dt; and redefine the cost as Z N(ffl;ffi)log ffi(fit)dt: (47) We can now restate the problem as follows. Find ffi 2 S such that 5 Error bound for slowly varying strategies Now we shall approximate the error bound (11), under the assumption that ffi 2 S. First, we obtain an asymptotic approximation for -(k; ffi), valid for fi - 1. To emphasize the fact that -(k; ffi) depends on fi, we denote it -(k; ffi; fi). To simplify the analysis we assume that the function ffi(x) is constant on [0; fi]. This assumption is not very restrictive since it requires only that we change the value of ffi 0 to equal . Moreover, since ffi k is slowly varying the impact of this change on the cost is negligible. The method we use is similar to the W.K.B method [13] for linear ordinary differential equations with a small parameter, and the ray method Keller [14] for linear partial differential equations with a small parameter. These methods have recently been adapted to linear difference equations with small parameters [12], [15]. We now obtain an approximate solution to equation (13) when belongs to S. Since we are looking for an asymptotic expansion of -(k; ffi; fi) for small fi, we introduce the new scaled variables Upon performing the change of variables (49) in (13), we obtain We seek an asymptotic expression for R(x; ffi; fi) for small fi, in the form The functions /(x; ffi), K(x; ffi), K 1 are to be determined to make R satisfy (50). Substitution of (51) into (50), and multiplication by e \Gamma/=fi yields e We now express each side of (52) in powers of fi, assuming that /(x etc. can be expanded in Taylor series in powers of fi. Then, we equate coefficients of powers of fi. The coefficients of fi 0 and of fi 1 yield tanh Solving (53) for / x yields with \Phi(ffi) given by (37). Integrating (55) yields, with a a constant of integration We now rewrite (54) as cosh / x sinh / x Integrating (57), with b a constant of integration, gives Now, we use expression (55) for / x in (58) to obtain To obtain the leading order term in -(k; ffi; fi), we substitute the two values (56) for / into (51) for R and add the two terms. Then, we use the result in (49) and set x j fik to find R fik\Phi(ffi(t))dt Here \Phi(ffi) is defined in (37) and K(x; ffi) is given by (59). The constants A and B are determined to make (60) satisfy the initial conditions (14): R fi R fik\Phi(ffi(t))dt Since ffi(x) is constant on [0; fi], (59) shows that K(0; R fi \Phi(ffi(0)). We substitute (61) into (60) to obtain, after some manipulation, sinh Z fik\Phi(ffi(t))dt R fik\Phi(ffi(t))dt When implies that / shows that K is also constant. Hence, (62) simplifies to the exact solution (36) of (13) and (14) when a constant. The exponentially decaying term in (62) can be neglected after a few outer iterations. Then we set in (62) and introduce the function sinh Now, we approximate -(k; ffi) by oe(k; ffi), and the bound for the error in the right hand side of (11) becomes In the next section, we shall analyze the validity of the approximation (64). 6 Validity of the asymptotic expansion Now we shall show that the leading order expression for -(k; ffi), given by (62), is indeed asymptotic to -(k; ffi) as fi ! 0. We denote this expression by - (k; ffi) and define the residual associated with it by r(k; ffi): To evaluate r(k; ffi) we substitute (60) for -(k; ffi) into (65) and then expand / and K in Taylor series, with remainders up to order fi 3 and fi 2 , respectively. We use (59) and (56) in the resulting expression to obtain, after some manipulation, Here and is independent of k and fi. The error in the asymptotic approximation, e(k; This equation is obtained by subtracting (13) from (65). The initial conditions for e(k; ffi) are Our goal is to show that for any constant C and all k - C To estimate the left side of (69), we obtain an explicit formula for e(k; ffi), by solving (67) and (68). We use the method of reduction of order [13]. Specifically, we seek a solution of the form where -(k; ffi) is the solution to equation (13), (14) and x k is to be determined. Upon substituting (70) into (69) we find that (69) will hold if We obtain an expression for x k by substituting (70) for e(k; ffi) into (67). Then, we eliminate from the resulting expression by using (13) and we find that Now, we introduce into (72) to obtain a linear first order equation for X k . The initial conditions (68) yield The solution of (72) and (74) is We take the absolute value of each side of (75) and use (66) to obtain e R k\Phi(ffi(fit))dt Here \Phi(ffi) is defined in (37). In lemma 1 we shall show that is bounded by a constant independent on k and fi. In lemma 2 we shall show that for a non-increasing strategy ffi(x) in S e where the constant P is independent of k and fi. We now use these bounds in the right side of (76) and conclude that for all k - 1 where the constant C is independent of k and fi. Equation (73) and the condition for x 1 in (74) determine x k through To derive the bound (71) for jx k j, we take the absolute value of each side of (79) and use (78) to obtain We summarize the above analysis in the following theorem: Theorem 2 Let -(k; ffi) satisfy (13) and (14). Let - (k; ffi; fi) be the expression on the right side of (62). Assume that ffi(x) is a non-increasing strategy and that ffi(x) 2 S with S defined in (46). Then, fi fi fi fi fi Furthermore, the coefficient of fi 2 in (81) is bounded by a linear function of k. We now briefly discuss the validity of the approximation (63). When is constant, (63) is exact up to an exponentially decaying term, and it is very accurate after a few iterations. When ffi is not a constant, the approximation is based on (62), which is valid for Therefore, the accuracy decreases as the number of outer iterations k !1, and for a fixed k, increases as fi ! 0. At the end of this section we present a few numerical calculations that demonstrate the accuracy of the expansion for a few variable strategies in S. As we shall see, even for large values of k, it is very accurate. and Proof: Inequality (82) is shown by induction. For it follows from initial conditions (14). Now assume by induction that (82) holds for all 1. Then from (13) By the induction hypothesis We use (85) in (84) to complete the induction. In order to prove (83), we recall from (46) that ffi k - j and we use this bound in (82) to obtain Furthermore, we note that Y We use (86) in (87) to obtain It follows that Inequality (83) follows from inequality (89). be a non-increasing strategy such that ffi(x) 2 S, with S defined in (46). Then e R k\Phi(ffi(fit))dt where the constant P is independent of k and fi. Proof: We note that when ffi(x) is a non increasing function of x, it follows from the monotonicity of \Phi(ffi) in (37) that We introduce the right side of (91) into (90) and use (37) for \Phi(x), to obtain e R k\Phi(ffi(fit))dt We now seek a lower bound on -(k; ffi). In view of the left condition in (14), we can as the product where It follows from (13) and (14) that ae j satisfies the equation with To obtain a lower bound for the product in (93), we introduce the sequence ae ae The number ae k is computed with the aid of the intermediate quantities ae as follows: ae ae We define ae In order to demonstrate (97), we show by induction on j that for all ae For it follows from (96), (98) and the fact that ffi(x) is non-increasing that ae Now, we assume that (101) is true for all it follows from (95), (99) and the fact that ffi(x) is non-increasing that ae ae ae The next step in the proof is to evaluate ae k explicitly and obtain a lower bound for it. This is done by solving the non-linear recurrence equation (99) for ae k;j , subject to the initial condition (98). We solve this equation with a method analogous to the one described in section 16.7 of [16] and obtain ae where From equation so that Furthermore, it follows from (105) and the definition of j in (46) where the equality on the right defines the constant -. We use (107) and (106) in (104) and obtain, in view of (100), ae Further manipulation of (108) yields ae Finally, we note that 1=(1 the latter inequality follows from (105). We use these bounds in (109) and use (97) to get We are now ready to prove the lemma. First, we substitute the right side of (93) for -(k; ffi) in (92). Then, we use (110) and (96) to find e The infinite product convergent because 1. Hence, the right side of (111) is bounded by a number P which is independent of fi and k. We now present a few numerical calculations that demonstrate the accuracy of the expansion derived in section 5. First, we solve (13) for -(k; ffi; fi) by iteration and then we compute the approximate solution oe(k; ffi) given by (63), for all 2 - k - 2000. We present the relative error in this approximation. We use strategies from the three parameter family A The minimal tolerance in (112) is In all our calculations ffi k AE j and for all practical purposes j can be neglected. The value of parameters A and fl is fixed at 1. The parameter B and the value of fi vary from one calculation to the other. The value of \Delta in (13) is set to 37. We performed analogous calculations with larger values of \Delta and with obtained similar results. In table 1, we present the maximum with respect to k, of the absolute value of the relative error in percent. Each entry in this table corresponds to a calculation with a different strategy. The strategy is determined by the parameters B and fi. Figure 1 depicts the relative error in percent between oe(k; ffi) and -(k; ffi; fi), for all 2 - k - 2000. Each graph corresponds to different values of B and fi. We note that the approximation is accurate even for large values of k. Relative error in 20000.020.06Relative error in -0.20.20.6Relative error in -0.020.020.06Relative error in Figure 1: The relative error j- (n; ffi; Each graph corresponds to different values of B and fi. 1:01 0:74 0:05 1:50 0:74 0:05 2:00 0:73 0:05 5:00 0:72 0:07 10:00 0:71 0:07 100:00 0:70 0:11 Table 1: The maximum over 2 - n - 2000 of j- (n; ffi; 7 Constant strategy is optimal using (47) and (64) we seek the optimal strategy for the Chebyshev iteration. The numbers N(ffl; ffi) and C(ffl; ffi) in (47), are hard to determine precisely. Therefore, we introduce the quantities NB (ffl; ffi) and CB (ffl; ffi), which are the number of outer iterations required to reduce the error bound (64) to ffl and the associated cost, respectively. The following theorem shows that a constant strategy is optimal. Theorem 3 Suppose that a linear system of equations is solved to accuracy ffl by the Chebyshev iteration using inner iterations with a sequence of tolerances fffi k g in S. There exists a constant strategy - ffi(ffi; ffl), for which the cost is smaller, i.e. Proof: Given the variable strategy ffi and the accuracy ffl used in the solution of the linear system, we define the associated constant strategy - R NB (ffl;ffi) In Lemma 3, we show that NB (ffl; - Therefore, In Lemma 4, we show that Using (115) in the right hand side of (114), proves the theorem. Lemma 3 Proof: By definition of NB (ffl; ffi) the bound for the error B(k; ffi) in (64) satisfies Therefore, to prove (116) it is sufficient to show that after NB (ffl; ffi) outer iterations, the bound for the error associated with the variable strategy is greater than the one associated with the constant strategy. Hence we need to show We see from (64) that (117) is equivalent to the inequality where oe is defined in (63). To prove (118) we begin by rewriting expression (63) for oe(k; ffi) with K(fiNB (ffl; ffi); ffi) sinh Then, we note from (37) that \Phi is monotonically increasing and that for all non-negative (46). Therefore, R NB (ffl;ffi) Furthermore, we see that K(fiNB (ffl; ffi); ffi)=K(0; ffi) - 1 from equation (59). Using this and (121) in the right hand side of (119) we obtain sinh@ NB (ffl; ffi)\Phi\Phi \Gamma1@ R NB (ffl;ffi) Lemma 4 Proof: The definition (37) of \Phi shows that \Phi . Therefore, strictly convex on the interval ffi)g. It follows from Jensen's inequality that log \Phi \Gamma1@ R NB (ffl;ffi) R NB (ffl;ffi) Multiplying (123) by NB (ffl; ffi) proves the lemma. We now show how to estimate the optimal constant - ffi. We note from (64) that for any iteration N Then, by equating the right side of (124) to ffl and using (37), we obtain log Re(cosh An estimate of the cost is then Re(cosh The right side of (126) can be minimized easily with respect to - using a standard minimization technique. The original variational problem (48) is thus reduced to a simple optimization problem. Since B(N; - ffi) approximates a bound for the error, the tolerance obtained by this method will be a lower bound for the optimal tolerance. The estimation of the optimal constant depends on the parameters - and ae in expression (126). These are often determined adaptively while solving the system [17]. 8 Numerical calculations We now present a few numerical calculations that verify the analysis of section 7. In each experiment, we solve a linear system with Chebyshev iteration to accuracy ffl, using a variable strategy ffi. Then, we solve the same system with the associated constant strategy defined in (113) with NB (ffl; ffi) replaced by N(ffl; ffi). We recall that N(ffl; ffi) is the exact number of outer iterations required to achieve an accuracy ffl, when solving the problem with strategy ffi. This number is obtained from our numerical experiment. Our goal is to verify that the predictions of lemma 3 and theorem 3 hold in practice. In section 4 we define the cost at outer iteration j by using log for the number of inner iterations required to achieve accuracy ffi j instead of d \Gamma log e. Here, ae is the convergence factor for the inner iteration. If ae is close to 1, then the relative error in using (127) is usually small and the cost (45) is truly proportional to the total number of inner iterations. In this case, we expect good agreement between the analysis and the numerical calculations. Moreover, we expect some fluctuations around the predicted behavior when ae - 1. We covered both cases in our experiments. We solve the symmetric system arising from the central difference discretization of the operator in the interval [0; 1] with homogeneous Dirichlet boundary conditions. The right side b in (128) is chosen at random. The splitting matrix M is obtained from the discretization of the operator with homogeneous Dirichlet boundary conditions. The mesh parameter in this discretization is 1=100. The tolerance for the outer iteration is . The initial iterates for both the inner and outer iterations are 0. In all our experiments, we use strategies from the family (112). The values of fl and A are fixed at 1. The parameter B and the value of fi vary from one experiment to the other. For each variable strategy ffi, the associated constant strategy - ffi is computed using (113) with NB (ffl; ffi) replaced by N(ffl; ffi). We note from (113) that \Phi depends on \Delta. We evaluate exactly but find that - ffi is not very sensitive to the value of \Delta. We performed calculations with various values of C in (129) and (130) and we shall report on a representative sample obtained with We use two methods for the inner iteration. The symmetric Gauss Seidel, with the convergence factor 0:993, close to 1, and the symmetric successive over relaxation method [18] (S.S.O.R) with the smaller convergence factor 0:925. In the S.S.O.R iteration, the relaxation parameter ! is the optimal parameter ! of S.O.R. In each experiment, we record the number of outer iterations and the total number of inner iterations for the variable and constant strategy cases. Tables 2-5 correspond to the case where the inner iteration is symmetric Gauss Seidel. In table 2 we report the difference in the total number of inner iterations between the variable strategy case and the associated constant strategy case. All entries in the table are in (%) and are computed from N in (ffl; ffi) \Gamma N in (ffl; - ffi) N in (ffl; - 100: (131) Here N in (ffl; ffi) is the total number of inner iterations performed when solving the system to accuracy ffl with strategy ffi. Each entry in table 2 corresponds to a different strategy. The strategy is determined by the parameters B and fi. Note that not all strategies are slowly varying since fi 6- 1 in the two rightmost columns of that table. The important thing to note in table 2 is that all entries are positive. Therefore, the number of inner iterations associated with the variable strategy is greater than or equal to the number of inner iterations with the constant strategy. Hence, there is agreement with Theorem 3. In table 3, we present the difference in the number of outer iterations between the variable strategy case and the constant strategy case i.e. N(ffl; We see that all entries are non-negative and there is very good agreement with Lemma 3. In table 4 we present the total number of inner iterations with the associated constant strategy. The lowest number of inner iterations is found at the top left entry. This entry corresponds to the lowest tolerance for the inner iteration. Table 5 presents the total number of outer iterations. We see that the top left entry maximizes the number of outer iterations. Hence, among all strategies considered in this table, the strategy which yields the lowest convergence rate also yields the lowest cost. Tables 6 and 7 present the difference in number of inner and outer iterations, respec- tively, when the inner iteration is S.S.O.R. Since the convergence factor is not close to 1 some fluctuations from the predicted behavior are expected. Indeed, two entries in table 6 are negative. However, the fluctuations are small and the constant strategy performs essentially as well as the variable one. In our numerical calculations we have used both slowly varying strategies, "rapidly" varying ones, Although our theory was developed for slowly varying strategies, the conclusion of theorem 3 is found to hold for all the strategies considered. 9 Generalization to other iterative procedures We now consider a general iterative algorithm in which, at iteration k, a subproblem is solved by an inner iteration to accuracy ffi k . The norm of the error at step k, e k , satisfies Table 2: The difference in number of inner iterations (N in (ffl; ffi) \Gamma N in (ffl; - ffi))=N in (ffl; - ffi) in (%). The tolerances ffi k and - are defined by (112) and (113), respectively. Inner iteration is symmetric Gauss Seidel. Table 3: The difference in number of outer iterations N(ffl; ffi). The tolerances are defined by (112) and (113), respectively. Inner iteration is symmetric Gauss Seidel. 1:50 4458 5144 5369 6203 2:00 4666 5143 5571 6324 5:00 5199 6208 6444 7447 10:00 5697 6722 7167 8093 100:00 8660 9423 10228 11137 Table 4: The number of inner iterations N in (ffl; - ffi) with - ffi given in (113). Inner iteration is symmetric Gauss Seidel. Table 5: Number of outer iterations given in (113). Inner iteration is symmetric Gauss Seidel. 1:01 2:36 8:60 4:29 8:62 1:50 2:26 3:66 4:34 3:42 2:00 0:98 3:43 4:79 4:76 5:00 0:64 5:87 4:64 5:44 10:00 \Gamma0:77 \Gamma0:67 5:93 0:14 100:00 0:42 0:60 0:78 1:03 Table The difference in number of inner iterations (N in (ffl; ffi) \Gamma N in (ffl; - ffi))=N in (ffl; - ffi) in (%). The tolerances ffi k and - are defined by (112) and (113), respectively. Inner iteration is S.S.O.R. Table 7: The difference in number of outer iterations N(ffl; and - ffi are defined by (112) and (113), respectively. Inner iteration is S.S.O.R. the relation In (132), ae(k; x convergence factor at step k, depends on the initial iterate x 0 and on the sequence of tolerance values ffi. We assume that ae(k; x product with Hence, the only tolerance upon which ae(k; x the tolerance at outer iteration k. Furthermore, the dependence of ae(k; x is the same at each iteration of the algorithm. We can prove a result similar to the one of section 7 for an iteration satisfying (133). Theorem 4 Consider an iterative algorithm in which at step k, a subproblem is solved by inner iteration to accuracy ffi k . Assume that the norm of the error satisfies (132), with of the form (133). Assume that g(OE) is a convex non decreasing function. Let be the reduction of the error after N outer iterations. Then, for any variable strategy ffi and any number of outer iterations N , there exists a constant - ffi(N; with the following properties. 1. After N outer iterations with the constant tolerance - ffi(N; ffi ) for the inner iteration, the error is reduced by exactly ffl(N; ffi ). 2. The cost (45) of performing N outer iterations with the constant tolerance - is lower than the cost of performing N outer iterations with the variable tolerance ffi. In other words, for such an iteration a constant strategy is optimal. Proof: From (132) and (133) we find that after N outer iterations of the algorithm with the variable tolerance ffi, the error is reduced by Let and - ffi(N; Then, it follows from equations (132) and (133) that after N iterations with the constant strategy - the error is reduced by The right hand side of equation (137) is exactly ffl(N; ffi ). Using (45) and (136) we find that the cost associated with N steps of the constant tolerance iteration is while the cost associated with the variable tolerance is Now, the right side of (138) is no greater then the right side of (139) since g is convex. The error bound (64) for the Chebyshev iteration is analogous to (135) with the sum over g(OE) replaced by an integral and the term e replaced by a function F (k; x 0 ), independent of ffi. Hence, theorem 3 is essentially a continuous version of theorem 4. The proof of the former is complicated by the presence of the amplitude term (59) in (63). Acknowledgements This work was supported in part by NSF under cooperative agreement no CCR-9120008 and grant CCR-9505393, ONR, and AFOSR. --R Chebyshev semi-iterative methods Hybrid Numerical Asymptotic Methods. On the interplay between inner and outer iterations for a class of iterative methods. The convergence of inexact chebyshev and richardson iterative methods for solving linear systems. On the local convergence of certain two step iterative procedures. Accelerating the convergence of discretization algorithms. On the convergence of two-stage iterative process for solving linear equa- tions Inexact and preconditioned uzawa algorithms for saddle point problems. The tchebyshev iteration for nonsymmetric linear systems. Eulerian number asymptotics. Advanced Mathematical Methods for Scientists and En- gineers Rays, waves and asymptotics. The wkb approximation to the g/m/m queue. Ordinary Differential Equations. Adaptive procedure for estimation of parameters for the nonsymmetric tchebychev iteration. Matrix iterative analysis. --TR

inexact iteration;iterative methods;inner iteration

275988

The Value Function of the Singular Quadratic Regulator Problem with Distributed Control Action.

We study the regularity properties of the value function of a quadratic regulator problem for a linear distributed parameter system with distributed control action. No definiteness assumption on the cost functional is assumed. We study the regularity in time of the value function and also the space regularity in the case of a holomorphic semigroup system.

Introduction . In this paper we are concerned with a general class of finite horizon linear quadratic optimal control problems for evolution equations with distributed control and non-definite cost. More precisely, we consider the following abstract differential equation over a finite interval [-; T where A is the infinitesimal generator of a strongly continuous semigroup e At on a Hilbert space X , B is a linear bounded operator from the control space U to X . With the dynamics (1.1), we associate the cost functional F is the mild solution to equation (1.1) and F is the quadratic (we denoted by h\Delta; \Deltai inner products in both the spaces X and U ). All the operators Q, S, R and P 0 contained in the functional (1.2) are linear bounded operators in the proper spaces, with . We define as usual the value function of the problem: The goal of the present work is ffl to characterize the property This research was supported by the Italian Ministero dell'Universit'a e della Ricerca Scientifica e Tecnologica within the program of GNAFA-CNR. y Dipartimento di Matematica Applicata, Universit'a di Firenze, Via S. Marta 3, 50139 Firenze, Italy (fbucci@dma.unifi.it). z Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy (lucipan@polito.it), partially supported also by HCM network CEC n. ERB-CHRX-CT93- ffl to study the regularity properties of the map on the interval [0; T ], when x 0 is fixed. We shall consider also the map x in a special case, see x6. It is well known that if the regulator problem is standard, i.e. then the solution to the operator Riccati equation corresponding to problem (1.1)- (1.2) provides the synthesis of the unique optimal control. This problem is well understood, both in finite and infinite dimensions, over a finite or infinite time horizon (compare [10], [2, 3]). The purpose of this paper is to examine the case when (1.5) fails, with special interest in non-coercive R. We shall see that in this case the function mild regularity properties, see x4. More regularity is obtained in the coercive case, see x5. The study of LQR problems with non-definite cost is related to a large variety of problems. Among them, we recall the study of dissipative systems (see [20]), the analysis of the stability of feedback systems ([14]), the analysis of second variations of non-linear optimization problems (see [5], [15]). When game theory is studied for linear systems then the quadratic form (1.3) is non-positive. In particular, the suboptimal H 1 -problem can be recast in this setting ([1]). Finally, very recently singular control theory has been used to obtain new results on regular control problems for some class of boundary control systems: systems with input delays first [16], and later systems described by wave- or plate-like equations with high internal damping [9]. We recall that the existing results for finite dimensional systems over an infinite time interval ([19, 21], see also [4]) were extended to distributed systems in [22, 23, 12, 13, 8]. If T ! +1 the only work we know in an infinite dimensional context, in which a nonpositive cost functional is studied, is [6]. This paper considers even time-varying systems, but under the restriction 2. A simple example. The interest of the results presented in this paper is justified by the possible applications that we already quoted, for instance to H 1 - control theory over a finite time interval, or to the analysis of the second variation of general cost functionals. However the following example may help the reader to understand our problem. The example is a bit artificial, since we want to present a very simple one. Nevertheless it is suggested by non trivial problems in network theory. A delay line in its simpler form is described by an input-output relation and the integral is a Stieltjes integral. For simplicity we assume that the input u(\Delta) is continuous, a condition that can be very much relaxed. The simplest case described by (2.1) is and corresponds to a jump function j, with jump at \Gamma1. If the system is started at then the input (2.1) is read only for t ? 0, so that the output v(\Delta) from (2.1) is given by The function OE describes the "initial state" of the system (quite often it will be In the case of equation (2.2) we have in particular Notice that if OE(\Delta) and u(\Delta) are regular then v(t; solves the first order hyperbolic equation The function v can be interpreted as a delayed potential at the output of the network produced by the potential u(\Delta) at the input. If the delay line is connected to a resistive load, it produces a current and the energy disspated by the load in time T is given by Z Since then The energy that the load can dissipate is at most u(\Delta) We see from this that the load dissipates a finite amount of energy V (OE) if T ! 1, described by the quadratic functional Z Otherwise, the load can dissipate as much energy as we want. Hence it makes sense to study the energy function E(T ) In this example the function E(T ) is finite only if T ! 1, and in this case E(T ) is the quadratic functional (2.3). In this paper we consider an analogous problem in more generality: we study the dependence on the interval [-; T ] of the "energy" dissipated by a certain linear time invariant system. 4 F. BUCCI AND L. PANDOLFI 3. Preliminary results. We recall that the solution to (1.1) is with e A(t\Gammas) Bu(s) ds; continuous Note that t ! (L - u)(t) is a X-valued continuous function. The adjoint L - of is given by (L e A (s\Gammat) f(s) ds; continuous Introduce also the bounded operator from U to X e A(T \Gammas) Bu(s) ds (which describes the map (3.1) from the input u to the solution of (1.1) at time with initial time - and x 0). The adjoint of L -;T is the map given by (L Using (3.1), one can easily show the following Lemma 3.1. The cost functional (1.2) can be rewritten as with selfadjoint, defined as follows (R We first state a Lemma, which will be useful later. Lemma 3.2. If there exists - 0 and a constant fl such that then R - fl I for any - 0 . Proof. It is sufficient to notice that if - 0 we can write where v(\Delta) is given by . Hence, from (3.7) it follows that R - fl I for any - 2 [- We shall use the following general result pertaining continuous quadratic forms in Hilbert spaces whose proof is given for the sake of completeness. Lemma 3.3. Let X and U be two Hilbert spaces, and consider with 1. If there exists x 2 X such that u2U 2. The infimum of f(x; \Delta) is attained if and only if the equation is solvable and in this case any solution u of (3.8) gives a minimum. 3. If for each x 2 X there exists a unique u x such that f(x; then R is invertible (the inverse R \Gamma1 may not be bounded) and u so that the transformation x ! u x is linear and continuous from X to U . 4. Let us assume that V (x) ? \Gamma1 for each x 2 X. Then there exists a linear bounded operator P 2 L(X) such that Proof. If there exists v such that hRv; vi ! 0 then f(x; This proves the first item of the Lemma. The second item is well known ([23, Lemma 2.3]). To prove the third item we use item 2: the minimum u x is characterized by (3.8). This equation is uniquely solvable for every x by assumption. Hence, ker and im N ' im R. Consequently, acts from the closure of the image of R. Hence, R \Gamma1 N is bounded since R \Gamma1 is closed and N is bounded. The proof of the fourth item follows an approach in [7]. If R is coercive, then it is boundedly invertible, so that f(x; \Delta) admits a unique minimum, namely \GammaR Hence, (3.9) holds true and we have obtained an explicit expression for P , i.e. If we simply have R - 0, we consider the function 6 F. BUCCI AND L. PANDOLFI Now n I is coercive, hence Vn with Pn 2 L(X). By construction is a decreasing numerical sequence for any x 2 X , and hence there exists P 2 L(X) such that To conclude, it remains to show that V (x) coincides with hx; Pxi for any x 2 X . Assume by contradiction that V (x) ! hx; Pxi for a given x 2 X , and let ff ? 0 such that there exists u 2 U such that Correspondingly, there exists an integer n 0 2 IN such that From (3.11) and (3.12) it follows which is a contradiction, compare (3.10). The above lemma and (3.3) imply a first necessary condition for finiteness of the value function. Lemma 3.4. If there exists x 0 such that V (-; x This observation is now used to obtain a necessary condition of more practical interest, which is well known in the finite dimensional case. The symbol I denotes the identity operator acting on a space which will be clear from the contest. Proposition 3.5. If there exists - 0 2 [0; T ) and a constant fl - 0 such that Consequently, if there exists x 0 and - 0 such that V (- (3. Proof. We first consider the case hence by assumption R -0 - 0. By contradiction, suppose that there exists a control u 0 2 U and a constant ff ? 0 such that \Gammaff. Given a small ffl ? 0, choose a control u as follows: ae and compute e A(t\Gammas) Bu 0 ds; e A(t\Gammas) Bu 0 dsi dt e A(T \Gammas) Bu 0 ds; e A(T \Gammas) Bu 0 dsi dt tends to 0: Since ffl can be taken arbitrarily small, (3.15) yields hR -0 u; ui ! 0, and this contradicts the assumption. Assume instead R -0 - fl I ? 0. By choosing direct computation yields which implies hRu Finally, if V (- , then from Lemma 3.4 it follows that R - is a non-negative operator for - 0 . Therefore from the previous part of the proof, R - 0. We now show that the value function satisfies the Bellmann's optimality principle which is known, in the context of linear-quadratic problems, as "Linear Operator Inequality" (LOI) or "Dissipation Inequality" (DI). We begin with the following Lemma 3.6. If for some number - and some x then we have also V (t; denotes the value at time t of the function given by (3.1), for any fixed control u(\Delta) on [-; t]. Proof. Let ds ds F now a control v j 0 on [-; t): then ds 8 F. BUCCI AND L. PANDOLFI and ds Conclusion immediately follows since in fact Theorem 3.7. Let - 2 [0; T ] and x 0 2 X be given. Let V be the value function of problem (1.1), (1.2) and assume that V (-; x ds for any u(\Delta) 2 L 2 (-; T ; U) and any t 2 (-; T ), with Moreover, the equality holds true if and only if the control u in(3.17) is optimal. Proof. We return to the conclusion of the preceding Lemma, and observe again that while hence plugging (3.18) into (3.16) and taking into account (3.19), we get F which is nothing but (3.17). Thus, if for a given initial datum x 0 there exists an optimal control u then we can rewrite (3.16) and (3.19) with is in fact an equality. Therefore (3.20) becomes an equality as well. For these arguments compare also [11]. Viceversa, assume that (3.17) is satisfied for any control u 2 L 2 (-; T ; U) and it is an equality for a given u . Then, passing to the limit, as and assuming for the moment that lim we readily get ds that is hence by definition u is optimal. To conclude, it remains to show that if (x ; u ) satisfies ds then (3.21) holds true. From (3.22) it follows that there exists lim and by the very definition of the value function it follows lim To see this rewrite the above limit as lim By contradiction, assume now that lim where fl is a suitable positive constant. Then, there exists for any t 2 Recall now that hence we can rewrite ds ds A3 Take a possibly smaller ffi, in order to get so that (3.23) yields Finally, let ffi such that hQe A(s\Gammat) x (t); e A(s\Gammat) x (t)ids Fix now t 2 (T \Gamma ffi; T ), so that (3.24) and (3.26) hold true. From (3.25) it follows that there exists a control such that that is, by means of (3.3), with M t , N t , R t defined in (3.4), (3.5) and (3.6), respectively. We know that R T ds. Thus we cancel the term A 1 , we take into account (3.26) and we obtain In particular this implies that v t 6= 0. Notice now that const \Deltajv t (\Delta)j 2 and therefore lim inf Hence there exists a sequence t n such so that we see from (3.27) for n largejv t n In other words J t n this is a contradiction since by assumption J - (0; u) is non-negative for any u 2 The next Proposition is an immediate consequence of Lemma 3.1 and of Lemma 3.3. We omit the proof. Proposition 3.8. Let - 2 [0; T ]. If there exists a selfadjoint operator W (\Delta) 2 L(X) such that W (T 4. Time regularity of the value function: the non-coercive case. In this section we investigate the regularity properties of V (-; x 0 ) with respect to the initial time - . We note that several regularity results are known for the value function even of non-linear systems, and with more general cost but under special boundedness properties, which are not satisfied in the present case, compare [11, Ch. 6]. Our first result is: Lemma 4.1. Let - be such that V (- upper semicontinuous at - 0 . Proof. Fix In order to show that lim sup we shall show that for any real number ff ? V (- is taken small enough. We first consider the case when - 0 . Let u be an admissible control such that and define e A(t\Gammas) Bu(s) ds: It is readily verified that 1: lim 2: lim ds so that if Finally, if - 0 , choose once more in such a way that (4.1) holds true. It is now sufficient to repeat tha same arguments used before, after replacing u with - u defined as follows: The proof is complete. As to lower semicontinuity, the following result holds true. Lemma 4.2. Let x 0 be such that is finite on [0; T ]. Then ffl the map lower semicontinuous at - 0 provided that for each element - n of a sequence f- n g which tends monotonically to - 0 there exists a control such that ii) there exists Proof. Let be given, and consider a sequence f- n gn2I N such that - n # - 0 . Introduce the inputs and define Notice that x -n x -n (t) ! 0, as n !1, for any t, and that its norm is uniformly bounded in L 2 (- lim Therefore lim inf where the last equality is due to i). On the other hand ii) implies the existence of an admissible such that as n !1. Now the map is convex continuous, hence weakly lower semi-continuous, so that To conclude the proof, we need to consider a sequence fr n gn2I N such that r n " - 0 . In this case, we introduce ~ ae Again from ii) it follows that there exists an input v 2 L 2 (- such that ~ in similar argument gives which finally yields Consequently, we can conclude Theorem 4.3. Under the same assumptions as Lemma 4.2, the map - ! continuous for any - 2 [ 0; T ]. In the case that an optimal control exists for each - near - 0 , Lemma 4.2 takes a simpler form. We state this form, under the assumption that an optimal control exists for each - . Corollary 4.4. Let x 0 2 X be fixed. Assume that there exists an optimal control u ii) there exists a constant fl ? 0, independent of - , such that (4. Under these conditions, the map We note explicitly that if there exists an optimal control u for J - each there exists an optimal control for J - 0 (x(- It has some interest to see that if the operator A generates a strongly continuous group then we can prove more: Theorem 4.5. Let us assume that for each - 2 [0; T ) and each x 0 2 X there exists a unique optimal control u At is a strongly continuous group then the value function is continuous from the right. Proof. We prove continuity from the right at a fixed - We know from Lemma 3.3 item 2 that x linear and continuous from X to L 2 (-; for each - 2 [0; T ). Now we consider points - 0 . We show that for each fixed - 0 there exists It is sufficient to see for this that there exists a solution x 1 of e A(- \Gammas) Bu If this is true, unicity of the optimal control shows that (4.5) holds. We noted above that ku so that the norm of the operator e A(- \Gammas) Bu ds can be estimated as follows: kT x 1 We write Eq. (4.6) in the form sufficiently small, is less then 1 hence Eq. (4.7) can be continuously solved for x 1 and gives a linear continuous transformation x which, of course, depends upon - . The vector x 1 , is continuous with respect to x 0 and also with respect to - if - is close to - 0 . In bounded in a neighborhood of - 0 . Therefore, Right continuity follows from Lemma 4.2. The previous theorem presents a case in which the quite involved condition of Lemma 4.2 is satisfied. The next example shows that the condition in that lemma cannot be avoided if we are to obtain continuity of the value function. We note first that the value function is not continuous in general, even for finite dimensional systems: if the cost is jx(T )j 2 and the system is controllable then the 14 F. BUCCI AND L. PANDOLFI value function has a jump at T . The following example shows that the value function may be discontinuous even at points - ! T . Example 4.6. Consider the delay system given by with initial datum OE 0 =col[x 0). The quadratic functional is Consequently In particular J 1 On the other hand, if and it can be arbitrarily fixed, by means of suitable choices of the control u, within the class of W 1;2 functions which are zero at 1. This set is dense in L suitable functions y can be found in order to drive x(t) to zero in time ffl ? 0, namely remaining uniformly bounded. Therefore we have that In conclusion, if and the value function is not continuous at Remark 4.7. The previous example shows that in the statement of Lemma 4.2- which concerns lower semicontinuity of V (-; x 0 )-assumption ii) cannot be dispensed with. In fact that assumption holds in the previous example for but not for 5. Time regularity of the value function: the coercive case. Let - 2 [0; T be given, and consider the operator R - - as defined in (3.6). Throughout this section we shall assume that R - is coercive, i.e. Our present goal is to show that under assumption (5.1) the value function V (-; x 0 ) displays better regularity properties with respect to - . We start by showing that the is continuous for any - 2 We recall that from (5.1), by virtue of Lemma 3.2, it follows that R - fl for any by continuity also on an interval (- Hence there exists a constant fl 0 such that Moreover (5.1) implies that for any initial time there exists a unique optimal control u - (\Delta) for short ), explicitly given in terms of the initial state by (compare item 3 of Lemma 3.3); and from (5.2) it follows independent of - : The following Theorem provides a simple explicit expression of the value function in terms of the optimal pair which will be useful in the next section. Theorem 5.1. Let R - be coercive, and let pair for problem (1.1)-(1.2). Then e A (t\Gamma- ) dt: Proof. Since the infimum of the cost is attained at - for short), plugging (5.3) into (3.3) we easily obtain The adjoint operator N )-function v in e A (t\Gamma- ) ((Q hence (5.5) follows from (5.6) by a direct computation. As a consequence of Corollary 4.4, we first have Theorem 5.2. Let x 2 X be given. Assume that(5.1) is satisfied. Then - ! is continuous on [- ; T ]. Actually we are able to show that the value function satisfies a further regularity property. Before we state a preliminary result. Lemma 5.3. Assume that R - is coercive. If w(\Delta) is a continuous function, then the function (R is continuous for any - - . In particular, if R - is coercive then the optimal control is continuous. Proof. Since R - - is coercive, R is coercive, so that we can assume that R = I . Moreover, for any - , R - is coercive, hence invertible. Let OE(t) := (R \Gamma1 - w)(t), with w(\Delta) continuous: we know that OE(\Delta) is at least an U - valued e A (s\Gammat) Q e A(s\Gammar) BOE(r) dr ds e A(t\Gammas) e A (s\Gammat) SOE(s) ds e A(T \Gammas) BOE(s) ds; and the second hand side is apparently an U -valued continuos function. The second statement follows from (5.3) since (N - x 0 )(\Delta) is a continuous function, compare (3.5). Theorem 5.4. Let x 2 D(A) be given. Assume that (5.1) is satisfied. Then the differentiable in [- ; T ]. Proof. Let x optimal control of problem (1.1)-(1.2), - - . As in (5.6) with M - and N - given by (3.4), (3.5), respectively. From the very definition of M - it readily follows that the derivative @ @- exists for any x 0 2 D(A). In order to show that the the second summand in (5.8), namely is differentiable with respect to - , we first observe that the factor (N - x 0 )(\Delta) is differ- entiable, with @ Moreover, again from (3.5) it follows that (5.10) is a continuos function. We next want to show that for each t ? - the U -valued function first derivative with respect to - and that this is continuous. Fix - 0 and consider first the case - 0 . Introduce the operator - defined as follows: By construction and for instance Moreover, we take into account (5.10) and we see that lim In fact it is sufficient to observe that Now we compute, via The first summand in (5.12) tends to @ '- , due to (5.11). As to the second summand, it can be rewritten in the following way: e A (s\Gammat) Q 1 e a(-;s) ds We rewrite, in turn, e (r)dr c(-;s) Observe now that as a consequence of Lemma 5.3 we have lim while lim - +c(-; Finally, since (-; s) ! a(-; s) is bounded, we can conclude that 1 converges to \GammaR e A (s\Gammat) Qe A(s\Gamma- as - tends to - . The convergence of the terms 2 and 3 can be proved even more easily. If - 0 we define instead and rewrite the term R \Gamma1 (R -0 (R -0 The rest of the proof is completely similar. Therefore we have proved that for each - there exists @ - (t) and that @ e A (s\Gammat) Qe A(s\Gamma- ds \GammaR In conclusion we saw that the function (N - x 0 )(t) - (t) is differentiable with respect to - , and moreover its derivative is a continuous function in [- Therefore (5.9) is differentiable, and @ @- We are now able to deduce a differential form of the Dissipation Inequality. Proposition 5.5. Assume that (5.1) holds true. Then there exists a selfadjoint operator W (\Delta) 2 L(X) such that d d- for any (a; v) 2 D(A) \Theta U , for any - 2 [- ; T ]. Proof. We fix a 2 D(A), v 2 U , and take a control u(\Delta) 2 C 1 ([-; T ]; U) such that u). It is well known (see for instance [2]) that in this case x is a strict solution to (1.1), that is x it satisfies (1.1) on [-; T ]. We write the dissipation inequality (3.17) for namely F If we divide in (5.15) by d ds To conclude, substitute We proved that if we replace an optimal pair in the left hand side of the dissipation inequality in integral form, then we get an equality. Hence we get an equality also in the differential form (5.14). In particular, we fix a 2domA and we see that a) is a minimum of the left hand side of inequality (5.14). Hence we find that Since R is coercive then R is coercive too and we see that the optimal control has the well known feedback form (if a 2 D(A) and, by continuity, for each a 2 X , see item 3 of Lemma 3.3). Moreover, as a)), the previous equality gives the feedback form of the optimal control on the interval [0; T ]. We replace this expression for the unique optimal control in the left hand side of (5.14) and we find a quadratic differential equation for W (-) which is the usual Riccati equation. Of course, the Riccati equation can be written provided that R \Gamma1 is a bounded operator. But, an example in [6] shows that if R is not coercive then the minimum of the cost may exist and be unique, in spite of the fact that the corresponding Riccati equation is not solvable on [-; T ]. 6. Space regularity of the value function. This section is devoted to the study of some space regularity properties of the value function in the case that the optimal control problem is driven by an abstract equation of parabolic type. See [17] for analogous arguments. More precisely, we shall make the following assumption: H1: A is the generator of an analytic semigroup e tA on X . It is well known (see for instance [18]) that in this case there exists a ! 2 IR such that the fractional powers are well defined for any ff 2 (0; 1), and moreover there exist constants M ff , fi such that the following estimates hold true (6. 20 F. BUCCI AND L. PANDOLFI For the sake of simplicity we assume that the semigroup is exponentially stable, i.e. that we can choose We associate the following output to system (1.1): where y belongs to a third Hilbert space Y and C 2 L(X; Y ), assume that the cost penalizes the output y i.e. that the quadratic functional F in (1.3) is given by so that special and important case is We now use similar arguments as in Lemma 3.3. Introduce a regularized optimal control problem with cost given by and observe that since the operator n I is coercive for each n, then there exists a unique optimal control u n and Vn Arguing as in the proof of statement 4 in Lemma 3.3 we know that (\Delta). Then we have the following Lemma 6.1. Let assume that there exists a number Then there exists a constant c such that Proof. The estimate is easily obtained as follows (note that 0 2 ae(A) since we assumed Remark 6.2. We stress that since the estimate (6.4) is uniform with respect to n and - . Lemma 6.3. Under the same assumptions of Lemma 6.1 there exists a constant k such that Proof. Let We recall that since by construction the operator R -;n relative to Jn (- 0 ; u) is coercive for each fixed n, then the regularized control problem admits a unique optimal pair Theorem 5.1 yields e A (t\Gamma- ) C y The regularity assumptions on C and P 0 imply that Wn (- Now, as a consequence of (6.4) there exists k such that uniformly in n. Conclusion follows immediately by choosing - Consequently we have the following Theorem 6.4. Under the same assumptions of Lemma 6.1 the operator admits a bounded extension to X for any fl ! . --R Representation and Control of Infinite Dimensional Systems The Riccati equation Singular Optimal Control: the Linear-Quadratic Problem Linear quadratic optimal control of time-varying systems with indefinite costs on Hilbert spaces: the finite horizon problem Spectral thoery of the linear quadratic optimal control problem: discrete-time single-input case Equivalent conditions for the solvability of the nonstandard LQ-Problem for Pritchard-Salamon systems A singular control approach to highly damped second-order abstract equations and applications Differential and Algebraic Riccati Equations with Application to Boundary/Point Control Problems: Continuous Theory and Approximation Theory Optimal Control Theory for Infinite Dimensional Systems The frequency theorem for equations of evolutionary type The Hilbert space regulator problem and operator Riccati equation under stabilizability Some nonlinear problems in the theory of automatic control Nonnegativity of a quadratic functional The standard regulator problem for systems with input delays: an approach through singular control theory Semigroups of Linear Operators and Applications to Partial Differential Equations Least squares stationary optimal control and the algebraic Riccati Equation The frequency theorem in control theory --TR

distributed systems;value function;quadratic regulator

276242

Two-Dimensional Periodicity in Rectangular Arrays.

String matching is rich with a variety of algorithmic tools. In contrast, multidimensional matching has had a rather sparse set of techniques. This paper presents a new algorithmic technique for two-dimensional matching: periodicity analysis. Its strength appears to lie in the fact that it is inherently two-dimensional.Periodicity in strings has been used to solve string matching problems. Multidimensional periodicity, however, is not as simple as it is in strings and was not formally studied or used in pattern matching. In this paper, we define and analyze two-dimensional periodicity in rectangular arrays. One definition of string periodicity is that a periodic string can self-overlap in a particular way. An analogous concept is true in two dimensions. The self-overlap vectors of a rectangle generate a regular pattern of locations where the rectangle may originate. Based on this regularity, we define four categories of periodic arrays--- nonperiodic, lattice periodic, line periodic, and radiant periodic---and prove theorems about the properties of the classes. We give serial and parallel algorithms that find all locations where an overlap originates. In addition, our algorithms find a witness proving that the array does not self-overlap in any other location. The serial algorithm runs in time O(m2) (linear time) when the alphabet size is finite, and in O(m2log m) otherwise. The parallel algorithm runs in time O(log m) using O(m2) CRCW processors.

Introduction String matching is a field rich with a variety of algorithmic ideas. The early string matching algorithms were mostly based on constructing a pattern automaton and subsequently using it to find all pattern appearances in a given text ([KMP-77, AC-75, BM-77]). Recently developed algorithms [G-85, V-85, V-91] use periodicity in strings to solve this classic string matching problem. Lately, there has been interest in various two-dimensional approximate matching problems, largely motivated by low-level image processing ([KS-87, AL-91, AF-91, ALV-90]). Unlike string matching, the methods for solving multidimensional matching problems are scant. This paper adds a new algorithmic tool to the rather empty tool chest of multidimensional matching techniques: two-dimensional periodicity analysis. String periodicity is an intuitively clear concept and the properties of a string period are simple and well understood. Two-dimensional periodicity, though, presents some difficulties. Periodicity in the plane is easy to define. However, we seek the period of a finite rectangle. We have chosen to concentrate on a periodicity definition that implies the ability for self-overlap. In strings such an overlap allows definition of a smallest period whose concatenation produces the entire string. The main contribution of this paper is showing that for rectangles also, the overlap produces a "smallest unit" and a regular pattern in which it appears in the array. The main differences are that this "smallest unit" is a vector rather than a sub-block of the array, and that the pattern is not a simple concatenation. Rather, based on the patterns of vectors that can occur, there are four categories of array periodicity: non-periodic, line periodic, radiant periodic and lattice periodic. As in string matching this regularity can be exploited. The strength of periodicity analysis appears to lie in the fact that it is inherently a two-dimensional technique whereas most previous work on two-dimensional matching has reduced the matrix problem to a problem on strings and then applied one-dimensional string matching methods. The two dimensional periodicity analysis has already proven useful in solving several multi-dimensional matching problems [ABF-94a, ABF-93, ABF-94b, KR-94]. We illustrate with two examples. The original motivation for this work was our research in image preserving compression. We wanted to solve the following problem: Given a two-dimensional pattern P and a two-dimensional text T which has been compressed, find all occurrences of P in T without decompressing the text. The goal is a sublinear algorithm with respect to the size of the original uncompressed text. Some initial success in this problem was achieved in [ALV-90], but their algorithm, being automaton based, seems to require a large amount of decompression. In [AB-92b, ABF-94b], we used periodicity to find the first optimal pattern matching algorithm for compressed two-dimensional texts. Another application is the two-dimensional exact matching problem. Here the text is not com- pressed. Baker [B-78] and, independently, Bird [Bi-77] used the Aho and Corasick [AC-75] dictionary matching algorithm to obtain a O(n 2 log j\Sigmaj) algorithm for this problem. This algorithm is automaton based and therefore the running time of the text scanning phase is dependent on the size of the alphabet. In [ABF-94a] we used periodicity analysis to produce the first two dimensional exact matching algorithm with a linear time alphabet independent text scanning phase. Since the work presented here first appeared [AB-92a], the analysis of radiant periodic patterns has been strengthened [GP-92, RR-93], and periodicity analysis has additionally proven useful in providing optimal parallel two dimensional matching algorithms [ABF-93, CCG+93], as well as in solving a three dimensional matching problem [KR-94]. This paper is organized as follows. In Section 2, we review periodicity in strings and extend this notion to two dimensions. In Section 3, we give formal definitions, describe the classification scheme for the four types of two-dimensional periodicity, and prove some theorems about the properties of the classes. In Section 4 we present serial and parallel algorithms for detecting the type of periodicity in an array. The complexity of the serial algorithm is O(m 2 ) (linear time) when the alphabet size is finite, and O(m 2 log m) otherwise. The parallel algorithm runs in time O(log m) with O(m 2 ) CRCW processors. In addition to knowing where an array can self overlap, knowing where it can not and why is also useful. If an overlap is not possible, then the overlap produces some mismatch. Our algorithms find a single mismatch location or witness for each self overlap that fails. 2 Periodicity in strings and arrays In a periodic string, a smallest period can be found whose concatenation generates the entire string. In two dimensions, if an array were to extend infinitely so as to cover the plane, the one-dimensional notion of a period could be generalized to a unit cell of a lattice. But, a rectangular array is not infinite and may cut a unit cell in many different ways at its edges. Instead of defining two-dimensional periodicity on the basis of some subunit of the array, we instead use the idea of self-overlap. This idea applies also to strings. A string w is periodic if the longest prefix p of w that is also a suffix of w is at least half the length of w. For example, if abcabcabcabcab, then abcabcab and since p is over half as long as w, w is periodic. This definition implies that w may overlap itself starting in the fourth position. The preceding idea easily generalized to two dimensions as illustrated in figure 1. A be a two-dimensional array. Call a prefix of A a rectangular subarray that contains one corner of A. (In the figure, the upper left corner.) Call a suffix of A a rectangular subbarray that contains the diagonally opposite corner of A (In the figure, the lower right corner). We say A is periodic if the largest prefix that is also a suffix has dimensions at least as large as some fixed percentage d of the dimensions of A. In the figure, if d - 5, then A is periodic. As with strings, if A is periodic, then A may overlap itself if the prefix of one copy of A is aligned with the suffix of a second copy of A. Notice that both the upper left and lower left corners of A can define prefixes, giving A two directions in which it can be periodic. As we will describe in the next section, the classification of periodicity type for A is based on whether it is periodic in either or both of these directions. a Figure 1: a) A periodic pattern. b) A suffix matches a prefix. 3 Classifying arrays Our goal here is classifying an array A into one of four periodicity classes. For clarity of presentation we concentrate on square arrays. We later show how to generalize all results to rectangles. We begin with some definitions of two-dimensional periodicity and related concepts (figure 2). be an m \Theta m square array. Each element of A contains a symbol from an alphabet \Sigma. A subarray of A is called a block. Blocks are designated by their first and last row and column. Thus, the block A[0::m \Gamma is the entire array. Each corner of A defines a quadrant. Quadrants are labeled counterclockwise from upper left, quadrants I , II , III and IV . Each quadrant has size q where 1 - q - d me. (Quadrants may share part of a row or column). Quadrant I is the block 1]. The choice of q may depend on the application. For this paper, Definition 3 Suppose we have two copies of A, one directly on top of the other. The copies are said to be in register because some of the elements overlap (in this case, all the elements) and overlapping elements contain the same symbol. If the two copies can be repositioned so that A[0; 0] overlaps and the copies are again in register, then we say that the array is quadrant I symmetric, that A[r; c] is a quadrant I source and that vector c~x is a quadrant I symmetry vector. Here, ~y is the vertical unit vector in the direction of increasing row index and ~x is the horizontal unit vector in the direction of increasing column index. If the two copies can be repositioned so that A[m \Gamma and the copies are again in register, then we say that the array is quadrant II symmetric, that A[r; c] is a quadrant II source and that c~x is a quadrant II symmetry vector. c a quadrant I source quadrant II source quadrant I symmetry vector quadrant II symmetry vector II III IV I Figure 2: Two overlapping copies of the same array. a) A quadrant I source. b) A quadrant II source. c) The symmetry vectors. Analagous definitions exist for quadrants III and IV , but by symmetry, if ~v is a quadrant III(IV ) symmetry vector, then \Gamma~v is a quadrant I(II) symmetry vector. We will usually indicate a vector c~x by the ordered pair (r; c). Note that symmetry vector (r; c) defines a mapping between identical elements, that is, (r; c) is a symmetry vector iff A[i; elements are defined. In particular, if (r; c) is a symmetry vector, then it maps the block A[i::j; k::l] to the identical block A[i In the remainder of this paper, we use the terms source and symmetry vector interchangeably. Definition 4 The length of a symmetry vector is the maximum of the absolute values of its coef- ficients. The shortest quadrant I (quadrant II) vector is the smallest one in lexicographic order first by row and then by column (first by column and then by absolute value of row). The basis vectors for array A are the shortest quadrant I vector (r 1 and the shortest quadrant II vector any). If the length of a symmetry vector is ! p where me then the vector is periodic. We are now ready to classify a square array A into one of four periodicity classes based on the presence or absence of periodic vectors in quadrants I and II . Following the classification we prove some theorems about the properties of the classes. In Section 4 we present algorithms for finding all the sources in an array. array. The four classes of two-dimensional periodicity are (figures 3 - 6): The array has no periodic vectors. ffl Lattice periodic - The array has periodic vectors in both quadrants. All quadrant I sources which occur in quadrant I fall on the nodes of a lattice which is defined by the basis vectors. The same is true for quadrant II sources in quadrant II . Specifically, let ~v 1 ) be the periodic basis vectors in quadrants I and II respectively. Then, an element in quadrant I is a quadrant I source iff it occurs at index A[ir 1 Lattice periodic array. for integers i; j. An element in quadrant II is a quadrant II source iff it occurs at index for integers -. ffl Line periodic - The array has a periodic vector in only one quadrant and the sources in that quadrant all fall on one line. ffl Radiant periodic-This category is identical to the line periodic category, except that in the quadrant with the periodic vector, the sources fall on several lines which all radiate from the quadrant's corner. We do not describe the exact location of the sources for this class, but see [GP-92] for a detailed analysis of the source locations. Next, we prove some theorems about the properties of the classes. All the theorems are stated in terms of square arrays for clarity. At the end of the theorems we explain how they can be modified to apply to any n \Theta m rectangular array. Line periodic array. In Lemmas 1-3, we establish the fact that if we have symmetry vectors for both quadrants I and II , and they meet a pair of constraints on the sum of their coefficients, then every linear combination of the vectors defines another symmetry vector. are symmetry vectors from quadrants I and II respectively, and is either a quadrant I symmetry vector or a quadrant II symmetry vector (r 1 Proof: We prove for the case r 1 - jr 2 j. The proof for the other case is similar. We show that ) is a quadrant I source. O O III O O O O O O O O I IV II Figure Radiant periodic array. Three non-colinear sources are starred. First, by the constraint on the c i , the fact that r 2 is negative and the assumption that r 1 - jr 2 j, S is an element of A. Next, we show via two pairs of mappings that the quadrant I prefix block is identical to the suffix block A[r 1 First maps the resultant block to block A[r 1 Second maps the resultant block to A[m are two quadrant k symmetry vectors and c 1 is also a quadrant k symmetry vector. Proof: We prove for quadrant I . The proof for the other quadrant is similar. We show that ) is a quadrant I source. First, by the restraints on the r i and the c i , S is an element of A. Next, by a pair of mappings, we show that the quadrant I prefix block is identical to the suffix block A[r 1 Recall that both r 1 and r 2 are postive. First mapping: (r 1 maps the block \Gamma 1] to the block maps the resultant block to the block Lemma 3 If ~v 1 are symmetry vectors from quadrants I and II respec- tively, and c 1 is an element of A, (ir 1 ) is a quadrant I symmetry vector. Similarly, for all - such that is an element of A, (-r 1 ) is a quadrant II symmetry vector. Proof: We prove for vector (ir 1 equivalent to source S The proof for the other vector is similar. Consider the lattice of elements in A defined by the quadrant I and II vectors and with one element at A[0; 0]. (The lattice elements correspond exactly to the elements S i;j .) Consider the line l that extends from element A[0; 0] through elements We prove the lemma only for those lattice elements on or to the right of l. The remaining elements are treated similarly. Case 1: S i;0 on line l. By induction on i. For S 1;0 ) is a symmetry vector by hypothesis. Now, assume that (ir 1 ) is a symmetry vector. For S are both quadrant I symmetry vectors, by Lemma ) is a quadrant I symmetry vector. Case 2: S i;j j - 1 to the right of line l. Elements S i;j fall on lines l j which are parallel to line l. We show that the uppermost element S i;j is a source. By application of Lemma 2, as in Case the remaining sources on l j are established. Consider a cell of the lattice with sides (r 1 corners with S the uppermost lattice element on line l j (figure 6): (j (j (j (j The following are always true: A l l 0 Figure 7: A candidate source S in lemma 3. Here jr is not an element of A. Otherwise e 2 - not S - is the top element on its line. is an element of A. Otherwise S is not in A, S is not to the right of line l or r 1 Two possibilities remain. Either e 1 is an element of A or it is not. Our proof is by induction on i and j. For the base cases we use ~v 1 which is either a quadrant I vector (r 1 - jr 2 j) or a quadrant II vector (r 1 Subcase A: r 1 j. is not an element of A. By the induction hypothesis, ~v e 4 is a symmetry vector. Since e 1 is not on A, r e 4 . That is, the row coefficient in ~v e 4 is smaller than the row coefficient in ~v 1: Apply Lemma 1 to ~v e 4 and ~v 2 and S is a source. is an element of A. By the induction hypothesis, ~v e 1 is a quadrant I symmetry vector. From the base case, ~v 3 is a quadrant I symmetry vector. Apply Lemma 2 to ~v e 1 and ~v 3 and S is a source. Subcase B: r 1 is not an element of A. Impossible, else S is not in A or S is not right of l. is an element of A. Note that S is above row r 1 or else e 2 is on the array. The vector is a quadrant II symmetry vector (because r e 2 is negative) by application of Subcase A to quadrant II . Now, r e 2 =(the row index of S)- 0 so r 1 or jr e 2 By hypothesis, r 1 Apply Lemma 2 to ~v 1 and ~v e 2 and S is a source. 2 The proof of Theorem 1 is simplified by the following easily proven observation. ) be symmetry vectors from quadrants I and II respectively, and c 1 let L be an infinite lattice of points on the xy-plane also with basis vectors (r ). If we put one copy of A on each lattice point by aligning element A[0; 0] with the lattice point, then the copies are in register and completely cover the plane. The next Lemma establishes that for a given lattice of elements in A, an element not on the lattice has a shorter vector to some lattice point than the corresponding basis vector for the lattice. (Note that a simplified version of the proof appeared in [GP-92] and we use essentially that same proof here.) Figure 8: One of the vectors from e 1 to S or S to e 2 is a quadrant I vector shorter than ~v 1 Lemma 4 Let L be an infinite lattice in the xy-plane with basis vectors ~v 1 (quadrants I and II symmetry vectors respectively) where all the r i and c i are integers. Then, for any point that is not a lattice element, where x and y are integers, there exists a lattice point e such that the vector ~v from e to S (or S to e) is a quadrant I vector shorter than ~v 1 or a quadrant II vector shorter than ~v 2 Proof: Let S be an element that does not fall on a lattice point. Consider the unit cell of the lattice containing S (figure 8) with nodes labeled e 1 Connect S to the four corners of the unit cell to get four triangles. At least one of these triangles has a right or obtuse angle. Wolog, let the triangle be on points e 1 and S. Then both the vector from e 1 to S and the vector from e 2 to S is shorter than the vector from e 1 to e 2 . Since at least one of the two is a quadrant I vector, we have a quadrant I vector shorter than ~v 1: 2 Our first main result is the following Theorem. It establishes that if an array has basis vectors in both quadrants, then in a certain block of the array, which depends on the coefficients of the basis vectors, all symmetry vectors are linear combinations of the basis vectors. We state the Theorem in terms of quadrant I for simplicity. Since the array can be rotated so that any quadrant becomes quadrant I , it applies to all quadrants. Theorem 1 Let A be an array with basis vectors (r in quadrants I and II respectively with c 1 m. Let L be an infinite lattice with the same basis vectors and containing the element A[0; 0]. Then, in the block an element is a quadrant I source iff it is a lattice element. Proof: By Lemma 3, if is a lattice element, then it is a source. Suppose that S is not a lattice element, but that it is a quadrant I source. We will show that S can not occur within block By way of contradiction, assume S does occur in prefix block There is a quadrant I vector ~v associated with S that is not a linear combination of ~v 1 and ~v 2 By Observation 1, copies of A can be aligned with the points of lattice L and the copies will be in register and cover the plane. Let A i.e. the suffix block originating at element Because S is a source, ~v maps . For each copy of A, remove all but A 0 . The copies of A 0 are in register. Since A 0 has dimensions at least r 1 , it is at least as large as a unit cell of the lattice and therefore, the copies of A 0 also cover the plane. Now every element of the plane is mapped by ~v from an identical element, and there is a complete copy of A at S. S falls within some cell of lattice L. By Lemma 4, there is a quadrant I or quadrant II vector ~v 3 from S to some corner e of the cell (or from e to S) which is shorter than the corresponding basis vector of L. Since there are complete copies of A at S and e, ~v 3 is a symmetry vector and therefore ~v 1 and ~v 2 are not both basis vectors of A as assumed. 2 Since our quadrants are of size d me \Theta d me, they are no greater in size than the smallest block that can contain only lattice point sources. The region that contains only lattice point sources can be larger than the block described in Theorem 1, see [GP-92]. Next, we prove the following important trait about radiant periodic arrays that facilitates their handling in matching applications [AB-92b, ABF-94b, KR-94]. Origins (A[0; 0]) of complete copies of a radiant periodic array A that overlap without mismatch can be ordered monotonically. Definition 5 A set of elements of an array B can be ordered monotonically if the elements can be ordered so that they have column index nondecreasing and row index nondecreasing (ordered monotonically in quadrant I) or row index nonincreasing (ordered monotonically in quadrant II). Our theorem is stated in terms of quadrant I , but generalizes to quadrant II . Theorem 2 Let A be a radiant periodic array with periodic vector in quadrant I . Let S 1 be quadrant I sources occuring within quadrant I . On each source, place one copy of A by aligning A[0; 0] with the source. If every pair of copies is in register, then the sources can be ordered monotonically in quadrant I . Proof: Suppose two sources A[c 1 cannot be ordered monotonically. That is, c 1 but . If there is no mismatch in the copies of A at these sources, then by the fact that ) is a periodic, quadrant II symmetry vector and by definition, A is lattice periodic, a contradiction. 2 As stated earlier, our classification scheme applies to any rectangular array. The major modification is a new definition of length. Definition 6 The length of a symmetry vector of a rectangular array is the maximum of the absolute values of its coefficients scaled to the dimensions of the array. Let A be n rows by m columns with m - n. Let c) be a symmetry vector in A. Then the length of ~v scaled to the dimensions of the array is max(r 4 Periodicity and Witness Algorithms In this section, we present two algorithms, one serial and one parallel for finding all sources in an array A. In addition, for each location in A which is not a source, our algorithms find a witness that proves that the overlapping copies of A are not in register. We want to fill out an array For each location A[i; j] that is a quadrant I source, some mismatch. Specifically A[r; c] 9). For each location A[i; j] that is a quadrant II source, where mismatch i+r Figure 9: The witness tables gives the location of a mismatch (if one exists) for two overlapping patterns: 4.1 The Serial Algorithm Our serial algorithm (Algorithm makes use of two algorithms (Algorithms 1 and 2) from [ML-84] which are themselves variations of the KMP algorithm [KMP-77] for string matching. Algorithm 1 takes as input a pattern string w of length m and builds a table lppattern[0::m \Gamma 1] where lppattern[i] is the length of the longest prefix of w starting at w i . Algorithm 2 takes as input a text string t of length n and the table produced by Algorithm 1 and produces a table lptext[0::n \Gamma 1] where lptext[i] is the length of the longest prefix of w starting at t i . The idea behind Algorithm A is the following: We convert the two-dimensional problem into a problem on strings (figure 10). Let the array A be processed column by column and suppose we are processing column j. Assume we can convert the suffix block A[0::m \Gamma string represents the suffix of row i starting in column j. This will serve as the text string. Assume also that we can convert the prefix block A[0::m \Gamma string represents the prefix of row i of length m \Gamma j. This will serve as the pattern string. Now, use Algorithm 1 to produce the table lppattern for W j and Algorithm 2 to produce the table lptext for T j . If a copy of the pattern starting at t i matches in every row to t then is a source. If the pattern doesn't match and the first pattern row to mismatch is row j] is not a source. The mismatch occurs between the prefix of pattern row k and the suffix of text row need merely locate the mismatch to obtain the witness. In order to treat the suffix and prefix of a row as a single character, we will build a suffix tree for the array. A suffix tree is a compacted trie of suffixes of a string Each node v has associated with it the indices [a,b] of some substring of S. If u is the Least Common Ancestor (LCA) of two nodes v and w, then S(u) is the longest common prefix of S(v) and S(w) [LV-85]. A tree can be preprocessed in linear time to answer LCA queries in constant m-j columns Figure 10: Representing a block of the array by a string. is the text and W is the pattern. time [HT-84]. Thus, we can answer questions about the length of S(u) in constant time. Algorithm A Serial algorithm for building a witness array and deciding periodicity class. Step A.1: Build a suffix tree by concatenating the rows of the array. Preprocess the suffix tree for least common ancestor queries in order to answer questions about the length of the common prefix of any two suffixes. Step A.2: For each column j, fill out Step A.2.1: Use Algorithm 1 to construct the table lppattern for w i is the prefix of row i of length j. We can answer questions about the equality of two characters by consulting the suffix tree. If the common prefix of the two characters has length at least m \Gamma j then the characters are equal. Step A.2.2: Use Algorithm 2 to construct the table lptext for is the suffix of row i starting in column j (also of length Again we test for equality by reference to the suffix tree. Step A.2.3: For each row i, if then we have found a quadrant I source and otherwise, using the suffix tree, compare the suffix of text row starting in column j with the prefix of pattern row lptext[i]. The length l of the common prefix will be less than Step A.3: Repeat step 2 for by building the automata and processing the columns from the bottom up. Step A.4: Select quadrant I and quadrant II basis vectors from Witness if they exist. Step A.5: Use the basis vectors to decide to which of four periodicity classes the pattern belongs. Theorem 3 Algorithm A is correct and runs in time O(m 2 log j\Sigmaj). Proof: The correctness of Algorithm A follows from the correctness of Algorithms 1 and 2 [ML-84]. The suffix tree construction [W-73] takes time O(m 2 log j\Sigmaj) while the preprocessing for least common ancestor queries [HT-84] can be done in time linear in the size of the tree. Queries to the suffix tree are processed in constant time. The tables lppattern and lptext can be constructed in time O(m) [ML-84]. For each of m columns, we construct two tables so the total time for steps 2 and 3 is O(m 2 ). Step 4 can be done in one scan through the witness array and step 5 requires comparing all vectors to the basis vectors in order to distinguish between the radiant and line periodic classes, so the time for steps 4 and 5 is O(m 2 ). The total complexity of the pattern preprocessing is therefore Recently, [GP-92] gave a linear time serial algorithm for the witness computation. 4.2 The Parallel Algorithm Our parallel algorithm (Algorithm B) makes use of the parallel string matching algorithm (Algo- rithm 3) from [V-85]. Algorithm 3 takes as input a pattern string w of length m and a text string t of length n and produces a boolean table true if a complete copy of the pattern starts at t i . Algorithm 3 first preprocesses the pattern and then processes the text. First, for a text of length m, we show how to modify Algorithm 3 to compute match[0::m \Gamma 1], is a prefix of the pattern. For simplicity, we assume m is a power of 2. Let Figure is a prefix of t i is a suffix of w 0 For example, is the prefix of w of length mand S 1 is a suffix of t of the same length. The following observation embodies the key idea (figure 11): is a prefix of w of length between m and m, then P 1 is a prefix of is a suffix of w Similarly, if is a prefix of w of length between mand m, then the prefix and suffix are P 2 , etc. Now, for each k - 1, we attempt to match P k in S k\Gamma1 and S k in P k\Gamma1 . If a matched prefix begins at t i and a matched suffix ends at is a prefix of w. Using Algorithm 3, we first preprocess the P k and S k as patterns and then use these to process the appropriate segments as text. We can additionally modify Algorithm 3 so that at every index where a prefix or suffix does not match, we obtain the location of a mismatch. Since the sum of the lengths of the P i and S i are no more than a linear multiple of the length of w, the modification does not increase the complexity of the algorithm and therefore the time complexity of the modified Algorithm 3 is O(log m) using O( m log m ) CRCW processors, the same as the unmodified algorithm [V-85]. In our parallel algorithm, only steps 2 differs from the serial algorithm. Algorithm B Parallel algorithm for finding sources and building a witness array Step B.2: For each column j, fill out Step B.2.1: For each Step B.2.1.1: Use W j to form P k and P k\Gamma1 and T j to form S k and S k\Gamma1 . Use modified Algorithm 3 to match P k in S k\Gamma1 and S k in P k\Gamma1 . As in the serial algorithm, use the suffix tree to answer questions about equality. Step B.2.1.2: For each row i for beginning at t i and S k matches ending at using the row r of mismatch from modified Algorithm 3, refer to the suffix tree to find the column c of mismatch and set Theorem 4 Algorithm B is correct and runs in time O(log m) using O(m 2 ) CRCW processors. Proof: The suffix tree construction [AILSV-87] and preprocessing for LCA queries [SV-88] is done in time O(log m) using O(m 2 ) CRCW processors. Step 2 is done in time O(log m) using O( m 2 log CRCW processors [V-85]. Finding the basis vectors is done by prefix minimum [LF-80] in time O(log m) using O( m 2 log m ) processors. Distinguishing the line and radiant periodic cases can be done in constant time using O(m 2 ) processors. The total complexity is therefore O(log m) time using processors. --R "Two-Dimensional Periodicity and its Application" "Efficient Two-Dimensional Compressed Matching" "Optimal Parallel Two Dimensional Text Searching on a CREW PRAM," "An Alphabet Independent Approach to Two-Dimensional Matching" "Optimal Two Dimensional Compressed Match- ing," "Efficient String Matching" "Efficient 2-dimensional Approximate Matching of Non-rectangular Figures" "Fast Parallel and Serial Multidimensional Approximate Array Matching" "Efficient Pattern Matching with Scaling" "Parallel Construction of a Suffix Tree with Applications" "A Technique For Extending Rapid Exact-Match String Matching to Arrays of More Than One Dimension" "Two Dimensional Pattern Matching" "A Fast String Searching Algorithm" "Optimally fast parallel algorithms for preprocessing and pattern matching inone and two dimensions," "Note on two dimensional string matching by optimal parallel algorithms," "Optimal Parallel Algorithms for String Matching" "Truly Alphabet Independent Two-Dimensional Pattern Match- ing," "Fast Algorithms for Finding Nearest Common Ancestors" "Alphabet Independent Optimal Parallel Search for 3-Dimensional Patterns," "Fast Pattern Matching in Strings" "Efficient Two Dimensional Pattern Matching in the Presence of "Parallel Prefix Computation" "Efficient string matching in the presence of errors" "An O(n log n) Algorithm for Finding all Repetitions in a String" "A Unifying Look at d-Dimensional Periodicities and Space Coverings," "On Finding Lowest Common Ancestors: Simplification and Parallelization" "Optimal Parallel Pattern Matching in Strings" "Deterministic new technique for fast pattern matching" "Linear Pattern Matching Algorithms" --TR --CTR Richard Cole , Zvi Galil , Ramesh Hariharan , S. Muthukrishnan , Kunsoo Park, Parallel two dimensional witness computation, Information and Computation, v.188 n.1, p.20-67, 10 January 2004 Amihood Amir , Gad M. Landau , Dina Sokol, Inplace 2D matching in compressed images, Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms, January 12-14, 2003, Baltimore, Maryland Amihood Amir , Gad M. Landau , Dina Sokol, Inplace 2D matching in compressed images, Journal of Algorithms, v.49 n.2, p.240-261, November Chiara Epifanio , Filippo Mignosi, A multidimensional critical factorization theorem, Theoretical Computer Science, v.346 n.2, p.265-280, 28 November 2005 Filippo Mignosi , Antonio Restivo , Pedro V. Silva, On Fine and Wilf's theorem for bidimensional words, Theoretical Computer Science, v.292 n.1, p.245-262, January Amihood Amir , Gad M. Landau , Dina Sokol, Inplace run-length 2d compressed search, Theoretical Computer Science, v.290 n.3, p.1361-1383, 3 January Chiara Epifanio , Michel Koskas , Filippo Mignosi, On a conjecture on bidimensional words, Theoretical Computer Science, v.299 n.1-3, p.123-150,

string matching;witness;parallel algorithm;periodicity;sequential algorithm;two-dimensional

276469

Convergence Analysis of Pseudo-Transient Continuation.

Pseudo-transient continuation ($\Psi$tc) is a well-known and physically motivated technique for computation of steady state solutions of time-dependent partial differential equations. Standard globalization strategies such as line search or trust region methods often stagnate at local minima. \ptc succeeds in many of these cases by taking advantage of the underlying PDE structure of the problem. Though widely employed, the convergence of \ptc is rarely discussed. In this paper we prove convergence for a generic form of \ptc and illustrate it with two practical strategies.

Introduction . Pseudo-transient continuation (\Psitc) is a method for computation of steady-state solutions of partial differential equations. We shall interpret the method in the context of a method-of-lines solution, in which the equation is discretized in space and the resulting finite dimensional system of ordinary differential equations is written as x @x @t and the the discretized spatial derivatives are contained in the nonlinear term F (x). Marching out the steady state by following the physical transient may be unnecessarily time consuming if the intermediate states are not of interest. On the other hand, Newton's method for F usually not suffice, as initial iterates sufficiently near the root are usually not available. Standard globalization strategies [12, 17, 25], such as line search or trust region methods often stagnate at local minima of kFk [20]. This is particularly the case when the solution has complex features such as shocks that are not present in the initial iterate (see [24], for example). \Psitc succeeds in many of these cases by taking advantage of the PDE structure of the problem. 1.1. The Basic Algorithm. In the simple form considered in this paper, \Psitc numerically integrates the initial value problem to steady state using a variable timestep scheme that attempts to increase the timestep as F (x) approaches 0. V is a nonsingular matrix used to improve the scaling of the problem. It is typically diagonal and approximately equilibrates the local CFL number (based on local cell diameter and local wave speed) throughout the domain. In a multicomponent system of PDEs, not already Version of April 20, 1997. y North Carolina State University, Department of Mathematics and Center for Research in Scientific Computation, Box 8205, Raleigh, N. C. 27695-8205 (Tim Kelley@ncsu.edu). The research of this author was supported by National Science Foundation grant #DMS-9321938. z Computer Science Department, Old Dominion University, Norfolk, VA 23519-0162 (keyes@cs.odu.edu) and ICASE, MS 132C, NASA Langley Research Center, Hampton, Virginia 23681-0001. The research of this author was supported by National Science Foundation grant #ECS-9527169, and NASA grants NAGI-1692 and NAS1-19480, the latter while the author was in residence at ICASE. properly nondimensionalized, V might be a block diagonal matrix, with blocksize equal to the number of components. We can define the \Psitc sequence fx n g by where F 0 is the Jacobian (or the Fr-echet derivative in the infinite-dimensional situation). Algorithmically ALGORITHM 1.1. 1. 2. While kF (x)k is too large. (a) Solve (ffi (b) (c) Evaluate F (x). (d) Update ffi. The linear equation for the Newton step that is solved in step 2a is the discretization of an PDE and is usually very large. As such, it is typically solved by an iterative method which terminates on small linear residuals. This results in an inexact method [11], and a convergence theory for \Psitc must account for this; we provide such a theory in x 3. We have not yet explained how ffi is updated, nor explored the reuse of Jacobian information from previous iterations. These options must be considered to explain the performance of \Psitc as practiced, and we take them up in x 3, too. In order to most clearly explain the ideas, however, we begin our analysis in x 2 with the most simple variant of \Psitc possible, solving (1.3) exactly, and then extend those results to the more general situation. As a method for integration in time, \Psitc is a Rosenbrock method ([14], p. 223) if ffi is fixed. One may also think of this as a predictor-corrector method, where the simple predictor (result from the previous timestep) and a Newton corrector are used. To see this consider the implicit Euler step from x n with timestep z In this formulation z n+1 would be the root of Finding a root of G with Newton's method would lead to the iteration If we take - the first Newton iterate is This leaves open the possibility of taking more corrector iterations, which would lead to a different form of \Psitc than that given by (1.2). This may improve stability for some problems [16]. PSEUDO-TRANSIENT CONTINUATION 3 1.2. Time Step Control. We assume that ffi n is computed by some formula like the "switched evolution relaxation" (SER) method, so named in [21], and used in, e.g. [19], [24], and [33]. In its simplest, unprotected form, SER increases the timestep in inverse proportion to the residual reduction. Relation (1.5) implies that, for n - 1, In some work [16], ffi n is kept below a large, finite bound ffi max . Sometimes ffi n is set to 1 (called "switchover to steady-state form" in [13]) when the computed value of In view of these practices, we will allow for somewhat more generality in the formulation of the sequence fffi n g. We will assume that ffi 0 is given and that for n - 1. The choice in [24] and [33] (equation (1.5)) is OE(-: Other choices could either limit the growth of ffi or allow ffi to become infinite after finitely many steps. Our formal assumption on OE accounts for all of these possibilities. ASSUMPTION 1.1. 1. 2. Either - and the timesteps are held bounded. If - then switchover to steady-state form is permitted after a finite number of timesteps. In [16] the timesteps are based not on the norms of the nonlinear residuals kF (x n )k but on the norms of the steps This policy has risks in that small steps need not imply small residuals or accurate results. However if the Jacobians are uniformly well conditioned, then small steps do imply that the iteration is near a root. Here formulae of the type are used, where OE satisfies Assumption 1.1. 1.3. Iteration Phases. We divide the \Psitc iteration into three conceptually different and separately addressed phases. 1. The initial phase. Here ffi is small and x is far from a steady state solution. This phase is analyzed in x 2.3. Success in this phase is governed by stability and accuracy of the temporal integration scheme and proper choice of initial data. 2. The midrange. This begins with an accurate solution x and a timestep ffi that may be small and produces an accurate x and a large ffi. We analyze this in x 2.2. To allow ffi to grow without loss of accuracy in x we make a linear stability assumption (part 3 of Assumption 2.1). 3. The terminal phase. Here ffi is large and x is near a steady state solution. This is a small part of the overall process, usually requiring only a few iterations. Aside from the attention that must be paid to the updating rules for ffi, the iteration is a variation of the chord method [25, 17]. We analyze the terminal phase first, as it is the easiest, in x 2.1. Unlike the other two phases, the analysis of the terminal phase does not depend on the dynamics of x (x). The initial and midrange phases are considered in x 2.3, with the midrange phase considered first to motivate the goals of the initial phase. This decomposition is similar to that proposed for GMRES and related iterations in [22] and is supported by the residual norm plots reported in [24, 10]. 2. Exact Newton Iteration. In this section we analyze the three phases of the solver in reverse order. This ordering makes it clear how the output of an earlier phase must conform to the demands of the later phase. 2.1. Local Convergence: Terminal Phase. The terminal phase of the iteration can be analyzed without use of the differential equation at all. LEMMA 2.1. Let fffi n g be given by either (1.6) or (1.8) and let Assumption 1.1 hold. Let be nonsingular, and F 0 be Lipschitz continuous with Lipschitz constant fl in a ball of radius ffl about x . Then there are then the sequence defined by (1.2) and (1.6) satisfies and x Proof. Let denote the error. As is standard, [12], [17], we analyze convergence in terms of the transition from a current iterate x c to a new iterate x+ . We must also keep track of the change in ffi and will let ffi c and ffi + be the current and new pseudo-timesteps. The standard analysis of the chord method ([17], p. 76) implies that there are ffl 1 - ffl and K C such that if The constant K C depends only ffl 1 , F , and x and does not increase if ffl 1 is reduced. Now let \Delta \Gamma1and ffl 1 be small enough to satisfy and, in particular, F increase if needed so that PSEUDO-TRANSIENT CONTINUATION 5 where - denotes condition number. Equations (2.1) and (2.2) imply that If fffi n g is computed with (1.6) we use the following inequality from [17] (p. 72) and (2.3) to obtain 2: We then have by Assumption 1.1 that t is from Assumption 1.1. If fffi n g is computed with (1.8), we note that and hence as before. In either case, Therefore we may continue the iteration and conclude that at least q-linearly with q-factor of 1=2. If we complete the proof by observing that since superlinear convergence follows from (2.1). The following simple corollary of (2.1) applies to the choice OE(-. COROLLARY 2.2. Let the assumptions of Lemma 2.1 hold. Assume that OE(-. 1. Then the convergence of fx n g to x is q-quadratic. 2.2. Increasing the Time Step: Midrange Phase. Lemma 2.1 states that the second phase of \Psitc should produce both a large ffi and an accurate solution. We show how this happens if the initial phase can provide only an accurate solution. This clarifies the role of the second phase in increasing the timestep. We do this by showing that if the steady state problem has a stable solution, then \Psitc behaves well. We now make assumptions that not only involve the nonlinear function but also the initial data and the dynamics of the IVP (1.1). ASSUMPTION 2.1. 1. F is everywhere defined and Lipschitz continuously Fr- echet differentiable, and there is 2. There is a root x of F at which F 0 then the solution of the initial value problem converges to x as t !1. 3. There are ffl 2 for all The analysis of the midrange uses part 3 of Assumption 2.1 in an important way to guarantee stability. The method for updating ffi is not important for this result. THEOREM 2.3. Let fffi n g be given by either (1.6) or (1.8) and let Assumption 1.1 hold. Let Assumption 2.1 hold. Let ffi max be large enough for the conclusions of Lemma 2.1 to hold. Then there is an ffl 3 ? 0 such that if or x Proof. Let 1 is from Lemma 2.1 and ffl 2 is from part 3 of Assumption 2.1. Note that Now there is a c ? 0 such that for all x such that kek - ffl 1 . Hence, reducing ffl 3 further if needed so that ffl 3 ! fi=(2c), we have If for all n - 1 and hence x n converges to x q-linearly with q-factor (1 This convergence implies that x (1.8) is used. This result says that once the iteration has found a sufficiently good solution, either convergence will happen or the the iteration will stagnate with inf latter failure mode is, of course, easy to detect. Moreover, the radius ffl 3 of the ball about the root in Theorem 2.3 does not depend on inf ffi n . 2.3. Integration to Steady State: Initial Phase. Theorem 2.3 requires an accurate estimate of x , but asks nothing of the timestep. In this section we show that if ffi 0 is sufficiently small, and (1.6) is used to update the timestep, then the dynamics of (1.1) will be tracked sufficiently well for such an approximate solution to be found. It is not clear how (1.8) allows for this. THEOREM 2.4. Let fffi n g be given by (1.6) and let Assumption 1.1 hold. Let Assumption 2.1 hold. There is a - ffi such that if ffi 0 - ffi then there is an n such that Proof. Let S be the trajectory of the solution to (2.5). By Assumption 2.1 x satisfies the assumptions [12, 17, 25], for local quadratic convergence of Newton's method and therefore there are ffl 4 and ffl f such that if then suffice for the conclusions of Theorem 2.3 to hold. Let We will show that if ffi 0 sufficiently small, then . By (1.7), if then as long as kF (x)k - ffl f and x k is within ffl 4 of the trajectory S. Let z be the solution of (2.5). Let be such that for all t ? T , Consider the approximate integration of (2.5) by (1.2). Set If holds. This cannot happen if n ? (which implies that t n ? T ). Therefore the proof will be complete if we can show that for all Note that (1.2) may be written as There is an m 1 such that the last term in (2.9) satisfies, for sufficiently small and )k. Then we have, by our assumptions on F , that there is an such that Finally, there is an m 2 such that for for sufficiently small and Setting we have for all n - 1 (as long as ffi n is sufficiently small and As long as (2.7) holds, this implies that Consequently, as is standard [14, 15], and using (2.8), then This completes the proof. The problem with application of this proof to the update given by (1.8) is that bounds on ffi like (2.7) do not follow from the update formula. 3. Inexact Newton Iteration. In this section we look at \Psitc as implemented in practice. There are two significant differences between the simple version in x 2 and realistic implementations: 1. The Fr-echet derivative recomputed with every timestep. 2. The equation for the Newton step is solved only inexactly. Before showing how the results in x 2 are affected by these differences, we provide some more detail and motivation. Item 1 is subtle. If one is solving the equation for the Newton step with a direct method, then evaluation and factorization of the Jacobian matrix is not done at every timestep. This is a common feature of many ODE and DAE codes, [30, 26, 27, 3]. Jacobian updating is an issue in continuation methods [31, 28], and implementations of the chord and Shamanskii [29] methods for general nonlinear equations [2, 17, 25]. When the Jacobian is slowly varying as a function of time or the continuation parameter, sporadic updating of the Jacobian leads to significant performance gains. One must decide when to evaluate and factor the Jacobian using iteration statistics and (in the ODE and DAE case) estimates of truncation error. Temporal truncation error is not of interest to us, of course, if we seek only the steady-state solution. CONTINUATION 9 In [16] a Jacobian corresponding to a lower-order discretization than that for the residual was used in the early phases of the iteration and in [19], in the context of a matrix-free Newton method, the same was used as a preconditioner. The risks in the use of inaccurate Jacobian information are that termination decisions for the Newton iteration and the decision to reevaluate and refactor the Jacobian are related and one can be misled by rapidly varying and ill-conditioned Jacobians into premature termination of the nonlinear iteration [30, 32, 18]. In the case of iterative methods, item 1 should be interpreted to mean that preconditioning information (such as an incomplete factorization) is not computed at every timestep. means that the equation for the Newton step is solved inexactly in the sense of [11], so that instead of where s is given by (1.3), step s satisfies for some small j, which may change as the iteration progresses. Item 1 can also be viewed as an inexact Newton method with j reflecting the difference between the approximate and exact Jacobians. The theory in x 2 is not changed much if inexact computation of the step is allowed. The proof of Lemma 2.1 is affected in (2.1), which must be changed to This changes the statement of the lemma to LEMMA 3.1. Let fffi n g be given by either (1.6) or (1.8) and let Assumption 1.1. Let F (x be nonsingular, and F 0 be Lipschitz continuous with Lipschitz constant fl in a ball of radius ffl about x . Then there are ffl 1 ? 0, - j for all n, and then the sequence defined by (3.1), (3.2), and (1.6) satisfies and x Corollary 2.2 becomes COROLLARY 3.2. Let the assumptions of Lemma 3.1 hold. Assume that OE(-. 1. Then the convergence of fx n g to x is q-superlinear if locally q-quadratic if The analysis of the midrange phase changes in (2.6), where we obtain for some K j ? 0. This means that - j must be small enough to maintain the q-linear convergence of fx n g during this phase. The inexact form of Theorem 2.3 is THEOREM 3.3. Let fx n g be given by (3.1) and (3.2) and let fffi n g be given by either (1.6) or (1.8). Let Assumption 1.1 hold. Let Assumption 2.1 hold. Let ffi max be large enough for the conclusions of Lemma 2.1 to hold. Then there are ffl 3 ? 0 and - j such that if j n - and or x Inexact Newton methods, in particular Newton-Krylov solvers, have been applied to ODE/DAE solvers in [1], [5], [4], [6], [7], and [9]. The context here is different in that the nonlinear residual F (x) does not reflect the error in the transient solution but in the steady state solution. The analysis of the initial phase changes through (2.10). We must now estimate and hence, assuming that the operators are uniformly bounded, there is m 3 such that ks and hence We express (3.5) as Hence, if and the inexact form of Theorem 2.4: THEOREM 3.4. Let fx n g be given by (3.1) and (3.2) and let fffi n g be given by (1.6) and let Assumption 1.1 hold. Let Assumption 2.1 hold. Assume that the operators are uniformly bounded in n. Let ffl ? 0. There are - ffi and - j such that if j then there is an n such that The restrictions on j in Theorem 3.4 seem to be stronger than those on the results on the midrange and terminal phases. This is consistent with the tight defaults on the forcing terms for methods when applied in the ODE/DAE context [1, 5, 6, 7, 9]. 4. Numerical Experiments. In this section we examine a commonly used \Psitc technique, switched evolution/relaxation (SER) [21], applied to a Newton-like method for inviscid compressible flow over a four-element airfoil in two dimensions. Three phases corresponding roughly to the theoretically-motivated iteration phases of x 2 may be identified. We also compare SER with a different \Psitc technique based on bounding temporal truncation error (TTE) [20]. TTE is slightly PSEUDO-TRANSIENT CONTINUATION 11 -0.4 Zoomed Grid FIG. 4.1. Unstructured grid around four-element airfoil in landing configuration - near-field view. more aggressive than SER in building up the time step in this particular problem, but the behavior of the system is qualitatively the same. The physical problem, its discretization, and its algorithmic treatment in both a nonlinear defect correction iteration and in a Newton-Krylov-Schwarz iteration - as well as its suitability for parallel computation - have been documented in earlier papers, most recently [10] and the references therein. Our description is correspondingly compact. The unknowns of the problem are nodal values of the fluid density, velocities, and specific total energy, at N vertices in an unstructured grid of triangular cells (see Fig. 4.1). The system F discretization of the steady Euler equations: r r r where the pressure p is supplied from the ideal gas law, and fl is the ratio of specific heats. The discretization is based on a control volume approach, in which the control volumes are the duals of the triangular cells - nonoverlapping polygons surrounding each vertex whose perimeter segments join adjacent cell centers to midpoints of incident cell edges. Integrals of (4.1)-(4.3) over the control volumes are transformed by the divergence theorem to contour integrals of fluxes, which are estimated numerically through an upwind scheme of Roe type. The effective scaling matrix V for the \Psitc term is a diagonal matrix that depends upon the mesh. The boundary conditions correspond to landing configuration conditions: subsonic (Mach number of 0.2) with a high angle of attack of (5 ffi ). The full adaptively clustered unstructured grid contains 6,019 vertices, with four degrees of freedom per vertex (giving 24,076 as the algebraic dimension of the discrete nonlinear problem). Figure 4.1 shows only a near-field zoom on the full grid, whose far-field boundaries are approximately twenty chords away. The initial pseudo- \Gamma4 corresponds to a CFL number of 20. The pseudo-timestep is allowed to grow up to six orders of magnitude over the course of the iterations. It is ultimately bounded at guaranteeing a modest diagonal contribution that aids the invertibility of (ffi The initial iterate is a uniform flow, based on the far field boundary conditions - constant density and energy, and constant velocity at a given angle of attack. The solution algorithm is a hybrid between a nonlinear defect correction and a full Newton method, a distinction which requires further discussion of the processes that supply F (x) and F 0 (x) within the code. The form of the vector-valued function F (x) determines the quality of the solution and is always discretized to required accuracy (second-order in this paper). The form of the approximate Jacobian matrix F 0 (x), together with the scaling matrix V and time step ffi, determines the rate at which the solution is achieved but does not affect the quality of a converged result, and is therefore a candidate for replacement with a matrix that is more convenient. In practice, we perform the matrix inversion in (1.2) by Krylov iteration, which requires only the action of F 0 (x) on a series of Krylov vectors, and not the actual elements of F 0 (x). The Krylov method was restarted preconditioned with 1-cell overlap additive Schwarz (8 subdomains). Following [5, 8], we use matrix-free Fr-echet approximations of the required action: However, when preconditioning the solution of (1.2), we use a more economical matrix than the Jacobian based on the true F (x), obtained from a first-order discretization of the governing Euler system. This not only decreases the number of elements in the preconditioner, relative to a true Jacobian, but also the computation and (in the parallel context) communication in applying the preconditioner. It also results in a more numerically diffusive and stable matrix, which is desirable for inversion. The price for these advantages is that the preconditioning is inconsistent with the true Jacobian, so more linear subiterations may be required to meet a given linear convergence tolerance. This has an indirect feedback on the nonlinear convergence rate, since we limit the work performed in any linear subiteration in an inexact Newton sense. In previous work on the Euler and Navier-Stokes equations [10, 23], we have noted that a \Psitc method based on a consistent high-order Jacobian stumbles around with a nonmonotonic steady-state residual norm at the outset of the nonlinear iterations for a typical convenient initial iterate far from the steady-state solution. On the other hand, a simple defect correction approach, in which is based on a regularizing first-order discretization everywhere it appears in the solution of (1.2), not just in the preconditioning, allows the residual to drop smoothly from the outset. In this work, we employ a hybrid strategy, in which defect correction is used until the residual norm has fallen by three orders of magnitude, and inexact Newton thereafter. As noted in x 3, inexact FIG. 4.2. SER convergence history iteration based on the true Jacobian and iteration with an inconsistent Jacobian can both be gathered under the j of (3.2), so the theory extends in principal to both. With this background we turn our attention to Fig. 4.2, in which are plotted on a logarithmic scale against the \Psitc iteration number: the steady-state residual norm jjF (x n )jj 2 at the beginning of each iteration, the norm of the update vector jjx and the pseudo-timestep ffi n . The residual norm falls nearly monotonically, as does the norm of the solution update. Asymptotic convergence cannot be expected to be quadratic or superlinear, since we do not enforce in (3.5). However, linear convergence is steep, and our experience shows that overall execution time is increased if too many linear iterations are employed in order to enforce asymptotically. In the results shown in this section, the inner linear convergence tolerance was set at 10 \Gamma2 for the defect correction part of the trajectory, and at 10 \Gamma3 for the Newton part. The work was also limited to a maximum of 12 restart cycles of 20 Krylov vectors each. Examination of the pseudo-timestep history shows monotonic growth that is gradual through the defect correction phase (ending at rapidly growing, and asymptotically at (beginning at show momentary retreats from ffi max in response to a refinement on the \Psitc strategy that automatically cuts back the pseudo-timestep by a fixed factor if a nonlinear residual reduction of less than 3is achieved at the exhaustion of the maximum number of Krylov restarts in the previous step (during the terminal Newton phase). Close examination usually reveals a stagnation plateau in the linear solver, and it is more cost effective to fall back to the physical transient to proceed than to grind on the ill-conditioned linear problem. These glitches in the convergence of jjF are not of nonlinear origin. Another timestep policy, common in the ODE literature, is based on controlling temporal truncation error estimates. Though we do not need to maintain temporal truncation errors at low levels when we are not attempting to follow physical transients, we may maintain them at high levels as a heuristic form of stepsize control. This policy seems rare in external aerodynamic simulations, but is popular in the combustion community and is implemented in [20]. The first FIG. 4.3. TTE convergence history neglected term in the Euler discretization of @x @t is reasonable mixed absolute-relative bound on the error in the i th component of x at the n th step is where can be approximated Taking - as 3and implementing this strategy in the Euler code in place of SER yields the results in Fig. 4.3. Arrival at ffi max occurs at the same step as for SER, and arrival at the threshold jjF occurs one iteration earlier. However, the convergence difficulties after having arrived at are slightly greater. 5. Conclusions. Though the numerical experiments of the previous section do not confirm the theory in detail, in the sense that we do not verify the estimates in the hypotheses, a reassuring similarity exists between the observations of the numerics and the conceptual framework of the theory, which was originally motivated by similar observations in the literature. There is a fairly long induction phase, in which the initial iterate is guided towards the Newton convergence domain by remaining close to the physical transient, with relatively small timesteps. There is a terminal phase which can be made as rapid as the capability of the linear solver permits (which varies from application to application), in which an iterate in the Newton convergence domain is polished. Connecting the two is a phase of moderate length during which the time step is built up towards the Newton limit of ffi max , starting from a reasonably accurate iterate. The division between these phases is not always clear cut, though exogenous experience suggests that it becomes more so when the corrector of x 1 is iterated towards convergence on each time step. We plan to examine PSEUDO-TRANSIENT CONTINUATION 15 this region of parameter space in conjunction with an extension of the theory to mixed steady/\Psitc systems (analogs of differential-algebraic systems in the ODE context) in the future. Acknowledgments . This paper began with a conversation at the DOE Workshop on Iterative Methods for Large Scale Nonlinear Problems held in Logan, Utah, in the Fall of 1995. The authors appreciate the efforts and stimulus of the organizers, Homer Walker and Michael Pernice. They also wish to thank Peter Brown, Rob Nance, and Dinesh Kaushik for several helpful discussions on this paper. This paper was significantly improved by the comments of a thoughtful and thorough referee. --R The Numerical Solution of Initial Value Problems in Differential-Algebraic Equations Some efficient algorithms for solving systems of nonlinear equations VODE: A variable coefficient ODE solver Using Krylov methods in the solution of large-scale differential-algebraic systems Hybrid Krylov methods for nonlinear systems of equations Pragmatic experiments with Krylov methods in the stiff ODE setting Numerical Methods for Nonlinear Equations and Unconstrained Opti- mization Towards polyalgorithmic linear system solvers for nonlinear elliptic problems Numerical Initial Value Problems in Ordinary Differential Equations Analysis of numerical methods Robust linear and nonlinear strategies for solution of the transonic Euler equations Iterative Methods for Linear and Nonlinear Equations Termination of Newton/chord iterations and the method of lines Aerodynamic applications of Newton-Krylov-Schwarz solvers A parallelized elliptic solver for reacting flows Experiments with implicit upwind methods for the Euler equations Convergence of Iterations for Linear Equations Application of Newton-Krylov methodology to a three-dimensional unstructured Euler code A Newton's method solver for the Navier-Stokes equations Iterative Solution of Nonlinear Equations in Several Variables A description of DASSL: a differential/algebraic system solver Description and use of LSODE Driven cavity flows by efficient numerical techniques A modification of Newton's method Implementation of implicit formulas for the solution of ODEs An error estimate for the modified newton method with applications to the solution of nonlinear two-point boundary value problems Accurate and economical solution of the pressure head form of Richards' equation by the method of lines Newton solution of inviscid and viscous problems --TR --CTR W. K. Anderson , W. D. Gropp , D. K. Kaushik , D. E. Keyes , B. F. Smith, Achieving high sustained performance in an unstructured mesh CFD application, Proceedings of the 1999 ACM/IEEE conference on Supercomputing (CDROM), p.69-es, November 14-19, 1999, Portland, Oregon, United States T. Coffey , R. J. McMullan , C. T. Kelley , D. S. McRae, Globally convergent algorithms for nonsmooth nonlinear equations in computational fluid dynamics, Journal of Computational and Applied Mathematics, v.152 n.1-2, p.69-81, 1 March D. Gropp , Dinesh K. Kaushik , David E. Keyes , Barry Smith, Performance modeling and tuning of an unstructured mesh CFD application, Proceedings of the 2000 ACM/IEEE conference on Supercomputing (CDROM), p.34-es, November 04-10, 2000, Dallas, Texas, United States Feng-Nan Hwang , Xiao-Chuan Cai, A parallel nonlinear additive Schwarz preconditioned inexact Newton algorithm for incompressible Navier-Stokes equations, Journal of Computational Physics, v.204 Howard C. 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pseudo-transient continuation;nonlinear equations;global convergence;steady state solutions

276471

Wavelet-Based Numerical Homogenization.

A numerical homogenization procedure for elliptic differential equations is presented. It is based on wavelet decompositions of discrete operators in fine and coarse scale components followed by the elimination of the fine scale contributions. If the operator is in divergence form, this is preserved by the homogenization procedure. For periodic problems, results similar to classical effective coefficient theory are proved. The procedure can be applied to problems that are not cell-periodic.

Introduction In many applications the problem and solution exhibit a number of different scales. In certain cases we are interested in finding the correct coarse-scale features of the solution without resolving the finer scales. The fine-scale features may be of lesser importance, or they may be prohibitively expensive to compute. However, the fine scales cannot be completely ignored because they contribute to the coarse scale solution: the high frequencies of solution may combine with the high frequencies of Research supported by Office of Naval Research grants N00014-92-J-1890 P00003 and N00014- 95-I-0272 y Numerical Analysis and Computing Science, Royal Institute of Technology, Stockholm (mihai@nada.kth.se). z Department of Mathematics, University of California, Los Angeles, California 90024 (engquist@math.ucla.edu) and Numerical Analysis and Computing Science, Royal Institute of Technology, Stockholm. the differential operator to yield low frequency components. The homogenization problem can be stated in various formulations. A classical formulation, see e.g. Bensoussan et al. [1], is the following: Consider a family of operators indexed by the small parameter ", and for a given f , let u " solve the problem Assume that homogenization problem is to find an operator L and f such that: For example, consider the following operator with oscillating coefficients d dx d dx where a(x) is a positive 1-periodic function bounded away from zero. Then it is easy to show that and the limit u satisfies a constant coefficients equation. The coefficient is not the average of a(x) over a period, but rather the harmonical average a also called the effective coefficient. The homogenized operator is since f: In practice, we often need to solve the equation (1) for a small but fixed ". Since close to u, we may solve the homogenized equation (2) instead of the original equation. The homogenized equation is usually much simpler to solve. In the case of effective coefficients, the solution of the homogenized equation contains no high frequency components and thus it is an approximation to the coarse scale behavior of In a very interesting paper, M. E. Brewster and G. Beylkin [3] describe a homogenization procedure based on a multi-resolution analysis (MRA) decomposition. They consider integral equations, which may arise, e.g., from the Method of Lines discretization of a PDE, and homogenize over the time-variable. In a MRA, the concept of different scales is contained in the nested spaces V j . Homogenization is reduced to projecting the solution of the original equation from the fine resolution space V n onto the coarse resolution space V 0 . The homogenized operator, if it exists, operates on the space V 0 , but in general it is not the projection of the original operator onto the coarse space. Many classical homogenization techniques are based on the essential assumption that the coefficients are periodic on the fine scale. However, this does not hold in many applications. The analytic expansions methods require an 'a priori known number of scales, which again may be a serious restriction, see e.g., L. Durlofsky [7]. For two-dimensional elliptic operators with cell-periodic coefficients @ the homogenized operator is The effective coefficients A ij are found by computing Z / is the solution to the cell-problem: @ @a ik Wavelet-based homogenization can deal with both non-periodic coefficients on the fine scale and use the all the scales involved from the fine-scale space V j to the coarse-scale space V 0 . Following the construction in [6], using the Haar system, we build a homogenized operator L J for the discrete operator 1 . The grid-size is are the forward and backward (undivided) difference operators. We show that the homogenized operator has a natural structure of the form 1 where H is well approximated by a band diagonal matrix. In some cases, we prove that H equals the effective coefficients predicted by the classical homogenization the- ory, modulo a small error-term. In the two-dimensions, we show that our technique preserves divergence form of operators. We are concerned with a model of fine- and coarse-resolution spaces. The framework is multi-resolution analysis or wavelet formalism. In this framework, we have the concepts of fine and coarse scales together with the locality properties needed in analyzing operators with variable coefficients. For the precise definitions of a MRA, we defer the reader to the books by I. Daubechies [4] and Y. Meyer [9]. We consider a ladder of spaces V J ae V J+1 which are spanned by the dilates and integer translates of one shape-function The functions ' J;k form an L 2 -orthonormal basis. The orthogonal complement of V J in V J+1 is denoted by W J and it is generated by another orthonormal basis is called the mother wavelet. The transformation that mapping the basis f' J+1;k g into f/ J;k ; ' J;k g is an orthogonal operator and we denote its inverse by W ? . The product W J called the wavelet transform and it can be optimally implemented (called the fast wavelet transform). We denote by P j and Q j the L 2 -projections onto If an operator L J+1 is acting on the space V J+1 , it can be decomposed into four operators L acting on the subspaces W J and V J , where As a shorthand notation we have that or simply Note that if evaluated on a basis, the operator notation becomes a legitimate block- matrix construction. The identification of a function f 2 V J with the sequence c of coefficients in the basis ' J;K is an isometry: If Unless otherwise specified, the jj:jj notation refers to the corresponding 2-norm (con- tinuous or discrete). The same holds for the inner-product notation. Our results are proven in the simplest multi-resolution analysis, the Haar system. The shape function is the indicator function of the interval [0 1] and the mother wavelet is The Haar system provides an orthonormal basis in both L 2 (R) and L 2 ([0 1]). The space V J consists of piece-wise constant functions on a grid with step-size It is identified with l 2 (or R 2 J in the finite case). The Haar transform from V J+1 to W J \Phi V J is simply: 3 Homogenization in the Haar Basis Discretize the equation d dx a d dx on a uniform grid with using finite volumes. Let diag(a) be the diagonal matrix containing the values of a(x) at the grid-points. As an operator on V J+1 , diag(a) represents multiplication by the grid-function a. The discrete equation is split by the natural decomposition V U l F l where the indices h and l denote the W J and V J components. The equation (3) is a discretization of the continuous equation (a(x)u 0 The coarse scale solution of the discrete equation (3) is the projection of U onto V J , i.e. U l . Eliminating U h yields the equation for U l : The homogenized operator is the Schur complement Let us make some preliminary remarks: ffl The homogenization procedure is in fact block Gaussian elimination. The idea is not new, it can be found, e.g., in odd-even reduction techniques. There is a real gain only if the homogenized operator L J can be well approximated by a sparse matrix. It is the compression properties of wavelets that maintain the homogenization procedure efficient, similar to the case of Calderon-Zygmund operators as seen in [2, 5]. ffl The experience with the non-standard form representation of elliptic operators indicate that A J has a strong diagonal dominance and thus its inversion will not be as difficult as inverting the operator L J+1 , see [6]. ffl We expect that the homogenized operator L J will have a similar structure as the operator L J . In fact we will see that if L J+1 is in divergence form, where H is a strongly diagonal dominant matrix. We will call H the homogenized coefficient matrix. ffl The homogenization procedure can be applied recursively. If we have the equation that produces the solution on the scale then we homogenize the operator L j+1 . This means that we produce the operator L j on and the right-hand side f j such that the solution of the homogenized equation is ffl If the homogenized operator has a rapid decay away from the diagonal, then it can be well approximated by a band-diagonal operator. The same applies for the matrix H. The structure of the homogenized operator is given by the decomposition of the discrete operators \Delta and diag(a). 3.1 multiplication operator We first examine the multiplication-by-functions operator diag(a). The following lemma is obvious: Lemma 1 If ' is the Haar system's shape function and / the mother wavelet, then For we use the notation W J a fi v denote the component-wise multiplication of vectors. We have the following point-wise multiplication rule: e a e a fi e Proof: Set a = P a J+1;k ' J+1;k and Using Lemma 1, we have k;l a J+1;k v J+1;l ' J+1;k ' J+1;l a J+1;k v J+1;k Thus point-wise multiplication of functions is the equivalent to component-wise multiplication of the coefficients. Then we have: e a k / J;k e which proves the statement.2 The high frequency components of a and v interact and contribute to the low frequency part of the product av. This is modeled in the Haar basis by correcting the product a fi v of the coarse scale coefficients with the fine scale contribution e a fi e v. The structure of the pointwise multiplication operator is given by the following statement: Proposition 1 If W J diag(a) diag(e a) diag(e a) diag(a)7 5 : The matrix diag(a) is the point-wise multiplication operator. We have the following amusing result: Proposition diag(a) be the multiplication-by-function operator on . The coarse-grid projection P J M J+1 P J is multiplication by the arithmetical averages (a 2k + a 2k+1 )=2 The homogenized operator M J is multiplication by the har- monical averages ff Proof: The coarse grid projection of diag(a) is 1= 2 diag(a), which is, in each component, the arithmetical average (a 2k + a 2k+1 )=2 of the corresponding fine-grid values. The homogenized operator ispi Component-wise this a k 2a 2k a 2k+1 which is the harmonical cell-average of the corresponding fine-grid values. 2 3.2 Decomposition of \Delta We start by computing the decomposition W J \Delta +W on the basis functions of W J \Phi Then we have Similar computations yield and then Let S n be the shift matrix S n defined by S (n) which is the projection of the shift operator We have the following proposition: Proposition 3 The decomposition of 1 in the Haar system is Obviously, the structure is repeated at each level j. Since , we have that Dropping the diag notation in Proposition 1, we have that p- A J where A 3.3 Boundary conditions The notations for the discrete difference operators and their corresponding matrices. They can describe periodic, Neumann or Dirichlet boundary conditions. They can also operate (as infinite matrices) on infinite sequences arising from discretizing problems on the whole real axis. The derivation of the decompositions of the L J+1 and the homogenized coefficient matrix H are formally the same. However, in the periodic case, the operator L J+1 is singular and it is not trivial that A J is invertible. In the periodic case, the matrices \Delta are circulant. This property is preserved by the transform W J . If we define the shift matrices S \Sigma1 as circulant matrices, then M is also circulant, and thus all the matrices corresponding to the level J have the same property. In the infinite case, are trivially circulant. With periodic boundary conditions, it is easy to show that L J+1 has a 1-dimensional null-space spanned by the constant grid-functions: Since vanishes only on con- stants, the ellipticity condition a ? 0 implies that any non-constant zero-eigenfunction must satisfy constant. It follows that v is monotone, which contradicts periodicity. The null-space of L J+1 is transformed by W J into the one-dimensional space N spanned by [0 is a constant grid-function. The quadratic form J x is positive whenever x 62 N . In particular, putting have [y for any y 6= 0. This proves that A J is positive definite and therefore invertible and thus the homogenized operator L J is well-defined even with periodic boundary conditions. Both the equations L J+1 u need extra conditions. If we decide e.g., to fix a boundary value, we can eliminate a row from both systems. This elimination can be done after the homogenized operator is produced. Thus we need not track the effect of the boundary condition through the homogenization process. Other type of conditions, such as Dirichlet boundary conditions in the non-periodic case or integral conditions can be handled in a similar fashion. 3.4 The homogenized coefficient matrix Let us consider a discretization on the whole real axis, i.e., the case where the matrices are infinite. The coarse-scale projection of L J+1 is L J is the "wrong" operator for an obvious reason: the average coefficient is obtained using only the even components of the the fine-scale coefficient a L J is insensitive to variations of the odd-components in the original problem. Even if the fine-scale is not present in a(x), i.e. e a = 0, L J still has the wrong coefficient We build the homogenized operator as the Schur complement L Proposition 4 The operator L J has a natural structure 1 a) (7) Definition 1 We call H the homogenized coefficient matrix of the operator The natural question to ask is if there is any connection between the homogenized coefficient matrix H and the classical homogenized equations. Proposition 2 gives that the Schur complement of the diagonal matrix a is the diagonal matrix ff containing the harmonical averages of neighboring values. This would then agree exactly with the classical homogenization theory, if the Schur complement of \Delta could be expressed in terms of the Schur complement of the middle factor a. Unfortunately, this is not the case, so we have to use the form given in (7). We look at the extreme case when is the sum of a constant and the highest frequency represented on the grid, i.e., a(xm We have that a and e a are represented as constant vectors in the bases of V J and W J . The fact that a(x) ? 0 implies je aj ! a. Since a and e a are constant vectors, we have a) where Simple computations yield and then The homogenized coefficient matrix defined by (7) is a) Classical homogenization theory yields the effective equation ff d 2 where the effective coefficient is given by the harmonical average: Z 2ha(x)dx a 0 In the rest of this section, we will be looking only at the coarse grid function space J . For simplicity, we will let denote the grid-size of V J . The following theorem shows that the numerically homogenized operator 1 equals the discrete form ff 1 of the classically homogenized equation, apart from a second-order term in h. Theorem 1 Let J+1 be such that a 2 V 0 is a constant and the oscillatory part e a 2 W J has constant amplitude and satisfies the condition j e aj ! a. Let L and ff be the harmonical average in (9). Then there exists a constant C independent of the grid-size h such that if v is the discretization of a function v(x) with a continuous and bounded fourth derivative, then Proof: Let us show first that the high-frequency operator A J is invertible by showing that it is diagonal dominant. We have and the tridiagonal structure of A J is clearly seen. The ellipticity of L J+1 implies that a. The diagonal entries of A J are larger then 4a while the sum of the off-diagonal terms is smaller then the same amount. The diagonal dominance of A J gives a rapid decay of the entries of A \Gamma1 J away from the diagonal. Indeed, we have e a \Gamma a 1=2, the Neumann expansion for A \Gamma1 J is convergent and (10) reveals the size of the off-diagonal entries: A I Next we compute the row-sum of H. Note first that since A J is circulant, it has an and the corresponding eigenvalue is 8a. A \Gamma1 J shares the eigenvector c, which shows that all its row-sums are 1=(8a). Note that c is also an eigenvector of I with the corresponding eigenvalue 2. Thus we have a) p2a Finally we estimate L J v. Note that since I Assuming v is a discrete smooth function, Taylor expansion around v Let us estimate the j component of Hv. Applying H to the first term in (12) produces just ffv j . Due to symmetry, we have that \Gamman ). Applying H to the odd-order terms of (12) shifts in the j component quantities with opposite signs and then adds them. The even-order terms contribute such that show later that the coefficients fl n have exponential decay and thus is convergent for any k. Applying 1 comparing to which in its turn gives the desired estimate with It remains to be shown that the constant C is independent of the grid-size h. The expansion (11) shows that A \Gamma1 J is generated infinite long stencil with exponential decay rate 1. To build H from A \Gamma1 J , we first apply I+S 1 and I+S \Gamma1 , which has the effect of adding together neighboring diagonals. Indeed, if A \Gamma1 P a n S n , we have a (a , the elements of the product (I are bounded by 4K ae n . H is then found by multiplication with a) 2 and the addition of a diagonal term. The decay away from the diagonal of the terms fl n is The exponential decay in fl n dominates the growth of n 2 and thus we find the constant C: ae (log ae) 2 CRemark: The fact that Hv - ffv for smooth functions v can be also seen from the Fourier analysis of H. By doing a discrete Fourier transform of H, we obtain a diagonal matrix diag(-g). The diagonal - g is given by the symbol of H which is Note that - g is just the Fourier transform of any row of H. It is therefore no surprise that - It turns out that d-g d! The approximation error is indeed quadratic in ! since - If the Fourier coefficients of v decay sufficiently fast, then we have - g-v - ff-v, and by the inverse transform, Hv - ffv. Note in Figure 1 that - g(!) has a moderate growth even for large !. Figure 1: The Fourier components of H and ffI (dashed line). - g(!) behaves like multiplication with - boundary conditions are assumed. In practice, we want to approximate the homogenized operator L J by a sparse approximation. Due to the diagonal decay, we can approximate L J by a band- diagonal matrix L J;- where - is the band-width. Let us consider the operator band defined by ae We have in fact two obvious strategies available for producing L J;- : We can set directly or use the homogenized coefficient form and build produce small perturbations of L J . However, important properties, such as divergence form, are lost in the first approach and numerical experiments show that - needs to be rather large to compensate for this. The second approach produces L J;- in divergence form. Moreover, the approximation error can be estimated, as in the following result: Theorem 2 If the conditions of Theorem 1 are valid, then If v is the discretization of a smooth function v(x), then Proof: The exponential decay from the diagonal in H, given in (13), yields If v is a smooth function, using the commutation property of H (and band(H)), we have where - is some point in R. Therefore we have Taking the supremum over all - and then the maximum over all j yields the desired estimate. 2 Remark: The above estimates hold also for Dirichlet boundary conditions. In the case of periodic boundary conditions, the meaning of "away from the diagonal" is different because the wrap-around effect. The diagonal band of width - is then defined by 2(ji \Gamma jj ( mod is the size of the matrix. 3.5 Numerical experiments We test the homogenization procedures on some examples. ffl With periodic coefficients, wavelet and classical homogenization produce the same discrete solution. With non-periodic variations of the coefficients, the effective equations cannot extract the local features of the solution. Due to the localization of the wavelet basis elements, such local features are preserved by wavelet homogenization. ffl Solution with several different scales. The test problem is (a(x; x=" 1. We project the solution on spaces that resolve either both the scales " or just the finest scale " 1 . ffl Comparison of the solutions of the homogenized forms using the two truncation strategies, band(L J ; -) and 1 different values for - We see that truncation of the homogenized coefficient matrix H performs much better. 3.5.1 Non-periodic variable coefficients First we compare the exact, classical-homogenized, and wavelet-homogenized solutions to a periodic problem. We consider the two-point boundary problem The exact solution solves the discrete equation We take a(x) to have alternating values 1 and 100 on a fixed grid. The classical and wavelet homogenized solutions are pictured in Figure 2. Exact solution Wavelet homog. Classical homog. solution Wavelet homog. Classical homog. Figure 2: Exact, classical homogenization and Haar basis homogenization solutions in the periodic case. grid-points. The plot on the right is a detail of the left image. The wavelet solution is computed using 3 levels, i.e. the coarse scale contains eight times fewer grid-points. The effective coefficient is 200=101 - 1:9802 and thus classical homogenization yields the approximation x: Note the detail in Figure 2 where the wavelet based solution u is essentially a shift of u eff , i.e. u contains no high frequencies. Now we take a(x) to be uniformly distributed in the interval [1 100], as plotted in Figure 3 (left). The classical homogenized coefficient (effective coefficient) is computed as a dx solution Wavelet homog. Classical homog. Figure 3: coefficients a(x) (left) and a comparison of the exact solution u, effective equations solution u eff and Haar basis homogenization solutions u. grid-points in these plots. Figure 3 (right) compares the exact solution u with the wavelet homogenized u and the result of classic homogenization u eff , where the effective coefficient is a 18:8404. The fine grid has 256 points. Both u eff and u are represented on the coarse grid using 32 points. However, the wavelet homogenized solution u captures much more coarse-scale detail then the classic solution u eff . 3.5.2 Homogenization over multiple scales We test a problem that contains three different scales: Let The coefficient has three scales, The solution of the equationh 2 contains all the three scales if h resolves the finest scale of a(x). Put resolve all the scales of the problem. Then we project the exact solution onto V 6 . Exact sol. on V9 Projection on V3 Projection on V6 -0.06 -0.02Exact sol. on V9 Projection on V3 Projection on V6 -0.06 Figure 4: Homogenization of several scales. Coefficient a Plot of u 9 , u 6 and (left). Details of plots (right) shows that u 6 averages the finest scale only and resolves the coarser scales. u 3 resolves only the coarsest scale. Figure 4 shows that the finest scale contribution is averaged out, but the coarser scales are resolved. Projection onto V 3 averages both the finer scales and the solution has the characteristics of a constant-coefficients problem. 3.5.3 Banding strategies We test the accuracy of approximating the homogenized operator by banded matrices using the two strategies described in Section 3.4. The coefficients a(x) are chosen at random, uniformly distributed in the interval (0:1 2). The boundary conditions are Exact homog. diagonals diagonals diagonals -0.4 -0.3 -0.2 -0.1Exact homog. 3 diagonals 5 diagonals 7 diagonals -0.4 -0.3 -0.2 Figure 5: The homogenized operator approximated by banded matrices. Banding the exact homogenized operator L J (left) needs a much larger band-width - as compared to banding the homogenized coefficient matrix H (right). 512 grid-points on the fine grid, 64 grid-points on the coarse grid. Figure 5 (left) is the plot the solutions of band(L J ; -)u To obtain even better accuracy, using the approximation of the homogenized coefficient considerably fewer diagonals are needed. Figure 5 (right) plots the solutions of 1 diagonals. 4 2-D Problems Numerical homogenization for multi-dimensional problems is of great interest since the analytic methods can only handle periodic micro-structures, see e.g., Bensoussan et al. [1]. The aim of this section is to show that if a 2-D fine-scale operator is in discrete divergence form then the homogenized operator L J acting on the coarser space has the same form. As we saw in the one-dimensional case, this property is important for efficient truncation strategies. 2 The operator L J+1 is called discrete elliptic if 1. L J+1 is symmetric, i.e., A 2. The spectrum of L J+1 lies in f0g [ [ffi; +1), where ffi ? 0, and 0 cannot be a multiple eigenvalue. 4.1 2-D tensor product wavelet spaces Let us make the notations precise. We consider the tensor product space V generated by the canonical basis The coarse space is V J\Omega V J and it is generated by ' J;k\Omega ' J;l . The orthogonal complement of V J in V J+1 is the wavelet space The wavelet transform maps the standard basis of V J+1 into the union of the standard bases of V J and the three components of W J . If L J+1 is the matrix of a linear operator on V J+1 , then the orthogonal basis transformation W J yields The operators A J , B J , C J and L J operate on subspaces: By elimination, we have that the homogenized operator is Note that in the finite case, dim(W J We can continue with the decomposition of V obtain in this manner a multi-resolution analysis on the product space. The product of the orthogonal transformations is the (orthogonal) wavelet transform that maps V J+1 into (\Phi 0-j-J W j 4.2 Invariance of divergence form The operator acts on V J+1 and is defined by \Delta x f)\Omega g, where is the 1-D forward difference operator. \Delta y are defined in a similar manner. We regard the operators A (ij) as multiplication by the discrete functions a (i;j) (x; y), i.e., A (i;j) (' l (y). In general, A (i;j) can be any operator on V J+1 , but then L J+1 is may no longer be a discretization of a differential operator. Let us formulate the result: Theorem 3 Let L J+1 be a discrete elliptic operator in divergence form (14). Assume periodic boundary conditions in the x and y directions and let L J be the homogenized operator using the Haar transform. Then L J is also in divergence form. Proof: We begin by observing that the orthogonal transform W V J can be written as W is a the corresponding 1-D transform in the x-direction, and W y is defined analogously. Remark also that W x W W y W x . Next we observe that \Delta x +\Omega I and \Delta y This gives that \Delta x and \Delta y . The next step is to compute the decomposition of \Delta x in (W Using the standard inner-product on tensor-product spaces, we apply \Delta x to a basis function and test it against another basis function: can be any ' J;k or / J;k . Note that the second inner-product is 0 if g 1 6= g 2 . The first inner-product gives the 1-D decomposition of \Delta + , as in Proposition 3. Using the notations of Proposition 3, we can synthesis the decomposition of in the following table: W \Theta W W \Theta V V \Theta W V \Theta V W \Theta W M\Omega I \Gamma\Delta +\Omega I M\Omega I \Gamma\Delta +\Omega I V \Theta W \Delta I \Delta +\Omega I I \Delta +\Omega I In a similar fashion, we obtain the decomposition of \Delta y I\Omega M I\Omega M I\Omega and \GammaM I \Delta \Gamma\Omega I \GammaM I \Delta \Gamma\Omega I \Gamma\Omega I \Delta \Gamma\Omega I \Gamma\Omega I \Delta The essential point is that the last block-row of the decomposition of \Delta x (or \Delta y contains only \Delta +\Omega I (or entries. For the \Delta x (or \Delta y the analogous holds for the last block-column. Noting that \Delta , on the coarse space V J , we have that the decomposition of the product 1 \Gamma is of the form4h 26 6 6 4 A are some arbitrary operators. Adding the contributions of all the terms in the form (14) of L J+1 yields:4h 26 6 6 4 A where D is in discrete divergence form. Since L J+1 is elliptic, periodic boundary conditions imply it has a one-dimensional null-space, spanned by the constant functions. This null-space is mapped by the transform W J into V J . Since the operator L J+1 has non-negative eigenvalues and A operates on the complement of V J , it follows that v ? Av ? 0 for any v 6= 0. Therefore A is invertible and we can build the homogenized operator by block Gauss elimination. This yields where \Delta (1) stands for \Delta x We have that L J is in divergence form on the coarse space V J . 2 Remark: The conservation of the divergence form of L J+1 under the 2-D Haar transformation has important consequences. In the multi-dimensional case, it is known that the problem r admits a homogenized equation but apart from the cell-periodic problem, there is no general algorithm for deriving the homogenized operator L. In fact, the nature of L is not known and numerical homogenization can therefore be used not only as a practical tool, but also to find information about the structure homogenized operator L. 4.3 A numerical example We chose a(x; 1. The classical homogenized equation is is the harmonical average of a in a cell with the length of a period and 2 is the (arithmetical) average in the same cell: The homogenized equation has constant coefficients but is strongly anisotropic. Figure 6 displays the exact and wavelet homogenized solutions. The domain is the unit square and there are Dirichlet boundary conditions on the coordinate axes and Neumann conditions on the other two sides. -5 -5 y Figure Fine scale (left) and homogenized solution (right). Note that the homogenized solution captures the effect of the coarse-scale strong anisotropy, averaging only the fine-scale variations. Extensions The homogenization procedure can be carried out on coarser and coarser levels to produce a sequence of homogenized equations If we solve the coarse scale problem exactly, then by block back-substitution we produce the exact solution: If no truncations are used in building the homogenized operators L j , the above strategy describes an exact, direct solver, which compares to the reduction techniques in computational linear algebra, see [8]. If truncations are used, the direct solver contains an approximation error. The homogenization procedure can be applied recursively on any number of lev- els, provided that the initial operator is in discrete divergence form and is elliptic. These two properties are sufficient for the existence of the Schur complement and they are are inherited by the homogenized operator. It is not necessary that L J+1 approximates a differential operator as long as it is elliptic and in divergence form. On coarser scales, the homogenized operator resembles the inverse of a differential operator and is expected to be dense. The use of wavelets with a high number of vanishing moments, known to compress well Calderon-Zygmund operators, could have better compression effects then the Haar system, in the spirit of the ideas presented in Beylkin, Coifman and Rokhlin's work [2]. In applications, if we want to use the homogenized coefficient matrices, we would not invert A j , but rather LU-factorize it in the prescribed bandwidth -. --R Asymptotic Analysis for Periodic Structures. Fast wavelet transform and numerical algorithms. A multiresolution strategy for numerical homoge- nization Ten Lectures on Wavelets. Wavelets and Singular Integrals on Curves and Surfaces. Numerical calculation of equivalent grid block permeability tensors for heterogenuous porous media. Matrix Computations. Ondelettes et Op'erateurs --TR --CTR Shafigh Mehraeen , Jiun-Shyan Chen, Wavelet-based multi-scale projection method in homogenization of heterogeneous media, Finite Elements in Analysis and Design, v.40 n.12, p.1665-1679, July 2004 Giovanni Samaey , Ioannis G. Kevrekidis , Dirk Roose, Patch dynamics with buffers for homogenization problems, Journal of Computational Physics, v.213 n.1, p.264-287, 20 March 2006 Assyr Abdulle , E. Weinan, Finite difference heterogeneous multi-scale method for homogenization problems, Journal of Computational Physics, v.191 n.1, p.18-39, 10 October Pingbing Ming , Xingye Yue, Numerical methods for multiscale elliptic problems, Journal of Computational Physics, v.214 n.1, p.421-445, 1 May 2006 Jiun-Shyan Chen , Hailong Teng , Aiichiro Nakano, Wavelet-based multi-scale coarse graining approach for DNA molecules, Finite Elements in Analysis and Design, v.43 n.5, p.346-360, March, 2007

wavelets;elliptic operators;homogenization

276474

Numerical Integrators that Preserve Symmetries and Reversing Symmetries.

We consider properties of flows, the relationships between them, and whether numerical integrators can be made to preserve these properties. This is done in the context of automorphisms and antiautomorphisms of a certain group generated by maps associated to vector fields. This new framework unifies several known constructions. We also use the concept of "covariance" of a numerical method with respect to a group of coordinate transformations. The main application is to explore the relationship between spatial symmetries, reversing symmetries, and time symmetry of flows and numerical integrators.

Introduction Recently there has been a lot of interest in constructing numerical integration schemes for ordinary differential equations (ODEs) in such a way that some qualitative geometrical property of the solution of the ODE is exactly preserved. This has resulted in much work on integration schemes that can preserve the symplectic structure for Hamiltonian ODEs [12, 13, 14, 23, 33]. Other authors have constructed volume-preserving integrators for divergence-free vector fields [3, 19, 30]. Other authors again have concentrated on preserving energy and other first integrals [14, 20, 21] or other mathematical properties [7]. In this paper we are interested in constructing integrators that preserve the symmetries and reversing symmetries of a given ODE. One reason this is important is that nongeneric bifurcations can become generic in the presence of symmetries, and vice versa. One can also consider the time step of the integration scheme as a bifurcation parameter, showing how vitally important it is to stay within the smallest possible class of systems. Reversing symmetries are particularly important, as they give rise to the existence of invariant tori and invariant cylinders [15, 22, 25, 28]. One of the first authors to study integrators that preserve a reversing symmetry was Scovel [24]. His method, as it stands, can only preserve a single reversing symmetry and no ordinary symmetries. In this paper we give a construction of integration schemes preserving an arbitrary number of symmetries and reversing symmetries. In exploring this question, it became clear that constructions were working because of the effect of certain operators on compositions of maps. This led us to express these effects in terms of automorphisms and antiautomorphisms of groups. In particular, it turns out that many different compositions designed to give methods particular features are all examples of the same general principle. We prepare the basic algebraic material in section 2. A curious structure emerges several times. Often a subset of a group is not closed under products AB, but is closed under symmetric products ABA. This is true for symmetric or antisymmetric matrices, reversible maps, and time-symmetric maps. We have called such a subset a 'pseudogroup', and it is very useful to us here in integration, although it remains to be seen whether it is an interesting algebraic object in its own right. Flows of vector fields have many nice properties, and many of these are desirable in an integrator as well. Some of these properties are defined for all vector fields, such as covariance under transformation groups, closure under differentiation [2], and time symmetry (see (27) below). These are discussed in section 3. Others are defined only for the flows of a certain class of vector fields, such as symplecticity for Hamiltonian systems, closure under restriction to closed subsystems [2], and energy conservation [14]. We believe that in the context of this new viewpoint of numerical methods, it is natural to try to 'lift' these properties so that they are naturally defined for all vector fields. For example, Ge's definition of invariance with respect to symplectic transformations [5] is only defined for Hamiltonian systems, but it can also be seen as a special case of general covariance. In section 4 we study this lift for symmetries and reversing symmetries, and also look at methods that make explicit use of the symmetries of the vector field. The automorphism point of view elucidates the relationship between different prop- erties, such as time-symmetry and reversibility, which are in fact independent, although related in some special cases considered previously (Runge-Kutta methods with a linear reversing symmetry by Stoffer [27], and splitting into reversible pieces by McLachlan [10]). For example, we show in section 4 how a map covariant under a group larger than strictly necessary can be more flexible under composition. Algebraic preliminaries Let G be a group with elements ' and consider functions G. We only consider functions which are either automorphisms of the group, that is, A+ is a bijection and or antiautomorphisms, that is, A \Gamma is a bijection and G: (2) Examples of automorphisms are the inner automorphisms G. Examples of antiautomorphisms are inverse and, if G is a linear group (a group of matrices), the transpose T '. (This can be generalized to the quantum case where ' is a linear operator on a Hilbert space.) Note that the set of automorphisms and antiautomorphisms itself also forms a group, where the group operation is composition. The automorphisms form a normal subgroup. More specifically, every such group G \Sigma is homomorphic to Z 2 there is an onto map We call the number oe(A) the grading of A. In the case at hand, the automorphisms have grading +1 and the antiautomorphisms have grading \Gamma1. Each such group G \Sigma is generated by, for example, its antiautomorphisms alone (if it has any), or by one antiautomorphism together with all the automorphisms. Define the fixed sets and Notice that but not. However, we do have that the set of group elements fixed by a given antiautomorphism is closed under the symmetric triple product: This concept seems to be useful in, for example, constructing integrators, and we shall call it a pseudogroup 2 . Example. Let G be the linear group GL(n) and consider the antiautomorphism T , transpose. The fixed set of this antiautomorphism is the pseudogroup of nonsingular symmetric matrices: if X and Y are symmetric matrices, then so is XY X. 1 Unless there are no antiautomorphisms in the group. A discussion of (what we call) the pseudogroup of maps with time-reversal symmetry, is given in [9]. Example. Let G be any group and consider the inner automorphism N / . The fixed set of this automorphism is the group of so-called /-equivariant elements: if ' 1 , ' 2 are /-equivariant (i.e., ' then so is ' 1 ' 2 . If the antiautomorphism A \Gamma is an involution (i.e., A 2 there is a 'projection' to because first introduced in a special case by Benzel, Ge, and Scovel [1].) This can be generalized to a group G \Sigma of automorphisms and (not necessarily involutory) antiautomorphisms. Unfortunately, it is difficult to project an arbitrary group element to fix(G \Sigma ); but QA \Gamma does map the fixed set of the subgroup G+ of automorphisms to the fixed set of the entire group. Proposition: (Generalized Scovel projection) Let G \Sigma be a group of automorphisms and antiautomorphisms, and let G+ be its subgroup of automorphisms. Let A \Gamma 2 G \Sigma be an antiautomorphism, so that G (The proof is given below.) This is not a true projection because it does not satisfy but it will have some useful properties in our applications. The 'projection' is not always surjective. For example, let take a group consisting of one automorphism (the identity) and one antiautomorphism (transpose), that is, G . Projection to the symmetric matrices produces symmetric matrices with nonnegative determinant. The projection Q and the pseudogroup property are in fact both examples of the following more general relationship, which is of central importance in this paper: Proposition: (Composition property) Let G \Sigma be a group of automorphisms fA j+ g and antiautomorphisms of the group G. Let ' 1 , for all A Proof: We only need consider the effect of an antiautomorphism; membership of fix(G will follow because the antiautomorphisms generate all of G \Sigma . A for all k. From this proposition the pseudogroup property follows by taking the generalized Scovel projection Q follows by taking ' 2 to be the identity. As an example of the composition property (12), take G g. All matrices B are in matrices A are in fix(G \Sigma ), and indeed, BAB T is symmetric. The reader is invited to check the consequences of the proposition for the sets of matrices given in table 1. We note that in most of this work, we can weaken the antiautomorphism requirement to G. This allows, for example, the operator A \Gamma the antisymmetric matrices. The composition property (12) is retained, but the projection (11) is lost if the identity is not in Projecting onto the fixed set of automorphisms is more difficult. However, one may have (as in the case of maps on R n ) that G is a near-ring: (G; +; :) is a near-ring if (G; +) is a group, (G; :) is a semigroup, and (' that maps under composition do not form a ring because they are not left-distributive.) If the automorphisms are linear with respect to addition in the near-ring, and G+ is a finite group with jG then we can use the following, which is a true projection: The half-sized group construction As mentioned above, groups G \Sigma are always homomorphic to Z 2 . In our application to reversing symmetries, we shall use groups G+ of automorphisms which themselves are homomorphic to Z 2 . From such a group we can construct a group H \Sigma of the same size as G+ in the following way. Let A+;oe denote an element of G+ of grading be a fixed antiautomorphism whose square A 2 \Gamma is an element of G+ with grading +1 which commutes with G+ . The automorphisms in H \Sigma are the elements A+;1 . The antiautomorphisms in H \Sigma are the elements . It is straightforward to check that this gives a group H \Sigma of the same size as G+ , i.e., half the size of G \Sigma . Example. Let G be GL(2) and G+ be the inner automorphisms corresponding to a subgroup of G homomorphic to Z 2 , say rotations by 3-=2. The antiau- tomorphism I (inverse) satisfies the above requirements: its square (the identity) is in G+ , does have grading +1, and does commute with G+ . The inner automorphisms N g;+1 together with the antiautomorphisms IN g;\Gamma1 form the group This group comprises two automorphisms and two antiautomorphisms-it is Z 2 \Theta Z 2 . Then in order for Q IN -=2 to be a projection to fix(H \Sigma ) we need ' to be fixed under all the automorphisms in H \Sigma . Here we require operator sign fixed set type 'projection' I anti IT auto AA orthogonal Table 1. Sets of matrices as fixed sets of (anti)automorphisms. Here I is the identity matrix and \GammaI 3 General properties of integration methods For simplicity we restrict our attention to systems of first-order autonomous ODEs on R n , i.e., ODEs of the form for some n. Let F be a set of vector fields on R n (each vector field need not be on a space of the same dimension), and let \Phi be a set of functions An element ' 2 \Phi is called an integrator of f if When we want to emphasize the functional dependence of ' on - and f we shall call it a "method;" at other times we may want to think of fixing - and f and looking at the resulting map. Examples. If all f 2 F are differentiable, then the exact flow ' - method Runge-Kutta method, and any one-step method which is functionally dependent only on f are possible members of \Phi. If we impose additional conditions on the set of vector fields F , then there is larger family of possible 's; e.g., if all f 2 F are C r , then the Taylor series method of order r may be in \Phi. Moreover, functions which are not integrators at all can also be in \Phi. Examples are ' - (f) : y 7! y, and, (if all f are C r ), elementary differentials such as - 2 f 0 f . To apply the results of the preceding section, a group is needed. To construct it we let the set of maps generate a group G by composition. For example, We will not specify the sets F and \Phi precisely, as we wish to make the constructions apply as generally as possible. Minimum requirements will be clear in each application. Not all the elements of G associate maps with vector fields. For example, let ' - (f) be an integrator of f . Then even though is an integrator of all vector fields not an operator on vector fields. In fact the group G may seem at first to be much bigger than necessary: one could also let, e.g., \Phi generate a group, with group operation But we wish to allow flexibility in the choice of time steps (for composition methods) and vector fields (for splitting methods). The functions A are now operators on the space of methods. They may or may not preserve the property of being an integrator, and if they do, they may not preserve the order of the integrator. The exact flows of differential equations are important elements of G. It is their properties that we are trying to mimic in constructing actual numerical methods. Some properties (e.g., the semigroup property ' - (f)' oe hold and are defined for all f ; others only hold and are defined for some f (e.g., volume-preservation det(d' - divergence-free f ). Where possible, we try and extend the definition of these restricted properties to all f ; but even with the guiding principle that they should hold for flows, this extension cannot be unique. In this section we consider properties defined for all f , in particular, those shared by fixed sets of automorphisms and antiautomorphisms of G. 3 Our first such property is "covariance". It is often useful to consider whether a method produces the same integrator in different coordinate systems. This concept was studied for symplectic transformations by Ge [5] with the aim of preserving integrals of Hamiltonian systems by symplectic integrators. It was also used (with general linear transformations and Runge-Kutta methods) by one of the present authors in [11]. We extend this concept to general integrators and arbitrary groups of transformations acting on phase space. Let H be a group with a left action (h; y) 7! hy on phase space R n 3 y. A method - (f) is covariant with respect to H if the following diagram commutes 8h 2 H: f=h f y x=hy \Gamma! ' - ( ~ The top arrow pushes forward the vector field f by h; the bottom arrow conjugates the Writing out the transformations through a clockwise loop shows that where 3 Note that if the antiautomorphism A \Gamma is such that flows are in is an integrator of f of a certain order, then so is A \Gamma ' - (f). This is useful when composing as in (12). (Compare the coordinate transformation automorphism K h to the inner automorphism whose fixed set is the equivariant maps.) Now that covariance is defined for \Phi by (24-26) (i.e., it is defined for the generators of G), the definition extends to all of G in the natural way. If ' - (f) is the time- flow of f , then ' is covariant with respect to the group of diffeomorphisms of R n . In general, however, discrete approximations will only be covariant with respect to the group of affine transformations or one of its subgroups. Different methods can then be classified by their covariance group. Many traditional methods, such as Euler's method, the midpoint and trapezoidal rules, and any Runge-Kutta or Taylor series method, are covariant with respect to the affine group (the semi-direct product of GL(n) and the translations R n ). (Some cases have been treated in [5, 27, 31].) Most methods are covariant with respect to translations, but some are not covariant with respect to the entire linear group but only with respect to a subgroup. For example, the known volume-preserving methods [3, 19] are covariant with respect to the subgroup of diagonal transformations. Splitting methods which partition the coordinates are covariant with respect to arbitrary transformations which preserve the partitioning. Our second important property is time-symmetry. A method ' is time-symmetric if This important property, also known as 'symmetry' or `self-adjointness' or 'reversibility', is considered in [4, 6, 17, 24, 32, 33]. (We reserve the name 'symmetry' for the spatial symmetries considered in the next section.) Because the parameter - is included in G, we can introduce the time-reversal automorphism so that time symmetry means being in the fixed set of the antiautomorphism IR. (One can also consider inverse-negative 's with ' - time-scaling 's with ' c- (f=c). However, since there is a simple projection onto the fixed set of the time-scaling operator, namely, ' - (f) 7! ' 1 (-f ), and the inverse-negative and time-symmetry properties are then equivalent, we will not emphasize these properties.) If our favourite integration method is not time-symmetric, we can use the projection \Gamma- to make it so. Now so the group G \Sigma generated by IR is just f1; IRg. It has no nontrivial automorphisms so all ' are in fix(G + ) and (29) gives a time-symmetric map for any '. (This projection was first given in [1].) We can also use the composition property (12) to get time-symmetric methods of the any method and ' 2 is time-symmetric, which has been used by many authors starting with Suzuki [29] and Yoshida [33] for constructing higher-order methods. Now consider the problem of constructing methods which are both covariant and time- symmetric. Take a group of coordinate transformation automorphisms and the antiautomorphism IR which together generate G (1) \Sigma . Firstly, if ' is covariant with respect to H (so that ' 2 fix(G for all h 2 H. Secondly, if H, and hence G+ is homomorphic to Z 2 , then the half-sized group construction gives a subgroup of G (1) G (2) with projection where g is an element with grading \Gamma1 and ' 2 fix(K h ) for all elements h with grading +1. All exact flows are in fix(G (1) are desirable properties for integrators as well. The advantage of also considering G (2) \Sigma is that then the projection requires less of '. 4 Integrators that preserve symmetries and reversing symmetries In this section we are interested in properties of integrators that do not hold for all vector fields, but only for those with some given property, such as possessing a symmetry or reversing symmetry. We first give some definitions: The ODE (17) is S-symmetric if the vector field f satisfies The ODE (17) is R-reversible if the vector field f satisfies Here S and R are arbitrary diffeomorphisms of phase space, i.e., R is not necessarily an involution. The map ' is S-symmetric if ' 2 fix(N S ), i.e. The map ' is R-reversible if ' 2 fix(IN R ), i.e. If f is S-symmetric, then S-symmetry of ' - (f) is equivalent to S-covariance (' 2 fix(K S )). If f is R-reversible, then R-reversibility of ' - (f) is equivalent to ' 2 fix(IRKR ). Of course, a dynamical system may possess more than one (reversing) symmetry. The set of all symmetries and reversing symmetries of a given dynamical system form a group under composition, the reversing symmetry group \Gamma [8, 22]. It is homomorphic to Z 2 : the symmetries have grading +1 and the reversing symmetries have grading \Gamma1. The flow exp(-f) of f has the same reversing symmetry group as f . The question we now address is: Given a vector field f , possessing a reversing symmetry group \Gamma, how does one construct integrators ' - (f) that possess the same reversing symmetry group? For the case of a single reversing symmetry R, Scovel [24] introduced the following projection: If R - R is R-reversible. This is another instance of the projection (11), in this case, Q IN R . (Just as for the antiautomorphism IR, the only generated automorphism is the identity.) For the case of a larger reversing symmetry group \Gamma, the half-sized group construction means that we can use 's which are not covariant with respect to all of \Gamma, but only with respect to its symmetries. Then for any reversing symmetry R in the group, the projection Q IN R R possesses the full reversing symmetry group \Gamma. There are two possible extensions of this projection to all vector fields f . The first, ', has the reversing symmetry group \Gamma, regardless of whether the vector field does. In particular, it is not an integrator of f even if ' is. Thus we prefer the following extension: The conceptual advantage is all flows are in fix(IRKR ). This is exactly the group and projection used in (33). A drawback of the method Q IRKR ' is that it is not time-symmetric, and that making it so by a second projection would destroy the desired reversibility: Q IRQ IRK R There are three ways out. The first is to use the composition property (12) to increase the order, ignoring time- symmetry completely. For example, if ' - is reversible and of order 1, then one can check that is reversible and of order 2. The second is to argue that time-symmetry does not affect the dynamics of the method as much as reversibility. So we could first construct a non-reversible high-order method (which may well be time-symmetric), and then form Q IRKR ', preserving the order but not time-symmetry. The third way is the most interesting. Abstractly, the problem is that we have two antiautomorphisms A \Gamma and B \Gamma and would like to construct an element invariant under them both. For the projection to work, the initial element ' must be invariant under all automorphisms generated by A \Gamma and B \Gamma . Here we have ' 2 not ' 2 does not map to the fixed set of the full group of anti- and automorphisms. However, if also ' 2 is easy to see that ' is fixed by all generated automorphisms, and that projection with any generated antiautomorphism gives the same Q'. In the case at hand, A . Thus we are interested in maps covariant under the reversing symmetry R as well, instead of just the ordinary symmetry R 2 as previously: Note that if f is R-reversible, its flow is R-reversible, R-covariant, and time-symmetric. If a map satisfies any two of time-symmetry, R-reversibility, and R-covariance, then it also satisfies the third. This gives a very flexible approach for constructing integrators, for R-covariance-invariance under an automorphism-is preserved under arbitrary composi- tions. At any stage one may project using Q IR to gain all three properties. The case of a reversing symmetry group \Gamma is similar. We would now require full \Gamma- covariance: reversing symmetries R in the group. (In practice, one only checks covariance under a set of generators of \Gamma, such as the reversing symmetries.) For example, affinely-covariant methods such as Runge-Kutta are covariant with respect to any affine reversing symmetry group. So by asking a little more of ' - (f ), we can directly apply the symmetric compositions of Yoshida [33] to increase the order while still preserving reversibility. Thus there are many routes to structure-preserving integrators. The preferred path will depend on what additional structure (e.g. volume-preservation, integrals) the ODE has, and on whether the symmetries and/or the reversing symmetries are linear. First find a first-order method ' - (f ). If it is \Gamma-covariant, we are done. If \Gamma is linear, (see (15)) is \Gamma-covariant, although this may break some structure in '. (For example, there is no known linearly-covariant volume-preserving integrator.) The case when ' is covariant only with respect to the symmetries is handled by the generalized Scovel projection, (38). Finally, splitting methods in which the constituent vector fields are not reversible introduce some new possibilities, discussed in the example below. Example 1: Affine method. One possible way of integrating an ODE whose reversing symmetry group is affine is well known. It is not difficult to show that the implicit midpoint rule is covariant with respect to the group of affine transformations (as indeed all Runge-Kutta methods are). It follows that if the vector field f has an affine reversing symmetry group \Gamma, then so does ' - (f ). Since the midpoint rule also has time symmetry (27), ' - (f) has reversing symmetry group \Gamma. (The midpoint rule is also symplectic when the ODE (17) is Hamiltonian ([23]), but not all Runge-Kutta methods with the time-symmetry property are symplectic, nor vice versa.) Example 2: Consider the divergence-free vector fields in R 4 with parameter ff @x (\Gammay 3 @y @ @z @ which possess the following reversing symmetry group the cyclic group generated by R 1 ). We want a method preserving the divergence-free structure, the symmetries, and the reversing symmetries. The midpoint rule would preserve the reversing symmetry group but is not, in general, volume preserving. It is shown in [17] that the midpoint rule is volume preserving if and only if P In R 4 , this requires the coefficient of - in P (-) to be zero. It can be checked that this coefficient is not zero here. Instead, we shall preserve volume by using a splitting method. Suppose each f i is divergence free and possesses all (nonreversing) symmetries (but not necessarily all reversing symmetries), and the solution of each - can be approximated by a preserving both volume and the symmetries and any reversing symmetries which f i may have. Then we only need to combine the approximations so as to recover the reversing symmetries. In general, there are three cases. Firstly, if each f i is reversible, then the ' - (f i ) are reversible by assumption, so the composition is reversible-and, if ' - (f) is time-symmetric, (44) is also time-symmetric. Secondly, if there is no structure when the f i are reversed, the best we can do is let and apply the generalized Scovel projection, giving / - R/ \Gamma1 is irrelevant here. (Volume is preserved because the reversing symmetry group preserves volume in this case.) Thirdly, and perhaps most efficiently, suppose we can split into three pieces, f 1 , f 2 , and with the following structure: (The reversible piece f 2 may be split further if desired.) Let ' - (f 2 ) be reversible, and integrator. Then \Gamma- is reversible. This situation occurs frequently when there is a reversing symmetry which is linear and satisfies R \Sigma1. For then let f 1 be any vector field and define f 3 := \GammaRf (because f is reversible by assumption.) We only need to find an f 1 such that this splitting is useful. In the present case we split into the planar vector fields @ @x @ @x @y @y @ @z so that each f i is divergence-free and hence Hamiltonian in two dimensions, and has symmetries (because their coefficients are odd functions). The f 's also satisfy (46) where the R i are given in (43). We let ' - be the midpoint rule, which is volume-preserving for these Hamiltonian systems, and get the reversible method This highlights the point that reversible integrators need not be time-symmetric, nor need the pieces in a splitting method possess the reversing symmetries. we have the nice situation of a splitting into two pieces, with the direct composition of their flows (or time-symmetric, \Gamma-covariant approximations of being reversible. Exactly the same constructions apply to a system with integrals, once basic maps preserving the integrals and symmetries have been found. Acknowledgement . G.S.T. is grateful to the Australian Research Council for partial support during the time this paper was written. --R Elementary construction of higher order Lie-Poisson integrators Dynamical systems and geometric construction of algorithms Equivariant symplectic difference schemes and generating functions Solving Ordinary Differential Equations I Chaotic Numerics Reversing symmetries in dynamical systems Symmetries and reversing symmetries in kicked systems On the numerical integration of ordinary differential equations by symmetric composition methods "Poisson schemes for Hamiltonian systems on Poisson mani- folds" The accuracy of symplectic integrators A survey of open problems in symplectic integration Lectures in Mechanics Convergent series expansions for quasi-periodic motions Coexistence of conservative and dissipative behavior in reversible dynamical systems Solving ODE's numerically while preserving a first integral Solving ODE's numerically while preserving all first integrals Chaos and time-reversal symmetry: order and chaos in reversible dynamical systems Symplectic numerical integration of Hamiltonian systems Reversible Systems Symmetric two-step algorithms for ordinary differential equations Variable steps for reversible integration methods Numerical analysis of dynamical systems Fractal decomposition of exponential operators with applications to many-body theories and Monte-Carlo simulations Representation of volume-preserving maps induced by solenoidal vector fields Construction of higher order symplectic integrators --TR --CTR G. R. W. Quispel , D. I. McLaren, Explicit volume-preserving and symplectic integrators for trigonometric polynomial flows, Journal of Computational Physics, v.186 n.1, p.308-316, 20 March R. I. McLachlan , M. Perlmutter , G. R. W. Quispel, Lie group foliations: dynamical systems and integrators, Future Generation Computer Systems, v.19 n.7, p.1207-1219, October

symmetries;automorphisms;numerical integrators

276485

Analysis of Algorithms Generalizing B-Spline Subdivision.

A new set of tools for verifying smoothness of surfaces generated by stationary subdivision algorithms is presented. The main challenge here is the verification of injectivity of the characteristic map. The tools are sufficiently versatile and easy to wield to allow, as an application, a full analysis of algorithms generalizing biquadratic and bicubic B-spline subdivision. In the case of generalized biquadratic subdivision the analysis yields a hitherto unknown sharp bound strictly less than 1 on the second largest eigenvalue of any smoothly converging subdivision.

Introduction The idea of generating smooth free-form surfaces of arbitrary topology by iterated mesh refinement dates back to 1978, when two papers [CC78], [DS78] appeared back to back in the same issue of Computer Aided Design. Named after their inventors, the Doo-Sabin and the Catmull-Clark algorithm represent generalizations of the subdivision schemes for biquadratic and bicubic B-splines, respectively. By combining a construction principle of striking simplicity with high fairness of the generated surfaces, both algorithms have since become standard tools in Computer Aided Geometric Design. However, despite a number of attempts [DS78], [BS86], [BS88], the convergence to smooth limit surfaces could not be proven rigorously so far. NSF National Young Investigator Award 9457806-CCR y BMBF Projekt 03-HO7STU-2 The proof techniques and actual proofs to be presented here are based on the concept of the characteristic map as introduced in [Rei95a]. The characteristic map is a smooth map from some compact domain U to R 2 which can be assigned to stationary linear subdivision schemes. It depends only on the structure of the algorithm and not on the data. If this map is both regular and injective, then the corresponding algorithm generates surfaces. It is shown in this paper that on the other hand non-injectivity at an interior point of the map implies non-smoothness of the limit surfaces. Further, we establish two sufficient conditions for regularity and injectivity of the characteristic map which allow a straightforward verification. The stronger one, however still applicable in many cases, only requires the sign of one partial derivative of one segment of the characteristic map to be positive. A careful analysis of the Doo-Sabin and the Catmull Clark algorithm yields the following results: ffl The Doo-Sabin algorithm in its general form uses weights for computing a new n-gon from an old one. Affine invariance and symmetry, i.e. imply that the discrete Fourier transform of ff is real and of the form - ff 1 is greater in modulus than the other entries except for 1 and if for certain values - then the limit surface is smooth. The bound - max (n) can be computed explicitly, see Table 1. If 1 ? - max (n), then the limit is a continuous, yet non-smooth surface. ffl In particular, the Doo-Sabin algorithm in its original form (5.1) complies with the conditions, hence generates smooth limit surfaces. ffl The Catmull-Clark algorithm in its general form uses three weights ff; fi; fl summing up to one for computing the new location of an extraordinary vertex from its predecessor and the centers of its neighbors. Iffi fi fi fi with c n := cos(2-=n), then the limit surface is smooth. If one of the two values on the left hand side exceeds the right hand side, then the limit surface is not smooth. ffl In particular, the Catmull-Clark algorithm in its original form (6.2) complies with the conditions and generates smooth limit surfaces. Generalized subdivision and the characteristic map In this section we briefly outline the results of subdivision analysis as developed in [Rei95a], and establish a new necessary condition for C 1 -subdivision schemes. Generalized B-spline subdivision generates a sequence Cm of finer and finer control polyhedra converging to some limit surface y. On the regular part of the mesh, standard B-spline subdivision is used for refinement, whereas special rules apply near extraordinary mesh points. Since all subdivision masks considered here are of fixed finite size, we can restrict ourselves to analyzing meshes with a single extraordinary mesh point of valence 4. The regular parts of the control polygons Cm correspond to B-spline surfaces ym which form an ascending sequence converging to the limit surface, With the prolongation of ym defined by xm := closure (ym+1nym the limit surface is the essentially disjoint 1 union The xm are ring-shaped surface layers which can be parametrized conveniently over a common domain U \Theta Z n ; Z n := Z mod n, consisting of n copies of the compact set see Figure 1. Each surface layer xm can be parametrized in terms of control points polynomial functions N ' according to Without loss of generality, we may assume that the functions N ' are linearly inde- pendent. Otherwise, the setup can be reduced without altering the properties of the scheme. The n parts x 0 forming xm are referred to as segments. Collecting 1 The intersection consists exclusively of points on the common boundary curve, which are identified. uvU x Figure 1: Domain U (left) and structure of surface layers xm (right). the functions N ' in the columns of a row matrix N and the control points in the rows of a column matrix Bm yields the vector notation xm (u; v; The schemes to be considered here are linear and stationary, i.e. there exists a square subdivision matrix A with Definition 2.1 Let the eigenvalues - of A be ordered by modulus, and denote by / the corresponding generalized real eigenvectors. If then the characteristic map of the subdivision algorithm is defined by or in complex form by Following (2.7), the segments \Psi j and \Psi j of the characteristic map are the restriction of \Psi and \Psi to U \Theta j, respectively. Figure 2: Injective (left) and non-injective (right) characteristic map Remark is a (L+ 1) \Theta 2-matrix. Its rows play the role of 2D control points. ii) Throughout, the subscript will indicate that we refer to the complexification of a two-dimensional real variable or function. We will switch between complex and real representation without further notice. On the left hand side, Figure 2 shows a typical example of a characteristic map for obtained for example by the Doo-Sabin algorithm. In order to guarantee affine invariance of the algorithm, the rows of A must sum to 1. Thus, an eigenvector of A to the eigenvalue 1. The following theorem establishes a sufficient condition for subdivision algorithms to generate smooth limit surfaces. Theorem 2.1 If - is a real eigenvalue with geometric multiplicity 2, and if the characteristic map is regular and injective, then the limit surface y is a regular C 1 -manifold for almost every choice of initial data B 0 . A proof of this theorem can be found in [Rei95a]. Generalizations, though not required here, are provided in [Rei95b] and [PR97]. Subsequently, it will be assumed that the eigenvalues of A satisfy the assumptions of Theorem 2.1, and - will be referred to as the subdominant eigenvalue. The following theorem states a necessary condition for the convergence of a subdivision scheme to smooth limit surfaces. Theorem 2.2 If the characteristic map of a subdivision scheme is non-injective, i.e. there exist (u; v; and if \Psi(u; v; j) is an interior point of \Psi(U; Z n ), then the limit surface y is not a regular -manifold for almost every choice of initial data B 0 . Proof Choose an "-neighborhood Then there exist neighborhoods V and V 0 of (u; v; respectively, with \Psi is a continuous map sufficiently close to \Psi, i.e. \Psi is also not injective. Now, express Bm in terms of the generalized eigenvectors / ' , Then for almost every choice of initial data B 0 , the coefficients b 1 and b 2 are linearly independent, and we can choose coordinates such that b is the origin and b are the first two unit vectors. A rescaling of the surface layers yields Now, assume that y is a regular C 1 -manifold. Then the latter equation implies that the tangent plane at the origin is the xy-plane. The projection ~ /m of ~ xm on the xy-plane is converging to \Psi, so ~ /m is non-injective for m sufficiently large. Consequently, the projection of the layers xm to the xy-plane are non-injective near the origin for almost all m. This contradicts the assumption since the projection of a regular C 1 -manifold on its tangent plane is locally injective. 2 Finally, we state two basic properties of characteristic maps. The first one is derived from the fact that \Psi and smoothly, The second one expresses continuity between segments, 3 Symmetry and Fourier Analysis This section examines the special structure of the characteristic map for subdivision schemes obeying generic symmetry assumptions, namely that subdivision is independent of the particular labeling of control points used for refining the control mesh. According to the split of xm into n segments, the vectors Bm of control points can be divided into equally structured blocks, and A is partitioned into n \Theta n square blocks A j;j 0 Definition 3.1 A subdivision algorithm is called symmetric if it is invariant both under a shift S and a reflection R of the labeling of the vector Bm of control points. S and R are permutation matrices characterized by Symmetry of the subdivision algorithm means that the subdivision matrix A commutes both with R and S, i.e. the segments of S and E the identity matrix of the same size as S j;j 0 , the shift matrix S is given by Comparison of SA and AS shows that the subdivision matrix of a symmetric scheme is block-cyclic, i.e. Thus, (2.8) becomes With define the discrete Fourier transform by is an n-vector in the generalized sense that its entries p j can be either scalars, or vectors, or matrices. It will always become clear from the context, what is meant. Applying the discrete Fourier transform to (3.7) yields see [Lip81] for a comprehensive introduction to the Fourier analysis of cyclic systems. Theorem 3.1 The characteristic map for a symmetric scheme is non-injective unless the subdominant eigenvalue - is an eigenvalue of - A 1 and - A Proof - is an eigenvalue of A if and only if it is an eigenvalue of - A k for some k 2 1g. If - is an eigenvalue of - A k then it is also an eigenvalue of - A n\Gammak since A is real and - A k . Let - /, then is a complex eigenvector of A. Consequently, the segments \Psi j of the complex characteristic map satisfy Now, the winding number of the closed curve is either k or the curve ff has self-intersections implying that \Psi is not injective. 2 The effect of the subdominant eigenvalue - stemming from the wrong Fourier component is depicted in Figure 2 on the right hand side. It shows the non-injective characteristic map for the Doo-Sabin algorithm for weights chosen such that - is an eigenvalue of - A 2 and - A 4 . As a consequence of Theorem 3.1, it will be assumed that - is an eigenvalue of - A 1 and - A n\Gamma1 from now on. So, (3.12) becomes The following lemma is the key to reducing the analysis of the characteristic map \Psi to the examination of a single segment, say \Psi . \Psi is called normalized if - / is scaled such that \Psi 0 (2; that normalization is always possible if \Psi is injective since then \Psi 0 (2; Lemma 3.1 If \Psi is a normalized characteristic map of a symmetric scheme, then and in particular Proof (3.2) and (3.3) yield N(u; v; j)S implying that S since the functions forming N are assumed to be linearly independent. From (3.4) one concludes that R/ is an eigenvector of A to the eigenvalue -, i.e. it can be written as R/ On using S/ one obtains from the latter two equations Since / and / are linearly independent, this implies a = 0, hence R/ . In order to determine b, consider \Psi (2; 2). By (3.3), thus Finally, we obtain Conditions for regularity and injectivity In this section we derive a sequence of lemmas resulting in two sufficient conditions for the regularity and injectivity of the characteristic map that can be verified efficiently. Throughout, it will be assumed that \Psi is a normalized characteristic map of a symmetric subdivision scheme. The first lemma states that for regular functions injectivity is equivalent to injectivity at the boundary. Lemma 4.1 Denote by @U the boundary of U and by \Psi 0 @ the restriction of \Psi 0 to @U . If then \Psi 0 is injective if and only if \Psi 0 @ is injective. Proof Assume that \Psi 0 is regular and \Psi 0 @ is injective. By the Inverse Function Theorem (IFT), points in the interior U of U are mapped to points in the interior of \Psi 0 (U ), i.e. Define the function - assigning the number of pre-images to the points in \Psi 0 (U ), Injectivity of \Psi 0 @ and (4.2) imply -(@ \Psi 0 1. - is upper semi-continuous by the IFT, hence -(\Psi 0 The second lemma gives a sufficient condition for \Psi 0 (U) being located in a sector of angle 2-=n in the complex plane. Lemma 4.2 If \Psi 0 is regular and for all t 2 [0; 1], then for all (u; v) 2 U . Proof By Lemma 3.1, we have and in particular \Psi 0 then p 2 is monotonically increasing because \Psi 0 we obtain for t 2 [1; 2] This implies either arg \Psi 0 -=n. The second case contradicts (4.7), thus -=n Figure 3: Curves p 1 . This means that \Psi 0 (t; 0) is a part of the straight half line monotonically increasing in both real and imaginary part, it has no intersections with h except for p (0), hence Using the scaling property (2.15) and symmetry with respect to the real axis, the latter two inequations imply that (4.5) holds for all (u; v) 2 @U . By the IFT, we have is compact, this implies \Gamma-=n - arg \Psi 0 (U) -=n as asserted. 2 The third lemma provides a condition on the partial derivatives of \Psi 0 that implies injectivity. Lemma 4.3 If \Psi 0 is regular and \Psi 0 Proof By Lemma 4.1 it suffices to show that the restriction \Psi 0 @ of \Psi 0 to the boundary of U is injective. Let (2t; for t 2 [0; 1], see Figure 3. Then with defined as in the proof of Lemma 4.2. p 2 and p 5 do not intersect since by (4.9). Both curves also do not have self-intersections since they are parametrized regularly and parts of straight lines. Next, we show that arg p (t) is monotonically increasing in t. By (4.7) and (4.9), By assumption, p 1 and p 2 are monotone increasing, thus This implies d dt as claimed. Monotonicity of arg p 6 has the following consequences: First, it guarantees that p 1 do not have self-intersections. Second, it excludes intersections of p 1 and p 3 since Analogously, p 4 and are disjoint. Third, the only intersections of p 1 and p 3 with are and analogously for p 4 . Fourth, the only intersections of p 1 and p 4 are and the proof is complete. 2 The following theorem establishes a sufficient condition on the partial derivatives of \Psi 0 that guarantees regularity and injectivity of the characteristic map. Its usefulness is due to the fact that it requires only estimates for the partial derivatives of the single segment \Psi 0 . Since for generalized B-spline subdivision schemes the functions in questions are piecewise polynomial, the condition can be verified numerically or even analytically using B-spline representations and the convex hull property. Theorem 4.1 If \Psi 0 is regular and \Psi 0 then the characteristic map \Psi is regular and injective. Figure 4: Mesh refinement by the Doo-Sabin algorithm. Proof By Lemma 4.3, \Psi 0 is regular and injective. (3.14) says that \Psi j is obtained from \Psi 0 by a 2-j=n-rotation about the origin. So, each \Psi is regular and injective. Further, the segments \Psi j do not overlap since Lemma 4.2 yields (4.22)The assumptions of the following Corollary are stronger than those of Theorem 4.1, but can be verified with less effort since no products of partial derivatives are involved in verifying that \Psi 0 is regular. Corollary 4.1 If \Psi 0 then the characteristic map \Psi is regular and injective. Proof The symmetry relation (4.6) yields so the determinant J 0 is positive if \Psi 0 5 The Doo-Sabin algorithm 5.1 Algorithm The Doo-Sabin algorithm is a generalization of the subdivision scheme for biquadratic tensor product B-splines. For each n-gon of the original mesh, a new, smaller n-gon is created and connected suitably with its neighbors, see Figure 4. Figure 5 shows the ff n\Gamma2 ff Figure 5: Masks for the Doo-Sabin algorithm. mask for generating a new n-gon from an old one for the regular case the general case (right). The weights suggested by Doo and Sabin in [DS78] are Below we analyze more general schemes assuming beforehand nothing but affine invariance and symmetry, 5.2 Characteristic map Each of the n segments x j of the surface layers generated by the Doo-Sabin algorithm consists of 3 biquadratic B-spline patches. Accordingly, the n blocks forming the vector of control points Bm consist of 9 elements, each. The labeling is shown in Figure 6. The 9 \Theta 9-matrices - introduced in (3.10) have the following structure, A k =B @ Figure Labeling of control points for the Doo-Sabin algorithm 1=16, the sub-matrices are given by r r q p The matrix A k 1;1 has eigenvalues 1=4; 1=8; 1=16, hence each of them is an n-fold eigenvalue of the subdivision matrix A. Further, A has a 5n-fold eigenvalue 0 stemming from the 5 \Theta 5-zero submatrix of - A k . Due to their high multiplicity, these eigenvalues cannot be playing the role of the subdominant eigenvalue -. The only eigenvalues left are the upper left entries - A k obtained by applying the discrete Fourier transform to the vector (ff of weights for the n-gon. Since the ff j sum up to 1, we have 1. Due to symmetry, the remaining eigenvalues are real and occur in pairs according to - ff n\Gammak . From the theory developed in the preceding sections we know that ff must satisfy ff n\Gamma2 The eigenvector of the matrix - A 1 corresponding to - is Note that the characteristic map depends only on - and n. That is, all masks ff with identical first Fourier component yield the same characteristic map. 5.3 Verification We start with briefly discussing the case as obtained in particular for the weights in (5.1). Rearranging the entries of the eigenvector / in the more convenient matrix form for tensor product B-spline coefficients, see Figure 6, yields The segment \Psi 0 of the characteristic map consists of three bi-quadratic patches, which can be expressed in Bernstein-B'ezier form with the following coefficients,2 28 s n 28 \Gamma7 s n \Gamma7 s n \Gamma14 s n \Gamma7 s n 28 \Gamma14 s n \Gamma14 s n 28 28 c n \Gamma28 s n (5.9) Computing the partial derivative \Psi 0 2;v with respect to v yields three quadratic-linear patches with coefficients2 Both the real and the imaginary part of the coefficients are positive. So, by the convex hull property and Corollary 4.1 the algorithm is verified to generate smooth limit surfaces. The situation for general - is more subtle, in particular as - ! 1. First, Corollary 4.1 turns out to be insufficient. Second, there exists a limit value - depending on n such that even the assumptions of Theorem 4.1 are not fulfilled for 1 - max . It will be shown in the next subsection that this is due to an actual loss of smoothness as - passes the bound. All formulas required here were derived using a computer algebra system. They are partially rather lengthy and will not be stated explicitly unless necessary. Rather, we depict the crucial results graphically. In order to apply Theorem 4.1, we have to compute J 0 , i.e. the determinant of the Jacobian of \Psi 0 . J 0 is a continuous, piecewise bi-cubic function over U which can be expressed in Bernstein-B'ezier form with 3 \Theta 16 coefficients J 0 depending on n and -. Explicit calculation shows that all coefficients J 0 - are of the form with polynomials of degree - 6 in -. We give the coefficient corresponding to The polynomials P 1 and Q 1 are In order to apply analytic tools, it is convenient to consider c n as a free variable varying in the interval c n 2 [\Gamma1=2; 1], which covers all possible values obtained for n - 3. For fixed there is at most one value ~ c n where J 0 - changes sign, Figure 7: Feasible set and functions R -). Figure 7 shows a plot of all these functions as well as a magnification of the significant region. From the analysis of the case we know that J 0 Thus, J 0 is positive as long as (-; c n ) lies in the shaded region, which is bounded by precisely, the feasible set for (-; c n ) providing positivity of J 0 is For the verification of the assumptions of Theorem 4.1 it remains to show that \Psi 0 . Note that both functions are linear in t. So, it suffices to check positivity for t 2 f0; 1g, which follows immediately from Finally, we summarize the results derived in this section. Theorem 5.1 Let ff 0 n be symmetric weights for the Doo-Sabin algorithm. If ff ff n\Gamma2 then the limit surface y is a regular C 1 -manifold for almost every choice of initial data Figure 8: Characteristic map for 5.4 Failure beyond the bound In contrast to the lower bound - ? 1=4, which appears naturally, the existence of an upper bound for - may come as a surprise. It is not an artifact of the particular type of sufficient conditions in Theorem 4.1, but a sharp bound beyond which the Doo-Sabin algorithm provably fails. If J 0 Consider the curve [g 1 (t); g 2 (t)] := \Psi 0 (t; t). Symmetry with respect to the x-axis implies For the first component we obtain hence for each sufficiently small " ? 0 there exists an " This implies the non-injectivity of the characteristic map \Psi, Moreover, for " sufficiently small, J 0 (1+ "; 1+ ") ! 0 by continuity. So, \Psi 0 (1+ "; 1+ ") is an interior point of \Psi(U; Z n ) by the IFT, and the assumptions of Theorem 2.2 are fulfilled proving sharpness of the bound. Figure 8 shows a magnification of the characteristic map in the vicinity of \Psi 0 (1; 1) for (right). The latter case corresponds to weights ff layers of a subdivision Figure 9: Non-smooth surface generated by the Doo-Sabin algorithm with surface generated by these weights are shown in Figure 9. The magnification on the right hand side is non-proportional, i.e. the 'height' of the surface has been expanded in order to depict its wavy shape. We conclude the discussion of the Doo-Sabin algorithm with a brief description of the qualitative and quantitative behavior of - max (n). As n !1, increasing monotonically towards 1. The asymptotic behavior for large n is The lowest bound occurs for namely cos arctan Table 1 lists the values of - max for 6 The Catmull-Clark algorithm 6.1 Algorithm The Catmull-Clark algorithm is a generalization of the subdivision scheme for bicubic tensor product B-splines. Each n-gon of the original mesh is subdivided into n quadrilaterals thus generating a purely quadrilateral mesh after the first step. There are three masks for subdividing such a mesh, namely one for computing a new centroid, one for 9 0.9829902941 Table 1: Values of the bound - max (n) for Figure 10: Mesh refinement by the Catmull-Clark algorithm. 1=4 1=4 1=4 1=4 1=32 fi=n fl=n fi=n fl=n fi=n fl=n fi=n fl=n fi=n fl=n Figure 11: Masks for the Catmull-Clark algorithm. Figure 12: Labeling of control points for the Catmull-Clark algorithm. a new edge point, and one for the new location of a former vertex, see Figure 11. So, the variables at our disposal are the weights In [CC78], Catmull and Clark suggest 6.2 Characteristic map Each of the n segments x j of the surface layers generated by the Catmull-Clark algorithm consists of 3 bicubic B-spline patches. Accordingly, the n blocks forming the vector of control points Bm consist of 13 elements, each. The labeling used here is shown in Figure 12. Note that the centroid is replaced by n identical copies in order to achieve the desired periodic structure. For all masks involving Mm we substitute Mm =n The 13 \Theta 13-matrices - turn out to have the following structure, With the sub-matrices are given by The eigenvalues 1=8; 1=16; 1=32; 1=64 of the sub-matrix - are n-fold eigenvalues of A. Other non-zero eigenvalues come only from - 0;0 . For we obtain the obligatory eigenvalue letting which might be either both real or complex conjugate. For k 6= 0, the non-zero eigen-values of - are where c n;k := cos(2-k=n). Let then straightforward calculus shows that for all n Consequently, - is subdominant if ff; fi; fl are chosen such that In particular, this inequality holds for the original weights of Catmull-Clark (6.2), as can be verified by inspection. A characterization of feasible positive weights can be found in [BS88] 2 . For computing the characteristic map, the eigenvector - / of - A 1 is partitioned into three blocks, - according to the special structure of - A 1 . / is equivalent to / can be computed conveniently starting from which solves the first eigenvector equation. Note that the characteristic map depends only on n, and not on the particular choice of weights ff; fi; fl provided that (6.11) holds. 6.3 Verification Corollary 4.1 is sufficient for verifying the algorithm. One proceeds as follows: 1. For given n - 3, compute the subdominant eigenvalue - according to (6.11) and the corresponding eigenvector - / according to (6.15). 2. Express the three patches of the segment \Psi 0 of the characteristic map in Bernstein- B'ezier form. 3. Compute the forward differences of B'ezier coefficients corresponding to the partial derivative with respect to v. 4. If all are positive in both components, then by the convex hull property of the Bernstein-B'ezier form the assumptions of Corollary 4.1 are fulfilled and the characteristic map is regular and injective. 2 The result in the reference is incorrect for yielding complex eigenvalues - 0 Figure 13: B'ezier coefficients of the partial derivatives \Delta - . This procedure can be run on a computer algebra system, but the resulting expressions are rather lengthy, and discussing them is not very instructive. A numerical treatment is more convenient and yields equally reliable results since only a finite number of quantities has to be checked for sign. The findings are summarized on Figure 13. The left and right hand side correspond to the two components of \Delta - . The top row shows the values of all \Delta - for 20. The bottom row shows the minimum of the \Delta - on a doubly- logarithmic scale for should cover most cases of practical relevance. The positivity of all differences is evident. By Corollary 4.1, this proves smooth convergence of the Catmull-Clark algorithm provided that the inequality (6.14) holds. --R A matrix approach to the analysis of recursively generated b-spline surfaces Conditions for tangent plane continuity over recursively generated b-spline surfaces Recursively generated B-spline surfaces on arbitrary topological meshes Behaviour of recursive subdivision surfaces near extraordinary points Elements of algebra and algebraic computing Necessary conditions for subdivision surfaces A unified approach to sudivision algorithms near extraordinary ver- tices Some new results on subdivision algorithms for meshes of arbitrary topology --TR --CTR Jorg Peters, Smooth patching of refined triangulations, ACM Transactions on Graphics (TOG), v.20 n.1, p.1-9, Jan. 2001 Jos Stam, Exact evaluation of Catmull-Clark subdivision surfaces at arbitrary parameter values, Proceedings of the 25th annual conference on Computer graphics and interactive techniques, p.395-404, July 1998 K. Kariauskas , J. Peters, Concentric tessellation maps and curvature continuous guided surfaces, Computer Aided Geometric Design, v.24 n.2, p.99-111, February, 2007 Jrg Peters , Ulrich Reif, Shape characterization of subdivision surfaces: basic principles, Computer Aided Geometric Design, v.21 n.6, p.585-599, July 2004 Jrg Peters , Ulrich Reif, The simplest subdivision scheme for smoothing polyhedra, ACM Transactions on Graphics (TOG), v.16 n.4, p.420-431, Oct. 1997 K. Kariauskas , Jrg Peters , U. Reif, Shape characterization of subdivision surfaces: case studies, Computer Aided Geometric Design, v.21 n.6, p.601-614, July 2004 M. K. Jena , P. Shunmugaraj , P. C. Das, A non-stationary subdivision scheme for generalizing trigonometric spline surfaces to arbitrary meshes, Computer Aided Geometric Design, v.20 n.2, p.61-77, May Jrg Peters, Patching Catmull-Clark meshes, Proceedings of the 27th annual conference on Computer graphics and interactive techniques, p.255-258, July 2000 Evelyne Vanraes , Adhemar Bultheel, A tangent subdivision scheme, ACM Transactions on Graphics (TOG), v.25 n.2, p.340-355, April 2006 I. P. Ivrissimtzis , H.-P. Seidel, Evolutions of polygons in the study of subdivision surfaces, Computing, v.72 n.1-2, p.93-103, April 2004 Barthe , Leif Kobbelt, Subdivision scheme tuning around extraordinary vertices, Computer Aided Geometric Design, v.21 n.6, p.561-583, July 2004 Henning Biermann , Adi Levin , Denis Zorin, Piecewise smooth subdivision surfaces with normal control, Proceedings of the 27th annual conference on Computer graphics and interactive techniques, p.113-120, July 2000 Ioana Boier-Martin , Denis Zorin, Differentiable parameterization of Catmull-Clark subdivision surfaces, Proceedings of the 2004 Eurographics/ACM SIGGRAPH symposium on Geometry processing, July 08-10, 2004, Nice, France Chhandomay Mandal , Hong Qin , Baba C. Vemuri, A novel FEM-based dynamic framework for subdivision surfaces, Proceedings of the fifth ACM symposium on Solid modeling and applications, p.191-202, June 08-11, 1999, Ann Arbor, Michigan, United States Georg Umlauf, Analysis and tuning of subdivision algorithms, Proceedings of the 21st spring conference on Computer graphics, May 12-14, 2005, Budmerice, Slovakia Yonggang Xue , Thomas P.-Y. Yu , Tom Duchamp, Jet subdivision schemes on the k-regular complex, Computer Aided Geometric Design, v.23 n.4, p.361-396, May 2006 Hong Qin , Chhandomay Mandal , Baba C. Vemuri, Dynamic Catmull-Clark Subdivision Surfaces, IEEE Transactions on Visualization and Computer Graphics, v.4 n.3, p.215-229, July 1998 Jrg Peters , Le-Jeng Shiue, Combining 4- and 3-direction subdivision, ACM Transactions on Graphics (TOG), v.23 n.4, p.980-1003, October 2004 Chhandomay Mandal , Hong Qin , Baba C. Vemuri, Dynamic Modeling of Butterfly Subdivision Surfaces, IEEE Transactions on Visualization and Computer Graphics, v.6 n.3, p.265-287, July 2000 Peter Oswald, Designing composite triangular subdivision schemes, Computer Aided Geometric Design, v.22 n.7, p.659-679, October 2005 Martin Bertram, Biorthogonal wavelets for subdivision volumes, Proceedings of the seventh ACM symposium on Solid modeling and applications, June 17-21, 2002, Saarbrcken, Germany Charles K. Chui , Qingtang Jiang, Matrix-valued subdivision schemes for generating surfaces with extraordinary vertices, Computer Aided Geometric Design, v.23 n.5, p.419-438, July 2006 Martin Bertram , Mark A. Duchaineau , Bernd Hamann , Kenneth I. Joy, Bicubic subdivision-surface wavelets for large-scale isosurface representation and visualization, Proceedings of the conference on Visualization '00, p.389-396, October 2000, Salt Lake City, Utah, United States Martin Bertram , Mark A. Duchaineau , Bernd Hamann , Kenneth I. 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arbitrary topology;b-spline;doo-sabin algorithm;characteristic map;catmull-clark algorithm;subdivision

276487

Discrete Shocks for Finite Difference Approximations to Scalar Conservation Laws.

Numerical simulations often provide strong evidences for the existence and stability of discrete shocks for certain finite difference schemes approximating conservation laws. This paper presents a framework for converting such numerical observations to mathematical proofs. The framework is applicable to conservative schemes approximating stationary shocks of one-dimensional scalar conservation laws. The numerical flux function of the scheme is assumed to be twice differentiable but the scheme can be nonlinear and of any order of accuracy. To prove existence and stability, we show that it would suffice to verify some simple inequalities, which can usually be done using computers. As examples, we use the framework to give an unified proof of the existence of continuous discrete shock profiles for a modified first-order Lax--Friedrichs scheme and the second-order Lax--Wendroff scheme. We also show the existence and stability of discrete shocks for a third-order weighted essentially nonoscillatory (ENO) scheme.

Introduction In this paper, we provide a general framework for proving the existence and stability of continuous discrete shock profiles for conservative finite difference schemes which approximate scalar conservation laws Research supported by ARPA/GNR grant N00014-92-J-1890. We consider schemes of conservative form: with Here \Delta) is the numerical flux of the scheme which satisfies g(u; u; (consistency) and is twice continuously differentiable w.r.t. its arguments; u n j is an approximation to u(j \Deltax; n\Deltat) is a constant integer such that (2p+1) is the stencil width of the scheme. Schemes with such flux functions include the first order Lax-Friedrichs scheme and some of its modified versions, the second order Lax-Wendroff scheme and a class of high resolution weighted ENO schemes[7]. Let be two constants such that Eq. (1.1) with the initial data admits an entropy satisfying shock given by where s is the shock speed and by Rankine-Hugoniot condition, In this paper, we will assume Let us clarify the concepts we will use frequently in the paper: Definition 1.1: For 1. sup 2. lim 3. we call ' an approximate stationary discrete shock with parameter q. If, furthermore, ' also satisfies 4. we call it an exact stationary discrete shock for scheme (1.2) with parameter q. Definition 1.2: If a function '(x)(x 2 R) is bounded, uniformly Lipschitz continuous in R and for any q 2 [0; 1], is an exact stationary discrete shock for scheme (1.2) with parameter q, then ' is called an continuous stationary discrete shock profile for scheme (1.2). Remark: An approximate discrete shock is not related to the scheme (1.2). However, we will only be interested in approximate discrete shocks which are so accurate that condition 4 in Definition 1.1 is almost satisfied. In the following discussions, we will omit "stationary" when referring to stationary discrete shocks. We will often refer to exact discrete shocks plainly by "discrete shocks". Existence and stability of discrete shocks are essential for the error analysis of difference schemes approximating (1.1). It is well known that solutions to (1.1) generally contain shocks and numerical schemes unavoidably commit O(1) error around the shocks. It is important to understand whether this O(1) error will destroy the accuracy of the scheme in smooth parts of the solution. For conservation laws whose solutions are sufficiently smooth away from isolated shocks, Engquist and Yu [4] proved that, the O(1) error committed by a finite difference scheme around shocks will not pollute the accuracy of the scheme in smooth regions up to a certain time, provided that, (i) the scheme is linearly stable and (ii) the scheme possesses stable discrete shocks. This paper was motivated by the work of Liu and Yu [9]. Their approach was to linearize the scheme around some constructed approximate discrete shocks. Existence and stability of exact discrete shocks were then obtained by proving that this linearized scheme defines a contractive mapping for small perturbations on the approximated discrete shock and the orginal scheme behaves closely like the linearized scheme. Our main observation is that, using computers, one can easily obtain approximate discrete shocks and most importantly, these approximate discrete shocks can be made as accurate as the machine limit allows. As we know, if a scheme possesses a discrete shock, it can be often observed from numerical experiments that the scheme converges quickly to a numerical discrete shock after a number of time iterations over an initial guess (this is often equivalently stated as "the residue quickly settles down to machine zero"). When it is linearized around such accurate approximate discrete shocks, the finite difference scheme can be expected to behave very closely like the linearized scheme (e.g. in terms of contractiveness of the induced mapping). The obvious advantage of using numerically computed approximate discrete shocks is that it can be applied to almost all schemes and all conservation laws with little efforts. A brief review on discrete shocks for finite difference schemes is as follows: The existence of a discrete shock was first studied by Jennings [6] for a monotone scheme by an L 1 - contraction mapping and the Brower's fixed point theorem. For a first order system, Majda and Ralston [10] used a center manifold theory and proved the existence of a discrete shock. Yu [15] and Michaelson [11] followed the center manifold approach and showed the existence and stability of a discrete shock for the Lax-Wendroff scheme and a third order scheme, resp. In [14], Smylris and Yu studied the continuous dependence of the discrete shocks by extending the functional space for finite difference schemes from L 1 (Z) to L 1 (R) and a fixed point theorem. All existence theorems above require an artificial assumption on the shock speed. In [8], Liu and Xin proved the existence and stability of a stationary discrete shock for a modified Lax-Friedrichs scheme. For a first order system, Liu and Yu [9] showed both the existence and stability of a discrete shock using a pointwise estimate and a fixed point theorem as well as the continuous dependence of the discrete shock on the end states Our paper is organized as follows: First we prove a basic fixed point theorem in Section 2. Then we show in Section 3 how the existence and stability problem can be formulated into a fixed point problem once an approximate discrete shock is available. In Section 4, we derive sufficient conditions for the scheme (1.2) to possess a single stable discrete shock and in Section 5, we derive sufficient conditions for scheme (1.2) to possess a continuous discrete shock profile. In Section 6, we discuss how to obtain approximate discrete shocks and how to verify the conditions derived in earlier sections by computers. In Section 7, we apply our framework to give a unified proof of the existence of continuous discrete shock profiles for a first order modified Lax-Friedrichs scheme and the second order Lax-Wendroff scheme. We will also show the proof of existence and stability of discrete shocks for a third order weighted ENO scheme. Some remarks will be given in Section 8. A Basic Fixed Point Theorem In this section, we prove a basic fixed point theorem. First let us define a weighted l 2 norm. Suppose that ff ? 1 and fi ? 1 are two constants. For any infinite dimensional vector becomes the regular l 2 norm. We denote the corresponding space by l 2 Because ff ? 1 and fi ? 1, it is easy to see that any vector v 2 l 2 lim We denote a closed ball with radius r around a vector v 0 in l 2 Let F be a mapping from l 2 ff;fi to l 2 ff;fi and have the form where 1. L is mapping which is linear, i.e. L[av ff;fi , and contractive, i.e. 2. N is generally a nonlinear mapping and N[- where - is the null vector. Moreover, v;w2Br (-) 3. E is a constant vector in l 2 independent of v. We have the following basic fixed point theorem: Theorem 2.1 fixed point theorem) If there exists oe ? 0 such that Then the mapping F 1. is contractive in B oe (-) under the norm jj \Delta jj ff;fi ; 2. has a unique fixed point, i.e. there exists only one - (-) such that - Moreover, Proof: For any v; w 2 B oe (-), we have Therefore Moreover, for any v 2 B oe (-), since From the condition (2.5), we obtain Condition (2.5) implies that jjLjj ff;fi ff;fi is strictly less than 1. Therefore (2.7) and (2.8) together imply that the mapping F maps B oe (-) into itself and is a contractive mapping. From Banach fixed point theorem, there exists a unique - (-) such that F In addition, we have from which (2.6) follows. Q.E.D. 3 Formulation of a Fixed Point Problem For a fixed q 2 [0; 1], assume - ' is an approximate stationary discrete shock with parameter q and is given. By Definition 1:1, we know lim Assume that there exists an exact discrete shock for scheme (1.2) with parameter q denoted as '. If we set then condition 2 in Definition 1.1 and Eq. (3.3), resp. imply lim Condition 4 in Definition 1.1 gives Here - u means a vector sum of - ' and - u, i.e. ( - we have - lim Using (1.3) to write (3.5) in terms of - v, we have io where we have used lim which is implied by (3.2), (3.6) and the consistency and continuity of the flux function g. Let us define the right-hand-side (RHS) of (3.7) as a mapping in some subspace of the space of infinite dimensional vectors, namely, let with v be any vector satisfying (3.6), i.e. lim 0: Then Eq. (3.7) gives which means that - v is a fixed point of F . If we reverse the above arguments, namely, assume that there exists an infinite dimensional vector - v such that it satisfies (3.6) and is a fixed point of the mapping F defined in (3.8), then it is easy to see that ' j - is an exact stationary discrete shock for scheme (1.2) with parameter q. So to prove the existence of a stationary discrete shock is equivalent to prove the existence of a fixed point for the mapping F . We will restrict the space for the search of fixed points of F to l 2 ff;fi in our study, where ff and fi are some suitable constants. We can rewrite the mapping F in (3.8) in the form of (2.2) with Here, are the first order partial derivatives of the numerical flux function g. It is easy to see that the mapping L is linear and the mapping N is generally nonlinear and satisfies N[-. In addition, E is a vector depending on - but independent of v. The linear mapping L is just the linearization of the mapping F around the approximate discrete shock - '. If - ' is accurate enough, we hope L becomes contractive under the weighted l 2 norm To see the mapping N would satisfy (2.4), we write, for any v; w Z 1Z 1g 00 j+l djd- where and g 00 are the second order derivatives of the flux function g. Notice that the summation on the RHS of (3.13) is a double summation over k; Introducing the shifting operator (or a mapping) which is defined as E k [v] vector v, we can rewrite (3.13) as Z 1Z 1g 00 j+l djd- If we apply the norm jj \Delta jj ff;fi on both sides of (3.15), due to the fact that v is O(r), the mapping would satisfy (2.4) provided that the second order derivatives of g are bounded. We will give precise estimates of jjN jj r ff;fi in the next two sections for different choices of approximate discrete shocks - ' and slightly different forms of the mapping N . Then we derive sufficient conditions on - ' and the first and second derivatives of g which will guarantee the existence and stability of exact discrete shocks. Our estimates will be based upon the following two bounds on the first two derivatives of g: The functions can be obtained analytically from the given flux function g. Without loss of generality, we assume that both \Gamma 1 (r) and \Gamma 2 (r) are non-decreasing function for r - 0. 4 A Single Discrete Shock Profile For any fixed q 2 [0; 1], we estimate the upper bounds of jjLjj ff;fi and jjN jj r assuming that an approximate discrete shock - ' with parameter q is known. We then give a sufficient condition which ensures the existence and stability of an exact discrete shock for scheme (1.2 with parameter q. The sense of stability will be made precise at the end of this section. First, we estimate the upper bound of jjLjj ff;fi . Let us write the linear mapping L in the form of matrix vector product: thinking v as a column vector with the j th row entry being v j (j 2 Z). According to (3.10), the infinite dimensional matrix A is given by refers to the entry of A on i th row and j th column. otherwise. We define D to be an infinite dimensional diagonal matrix with the i th diagonal entry being Use jj \Delta jj 2 and (\Delta; \Delta) 2 to stand for the norm and the inner product in l 2 . It is easy to see that, for any v 2 l 2 ff;fi , we have jjvjj (D \Gamma1 is the inverse of D), we have A T ~ A T ~ A T ~ Since ~ A T ~ A is symmetric, its l 2 norm is just its spectral radius, ae( ~ A T ~ A). Using Gerschgorin Circle Theorem from matrix theory, we have ae( ~ A T ~ A T ~ Notice that A is banded with bandwidth 2p + 1. Therefore ~ A T ~ A is also banded with band-width not more than 4p + 1. We have the following upper bound for jjLjj Lemma 4.1 ~ A T ~ A For later use, we define ~ A T ~ A Here ~ with A and D given by (4.1) and (4.2). Next we estimate the upper bound of jjN jj r We start with a simple lemma on the norm of the shifting operator Z). Due to the non-unitary weight in the norm jj \Delta jj ff;fi , the shifting operator is not unitary. Lemma 4.2 Proof: Assume k ? 0, for any v 2 l 2 last equality implies Similarly, for k ! 0, we have and (4.5) follows by combining the above two inequalities. Q.E.D. Recall (3.14), we have for any v; w sup k is defined in (3.1). Thus we have j+l is the function defined in (3.17). From (3.15) and Lemma 4.2, we have k=\Gammap Thus we obtain the following upper bound for jjN jj r Lemma 4.3 are defined in (3.1) and (3.17), resp. Combining Theorem 2.1 and Lemma 4.1 and 4.3, we obtain: Theorem 4.1 (Existence and stability of a single discrete shock) If there exist ff ? and an approximate discrete shock - ' with parameter q such that oen where ffi is defined in (4.4), then 1. the finite difference scheme (1.2) possesses an exact stationary discrete shock ' with parameter q; Moreover, k j sup 2. ' is stable in B oe(-) in the following sense: For any v 2 B oe(-), lim under maximum norm. Here L n [\Delta] means iterating the finite difference scheme (1.2) n times using its argument as the initial vector. Proof: 1. Based on the discussions in Section 3, the existence of an exact discrete shock for scheme (1.2) is equivalent to the existence of a fixed point for the mapping F in (3.8). This mapping can be put in the form of L[v] +N [v] (3.12). By Lemma 4.1 and 4.3, condition (4.7) implies condition (2.5) in Theorem 2.1 (with an extra factor of 1 therefore the mapping (3.8) is contractive in B oe (-) and, as a result of this, possesses a fixed point - v. It is easy to see that ' is an exact stationary discrete shock for scheme (1.2) with parameter q. Moreover, Due to the extra factor 1in (4.7), by the second conclusion in Theorem 2.1, namely, (2.6), we have jj-vjj ff;fi - oe 2. For any v 2 B oe(-), we write oe and the fact that the mapping F is contractive in B oe (-), we know that, after applying the mapping F infinitely many times on - v, the mapping will converge to the fixed point - v. By the equivalence between the application of the mapping F on iteration of the scheme (1.2) with initial vector established in Section 3, we conclude that ' is stable in B oe(-). 5 Continuous Discrete Shock Profiles In this section, we derive sufficient conditions for the existence of a continuous discrete shock profile for the scheme (1.2). For any fixed q 0 2 [0; 1], assume now the conditions in Theorem 4.1 are satisfied. Namely, there exist ff ? and an approximate discrete shock - ' q0 with parameter q 0 such that condition (4.7) is true. Then by Theorem 4.1, there exists an exact discrete shock Here we have added superscripts and subscripts q 0 to indicate the dependence on parameter . The constants ff and fi, however, will be chosen to be independent of q 0 . For proper choices of ff and fi, we are interested in finding the conditions on the approximate discrete shock - and the first two derivatives of the numerical flux function g such that an exact discrete shock ' q for the scheme (1.2) is guaranteed to exist for any q in a small neighborhood of q 0 , say [q Once such conditions are found and satisfied for a finite number of values of q 0 , e.g. it becomes clear that for any q 2 [0; 1], there exists an exact discrete shock for scheme (1.2). A continuous discrete shock profile is then obtained by properly arranging the family of exact discrete shocks which are parameterized by q 2 [0; 1]. Let us take an approximate discrete shock with parameter q 2 [q 2M ] to be Zg. It is easy to check that conditions 1,2,3 in Definition 1.1 are satisfied. We have sup Define a mapping based on the approximate discrete shock - ' q in (5.1): where v is any vector in l 2 ff;fi . We can rewrite the mapping F in the form of (2.2) with It is easy to see L is a linear mapping and its norm can be estimated similarly as in Lemma 4.1, namely, we have A T ~ A q0 and similar to (4.4), we define A T ~ A q 0 Here ~ is defined in (4.2); A q0 is given in (4.1) with - ' replaced by - The mapping N satisfies N[- and its norm jjN jj r ff;fi can be estimated as follows: For any v; w same as (3.15), we have Z 1Z 1g 00 j+l djd- but with (different from (5.11)) Using (5.2) and (5.3), we have sup sup Thus similar to Lemma 4.3, we have Now we estimate jjE jj ff;fi . Since ' q0 is an exact discrete shock profile, we have ' which implies We can write In the third equality above, we have used the definition of - ' q in (5.1) and in the second j has the bound sup Recall the function \Gamma 1 (\Delta) defined in (3.16), we get Using the fact that jq \Gamma q 2M , we get a larger upper for jjE jj ff;fi which does not depend on q, We want to note that the factor jq \Gamma q 0 j on the RHS of (5) will enable us to prove uniform Lipschitz continuity of a family of exact discrete shocks when they exist. Based on the bounds found in (5.8), (5.11) and (5.13), we obtain the following sufficient condition for the existence of a continuous discrete shock profile for scheme (1.2): Theorem 5.1 (Existence of a continuous discrete shock profile) If there exist ff ? (an integer) such that for each q there exist oe q0 ? 0 and an approximate discrete shock profile - ' q0 for which the following inequality is true: oe q0( Here E q0 is given by is given in are two functions defined in (3.16) and (3.17). Then 1. for any q 2 [0; 1], there exists an exact discrete shock ' q for scheme (1.2); 2. if for any x 2 R, we define '(x) are uniquely determined by is a continuous discrete shock profile for scheme (1.2). Schematic proof: 1. Condition (5.14) clearly implies condition (4.7) in Theorem 4.1 for Therefore there exists an exact discrete shock for scheme (1.2) with parameter q 0 . For any ], we can define an approximate discrete shock - ' q as in (5.1). According to the estimates (5.8), (5.11) and (5.13), condition (5.14) implies (2.5) for the mapping F in (5.4). By the same logic used in Theorem 4.1, we see that there exists an exact discrete shock ' q for scheme (1.2) for any q 2 [q condition (5.14) is true for all q 2. We only need to check that '(x) is bounded and uniformly Lipschitz continuous. For all are uniformly bounded by the definition of approximate discrete shocks and the finiteness of M . Each ' q0 differs from - ' q0 by a vector whose maximum can be bounded by oe q 0 , so does ' q differ from - . Due to choice of - ' q in (5.1), it is easy to see that ' q is uniformly bounded for q 2 [0; 1]. To prove that '(x) is uniformly Lipschitz continuous, we first give two observations which can be shown easily: Observation #1: For any q 2 [0; 1], there exists q M for some integer i between 0 and M , such that sup where the constant C are independent of q and q 0 . This is a result of the estimate (5.12), the bound (2.6) in Theorem 2.1 and condition (5.14). Observation #2: This is due to the way we parameterize the family of discrete shocks, namely the parameter q in Definition 1.1. An easy generalization of Observation #1 is that for any q 1 or for any j 2 Z. Let x only need to consider the case which we have Because each term on the RHS of the last equality is of the form of (5.16), the uniform Lipschitz continuity of '(x) follows. 6 Algorithms for Computer Verification In this section, we discuss how to use a computer to verify condition (4.7) in Theorem 4.1 to prove the existence and stability of an exact discrete shock or condition (5.14) in Theorem 5.1 to prove the existence of a continuous discrete shock profile, for scheme (1.2). 6.1 Computing an approximate discrete shock We start with providing a method of obtaining an accurate approximate discrete shock - for any fixed q 2 [0; 1] using scheme (1.2). Let J be an integer. Set where - j is given by (1.4) and f- are chosen such that For example, we can take We then apply the finite difference scheme (1.2) to the initial data u 0 repeatedly for sufficiently many times. Note that when the scheme is applied to u n need values of u n in order to compute values of u n+1 j for all jjj - J . We can simply set u p. Although this makes the scheme non-conservative in the bounded region (i.e. jjj - J), it actually does not make an error much bigger than the machine accuracy if J is taken to be large enough. This is because an exact discrete shock is generally believed to be converging to the two end states exponentially fast. For the sake of rigorousness, we can modify the value of u n 0such that to make the procedure conservative in the bounded region. Here n 0 is assumed to the number of applications of the scheme on u 0 . To determine how large n 0 should be, we can monitor sup is defined in (3.12) with - ' replaced by u n ) to see if it is small enough, say close to machine accuracy, for the purpose of our verification of the conditions in Theorem 4.1 or Theorem 5.1. Finally, we can set the approximate discrete shock to be It is easy to check that - ' q satisfies the conditions 1 to 3 in Definition 1.1. Remark: Theoretically the larger J is, the more accurate - ' q one can get. However, larger J means longer computer time that it takes to verify the conditions. If to 80 is believed to be good enough. If very small, J needs to be very large and it may even exceed the computer power. In the latter case, the framework in this paper may be improper. 6.2 Choosing the constants ff and fi Once we have an approximate discrete shock - ' q , we can decide what to choose for ff and fi. The criterion for this is to make the norm jjLjj ff;fi as much below 1 as possible, or in our estimates, to maximize ffi in (4.4). The range of possible values of ff can be obtained by studying the linearized scheme of (1.2) around u Similarly, fi can be obtained by studying the linearized scheme of (1.2) around u \Gamma . See Smylris [13] for details. We can use one approximate discrete shock or a few such approximate discrete shocks (corresponding to different values of q in [0; 1]) to choose the constants ff and fi. In the latter case, we should make the minimum of ffi (over different values of q) above 0 as far as possible. Note that the matrix A in (4.4), which is given by (4.1) with - ' replaced by - essentially finite due to constancy of - we have a finite row sums to take a maximum of in (4.4). 6.3 Strategy for verification We suggest the following strategy for verifying the condition (4.7) in Theorem 4.1. For a given q 2 [0; 1], 1. Find the function \Gamma 2 (r). Make sure it is non-decreasing in r. 2. Compute an approximate discrete shock following the method described in subsection 6.1. Make it as accurate as possible should the condition (4.7) seems very demanding 3. Find the range of the constants ff and fi from the linear analysis of the scheme and choose ff ? in (4.4) is positive and maximized. 4. Find the largest possible value for oe for which condition (4.7) is true. Mostly, we can take If the above four steps are through, we can conclude that scheme (1.2) does possess an exact discrete shock with parameter q and it is stable in the sense stated in the second part of Theorem 4.1. To verify condition (5.14), we suggest the following steps: 1. Find the functions 2. For several q values, compute an approximate discrete shock - ' q0 for each q 0 . Find the proper constants ff ? 1 and fi ? 1 such that minimum of the values of ffi q 0 based on each - positive and maximized. We then set M - oe \Gamma1 q0 and find oe q 0 such that the RHS of (5.14) is maximized. Using this oe q0 and replacing - k q0 by can obtain an estimate of the size of M by requiring the second term on the LHS of (5.14) to be less than the RHS of (5.14). One can even replace M inside the functions q0 as long as we eventually use an M - oe \Gamma1 q0 for all values of q 0 sampled (this is due to the monotonicity of \Gamma 1 (\Delta) and \Gamma 2 (\Delta)). We suggest one always use a bigger M than necessary to attain a bigger margin of the RHS of (5.14) over the LHS. Usually, one can take 3. For each q we compute an approximate discrete shock - parameter q 0 and check if for this - q0 , there exists oe q0 ? 0 such that (5.14) is true. If (5.14) is true for every q conclude that the finite difference scheme does have a continuous discrete shock profile. 7 Some Examples In this section, we apply Theorem 5.1 to give an unified proof of the existence of a continuous discrete shock profile for a modified Lax-Friedrichs scheme and the Lax-Wendroff scheme. For a third order WENO scheme, we apply Theorem 4.1 to prove the existence and stability of an exact discrete shock for some sample values of q in [0; 1]. As an example, we take the conservation law to be the Burgers' Eq.: x and take the end states \Gamma1. It is well known that an entropy satisfying stationary shock exists for these two end states. 7.1 A modified Lax-Friedrichs scheme The flux function for a modified Lax-Friedrichs scheme is It is clear that the stencil width constant We take the upper bound of the first and second derivatives of this flux function to be \Gamma 1 Appendix A.1). For 0:5, we use 41 points (i.e. to compute approximate discrete shocks and choose to define the weighted l 2 norm. Roughly oe q 0 12ff maximizes the RHS of and M can be estimated by 64ff 2 . We take 10000 and is able to verify condition . Thus this modified Lax-Friedrichs scheme possesses a continuous stationary discrete shock profile for Burgers' Eq. with end states \Gamma1. The constant oe 0 , which represents the size of the stability region for the discrete shocks (see conclusion 2 in Theorem 4.1), is approximately 7 \Theta 10 \Gamma3 . We plot values of the LHS and RHS of (5.14) for 100 even spaced samples of q 0 in [0; 1] in Figure 1a. The discrete shock profile is plotted in Figure 1b. 7.2 The Lax-Wendroff scheme The flux function for the Lax-Wendroff scheme is The stencil width constant Appendix A.2). For 0:5, we use 61 points (or to compute approximate discrete shocks. For the weighted l 2 norm, we take Roughly oe q 0 24ff maximizes the RHS of and M can be estimated by 96ff 2 . We take and is able to verify (5.14) for . Thus the Lax-Wendroff scheme possesses a continuous stationary discrete shock profile for Burgers' Eq. with end states \Gamma1. The constant for the size of the stability region, namely, oe 0 , is approximately 2:3 \Theta 10 \Gamma3 . We plot the LHS and RHS of (5.14) for 100 even spaced samples of q 0 2 [0; 1] in Figure 1c. The discrete shock profile is plotted in Figure 1d. 7.3 The third order WENO scheme The WENO (weighted essentially non-oscillatory) schemes [7] are variations of the ENO (esstially non-oscilltory) schemes [12]. They both achieve essentially non-oscillatory property by favoring information from the smoother part of the stencil over that from the less smooth (a) (b) -20 (d) Figure 1: (a) The modified Lax-Friedrichs scheme. Solid line: the value of the LHS of (5.14); Dashed line: the value of the RHS of (5.14). (b) A discrete shock profile for the modified Lax-Friedrichs scheme. (c) Same as (a) but for the Lax-Wendroff scheme. (d) Same as (b) but for the Lax-Wendroff scheme. or discontinuous part. However, the numerical flux function of the ENO schemes is at most Lipschitz contiuous while the numerical flux function of the WENO schemes are infiinitely smooth (if one takes ffl w appearing below to be nonzero). The numerical flux function for the third order WENO scheme with global Lax-Friedrichs flux splitting is and f \Sigma (z) =2 (f(z) \Sigma \Lambdaz) Here, ffl w is a small constant to avoid the denominator to be zero and is taken as ffl is a constant which is the maximum of jf 0 (u)j over all possible values of u. In our case, we take to be slightly above the maximum of the modulus of the two end states, Namely, we set 1:1. The first and second order partial derivatives of the numerical flux g are shown in the Appendix . This scheme is third order accurate in space where the solution is monotone and smooth. It degenerates to second order at smooth extrema. See [7] for details. If we use Euler Forward in time, the scheme has the form However this scheme is linearly unstable for any constant - ? 0. Abbreviate the RHS of express the scheme with the third order Runge-Kutta scheme [12] in time as We abbreviate (7.6) as We have two observations: (i) if f' as in (3.8), if this mapping is contractive under jj \Delta jj ff;fi for some ff ? 1 and fi ? 1, then the mapping derived from contractive under the same norm. The first observation is obvious. The second one is due to the fact that each stage in the third order Runge-Kutta scheme (7.6) is a convex combination of u n and E[u] where u is u n in stage one, u (1) in stage two and u (2) in the final stage. See [12] for details. Therefore, in order to prove the existence and stability of exact discrete shocks for (7.6), it suffices to prove the existence and stability of exact discrete shocks for (7.4). We have attempted to apply Theorem 5.1 to prove the existence of a continuous discrete shock profile for (7.4) and found that we need sample roughly 10 19 even spaced values of q 0 in [0; 1] which is far beyond the computer power. Nevertheless, we are able to use Theorem 4.1 to prove the existence and stability of exact discrete shocks for (7.4) for many sample values of q 0 2 [0; 1]. Our computer verification strongly indicates that a continuous discrete shock profile does exist for this scheme. Here are the details of the computer verification: It is clear that the stencil width constant p equals 2. We take the upper bound for the second derivatives of the numerical flux function to be Its derivation is detailed in Appendix A.3. We have taken used 161 points (or to compute the approximate discrete shocks. ff and fi are both taken to be 1:8. The condition in Theorem 4.1 is verified for q This verification is done on CRAY C-90 using double precision. We plot the LHS and RHS of (4.7) for the 1000 even spaced samples of q 0 in Figure 2a. It can be seen that the curve for the LHS is properly below the curve for the RHS. We believe it is true for all q 0 2 [0; 1]. The discrete shock profile with 40 even spaced samples of q 0 is plotted in Figure 2b. The discrete shock profile appears to be monotone. In Figure 2cd, we show the profile zoomed around between the grid points indexed from \Gamma10 to 0 and around between points indexed from 0 to 10, resp. Notice that the profile contains very small oscillations of magnitude around 10 \Gamma4 on both sides of the shock. However the profile looks very smooth, which leads us to believe that a continuous discrete shock profile does exist for this third order WENO scheme. We have yet been able to find a less stringent condition than that in Theorem 5.1 in order to prove this by computer. Concluding Remarks We have provided sufficient conditions for a conservative scheme (1.2) to have a single discrete shock (Theorem 4.1) or a continuous discrete shock profile (Theorem 5.1) for a scalar conservation law in one dimension. These conditions can usually be verified by computers as (a) (b) -80 -60 -40 -20 -0.50.5(c) (d) Figure 2: The third order WENO scheme. (a) Solid line: the value of LHS of (4.7); Dashed line: the value of the RHS of (4.7). (b) The discrete shock profile. (c) Zoom of (b) around Zoom of (b) around demonstrated in the last section. The key idea here was to linearize the scheme around accurate numerical approximations of the discrete shocks and find suitable weighted l 2 norms for this linear part to define a contractive mapping. If we can find sufficiently accurate approximate discrete shocks, the orginal scheme behaves closely like the linearized scheme around this approximate discrete shock in terms of contractiveness of the induced mapping. Several generalizations or implications of Theorem 4.1 or Theorem 5.1 are immediate. For example, we can find sufficent conditions, in the form of a single inequality, which assure the existence of discrete shocks or a continuous discrete shock profile for a range of the time-space ratio - and the end states Using the result in [4], we can generalize the theorems to non-stationary shocks with sufficiently small shock speed. Pointwise convergence rate estimates can also be obtained for schemes that possess stable discrete shocks and are linearly stable, when used to approximate scalar conservation laws whose solution is smooth except for some isolated shocks. However, we will not elaborate on such extensions. 9 Acknowledgments We would like to thank Bj-orn Engquist and Stanley Osher for their support in this research. The first author wants to thank Chi-Wang Shu for valuable suggestions. Functions Let jzj-r where f is the flux function of the conservation law (1.1). We derive the functions \Gamma 1 (r) and for the numerical flux functions of the schemes discussed in Section 7. For simplicity, we use g 0 k to denote the first order derivative of g(z and use g 00 k;l to denote its second order derivative w.r.t z k and z l , where k; A.1 The modified Lax-Friedrichs scheme The numerical flux function is given in (7.1). Its first order derivatives are Its second order derivatives are Therefore, we have A.2 Lax-Wendroff scheme The numerical flux function is given in (7.2). Its first order derivatives are Its second order derivatives are 1). Therefore, we have A.3 The third order WENO scheme The flux function for the third order WENO scheme is given in (7.3). To obtain \Gamma 1 (r) and first find the first two derivatives of the flux function. The derivatives of the function are: r 00 r 00 r 00 The derivatives of the function a / a a aa /) a a r 0 ab /) Let 2. We define Similarly we define by / \Sigma aa / \Sigma ab bb the first and second order derivatives of / with arguments, resp. same as / \Sigma . We drop 0 and 00 in the notation for derivatives of / for simplicity. The first order derivatives of the numerical flux function (7.3) are a a 2. The second order derivatives are aa ab ab ab bb ab ab bb ab aa Other second order derivatives are know by symmetry, i.e. g 00 l;k . We have the following simple observations: rr 00 where r 0 can be r 0 a or r 0 can be r 00 aa or r 00 bb or r 00 ab and similarly, / 0 can be / 0 a or / 0 be / 00 aa or / 00 bb or / 00 ab . Based on the above observations, we have --R Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics III Convergence of difference scheme with high resolution to conservation laws. Courant and Friedrichs Convergence of a finite difference scheme for a piece-wise smooth solution Viscous limits for piecewise smooth solution to system of conservation laws Gray Jennings Discrete shocks Efficient implementation of Weighted ENO schemes Construction and nonlinear stability of shocks for discrete conservation laws Discrete shock profiles for systems of conservation laws Discrete shocks for difference approximations to system of conservation laws Efficient Implementation of essentially non-oscillatory shock-capturing schemes Existence and Stability of Stationary profiles of the LW Scheme On the existence and stability of a discrete shock profile for a finite difference scheme --TR --CTR Hailiang Liu, The l1 global decay to discrete shocks for scalar monotone schemes, Mathematics of Computation, v.72 n.241, p.227-245, 01 January

conservation law;weighted ENO;discrete shock

276491

A New Spectral Boundary Integral Collocation Method for Three-Dimensional Potential Problems.

In this work we propose and analyze a new global collocation method for classical second-kind boundary integral equations of potential theory on smooth simple closed surfaces $\Gamma \subset {\Bbb R}^3$. Under the assumption that $\Gamma$ is diffeomorphic to the unit sphere $\partial B$, the original equation is transferred to an equivalent one on $\partial B$ which is solved using collocation onto a nonstandard set of basis functions. The collocation points are situated on lines of constant latitude and longitude. The interpolation operator used in the collocation method is equivalent to a certain discrete orthogonal (pseudospectral) projection, and this equivalence allows us to establish the fundamental properties of the interpolation process and subsequently to prove that our collocation method is stable and super-algebraically convergent. In addition, we describe a fast method for computing the weakly singular collocation integrals and present some numerical experiments illustrating the use of the method. These show that at least for model problems the method attains an exponential rate of convergence and exhibits a good accuracy for very small numbers of degrees of freedom.

Introduction In this work we present a new spectrally convergent collocation method for second-kind boundary integral equations of potential theory. Our method is the result of a search for a three-dimensional analogue of well-known discrete global Galerkin or "pseudospec- tral" schemes for integral equations on planar contours (see, e.g., [15], [19], [4] or [14]). In these two-dimensional methods the contour is parametrized by a periodic univariate function and the basis functions used to approximate the solution are constructed from trigonometric polynomials with respect to this parameter. With an appropriate choice of quadrature rule the discrete orthogonal projection is then also an interpolation operator, the discrete Galerkin method is equivalent to a collocation scheme and hence is relatively cheap to implement. Stability and spectral orders of convergence are then proved using reasonably standard arguments. It is interesting to consider the extension of such ideas to three dimensions. In that case a spectral method may be quite attractive since it has the potential to achieve an acceptable accuracy with a significantly smaller linear algebra overhead than that required by conventional (piecewise) methods. In line with spectral methods in general, we consider here only problems with smooth solutions and hence (in common with the two-dimensional case) we assume that our boundary integral equation is posed on a smooth surface. Although most boundaries of interest in structural or mechanical engineering are not smooth, there are many applications of smooth boundaries in other branches of science - such as biology and medicine, where problems with corners and edges are rare. We make the further assumption that our boundary is diffeomorphic to the unit sphere, which means that the sphere may be used as a parametrization of the boundary. Although this does restrict the range of problems which can be treated, it is in some sense a natural generalisation of the fact that in two dimensions all simple closed contours may be parametrized by the unit circle. Moreover there exists quite a lot of interest already in the literature on problems of this form. Applications of boundary value problems on spherical or nearly spherical geometries are found for example in global weather prediction and in geodesy see, e.g. [21], [11]. Numerical analyses of various methods for such problems and further applications can be found in [2], [1], [5], [7, x3.6], [8], [9], [12], [16], [23], [24], [25] and the associated references. For a recent review of spectral methods in general see [10]. For boundary integral equations of the type considered in this paper, nondiscrete (and therefore not fully practical) Galerkin methods using spherical harmonic basis functions are the only global approximation schemes whose convergence has been proved ([2], [1], [16]). Two questions then naturally arise: (a) whether these (or related) Galerkin methods still converge when quadrature rules are used to compute the Galerkin integrals, and (b) whether any such discrete Galerkin method is equivalent to a collocation scheme. A discrete method using spherical harmonics and closely related to those used for implementation (but not covered by the proofs) in [2], [3], [1] and [16] is analysed completely in [12] for boundaries which are exactly spherical. However the convergence proof used there fails when the boundary is merely diffeomorphic to the sphere. This is essentially because the corresponding discrete Galerkin projection has a bound which grows so fast that the usual stability proof fails. Thus the answer to (a) is still unknown in this case. With regard to (b), it is known from [22] that the answer is "no" when the underlying basis functions are spherical harmonics. The very interesting unpublished Ph.D. thesis of Wienert [25] (see also [7]) suggested some fully discrete variants of Galerkin's method with both spherical harmonics and other basis sets and proved some new approximation theory (yielding exponential convergence rates) for the discrete integral operators. However Wienert did not give a convergence analysis for the corresponding approximate integral equations and thus does not provide an answer to (a). Nevertheless one of his methods suggests the use of a discrete orthogonal projection, with basis functions chosen as Fourier modes on R 2 (mapped to the unit sphere with polar coordinates). This happens also to be an interpolatory projection and hence satisfies (b). Our method takes Wienert's idea as starting point. We construct a basis set by choosing a distinguished set of Fourier modes on the plane and mapping them to the unit sphere, again with spherical polar coordinates. However unlike Wienert, we only choose those Fourier modes which map to continuous functions on the sphere and which have a certain reflectional symmetry. (Similar ideas are in [21].) Our choice ensures that the spherical polar coordinate transformation forms a bijection between our basis set and the distinguished set of Fourier modes and that the usual discrete orthogonal projection onto these Fourier modes corresponds to a discrete orthogonal projection on the sphere. Both are interpolatory and have norm growing with O(log 2 (N)), when the number of degrees of freedom is O(N 2 ). This growth is slow enough to allow stability of the corresponding discrete Galerkin method to be proved. So the answer to both (a) and (b) is "yes" for this new method. These results and the implementation of the method are the principal aim of this paper. The layout of the paper is as follows. In Section 2 we describe our basis set on the sphere and prove the required properties of the corresponding discrete orthogonal projection. The space spanned by our basis functions has dimension 2. It contains, as a proper subset, spherical polynomials (which are smooth on the sphere) as well as extra functions which are continuous on the sphere but lack smoothness at the poles. Once Section 2 is done the convergence of the collocation method is obtained very easily in Section 3. The method is proved to be super-algebraically convergent. It should be possible also to show exponential convergence in an appropriate analytic setting (similar to that in [25]), but we do not attempt that here. Computation of the collocation integrals is considered in Section 4. Here were are again influenced by [25]. If the basis functions were smooth on the sphere there is a very nice way of evaluating the required weakly singular integrals using a rotational change of variables and then polar coordinates, which eliminates the weak singularity in the integrand (see also [7, x3.6]). In Section 4 we consider this transformation applied to our collocation integrals and we show that it yields integrands on polar coordinate space ('; OE) 2 [0; -] \Theta [0; 2-] with a mild singularity in the second derivative at certain points. Fortunately this singularity seems not to affect the empirical convergence of standard quadrature rules for the collocation integrals. These perform as if the kernel were smooth, as is seen in Section 5 where we use the tensor product Gauss-Legendre rule for the integrals (after having transformed the singularities to the edges of the domain of in- tegration) and observe exponential convergence. This would be difficult to prove because of the mild singularity in the integrands. We also observe exponential convergence of the solutions to the integral equation and to the corresponding solution to the underlying harmonic boundary value problem, provided the number of quadrature points increases appropriately with the number of degrees of freedom. For regular problems the method is extremely accurate: relative errors of about 10 \Gamma3 are obtained by inverting linear systems of dimension about 40. Because of the relatively small linear algebra requirements of this method we are able to code it using MATLAB ([18]). On modern machines this has been found to yield accurate solutions of potential problems quickly enough to be usable as an interactive tool. We finish this section by describing in more detail the problem to be solved. Let \Omega ae R 3 be a bounded domain with simple smooth closed boundary \Gamma. For any j 2 \Gamma let denote the outward unit normal to \Gamma and let d\Sigma(j) denote surface measure on \Gamma. denote the Euclidean norm on R 3 . We are primarily concerned in this paper with fast (spectrally accurate) solutions of the boundary integral equation Z \Gamma2- This is the classical second kind equation which arises when the indirect boundary integral method with a double layer potential ansatz is used to solve the Dirichlet problem for Laplace's equation on \Omega\Gamma The method of this paper can also be used to solve the Neumann problem on\Omega or the corresponding (exterior) problems Our method for solving (1.1) is based upon expanding the solution u in terms of a certain set of global basis functions, with the coefficients in the expansion determined by collocation. We make the following assumptions which are fundamental to the construction of this collocation method. There exists a smooth map q : D ae R 3 (where D is a domain containing the unit sphere bijective and its Jacobian dq is assumed to satisfy det dq(x) 6= 0; x Since @B ae D is compact it follows from (A.1) that: All derivatives of q are bounded on @B : (1.2) With the change of variable the directed measures on \Gamma and @B are related by Nanson's formula ([20, p.88]): where doe(y) denotes surface measure on @B at y 2 @B, y is the unit normal to @B at y and Adj(M) denotes the adjugate of the 3 \Theta 3 matrix M (i.e. the transposed matrix of its cofactors). Now, with the substitution using (1.3), we can transplant onto @B to obtain Z where We will be concerned with the solution of (1.4), which is equivalent to (1.1). Thus we rewrite u ffi q as u and f ffi q as f and we abbreviate (1.4) as where K is the integral operator on @B induced by the kernel k. It is well known that (1.6) has a unique solution u which is smooth when f is smooth. series and interpolation on the unit sphere 2.1 Functions on the unit sphere Let C r (@B); r - 0, denote the usual space of r-times continuously differentiable functions on the unit sphere @B with norm jj:jj 1;r;@B . Let denote the space of r-times continuously differentiable functions on R 2 which are 2- periodic in each argument with norm jj:jj 1;r;D . In addition, for r we let C r;ff (@B) denote the functions in C r (@B) whose rth partial derivatives (defined with respect to a suitable chart) satisfy a H-older condition of order ff. (C r;0 (@B) j C r (@B).) Also, let C r;ff (D) denote the analogous space on the Euclidean domain D. Let k:k 1;r;ff;@B and k:k 1;r;ff;D denote the corresponding norms. When we drop the index r in the above notations. The standard mapping from R 2 to @B is the spherical polar coordinate transformation This is invertible when considered as a map from (0; -) \Theta [0; 2-) onto the punctured sphere @Bnfn; sg, where (for any OE 2 R) are the north and south poles, respectively. On @Bnfn; sg, p has a continuous inverse given by p is the solution of the equations The function p is smooth and 2-periodic in ('; OE) and any v 2 C(@B) thus induces a function J v 2 C(D) given by Analogously, for all r C r;ff (D) and It is clear that J is an isometry from C(@B) into C(D), i.e. Since p has reflectional symmetry: and is independent of OE 2 R at it is natural to introduce the proper subspaces of C(D): are independent of OE 2 Rg: Then it is easy to show that J : C(@B) ! S(D) is an isometric isomorphism with inverse given by J w(0; (w w(-; s: We shall use (2.6) to define our approximation procedure in C(@B). We do this by first recalling the discrete Fourier projection on C(D). Provided this is carefully defined it turns out also to be a projection operator in S(D). Using the isometry (2.6) this gives us an analogue of discrete Fourier approximation in C(@B). The details are explored in the following subsections. 2.2 Discrete Fourier series in C(D) The Fourier modes are orthonormal with respect to the usual inner product We shall use a corresponding discrete inner product defined using the quadrature points Throughout the paper we assume that N is odd, so that -=2 is never a quadrature point. The commonly used discrete analogue of (2.8) is then obtained by applying the tensor product trapezoidal rule based at the nodes (2.9) to obtain for Throughout the paper means the usual summation with only half the first and last terms included and C denotes a generic constant which is independent of N . The truncated discrete Fourier series of a function w 2 C(D) is then defined by m=\GammaN The following lemma discusses various properties of EN . Before we prove it we recall the following useful formula2N Lemma 2.1 (iii) log 2 N Proof Let EN denote the set defined on the right hand side of (2.12) and let w 2 C(D). Note first that Hence, for m; \GammaN thus it follows that hw; e m \GammaN Hence from the definition of EN , we get Moreover, using the identity he m we obtain, for together with EN (e N Hence a short calculation shows that Thus (i) and (ii) follow from (2.15) and (2.16). Also observe that for m=\GammaN Then (iii) follows on using (2.12). For (iv), let w 2 C(D) and ('; OE) 2 D. Then where D N is the modified Dirichlet kernel [26, Vol I, p.50] given by Hence sup We show in the Appendix that sup proof is obtained by modifying the argument in [26, Vol.II, p.19]). Hence (iv) follows from (2.19). Finally, to obtain (v), let w 2 C r;ff (D); r - 0; ff 2 (0; 1]: Then recall that for each there exists a trigonometric polynomial T n ('; OE) of degree n in each variable such that [17, p.89, Theorem 7]. Since EN TN Using (iv) and (2.20) in (2.21), we get (v). 2.3 Discrete Fourier series in R(D) and C(D) The next result shows that the discrete orthogonal projection EN maps R(D) into itself with range a subspace of EN (C(D)). To show this, it is convenient to introduce functions \Gamman Observe that r m Lemma 2.2 m=\GammaN and hence EN Proof Let RN denote the set defined by the right hand side of (2.23). We prove first that EN (R(D)) ' RN , using essentially the method of [25, p.26] but in a slightly different setting. Let w 2 R(D). We write where m=\GammaN and w (2) m=\GammaN m=\GammaN Using the fact that w 2 R(D), and the 2- periodicity of w and e m n we have w (2) m=\GammaN m=\GammaN Using this together with (2.24) and (2.25), we have m=\GammaN n , this yields m=\GammaNN where By easily that r N Z. Hence we have . Also for 0 - jmj - N with m odd we have which implies ff m Thus we have shown The fact that RN ' EN (R(D)) follows from the following four identities, which can be proved using standard arguments. EN r q EN (r \GammaN EN r q r q EN (r \GammaN r \GammaN The next result shows that in fact EN leaves S(D) invariant. Lemma 2.3 For all w 2 S(D), and all OE 2 R, we have Proof Let w 2 S(D) and set w(0; (2.12) we have, for all OE 2 R, m=\GammaN m=\GammaN m=\GammaN Similarly Lemma 2.2, we have The previous lemmas show that EN is an interpolatory projection operator on S(D). Its image EN (S(D)) constitutes a distinguished subspace of the space spanned by the Fourier modes. It is from this subspace that a basis of interpolating trigonometric polynomials in S(D) can be constructed. Using J \Gamma1 given by (2.6) this leads directly to an interpolatory basis in C(@B). To define the basis in S(D), define a function \Theta R! R by f0g. Moreover the sets fs m are orthogonal with respect to the inner product (2.8). The spaces are mutually orthogonal and have dimension Hence the space is a subspace of S(D) with dimension 2N 2. We shall show in Theorem 2.6 below that for each w 2 S(D), EN w is the unique element of N which interpolates w at a certain set of 2N points. Thus EN is an interpolation operator. The proof is obtained with the help of the following two lemmas. Lemma 2.4 where Proof Observe first that, since the basis functions for N are independent of those for f0g. Now, by examining each of the basis functions for N \Phi \Delta N in turn and using the characterisation (2.13) and the definition of R(D) we can show that Hence if wN 2 N \Phi \Delta N then by Lemma 2.1 (ii), Conversely, suppose wN 2 EN (R(D)). Then from Lemma 2.2 there exist constants ff m with ff \GammaN n and ff m when jmj is odd such that m=\GammaN Thus, recalling (2.22), wN ('; OE) =- m=\GammaN even m=\GammaN sin n' exp imOE: where and w (2) Since w (1) it remains to show w (2) To obtain (2.35) let m 2 f0; :::; Ng be even. Then observe that for n 2 f1; :::; Ng with n odd, cos n' cos sin j' sin ' cos mOE In addition if n 2 f0; :::; Ng with n even then cos n' cos sin j' sin ' cos mOE even and n 2 f0; :::; Ng cos n' sin mOE 2 N \Phi Now (2.36)-(2.38) imply (2.35) and wN 2 N \Phi \Delta N . Hence and, together with (2.34) this proves the result. Lemma 2.5 EN Proof Let wN 2 N . It is easily checked that wN 2 S(D). Since by (2.33), wN 2 EN (C(D)), Lemma 2.1 (ii) yields wN 2 EN (S(D)) and so On the other hand suppose wN 2 EN (S(D)). Then also wN 2 EN (R(D)) and by Lemma 2.4 there exist unique w (1) N . Also, by Lemma 2.3, wN 2 S(D) and, since N ' S(D), we have w (2) Hence there exist C w (2) for all OE 2 R. From the definition of \Delta N it follows easily that w (2) Hence EN (S(D)) ' N and the result follows from this and (2.39). These lemmas then lead to the following theorem describing the interpolatory properties of EN . Theorem 2.6 For all w 2 S(D), EN w is the unique element of N which satisfies Proof Let w 2 S(D). From Lemmas 2.1 and 2.5, EN w 2 N has the interpolation property (2.40) -(2.42). To establish uniqueness, suppose wN 2 N satisfies wN (0; Then since N ae S(D) we have in fact wN thus we have (using Lemma 2.1 (ii)), as required. 2.4 A discrete orthogonal projection on C(@B) Finally in this section we study the interpolatory operator on C(@B) induced by the operator EN on S(D). That is we set is the isometric isomorphism introduced in (2.3). We introduce the interpolation points on the sphere Together with the north pole n and the south pole s these form a set of 2N points on @B. Similarly we introduce the interpolation space Theorem 2.7 (ii) For v 2 C(@B); EN v is the unique element of SN with the property (iv) For log 2 N Proof This is immediate from Lemma 2.1, Theorem 2.6 and the properties of J . 3 The Collocation (Pseudospectral) Method In this section we introduce our pseudospectral method for (1.6) and prove its convergence. First we give an algorithmic description of the method. Let the dimension of the space introduced in (2.30) and (2.43). A basis for this space is For convenience we denote this basis f/ dg. It is natural to choose the ordering implicit in (3.1), i.e. to choose the first 2N basis functions 2N to be are and so on with / being the last two basis functions J N+1 . The collocation points are the set These have a corresponding natural ordering x 1k and so on, with n and s being the last two collocation points. Our method for (1.6) then consists of seeking a numerical solution d a where the fa p g are scalars chosen so that or equivalently d a We discuss the calculation of the matrix entries in the next section. The proof of convergence of (3.3) is straightforward from the results of x2. Theorem 3.1 Let f 2 C(@B). For N sufficiently large (3.3) has a unique solution . If, in addition, f 2 C r (@B) with r - 1 then, for N sufficiently large, log 2 N Proof Observe that by Theorem 2.7, uN satisfies (3.3) if and only if It is well known e.g. [13, x6.4] that exists and is bounded on C(@B). Moreover [6, is bounded for ff 2 (0; 1). Hence, for all u 2 C(@B), Theorem 2.7 (iv) yields log 2 N log 2 N for any ff 2 (0; 1). Hence perturbation theory shows that exists for N sufficiently large and is uniformly bounded. Then, since then since I \Gamma K is also invertible on C r (@B); u 2 C r (@B) and the result follows from Theorem 2.7 (iv). 4 Collocation Integrals To implement the collocation method we must calculate integrals of the form Z where v is one of the basis functions for SN introduced in Section 2 and k is given by (1.5). To compute the weakly singular integrals (4.1) we follow the approach of Wienert ([25]) and introduce the Householder matrix defined for x 2 @B by where and and Moreover, since H(x) has eigenvalues \Gamma1 and 1 in directions u(x) and fu(x)g ? , it follows that geometrically, multiplication by H(x) represents reflection in the plane perpendicular to the line joining x to (0; 0; - (x)) T , passing through the mid-point of that line. Since surface measure on @B is unaffected by an orthogonal change of variable, we may rewrite (4.1) as Z By virtue of (4.5) the integrand in (4.7) now has a singularity at We handle this by taking polar coordinates to obtain Z -Z 2--(x; '; OE)v(H(x)p('; OE))dOEd' (4.8) with We shall approximate the integrals (4.8) by quadrature. As we shall see in Lemmas 4.1- 4.3 below, -(x; '; OE) is smooth on ('; OE) 2 D with all its derivatives uniformly bounded in x 2 @B. This is obtained using the arguments of Wienert [25] (see also [7]) but in the context of C 1 rather than analytic function spaces. We shall also examine below the behaviour of v(H(x)p('; OE)). As a function of ('; OE) 2 D this may have a very mild singularity when p('; We characterise this behaviour in results from Lemma 4.4 below onwards. Together these results show that the integrand in (4.8) is smooth except for a mild singularity at a single point. We devise a suitable method for approximating these integrals in the context of a model example in Section 5. When examining the properties of - it is useful to define the function We can then write with (for x 3 - sin ' and then we replace 0 by - in (4.11) and (4.12).) Before we analyse - 1 and - 2 we point out a useful simpler expression for the last factor on the right-hand side of (4.12) This is obtained by recalling the formula (valid for any n \Theta n matrix A and any two vectors v; w 2 R n Then (as is easily verified), and using (4.4), (4.6) we have Hence (4.4) and (4.13)-(4.15) yield sin ' ('; OE) Using (4.13) again together with the chain rule, we have sin ' ('; OE) To examine the smoothness of the first factor in the integrand of (4.8) it is convenient to make the following definition. Definition Given v : @B \Theta D ! R, we say v(x; '; OE) is C 1 (D) uniformly on @B if the function ('; OE) 7! v(x; '; OE) is in C 1 (D) and all its derivatives are uniformly bounded in Our first lemma is the following. Lemma 4.1 Q(x; '; OE) is C 1 (D) uniformly on @B. Proof From (1.2) it is clear that the function ('; OE) 7! Q(x; '; OE) is in C 1 (D). By the chain rule, its ('; OE) derivatives are products of the Euclidean derivatives of q evaluated at H(x)p('; OE) 2 @B and ('; OE) derivatives of H(x)p('; OE). The former are bounded independently of x (by (1.2)). For the latter, observe that for all x 2 @B, we have, using (4.4), d k+l d k+l p ('; OE) d k+l p ('; OE) and the results follow. The next two lemmas treat the two factors on the right-hand side of (4.10). Lemma 4.2 - 1 (x; '; OE) is C 1 (D) uniformly on @B. Proof Let x 2 @B and without loss of generality assume x 3 - 0. Then using (4.11), we have The result then follows from the following properties of - Q which we shall establish below uniformly in x ; (4.18) and proceed analogously.) To obtain (4.18), observe that Z 1d dt Z 1@Q Hence Z 1@Q Z 1@Q and (4.18) follows from Lemma 4.1. To obtain (4.19) observe that - Since D is compact we simply have to show that lim However by (4.21), - Q(x; 0; OE) is well-defined and non-negative. Thus, if (4.22) is not true, then (4.21) implies for some OE 2 [0; 2-]. By (4.9) and (4.5) this implies cos OE sin OE3 cos OE sin OE3 Hence dq(x) has a zero eigenvalue. However this is ruled out by assumption (A.2) of the Introduction and so (4.19) (and hence the result) follows. Lemma 4.3 - 2 (x; '; OE) is C 1 (D) uniformly on @B. Proof As in Lemma 4.2, without loss of generality, let x 2 @B with x 3 - 0. By (4.20), we have Z 1( Z 1Z 1d ds ds ds Thus, in view of (4.16), ds \Theta Hence dividing (4.12) top and bottom by ' 2 we have a numerator which is C 1 (D) uniformly on @B and a denominator which is precisely - complete the proof. Our next set of results concerns the second factor in the integrand of (4.8), namely the function where v is one of the basis functions for the space SN defined in (2.43), i.e. where w is one of the basis functions for N listed in (2.29). It turns out to be convenient to study the function v(x; '; OE) as ('; OE) ranges over all of R 2 . Then, recalling (2.6) motivates us to introduce the set ('; OE) 2 P(x) if and only if p('; OE) 2 f\Gammax; xg. If, in particular, then it is easy to see that Then by (2.43) and (2.6), we have, for ('; OE) 2 R 2 nP(x), Moreover if ('; OE) 2 P(x), then either H(x)p('; or s in which case It follows easily that ('; OE) 7! v(x; '; OE) defines a function in S(D) for each fixed x 2 @B. It is convenient to write (4.27) as where The first step in characterising the smoothness of v is to examine the properties of G. Here some care must be taken, since (as well as being undefined at n and s), the second component of p \Gamma1 suffers a jump discontinuity of amplitude 2- along the line (0; -)g on @B, This motivates the introduction of the set Then G(x; '; OE) is well-defined for ('; OE) 2 R 2 nZ(x), where and in fact is smooth there, as is shown in the following lemma. Lemma 4.4 Let x 2 @B. Then the function ('; OE) 7! G(x; '; OE) is in (C 1 (R 2 nZ(x))) 2 . Proof If ('; OE) a solution of the parameter-dependent problem: F is infinitely continuously differentiable in y and ('; OE). The Jacobian of F with respect to y 2 R 2 is the 3 \Theta 2 matrix which may easily be shown to have rank 2 when evaluated at G(x; '; OE) for all ('; OE) 2 R 2 nZ(x). Hence the result follows from the implicit function theorem. Our next result obtains formulae for the first partial derivatives of G. Notation If a; b are vectors in R n then [a; b] denotes the n \Theta 2 matrix which has a as its first column and b as its second column. Lemma 4.5 Let x 2 @B. Then, for ('; OE) 2 R 2 nZ(x), we have sin @' sin g 1 cos sin ('; OE) where the components of G and their derivatives are evaluated at (x; '; OE). Moreover extend as continuous functions to ('; OE) 2 R 2 nP(x). Proof Since p(G(x; '; we can differentiate this with respect to ' first and then OE to obtainB @ cos cos sin ('; OE) Then, multiplying both sides of (4.32) by the matrix cos \Gammasin yields (4.31). Since the right-hand side of (4.31) is continuous across L(x) the result follows. Our next result concerns the smoothness of the function v(x; '; OE) given by (4.29). We use the following standard multi-index notation : Given denotes the partial derivative @ jffj /=@' ff 1 @OE ff 2 where Theorem 4.6 (i) For fixed x 2 @B the function ('; OE) 7! v(x; '; OE) is infinitely continuously differentiable on R 2 nZ(x). (ii) If x 2 fn; sg all the partial derivatives of v(x; \Delta) on R 2 nZ(x) have a continuous extension to R 2 nP(x) and are bounded on R 2 nP(x). (iii) If x 2 @Bnfn; sg then for each ff with jffj - 1, D ff v(x; \Delta) has a continuous extension to R 2 nP(x) and there exists a function Rx;ff 2 C 1 (R 4 ) which is 2-periodic in each argument such that where v; g 1 and g 2 are evaluated at (x; '; OE). Proof By (4.29), Lemma 4.4 and the fact that w 2 C 1 (D), part (i) follows directly. To obtain (ii), observe that if x 2 fn; sg then, for ('; OE) 2 (0; -) \Theta [0; 2-), H(x)p('; Hence, we have v(x; '; which is smooth and 2-periodic on (0; -) \Theta R. Also, since v(x; \Delta) 2 S(D), if ('; OE) 2 (\Gamma-; 0) \Theta R, then Then v(x; '; OE) is defined by extending these formulae 2-periodically and part (ii) follows. Part (iii) is proved by induction on first examine the case simple analysis of (2.29) shows that any basis function w 2 N satisfies either or where functions which are 2-periodic in each argument. We shall consider only the case (4.34). The case (4.35) is similar but slightly simpler. Substituting (4.34) into (4.29) and differentiating with respect to ' gives cos where v; g 1 and g 2 are evaluated at (x; '; OE). Now substituting the expressions for @g 1 and sin found in (4.31) yields an expression of the form (4.33) in the case identical argument can be used to obtain an analogous expression for @v . This completes the proof of (4.33) when Now suppose (4.33) holds for all By the inductive hypothesis we have Differentiating with respect to ' yields sin sin sin with Rx;~ ff evaluated at (g 1 substituting the expressions for @g 1 =@' and sin found in (4.31) yields an expression for D - ff v of the form (4.33). If - then we set ~ proceed analogously with ' replaced by OE. Hence (4.33) holds when and the assertion follows. Theorem 4.6 (iii) allows the higher derivatives of v(x; \Delta) to blow up as ('; OE) approaches any point in P(x). In the following corollary we give more detail of the blow-up behaviour which may arise. Recall that P(x) is characterised by (4.26) when x satisfies (4.25). Corollary 4.7 Let x 2 @Bnfn; sg. For any multi-index ff, D ff v(x; '; OE) has a continuous extension to ('; OE) 2 R 2 nP(x). Moreover there exists a constant M ff such that, if for all ('; OE) 2 R 2 sufficiently close (but not equal) to ( - OE). (Here k \Delta k denotes the Euclidean norm in R 2 .) Proof Since Rx;ff defined in Theorem 4.6(iii) is 2- periodic in its second argument and 2 has a jump discontinuity of amplitude 2- along the curves in L(x) the first assertion follows directly from (4.33). This yields the estimate for ('; OE) sufficiently close to ( - OE), where To obtain the second assertion from the bound (4.37), we observe that when ('; OE) is near to but not equal to ( - OE) then ('; 2 P(x) and g 1 2 (0; -) is uniquely defined by cos Suppose x is given by (4.25) and recall (4.26). It is then a simple calculation to show that A Taylor expansion about ( - for some constant C and all ('; OE) sufficiently close to ( - OE). Hence from (4.38) sin which, together with (4.37), yields (4.36). Remark 4.8 It is not difficult to find an illustration of the sharpness of the estimates in Corollary 4.7. For example OE) with ( ~ sin ' sin OE sin ' cos When examining the smoothness of G and v we can consider the limit ('; OE) ! ( ~ '; ~ along infinitely many paths. Two such paths are Then a simple calculation shows that G(x; '; (ffi; -=2) on Path 2. Making use of this and (4.31) shows that as ('; OE) ! ( ~ OE) on Path 1, OE) on Path 2 then Now let us consider, for example, the basis function Using (4.39), (4.40), it is then straightforward to check that v(x; '; OE) on Path 1 whereas OE) on Path 2: So @v=@' is discontinuous at ('; OE). In this case the limiting values of @ 2 v=@' 2 are actually equal along each of Paths 1 and 2. However if we consider instead v(x; '; then @v has the same limiting value along Paths 1 and 2 but an elementary calculation shows OE) along Path 1 and OE) along Path 2: So the estimate O((sin g 1 ) 1\Gammajffj ) of Corollary 4.7 cannot, in general, be strengthened. 5 Numerical Experiments As a model case we solve (1.1) where \Gamma is the ellipsoid and\Omega is the interior of \Gamma. This was converted to (1.6) using the mapping We used the method in x3 to solve (1.6) yielding the solution uN on @B which is identified with a function on \Gamma (also called uN ), defined using the inverse of q. Having found this we can compute the potential in \Omega\Gamma Z which approximates the solution U of the boundary value problem on \Gamma. We consider two cases 2. In Case A, (1.1) has the exact solution u and the solution to the boundary value problem is U j 1. Because of the quasioptimal estimate (3.5) and since the constant functions lie in the basis space for the collocation method we have uN j 1 for all N provided the collocation integrals are computed exactly. Thus Case A can be used as a test of the accuracy of the quadrature method which we shall use for the collocation integrals. In Case B the solution of (1.1) is not known analytically but which can be used for comparison with the computed value UN (-). In all the tables of results given below kuN \Gamma 1k1 is computed by finding the maximum of at the collocation points on @B, and - is the (randomly chosen) point x on the unit sphere. For any table of results containing a sequence (a N ) which tends to 0 as N ! 1, exponential convergence is tested by conjecturing that and then computing ff from results for two different values of N by oe log(a It remains to explain how we use the information derived in x4 to devise an accurate method of computing the collocation integrals. Since the collocation method has a spectral convergence rate we do not want to use any quadrature rule which would destroy that fast rate and so this question is vitally important. Recall that the collocation integrals are given by (4.8). The integrands have singularities (described by Corollary 4.7) if x 6= n or s. So suppose OE) with ~ and ~ Then we are concerned with singularities in the integrand of (4.8) at any point in the region [0; -] \Theta [0; 2-]. By Corollary 4.7 these can only happen at points and at their translates through 2- in the OE direction. Since the integrand in (4.8) is 2-periodic in OE we can write where the integrand now has singularities at three points in the domain of integration: We split (5.2) into the sum of integrals over [0; -] \Theta [0; -] and [0; -] \Theta [-; 2-] and apply the tensor product Gauss-Legendre rule with M points in each coordinate direction to each integral. Since the singularities lie on the boundaries of the integration domains we expect this to converge reasonably well. We solved (1.1) with above. The results are given in Table 1 and they indicate that in fact the quadrature rule yields solutions to the integral equation and approximate potentials which converge with order roughly O(exp(\Gamma1:2M)). The convergence is less regular for the potential at the arbitrarily chosen point - . These results are surprisingly good (we have no proof of the exponential convergence), and they indicate that we have no problems from the weak singularities in (4.8). Table A. Next, to check the performance of the collocation method in the absence of quadrature errors we solved (1.1) with From Table 1, we expect quadrature error to pollute at worst (about) the 11th decimal place of our solution. The results are in Table 2 and show a convergence rate for approximate potentials which is roughly of order O(exp(\Gamma2:5N)). 9 Table Table 2 shows that good solutions to the boundary value problem are obtained by solving a system of very small dimension (e.g. when are 86 degrees of freedom). A similar accuracy is reported in [2]. However since the method used there is a Galerkin method more quadrature is needed to assemble the stiffness matrix (for the same number of degrees of freedom). The program used to compute these solutions is very simple and compact, was written in MATLAB ([18]) and runs interactively on a Sun 4 workstation with many concurrent users and also on stand alone PC-486. Finally it is interesting to try to reduce the overall cost of the method by balancing the quadrature error with the underlying error in the collocation method. To do this let u e denote the (theoretical) collocation solution (if the collocation integrals are done exactly) and let uN denote the corresponding solution in the presence of quadrature. Then We know that the first term on the right-hand side of (5.3) converges exponentially. We want to ensure the second does also. Recall that from x3 suppose that denotes the quadrature approximation to ENK (using the M point Gauss rule described above). Then a simple manipulation shows Thus, assuming stability for using the fact that u e u, we expect that where SN is as in x3. The right-hand side of (5.4) is the uniform norm of a matrix with dimension O(N 2 ), each entry of which is converging exponentially. Thus it is reasonable to conjecture for some fi. We checked the sharpness of the N-dependence in this estimate by solving Case A. The results are in Table 3 and confirm the growth conjectured in (5.5). (Note u e u in this example.) 9 Table A. To ensure exponential convergence for u e for some ff 0 ? 0) it is then clear from (5.5) that we must choose In principle we would choose ff 0 equal to ff where and then M by (5.6), but in practice ff, and fi are unknown in general, and ff 0 , and fi in would have to be chosen by experiment. Here we can verify this analysis since Table In (5.6) we put 2. The results are in Table 4 and exhibit a very similar convergence and accuracy to Table 2, but much less quadrature is needed. 9 23 7.917(-10) 2.7 Table Finally, we tried the method on the long slender ellipsoid 10. The results are in Table 5 and exhibit similar convergence to Table 4 but with a somewhat larger asymptotic constant. The accuracy of 3 \Theta 10 \Gamma6 with 86 degrees of freedom is still very good. 9 23 1.521(-7) 1.6 Table --R The numerical solution of Laplace's equation in three dimensions II. The numerical solution of Laplace's equation in three dimensions. Algorithm 629: An integral equation program for Laplace's equation in Three dimensions. A discrete Galerkin method for first kind integral equations with a logarithmic kernel. Galerkin methods for solving single layer integral equations in three dimen- sions Integral Equation Methods in Scattering Theory. Inverse acoustic and electromagnetic scattering theory. Spectral algorithms for vector elliptic equations in a spherical gap. A pseudospectral approach for polar and spherical geometries. A review of pseudospectral methods for solving partial differential equations. Geophysical data inversion methods and applications. A pseudospectral 3D boundary integral method applied to a nonlinear model problem from finite elasticity. Linear Integral Equations. On the numerical solution of a logarithmic integral equation of the first kind for the Helmholtz equation. The fast Fourier transform and the numerical solution of one-dimensional boundary integral equations The numerical solution of Helmholtz's equation for the exterior Dirichlet problem in three dimensions. Approximation of Functions. Mathworks Inc. A spectral Galerkin method for a boundary integral equation. Nonlinear Elastic Deformations. Fourier series on spheres. Polynomial interpolation and hyperinterpolation over general regions. On the spectral approximation of discrete scalar and vector functions on the sphere. Cubature for the sphere and the discrete spherical harmonic transform. Die numerische approximation von randintegraloperatoren f? Trigonometric Series. --TR --CTR M. Ganesh , I. G. Graham, A high-order algorithm for obstacle scattering in three dimensions, Journal of Computational Physics, v.198 n.1, p.211-242, 20 July 2004 Lexing Ying , George Biros , Denis Zorin, A high-order 3D boundary integral equation solver for elliptic PDEs in smooth domains, Journal of Computational Physics, v.219 n.1, p.247-275, 20 November 2006 Piotr Boronski, Spectral method for matching exterior and interior elliptic problems, Journal of Computational Physics, v.225 n.1, p.449-463, July, 2007

fourier approximation;pseudospectral method;boundary integral equation;three-dimensional potential problems;collocation

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Incorporating speculative execution into scheduling of control-flow intensive behavioral descriptions.

Speculative execution refers to the execution of parts of a computation before the execution of the conditional operations that decide whether it needs to be executed. It has been shown to be a promising technique for eliminating performance bottlenecks imposed by control flow in hardware and software implementations alike. In this paper, we present techniques to incorporate speculative execution in a fine-grained manner into scheduling of control-flow intensive behavioral descriptions. We demonstrate that failing to take into account information such as resource constraints and branch probabilities can lead to significantly sub-optimal performance. We also demonstrate that it may be necessary to speculate simultaneously along multiple paths, subject to resource constraints, in order to minimize the delay overheads incurred when prediction errors occur. Experimental results on several benchmarks show that our speculative scheduling algorithm can result in significant (upto seven-fold) improvements in performance (measured in terms of the average number of clock cycles) as compared to scheduling without speculative execution. Also, the best and worst case execution times for the speculatively performed schedules are the same as or better than the corresponding values for the schedules obtained without speculative execution.

Introduction Speculative execution refers to the execution of a part of a computation before it is known if the control path to which it belongs will be executed (for example, execution of the code after a branch statement before the branch condition itself is evaluated). It has been used to overcome, to some extent, the scheduling bottlenecks imposed by control-flow. There has been previous work on speculative execution in the areas of high-level synthesis [1, 2, 3] as well as high-performance compilation [4, 5]. Previous work [1, 2, 3] in high-level synthesis has attempted to locate single or multiple paths for speculation prior to schedul- ing. This paper presents techniques to integrate speculative execution into scheduling during high-level synthesis of control-flow intensive designs. In that context, we demonstrate that not using information such as resource constraints and branch probabil- * This work was supported in part by NSF under Grant No. 9319269 and in part by Alternative System Concepts, Inc. under an SBIR contract from Air Force Rome Laboratories. Permissions to make digital/hard copy of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage, the copyright notice, the title of the publication and its date appear, and notice is given that copying is by permission of ACM, Inc. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. DAC 98, San Francisco, California ities while deciding when to speculate can lead to significantly sub-optimal performance. We also demonstrate that it is necessary to perform speculative execution along multiple paths at a fine-grain level during the course of scheduling, in order to obtain maximal benefits. In addition, we present techniques to automatically manage the additional speculative results that are generated by speculatively executed operations. We show how to incorporate speculative execution into a generic scheduling methodology, and in particular present the results of its integration into an efficient scheduler Wavesched [6]. Experimental results for various benchmarks and examples are presented that indicate upto seven-fold improvement in performance (average number of clock cycles required to perform the computation). Background and Motivation Scheduling tools typically work using one or more intermediate representations of the behavioral description, such as a data flow graph (DFG), control flow graph (CFG), or control-data flow graph (CDFG). In this paper, we use the CDFG as the intermediate representation of a behavioral description, and state transition graphs (STGs) to represent the scheduled behavioral description, as explained in later sections. In addition to the behavioral description, our scheduler also accepts the following information: ffl A constraint on the number of resources of each type available (resource allocation constraints). ffl The target clock period for the implementation, or constraints that limit the extent of data and control chaining allowed. ffl Profiling information that indicates the branch probabilities for the various conditional constructs present in the behavioral description. We now present some motivational examples to illustrate the use of speculative execution during scheduling. Consider a part of a behavioral description and the corresponding CDFG fragment shown in Figure 1, that contains a while loop. The CDFG contains vertices corresponding to operations of the behavioral description, where solid lines indicate data dependencies, and dotted lines indicate control dependencies. Control edges in the CDFG are annotated with a variable that represents the result of the conditional operation that generates them. For ex- ample, the control edges fed by operation > 1 are marked c in Figure 1. The initial values of variables i and t 4 used in the loop body are indicated in parentheses beside the corresponding CDFG data edges. Let us now consider the task of scheduling the CDFG shown in Figure 1. Suppose we have the following constraints to be used during scheduling. while (k > t4) { c c Figure 1: A CDFG to illustrate speculative execution S5 S5 ++1_5, M1_4, *1_2, *1_3, *2_1, *2_2, +1_0 *2_2, *2_3, +1_1, M2_0 *2_3, *2_4, +1_2, M2_1 M1_1/c_1, *1_0 (a) (b) Figure 2: (a) Non-speculative schedule for the CDFG of Figure 1, and (b) schedule incorporating speculative execution ffl The target clock period allows the execution of +, ++, >, and memory access operations in one clock cycle, while the * operation requires two clock cycles. In addition, we assume that the * operation will be implemented using a 2-stage pipelined multiplier. chaining is allowed, since it leads to a violation of the target clock period constraint (in general, however, our algorithm can handle chaining). ffl The aim is to optimize the performance of the design as much as possible. Hence, no resource constraints are specified for the purposes of illustration for this example. This is not a limitation of our scheduling algorithm, which does handle resource constraints as described in later sections. A schedule for the CDFG that does not incorporate speculative execution is shown in Figure 2(a). This schedule can be obtained by applying either the loop-directed scheduling [7] technique or the Wavesched [6] technique to the CDFG. Vertices in the STG represent schedule states, that directly correspond to states in the controller of the RTL implementation. Each state is annotated with the names of the CDFG operations that are performed in that state, including a suffix that represents a symbolic iteration index of the CDFG loop that the operation belongs to. For example, consider operation > 1 of the CDFG. When > 1 is encountered the first time during scheduling, it is assigned a subscript 0, resulting in operation in the STG of Figure 2(a). In general, multiple copies of an operation may be generated during scheduling, corresponding to different conditional paths, or different iterations of a loop. For ex- ample, operation > 1 1 in the STG of Figure 2(a) corresponds to the execution of the first unrolled instance of CDFG operation > 1. An edge in the STG represents a controller state transition, and is annotated with the conditions that activate the transition. Each iteration of the loop in the scheduled CDFG requires eight clock cycles. For this example, the data dependencies among the operations within the loop require them to be performed serially. In addition, the control dependencies between the comparison operation together with the inter-iteration data dependency 1 from +1 to > 1, prevent the parallel computation of multiple loop iterations, even when loop unrolling is employed. A schedule for the CDFG of Figure 1 that incorporates speculative execution is shown in the STG of Figure 2(b). This schedule was derived by techniques we present in later sections. Speculatively executed operations are annotated with the conditional operations whose results they depend upon, using the following tion. op/cond represents an operation op that is executed assuming that the speculation condition cond will evaluate to t rue. The speculation condition cond could, in general, be an expression that is a conjunction of the results of various conditional operations in the STG. For example, consider operation ++1 1/c 1 in state S1 of Figure 2(b). This is a speculatively executed operation, that corresponds to the second instance of CDFG operation in the schedule, and assumes that the result of conditional operation > 1 1, which is executed only in state S7, is going to be true. States S7 and S8 represent the steady state of the schedule. Note that, when in the steady state, a new iteration is initiated every cycle, as opposed to once in eight cycles. The following example illustrates the impact of branch probabilities and resource constraints on the performance of speculatively derived schedules and makes a case for the integration of speculation into the scheduling process. Consider the example CDFG shown in Figure 3. The select operation Sel1 selects the data operand at its l (r) port if the value at its s port is 1 (0). Figure 4 shows three different schedules that use speculative execution, that were generated using different resource constraints and branch probabilities. The STG of Figure 4(a) was generated assuming the following information. Available resources consist of one incrementer (++), one adder(+), An intra-iteration data or control dependency is between operations that correspond to the same iteration of a loop, while an inter-iteration dependency is between operations in different (e.g., consecutive) iterations. We refer to intra-iteration data and control dependencies simply as data and control dependencies. a out >>1c d e s l r Figure 3: CDFG demonstrating the effect of resource constraints and branch probabilities on speculative execution (a) (b) (c) Figure 4: Three speculative schedules derived using different resource constraints or branch probabilities one comparator (>), one shifter (>>), and one multiplier (*), all of which require one cycle. Also, the probability of comparison > 1 evaluating to f alse is higher than it evaluating to true. Since the result of > 1 evaluates to f alse more often, the schedule of Figure 4(a) gives preference to executing operations from the corresponding control path (e.g., +2). As a result, +2 is scheduled to be performed on the sole adder in state S0, as opposed to +1, even though the data operands for both operations are available. The average number of clock cycles, CC a , required for the STG in Figure 4(a) can be calculated as follows. In the above equation, P(c1) represents the probability that the result of comparison > 1 evaluates to true. The STG of Figure 4(b) was derived with the same information above, except that it was assumed that comparison > 1 evaluates to true more often than it evaluates to f alse. Hence, operation +1 is given preference over operation +2 and is scheduled in S0. The average number of clock cycles, CC b , required for the STG in Figure 4(b) is given by the following expression Suppose the resource constraints were relaxed to allow two adders. The speculative schedule that results is shown in Figure 4(c). The average number of clock cycles, CC c , required for the STG in Figure 4(c) is given by the following expression. Expected Number of Cycles CC a Figure 5: Comparison of the speculative schedules The values of CC a , CC b , and CC c for various values of P ranging from 0 to 1 are plotted in Figure 5. As expected, the schedule of Figure 4(a) outperforms the schedule of Figure 4(b) when P(c1) < 0.5, and the schedule of Figure 4(b) performs better when 0.5. Moreover, the schedule of Figure 4(c), which was derived using one extra adder, outperforms the other two schedules for all values of P(c1). Thus, we can conclude that branch probabilities and resource constraints do influence the trade-offs involved in deciding which conditional paths to speculate upon, making the case for the integration of speculative execution into the scheduling step where such information is available. The following example illustrates that it is necessary to perform speculative execution along multiple paths, in a fine-grained man- ner, in order to obtain maximal performance improvements. schedules shown in Figure 4 were all generated S5 Figure Speculation along a single path by speculatively executing operations from both the conditional paths of the CDFG in a fine-grained manner, as allowed by the resource constraints. For the purpose of comparison, we scheduled the CDFG shown in Figure 3, assuming the same scheduling information that was assumed to derive the schedule of Figure 4(b). However, in this case, we restricted the scheduler to allow speculative execution along only one path. The resulting schedule is shown in Figure 6. The average number of clock cycles, CC d , required for the STG in Figure 6 is given by the following expression. Comparing the expression for CC d to the expression for CC b from the previous example indicates that CC d CC b for all feasible values of P(c1). Thus, in this example, simultaneously speculating along multiple paths according to resource availability results in a schedule that is provably better than one derived by speculating along only the most probable path. Our scheduling algorithm automatically decides the best paths to speculate upon for the given resource constraints and branch probabilities. 3 The Algorithm In this section, we present the changes that need to be made to a generic scheduling algorithm to support speculative execution. 3.1 A generic scheduling algorithm Figure 7 shows the pseudocode for a generic scheduling algo- rithm. The inputs to the scheduler are a CDFG, G, to be sched- Generic scheduler (CDFG G, ALLOCATION CONSTRAINT K, MODULE SELECTION INFO M inf , CLOCK PERIOD clk) f SET Unscheduled operations; SET Schedulable operations; while (jUnscheduled operationsj > schedulable operation (Schedulable operations, K, M inf , clk); //Select an operation for scheduling. The selected // operation must honor allocation and clock cycle constraints Unscheduled operations.remove operation(op); 5 Schedulable operations.remove operation(op); 6 SET schedulable successors = Compute- schedulable successors(op);//Find the set of operations // in op's fanout which become schedulable when op is scheduled 7 Schedulable operations.append(schedulable successors); //Augment Schedulable operations by addition of //operations in schedulable successors gg Figure 7: Pseudocode for a generic scheduling algorithm uled, the target clock period of the design, allocation constraints, which specify the numbers and types of functional units available, and module selection information, which gives the type of functional unit an operation is mapped to. The output of the scheduler is an STG which describes the schedule. At any point, a generic scheduler maintains (a) the set of unscheduled operations whose data and control dependencies have been satisfied, and can therefore be scheduled (Schedulable operations), and (b) the set of operations which are unscheduled (Unscheduled operations). The scheduling process proceeds as follows: an operation from Schedulable operations is selected for scheduling in a given state (statement 2). The selection should honor allocation and clock cycle constraints. The manner in which the selection is done varies from one scheduling algorithm to another. The selected operation, op, is scheduled in the state. Since op no longer belongs to either Schedulable operations or Unscheduled operations, it is removed from these sets (statements 4 and 5). Also, the scheduling of op might render some of the operations in its fanout schedu- lable. The routine Compute schedulable successors (statement identifies such operations, and these operations are subsequently included in the set Schedulable operations (statement 7). 3.2 Incorporating speculative execution into a generic scheduler: An overview We now provide an overview of the changes that need to be made to incorporate speculative execution into the framework of the generic scheduler shown in Figure 7. To support speculative execution, the generic scheduler shown in Figure 7 needs to be modified as follows (the details of these steps are provided in Section 3.3). 1. When an operation is scheduled, one needs to recognize all its schedulable successors, including the ones which can be s r Figure 8: A CDFG fragment illustrating speculative execution speculatively scheduled. In addition, speculatively executed operations and their successors need to be specially marked. Clearly, procedure Compute schedulable successors needs to be augmented to consider such cases. Note that, at any stage, every speculatively schedulable operation is added to the list of schedulable operations. However, few of them are actually scheduled. Operations which are not worth being speculated on are ignored, and eventually removed from the list of schedulable operations, using procedures described later in this section. Example 4: Consider the CDFG fragment shown in Figure 8. We assume that operation op0 is scheduled, operation op2 has just been scheduled, and operations op1, op3, Sel1, and op4 are unscheduled. The output of the routine Compute schedulable successors(op2) must include operation op4, which can now be speculatively executed, i.e., its operands can be assumed to be the results of operations op2 and op0. 2. When operations are scheduled, control and data dependencies of speculatively executed operations are resolved. This would potentially validate or invalidate speculatively performed operations. Operations which are validated should be considered "normal", i.e., they need not be specially marked any longer. Operations in Unscheduled operations and Schedulable operations which are invalidated need no longer be considered for scheduling. They can, therefore, be removed from these sets. In general, the resolution of the control or data dependencies of a speculatively performed operation creates two separate threads of execution, which correspond to the success and failure of the speculation. Example 5: Consider again, the CDFG fragment shown in Figure 8. Suppose operations op0, op2 and op4 have been scheduled, and operation op3 is unscheduled. Operation op4 uses as its operands, the results of operations op2 and op0. Assume that operation op1 has just been scheduled. If op1 evaluates to true, then the execution of op4 can be considered fruitful, because the operands chosen for its computation are correct. Therefore, op4, and its scheduled and schedulable successors need not be considered conditional on the result of op1 anymore, and the data structures can be modified to reflect this fact. If, however, op1 evaluates to false, then op4 should use as its operands, the results of operations op3 and op0, thus invalidating the result of our speculation. There- fore, schedulable operations, whose computations are influenced by the result computed by op4 are invalid, and can be removed from the set Schedulable operations. 3. The set, Schedulable operations, from which an operation is selected for scheduling, contains operations whose execution is speculative, i.e., whose results are not always use- ful. The selection procedure, represented by the routine Select schedulable operation() (statement 2), needs to be modified to account for this fact. For example, operations, whose execution is extremely improbable, would make poor selection candidates, as the resources consumed by them might be better utilized by operations whose execution is more proba- ble. Also, operations, which fall on critical paths, would be better candidates for selection than those on off-critical paths. 3.3 Incorporating speculative execution into a generic scheduler: A closer look In this section, we fill in the details of the changes outlined in Section 3.2. This is preceded by a formal treatment of concepts related to speculative execution. A scheduler which supports speculative execution works with conditioned operations as its atomic schedulable units, just as a normal scheduler uses operations. Therefore, the fanin-fanout relationships between operations, captured by the CDFG, need to be defined for conditioned operations. Since all speculatively performed operations are conditioned on some event, the adjective "speculatively performed" when applied to an operation, implies that it is conditioned on some event or combination of events. As mentioned in Section 3.2, when an operation is scheduled, its schedulable successors need to be computed. s r l s r Figure 9: Illustrating the scheduling of successors of speculatively performed operations Consider the CDFG fragment shown in Figure 9. Assume that operations op5 and op6 have been scheduled, operations op1, op3, and op4 are unscheduled, and op2 has just been sched- uled. It is now possible to schedule two versions of operation op7, with the first version, op7 0 , using op2 and op5 as its operands, and the second, op7 00 using op2 and op6. op7 0 is conditioned on c(op1) c(op4), and op7 00 is conditioned on c(op1) c(op4). The following analysis presents a structured means of identifying such relationships. We now present a result which helps derive fanin-fanout relationships among speculatively performed operations. Lemma 1: Consider an operation, op, whose fanins are op1, op2, ., opn. If the fanins of op have been speculatively scheduled, so can op. In particular, if the ith fanin, opi, is conditioned on C i , then op would be conditioned on V n We now present details of Steps 1, 2, and 3, outlined in Section 3.2. Step 1: This step addresses the issue of deriving all schedulable successors of a scheduled operation, op0. The result of Lemma 1 is used for this procedure. scheduled op- erations, which satisfies the following condition sources a schedulable operation. Condition: There exits an operation, fanout, in the CDFG, all of whose fanins are reachable from the outputs of the operations in S through paths which consist exclusively (if at all) of select opera- tions. The path connecting the output of an operation opj in S to an input of fanout is denoted by Pj, and the operations on Pj are . Note that aj can equal represents the condition that path Pj is selected, i.e., the result of operation op j is propagated through path Pj to the appropriate input of fanout. Operation fanout is conditioned on V represents the expression opk is conditioned on. Observation 1 can be used to infer the schedulable successors of an operation. The procedure Compute schedulable successors, which is called in statement 6 of the pseudocodeshown in Figure 7, is appropriately augmented. So far, we have described the technique used to identify all schedulable successors of an operation. This was accomplished by tagging operations with the conditions under which their results would be valid. Note that our procedure allows us to speculate on all possible outcomes of a branch, and arbitrarily deeply into nested branches. If integrated with a scheduler which supports loop unrolling, the speculation could also cross loop boundaries. We now present the technique used to validate or invalidate speculatively performed operations whose dependencies have just been resolved. Step 2: Suppose operation op s , which resolves a condition c, has just been scheduled. The resolution of c results in the creation of two different threads of execution, where (i) true, and (ii) false. The following procedure is carried out for every operation, which belongs either to the set, Schedulable operations, or the set of scheduled operations. Let op be conditioned on In the true (false) branch, C is evaluated assuming a value of 1 (0) for c, and the resultant expression is the new expression that op is conditioned on. Step 3: We now describe the procedure employed by the scheduler to select an operation to schedule, from a pool of schedulable oper- ations, Schedulable operations. Schedulable operations can contain operations which are conditioned on different sets of events, i.e., we can choose different paths to speculate upon. We need to decide the "best" candidate to map to a given resource, where, by best, we mean the operation whose mapping on the given resource would minimize the expected number of cycles for the schedule. Formally, the problem can be stated as follows: given (i) a partial schedule, (ii) a functional unit, fu, (iii) a set of operations, S (some of which may be speculative), which can execute on the functional unit, and (iv) typical input traces, select the operation, which, if mapped to fu, would minimize the expected number of cycles. The above problem has been proven to be NP-complete, even for conditional- and loop-free behavioral descriptions [8]. We, therefore, use the following heuristic, whose guiding principle has been successfully employed by several scheduling algorithms [9]. The heuristic is based on the following premise: operations in the CDFG which feed primary outputs through long paths are more critical than operations which feed primary outputs through short paths and, therefore, need to be scheduled earlier. The rationale behind this heuristic is that operations which belong to short paths are more mobile than those on long paths, i.e., the total schedule length is less sensitive to variations in their schedules. The length of a path is measured as the sum of the delays of its constituent operations. In data-dominated descriptions, with no loops and conditional operations, the longest path between any pair of operations is fixed. In control-flow intensive descriptions, some paths could be input- dependent. Therefore, the longest path between a pair of operations must be defined with respect to a given input. For example, for the CDFG shown in Figure 3, the longest path connecting primary input c with output out depends upon the value taken by operation > 1. Since our scheduling algorithm is geared towards minimizing the average execution time, we use the expected length of the longest path from an operation to a primary output as a metric to rank different operations. We use the notation l(op) to denote this quantity for operation op. Speculation adds a new dimension to this problem: the result computed by an operation is not guaranteed to be useful. For an Table 1: Expected number of cycles, number of states, best- and worst-case number of cycles results Circuit E.N.C. #states bc wc WS SP WS SP WS SP WS SP Barcode GCD 95 Findmin 522 265 4 Table 2: Allocation constraints for the examples in Table 2 Circuit add1 sub1 mult1 comp1 eqc1 inc1 Findmin operation, op, we account for this effect by multiplying the probability that an operation's output is utilized with l(op) to derive a metric of an operation's criticality. This is expressed by means of the following equation: where criticality(op) measures the desirability of scheduling op, is the product of the probabilities of the events that op is conditioned on, and l(op) is as defined above. 4 Experimental Results The techniques described in this paper were implemented in a program called Wavesched-spec, written in C++. We evaluated this program by using it to produce schedules for several commonly available benchmarks. These schedules were compared against those produced by the scheduling algorithm, Wavesched [6], without the use of speculative execution, with respect to the following metrics: (a) expected number of cycles, (b) number of states in the STG produced, (c) the smallest number of cycles taken to execute the behavioral description, and (d) the largest number of cycles taken to execute the behavioral description. In general, finding the largest number of cycles taken to execute a behavioral description is a hard problem. However, for the examples considered in this paper, static analysis of the description was sufficient to find the number. Table 1 summarizes the results obtained. The columns labeled #states, bc, and wc represent, respectively, the expected number of cycles, the number of states in the STG produced, smallest number of cycles taken to execute the STG, and the largest number of cycles taken to execute the STG. Minor columns WS and produced by Wavesched and Wavesched- spec, respectively. We used a library of functional units which consisted of (a) an adder, add1, (b) a subtracter, sub1, (c) a mul- tiplier, mult1, (d) a less-than comparator, comp1, (e) an equality comparator, eqc1, and (f) an incrementer, inc1. Unlimited numbers of single-input logic gates (OR, AND, and NOT) were assumed to be available. All functional units except mult1, which executes in two cycles, take one cycle to execute. The allocation constraints for an example can be found by looking up the entry corresponding to the example in Table 2. For example, the allocation constraints for GCD are two sub1, one comp1, and two eqc1. The expected number of cycles for the final design was measured by simulating a VHDL description of the schedule using the SynopsysVSS simulator. The input traces used for simulation were obtained as zero-mean Gaussian sequences. Of our examples, Barcode, GCD, TLC, and Findmin are borrowed from the literature. Test1 is the example shown in Figure 1. Barcode represents a barcode reader, GCD computes the greatest common divisor of its inputs, TLC represents a traffic light con- troller, and Findmin returns the index of the minimum element in an array. The results obtained indicate that Wavesched-spec produced an average expected schedule length speedup of 2.8 over schedules obtained using Wavesched. Note that Wavesched [6] was reported to have achieved an average speedup of 2 over schedules produced by existing scheduling algorithms, such as path-based scheduling [10], and loop-directed scheduling [7]. To get an idea of the area overhead of this technique, we obtained a 16-bit RTL implementation for the GCD example using an in-house high-level synthesis system, for the schedules produced by Wavesched-spec and Wavesched. These RTL circuits were technology-mapped using the MSU library, and the area of the gate-level circuits were obtained. The area overhead for the circuit produced from Wavesched-spec was found to be only 3.1%. We also note that for Wavesched-spec, the number of cycles in the shortest and longest paths is smaller than or equal to the corresponding number for Wavesched. Conclusions In this paper, we presented a technique for incorporating speculative execution into scheduling of control-flow intensive designs. We demonstrated that in order to fully exploit the power of speculative execution, one needs to integrate it with scheduling. We introduced a node-tagging scheme for the identification of operations which can be speculatively scheduled in a given state, and a heuristic to select the "best" operation to schedule. Our techniques were fully integrated into an existing scheduling algorithm which can support implicit unrolling of loops, functional pipelining of control-flow intensive behaviors, and can parallelize the execution of independent loops whose bodies share resources. Experimental results demonstrate that the presented techniques can improve the performance of the generated schedule significantly. Schedules produced using speculative execution were, on an average, 2.8 times faster than schedules produced without its benefit. --R "Experiments with low-level speculative computation based on multiple branch prediction," "Global scheduling independent of control dependenciesbased on condition vectors," "Combining MBP-speculative computation and loop pipelining in high-level synthesis," "Trace scheduling: A technique for global microcode compaction," "Sentinel scheduling: A model for compiler-controlled speculative execution," "Wavesched: A novel scheduling technique for control-flow intensive behavioral de- scriptions," "Performance analysis and optimization of schedules for conditional and loop-intensive specifica- tions," Computers and Intractibility. "Empirical evaluation of some high-level synthesis scheduling heuristics," "Path-based scheduling for synthesis," --TR Global scheduling independent of control dependencies based on condition vectors Empirical evaluation of some high-level synthesis scheduling heuristics Sentinel scheduling Performance analysis and optimization of schedules for conditional and loop-intensive specifications <italic>Wavesched</italic> Computers and Intractability Combining MBP-speculative computation and loop pipelining in high-level synthesis --CTR Sumit Gupta , Nick Savoiu , Sunwoo Kim , Nikil Dutt , Rajesh Gupta , Alex Nicolau, Speculation techniques for high level synthesis of control intensive designs, Proceedings of the 38th conference on Design automation, p.269-272, June 2001, Las Vegas, Nevada, United States Sumit Gupta , Nikil Dutt , Rajesh Gupta , Alex Nicolau, Dynamic Conditional Branch Balancing during the High-Level Synthesis of Control-Intensive Designs, Proceedings of the conference on Design, Automation and Test in Europe, p.10270, March 03-07, Sumit Gupta , Nick Savoiu , Nikil Dutt , Rajesh Gupta , Alex Nicolau , Timothy Kam , Michael Kishinevsky , Shai Rotem, Coordinated transformations for high-level synthesis of high performance microprocessor blocks, Proceedings of the 39th conference on Design automation, June 10-14, 2002, New Orleans, Louisiana, USA Satish Pillai , Margarida F. Jacome, Compiler-Directed ILP Extraction for Clustered VLIW/EPIC Machines: Predication, Speculation and Modulo Scheduling, Proceedings of the conference on Design, Automation and Test in Europe, p.10422, March 03-07, Sumit Gupta , Nick Savoiu , Nikil Dutt , Rajesh Gupta , Alex Nicolau, Conditional speculation and its effects on performance and area for high-level snthesis, Proceedings of the 14th international symposium on Systems synthesis, September 30-October 03, 2001, Montral, P.Q., Canada Soha Hassoun, Fine grain incremental rescheduling via architectural retiming, Proceedings of the 11th international symposium on System synthesis, p.158-163, December 02-04, 1998, Hsinchu, Taiwan, China Luiz C. V. dos Santos , Jochen A. G. Jess, Exploiting state equivalence on the fly while applying code motion and speculation, Proceedings of the conference on Design, automation and test in Europe, p.120-es, January 1999, Munich, Germany Darko Kirovski , Miodrag Potkonjak, Engineering change: methodology and applications to behavioral and system synthesis, Proceedings of the 36th ACM/IEEE conference on Design automation, p.604-609, June 21-25, 1999, New Orleans, Louisiana, United States Srivaths Ravi , Ganesh Lakshminarayana , Niraj K. Jha, Removal of memory access bottlenecks for scheduling control-flow intensive behavioral descriptions, Proceedings of the 1998 IEEE/ACM international conference on Computer-aided design, p.577-584, November 08-12, 1998, San Jose, California, United States Sumit Gupta , Nikil Dutt , Rajesh Gupta , Alexandru Nicolau, Loop Shifting and Compaction for the High-Level Synthesis of Designs with Complex Control Flow, Proceedings of the conference on Design, automation and test in Europe, p.10114, February 16-20, 2004

high-level synthesis;telecommunication

277563

Minimization of Communication Cost Through Caching in Mobile Environments.

AbstractUsers of mobile computers will soon have online access to a large number of databases via wireless networks. Because of limited bandwidth, wireless communication is more expensive than wire communication. In this paper, we present and analyze various static and dynamic data allocation methods. The objective is to optimize the communication cost between a mobile computer and the stationary computer that stores the online database. Analysis is performed in two cost models. One is connection (or time) based, as in cellular telephones, where the user is charged per minute of connection. The other is message based, as in packet radio networks, where the user is charged per message. Our analysis addresses both the average case and the worst case for determining the best allocation method.

Introduction Users of mobile computers, such as palmtops, notebook computers and personal communication systems, will soon have online access to a large number of databases via wireless networks. The potential market for this activity is estimated to be billions of dollars annually, in access and communication charges. For example, while on the road, passengers will access airline and other carriers schedules, and weather information. Investors will access prices of financial instruments, salespeople will access inventory data, callers will access location dependent data (e.g. where is the nearest taxi-cab, see [10, 24]) and route-planning computers in cars will access traffic information. Because of limited bandwidth, wireless communication is more expensive than wire commu- nication. For example, a cellular telephone call costs about $0.35 cents per minute. As another example, RAM Mobile Data Corp. charges on average $0.08 per data message to or from the mobile computer (the actual charge depends on the length of the message). It is clear that for users that perform hundreds of accesses each day, wireless communication can become very ex- pensive. Therefore, it is important that mobile computers access online databases in a way that minimizes communication. We assume that an online database is a collection of data items, where a data item is, for example, a web-page or a file. Users access these data items by a unique id, such as a key, one at a time. We minimize communication using an appropriate data-allocation scheme. For example, if a user frequently reads a data-item x, and x is updated infrequently, then it is beneficial for the user to allocate a copy of x to her/his mobile computer. In other words, the mobile user subscribes to receive all the updates of x. This way the reads access the local copy, and do not require communication. The infrequent updates are transmitted from the online database to the mobile computer. In contrast, if the user reads x infrequently compared to the update rate, then a copy of x should not be allocated to the mobile computer. Instead, access should be on-demand; every read request should be sent to the stationary computer that stores the online database. Thus, one-copy and two-copies are the two possible allocation schemes of the data item x to a mobile computer. In the first scheme, only the stationary computer has a copy of x, whereas in the second scheme both, the stationary and the mobile computer have a copy of x. An allocation method determines whether or not the allocation scheme changes over time. In a static allocation method the allocation scheme does not change over time, whereas in a dynamic one it does. The following is an example of a dynamic allocation method. The allocation scheme changes from two-copies to one-copy as a result of a larger number of writes than reads in a window of four minutes. In mobile computing the geographical area is usually divided into cells, each of which has a stationary controller. Our stationary computer should not be confused with the stationary controller. The stationary computer is some node in the stationary network that is fixed for a given data item, and it does not change when the mobile computer moves from cell to cell. In this paper we analyze two static allocation methods, namely the one that uses the one-copy scheme and the one that uses the two-copies scheme; and a family of dynamic data allocation methods. These methods are suggested by the need to select the allocation scheme according to the read/write ratio: if the reads are more frequent then the methods use the two-copies allocation scheme, otherwise they use the one-copy scheme. The family consists of all the methods that allocate and deallocate a copy of a data item to the mobile computer based on a sliding window of k requests. For every read or update (we often refer to updates as writes) the latest k requests are examined. If the number of reads is higher than the number of writes and the mobile computer does not have a copy, then such a copy is allocated to the mobile computer; if the number of writes is higher than the number of reads and the mobile computer does have a copy, then the copy is deallocated. Thus, the allocation scheme is dynamically adjusted according to the relative frequencies of reads and writes. The algorithms in this family are distributed, and they are implemented by software residing on both, the mobile and the stationary computers. The different algorithms in this family differ on the size of the window, k. Our analysis of the static and dynamic algorithms addresses both worst-case, and the expected case for reads and writes that are Poisson-distributed. Furthermore, this analysis is done in two cost models. The first is connection (or time) based, where the user is charged per minute of cellular telephone connection. In this model, if the mobile computer reads the item from the stationary database computer, then the read-request as well as the response are executed within one connection of minimum length (say one minute). If writes are propagated to the mobile computer, then this propagation is also executed within one minimum-length connection. The second cost model is message based. In this model the user is charged per message, and the exact charge depends on the length of the message. Therefore, in this model we distinguish between data-messages that are longer, and control-messages that are shorter. Data- messages carry the data-item, and control messages only carry control information, specifically read-requests (from the mobile computer to the stationary computer) and delete-requests (the delete-request is a message that deallocates the copy at the mobile computer). Thus a remote read-request necessitates one control message, and the response necessitates a data message. A write propagated to the mobile computer necessitates a data-message. The rest of the paper is organized as follows. In the next section we present a summary of the results of this paper. In section 3 we formally present the model, and in section 4 we precisely present the sliding-window family of dynamic allocation algorithms. In section 5 we develop the results in the connection cost model, and in section 6 we develop the results in the message model. In section 7 we discuss some other dynamic allocation methods, and extensions to handle read, write operations on multiple data items. In section 8 we compare our work to relevant literature. In section 9 we discuss the conclusions of our analysis. 2 Summary of the results We consider a single data item x and a single mobile computer, and we analyze the static allocation methods ST 1 (mobile computer does not have a copy of x) and ST 2 (mobile computer does have a copy of x), and the dynamic allocation methods SW k (sliding-window with windowsize k). We assume that reads at the mobile computer are issued according to the Poisson distribution with the parameter - r , namely in each time unit the expected number of reads is - r . The writes at the stationary computer are issued independently according to the Poisson distribution with the parameter -w . Other requests are ignored in this paper since their cost is not affected by the allocation scheme. We let ' denote -w -r+-w . Our analysis of each one of the algorithms uses three measures. The first, called expected cost and denoted EXP , gives the expected cost of a read/write request in the case that ' is known and fixed. The second, called average expected cost and denoted AV G, is important for the case ' is unknown or it varies over time with equal probability of having any value between 0 and 1. It gives average the expected cost of a request over all possible values of '. Our third measure is for the worst case, and it is based on the notion of competitiveness 1 (see [9, 18, 23, 29, 32]) of an on-line algorithm. Intuitively, a data allocation algorithm A is said to be c-competitive if for any sequence s of read-write requests, the cost of A on s is at most c times as much as the minimum cost, namely the cost incurred by an ideal offline algorithm that knows the whole sequence of requests in advance (in contrast our algorithms are online, in the sense that they service the current request without knowing the next request). In the remainder of this section we summarize the results for each one of the two cost models discussed in the introduction. These results will be interpreted and discussed at the intuitive level in the conclusion section. 2.1 Summary of results in the connection model In the connection model our results are as follows. For ST 1 the expected cost (i.e. expected number of connections) per request is the expected number of connections per request is '. For SW k the expected cost per request is ' ff k is the probability that the majority of k consecutive requests are reads (the formula for this probability is in equation 5). Furthermore, we show that for any fixed k, SW k is not lower than 'g. Thus, if ' - 1, then the static allocation method ST 1 has the best expected cost per request, and if ' - 1, then the static allocation method ST 2 has the best expected cost per request. Next consider the average expected cost. SW k has the best average (over the possible values of ') expected cost per request. This cost is 1+ 1 , and it decreases as k increases, coming within 6% of the optimum for In contrast, ST 1 and ST 2 , both have an average expected cost of 1For the worst case, we show that ST 1 and ST 2 are not competitive, i.e., the ratio between their performance and the performance of the optimal, perfect-knowledge algorithm is unbounded. In contrast, we show that SW k is 1)-competitive, and this competitiveness factor is tight. In summary, in the worst case the cost of the SW k family of allocation algorithms increases as k increases, whereas the average expected cost decreases as k increases. The window size k should be chosen to strike a balance between these two conflicting requirements. For example, k may provide a reasonable compromise. 1 The traditional worst case complexity as a function of the size of the input is inappropriate since all the algorithms discussed in this paper have the same complexity under this measure. For example, in the connection model, for each algorithm there is a sequence of requests of size m on which the algorithm incurs cost m. 2.2 Summary of results in the message passing model In this model our results are as follows. Let the cost of a data message be 1 and the cost of a control message be !, where 0 - 1. For ST 1 , the expected cost per request is (1+!) and for ST 2 the expected cost is '. For SW 1 , the expected cost is ' for derived the expected cost as a function of ! and ' as shown in equation 15 of section 6.3 2 . From these formulae of the expected costs, we conclude the following. If , then ST 1 has the best expected cost; if ' ! 2\Delta! , then ST 2 has the best expected cost; otherwise, namely if 2\Delta! , the SW 1 algorithm has the best expected cost. The dominance graph of these three strategies is shown in the following figure 1. It indicates the superior algorithm for each value of ' and !. Figure 1: Superiority coverage in message model Next we consider the average expected cost, and we obtain the following results. ST 1 has an average expected cost of 1+!; ST 2 has an average expected cost of 1; SW 1 has an average expected cost of 1+2\Delta!; and the average expected cost of SW k (for k 6= 1) is given by equation of section 6.3, and it has a lower bound of 2+!. Then we conclude that, if ! - 0:4, then SW 1 has the best average expected cost; if ! ? 0:4, then the average expected cost decreases as the window size k increases (see corollary 2 in section 6.3). For the worst case we show that, as in the connection cost model, neither ST 1 nor ST 2 are competitive. Similarly, we show that the sliding-window algorithm SW 1 is (1+2 \Delta !)-competitive, and SW k (for k ? 1) is [(1 In summary, the trade-off between the average expected cost and the worst case is similar to the connection model. Namely, a dynamic allocation algorithm is superior to the static ones, with the worst case improving with a decreasing window size; whereas the average expected cost decreases as the window size increases. 2 The SW 1 algorithm is not a special case of the SW k algorithms, as pointed out at the end of section 4 3 The Model A mobile computer system consists of a mobile computer MC and a stationary computer SC that stores the online database. We consider a data item x that is stored at the stationary computer at all times. Reads and writes are issued at the mobile or stationary computers. Actually, the reads and writes at the stationary computer may have originated at other computers, but the origin is irrelevant in our model. Furthermore, we ignore the reads issued by the stationary computer and the writes issued by the mobile computer, since the cost of each such request is fixed (zero and one respectively), regardless of whether or not MC has a copy of the data item. Thus, the relevant requests are writes that are issued by the stationary computer, and reads that are issued by the mobile computer. A schedule is a finite sequence of relevant requests to the data item x. For example, w; w is a schedule. When each request is issued, either the MC has a copy of the data item, or it does not. For the purpose of analysis we assume that the relevant requests are sequential. In practice they may occur concurrently, but then some concurrency control mechanism will serialize them, therefore our analysis still holds. We assume that messages between the stationary computer and each mobile computer are delivered in a first-in-first-out order. We consider the following two cost models. The first is called the connection model. In this model, for each algorithm (static or dynamic) the cost of requests is as follows. If there does not exist a copy of the data item at the MC when a read request is issued, then the read costs one connection (since the data item must be sent from the SC). Otherwise the read costs zero. For a write at the SC, if the MC has a copy of the data item, then the write costs one connection; otherwise the write costs zero. The total cost of a schedule /, denoted by COST (/), is the sum of the costs for all requests in /. The second model is called the message cost model. In this model, we assume that a data message cost is 1, and a control message cost is !. Since the length of a control message is not higher than the length of a data message, 0 - 1. In this model the cost of requests is as follows. For a read request, if there exists a copy at the MC, then the read does not require communication; otherwise, it necessitates a control message (which forwards the request to the SC) and a data message (which transfers the data to the MC) with a total cost of 1 For a write request, if the MC does not have a copy of the data item, then the write costs Otherwise the write costs 1, !, or 1 !, depending on the algorithm and on the result of the comparison of reads and writes executed by the MC in response to the write request. If the write is propagated to the MC and the MC does not deallocate its copy in response, then the cost is 1; if the MC deallocates its copy in response then the cost is accounts for the deallocate request). Finally, as will be explained in the next section, SW 1 does not propagate writes to the MC; it simply deallocates the copy at the MC at each write request. Then the cost of the write is !. We assume that the reads issued from the MC are Poisson distributed with parameter - r , and the writes issued from the SC are Poisson distributed with parameter -w . Denote -w -w+-r by '. Observe that, since the Poisson distribution is memoryless, at any point in time ' is the probability that the next request is a write, and -w+-r is the probability that the next request is a read. Suppose that A is a data allocation algorithm, and - r and -w are the read and write distribution parameters, respectively. We denote by EXPA (') the expected cost of a relevant request. Suppose now that ' varies over time with equal probability of having any value between 0 and 1. Then we define the average expected cost per request, denoted AV GA , to be the mean value of EXPA (') for ' ranging from 0 to 1, namely Z 1EXPA (')d' (1) The average expected cost should be interpreted as follows. Suppose that time is subdivided into sufficiently large periods, where in the first period the reads and writes are distributed with parameters - 1 r and - 1 w , and ' r ; in the second period the reads and writes are distributed with parameters - 2 r and - 2 w , and ' etc. Suppose further that each ' i has equal probability of having any value between 0 and 1 (i.e. the probability densisty function of ' has value 1 everywhere between 0 and 1, and is 0 everywhere else). In other words, each ' i is a random number between 0 and 1. Then, when using the algorithm A, the expected cost of a relevant request over all the periods of time is the integral denoted AV GA . In other words, AV GA is the expected value of the expected cost. One can also argue that AV GA is the appropriate objective cost function when ' is unknown and it has equal probability of having any value between 0 and 1. For the worst-case study, we take competitiveness as a measure of the performance of an on-line data allocation algorithm. Formally, a c-competitive data allocation algorithm A is defined as follows. Suppose that M is the perfect data allocation algorithm that has complete knowledge of all the past and future requests. Data allocation algorithm A is c-competitive if there exist two numbers c (- 1), and b (- 0), such that for any schedule /, We call c the competitiveness factor of the algorithm A. A competitive algorithm bounds the worst-case cost of the algorithm to be within a constant factor of the minimum cost. We say an algorithm A is tightly c-competitive if A is c-competitive, and for any number A is not d-competitive. 4 Sliding-window algorithms The Sliding-Window(k) algorithm allocates and deallocates a copy of the data item x at the mobile computer. It does so by examining a window of the latest relevant read and write requests. The window is of size k, and for ease of analysis we assume that k is odd. Recall, the reads are issued at the mobile computer, and the writes are issued at the stationary computer. Observe that at any point in time, whether or not the mobile computer has a copy of x, either the mobile computer or the stationary computer is aware of all the relevant requests. If the mobile computer has a copy of x, then all the reads issued at the mobile computer are satisfied locally, and all the writes issued at the stationary computer are propagated to the mobile computer; thus the mobile computer receives all the relevant requests. Else, i.e. if the mobile computer does not have a copy, then all reads issued at the mobile computer are sent to the stationary computer; thus the stationary computer receives all the relevant requests. Thus, either the mobile computer or the stationary computer (but not both) is in charge of maintaining the window of k requests. The window is tracked as a sequence of k bits (e.g. 0 represents a read and 1 represents a write). At the receipt of any relevant request, the computer in charge drops the last bit in the sequence and adds a bit representing the current operation. Then it compares the number of reads and the number of writes in the window. If the number of reads is bigger than the number of writes and there is a copy of x at the mobile computer, then the SW k algorithm simply waits for the next operation. If the number of reads is bigger than the number of writes and there is no copy at the mobile computer (i.e. the stationary computer is in charge), then such a copy is allocated as follows. Observe that the last request must have been a read. The stationary computer responds to the read request by sending a copy of x to the mobile computer. The SW k algorithm piggybacks on this message (1) an indication to save the copy in MC's local database, in which the SC also commits to propagate further writes to the MC, and (2) the current window of requests. From this point onwards, the MC is in charge. If the number of writes is bigger than the number of reads and there is no copy of x at the MC, then the SW k algorithm waits for the next request. If the number of writes is bigger than the number of reads and there is a copy of x at the MC (i.e. the MC is in charge), then the copy is deallocated as follows. The SW k algorithm sends to the SC (1) an indication that the SC should not propagate further writes to the MC, and (2) the current window of requests. From this point onwards the SC is in charge. This concludes the description of the algorithm, and at this point we make two remarks. First, when the window size is 1 and the MC has a copy of x, then a write at the SC will deallocate the copy (since the window will consist of only this write). Therefore, instead of sending to the MC a copy of x, the SC simply sends the delete-request that deallocates the copy at the MC. Thus, SW 1 denotes the algorithm so optimized. Observe that SW 1 the classic write-invalidate protocol. 5 Connection cost model In this section we analyze the algorithms in the connection cost model. The section is divided into 3 subsections. In the first subsection, we probabilistically study the static data allocation algorithms, and in the second we study the family of sliding window algorithms. In each of these subsections we derive the expected cost first, then the average expected cost, and then we compare the algorithms based on these measures. Finally, in section 5.3 we analyze the worst case performance of all the algorithms. 5.1 Probabilistic analysis of the static algorithms For the ST 1 algorithm, a write request costs 0, and a read request cost 1 connection. For the algorithm, every write costs 1, and every read costs 0. Hence, EXP ST 1 (') and EXP ST 2 are simply equal to the probabilities that a request is a read and a write, respectively. Thus, Concerning the average expected cost, by equation 1 and equation 2 we obtain and AV G ST 2 5.2 Probabilistic analysis of the SW k algorithms In this section we derive the expected cost of the SW k algorithms, and we show that for each k and for each ', the SW k algorithm has a higher expected cost than one of the static algorithms. Then we derive the average expected cost of the SW k algorithms, and we show that for any k the SW k algorithm has a lower average expected cost than both static algorithms. Also, we show that the average expected cost of the SW k algorithms decreases when k increases. Recall that we are assuming that the size of the window k (= 2 is an odd number. At any point in time, the probability that there exists a copy at the MC (which we denote by is the probability that the majority among the preceding k requests are reads, and this is the same as the probability that the number of writes in the preceding k requests is less than or equal to n, namely Theorem 1 For every k and for every ', the expected cost of the SW k algorithm is Proof: Let us consider a single request, q. When there is a copy at the MC, then the expected cost of q is equal to the probability that q is a write operation, and it equals '. When there is no copy at the MC, the expected cost of q is '. The expected cost of q is the probability that there is a copy at the MC times the expected cost of q when there is a copy at the MC, plus the probability that there is no copy at the MC times the expected cost of q when there is no copy at the MC. Thus, we conclude the theorem. 2 The next theorem compares the expected costs of the SW k and the static algorithms. Theorem 2 For every k and every ', EXP SW k (')g Proof: From equations 2, 5 it follows that EXP SW k theorem follows due to the fact that the weighted average of two values is not smaller than the minimum of the two values. 2 Now let us consider the average expected costs. Theorem 3 For the sliding-window algorithm with window size k, SW k , the average expected cost per request is Z 1EXP SW k Our derivation of equation 6 uses the following identity for positive integers a and b, Z 1x a Using equation 5, it is straightforward to show that Using equation 4 and the identity given by equation 7 and after some algebraic simplifications, it can be shown that Z 1ff k \Delta and Z 1ff k Substituting for in equation 8, and after some simplification, we get the result given by equation 6. 2 Corollary 1 The average expected cost of the SW k algorithms decreases when the window size k increases, and AV G SW k for any k - 1. Proof: From theorem 3, it is easy to see that AV G SW k decreases when k increases, and 1= 1. From equations 3 in section 5.1, we conclude the corollary.5.3 Worst case analysis in connection model In this section we show that the static algorithms, ST 1 and ST 2 , are not competitive. Then we show that the SW k algorithm is (k 1)-competitive. Therefore, our competitiveness study suggests that for optimizing the worst case, one has to choose the sliding window algorithm with a small window size k. First, let's consider the two static strategies. For the ST 1 algorithm, we can pick a long schedule which consists of only reads. Then the cost of the ST 1 algorithm is un-boundedly higher than the cost of the optimal algorithm on this schedule (which is 0 if we keep a copy at the MC). For the ST 2 algorithm, we can also pick a long schedule which consists of only writes. Then the cost of the ST 2 algorithm on this schedule is also un-boundedly higher than the optimal cost (which is 0 if we do not keep a copy at the MC). Therefore, the static algorithms, ST 1 and are not competitive. Theorem 4 The sliding-window algorithm SW k is tightly We prove this by showing that for any schedule / of requests, COST SW k is the number of read requests in / that occur immediately after a write request. We will also exhibit a schedule / 0 for which COST SW k Since it can be shown that the cost of an optimal off-line algorithm on a schedule / is N / , it follows that SW k is tightly 1)-competitive. As before, we assume throughout the proof that First we prove that COST SW k 1). Let / be a schedule consisting of read and write requests. Let N / be the number of read requests in / that occur immediately after a write request. We divide the schedule / into maximal blocks consisting of similar requests. Formally, let r be the division of / into blocks such that the requests in any block are all reads or they are all writes, and successive blocks have different requests. It should be easy to see that the total number of read blocks in /, i.e. blocks that only contain read requests, is less than or equal to (N / 1). Similarly, the total number of write blocks in / is less than or equal to (N / 1). Now, we analyze the cost of read and write requests separately. Consider any read block B i . It should be easy to see that only the first n+1 reads in may each incur a connection. After the first n reads the window will definitely have more reads than writes, and the algorithm will maintain two copies (consequently further reads in the block do not cost any connections). Thus the cost of executing all the reads in B i is bounded by (n 1). Hence the cost of all the reads in / is bounded above by (n 1). By a similar argument, it can be shown that the cost of all the writes in a write block is bounded by (n+ 1). As a consequence, the cost of all the writes in / is bounded by (n+ 1) \Delta (N rearranging the terms, we get COST SW k To show that the above bound is tight, assume that initially there is a single copy of the data item. Consider a schedule / 0 that starts with a block of read requests, ends with a block of write requests, and in each block there are exactly k requests. It should be easy to see that 6 Message cost model This section is divided into 4 subsections. In the first subsection we probabilistically analyze the static algorithms, in the second we analyze SW 1 , and in the third we analyze the family of sliding window algorithms SW k for k ? 1. 3 In each one of the first three subsections we study the algorithm's expected cost first, then the average expected cost. We also study the relation among the expected costs of all the static and dynamic algorithms; and the relation among the average expected costs. In subsection 6.4, we study the worst case of all the algorithms. Recall that in this model we assume that a data message cost is 1, and a control message cost is !, where ! ranges from 0 to 1. 6.1 Probabilistic analysis of the static algorithms For the ST 1 algorithm, the write does not require any communication, whereas the read costs for the ST 2 algorithm, every write costs 1, the read costs 0. So, 3 As mentioned at the end of section 4, SW 1 is not simply SW k with 1. In this cost model this difference in the algorithms results in a different analysis, thus the need for a separate subsection dedicated to the analysis of SW 1 . 6.2 Probabilistic analysis of the SW 1 algorithm First we derive the expected cost of a relevant request. Theorem 5 The expected cost of the SW 1 algorithm is Proof: For the SW 1 algorithm, a read that immediately follows a write costs the control message that conveys the read request, and 1 for the data message); a write that immediately follows a read costs an ! (the cost of a control message deallocating the copy at the MC). No other relevant requests cost any communication. Therefore, the expected cost of a request q is the expected cost of a read that immediately follows a write times the probability of q being such a read, plus the expected cost of a write that immediately follows a read times the probability of q being such a write, namely, EXP SW 1 In the next theorem we study the relation of the expected costs of three algorithms, i.e., ('), and EXP SW 1 ('). The results of this theorem is graphically illustrated in Figure 1. Theorem 6 The expected costs EXP SW 1 ('), and EXP ST 2 are related as follows, depending on ' and !. , then EXP ST 1 , then EXP SW 1 , then EXP ST 2 Proof: This is a straight-forward algebraic derivation, that uses equations 11, 13, and the fact Now we are ready to consider the average expected cost. Theorem 7 The average expected cost of the SW 1 algorithm is and AV G SW 1 Proof: Equation 14 can be easily obtained from equation 13, based on the definition of the average expected cost (equation 1). Since 0 - 1, we obtain 1+2\Delta!- 1- 1+!. From the equations 12 in section 6.1, we conclude the theorem. 2 6.3 Probabilistic analysis of SW k In this section we consider the SW k algorithms, for First, we derive the formula of the expected cost for SW k . Then we show that for any k and for any ' the expected cost of SW k is higher than the minimum of the expected costs of SW 1 , ST 1 , and ST 2 . Thus we conclude that for a known fixed ', SW k is inferior to the other algorithms. Then we derive the formula of the average expected cost for SW k . Then we show that SW k has the best average expected cost for some k - 1, and we determine this optimal k as a function of !, the cost of a control message. Theorem 8 For every k ? 1, the expected cost of the SW k algorithm is Consider a write request w. It costs a data message if there exists a copy at the MC when the request is issued. The probability of having a copy at the MC is ff k . Additionally, if the MC deallocates its copy as a result of this write, the write will necessitate a delete message sent from the MC to the SC, It can be argued (and we omit the details) that this occurs if and only if the sequence of k requests immediately preceding w, starts with a read and has exactly writes. Therefore, the expected cost of w is Now consider a read request r. It does not require communication if there is a copy at the MC when the request is issued. Otherwise, it costs a control message for the request, and a data message for the response. Thus, the expected cost of r is Therefore, the expected cost of a request is the expected cost of a write times the probability that the request is a write, plus the expected cost of a read times the probability that the request is a read, namely, EXP SW k A simple algebraic manipulation of the above expression leads to equation 15. 2 Theorem 9 For any ' and for any k ? 1, the expected cost of algorithm SW k is higher than the expected cost of at least one of the algorithms SW 1 , ST 1 , and ST 2 . Namely, EXP SW k (')g In order to prove this theorem, we need the following three lemmas. Lemma 1 For any k ? 1, if ' - 0:5, then EXP SW k ('). Proof: From equations 11 and 15 we derive decreases when k increases, and From the definition of ff k (see equation 4), we can derive are omitted). From this formula, we see that ff k+2 \Gamma ff k is negative for all k - 1. Hence ff k decreases with k. As a consequence ff for any k ? 1. 2 Lemma 3 For any k ? 1 and any ' ? 0:5, , then EXP SW k , then EXP SW k ('). Proof: EXP SW k equations 11 and 15) ff k namely, Base on the above inequality, it is easy to show that if , then EXP SW k Thus, we have proved the first claim of the lemma. namely, EXP SW k Based on the above inequality and lemma 2, it is easy to show that if ! - , then EXP SW k Proof of theorem 9: If ' - 0:5, then lemma 1 indicates that EXP SW k (')g. The theorem follows. 2 Now, let's consider the average expected cost of the SW k algorithms for k ? 1. Theorem 10 For the SW k algorithm with window size k ? 1, the average expected cost is Z 1EXP SW k From the definition of ff k in section 5.2, equation 7 and equation 15, we can derive the equation 16. The tedious intermediate derivation steps are omitted. 2 Corollary 2 For k ? 1, AV G SW k decreases when k increases, and AV G SW k This corollary is straight forward from equation 16. 2 In theorem 7 we have shown that the average expected cost of the SW 1 algorithm is better (i.e. lower) than that of the static algorithm. In the following corollaries, we analyze when the average expected cost of SW k (for k ? 1) is lower than the average expected cost of SW 1 based on the two formulae 14 and 16. In corollary 3 below, we show that when ! - 0:4, the average expected cost of SW k is always higher than that of SW 1 . Corollary 3 If ! - 0:4, then AV G SW k for any k ? 1. Thus, by theorem 7 and corollary 2, we conclude this corollary. 2 In the next corollary, we study the case where ! ? 0:4. We show that for a given ! ? 0:4, there is some k 0 , such that if k - k 0 , then the average expected cost of SW k is lower than that of SW 1 . The following figure illustrates the results of the corollaries 3 and 4. For example, if only when k - 39, the SW k algorithm has a lower expected cost than that of only when k - 7, the SW k algorithm has a lower expected cost than that of SW 1 . Corollary 4 If ! ? 0:4, then AV G SW k for any k which satisfies manipulation using equations 14 and 16. 2 6.4 Worst case in message model In this section, we study the competitiveness of the algorithms ST 1 , ST 2 , and SW k for k - 1, in the message cost model. The result for SW 1 is stated separately, since it is a special case (see section 4). We conclude that the static algorithms are not competitive as is the case in the connection model. Then we show that SW 1 is more competitive than SW k for k ? 1, and we show that the competitiveness factor of the SW k algorithms deteriorates when k increases, thus performs the best in the worst case. As in the connection model, we can easily derive that the static algorithms are not competitive in the message model. Theorem 11 The algorithm SW 1 is tightly !)-competitive in the message cost model, where !(! 1) is the ratio of control message cost to data message cost. Similarly to the proof of theorem 4, we let N / be the number of reads in / that occur immediately after a write, where / is an arbitrary schedule of requests. It is easy to see that N / is the minimum cost to satisfy all the requests in /. Let r be the division of / into blocks such that the requests in any block are all reads or they are all writes, and successive blocks have different requests. It should be easy to see that the total number of read blocks in / is less than or equal to (N / +1), and a read block costs at most (1+!) since after the first read the mobile computer will keep a copy of the data item. The total cost of reads is bounded by (N Similarly, the total number of write blocks in / is less than or equal to (N / + 1), and a write block costs only ! since the first write in the block will invalidate the copy at the MC. Thus, the total cost of writes in / is bounded by (N To show that the above bound is tight, assume that initially there is a single copy of the data item. Consider a schedule / 0 that starts with a read request, ends with a write request, and in each block there is exactly 1 request. It should be easy to see that COST SW k Theorem 12 The algorithm SW k (for k ? 1) is tightly [(1 +!]-competitive in the message cost model, where !(! 1) is the ratio of control message cost to data message cost. Similarly to the proofs of theorems 4 and 11, we prove that for any schedule / of requests, COST SW k is the number of read requests in / that occur immediately after a write request. We will also exhibit a schedule / 0 for which COST SW k is a constant. Since it can be shown that the cost of an optimal off-line algorithm on / is N / , it follows that SW k is tightly [(1 !]-competitive. As before, we assume throughout the proof that Let / be a schedule consisting of read and write requests. We prove that COST SW k We divide the schedule / into maximal blocks consisting of similar requests. Formally, let r be the division of / into blocks such that the requests in any block are all reads or they are all writes, and successive blocks have different requests. It should be easy to see that the total number of read blocks in /, i.e. blocks that only contain read requests, is less than or equal to (N / 1). Similarly, the total number of write blocks in / is less than or equal to (N / 1). Now, we analyze the cost of read and write requests separately. Consider any read block B i . It should be easy to see that only the first may each cost (1 !). After the first n reads the window will definitely have more reads than writes, and the algorithm will maintain two copies and further reads in the block do not cost any communication. Thus the cost of executing all the reads in B i is bounded by (n Hence the cost of all the reads in / is bounded above by 1). Now consider a . It is easy to see that B j will cost at most (n data message, since after the first n the window will definitely have more writes than reads and the copy at the MC will be deallocated, and this deallocation may cost this block an additional control message. Thus, the cost of a write block is bounded by (n+1+!). As a consequence, the cost of all the writes in / is bounded by (n+1+!) \Delta (N Hence, COST SW k rearranging the terms, we get COST SW k To show that the above bound is tight, assume that initially there is a single copy of the data item. Consider a schedule / 0 that starts with a block of read requests, ends with a block of write requests, and in each block there are exactly k requests. It should be easy to see that Extensions In this section we discuss various extensions to the previous methods. In particular, in the first subsection we show how to modify the static algorithms to make them competitive, and in the second subsection we discuss extensions of the algorithm to optimize the case where multiple data items can be read and written in a single operation. 7.1 Modifications to the Static Methods We have presented two simple static methods that use the one-copy and two-copies schemes. The static methods can be chosen if the value of ' is known in advance. For example, in the connection model, the static method using a single copy at the stationary computer has the best expected cost if ' ? 0:5. Similarly, the static method using the two-copy scheme has the best expected cost when ' - 0:5. However, the static methods do not have a good worst case behavior, i.e. they are not competitive. For example, a static method using a single copy will incur a high cost on a sequence of requests consisting of only reads from the mobile computer. This cost can be arbitrarily large, depending on the length of the sequence. Even though such a sequence is highly improbable, it can occur with nonzero probability. We can overcome this problem by simple modifications to the static methods, that actually make them dynamic. For example, we can modify the one-copy static method as follows. It will normally use the one-copy scheme until m consecutive reads occur; then it changes to the two-copies scheme and uses this scheme until the next write. Then it reverts back to one-copy scheme and repeats this process. We refer to this algorithm as T1m . It can be shown that T1m 1-competitive and that its expected cost is in the connection model. Note that the second term is the additional expected cost over the static method (it can be shown that for each ' ? 0:5 T1m has a lower expected cost than SWm and they are both equally competitive). This is the price of competitiveness. Thus, if we know that ' ? 0:5 then we can choose the T1m algorithm instead of ST 1 , for an appropriate value of m. Similarly, we can modify the ST 2 algorithm to obtain the T2m algorithm that has almost the same expected cost as ST 2 , and is (m 7.2 Multiple Data Items In this paper we have addressed the problem of choosing an allocation method for a single data items. These results can be extended to the case where multiple data items can be read and written in a single operation. We will sketch an algorithm that gives an optimal static allocation method, in the connection model, for multiple data items, when the frequencies of operations on the data items are known in advance. Assume that multiple data items can be remotely read in one connection; similarly for the remote writes. We present the algorithm for the case when we have only two data items x and y. This can be generalized to more than two data items. Also, we discuss how this approach can be extended to the dynamic window based algorithms. Assume that we have two data items x and y. We classify the read operations into three classes. reads of x only, reads of y only, and reads that access both x and y. We assume that these three different reads occur according to independent Poisson distributions with frequencies respectively. We classify the writes similarly and assume that these writes occur with frequencies -w;x ; -w;y and -w; , respectively. It is to be noted that - denote the frequencies of joint reads and writes respectively. Now, we have four possible allocation methods for x and y: ST 1 (both x and y have only one copy), ST 2 ( both x and y have two copies), ST 1;2 has one copy and y has one copy) and ST 2;1 (x has two copies and y has only one). For each of these allocation methods we can obtain the expected cost of a single operation using the above frequencies, and then choose the one with the lowest expected cost. For example, the expected cost for ST 1 is (- r;x and that of ST 1;2 is (- r;x the sum of all the read and write frequencies. The above method can be generalized to any finite set of data items. We need the frequencies of various joint operations on these data items. To use the method given above we need to know the frequencies of various operations in advance. If these frequencies are not known in advance, then we can use the window based approach that dynamically calculates these frequencies. In this case, we need to keep track of the number of operations of different kind (i.e. the joint/exclusive read/write operations for multiple data items) in the window. From these numbers, we can calculate the frequencies of these operations, compute their expected costs (similar to the static methods given in the previous using these frequencies, and choose an appropriate future allocation method. To avoid excessive overhead, this recomputation can be done periodically instead of after each operation. Future work will address the performance analysis of this method. 8 Comparison with Relevant Literature As far as we know this is the first paper to study the communication cost of static and dynamic allocation in distributed systems using both, average case and worst case analysis. There are two bodies of relevant work, each of which is discussed in one of the following two subsections. In the first subsection we compare this paper with database literature on data allocation in distributed systems. In the second subsection we compare this paper to literature on caching and distributed virtual memory. 8.1 Data allocation in distributed systems Data allocation in distributed systems is either static or dynamic. In [35] and [36], we considered dynamic data allocation algorithms, and we analyzed them using the notion of convergence, which is different than the measures used in this paper, namely expected case and worst case. Additionally, the algorithms in those works are different than the ones discussed here. Further- more, in [35] and [36] we did not consider static allocation algorithms, and we did not consider the connection cost model. Other dynamic data allocation algorithms were introduced in [22] and [23]. Both works analyze dynamic data allocation in the worst case only. Actually, the SW 1 algorithm was first analyzed in [23]. However, the model there requires a minimum of two copies in the system, for availability purposes. Thus even for the worst case the results are different. In contrast, in this paper we assume that availability constraints are handled exclusively within the stationary system, independently of the mobile computers. There has also been work addressing dynamic data allocation algorithms in [9]. This work also addresses the worst case only. Additionally, the model there does not allow concurrent requests, and it requires centralized decision making by a processor that is aware of all the requests in the network. In contrast, our algorithms are distributed, and allow concurrent read-write requests. Static allocation was studied in [37, 14]. These works address the following file-allocation problem. They assume that the read-write pattern at each processor is known a priori or it can be estimated, and they find the optimal static allocation scheme. However, works on the file-allocation problem do not compare static and dynamic allocation, and they do not quantify the cost penalty if the read-write pattern deviates from the estimate. Many works on the data replication problem (such as [4, 6, 11, 13, 17, 21, 34]) and on file systems (such as CODA [30, 33]) address solely the availability aspect, namely how to ensure availability of a data item in the presence of failures. In contrast, in this paper we addressed the communication cost issue. The works done by Alonso and Ganguly in [5, 20] are also related to the present paper in the sense that they also address the optimization issue for mobile computers. However, their optimization objective is energy, whereas ours is communication. The work on broadcast disks ([2]) also addresses peformance issues related to push/pull of data in a mobile computing environment. However, this work assumes read-only data items and it does not perform the type of analytical performance evaluation present in this paper. 8.2 Caching and virtual memory In the computer architecture and operating systems literature there are studies of two subjects related to dynamic data allocation, namely caching and distributed virtual memory (see [1, 3, 7, However, there are several important differences between Caching and Distributed Virtual Memory (CDVM) on one hand, and replicated data in distributed systems on the other. There- fore, our results have not been obtained previously. First, many of the CDVM methods do not focus on the communication cost alone, but consider the collection of factors that determine performance; the complexity of the resulting problem dictates that their analysis is either experimental or it uses simulation. In contrast, in this paper we assumed that since in mobile computing wireless communication involves an immediate out-of-pocket expense, the optimization of wireless communication is the sole caching objective; and we performed a thorough analytical cost evaluation. Second, in CDVM the size of the cache is assumed to be limited. Thus, the important issues in CDVM literature are cache utilization, and the page replacement strategy (e.g. LRU, MRU, etc.), namely which page to replace in the cache when the cache is full and a new page has to be brought in. In other words, in contrast to replicated data in distributed systems, which may reside on secondary and even tertiary storage, in CDVM a page may be deleted from the cache as a results of limited storage. One may argue whether or not limited storage is a major issue in distributed databases, however, in this paper we assumed that storage at the mobile computer is abundant. There have been some CDVM methods which consider communication cost as one of the optimization criteria (e.g. TreadMarks [26]). However, they do not use dynamic allocation schemes. 9 Conclusions In this paper we have considered several data allocation algorithms for mobile computers. In particular, we have considered the one-copy and the two-copies allocation schemes. We have investigated static and dynamic allocation methods using the above schemes. In a static method the allocation scheme remains unchanged throughout the execution. In a dynamic method the allocation scheme changes dynamically based on a window consisting of the last k requests; if in the window there are more reads at the mobile computer than writes at the stationary computer, then we use the two-copy scheme, otherwise we use the one-copy scheme. We get different dynamic methods for different values of k. For the dynamic method is simply the classic write-invalidate protocol. We have considered two cost models - the connection model and the message model. In the connection model, the cost is measured in terms of the number of (wireless telephone) connec- tions, where as in the message model the cost is measured in terms of the number of control and data messages. We have considered three different measures- expected cost, average expected cost, and the worst case cost which uses the notion of competitiveness. Roughly speaking, an algorithm A is said to be k-competitive if for every sequence s of read-write requests the cost of A on the sequence s is at most k times the cost of an ideal off-line algorithm on s which knows s in advance. An algorithm A is said to be competitive, if for some k ? 0, A is k-competitive. The expected cost is the standard expected cost per request assuming fixed probabilistic distributions for reads and writes. We believe that an allocation method should be chosen based on the expected cost as well as the worst case cost. Specifically, we think that an allocation method should be chosen to minimize the expected cost, provided that it has some bound on the worst case behavior. Now we explain the difference between the expected cost and the average expected cost. We have assumed that both, reads at the mobile computer and writes at the stationary computer, occur according to independent Poisson distributions with frequencies - r and -w respectively. When the values of - r and -w are known (more specifically, when the value of -r+-w is known), then an allocation method should be chosen based on the expected cost and the competitiveness. However, when ' varies and it is equally likely to have any value between 0 and 1, then an allocation method should be chosen based on the average expected cost (in addition to competitiveness). The average expected cost is the integral of the expected cost over ' from 0 to 1. An allocation method with a lower average expected cost will have a lower average cost per request, in a sequence of requests in which the frequencies of reads and writes vary over time. Furthermore, the average expected cost can also provide an insight and/or a measure for selecting an allocation method in the case when ' is unknown, but it is equally likely to have any value between 0 and 1 4 . In the connection model, if ' is greater than 0:5, i.e., the read frequency is lower than the write frequency, then the static allocation method using only one copy at the stationary computer has 4 If ' is not uniformly distributed, then the average expected cost should be defined as the integral of the expected cost multiplied by the density function for '. the best expected cost. Similarly, if ' is smaller than 0:5, then the static allocation method using one copy at the stationary computer and one at the mobile computer has the best expected cost. When - r and -w change over time (i.e. ' changes over time), then one of the dynamic methods SW k for an appropriate value of k should be chosen. This is due to the fact that the average expected cost of the SW k algorithms is lower than either one of the static methods. The value of the window size k should be chosen to strike a balance between the average expected cost (which decreases as k increases, see equation 6) and competitiveness (the SW k algorithm 1)-competitive, thus competitiveness becomes worse as k increases). For example, for 9 the sliding-window algorithm will have an average expected cost that is within 10% of the optimum, and in the worst case will be at most 10 times worse than the optimum offline algorithm. In the message model, the static allocation methods are still not competitive; and the dynamic allocation methods SW k are again competitive, although with a different competitiveness factor. For a given ', the expected cost of one of the three methods ST 1 , ST 2 and SW 1 is lowest; the particular one depends on the values of ' and ! (the ratio between the control message cost and the data message cost). The lowest expected-cost algorithm as a function of ' and ! is given in figure 1. If ' is unknown or it varies over time, then one of the sliding window methods provides the optimal average expected cost. The particular one depends on the value of !. If ! - 0:4 then the SW 1 algorithm should be chosen as it has the least average expected cost; for other values of !, the higher the value of k the lower the average expected cost of the SW k algorithm (see figure 2). Again, the appropriate value of k should be chosen to strike a balance between average expected cost and competitiveness. The data allocation methods and the results of this paper pertain to applications where the data items accessed by the various mobile computers are mostly disjoint, and the read requests must be satisfied by the most-up-to-date version of the data item. For applications that do not satisfy these assumptions, techniques that use data broadcasting and batching of updates may be appropriate, and our results need to be extended. This is the subject of future work. ACKNOWLEDGEMENT We wish to thank the referees for their insightful comments. --R "An Evaluation of Cache Coherence Solutions in Shared-Bus Multiprocessors" "Balancing Push and Pull for Data Broadcast" "An Evaluation of Directory Schemes for Cache Coherence" "Multidimensional Voting" "Query Optimization for Energy Efficiency in Mobile Enviro- ments" "The Tree Quorum Protocol: An Efficient Approach for Managing Replicated Data" "Adaptive Software Cache Management for Distributed Shared Memory Architectures" shared memory based on type-specific memory coherence" "Competitive Algorithms for Distributed Data Management" "Replication and Mobility" "The Grid Protocol: A High Performance Scheme for Maintaining Replicated Data" "Data Caching Tradeoffs in Client-Server BDMS Architectures" "Distributed concurrency control performance: A study of algorithms, distribution and replication" "A Characterization of Sharing in Parallel Programs and Its Application to Coherency Protocol Evaluation" "Evaluating the Performance of Four Snooping Cache Coherency Protocols" "Achieving Robustness in Distributed Database Systems" "Competitive paging algorithms" "Client data Caching: A foundation for high performance object database systems" "Query Optimization in Mobile Enviroments" "Weighted Voting for Replicated Data" "A Competitive Dynamic Data Replication Algorithm" "Dynamic Allocation in Distributed System and Mobile Com- puters" "Querying in highly mobile distributed environments" "Shared Virtual Memory on Loosely Coupled Multiprocessors" "TreadMarks: Distributed Shared Memory on Standard Workstations and Operating Systems" "Memory Coherence in Shared Virtual memory systems" "Synchronization with Multiprocessor Caches" "Competitive Snoopy Caching" "Disconnected Operation in the Coda File System" "Disk Cache Performance for Distributed Systems" "Competitive algorithms for online problems" "Coda: A Highly Available File System for a Distributed Workstation Environment" "A Majority Consensus Approach to Concurrency Control for Multiple Copy Database" "Distributed Algorithms for Dynamic Replication of Data" "An Algorithm for Dynamic Data Distribution" "The Multicast Policy and Its Relationship to Replicated Data Placement" "Cache Consistency and Concurrency Control in a Client/Server DBMS Architecture" --TR --CTR Sandeep K. 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Srimani, Data management in wireless mobile environments, Handbook of wireless networks and mobile computing, John Wiley & Sons, Inc., New York, NY, 2002 Wang , Ing-Ray Chen , Chih-Ping Chu , I-Ling Yen, Replicated Object Management with Periodic Maintenance in Mobile Wireless Systems, Wireless Personal Communications: An International Journal, v.28 n.1, p.17-33, January 2004 Ycel Saygin , zgr Ulusoy, Exploiting Data Mining Techniques for Broadcasting Data in Mobile Computing Environments, IEEE Transactions on Knowledge and Data Engineering, v.14 n.6, p.1387-1399, November 2002 James Jayaputera , David Taniar, Data retrieval for location-dependent queries in a multi-cell wireless environment, Mobile Information Systems, v.1 n.2, p.91-108, April 2005 Isabel Cruz , Ashfaq Khokhar , Bing Liu , Prasad Sistla , Ouri Wolfson , Clement Yu, Research activities in database management and information retrieval at University of Illinois at Chicago, ACM SIGMOD Record, v.31 n.3, September 2002 W. J. Lin , B. Veeravalli, A dynamic object allocation and replication algorithm for distributed systems with centralized control, International Journal of Computers and Applications, v.28 n.1, p.26-34, January 2006 Bharadwaj Veeravalli, Network Caching Strategies for a Shared Data Distribution for a Predefined Service Demand Sequence, IEEE Transactions on Knowledge and Data Engineering, v.15 n.6, p.1487-1497, November Bharadwaj Veeravalli, Network Caching Strategies for a Shared Data Distribution for a Predefined Service Demand Sequence, IEEE Transactions on Knowledge and Data Engineering, v.15 n.6, p.1487-1497, November Anurag Kahol , Sumit Khurana , Sandeep K.S. Gupta , Pradip K. Srimani, A Strategy to Manage Cache Consistency in a Disconnected Distributed Environment, IEEE Transactions on Parallel and Distributed Systems, v.12 n.7, p.686-700, July 2001 Xiao-Hui Lin , Yu-Kwong Kwok , Vincent K. N. 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probabilistic analysis;mobile computing;communication cost;wireless communication;dynamic data allocation;caching

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Concurrency Control and View Notification Algorithms for Collaborative Replicated Objects.

AbstractThis paper describes algorithms for implementing a high-level programming model for synchronous distributed groupware applications. In this model, several application data objects may be atomically updated, and these objects automatically maintain consistency with their replicas using an optimistic algorithm. Changes to these objects may be optimistically or pessimistically observed by view objects by taking consistent snapshots. The algorithms for both update propagation and view notification are based upon optimistic guess propagation principles adapted for fast commit by using primary copy replication techniques. The main contribution of the paper is the synthesis of these two algorithmic techniquesguess propagation and primary copy replicationfor implementing a framework that is easy to program to and is well suited for the needs of groupware applications.

Introduction Synchronous distributed groupware applications are finding larger audiences and increased interest with the popularity of the World Wide Web. Major browsers include loosely integrated groupware applications like chat and whiteboards. With browser functionality extensible through programmability (Java applets, plug-ins, ActiveX), additional groupware applications can be easily introduced to a large community of potential users. These applications may vary from simple collaborative form filling to collaborative visualization applications to group navigation tools. Synchronous collaborative applications can be built using either a non-replicated application architecture or a replicated application architecture. In a non-replicated architecture, only one instance of the application executes, and GUI events are multicast to all the clients, via systems such as shared X servers [1]. In a replicated architecture, each user runs an appli- cation; the applications are usually identical, and the state or the GUI is "shared" by synchronously mirroring changes to the state of one copy to each of the others [6, 13]. Research Center, Hawthorne, NY 10532, USA. E-mail: fstrom, banavar, klm, mjwg@watson.ibm.com 2 Department of Electrical Engineering and Computer Sci- ence, University of Michigan, Ann Arbor, MI, USA. E-mail: aprakash@umich.edu In this paper, we assume that replicated architectures are used because they generally have the potential to provide better interactive responsiveness and fault-tolerance, as users join and leave collaborative sessions. However, the domain of synchronous collaborative applications is broader than those supported by a fully replicated application architecture. For example ffl the applications may have different GUIs and even different functionality, sharing only the replicated state, ffl the shared state may not be the entire application state, and ffl an application may engage in several independent collaborations, e.g., one with a financial planner, another with an accountant, and each collaboration may involve replication of a different subset of the application state. In order to support the development of such a large variety of applications, it is clearly beneficial to build a general application development framework. We have identified the following requirements for such a frame-work From the end-user's perspective, collaborative applications built using the framework must be highly responsive. That is, the GUI must be as responsive as a single user GUI at sites that initiate updates, and the response latency at remote sites must be minimal. Second, collaborative applications must provide sufficient awareness of ongoing collaborations. From the perspective of the developer of collaborative applications, the framework must be application-independent and high-level. That is, it must be capable of expressing a wide variety of collaborative appli- cations. Second, the developer should not be required to be proficient in distributed communication proto- cols, thread synchronization, contention, and other complexities of concurrent distributed programming. We have implemented a framework called Decaf (Distributed, Extensible Collaborative Application Framework) that meets the above requirements. Our framework extends the well-known Model-View- Controller paradigm of object-based application development [10]. In the MVC paradigm, used in GUI- oriented systems such as Smalltalk and InterViews [12], view objects can be attached to model objects in order to track changes to model objects. Views are typically GUI components (e.g., a graph or a win- dow) that display the state of their attached model objects, which contain the actual application data. Controllers, which are typically event handlers, receive input events from GUI components and, in response, invoke operations to read and write the state of model objects. Updated model objects then notify their attached views of the change, so that the view may re-compute itself based on the new values. The MVC paradigm has several beneficial properties such as (1) modular separation of application state components from presentation components and (2) the ability to incrementally track dependencies between such components To support groupware applications, Decaf extends the MVC paradigm as indicated in Figure 1. First, the framework supplies generic collaborative model objects, such as Integers, Strings, Vectors, etc. to application developers. These model objects can have replica relations with model objects across applica- tions, so that replicated groupware applications can be easily built. Second, it provides atomicity guarantees to updates on model objects, even if multiple objects are modified as part of an update. The framework automatically and atomically propagates all updates to replicas of model objects and their attached views. Third, writers can choose whether views see updates to model objects as they occur (optimistic) or only after commit (pessimistic). Fourth, applications can dynamically establish collaborations between selected model objects at the local site and model objects in remote applications. Finally, users may also code authorization monitors to restrict access to sensitive objects In this paper, we first introduce the basic concepts of the Decaf framework (Section 2). Next, we describe the distributed algorithms that implement consistent update propagation (Section 3) and view notification (Section 4). Then, we present comparison with related work, our experience with using Decaf (Section 5) and finally, some concluding remarks (Sec- tion 6). 2 The DECAF Framework As mentioned earlier, Decaf extends the Model- View-Controller paradigm [10]. Decaf model object classes are supplied by the framework; the application programmer simply instantiates them (model objects Replica update notifications updates reads/ A Replica Model Objects Transaction View Collaborative Application A2 reads Application Framework Figure 1: Typical structure of Decaf applications. are shown below the horizontal line that separates the framework from the application in Figure 1). The application programmer writes views and controllers, which initiate transactions (these are shown above the horizontal line in Figure 1). In the following subsec- tions, we describe the key concepts in the framework and the atomicity guarantees on access to model objects provided by the Decaf infrastructure. 2.1 Model objects Model objects hold application state. All model objects allow reading, writing, and attaching views. There are three kinds of model objects: (1) Scalar model objects, which currently are of types integer, real, and string; (2) Composite model objects, which support operations to embed and to remove other model objects called children, and which may be either lists (linearly indexed sequences of children) or tuples (collections of children indexed by a Association model objects, which are used to track membership in collaborations. Model objects can join and leave replica relationships with other model objects. The value of an association object is a set of replica relationships which are bundled together for some application purpose. For each replica relationship, the association object contains the set of model objects which have joined, together with their sites and object descriptions. The operations on association objects relevant to this paper are join and leave, by which a model object joins or leaves a particular replica relationship as described in Section 2.7. 2.2 Replica Relationships A replica relationship is a collection of model ob- jects, usually spanning multiple applications, which are required to mirror one another's value. Replica relationships are symmetric and transitive. 2.3 Controllers A controller is an object which responds to end-user initiated actions, such as typing on the keyboard, clicking or dragging the mouse, etc. A controller may initiate transactions to update collaborative model ob- jects. A controller may also perform other external interactions with the end user. 2.4 Transactions Transactions on model objects are executed by invoking an execute method on a transaction object. Application programmers may define transaction objects, with their associated execute method, for actions that need to execute atomically with respect to updates from other users. The execute method may contain arbitrary code to read and write model objects within the application. Any changes to model objects will be automatically propagated to their replicas. The execution of a transaction is an atomic ac- tion. That is, it behaves as if all its operations - those of the execute method and those which propagate changed values to replicas - take place at a single instant of time, totally ordered with respect to the times of all the other atomic actions in the system. Atomicity is implemented optimistically in Decaf. Transactions may abort, e.g., if two transactions originated at different sites and each transaction guessed that it read and updated a certain value of an object before the other transaction did, then one of the transactions will abort. Aborted transactions are re-executed at the originating site. The effects of aborted transactions will be invisible to pessimistic views, and automatically undone as seen by optimistic views. 2.5 View Objects A view object is a user-defined object which can be dynamically attached to one or more model objects. When a view is attached to a model object, that view object will be able to track changes to the model object by receiving update notifications, as calls to its update method. If a view object is attached to a composite model object, it will receive notifications for changes to the composite as well as any of its children. The purpose of a view object is to compute some function, e.g., a graphical rendering, of some or all of the model objects it is attached to. When the view object receives an update notification, its update method may take a state snapshot by reading any of the model objects that it is attached to. State snapshots are guaranteed by the infrastructure to be atomic actions - behaving as if they are instantaneous. Besides taking a state snapshot, the update method may initiate new transactions and perform arbitrary external in- teractions, such as rendering on the display, printing, playing audio data, etc. Each update notification contains a list of all ob- jects, and only such objects, that have changed value since the last notification. Objects not on this list may be assumed not to have changed value. This information allows view objects to recompute its function more efficiently when only a part of a large value has changed. For example, when large composite objects are changed, the update notification will not only specify which composite object has changed, but also which parts have changed, to allow incremental recalculation 2.6 Optimistic and Pessimistic Views View objects can be either optimistic or pessimistic. Optimistic and pessimistic views differ in the protocols for delivery of update notifications. An optimistic view will receive an update notification as soon as possible after any of its attached model objects has changed. However, the state snap-shot may be inaccurate or inconsistent because messages have arrived out of order or because transactions will abort. If an optimistic view ever takes an incorrect snapshot, the infrastructure will eventually execute a superseding update notification. Therefore, so long as the system eventually quiesces, the final snapshot taken before the system quiesces will be correct. An optimistic view will receive a commit notification (as a call to its commit method) whenever its most recent update notification is known to be correct. Committed state snapshots are always correct and always occur in monotonic order. An optimistic view therefore trades off accuracy and the risk of wasted work in exchange for responsiveness. Pessimistic views receive update notifications only when transactions updating attached model objects have committed and when the view will be able to see consistent values. The system makes two guarantees to a pessimistic view: (1) never to show any uncommitted or inconsistent values, and (2) to show all committed values in monotonic order of applied updates. 2.7 Collaboration Establishment Users may create replica relationships and cause model objects to join or leave replica relationships dy- namically. In order for an object A at one site to join a replica relationship involving an object B at another site, the following steps must occur: ffl B's owner must create an association object BAs- soc containing at least one replica relationship joined by B. ffl B's owner must then publicize the right to collaborate with B by creating an external token called an invitation including a reference to BAssoc and export it somewhere where A's owner can access it (e.g., on a bulletin board). ffl A's owner must then import this invitation and use it to instantiate his own association object AAssoc. Object AAssoc must then be authorized to reveal BAssoc's replica relationships. ffl A's owner can then read AAssoc, discover the existence of a replica relationship involving B that it wishes to join, and issue a join of A to that relationship. Since association objects are also model objects, and can have views attached to them, changes in membership in associations are signalled in exactly the same way as changes in values of data objects. 3 Concurrency control This section describes the optimistic concurrency control algorithms for propagating updates among model objects in replica relationships. Each transaction is started at some originating site, where it is assigned a unique virtual time to execution. The V T is computed as a Lamport time [11], including a site identifier to guarantee uniqueness When a transaction is initiated, a transaction implementation object is created at the originating site. When updates are propagated to remote replicas, transaction implementation objects are created at those sites. Each transaction implementation object at a site contains: the V T of the transaction, references to all model objects updated by the transaction at that site, and additional state information. Each model object holds: ffl Value History: The value history is a set of pairs of values and V T 's, sorted by V T . The value with the latest V T is called the current value. ffl Replication Graph History: This is a similarly indexed set of replication graphs. A replication graph is a connected multigraph whose nodes are references to model objects, and whose multi- edges are the replication relations built by the users. It includes the current model object, and all other model objects which are directly or indirectly required to be replicas of the current model object as a result of replication relations. Since replication graphs change very infrequently, in practice this history will most frequently contain only a single graph. Histories are garbage-collected as transactions com- mit. Committal makes old values no longer needed for view snapshots or for rollback after abort, thus they are discarded. There is a function which maps replication graphs to a selected node in that graph. The node is called the primary copy and the site of that node is called the primary site, adapting a replication technique used by Chu and Hellerstein [4] and others. The correctness of the concurrency control algorithm is based upon the fact that the order of all reads and updates of replicas is guaranteed to match the order of the corresponding reads and updates at the primary copy. Whenever the originating site of a transaction is not the primary site of some of the objects read or written by a transac- tion, the transaction executes optimistically, guessing that the reads and updates performed at the originating site will execute the same way when re-executed at the primary sites. If these guesses are eventually confirmed, the transaction is said to commit. If not, the effects of the transaction are undone at all sites and the transaction is retried at the originating site. 3.1 Concurrency Control for Scalar Model Objects When a transaction T is first executed, it is assigned a V T which we call t T . As it executes, the transaction reads and/or modifies one or more model objects at the originating site. Each model object records each operation in the transaction implementation object. For each model object M read, the transaction implementation object records the read R , where t M R is defined as the V T when the current value was written. For "blind writes" (writes into objects not read by the transaction), t M R is defined as equal to t T . The transaction object additionally records the graph time t M G , defined as the V T that 's replication graph was last changed. Consider a transaction T given in Figure 2 that is originated at some site. T is assigned a V T t The current values of W, X, Y, and Z at t T are 4, written at V T 80, 3, written at V T 70, and Assume all replication graphs were initialized at V T (this is not shown in the figure). At the end of the transaction execution, the transaction implementation object records the following: - Read object W, t W - Read object X, t X Y := X; then Z := Z Transaction T Figure 2: Example of transaction execution. Update object Y, t Y Update object Z, t Z Observe that the update to Y is a "blind write," since Y was not read in this transaction; hence t Y The transaction implementation object next distributes the modifications to all replicas of the above model objects. The transaction requests each primary copy to "reserve" a region of time between t M R and t T as write-free. Since replica graphs can also change (al- beit slowly), the transaction must also reserve a region of time between t M G and t T as free of graph updates. As mentioned earlier, the originating site of the transaction has executed optimistically, guessing that its reads and updates will be conflict-free at each primary copy. Specifically, the validity of the transaction depends upon the following types of guesses: ffl "Read committed" (RC) guesses: That each model object value (or graph) read by the trans-action was written by a committed transaction. ffl "Read latest" (RL) guesses: That for each value (or graph) of model object M read by trans-action T , no write of M by another transaction occurred (or will occur) at the primary copy between R (or t M G ) and the transaction's t T . This guess implies that the primary site would have read the same version of the object had the trans-action executed pessimistically. ffl "No conflict" (NC) guesses: That for each model object value (or graph) written by the transaction, no other transaction reserved at the primary copy a write-free region of time containing the transaction's V T . This guess implies that the primary site would not invalidate previous reads by confirming this write. CONFIRM Y Z CONFIRM-READ COMMIT COMMIT CONFIRM Figure 3: Example of update propagation. The execution of the transaction takes place op- timistically, using strategies derived from the optimistic guess propagation principles defined in Strom and Yemini [15], and applied to a number of distributed systems (e.g. optimistic call streaming [2] and HOPE [5]). However, our algorithm makes certain specializations to reduce message traffic. For RC guesses, the originating site simply records the V T of the transaction which wrote the uncommitted value that was read. The originating site will not commit its transaction until the transaction at the recorded V T commits. For each uncommitted trans-action T at a site, a list of other transactions at that site which have guessed that T will commit is maintained The RL and NC guesses are all checked at the site of the primary copy of an object M . The RL guess checks that no value (or graph) update has occurred R (or t M G ) and t T , and if this check succeeds, creates a write-free reservation for this interval so that no conflicting write will be made in the future; the NC guess checks that no write-free reservation has been made for an interval including t T . For each object M read but not written, a message is sent to the primary copy (if it is at a remote site). This message contains t M G , and t T . Each primary copy object then verifies the RL guesses for values and graphs. A confirmation message is then is- sued, confirming or denying the guess. In the general approach, or in Hope, this confirmation message would be broadcast to all sites. But in the Decaf implementation, this confirmation is sent only to the originating site. It is a property of our Decaf implementation that the originating site always knows the totality of sites affected by its transaction by com- mit/abort time. Therefore, the originating site is in a position to wait for all confirmations to arrive and then to forward a summary commit or abort of the transaction as a whole to all other sites. This avoids the need for each primary copy to communicate with all non-primary sites, and it avoids the needs for non-primary remote sites to be aware of guesses other than the summary guess that "the transaction at virtual For each object M modified by T we send a message to all relevant sites, containing t M the new value. However while all sites other than the primary site simply apply the update at the appropriate primary site additionally performs the RL and NC guess checks and then sends a confirmation message to the originating site. The originating site waits for confirmations of guesses from remote primary sites. If all guesses for a transaction are confirmed, the originating site commits the transaction and sends a commitmessage to all remote sites which received update messages. If any guess is denied, the originating site aborts the trans-action and sends an abort message to all remote sites. The originating site then re-executes the transaction. If a site detects that a transaction at V T has com- mitted, the modified model objects at that site are in- formed. This notification can be used to schedule view notifications and eventually to garbage-collect histo- ries. The site retains the fact that the transaction has committed so that if any future update messages arrive, the updates are considered committed. If a transaction is aborted, the modified model objects are informed so that the value at V T can be purged from the history. The site retains the fact that the transaction has aborted so that if any future up-date messages arrive, the updates are ignored. Let us examine how these algorithms would apply in our example, shown in Figure 3. Suppose there are four sites, and that W and X are replicated at sites 1, 2, and 3, while Y and Z are replicated at sites 2, 3, and 4. Suppose that T is initiated at site 2. Suppose further that the primary site of W and X is 1, and of Y and Z is 4. Ignoring graph times and graph updates for now, and assuming that the three current values read by the transaction were committed (hence there are no RC guesses), the following messages are sent from site 2 after the transaction is applied locally (we perform the obvious optimization of sending messages only to relevant sites): 1. To site 1: CONFIRM-READ t 2. To sites 3 and 4: checks that W is write-free for the V T range from 80 to 100 (RL guess check) and that X is write- free for the V T range from 60 to 100 (RL guess check). If so, it reserves those times as write-free and sends a CONFIRM to site 2. simply applies the updates to its Y and Z replica objects. Site 4 checks that Z is write-free for the V T range from 40 to 100 (RL guess check). If so, it reserves those times as write-free. It also checks that writing Y or Z at V T 100 does not conflict with any previously made read reservations (NC guess checks). If all the above checks succeed, it applies the updates and sends a CONFIRM to site 2. responses. If both are confirma- tions, it sends COMMIT 100 to all other sites involved. 3.2 Concurrency Control for Composite Model Objects Although the concurrency control algorithm is the same for composite objects as for scalar objects, it is desirable to save space by not keeping a separate replication graph for each object inside a composite. That is, if composite A is a replica of composite A 0 and A 00 (see Figure 4), we wish to avoid encoding inside object A[103] that it is a replica of objects A 0 [103] and A 00 [103]. Our approach is that by default, an object embedded within a composite inherits the replication graph of its root; e.g. A[103]'s replicas would be at the same sites as A's replicas, at the corresponding index (103). Similarly, if A[103] is itself a composite object, its embedded objects, e.g. A[103]["John"][12] would be replicated at the same sites.JohnA' JohnA" Figure 4: Replicas of composite model objects. The set of indices between the root and a given object is called its path; when an object such as A[103]["John"][12] is modified, the change and the path to A are then sent by A to its replicas A 0 and A 00 , which then use the same path name, [103]["John"][12], to propagate the update to their corresponding components A 0 [103]["John"][12] and A 00 [103]["John"][12]. We call this technique indirect propagation of updates, in contrast to the direct propagation technique discussed earlier, in which each object holds its own replication graph and communicates directly to its replicas. In addition to saving space, indirect replication avoids the problem in direct replication that small changes to the embedding structure could end up changing a large number of objects. For example, if indirect replication were not used, adding a new replica A 000 to the set fA; A 0 ; A 00 g would entail updating the replication graph for every object embedded within A and its replicas. Similarly, removing A[103] from A would entail updating the replication graph for every object embedded within A[103]. 3.2.1 Adjustments to support indirect propa- gation There are two adjustments which have to be made to ensure the correctness of the concurrency control algorithm in the presense of indirect propagation. The first has to do with the relative order of list items. A transaction at V T 100 may modify having seen that an earlier transaction at V T 80 deleted A[5] so that what the originating site thinks of as A[103] may appear to some other sites to be A[102]. This is not a concurrency control conflict - it is simply a consequence of the fact that path names like [103]["John"][12] are fragile. To overcome this, rather than using the actual list index in a path name, the propagation algorithm uses the transaction V T as a unique identifier - e.g. if A[103] was embedded in A at V T 40, then 40 is used as an index. If several embeds were performed at V T 40, they are distinguished by a "subtime", e.g. 40.1 or 40.8. A composite object receiving such an indirect propagation message can always propagate it down the tree regardless of the order in which it has received other structure-changing operations. During such a propagation, if it is determined (using the transaction that an earlier path changing update has not yet been received, the propagation will block until the earlier update is received. The second adjustment has to do with guesses associated with the paths for indirect object propagation. The updated model objects must make RC guesses to ensure that transactions that created their paths have committed and RL guesses that no straggling transactions have removed any component of their paths. 3.2.2 When indirect propagation is not possi- ble Indirect propagation is the default mode of propagating value updates to objects within composites. However indirect propagation is not always possible. Consider the configuration in Figure 5. In this case, node C can indirectly propagate changes to C 0 , but node B cannot because it has a different set of replicas than the rest of the tree. We therefore use direct replication for objects B, B 0 , and B 00 . A' A B" Figure 5: Indirect propagation not possible for this case. 3.3 Dynamic Collaboration Establish- ment The set of replica relations between objects remains relatively static. Most transactions change the values of objects rather than the replication graphs. But replication graphs do change, as users join and leave collaborations. Direct propagation graphs for embedded objects inside composites can also change as a result of deleting objects from composites and embedding them elsewhere. Dynamic collaboration establishment transactions need not be especially fast, but they must work correctly in conjunction with all the other transactions. We have already seen some of the effects of dynamic collaboration establishment in the algorithms described above. Replication multi-graphs are time-stamped with the transaction V T which changed them. There is no "negotiation" for primary copy; each node is able to map a given multi-graph to the identity of the primary site for that configuration. A primary copy always confirms the RL guess that the graph hasn't changed as well as confirming whatever else it is being asked to check: this guards against the possibility that the originating site is propagating to the wrong set of sites or that it is asking the wrong primary copy because of a graph change that it hasn't seen yet. A primary copy always reserves the graph against changes during a region of time that a previously confirmed transaction has assumed to be change-free. 4 View Notification This section describes the algorithms for implementing the view notification semantics given in Section 2.5. When a transaction implementation object completes executing at a site, the Decaf infrastructure initiates view notifications to be sent to all the view objects attached to the model objects updated in the transaction. View object attachments are always lo- cal, i.e., views are always attached to model objects at the same site. Thus, a view notification is simply a method call to the update method implemented by the view object. The update method can contain arbitrary code that takes a state snapshot by reading the view's attached model objects and recomputes its dis- play. The infrastructure guarantees that such a state snapshot is implicitly a consistent atomic action. For every view notification initiated, a snapshot object is created internal to the Decaf infrastructure. All the snapshots associated with a particular user level view object are managed internally by a view proxy object. Each model object contains the set of view proxies corresponding to its views, which it notifies upon receiving an update or a commit. Since a snapshot is an atomic action, it is assigned a virtual time t S . Each snapshot is assumed to read all the model objects attached to the view at V T t S . These reads may be optimistic; hence, as described in Section 3.1, their validity depends upon the read values being the latest (RL guess) and the read values being committed (RC guess). Confirming RL guesses involves remote communication with the primary copies of the objects read in the snapshot. If the RL and RC are confirmed, the snapshot is said to commit. Optimistic and pessimistic views differ in two re- spects. First, they differ in the time at which view notifications are scheduled. Optimistic notifications are scheduled as early as possible, i.e., as soon as a model is updated and a snapshot thus initiated, whereas pessimistic notifications are scheduled after it is known that the snapshot is valid, i.e., that the view will read consistent committed values. Second, they differ in the lossiness of notifications. Pessimistic views are notified losslessly of every single update in monotonic order of updates whereas optimistic views are notified only of the latest update. Subsections 4.1 and 4.2 describe these behaviors in more detail. As described in Section 2.5, view notifications are incremental, i.e., each notification provides only that part of the attached model object state that has changed since the last notification. However, for the sake of simplicity, the algorithms presented in this section do not incorporate incrementality; each snapshot is assumed to read the set of attached model objects in its entirety. Furthermore, notifications may be bundled to enhance performance, i.e., a single view notification may be delivered for multiple model objects that were updated in a single transaction. 4.1 Optimistic Views Figure 6 shows an optimistic view V attached to model objects A and B. The view proxy object V P represents V internally. A and B have committed current values (i.e., values with the latest V T ) at V T 's 100 and 80 repectively. A transaction T runs at V T 110 and updates A, which notifies its view proxy V P . The primary requirement of optimistic views is fast response. Consequently, as soon as V P is notified, it performs the following: 1. It creates a snapshot object and assigns it a V T equal to the greatest of the V T 's of the current values of all attached model objects. In this case, 2. It schedules a view notification, i.e., calls the view's update method. A Figure View notification. At the end of the snapshot, the snapshot object records that all attached model objects were read at t S . In order for the snapshot in this example to com- mit, two guesses must be confirmed (as before, we ignore guesses related to the graph): 1. An RC guess that the update by transaction T at T 110 has committed. This requires receiving a COMMIT message from the site that originated transaction T . 2. An RL guess that the V T interval from 80 to 110 is update free for B. This requires sending a CONFIRM-READ message to B's primary copy and waiting for the response. Eventually, if these guesses are confirmed, then the snapshot commits, and a commit notification is sent to V , i.e., its commit method is called. If, on the other hand, an RC guess turns out to be false, the view proxy re-runs the snapshot with a new t S . In the example of Figure 6, if the RC guess was denied as a result of transaction T at V T 110 aborting, a new snapshot is run. This snapshot will have since that is now the greatest V T of the current values of all attached model objects. Notice from this example that optimistic view notifications are not necessarily in monotonic V T order. In the case that an RL guess is denied by the primary copy, that means that the requested interval is not update free, and thus a straggler update is yet to arrive at the guessing site. In this case, the straggler itself will eventually arrive and cause a rerun of the view notification. In the example in Figure 6, if the RL guess was denied as a result of a straggler update to B at V T 105, the update at V T 105 will trigger a new view notification at t This algorithm implements the liveness rule for optimistic views that an update notification is followed either by a commit notification or, in the case of an invalid optimistic guess or a subsequent update, a new update notification. An optimistic view proxy maintains at most one uncommitted snapshot - the one with the latest t S - at any given time. If a new update arrives before the current snapshot has committed, then we're obliged to notify the new update to the view due to the responsiveness requirement. The system may as well discard the old snapshot since there is no way to notify the view of its commit (as we don't expose V T 's to views). As a result, an optimistic view gets a commit notification only when the system quiesces, that is, when no new updates are initiated in the system before existing updates are committed. 4.2 Pessimistic Views Recall that the system makes two guarantees to a pessimistic view: (1) never to show any uncommitted values, and (2) to show all committed values in monotonic order of applied updates. A pessimistic view proxy initiates a snapshot at every V T that one or more of its attached model objects receive a committed update. However, it does- n't schedule a view notification for the snapshot until the snapshot commits. Snapshot committal depends on (1) the validity of model object reads at the snap- shot's t S , and (2) whether its preceding snapshots have already committed (this is due to the monotonicity requirement). When one or more snapshots commit, the view is notified, once for each committed snapshot, in V T sequence. Thus, unlike an optimistic proxy, a pessimistic proxy manages several uncommitted snap-shots A pessimistic view proxy thus contains a list of snapshot objects sorted by V T . It also contains a field lastNotifiedVT which is the V T of the last up-date notification. To illustrate pessimistic view notification, let us say that the view V in the example of Figure 6 is a pessimistic view. Suppose that lastNotifiedVT = 80. Suppose further that the snapshot at V T 100 is as yet uncommitted and thus A's committed update at V T 100 is not yet notified. When the transaction at V T 110 commits, it informs the model object A, which in turn informs the pessimistic view proxy V P . V P creates a snapshot object, assigns it a t records the following guesses: 1. A "snapshot committed" guess (SC guess) that the preceding snapshot at V T 100 will commit. This stems from the monotonicity requirement. 2. An RL guess that the V T interval from 100 to committed updates for A. This also stems from the monotonicity requirement. This requires sending a CONFIRM-READ message to A's primary copy and waiting for a response. 3. An RL guess that the V T interval from 100 to 110 is free of committed updates for B. This requires a CONFIRM-READ message as above. Eventually, if all the guesses made by a particular snapshot object are confirmed, it commits, and it can confirm the SC guess of its successor snapshot. When any snapshot commits, all contiguous committed snapshots after lastNotifiedVT are notified, and lastNotifiedVT is updated. A straggling committed update, say at for B in the example, may cause an RL guess to be negated. In this case, when the straggling committed update is notified to the proxy, a new snapshot is created at V T 105 as given above. Additionally, the RL guess made by the succeeding snapshot at V T 110 (guess (3) above) is revised to be for the V T interval from 105 to 110 for B. This algorithm implements the consistency and monotonicity requirements for pessimistic views. 5.1 Related Work The Decaf framework is designed for collaborative work among a small and possibly widely distributed collection of users. Consistency, responsiveness, and ease of programming are important objectives. ISIS [3] provides programming primitives for consistent replication, although its implementation strategies are pessimistic. Interactive groupware systems have different performance requirements and usage characteristics from databases, leading to different choices for concurrency control algorithms. First, almost all databases use pessimistic concurrency control because it gives much better throughput, a major goal of databases. In interactive groupware systems, on the other hand, pessimistic concurrency control strategies are not always suitable because of impact on response times to user actions - ensuring interactive response time is often more important than throughput. Second, possibilities of conflicts among transactions is lower in groupware systems because people typically use social protocols to avoid most of the conflicts in parallel work. Optimistic protocols based on Jefferson's Time Warp [8] were originally designed for distributed simulation environments. They have been successfully applied in other application areas as well[7]. However, one important characteristic of distributed simulation is that there is usually an urgency to compute the final result, but not necessarily to commit the intermediate steps. In these protocols, the primary purpose of "committing" is to free up space in the logs, not to make the system state accessible to view. But in a co-operative work environment such as ours, fast commit is essential. The delay associated with waiting for at most a single primary site per model object in Decaf is typically considerably less than a Time Warp style global sweep of the system would be. The ORESTE [9] implementation provides a useful model in which programmers define high-level operations and specify their commutativity and masking re- lations. One drawback is that there are no high-level operations on multiple objects, nor are there ways of combining multiple high-level operations into transac- tions. To get the effect of transactions, one must combine what are normally thought of as multiple objects into single objects and then define new single-operand operations whose effects are equivalent to the effects of the transaction. One must then explicitly specify the interactions between the new operations and all the other operations. There is also a subtle difference between the correctness requirements in Decaf and in ORESTE. This difference results from the fact that ORESTE only considers quiescent state - the analysis does not consider "read transactions" (e.g., snapshots) which can coexist with "update transactions". For instance, in the ORESTE model, a transaction which changes an object's color can reasonably be said to commute with a transaction which moves an object from container A to container B, since for example, starting with a red object at A and applying both "change to blue" and "move to B" yields a blue object at B regardless of the order in which the operations are ap- plied. But once viewers or read-only transactions or system state in non-quiescent conditions is taken into account, some sites might see a transition in which a blue object was at A and others a transition in which a red object was at B. Finally, in ORESTE a state cannot be committed to an external viewer until it is known that there is no straggler; this involves a global sweep analogous to Jefferson's Global Virtual Time algorithm. In a system with a single group of collaborating applications, this may not be too severe a problem. In a world-wide web in which sites A, B, and C are collaborating, and independently sites C, D, and E are collaborating, and and F are collaborating, etc., it is preferable not to have commit depend on the global state of the net- work, but rather on a small number of objects. A recent system, COAST [14], also attempts to use optimistic execution of transactions with the MVC paradigm for supporting groupware applications. Key differences with our system are the following. First, COAST only supports optimistic views. Second, concurrency algorithms used in COAST assume that all model objects in the application are shared among all participants. Furthermore, the optimistic algorithm implemented in COAST is based on a variation of the algorithm discussed above. 5.2 Status and Experience A substantial implementation of the Decaf frame-work has been completed in the Java programming language. The framework currently supports scalar model objects, transactions, and optimistic and pessimistic views. The implementations of these objects use the algorithms described in this paper. Several optimizations are forthcoming, including commit del- egation, faster commit of snapshots, and incremental propagation. We are currently implementing composite model objects. Several collaborative applications have been successfully built using the current prototype implemen- tation. These include several groupware applications that allow an insurance agent to help clients understand insurance products via data visualization and fill out insurance forms, a multi-user chat program, and simple games. Our preliminary experience is that it is easy to write new applications or to modify existing programs to use our MVC programming para- digm. Optimistic views have been very useful due to their fast response, and also due to the low conflict rate in typical use. Pessimistic views have also been useful for viewers that want to track all changes to the values of model object. 6 Conclusions The Decaf framework's major objectives are ease of programming, and responsiveness in the context of systems of collaborating applications. The ease of programming is achieved primarily through hiding all concerns about distribution, multi- threading, and concurrency from users. Programmers write at a high level, using the Model-View-Controller paradigm, and our implementation transparently converts operations on model objects to operations on distributed replicated model objects. The View Notification algorithm automatically schedules view snap-shots at appropriate times and also allows viewers to respond efficiently to small changes to large ob- jects. Model objects for standard data types (Inte- gers, Strings, etc.) and collections (e.g., Vectors) are provided as part of the Decaf infrastructure. The responsiveness results from the use of optimism combined with the fast commit protocol of the primary copy algorithm. If a transaction updates objects A and B, then a viewer of B 0 , a replica of B, sees the commit as soon as A's primary site and B's primary site have each notified the originating site that the updates are non-conflicting, and the originating site has notified B 0 's site that the transaction has com- mitted. This is a small delay, even for a pessimistic view. Users can get even more rapid response time using optimistic views, and most of the time their optimistic view will later be committed with the same speed as the pessimistic view. Our experience with using Decaf has shown the architecture and algorithms to be well suited for a variety of groupware applications. Acknowledgements We gratefully acknowledge Gary Anderson's input to the design of our framework. He has also built several collaborative applications and components on top of our framework. --R XTV: A frame-work for sharing X window clients in remote synchronous collaboration Optimistic parallelization of communicating sequential processes. Pat Stephen- son The exclusive- writer approach to updating replicated files in distributed processing systems Concurrency control in groupware systems. The time warp mechanism for database concurrency control. Virtual time. An algorithm for distributed groupware applications. A Cookbook for Using the Model-View-Controller User Interface Paradigm in Smalltalk-80 Support for building efficient collaborative applications using replicated objects. Jan Schum- mer Synthesizing distributed and parallel programs through optimistic transformations. --TR --CTR Sumeer Bhola , Mustaque Ahamad, 1/k phase stamping for continuous shared data (extended abstract), Proceedings of the nineteenth annual ACM symposium on Principles of distributed computing, p.181-190, July 16-19, 2000, Portland, Oregon, United States Guruduth Banavar , Sri Doddapaneni , Kevan Miller , Bodhi Mukherjee, Rapidly building synchronous collaborative applications by direct manipulation, Proceedings of the 1998 ACM conference on Computer supported cooperative work, p.139-148, November 14-18, 1998, Seattle, Washington, United States Christian Schuckmann , Jan Schmmer , Peter Seitz, Modeling collaboration using shared objects, Proceedings of the international ACM SIGGROUP conference on Supporting group work, p.189-198, November 14-17, 1999, Phoenix, Arizona, United States James Begole , Randall B. Smith , Craig A. Struble , Clifford A. Shaffer, Resource sharing for replicated synchronous groupware, IEEE/ACM Transactions on Networking (TON), v.9 n.6, p.833-843, December 2001 Wanlei Zhou , Li Wang , Weijia Jia, An analysis of update ordering in distributed replication systems, Future Generation Computer Systems, v.20 n.4, p.565-590, May 2004

optimistic concurrency control;pessimistic views;optimistic views;groupware;replicated objects;model-view-controller programming paradigm

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Complete removal of redundant expressions.

Partial redundancy elimination (PRE), the most important component of global optimizers, generalizes the removal of common subexpressions and loop-invariant computations. Because existing PRE implementations are based on code motion, they fail to completely remove the redundancies. In fact, we observed that 73% of loop-invariant statements cannot be eliminated from loops by code motion alone. In dynamic terms, traditional PRE eliminates only half of redundancies that are strictly partial. To achieve a complete PRE, control flow restructuring must be applied. However, the resulting code duplication may cause code size explosion.This paper focuses on achieving a complete PRE while incurring an acceptable code growth. First, we present an algorithm for complete removal of partial redundancies, based on the integration of code motion and control flow restructuring. In contrast to existing complete techniques, we resort to restructuring merely to remove obstacles to code motion, rather than to carry out the actual optimization.Guiding the optimization with a profile enables additional code growth reduction through selecting those duplications whose cost is justified by sufficient execution-time gains. The paper develops two methods for determining the optimization benefit of restructuring a program region, one based on path-profiles and the other on data-flow frequency analysis. Furthermore, the abstraction underlying the new PRE algorithm enables a simple formulation of speculative code motion guaranteed to have positive dynamic improvements. Finally, we show how to balance the three transformations (code motion, restructuring, and speculation) to achieve a near-complete PRE with very little code growth.We also present algorithms for efficiently computing dynamic benefits. In particular, using an elimination-style data-flow framework, we derive a demand-driven frequency analyzer whose cost can be controlled by permitting a bounded degree of conservative imprecision in the solution.

Introduction Partial redundancy elimination (PRE) is a widely used and effective optimization aimed at removing program statements that are redundant due to recomputing previously produced values [27]. PRE is attractive because by targeting statements that are redundant only along some execution paths, it subsumes and generalizes two important value-reuse techniques: global common subexpression elimination and loop-invariant code motion. Consequently, PRE serves as a unified value-reuse optimizer. Most PRE algorithms employ code motion [11, 12, 14, 15, 16, 17, 25, 27], a program transformation that reorders instructions without changing the shape of the control flow graph. Unfortunately, code-motion alone fails to remove routine redundancies. In practice, one half of computations that are strictly partially redundant (not redundant along some paths) are left unoptimized due to code-motion obstacles. In theory, even the optimal code-motion algorithm [25] breaks down on loop invariants in while-loops, unless supported by explicit do-until conversion. Recently, Steffen demonstrated that control flow restructuring can remove from the program all redundant computations, including conditional branches [31]. While his property-oriented expansion algorithm (Poe) is complete, it causes unnecessary code duplication. As the first step towards a complete PRE with affordable code growth, this paper presents a new PRE algorithm based on the integration of code motion and control flow restructuring, which allows a complete removal of redundant expressions while minimizing code duplication. No prior work systematically treated combining the two trans- formations. We control code duplication by restricting its scope to a code-motion preventing (CMP) region, which localizes adverse effects of control flow on the desired value reuse. Whereas the Poe algorithm applied to expression elimination (denoted PoePRE) uses restructuring to carry out the entire transformation, we apply the more economical code-motion transformation to its full extent, resorting to restructuring merely to enable the necessary code motion. The resulting code growth is provably not greater than that of PoePRE; on spec95, we found it to be three times smaller. Second, to answer the overriding question of how complete a feasible PRE algorithm is allowed to be, we move from theory to practice by considering profile information. Using the dynamic amount of eliminated computations as the measure of optimization benefit, we develop a profile-guided PRE algorithm that limits the code growth cost for (;;) { else R; if (O) . = c+d; else if (P) break; En En duplicated to make [a+b] fully redundant duplicated to allow code motion of [a+b] En En code motion En duplicated for complete optimization of [c+d] duplicated for partial optimization of [c+d] R O R O c+d c+d c+d c+d a+b R c+d u O O a+b c+d O R O c+d c+d R O c+d c+d O a+b a+b R O c+d a+b a+b O d) our optimization of [c+d] a) source program b) PoePRE of [a+b] c) our optimization of [a+b] e) trade-off variant of d) R Figure 1: Complete PRE through integration of code motion and control flow restructuring. by sacrificing those value-reuse opportunities that are infrequent but require significant duplication. Third, we describe how and when speculative code motion can be used instead of restructuring, and how to guarantee that speculative PRE is profitable. Finally, we demonstrate that a near-complete PRE with very little code growth can be achieved by integrating the three PRE transformations: pure code motion, restructuring, and speculative code motion. All algorithms in this paper rely in a specific way on the notion of the CMP region which is used to reduce both code duplication and the program analysis cost. Thus, we make the PRE optimization more usable not only by increasing its effectiveness (power) through cost-sensitive restruc- turing, but also by improving its efficiency (implementa- tion). We develop compile-time techniques for determining the impact of restructuring a program region on the dynamic amount of eliminated computations. The run-time benefit corresponds to the cumulative execution frequency of control flow paths that will permit value reuse after the restructuring. We describe how this benefit can be obtained either using edge profiles, path-profiles [7], or through data-flow frequency analysis [28]. As another contribution, we reduce the cost of frequency analysis by presenting a frequency analyzer derived from a new demand-driven data-flow analysis framework. Based on interval analysis, the framework enables formulation of analyzers whose time complexity is independent of the lattice size. This is a requirement of frequency analysis whose lattice is of infinite-height. Due to this requirement, existing demand frameworks are unable to produce a frequency analyzer [18, 23, 30]. Furthermore, we introduce the notion of approximate data-flow frequency information, which conservatively underestimates the meet-over-all-paths solution, keeping the imprecision within a given degree. Approximation permits the analyzer to avoid exploring program paths guaranteed to provide insignificant contribution (frequency- wise) to the overall solution. Besides PRE, the demand-driven approximate frequency analysis is applicable in interprocedural branch correlation analysis [10] and dynamic optimizations [5]. Let us illustrate our PRE algorithms on the loop in Figure 1(a). Assume no statement in the loop defines variables or d. Although the computations [a+b] and [c+d] are loop-invariant, removing them from the loop with code motion is not possible. Consider first the optimization of [a+b]. This computation cannot be moved out of the loop because it would be executed on the path En; O;P;Ex, which does not execute [a + b] in the original program. Because this could slow down the program and create spurious excep- tions, PRE disallows such unsafe code motion [25]. The desired optimization is only possible if the CFG is restructured. The PoePRE algorithm [31] would produce the program in Figure 1(b), which was created by duplicating each node on which the value of [a+b] was available only on a subset of incoming paths. While [a + b] is fully opti- mized, the scope of restructuring is unnecessarily large. Our complete optimization (ComPRE) produces the program in Figure 1(c), where code duplication is applied merely to enable the necessary code motion. In this example, to move [a out of the loop, it is sufficient to separate out the offending path En;O; P;Ex which is encapsulated in the CMP region highlighted in the figure. As no opportunities for value reuse remain, the resulting optimization of [a is complete. Because restructuring may generate irreducible programs, as in Figure 1(c), we also present a restructuring transformation that maintains reducibility. Hoisting the loop invariant [a out of the loop was prevented by the shape of control flow. Our experiments show that the problem of removing loop invariant code (LI) has not been sufficiently solved: a complete LI is prevented for 73% of loop-invariant expressions. In some cases, a simple transformation may help. For example, [a can be optimized by peeling off one loop iteration and performing the traditional LI [1], producing the progra Figure 1(b). In while-loops, LI can often be enabled with more economical do-until conversion. The example presented does not allow this transformation because the loop exit does not post-dominate the loop entry. In effect, our restructuring PRE is always able to perform the smallest necessary do-until conversion for an arbitrary loop. Next, we optimize the computation [c+d] in Figure 1(c). Our optimization performs a complete PRE of [c +d] by duplicating the shaded CMP region and subsequently performing the code motion (Figure 1(d)). The resulting program may cause too much code growth, depending on the sizes of duplicated basic blocks. Assume the size of block S outweighs the run-time gains of eliminating the upper [c In such a case, we select a smaller set of nodes to duplicate, as shown in Figure 1(e). When only block Q is duplicated, the optimization is no longer complete; however, the optimization cost measured as code growth is justified with the corresponding run-time gain. In Section 3.2, speculative code motion is used to further reduce code duplication. In summary, this paper makes the following contributions: ffl We present an approach for integrating two widely used code transformation techniques, code motion and code restructuring. The result is an algorithm for PRE that is complete (i.e., it exploits all opportunities for value reuse) and minimizes the code growth necessary to achieve the code motion. ffl We show that restricting the algorithm to code motion produces the traditional code-motion PRE [17, 25]. ffl Profile-guided techniques for limiting the code growth through integration of selective duplication and speculative code motion are developed. ffl We develop a demand-driven frequency analyzer based on a new elimination data-flow analysis framework. ffl The notion of approximate data-flow information is defined and used to improve analyzer efficiency. ffl Our experiments compare the power of code-motion PRE, speculative PRE, and complete PRE. Section 2 presents the complete PRE algorithm. Section 3 describes profile-guided versions of the algorithm and Section 4 presents the experiments. Section 5 develops the demand-driven frequency analyzer. The paper concludes with a discussion of related work. Complete PRE In this section, we develop an algorithm for complete removal of partial redundancies (ComPRE) based on the integration of code motion and control flow restructuring. Code motion is the primary transformation behind ComPRE. To reduce code growth, restructuring is used only to enable hoisting through regions that prevent the necessary code motion. The smallest set of motion-blocking nodes is identified by solving the problems of availability and anticipabil- ity on an expressive lattice. We also show that when control flow restructuring is disabled, ComPRE becomes equivalent to the optimal code-motion PRE algorithm [25]. An expression is partially redundant if its value is computed on some incoming control flow path by a previous expression. Code-motion PRE eliminates the redundancy by hoisting the redundant computation along all paths until it reaches an edge where the reused value is available along either all paths or no paths. In the former case, the computation is removed; in the latter, it is inserted to make the original computation fully redundant. Unfortunately, code motion may be blocked before such edges are reached. Nodes that prevent the desired code motion are characterized by the following set of conditions: 1. hoisting of expression e across node n is necessary when a) an optimization candidate follows n: there is a computation of e downstream from n on some path, and b) there is a value-reuse opportunity for e at node n: a computation of e precedes n on some path. 2. hoisting of e across n is disabled when c) any path going through n does not compute e in the source program: such path would be impaired by the computation of e. All three conditions are characterizable via solutions to the data-flow problems of anticipability and availability, which are defined as follows. be any path from the start node to a node n. The expression e is available at n along p iff e is computed on p without subsequent redefinition of its operands. Let r be any path from n to the end node. The expression e is anticipated at n along r iff e is computed on r before any of its operands are defined. The availability of e at the entry of n w.r.t. the incoming paths is defined as: AVAIL in [n; Must all available along no paths. May some Anticipability (ANTIC ) is defined analogously. Given this refined value-reuse definition, code motion is necessary when a) and b) defined above hold mutually. Hence, AVAIL in [n; e] 6= No: Code motion is disabled when the condition c) holds: Must - AVAIL in [n; e] 6= Must: A node n prevents the necessary code motion for e when the motion is necessary but disabled at the same time. By way of conjunction, we get the code motion-preventing condition: AVAIL in [n; The predicate Prevented characterizes the smallest set of nodes that must be removed for code motion to be enabled. a) code motion prevented by CMP region b) CMP region diluted via code duplication c) complete PRE of [a+b] code motion becomes possible code motion ANTIC=No ANTIC=Must AVAIL=No AVAIL=Must R a+b R a+b a+b a+b R AVAIL=May ANTIC=May ANTIC=May ANTIC=May ANTIC=May AVAIL=No AVAIL=Must AVAIL=No AVAIL=Must ANTIC=May Figure 2: Removing obstacles to code motion via restructuring. Motion Preventing region, denoted CMP [e], is the set of nodes that prevent hoisting of a computation e: CMP [e] = fn j ANTIC in [n; AVAIL in [n; Mayg. To enable code motion, ComPRE removes obstacles presented by the CMP region by duplicating the entire region, as illustrated in Figure 2. The central idea is to factor the May-availability that holds in the entire region into Must- and No-availability, to hold respectively in each region copy. An alternative view is that we separate within the region the paths with Must- and No-availability. To achieve this, we can observe that a) no region entry edge is May-available, and b) the solution of availability within the region depends solely on solutions at entry edges (the expression is neither computed nor killed within the region). Hence, the desired factoring can be carried out by attaching to each region copy the subset of either Must or No entry edges, as shown in Figure 2(c). After the CMP is duplicated, the condition Prevented is false on each node, enabling code motion. The ComPRE algorithm, shown in Figure 3, has the following three steps: 1. Compute anticipability and availability. The problems use the lattice the flow functions are distributive under the least common element operator -, which is defined using the partial order v shown below. Distributivity property implies that data-flow facts are not approximated at control flow merge points. Intuitively, this is because L is the powerset lattice of fNo; Mustg, which are the only facts that may hold along an individual path. The partial order v: Must May 2. Remove CMP regions via control flow restructuring. Given an expression e, the CMP region is identified by examining the data-flow solutions locally at each node. Line 2 in Figure 3 duplicates each CMP node and line 3 adjusts the control flow edges, so that the new copy of the region hosts the Must solution. Restructuring necessitates updating data-flow solutions within the CMP region (lines 4-12). While the ANTIC solution is not altered, the previously computed AVAIL solution is invalidated because some paths flowing into the region were eliminated when region entry edges were discon- nected. For the expression e, AVAIL becomes either Must or No in the entire region. For other expressions, the solution may become (conservatively) imprecise. In other words, splitting a May path into Must/No paths for e might have also split a May path for some other expression. Therefore, line 6 resets the initial guess and lines 10-12 recompute the solution within the CMP. 3. Optimize the program. The code motion transformation is carried out by replacing each original computation e with a temporary variable te . The temporary is initialized with a computation inserted into each No-available edge that sinks either into a May/Must- availability path or into an original computation. The insertion edge must also be Must-anticipated, to verify hoisting of the original computation to the edge. Theorem 1 (Completeness). ComPRE is optimal in that it minimizes the number of computations on each path. Proof. First, each original computation is replaced with a temporary. Second, no computation is inserted where its value is available along any incoming path. Hence, no additional computations can be removed. Within the domain of the Morel and Renviose code- motion transformation, where PRE is accomplished by hoisting optimization candidates (but not other statements) [27], ComPRE achieves minimum code growth. 1 This follows from the fact that after CMP restructuring, no program node can be removed or merged with some other node without destroying any value reuse, as shown by the following observations. Prior to Step 2, each node n may belong to CMP regions of multiple offending expressions. Duplication of n during restructuring can be viewed as partitioning of control flow paths going through n: each resulting copy of n is a path partition that does not contain both a Must- and a No-available path, for any offending expression. The 1 Outside this domain, further code growth reduction is possible by moving instructions out of the CMP before its duplication. Step 1: Data-flow analysis: anticipability, availability. ffl Input: control flow graph each node contains a single assignment x := e, ffl Comp(n; e): node n computes an expression e, ffl Transp(n; e): node n does not assign any variable in e, ffl boundary conditions: for each expression e ANTIC out [end; e] := AVAIL in [start; e] := No, ffl initial guess: set all vectors to ? S , where S is the number of candidate expressions. Solve iteratively. ANTIC in [n; e] :=? ! Must if Comp(n; e), ANTIC out [n; e] otherwise. ANTIC out [n; e] := ANTIC in [m; e] AVAIL in [n; e] := AVAIL out [m; e] AVAIL out [n; e] := f e n (AVAIL in [n; e]) Must if Comp(n; e) -Transp(n; e), x otherwise. Step 2: Remove CMP regions: control flow restructuring. ffl modify G so that no CMP nodes exists, for any expression e. 1 for each expression e do duplicate all CMP[e] nodes to create a copy of the CMP. n Must is a copy of node n hosting Must attach new nodes to perform the restructuring Must Must Must update data-flow solutions within CMP and its copy 4 for each node n 2 CMP[e] do 5 ANTIC in [n Must ] := ANTIC in [n] ANTIC out [n Must ] := ANTIC out [n] 6 AVAIL in [n Must ] := AVAIL in [n] := ? S AVAIL out [n Must 7 AVAIL in [n Must ; e] := AVAIL out [n Must ; e] := Must 8 AVAIL in [n; e] := AVAIL out [n; e] := No 9 end for reanalyze availability inside both CMP copies for each expression e 0 not yet processed do 12 end for 13 end for Step 3: Optimize: code motion. Insert[(n; m); e] , ANTIC in [m; Must - AVAIL out [n; (AVAIL in [m; Replace[n; e] , Comp(n; e) Figure 3: ComPRE: the algorithm for complete PRE. following properties of Step 2 can be verified: 1) the number of path partitions (node copies) created at a given node is independent of the order in which expressions are considered (in line 1), 2) each node copy is reachable from the start node, and 3) for any two copies of n there is an expression e such that remerging the two copies and their incoming paths will prevent code motion of e across the resulting node. To compare ComPRE with a restructuring-only PRE, we consider PoePRE, a version of Steffen's complete algorithm [31] that includes minimization of duplicated nodes but is restricted in that only expressions are eliminated (as is the case in ComPRE). Elimination is carried out using a temporary, as in Step 3. Theorem 2 ComPRE does not create more new nodes than PoePRE. Proof outline. The proof is based on showing that the PoePRE-optimized program after minimization has no less nodes than the same program after CMP restructuring. It can be shown that, given an original node n, for any two copies of n created by CMP restructuring, there are two distinct copies of n created by PoePRE such that the minimization cannot merge them without destroying some value reuse opportunity. In fact, PoePRE can be expressed as a form of Com- PRE on a (non-strictly) larger region: for each computation e, PoePRE duplicates fnjANTIC in [n; e] 2 fMust; Mayg - AVAIL in [n; which is a superset of CMP [e]. Algorithm complexity. Data-flow analysis in Step 1 and in lines 10-12 requires O(NS) steps, where N is the flow graph size and S the number of expressions. The restructuring in Step 2, however, may cause N to grow exponentially, as each node may need to be split for multiple expressions. Because in practice a constant-factor code-growth budget is likely to be defined, the asymptotic program size will not change. Therefore, the running time of Step 2, which dominates the entire algorithm, is O(NS 2 ). 2.1 Optimal Code-Motion PRE Besides supporting a complete PRE, the notion of the CMP region also facilitates an efficient formulation of code-motion PRE, called CM-PRE. In this section, we show that our complete algorithm can be naturally constrained by prohibiting the restructuring, and that such modification results in the same optimization as the optimal motion-only PRE [17, 25]. In comparison to ComPRE, the constrained CM-PRE algorithm bypasses the CMP removal; the last step (trans- formation) is unchanged (Figure 3). The first step (data- flow analysis) is modified with the goal to prevent hoisting across a node n when such motion would subsequently be blocked by a CMP region on each path flowing into node n. First, anticipability is computed as in ComPRE. Second, availability is modified to include detection of CMP nodes. When a CMP node is found, instead of propagating forward May-availability, the solution is adjusted to No. Such adjustment masks those value reuse opportunities that cannot be exploited without restructuring. The result of masking is that code motion is prevented from entering paths that cross a CMP region (see predicate Insert in Step 3 of Figure 3). The modified flow function for the AVAIL problem fol- lows. The third line detects a CMP node. No-availability is now extended to mean that the value might be available along some path but value reuse is blocked by a CMP region along that path. Must if Comp(n; e) - Transp(n; e), x otherwise. Given a maximal fixed point solution to redefined AVAIL, CM-PRE performs the unchanged transformation phase Figure 3, Step 3). It is easy to show that the resulting optimization is complete under the immutable shape of the control flow graph. The proof is analogous to that of Theorem 1: all original computations are removed and no computation has been inserted where an optimization opportunity not blocked by a CMP exists. Besides exploiting all opportunities, a PRE algorithm should guarantee that the live ranges of inserted temporary variables are minimal, in order to moderate the register pres- sure. The live range is minimal when the insertion point specified by the predicate Insert cannot be delayed, that is, moved further in the direction of control flow. Theorem 3 (Shortest live ranges). Given the CMP- restructured (or original) control flow graph, ComPRE (CM-PRE) is optimal in that it minimizes the live range lengths of inserted temporary variables. Proof. An initialization point Insert cannot be delayed either because it would become partially redundant, destroying completeness, or because its temporary variable is used in the immediate successor. Existing PRE algorithms find the live-range optimal placement in two stages. First, computations are hoisted as high as possible, maximizing the removal of redundancies. Later, the placement is corrected through the computation of delayability [25]. Our formulation specifies the optimal placement directly, as we never hoist into paths where a blocking CMP will be subsequently encountered. However, note that after the above redefinition, f e n is no longer monotone: given ANTIC in [n; Must, we have x1 v x2 but f e Must. Although a direct approach to solving such system of equations may produce conservatively imprecise solution, the desired maximal fixed point is easily obtained using bit-vector GEN/KILL operations as follows. First, compute ANTIC as in Figure 3. Second, solve the well-known availability property, denoted AV all , which holds when the expression is computed along all incoming paths: Must. Finally, we compute AV some which characterizes some-paths availability and also encapsulates CMP detection: AV some , AVAIL 6= No. The pair of solutions some ) can be directly mapped to the desired solution of AVAIL. The GEN and KILL sets [1] for the some problem are given below. The value of the initial guess is false, the meet operator is the bit-wise or. Must - ANTIC 6= Must) The condition (AVAIL 6= Must - ANTIC 6= Must) detects the CMP node. While it is less strict than that in Definition 2, it is equivalent for our purpose, as it is safe to kill single loop copied for reducibility entry node En En a+b O c+d O R O c+d c+d a) source program c+d a+b R c+d reducible ComPRE of [a+b] Figure 4: Reducible restructuring. (See Figure 1(c)) when there is no reuse or when there is no hoisting No). The less strict condition is beneficial because computing and testing Must requires one bit per expression, while two bits are required for May. Con- sequently, we can substitute ANTIC 6= Must with :AN all , where AN all is defined analogously to AV all . As a result, we obtain the same implementation complexity as the algorithms in [17, 25]: three data-flow problems must be solved, each requiring one bit of solution per expression. In conclusion, the CMP region is a convenient abstraction for terminating hoisting when it would unnecessarily extend the live ranges. It also provides an intuitive way of explaining the shortest-live-range solution without applying the corrective step based on delayability [25]. Furthermore, the CMP-based, motion-only solution can be implemented as efficiently as existing shortest-live-range algorithms. 2.2 Reducible Restructuring Duplicating a CMP region may destroy reducibility of the control flow graph. In Figure 1(c), for example, ComPRE resulted in a loop with two distinct entry nodes. Even though PoePRE preserves reducibility on the same loop Figure 1(b)), like other restructuring-based optimizations [4, 10, 31], it is also plagued by introducing irreducibility. One way to deal with the problem is to perform all optimizations that presuppose single-entry loops prior to PRE. However, many algorithms for scheduling (which should follow PRE) rely on reducibility. After ComPRE, a reducible graph can be obtained with additional code duplication. An effective algorithm for normalizing irreducible programs is given in [24]. To suppress an unnecessary invocation of the algorithm, we can employ a simple test of whether irreducibility may be created after a region duplication. The test is based upon examining only the CMP entry and exit edges, rather than the entire program. Assuming we start from a reducible graph, re-structuring will make a loop L irreducible only if multiple CMP exit edges sink into L, and at least one region entry is outside L (i.e., is not dominated by L's header node). If such a region is duplicated, target nodes of region exit edges may become the (multiple) loop entry nodes. Consider the loop in Figure 4(a). Two of the three exits of CMP [a fall into the loop. After restructuring, they will become loop entries, as shown in Figure 1(c). Rather than applying a global algorithm like [24], a straightforward approach to make the affected loop reducible is to peel off a part of its body. The goal is to extend the replication scope so that the region exits sink onto a single loop node, which will then become the new loop entry. Such a node is the closest common postdominator (within the loop) of all the offending region exits and the original loop entry. Figure 4(a) highlights node c+d whose duplication after CMP restructuring will restore reducibility of the loop. The postdominator of the offending exits is node Q, which becomes the new loop header. 3 Profile-Guided PRE While the CMP region is the smallest set of nodes whose duplication enables the desired code motion, its size is often prohibitive in practice. In this section, relying on the profile to estimate optimization benefit, complete PRE is made more practical by avoiding unprofitable code replication. First, we extend ComPRE by inhibiting restructuring in response to code duplication cost and the expected dynamic benefit. The resulting profile-guided algorithm duplicates a CMP region only when the incurred code growth is justified by a corresponding run-time gain from eliminating the redundancies. Second, the notion of the CMP region is combined with profiling to formulate a speculative code-motion PRE that is guaranteed to have a positive dynamic effect, despite impairing certain paths. The third algorithm integrates both restructuring and speculation and selects a profitable subgraph of the CMP for each. While profitably balancing the cost and benefit under a given profile is NP- hard, the empirically small number of hot program paths promises an efficient algorithm [4, 19]. Finally, to support profile guiding, we show how an estimate of the run-time gain thwarted by a CMP region can be obtained using edge profiles, frequency analysis [28], or path profiles [7]. 3.1 Selective Restructuring We model the profitability of duplicating a CMP region R with a cost-benefit threshold predicate T (R), which holds if the region optimization benefit exceeds a constant multiple of the region size. Our metric of benefit is the dynamic amount of computations whose elimination will be enabled after R is duplicated, denoted Rem(R). That is, true for each region R, the algorithm is equivalent to the complete Com- PRE. When T (R) = false for each region, the algorithm reduces to the code-motion-only CM-PRE. Obviously, predicate determines only a sub-optimal tradeoff between exploiting PRE opportunities and limiting the code growth. In particular, it does not explicitly consider the instruction cache size and the increase in register pressure due to introduced temporary variables. We have chosen this form of T in order to avoid modeling complex interactions among compiler stages. In the implementation, T is supplemented with a code growth budget (for example, in [6], code is allowed to grow by about 20%). First, we present an algorithm for computing the optimization benefit Rem(R). The method is based on the fact Step 1: compute anticipability and availability. (unchanged) Step 2: Partial restructuring: remove profitable CMP regions. 1 for each computation e do 2 for each disconnected subregion R i of CMP[e] do build the largest connected subregion 3 select a node from R and collect all connected CMP nodes determine optimization benefit Rem(R i ) 4 carry out frequency analysis of AVAIL on R i if profitable, duplicate (lines 2-12 of Fig. 6 end for 7 end for 8 recompute the AVAIL solution, using f e n from Section 2.1 Step 3: Optimize: code motion. (unchanged) Figure 5: PgPRE: profile-guided version of ComPRE. that the CMP scope localizes the entire benefit thwarted by the region: to compute the benefit, it suffices to examine only the paths within the region. Consider an expression e and its CMP region [e]. For each region exit edge [e]), the value of ANTIC in [m; e] is either Must or No, otherwise m would be in CMP [e]. Let ExitMust (R) be the set of the Must exit edges. The dynamic benefit is derived from the observation that each time such an edge is executed, any outgoing path contains exactly one computation of e that can be eliminated if: i) R is duplicated and ii) the value of e is available at the exit edge. Let ex(a) be the execution frequency of edge a and p(AVAIL out [n; the probability that the value e is available when n is executed. After the region is dupli- cated, the expected benefit connected with the exit edge a is ex(a):p(AVAIL out [n; which corresponds to the number of computations removed on all paths starting at a. The benefit of duplicating the region R is thus the sum of all exit edge benefits a=(n;m)2Exits Must (R) ex(a):p(AVAILout [n; The probability p is computed from an edge profile using frequency analysis [28]. In the frequency domain, the probability of each data-flow fact occurring, rather than the mere boolean meet-over-all-paths existence, is computed by incorporating the execution probabilities of control flow edges into the data-flow system. Because the frequency analyzer cannot exploit bit-vector parallelism, but instead computes data-flow solutions on floating point numbers, it is desirable to reduce the cost of calculating the probabili- ties. The CMP region lends itself to effectively restricting the scope of the program that needs to be analyzed. Because all region entry edges are either Must- or No-available, the probability of e being available on these edges are 1 and 0, respectively. Therefore, the probability p at any exit edge can only be influenced by the paths within the region. As a result, it is sufficient to perform the frequency analysis for expression e on CMP [e], using entry edges as a precise boundary condition for the CMP data-flow equation system. In Section 5 we reduce the cost of frequency analysis through a demand-driven approach. The algorithm (PgPRE) that duplicates only profitable CMP regions is given in Figure 5. It is structured as its complete counterpart, ComPRE: after data-flow analysis, we proceed to eliminate CMP regions, separately for each expression. While in ComPRE it was sufficient to treat all nodes from a single CMP together, selective duplication benefits from dividing the CMP into disconnected subregions, if any exist. Intuitively, hoisting of a particular expression may be prevented by multiple groups of nodes, each in a different part of the procedure. Therefore, line 3 groups nodes from a connected subregion and frequency analysis determines the benefit of the group (line 4). After all profitable regions are eliminated, the motion-blocking effect of CMP regions remaining in the program must be captured. All that is needed is to apply the CM-PRE algorithm from Section 2.1 on the improved control flow graph. Blocked hoisting is avoided by recomputing availability (line using the re-defined flow function f e n from Section 2.1, which asserts No-availability whenever a CMP is detected. 3.2 Speculative Code-Motion PRE In code-motion PRE, hoisting of a computation e is blocked whenever e would need to be placed on a control flow path p that does not compute e in the original program. Such speculative code motion is prevented because executing e along path p could a) raise spurious exceptions in e (e.g., over- flow, wrong address), and b) outweigh the dynamic benefit of removing the original computation of e. The former restriction can be relaxed for instruction that cannot except, leading to safe speculation. New processor generations will support control-speculative instructions which will suppress raising the exception until the generated value is eventually used, allowing unsafe speculation [26]. The latter problem is solved in [20], where an aggressive code-motion PRE navigated by path profiles is developed. The goal is to allow speculative hoisting, but only into such paths on which dynamic impairment would not outweigh the benefit of eliminating the computation from its original position. Next, we utilize the CMP region to determine i) the profitability of speculative code motion and ii) the positions of speculative insertion points that minimize live ranges of temporary variables. Figure 6 illustrates the principle of speculative PRE [20]. Instead of duplicating the CMP region, we hoist the expression into all No-available entry edges. This makes all exits fully available, enabling complete removal of original computations along the Must exits. In our example, moved into the No-available region entry edge e2 . This hoisting is speculative because [a+b] is now executed on each path going through e2 and e3 , which previously did not contain the expression. The benefit is computed as follows. The dynamic amount of computations is decreased by the execution frequency ex(e4) of the Must-anticipable exit edge (following which a computation was removed), and increased by the frequency ex(e2) of the No-available entry edge (into which the computation was inserted). Since speculation is always associated with a CMP region, we are able to obtain a simple (but precise) profitability test: speculative PRE of an expression is profitable if the total execution frequency of Must-anticipable exit edges exceeds that of No-availaible entry edges. Note that the benefit is calculated locally by examining only entry/exit edges, and not the paths within the region, which was necessary in selective restructuring. Hence, the speculative benefit is independent from branch correlation and edge profiles are as precise as path profiles in the case of speculative-motion PRE. As far as temporary live ranges are concerned, insertion into entry edges results in a shortest-live-range solution, and Theorem 3 still holds. code motion AVAIL=Must AVAIL=No ANTIC=Must speculative ANTIC=No removal insertion Optimization benefit: a+b a+b Figure Speculative code-motion PRE. 3.3 Partial Restructuring, Partial Speculation In Section 3.1, edge profiles and frequency analysis were used to estimate the benefit Rem of duplicating a region. An alternative is to use path profiles [3, 7], which are convenient for establishing cost-benefit optimization trade-offs [4, 19, 20]. To arrive at the value of the region benefit with a path profile, it is sufficient to sum the frequencies of Must- Must paths, which are paths that cross any region entry edge that is Must-available and any exit edge that is Must- anticipated. These are precisely the paths along which value reuse exists but is blocked by the region. While there is an exponential number of profiled acyclic paths, only 5.4% of procedures execute more than 50 distinct paths in spec95 [19]. This number drops to 1.3% when low-frequency paths accounting for 5% of total frequency are removed. Since we can afford to approximate by disregarding these infrequent paths, summing individual path frequencies constitutes a feasible algorithm for many CMP regions. Furthermore, because they encapsulate branch correlation, path profiles compute the benefit more precisely than frequency analysis based on correlation-insensitive edge profiles. Moreover, the notion of individual CMP paths leads to a better profile-guided PRE algorithm. Considering the CMP region as an indivisible duplication unit is overly conserva- tive. While it may not be profitable to restructure the entire region, the region may contain a few paths Must-Must paths that are frequently executed and are inexpensive to dupli- cate. Our goal is to find the largest subset (frequency-wise) of region paths that together pass the threshold test T (R). Similarly, speculative hoisting into all entry edges may fail the profitability test. Instead, we seek to find a subset of entry edges that maximizes the speculative benefit. In this section, we show how partial restructuring and speculation are carried out and combined. Partial speculation selects for speculative insertion only a subset I of the No region entries. The selection of entries influences which subset R of region exits will be able to exploit value reuse. R consists of all Must exits that will become Must-available due to the insertions in I. The rationale behind treating entries separately is that some entries may enable little value reuse, hence they should not be speculated. Note that No entry edges are the only points where speculative insertion needs to be considered: insertions inside the region would be partially redundant; insertions outside the region would extend the live-ranges. Partial speculation is optimal if the difference of total frequencies of R and I is maximal (but non-negative). As pointed out in [22], this Y Y A A O c+d R a) source program b) speculation made profitable O c+d RT No-path peeled off not profitable profitable speculation1001000901000 100 Figure 7: Integrating speculation and restructuring. problem can be solved as a maximum network flow problem. An interesting observation is that to determine optimal partial speculation, a) edge profiles are not inferior to path profiles and b) frequency analysis is not required. Therefore, to exploit the power of path profiles, partial restructuring, rather than (speculative) code motion alone, must be used. This becomes more intuitive once we realize that without control flow restructuring, one is restricted to consider only an individual edge (but not a path) for expression insertion and removal. To compare the CMP-based partial speculation with the speculative PRE in [20], we show how to efficiently compute the benefit by defining the CMP region and how to apply edge profiles with the same precision as path profiles. In acyclic code, we achieve the same preci- sion; in cyclic code, we are more precise in the presence of loop-carried reuse. The task of partial restructuring is to localize a subgraph of the CMP that has a small size but contains many hot Must-Must paths. By duplicating only such a subregion, we are effectively peeling off only hot paths with few in- structions. In Figure 1(e), only the (presumably hot) path through the node Q was separated. Again, the problem of finding an optimal subregion, one whose benefit is maximized but passes the T (R) predicate and is smaller than a constant budget, is NP-hard. However, the empirically very small number of hot paths promises an efficient exhaustive-search algorithm. Integrating partial speculation and restructuring offers additional opportunities for improving the cost-benefit ra- tio. We are no longer restricted to peeling off hot Must-Must paths and/or selecting No-entries for speculation. When the high frequency of a No entry prevents speculation, we can peel off a hot No-available path emanating from the thereby reducing entry edge frequency and allowing the speculation, at the cost of some code duplication. Figure 7(a) shows an example program annotated with an edge profile. Because peeling hot Must-Must paths from the high-lighted CMP ([c+d]) would duplicate all blocks except S, we try speculation. To eliminate the redundancy at the CMP exit edge Y with frequency ex(Y computation must be inserted into No-entries B and C. While B is low-frequency (10), C is not (100), hence the speculation is dis- advantageous, as ex(Y Now assume that the exit branch in Q is strongly biased and the path C; Q;X has a frequency of 100. That is, after edge C is executed, the execution will always follow to X. We can peel off this No-available path, as shown in (b), effectively moving the speculation point C off this path. After peeling, the frequency of C becomes 0 and the speculation is profitable, ex(Y 4 Experiments We performed the experiments using the HP Labs VLIW back-end compiler elcor, which was fed spec95 benchmarks that were previously compiled, edge-profiled, and inlined (only spec95int) by the Impact compiler. Table 1 shows program sizes in the total number of nodes and expres- sions. Each node corresponds to one intermediate state- ment. Memory requirements are indicated by the column space, which gives the largest nodes-expressions product among all procedures. The running time of our rather inefficient implementation behaved quadratically in the number of procedure nodes; for a procedure with 1,000 nodes, the time was about 5 seconds on PA-8000. Typically, the complete PRE ran faster than the subsequent dead code elimination. Experiment 1: Disabling effects of CMP regions. The column labeled optimizable gives the percentage of expressions that reuse value along some path; 13.9% of (static) expressions have partially redundant computations. The next column prevented-CMP reports the percentage of optimizable expressions whose complete optimization by code motion is prevented by a CMP region. Code-motion PRE will fail to fully optimize 30.5% of optimizable expressions. For comparison, column prevented-POE reports expressions that will require restructuring in PoePRE. Experiment 2: Loop invariant expressions. Next, we determined what percentage of loop invariant (LI) expressions can be removed from their invariant loops with code motion. The column loop invar shows the percentage of optimizable expressions that pass our test of loop-invariance. The following column gives the percentage of LI expressions that have a CMP region; an average of 72.5% of LI computations cannot be hoisted from all enclosing invariant loops without restructuring. Experiment 3: Eliminated computations. The column global CSE reports the dynamic amount of computations removed by global common subexpression elimination; this corresponds to all full redundancies. The column complete PRE gives the dynamic amount of all partially redundant statements. The fact that strictly partial redundancies contribute only 1.7% (the difference between complete PRE and global CSE) may be due to the style of Impact 's intermediate code (e.g., multiple virtual registers for the same variable). We expect a more powerful redundancy analysis to perform better. Figure 8 plots the dynamic amount of strictly partial redundancies removed by various PRE tech- niques. Code-motion PRE yields only about half the benefit of a complete PRE. Furthermore, speculation results in near-complete PRE for most benchmarks, even without special hardware support (i.e., safe speculation). Speculation was carried out on the CMP as whole. Note that the graph accounts for the dynamic impairment caused by speculation. benchmark program size E-1: CM prevented E-2: loop inv E-3: dynamic E-4: code growth spec95int spec95fp procedures nodes (k) expressions space (M) optimizable (% of prevented-CMP (% of prevented-POE (% of loop invar of invar-prevent (% of global of all) complete of all) ComPRE (% increase) PoePRE (% increase) 099.go 372 153.6 37.3 5.8 10.2 29.6 45.4 7.1 83.4 9.5 11.7 49.9 90.2 126.gcc 1661 917.2 158.2 38.0 8.0 34.2 45.0 2.5 69.8 3.7 4.6 33.9 36.7 129.compress 130.li 357 37.4 8.4 2.0 11.8 22.4 34.4 10.4 69.9 6.8 8.0 21.5 35.1 132.ijpeg 472 81.8 22.8 1.2 13.9 24.1 45.3 5.1 78.1 4.3 5.1 48.8 104.7 134.perl 276 135.0 25.5 40.4 9.6 39.5 51.8 11.9 93.5 4.8 6.8 31.2 50.0 147.vortex 923 325.9 65.7 5.8 16.6 29.5 36.1 6.3 81.6 11.1 13.0 35.7 55.4 Avg: spec95int 542.1 216.7 42.0 12.2 12.1 29.1 43.4 8.2 75.0 7.4 9.1 33.3 56.5 103.su2cor 37 10.6 3.9 2.5 15.3 29.8 53.8 14.5 43.7 12.8 13.0 42.1 142.0 104.hydro2d 43 8.5 2.4 0.4 16.8 21.7 42.7 5.9 41.7 1.9 6.0 43.9 141.7 145.fpppp 37 13.6 6.7 19.6 14.6 52.2 57.7 43.0 91.9 7.1 7.7 2.4 18.2 146.wave5 110 33.3 12.3 5.3 12.4 34.8 47.8 4.9 66.2 7.1 7.8 36.6 107.6 Avg: spec95fp 39.2 11.4 4.4 4.7 16.2 32.4 49.8 15.3 69.2 8.3 10.0 94.3 313.0 Avg: spec95 326.6 128.7 25.9 9.0 13.9 30.5 46.1 11.3 72.5 7.8 9.5 59.5 166.4 Table 1: Experience with PRE based on control flow restructuring. go compress li ijpeg perl vortex AVG-int tomcatv swim su2cor hydro2d fpppp wave5 AVG-fp AVG1.03.05.0 Dynamic computations eliminated code-motion PRE safe speculative PRE unsafe speculative PRE complete PRE Figure 8: Benefit of various PRE algorithms: dynamic op- count decrease due to strictly partial redundancies. Each algorithm also completely removes full redundancies. The measurements indicate that an ideal PRE algorithm should integrate both speculation and restructuring. Using restructuring when speculation would waste a large portion of benefit will provide an almost complete PRE with small code growth. Experiment 4: Code growth. Finally, we compare the code growth incurred by ComPRE and PoePRE. To make the experiment feasible, we limited procedure size by 3,000 nodes and made the comparison only on procedures that did not exceed the limit in either algorithm. Overall, ComPRE created about three times less code growth than PoePRE. 5 Demand-Driven Frequency Analysis Not amenable to bit-vector representation, frequency analysis [28] is an expensive component of profile-guided opti- mizers. We have shown that ComPRE allows restricting the scope of frequency analysis within the CMP region without a loss of accuracy. However, in large CMP regions the cost may remain high, and path profiles cannot be used as an efficient substitute when numerous hot paths fall into the region. One method to reduce the cost of frequency analysis is computing on demand only the subset of data flow solution that is needed by the optimization. In this section, we develop a demand-driven frequency analyzer which reduces data-flow analysis time by a) examining only nodes that contribute to the solution and, option- ally, b) terminating the analysis prematurely, when the solution is determined with desired precision. Besides PRE, the analyzer is suitable for optimizations where acceptable running time must be maintained by restricting analysis scope, as in run-time optimizations [5] or interprocedural branch removal [10]. Frequency analysis computes the probability that a data-flow fact will occur during execution. Therefore, the probability "lattice" is an infinite chain of real numbers. Because existing demand-driven analysis frameworks are built on iterative approaches, they only permit lattices of finite size [18] or finite height [23, 30] and hence cannot derive a frequency analyzer. We overcome this limitation by designing the demand-driven analyzer based upon elimination data-flow methods [29] whose time complexity is independent of the lattice shape. We have developed a demand-driven analysis framework motivated by the Allen-Cocke interval elimination solver [2]. Next, using the framework, a demand-driven algorithm for general frequency data-flow analysis was derived [8]. We present here the frequency solver specialized for the problem of availability. Definitions. Assume a forward data-flow problem specified with an equation system Vector n ) is the solution for a node n, variable n denotes the fact associated with expression e. The equation system induces a dependence graph EG whose nodes are variables x e n and edges represent flow functions an edge exists if the value of x e n is computed from x d pred(n). The graph EG is called an exploded graph [23]. The data flow problems underlying ComPRE are separable, hence x e only depends on x e m . In value-based PRE [9], constant propagation [30], and branch correlation analysis [10], edges e, may exist, complicating the analysis. The analyzer presented here handles such general exploded graphs. Requirements. The demand-driven analyzer grew out of four specific design requirements: 1. Demand-driven. Rather than computing xn for each node n, we determine only the desired x e n , i.e. the solution for expression e at a node n. Analysis speed-up is obtained by further requiring that only nodes transitively contributing to the value of x e n are visited and examined. To guarantee worst-case behavior, when solutions for all EG nodes are desired, the solver's time complexity does not exceed that of the exhaustive Allen-Cocke method, O(N 2 ), where N is the number of EG nodes. 2. Lattice-independent. The amount of work per node does not depend on lattice size, only on the EG shape. 3. On-line. The analysis is possible even when EG is not completely known prior to the analysis. To save time and memory, our algorithm constructs EG as analysis progresses. The central idea of on-demand construction is to determine a flow function f e only when its target variable x e n is known to contribute to the desired solution. Furthermore, the solver must produce the solution even when EG is irreducible, which can happen even when the underlying CFG is reducible. 4. Informed. In the course of frequency analysis, the contribution weight of each examined node to the desired solution must be known. This information is used to develop a version of the analyzer that approximates frequency by disregarding low-contribution nodes with the goal of further restricting analysis scope. The exhaustive interval data-flow analysis [2] computes xn for all n as follows. First, loop headers are identified to partition the graph into hierarchic acyclic subregions, called intervals. Second, forward substitution of equations is performed within each interval to express each node solution in terms of its loop header. The substitution proceeds in the interval order (reverse postorder), so that each node is visited only once. Third, mutual equation dependences across loop back-edges are reduced with a loop breaking rule L: )). The second and third step remove cyclic dependences from all innermost loops in EG; they are repeated until all nesting levels are processed and all solutions are expressed in terms of the start node, which is then propagated to all previously reduced equations in the final propagation phase [2]. The demand-driven interval analysis substitutes only those equations and reduces only those intervals on which the desired x e n is transitively dependent. To find the relevant equations, we back-substitute equations (flow functions) into the right-hand side of x e n along the EG edges. The edges are added to the exploded graph on-line, whenever a new EG node is visited, by first computing the flow function of the node and then inserting its predecessors into the graph. As in [2], we define an EG interval to be a set of nodes dominated by the sink of any back-edge. In an irreducible EG, a back-edge is each loop edge sinking onto a loop entry node. Because the EG shape is not known prior to analysis, on-line identification of EG intervals relies only on the structure of the underlying control flow graph. When the CFG node of an EG node x is a CFG loop entry, then x may be an EG loop entry, and we conservatively assume it is an interval head. Within each interval, back-substitutions are performed in reversed interval order. Such order provides lattice-independence, as each equation needs to be substituted only once per interval reduction, and there are at most reductions. To find interval order on an incomplete EG, we observe that within each EG interval, the order is consistent with the reverse postorder CFG node numbering. To loop-break cyclic dependencies along an interval back- edge, the loop is reduced before we continue into the preceding interval, recursively invoking reductions of nested loops. This process achieves demand analysis of relevant intervals. The desired solution is obtained when x e n is expressed exclusively using constant terms. At this point, we have also identified an EG subgraph that contributes to x e n , and removed from it all cyclic dependences. A forward substitution on the subgraph will yield solutions for all subgraph nodes which can be cached in case they are later desired (worst-case running time). This step corresponds to the propagation phase in [2], and to caching in [18, 30]. The framework instance calculates the probability of expression e being available at the exit of node n during the execution: x e denote the probability of edge a being taken, given its sink node is executed. We relate the edge probability to the sink (rather than the source, as in exhaustive analysis [28]) because the demand solver proceeds in the backward direction. The frequency flow function returns probability 1 when the node computes the expression e and 0 when it kills the ex- pression. Otherwise, the sum of probabilities on predecessors weighted by edge execution probabilities is returned. Predicates Comp and Transp are defined in Figure 3. 1:0 if Comp(n; e) - Transp(n; e), 0:0 if :Transp(n; e), p((m; n)):x e otherwise. The demand frequency analyzer is shown in Figure 9. Two data structures are used : sol accumulates the constant terms of the desired probability x e rhs is the current right-hand side of x e n after all back-substitutions. The variables sol and rhs are organized as a stack, the top being used in the currently analyzed interval. The algorithm treats rhs both as a symbolic expression and as a working set of pending nodes (or yet unsubstituted variables, to be precise). For example, the value of rhs may be 0:25 m+0:4 k, where the weights are contributions of nodes m and k to the desired probability x e n . If e is never available at m, and is available at k with probability 0.5, then it is available at node n with probability 0:25 0+0:4 formally, the contribution weight of a node represents the probability that a path from that node to n without a computation or a kill of the expression e will be executed. First, the rhs is set to 1:0 n in line 1. Then, flow functions are back-substituted into rhs in post-order (line 3). Substitutions are repeated until all variables have been replaced with constants (line 2), which are accumulated in sol. If a substituted node x computes the expression e, its weight rhs[x] is added to the solution and x is removed from the rhs by the assignment rhs[x] := 0:0 (line 6). In the simple case when x is not a loop entry node (line 12), its contribution c is added to each predecessor's contribution, weighted by the edge probability p. If x is a loop entry node (line 8), then before continuing to the loop predecessor, all self-dependences of x are found in a call to reduce loop. The procedure reduce loop mimics the main loop (lines 1- but it pushes new entries on the stacks to initiate a reduction of a new interval and also marks the loop entry node to stop when back-substitution collected cyclic dependences along all cyclic paths on the back-edge edge (y; x). The result of reduce loop is returned in a sol-rhs pair (s; r), where s is the constant and r the set of unresolved vari- ables, e.g. 0:1. If EG is reducible, the set r contains only x. The value 0:3 is the weight of the x's self-dependence, which is removed by the loop breaking rule derived for frequency analysis from the sum of infinite geometric sequence (lines 10-11). After the algorithm terminates, the stack visited (line 14) specifies the order in which forward substitution is performed to cache the results. Also shown in Figure 9 is an execution trace of the demand-driven analysis. It computes the probability that the expression computed in nodes F , H, and killed in A, D, is available at node C. All paths where availability holds are highlighted. Approximate Data-Flow Analysis. Often, it is necessary to sacrifice precision of the analysis for its speed. We define here a notion of approximate data flow information, which allows the analyzer a predetermined degree of conservative imprecision. For example, given a 5% imprecision level 0:05), the analyzer may output "available: 0.7," when the maximal fixed point solution is "available: 0.75." The intention of permitting underestimation is to reduce the analysis cost. When the analyzer is certain that the contribution of a node (and all its incoming paths) to the overall solution is less than the imprecision level, it can avoid analyzing the paths and assume at the node the most conservative data-flow fact. Because the algorithm in Figure 9 was designed to be informed, it naturally extends to approximate analysis. By knowing the precise contribution weight of each node as the analysis progresses, whenever the sum of weights in rhs at the highest interval level falls below ffl (the while-condition in line 2), we can terminate and guarantee the desired pre- cision. An alternative scenario is more attractive, however. When a low-weight node is selected in line 3, we throw it away. We can keep disregarding such nodes until their total weights exceed ffl. In essence, this approach performs analysis along hot paths [4], and on-line region formation [21]. The idea of terminating the analysis before it could find the precise solution was first applied in the implementation of interprocedural branch elimination [10]. Stopping after visiting a thousand nodes resulted in two magnitudes of analysis speedup, while most optimization opportunities were still discovered. However, without the approximate frequency analyzer developed in this paper, we were unable to a) determine the benefit of restructuring, b) select a profitable subset of nodes to duplicate, and c) get a bound on the amount of opportunities lost due to early termination. Algorithm complexity. In an arbitrary exploded graph, reduce loop may be (recursively) invoked on each node. Hence, each node may be visited at most NE times, where is the number of EG nodes, N the number of CFG nodes, and S the number of optimized expressions. With caching of results, then each node is processed in at most one invocation of the algorithm in Figure 9, yielding worst-case time complexity of O(N 2 real programs have loop nesting level bound by a small con- stant, the expected complexity is (NS), as in [2]. Although most existing demand-driven data-flow algorithms ([18, 23], [30] in particular) can be viewed (like ours) to operate on the principle of back-substituting flow functions into the right-hand side of the target variable, they do not focus on specifying a profitable order of substitutions (unlike ours) but rely instead on finding the fixed point it- eratively. Such an approach fails on infinite-height lattices where CFG loops keep always iterating towards a better approximation of the solution. Note that breaking each control flow cycle by inserting a widening operator [13] does not appear to resolve the problem because widening is a local adjustment primarily intended to approximate the solution. Therefore, in frequency analysis, too many iterations would be required to achieve an acceptable approximation. Instead of fixing the equation system locally, a global approach of structurally identifying intervals and reducing their cyclic dependences seems necessary. We have shown how to identify intervals and perform substitutions in interval order on demand, even when the exploded graph is not known prior to the analysis. We believe that existing demand methods can be extended to operate in a structural manner, enabling the application of loop-breaking rules. This would make the methods reminiscent of the elimination algorithms [29]. 6 Conclusion and Related Work The focus of this paper is to improve program transformations that constitute value-reuse optimizations commonly known as Partial Redundancy Elimination (PRE). In the long history of PRE research and implementation, three distinct transformations can be identified. The seminal paper by Morel and Renviose [27] and its derivations [11, 14, 15, 16, 17, 25] employ pure, non-speculative code motion. Second, the complete PRE by Steffen [31] is based on control flow restructuring. Third, navigated by path profile information, Gupta et al apply speculative code motion in order to avoid code-motion obstacles by controlled impairment of some paths [20]. In this work, we defined the code-motion-preventing (CMP) region, which is a CFG subgraph localizing adverse effects of control flow on the desired value reuse. The notion of the CMP is applied to enhance and integrate the three existing PRE transformations in the following ways, 1. Code motion and restructuring are integrated to remove all redundancies at minimal code growth cost (ComPRE). 2. Morel and Renviose's original method is expressed as a restricted (motion-only) case of the complete algorithm (CM-PRE). 3. We develop an algorithm whose power adjusts contin- Input: node n, expression e. Output: in sol , the probability of e being available at the exit of n. stack of reals (names sol , rhs refer always to top of stack) stack of sets of unsubstituted nodes n with weights rhs [n] post-order numbering of CFG nodes while rhs not empty do 3 select from rhs a node x with smallest post-dfs(x) 5 end while procedure substitute(node x) if x has not been visited, determine its flow function if x computes or kills e, adjust sol and remove x from rhs if Comp(x; e) - Transp(x; e) then 7 else if :Transp(n; e) then rhs [x] := 0:0; return back-edge is each edge that meets a loop-entry edge 8 if back-edge (y; x) exists then assume one back-edge per node substitute for y until x occurs on the r.h.s. 9 (s; r) := reduce loop(y; x) apply loop breaking rule: sum of infinite geom. sequence substitute "acyclic" predecessors for each non-backedge node z 2 pred(x) do end for x is now fully substituted 14 rhs [x] := 0:0; visited:push(x) substitute function reduce loop(node u, node v) while rhs contains unmarked nodes do 17 select from rhs an unmarked node x with lowest post-dfs(x) 19 end while reduce loop F post-dfs: H; G; 9 reduce loop(E; B) 14 rhs := 0:391 A 7 sol := 0:2818 unchanged / Final probability 7 rhs := 0:0 Figure 9: Demand-driven frequency analysis for availability of computations, and a trace of its execution. ually between the motion-only and the complete PRE in response to the program profile (PgPRE). 4. We demonstrate that speculation can be navigated precisely by edge profiles alone. 5. Path profiles are used to integrate the three transformations and balance their power at the level of CMP paths. While PRE is significantly improved through effective program transformations presented in this paper, a large orthogonal potential lies in detecting more redundancies. Some techniques have used powerful analysis to uncover more value reuse than the traditional PRE analysis [9, 11]. However, using only code motion, they fail to completely exploit the additional reuse opportunities. Thus, the transformations presented here are applicable in other styles of PRE as well, for example in elimination of loads. Ammons and Larus [4] developed a constant propagation optimization based on restructuring, namely on peeling of hot paths. In their analysis/transformation framework, re-structuring is used not only to exploit optimization opportunities previously detected by the analysis, as is our case, but also to improve the analysis precision by eliminating control flow merges from the hot paths. Even though our PRE cannot benefit from hot path separation (our distributive data-flow analysis preserves reuse opportunities across merges), a more complicated analysis (e.g., redundancy of array bound checks) would be improved by their approach. After the analysis, their algorithm recombines separated paths that present no useful opportunities. It is likely that path recombination can be integrated with code motion, as presented in this paper, to further reduce the code growth. In a global view, we have identified four main issues with path-sensitive program optimizations [8]: a) solving non-distributive problems without conservative approximation (e.g. non-linear constant propagation), b) collecting distinct opportunities (e.g., variable has different constant along each path), c) exploiting distinct opportunities (e.g., enabling folding of path-dependent constants through re- structuring), and d) directing the analysis effort towards hot paths. In the approach of Ammons and Larus, all four issues are attacked uniformly by separation of hot paths, their subsequent individual analysis, and recombination. Our approach is to reserve restructuring for the actual transforma- tion. This implies a different overall strategy: a) we solve non-distributive problems precisely along all paths by customizing the data-flow name space [9], b) we collect distinct opportunities through demand-driven analysis as in branch elimination [10], which is itself a form of constant propa- gation, c) we exploit all profitable opportunities with economical transformations, and d) avoid infrequent program regions using the approximation frequency analysis (the last three presented in this paper). Acknowledgments We are indebted to the elcor and Impact compiler teams for providing their experimental infrastructure. Sadun Anik, Ben-Chung Cheng, Brian Dietrich, John Gyllenhaal, and Scott Mahlke provided invaluable help during the implementation and experiments. Comments from Glenn Am- mons, Evelyn Duesterwald, Jim Larus, Mooly Sagiv, Bernhard Steffen, and the anonymous reviewers helped to improve the presentation of the paper. This research was partially supported by NSF PYI Award CCR-9157371, NSF grant CCR-9402226, and a grant from Hewlett-Packard to the University of Pittsburgh. --R Compilers Principles A program data flow analysis procedure. Exploiting hardware performance counters with flow and context sensitive profiling. Improving data-flow analysis with path profiles Aggressive inlining. Efficient path profiling. Effective partial redundancy elimination. A new algorithm for partial redundancy elimination based on SSA form. Abstract intrepretation: a unified lattice model for static analysis of programs by construction or approximation of fixpoints. How to analyze large programs efficiently and infor- matively Practical adaptation of the global optimization algorithm of Morel and Renvoise. "global optimization by suppression of partial redundancies" A variation of Knoop A practical framework for demand-driven interprocedural data flow analysis Path profile guided partial redundancy elimination using speculation. Partial redundancy elimination based on a cost-benefit analysis Demand Interprocedural Dataflow Analysis. Controlled node split- ting Optimal code motion: Theory and practice. Sentinel scheduling for VLIW and superscalar processors. Global optimization by supression of partial redundancies. Data flow frequency analysis. Elimination algorithms for data flow analysis. Precise interprocedural dataflow analysis with applications to constant propagation. Property oriented expansion. --TR Compilers: principles, techniques, and tools Elimination algorithms for data flow analysis How to analyze large programs efficiently and informatively A variation of Knoop, RuMYAMPERSANDuml;thing, and Steffen''s <italic>Lazy Code Motion</italic> Sentinel scheduling Effective partial redundancy elimination Optimal code motion A solution to a problem with Morel and Renvoise''s MYAMPERSANDldquo;Global optimization by suppression of partial redundanciesMYAMPERSANDrdquo; Practical adaption of the global optimization algorithm of Morel and Renvoise Demand interprocedural dataflow analysis Region-based compilation Fast, effective dynamic compilation Data flow frequency analysis Precise interprocedural dataflow analysis with applications to constant propagation Efficient path profiling Exploiting hardware performance counters with flow and context sensitive profiling Aggressive inlining Interprocedural conditional branch elimination A new algorithm for partial redundancy elimination based on SSA form Resource-sensitive profile-directed data flow analysis for code optimization Path-sensitive value-flow analysis A practical framework for demand-driven interprocedural data flow analysis Improving data-flow analysis with path profiles Global optimization by suppression of partial redundancies A program data flow analysis procedure Abstract interpretation Property-Oriented Expansion Controlled Node Splitting Path Profile Guided Partial Redundancy Elimination Using Speculation --CTR Daniel A. Connors , Wen-mei W. Hwu, Compiler-directed dynamic computation reuse: rationale and initial results, Proceedings of the 32nd annual ACM/IEEE international symposium on Microarchitecture, p.158-169, November 16-18, 1999, Haifa, Israel Jin Lin , Tong Chen , Wei-Chung Hsu , Pen-Chung Yew , Roy Dz-Ching Ju , Tin-Fook Ngai , Sun Chan, A compiler framework for speculative optimizations, ACM Transactions on Architecture and Code Optimization (TACO), v.1 n.3, p.247-271, September 2004 Zhang , Rajiv Gupta, Whole Execution Traces, Proceedings of the 37th annual IEEE/ACM International Symposium on Microarchitecture, p.105-116, December 04-08, 2004, Portland, Oregon Bernhard Scholz , Nigel Horspool , Jens Knoop, Optimizing for space and time usage with speculative partial redundancy elimination, ACM SIGPLAN Notices, v.39 n.7, July 2004 Rei Odaira , Kei Hiraki, Sentinel PRE: Hoisting beyond Exception Dependency with Dynamic Deoptimization, Proceedings of the international symposium on Code generation and optimization, p.328-338, March 20-23, 2005 Timothy Heil , James E. Smith, Relational profiling: enabling thread-level parallelism in virtual machines, Proceedings of the 33rd annual ACM/IEEE international symposium on Microarchitecture, p.281-290, December 2000, Monterey, California, United States David Gregg , Andrew Beatty , Kevin Casey , Brain Davis , Andy Nisbet, The case for virtual register machines, Science of Computer Programming, v.57 n.3, p.319-338, September 2005 Dhananjay M. Dhamdhere, E-path_PRE: partial redundancy elimination made easy, ACM SIGPLAN Notices, v.37 n.8, August 2002 Eduard Mehofer , Bernhard Scholz, Probabilistic data flow system with two-edge profiling, ACM SIGPLAN Notices, v.35 n.7, p.65-72, July 2000 Jin Lin , Tong Chen , Wei-Chung Hsu , Pen-Chung Yew , Roy Dz-Ching Ju , Tin-Fook Ngai , Sun Chan, A compiler framework for speculative analysis and optimizations, ACM SIGPLAN Notices, v.38 n.5, May Spyridon Triantafyllis , Matthew J. Bridges , Easwaran Raman , Guilherme Ottoni , David I. August, A framework for unrestricted whole-program optimization, ACM SIGPLAN Notices, v.41 n.6, June 2006 Max Hailperin, Cost-optimal code motion, ACM Transactions on Programming Languages and Systems (TOPLAS), v.20 n.6, p.1297-1322, Nov. 1998 Ran Shaham , Elliot K. Kolodner , Mooly Sagiv, Heap profiling for space-efficient Java, ACM SIGPLAN Notices, v.36 n.5, p.104-113, May 2001 Reinhard von Hanxleden , Ken Kennedy, A balanced code placement framework, ACM Transactions on Programming Languages and Systems (TOPLAS), v.22 n.5, p.816-860, Sept. 2000 Jingling Xue , Qiong Cai, A lifetime optimal algorithm for speculative PRE, ACM Transactions on Architecture and Code Optimization (TACO), v.3 n.2, p.115-155, June 2006 Zhang , Rajiv Gupta, Whole execution traces and their applications, ACM Transactions on Architecture and Code Optimization (TACO), v.2 n.3, p.301-334, September 2005 Uday P. Khedker , Dhananjay M. Dhamdhere, Bidirectional data flow analysis: myths and reality, ACM SIGPLAN Notices, v.34 n.6, June 1999 Keith D. Cooper , Li Xu, An efficient static analysis algorithm to detect redundant memory operations, ACM SIGPLAN Notices, v.38 n.2 supplement, p.97-107, February Keith D. Cooper , L. Taylor Simpson , Christopher A. Vick, Operator strength reduction, ACM Transactions on Programming Languages and Systems (TOPLAS), v.23 n.5, p.603-625, September 2001 eliminating array bounds checks on demand, ACM SIGPLAN Notices, v.35 n.5, p.321-333, May 2000 Litong Song , Krishna Kavi, What can we gain by unfolding loops?, ACM SIGPLAN Notices, v.39 n.2, February 2004 Glenn Ammons , James R. Larus, Improving data-flow analysis with path profiles, ACM SIGPLAN Notices, v.33 n.5, p.72-84, May 1998 Phung Hua Nguyen , Jingling Xue, Strength reduction for loop-invariant types, Proceedings of the 27th Australasian conference on Computer science, p.213-222, January 01, 2004, Dunedin, New Zealand Youtao Zhang , Rajiv Gupta, Timestamped whole program path representation and its applications, ACM SIGPLAN Notices, v.36 n.5, p.180-190, May 2001 K. V. Seshu Kumar, Value reuse optimization: reuse of evaluated math library function calls through compiler generated cache, ACM SIGPLAN Notices, v.38 n.8, August Sriraman Tallam , Xiangyu Zhang , Rajiv Gupta, Extending Path Profiling across Loop Backedges and Procedure Boundaries, Proceedings of the international symposium on Code generation and optimization: feedback-directed and runtime optimization, p.251, March 20-24, 2004, Palo Alto, California Qiong Cai , Jingling Xue, Optimal and efficient speculation-based partial redundancy elimination, Proceedings of the international symposium on Code generation and optimization: feedback-directed and runtime optimization, March 23-26, 2003, San Francisco, California Mary Lou Soffa, Load-reuse analysis: design and evaluation, ACM SIGPLAN Notices, v.34 n.5, p.64-76, May 1999 Matthew Arnold , Michael Hind , Barbara G. Ryder, Online feedback-directed optimization of Java, ACM SIGPLAN Notices, v.37 n.11, November 2002 Glenn Ammons , James R. Larus, Improving data-flow analysis with path profiles, ACM SIGPLAN Notices, v.39 n.4, April 2004 Motohiro Kawahito , Hideaki Komatsu , Toshio Nakatani, Partial redundancy elimination for access expressions by speculative code motion, SoftwarePractice & Experience, v.34 n.11, p.1065-1090, September 2004

speculative execution;demand-driven frequency data-flow analysis;control flow restructuring;partial redundancy elimination;profile-guided optimization

277715

An implementation of complete, asynchronous, distributed garbage collection.

Most existing reference-based distributed object systems include some kind of acyclic garbage collection, but fail to provide acceptable collection of cyclic garbage. Those that do provide such GC currently suffer from one or more problems: synchronous operation, the need for expensive global consensus or termination algorithms, susceptibility to communication problems, or an algorithm that does not scale. We present a simple, complete, fault-tolerant, asynchronous extension to the (acyclic) cleanup protocol of the SSP Chains system. This extension is scalable, consumes few resources, and could easily be adapted to work in other reference-based distributed object systems---rendering them usable for very large-scale applications.

Introduction Automatic garbage collection is an important feature for modern high-level languages. Although there is a lot of accumulated experience in local garbage collection, distributed programming still lacks effective cyclic garbage collection. A local garbage collector should be correct and complete. A distributed garbage collector should also be asynchronous (other spaces continue to work during a local garbage collection in one space), fault-tolerant (it works even with unreliable communications and space crashes), and scalable (since networks are connecting larger numbers of computers over increasing distances). Previously published distributed garbage collection algorithms fail in one or more of these requirements. In this paper we present a distributed garbage collector for distributed languages that provides all three of these desired properties. Moreover, the algorithm is simple to implement and consumes very few resources. The algorithm described in this paper was developed as part of a reference-based distributed object system for (a dialect of ML with object-oriented ex- Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. SIGPLAN PLDI'98 Montreal Canada tensions). Remote references are managed using the Stub- Scion Pair Chains (SSPC) system, extended with our cyclic detection algorithm. Although our system is based on transparent distributed references, our design assumptions are weak enough to support other kinds of distributed languages; those based on channels, for example (-calculus [8], join-calculus [3], and others). The next two sections of the paper introduce the basic mechanisms of remote references and the SSPC system for acyclic distributed garbage collection. Section 4 describes our cycle detection algorithm, and includes a short example showing how it works. Section 5 briefly investigates some issues related to our algorithm. Sections 6 and 7 analyze the algorithm in greater depth, and discuss some of the implementation issues surrounding it. The final two sections compare our algorithm with other recent work in distributed garbage collection and present our conclusions. Basics We consider a distributed system consisting of a set of spaces. Each space is a process, that has its own memory, its own local roots, and its own local garbage collector. A space can communicate with other spaces (on the same computer or a different one) by sending asynchronous messages. These messages may be lost, duplicated or delivered out of order. Distributed computation is effected by sending messages that invoke procedures in remote objects. These remote procedure calls (RPCs) have the same components as a local procedure call: a distinguished object that is to perform the call, zero or more arguments of arbitrary (includ- ing reference) type, and an optional result of arbitrary type. The result is delivered to the caller synchronously; in other words, the caller blocks for the duration of the procedure call. Encoding an argument or result for inclusion in a message is called marshaling; decoding by the message recipient is called unmarshaling. When an argument or result of an RPC has a reference type (i.e. it refers to an object) then this reference can serve for further RPCs from the recipient of the reference back to the argument/result object. The object is also protected from garbage collection while it remains reachable; i.e. until the last (local or remote) reference to it is deleted. In the following sections we will write nameX (A) to indicate a variable called name located on space X that contains information about object A. We will write a is increased to b to mean the variable a is set to the maximum of variable a and variable b. A stub(R) R Figure 1: A reference from A in space X to B in space Y . 2.1 Remote references Marshaled references to local or remote objects are sent in messages to be used in remote computations (e.g. for remote invocation). Such a reference R from object A in space X to object B in space Y is represented by two objects: stubX (R) and scionY (R). These are represented concretely by: ffl a local pointer in X from A to stubX (R); and ffl a local pointer in Y from scionY (R) to B. A scion corresponds to an incoming reference, and is treated as a root during local garbage collection. An object having one or more incoming references from remote spaces is therefore considered live by the local garbage collector, even in the absence of any local reference to that object. The stub is a local "proxy" for some remote object. It contains the location of its associated matching scion. Each scion has at most one matching stub, and each stub has exactly one matching scion. If several spaces contain stubs referring to some object B, then each will have a unique matching scion in B's space: one scion for each stub. A reference R is created by space Y and exported to some other space X as follows. First a new scion scionY (R) is created and marshaled into a message. The marshaled representation encodes the location of scionY (R) relative to X. The message is then sent to to X, where the location is unmarshaled to create stubX (R). 3 Stub-Scion Pair Chains The SSPC system [13] is a mechanism for distributed reference tracking similar to Network Objects [1] and supporting acyclic distributed garbage collection. It differs from Net-work Objects in several important respects, such as: a reduction of the number of messages required for sending a reference, lower latencies, fault-tolerance, and support for object migration. However, we will only describe here the part needed to understand its garbage collector. The garbage collector is based on reference listing (an extension of reference counting that is better suited to unreliable communications), with time-stamps on messages to avoid race conditions. The following explanation is based on the example shown in Figure 1. This simple example is easily generalizable to situations having more references and spaces. Each message is stamped by its sender with a monotonically increasing time. When a message containing R is sent by Y to X, the time-stamp of the message is stored in scionY (R) in a field called scionstamp. When the message is received by X, a field of stubX (R) called stubstamp is increased to the time-stamp of the message. For stubX (R), stubstamp contains the time-stamp of the most recent message containing R that was received from Y . Similarly for scionY (R), scionstamp is the time-stamp of the last message containing R that was sent to X. When object A becomes unreachable, stubX (R) is collected by the local garbage collector of space X. When stubX (R) is finalized, a value called increased to the stubstamp field of X. thresholdX [Y ] therefore contains the time-stamp of the last message received from Y that contained a reference to an object whose stub has since been reclaimed by the local garbage collector. After each garbage collection in space X, a message LIVE is sent to all the spaces in the immediate vicinity. The immediate vicinity of space X is the set of spaces that have stubs and scions whose associated scions and stubs are in X. The LIVE message sent to space Y contains the names of all the scions in Y that are still reachable from stubs in X. The value of thresholdX [Y ] is also sent in the LIVE message to Y . This value allows space Y to determine the most recent message that had been received by X from Y at the time the LIVE message was sent. Space Y extracts the list of scion names on receipt of the LIVE message. This list is compared to the list of existing scions in Y whose matching stubs are located in X. Any existing scions that are not mentioned in the list are now known to be unreachable from X, and are called suspect. A suspect scion can be deleted, provided there is no danger that a reference to it is currently in transit between X and Y . To prevent an incorrect deletion of a suspect scion, the scionstamp field of suspect scions is compared to the contained in the LIVE message. If then some stub referred to by a message sent after the last one containing R has been collected. This implies that the last message containing R was received before the LIVE was sent, and so any stub created for R from this message must no longer exist in space X. The suspect scion can therefore be deleted safely. To prevent out-of-order messages from violating this last condition, any messages from Y marked with a time-stamp smaller than the current value of thresholdX [Y ] are refused by space X. (thresholdX [Y ] must therefore be initialized with a time-stamp smaller than the time-stamp of the first messages to be received.) This mechanism is called threshold-filtering. The LIVE message can be extended by a "missing time- stamps" field, to inform the space Y of the time-stamps which are smaller than thresholdX [Y ] and which have not been received in a message yet. Y then has the possibility of re-sending the corresponding messages using a new time-stamp and newly-created scions, since older messages will be refused by the threshold-filtering. The above algorithm does not prevent premature deletion of scions contained in messages that are delayed in tran- sit. These deletions are however safe, since such delayed messages will be refused by the threshold-filtering. This situation can occur only if a more recent message arrives before some other delayed message, and the more recent message causes the creation of stubs that are subsequently deleted by a local garbage collection before the arrival of the delayed message. This can not happen with FIFO communications (such as TCP/IP). Moreover, threshold- filtering of delayed messages is not problematical for applications using unreliable communications (such as UDP), since these applications should be designed to function correctly even in the presence of message loss. Threshold-filtering and message loss due to faulty communication are indistinguishable to the application. The above distributed garbage collection mechanism is fault-tolerant. Unreliable communications can not create dangling pointers, and scions are never deleted in the case of crashed spaces that contain matching stubs (which supports extensions for handling crash recovery). Moreover, it is scalable because each space only sends and receives messages in its immediate vicinity, and asynchronous because local garbage collections in each space are allowed at any time with no need to synchronize with other spaces. However, the mechanism is not complete. Distributed cycles will never be deleted because of the use of reference- listing. The remainder of this paper presents our contri- bution: an algorithm to detect and cut distributed cycles, rendering the SSPC garbage collector complete. 4 Detection of free distributed cycles The detector of free distributed cycles is an extension to the SSPC garbage collector. Spaces may elect to use the acyclic SSPC GC without the detector extension (e.g. for scalability reasons). Spaces that choose to be involved in the detection of cycles are called participating spaces; other spaces are called non-participating spaces. Our detector will only detect cycles that lie entirely within a set of participating spaces. 4.1 Overview The algorithm is based on date propagation along chains of remote pointers. The useful property of this propagation is that reachable stubs receive increasing dates, whereas unreachable stubs (belonging to a distributed cycle) are eventually marked with constant dates. A threshold date is computed by a central server. Stubs marked with dates inferior to this threshold are known to have constant dates, and are therefore unreachable. Each participating space sends the minimum local date that it wishes to protect to the central server (stubs with these dates should not be collected). This information is based not only on the dates marked on local stubs, but also on the old dates propagated to outgoing references. The algorithm is asynchronous (most values are computed conservatively), tolerant to unreliable communications (using old values in computations is always safe, since most transmitted values are monotonically increasing) and benign to the normal SSPC garbage collector (non-participating spaces can work with an overlapping cluster of participating spaces, even if they do not take part in cycle detection). 4.2 Data structures and messages Stubs are extended with a time-stamp called stubdate. This is the time of the most recent trace (possibly on a remote site) during which the stub's chain was found to be rooted. Stubs have a second time-stamp, called olddate, which is the value of stubdate for the previous trace. Scions are extended with a time-stamp called sciondate. This is a copy of the most recently propagated stubdate from the scion's matching stub-i.e. the time of the most recent remote trace during which the scion's chain was found to be rooted. The stubdates from a space are propagated to their matching scions in some other space by sending a STUBDATES message. STUBDATES messages are stamped with the time of the trace that generated them. Each site has a vector, called cyclicthreshold, containing the time-stamp of the last STUBDATES message received from each remote space. The cyclicthreshold value for a remote space is periodically propagated back to that space by sending it a THRESHOLD message. The emission of THRESHOLD messages can be delayed by saving the cyclicthreshold values for a given time in a set called CyclicThresholdToSend until a particular event. Each site can protect outgoing references from remote garbage collection. For this, it computes a time called lo- calmin, which is sent in a LOCALMIN message to a dedicated site, the Detection Server, where the minimum localmin of all spaces is maintained in a variable called globalmin. LO- CALMIN messages are acknowledged by the Detection Server by sending back ACK messages. Finally, to compute localmin, each site maintains a per- space value, called ProtectNow, containing the new dates to be protected at next local garbage collection. These values are saved in a per-space table, called Protected Set, to be re-used and thus protected for some other local garbage collections. 4.3 The algorithm A Lamport clock is used to simulate global time at each participating space. 1 4.3.1 Local propagation The current date of the Lamport clock is incremented before each local garbage collection and used to mark local roots. Each scion's sciondate is marked with a date received from its matching stub. These dates are propagated from the local roots and scions to the stubdate field of all reachable stubs during the mark phase of garbage collection. If a stub is reachable from different roots marked with different dates then it is marked with the largest date. Such propagation is easy to implement with minor modifications to a tracing garbage collector. The scions are sorted by decreasing sciondate, and the object memory traced from each scion in turn. During the trace, the stubdate for any visited unmarked stub is increased to the sciondate of the scion from which the trace began. 4.3.2 Remote propagation A modified LIVE message, called STUBDATES, is sent to all participating spaces in the vicinity after a local garbage col- lection. This message serves to propagate the dates from all stubs to their matching scions. These dates will be propagated (locally, from scions to stubs) by the receiving space at next local garbage collection in that space. clock is implemented by sending the current date in all messages. (In our case, only those messages used for the detection of free cycles are concerned). When such a message is received, the current local date is increased to be strictly greater than the date in the message. increment current date; FIFO add(cyclicthresholdtosend set, (current date,cyclic threshold[])); Mark from root(local roots,current date); if scion.scion date ! globalmin then scion.pointer := NULL; else if scion.scion date = NOW then Mark from root(scion.pointer,current date); else Mark from root(scion.pointer,scion.scion date); if stub.stub date ? stub.olddate then decrease protect now[space] to stub.olddate; stub.olddate := stub.stub date;g FIFO add(protected set[space], (protect now[space],current date)); protect now[space] := current date; Send(space,STUBDATES,current date, (stub.stub id,stub.stub date)g); localmin := min(protected set[]) Send(server,LOCALMIN,current date,localmin); Figure 2: Pseudo-code for a local garbage collec- tion. The Protected Sets and Cyclicthreshold- ToSend Set are implemented by FIFO queues with three functions (add, head and remove). 4.3.3 Characterisation of free cycles Local roots are marked with the current date, which is always increasing. Reachable stubs are therefore marked with increasing dates. On the other hand, the dates on stubs included in unreachable cycles evolve in two different phases. In the first phase, the largest date on the cycle is propagated to every stub in the cycle. In the second phase, no new date can reach the cycle from a local root, and therefore the dates on the stubs in the cycle will remain constant forever. Since unreachable stubs have constant dates, whereas reachable stubs have increasing dates, it is possible to compute an increasing threshold date called globalmin. Reachable stubs and scions are always marked with dates larger than globalmin. On the other hand, globalmin will eventually become greater than the date of the stubs belonging to a given cycle. Scions whose dates are smaller than the current glob- almin are not traced during a local garbage collection. Stubs which were only reachable from these scions will therefore be collected. The normal acyclic SSPC garbage collector will then remove their associated scions, and eventually the entire cycle. 4.3.4 Computation of globalmin globalmin is computed by a dedicated space (the Detection Server) as the minimum of the localmin values sent to it by each participating space. 2 The central server always com- globalmin could be computed with a lazy distributed consensus. However, a central server is easier to implement (it can simply be Receive(space,STUBDATES, gc date ,stub set, increase cyclicthreshold[space] to gc date; old scion set := space.scions; space.scions := fg; old scion set, f find(scion.scion id,stub set, found, stub date); if found or scion.scionstamp ? threshold then f threshold then increase scion.scion date to stub date; space.scions := space.scions U fsciong gg Figure 3: Pseudo-code for the STUBDATES handler. The find function looks for a scion identifier in the set of stubs received in the message. If the stub is found in the set then found is set to true, and stub date is set to the date on the associated stub. If the scionstamp is greater than the threshold in the message then the scion is kept alive and its date is not set. if gc date?threshold date[space] then f increase threshold date[space] to current date; localmin[space]:=localmin; Figure 4: Pseudo-code for the Detection Server. The message is treated only if garbage collection date is the lattest date received from the space. putes globalmin from the most recently received value of localmin sent to it from each space. (See the pseudo-code in Figure 4.) 4.3.5 Computation of localmin localmin is recomputed after each local garbage collection in a given participating space. (The pseudo-code is shown in Figure 2.) We now introduce the notion of a probably-reachable stub. A stub is probably-reachable either when it has been used by the mutator for a remote operation (such as an invocation) since the last local garbage collection, or when its stubdate is increased during the local trace. This notion is neither a lower nor an upper approximation of reachability. A stub might be both reachable and not probably-reachable at the same time; it might also be probably-reachable and not reachable at some other time. However, on any reachable chain of remote references there is at least one probably-reachable stub for each different date on the chain. Therefore, since each space will "protect" the date of its probably-reachable stubs, all dates on the chain will be "protected". To detect probably-reachable stubs after the local trace, the previous stubdate of each stub (stored the olddate one of the participating spaces), and local networks (where such a collector is most useful) often have a centralized structure. FIFO head(cyclicthresholdtosend set, (date,cyclic thresholds to send[])); if date - gc date then f repeat f FIFO remove(cyclicthresholdtosend set, (date,cyclic thresholds to send[])); until (date == gc date); cyclic thresholds to send[space]); Figure 5: Pseudo-code for the ACK message handler. Old values in the CyclicthresholdToSend Set can be discarded, since they are smaller than those which will be sent in the THRESHOLD messages. Their corresponding ProtectNow values in the Protected Sets will therefore also be removed when the THRESHOLD messages is received. field), is compared to the newly-propagated stubdate. For each participating space in the immediate vicinity, a date (called ProtectNow) contains the minimum olddate of all stubs which have been detected as probably-reachable since the last local garbage collection. The value of ProtectNow for each space is saved in a per-space set, called Protected Set, after each garbage col- lection. ProtectNow is then re-initialized to the current date. The localmin for the space is then computed as the minimum of all ProtectNow values in all the Protected Sets. This new value of localmin is sent to the detection server in a LOCALMIN message. The next value of globalmin will be smaller than these olddates. All olddates associated with stubs that were detected probably-reachable since some of the latest garbage collections will therefore be protected by the new value of globalmin: stubs and scions marked with those dates will not be collected. 3 globalmin must protect the olddates rather than the stubdates. This is because the scions associated with probably- reachable stubs must be protected against collection, and these scions are marked with the olddate of their matching stub. In fact globalmin not only protects the associated scions, but also all references that are reachable from probably-reachable stubs and which are marked with the olddates of these stubs. 4.3.6 Reduction of the Protected Set STUBDATES and LOCALMIN messages both contain the date of the local garbage collection during which they were sent. When a STUBDATES message is received (see Figure 3), the per-space threshold CyclicThreshold is increased to the GC date contained in the message. The CyclicThreshold for each participating space is saved in the CyclicThresh- oldToSend Set before each local garbage collection. Each LOCALMIN message received by the Detection Server is acknowledged by a ACK message containing the same GC date. When this ACK message is received (see Figure 3 The slightly cryptic phrase "some of the latest garbage collec- tions" will be explained in full in the next section. Receive(space,THRESHOLD,cyclic FIFO head(protected set[space], (protect now,gc date)); while (gc date - cyclic threshold) f FIFO remove(protected set[space], (protect now, gc date)); FIFO head(protected set[space], (protect now, gc date)); Figure Pseudo-code for the THRESHOLD handler. , the CyclicThresholds saved in the CyclicThreshold- ToSend Set for the local garbage collection started at the GC date of the ACK message are sent to their associated space in THRESHOLD messages. Older values (for older local garbage collections) in the CyclicThresholdToSend Set are discarded (This is perfectly safe. When a space receives a THRESHOLD message it will perform all of the actions that should have been performed for any previous THRESHOLD messages that were lost). When a CyclicThreshold date is received in a THRESHOLD message, all older ProtectNow values in the Protected Set associated with the sending space are removed. (See Figure 6.) These values will no longer participate in the computation of globalmin. We can now explain the cryptic phrase "some of the latest garbage collections" that appeared in the previous section The olddate on a probably-reachable stub is protected by a ProtectNow in a Protected Set. It will continue to be protected for a certain time, until several events have oc- curred. The new stubdate must first be sent to the matching scion in a STUBDATES message. From there it is propagated from by a local trace to any outgoing stubs (new probably-reachable stubs in that space will be detected during this trace). The new localmin for that must then be received and used by the detection server (ensuring that the olddates on the newly detected probably-reachable stubs are protected by next values of globalmin). After this, the ACK message received from the detection server will trigger a THRESHOLD message containing a Cyclicthreshold equal to the GC date of the STUBDATES message (or greater if other STUBDATES messages have been received before the local garbage collection). Only after this THRESHOLD message is received will the the ProtectNow be removed from its Protected Set. 4.4 Example Figures 7, 8 and 9 show a simple example of distributed detection of free cycles. Spaces A and B are participating spaces; space C is the detection server. The system contains two distributed cycles C(1) and C(2), each containing two objects: OA(1) and OB (1) for C(1), OA(2) and OB (2) for C(2). C(1) is locally reachable in A, whereas C(2) has been unreachable since date 2. A local garbage collection in A at date 6 has propagated this date to stubA (1), which was previously marked with date 2. The Protected Set associated with B contains a single entry: a ProtectNow 2 at date 6. In figure 7, a local garbage collection occurs in B at date 8. The date 6, marked on scionB (1), is propagated to stubB (1) which was previously marked with 2. B saves the localmins: A -> 2 globalmin gc_date gc_date dates A A A A A Figure 7: After a local garbage collection at date 8 on space B, the new localmin 2 is sent to the detection server C. After the acknowledgment, the cyclic threshold 6 message is sent to A, which will remove this entry from its protected set. new ProtectNow 2 associated with A in its Protected Set. It then sends a STUBDATES message with the new stub- dates to A, and a LOCALMIN message with its new localmin 2 to the detection server. After saving this new localmin, the detection server sends an ACK message to B containing the same date as the original LOCALMIN message. A glob- almin value (possibly not up-to-date) can be piggybacked on this message. After reception of this ACK message, B sends a THRESHOLD message to A containing the date of the last STUBDATES message received from A. A consequently removes the associated ProtectNow entry from its protected set, which is now empty. In figure 8, a local garbage collection occurs in A at date 10. The current date 10 is propagated to stubA (1), previously marked with 6. The ProtectNow associated with B is therefore decreased to 6. stubA(2) does not participate in the computation of ProtectNow, since is still marked with 2. This ProtectNow is then saved in the Protected Set, and the new localmin (6) is sent to the detection server. After the reception of the ACK message from C, a THRESHOLD message is sent ot B which removes the associated entry from its Protected Set. However, its localmin on the detection server is still equal to 2, thus, preventing globalmin from increasing. In figure 9, a local garbage collection occurs in B at date 12. The new localmin computed in B is equal to 6. The new globalmin is therefore increased to 6. All scions marked with smaller dates will not be traced, starting from the moment that A and B receive this new value of globalmin. Consequently scionA(2) and scionB (2) will not be traced in subsequent garbage collections, and OA (2), OB (2), stubB (2) and stubA(2) will be collected by local garbage collections. At the same time, scionA(2) and scionB (2) will be collected by the SSPC garbage collector when STUBDATES messages that do not contain stubB (2) and stubA(2) are received by A and B respectively. The cycle C(2) has now been entirely collected. 5 Related issues 5.1 New remote references and non-participating spaces When a new remote reference is created, the stub olddate is set to the current date and the sciondate is initialized with a special date called NOW. Moreover, each time a scion location is resent to its associated space, a new stub may be created if the previous one had already been collected. sciondate is therefore re-initialized to NOW each time its scion's location is resent in a message. Scions marked with NOW propagate the current date at each garbage collection. A newly-created scion therefore behaves as a normal local root, until a new date is propagated by a STUBDATES message from its matching stub. The SSPC threshold is then compared to the scionstamp to ensure that all messages containing the scion have been received before fixing the sciondate. This mechanism is also used to allow incoming references from non-participating spaces. (STUBDATES messages will never be received from non-participating spaces.) The sciondates of their associated scions will therefore remain at NOW forever, and they will act as local roots. Distributed cycles that include these remote references will never be collected. This is safe, and does not impact the completeness of the algorithm for participating spaces. We must also cope with outgoing references to non-participating spaces. We must avoid putting entries in the Pro- localmins: A -> 6 Figure 8: After a new local garbage collection in A, localmin A is increased to 6. tected Sets for non-participating spaces, since no THRESHOLD messages will be received to remove such entries. (This would prevent localmin and hence globalmin from increas- ing, thus stalling the detection process.) A space must therefore only send STUBDATES messages to, and create entries in the protected sets for, known participating spaces. The list of participating spaces is maintained by the detection server, and is sent to other participating spaces whenever necessary (when new participating space arrives, when a space quits the detection process, or if a space is suspected of having crashed or is being too slow to respond). 5.2 Coping with mutator activity The mutator can create and delete remote references in the interval between local garbage collections. Dates on a remotely-reachable object might therfore never increase because of a "phantom reference": each time a local garbage collection occurs in a space from which the object is reach- able, the mutator gives the reference on the object to another space and deletes the local reference - just before the collection. Greater dates might therefore never be propagated to the object and the object would be detected as a it is still reachable (see Figure 10 for an example). Such transient references may move from stubs to scions (for invocation) or from scions to stubs (by reference pass- ing). In the first case, we mark the invoked scions with the current date (This prevents globalmin from stalling). In the second case, we ensure that each time a stub is used by the mutator (for invocation, or copy to/from another space) its olddate is used to increase the ProtectNow associated with the space of its matching scion. The date of the ProtectNow therefore always contains the minimum olddate of all the stubs that have been used in the interval between two local current stub R O 11O Figure 10: With its local reference to O 1 , A invokes stub 1 which creates a new local reference in B to O 2 . A deletes its local reference R 1 , and performs a new local garbage collection. stub 1 is therefore re-marked with 2, and localmin A is increased to 5. This is incorrect, since the cycle is reachable from B. This is the reason why the external mutator activity must be monitored by the detector of free cycles. garbage collections. This protects any object reachable from these stubs against such transient "phantom references". 5.3 Fault tolerance Our algorithm is tolerant to message loss and out-oforder delivery. The STUBDATES, THRESHOLD, LOCALMIN and ACK messages are only accepted if their sates are greater than those of the previously received such message. More- over, the computations are always conservative when using old values. Even LOCALMIN messages may be lost: no ACK messages will be sent and therefore no THRESHOLD will be localmins: A -> 6 Figure 9: After a new local garbage collection in B, localmin B is set to 6, and globalmin is increased to 6. Thus, the free cycle marked with 2 will be collected since its date is now smaller than globalmin. to other spaces, which will continue to protect the dates that the lost LOCALMIN messages would have protected. Crashed spaces (or spaces that are too slow to respond) are handled by the detection server, which can exclude any suspected space from the detection process by sending a special message to all participating spaces. The participating spaces set the sciondates for scions whose matching stubs are in the suspect space(s) to NOW, and remove all entries for the suspected spaces in their Protected Sets. Finally, the detection server may also crash. This does not stop acyclic garbage collection, and only delays cyclic garbage collection. A detection server can be restarted, and dynamically rebuild the list of participating spaces through some special recovery protocol. It then waits for each participating space to send a new localmin value before computing a new up-to-date value for globalmin. 6 Analysis We can estimate the worst-case time needed to collect a newly unreachable cycle. It is the time needed to propagate dates greater than those on the cycle to all reachable stubs. Assuming that spaces perform local garbage collections at approximately the same rate, we define a period to be the time necessary for spaces to perform a new local garbage collection. The time needed to collect the cycle is equal to the product of the length of the largest chain of reachable references by the period: We can also estimate the number and the size of the messages that are sent after a local garbage collection. There is one LIVE message (sent by the SSPC garbage collector), plus one STUBDATES message and one THRESHOLD message sent for each space in the immediate vicinity. The first two messages can be concatenated into a single network message. Hence there are only two messages sent for each space in the vicinity. The STUBDATES message contains one identifier and one date for each live stub referring to the destination space, plus the SSPC threshold time-stamp. The THRESHOLD message contains only the CyclicThreshold value for the destination space. One LOCALMIN message is also sent to the detection server, and one ACK message sent back from the server. The Protected Set contains triples for each space in the vicinity. For a space X in the vicinity of Y , the number of triples for X in the Protected Set of Y is equal to the number of local garbage collections that have occurred on Y since the last garbage collection on X. If the frequencies of the garbage collections in the different participating spaces are similar, the Protected Set should not grow too much. If one space requires too many garbage collections, and its Protected Set becomes too large, it should avoid performing cyclic detection after each garbage collection (but not stop garbage collections) until sufficient entries in its Protected Set have been removed. Finally, a very large number of spaces may use the same detection server. The server only contains two dates per participating space, and the computation of the minimum of this array should not be expensive. 7 Implementation Our algorithm has been incorporated into an implementation of the SSP Chains system written in Objective-CAML [5], using the Unix and Thread modules [6]. The Objective-Caml implementation of SSPC consists of 1300 lines of code, of which 200 are associated with the cyclic GC algorithm. The propagation of dates by tracing was implemented as a minor modification to the existing Caml garbage collector [2]. The Mark from root(roots) function was changed into Mark from root(roots,date), which marks stubs reachable from a set of roots with the given date. This function is then applied first to the normal local roots with the current date (which is always greater than all the dates on scions), and then to sets of scions sorted by decreasing dates. Each reachable stub is therefore only marked once, with the date of the first root from which it is reachable. Finalization of stubs (required for updating the threshold when they are collected) is implemented by using a list of pairs. Each pair contains a weak pointer to a stub and a stubstamp field. After a garbage collection, the weak pointers are tested to determine if their referent objects are still live. The stubstamp field is used to update the threshold if the weak pointer is found to be dangling. The Protected Set is implemented as a FIFO queue for each participating space. The head of the queue contains the ProtectNow value, which can be modified by the mutator between local garbage collections. When a THRESHOLD message is received, entries are removed from the tail of the queue until the last entry has a date greater than the one in the message. Finally, localmin is computed as the minimum of all entries in all queues. Objective-CAML has high-level capabilities to automatically marshal and unmarshal symbolic messages, easing the implementation of complex protocols. Some modification of the compiler and the standard object library was needed to enable dynamic creation of classes of stubs and dynamic type verification for SSPC. However, these modifications are not related to either the acyclic GC or the cycle detector algorithm 8 Related work 8.1 Hughes'algorithm Our algorithm was inspired Hughes' algorithm. In Hughes' algorithm, each local garbage collection provokes a global trace and propagates the starting date of the trace. How- ever, the threshold date is computed by a termination algorithm (due to Rana [11]). The date on a stub therefore represents the starting date of the most recent global trace in which the stub was detected as reachable. If the threshold is the starting date of a terminated global trace, then any stub marked with a strictly smaller date has not been detected as reachable by this terminated global trace. It can therefore be collected safely. However, the termination algorithm used in this algorithm requires a global clock, instantaneous communication, and does not support failures. Moreover, each local garbage collection in one space triggers new computations in all of the participating spaces. Such behavior is not suitable for a large-scale fault-tolerant system. 8.2 Recent work Detecting free cycles has been addressed by several researchers. A good survey can be found in [10]. We will only present more recent work below. All three of the recent algorithms are based on partitioning into groups of spaces or nodes. Cycles are only collected when they are included entirely within a single partition. Heuristics are used to improve the partitioning. These algorithms are complex, and may be difficult to implement. Moreover, their efficiency depends greatly on the choice of heuristic for selecting "suspect objects". Maheshwari and Liskov's [7] work is based on back-tracing. The group is traced in the opposite direction to references, starting from objects that are suspected to belong to an unreachable cycle. An heuristic based on distance selects "suspected objects". If the backward trace does not encounter a local root, the object is on a free cycle. Their detector is asynchronous, fault-tolerant, and well-adapted to large-scale systems. Nevertheless, back-tracing requires extra data structures for each remote reference. Further- more, every suspected cycle needs one trace, whereas our algorithm collects all cycles concurrently. Rodrigues and Jones's [12] cyclic garbage collector was inspired by Lang et al.[4], dividing the network into groups of processes. The algorithm collects cycles of garbage contained entirely within a group. The main improvement is that only suspect objects (according to an heuristics such as Maheshwari and Liskov's distance) are traced. Global synchronization is needed to terminate the detection. It is difficult to know how the algorithm behaves when the group becomes very large. The DMOS garbage collector [9] has some desirable prop- erties: safety, completeness, non-disruptiveness, incremen- tality, and scalability. Spaces are divided into a number of disjoint blocks (called "cars"). Cars from different spaces are grouped together into trains. Reachable data is copied from cars in one train to cars in other trains. Unreachable data and cycles contained in one car or one train are left behind and can be collected. Completeness is guaranteed by the order of collections. This algorithm is highly complex and has not been implemented. Moreover, problems relating to fault-tolerance are not addressed by the authors. 9 Conclusion We have described a complete distributed garbage collector, created by extending an acyclic distributed garbage collector with a detector of distributed garbage cycles. Our garbage collector has some desirable properties: asynchrony between participating spaces, fault-tolerance (messages can be lost, participating spaces and servers can crash), low resource requirements (memory, messages and time), and finally ease of implementation. It seems well adapted to large-scale distributed systems since it supports non-participating spaces, and consequently clusters of cyclically-collected spaces within larger groups of interoperating spaces. We are currently working on a new implementation for the Join-Calculus language. Future work includes the handling of overlapping sets of participating spaces, protocols for server recovery, and performance mesurements. Acknowledgments The authors would like to thank Neilze Dorta for her study of recent cyclic garbage collectors. We also thank Jean-Jacques Levy and Damien Doligez for their valuable comments and suggestions on improving this paper. --R Network objects. A concurrent The reflexive chemical abstract machine and the join-calculus Garbage collecting the world. The objective-caml system software Unix system programming in caml light. Collecting cyclic distributed garbage by back tracing. A calculus of mobile processes I and II. A survey of distributed garbage collection techniques. A distributed solution to the distributed termination problem. A cyclic distributed garbage collector for Network Ob- jects chains: Robust --TR Garbage collecting the world A concurrent, generational garbage collector for a multithreaded implementation of ML A calculus of mobile processes, II Network objects Collecting distributed garbage cycles by back tracing Garbage collecting the world A Survey of Distributed Garbage Collection Techniques A Cyclic Distributed Garbage Collector for Network Objects A Calculus of Mobile Agents --CTR Fabrice Le Fessant, Detecting distributed cycles of garbage in large-scale systems, Proceedings of the twentieth annual ACM symposium on Principles of distributed computing, p.200-209, August 2001, Newport, Rhode Island, United States Stephen M. Blackburn , Richard L. Hudson , Ron Morrison , J. Eliot B. Moss , David S. Munro , John Zigman, Starting with termination: a methodology for building distributed garbage collection algorithms, Australian Computer Science Communications, v.23 n.1, p.20-28, January-February 2001 Abhay Vardhan , Gul Agha, Using passive object garbage collection algorithms for garbage collection of active objects, ACM SIGPLAN Notices, v.38 n.2 supplement, February Michael Hicks , Suresh Jagannathan , Richard Kelsey , Jonathan T. Moore , Cristian Ungureanu, Transparent communication for distributed objects in Java, Proceedings of the ACM 1999 conference on Java Grande, p.160-170, June 12-14, 1999, San Francisco, California, United States Luc Moreau , Peter Dickman , Richard Jones, Birrell's distributed reference listing revisited, ACM Transactions on Programming Languages and Systems (TOPLAS), v.27 n.6, p.1344-1395, November 2005 Laurent Amsaleg , Michael J. Franklin , Olivier Gruber, Garbage collection for a client-server persistent object store, ACM Transactions on Computer Systems (TOCS), v.17 n.3, p.153-201, Aug. 1999

distributed object systems;storage management;garbage collection;reference tracking

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The implementation of the Cilk-5 multithreaded language.

The fifth release of the multithreaded language Cilk uses a provably good "work-stealing" scheduling algorithm similar to the first system, but the language has been completely redesigned and the runtime system completely reengineered. The efficiency of the new implementation was aided by a clear strategy that arose from a theoretical analysis of the scheduling algorithm: concentrate on minimizing overheads that contribute to the work, even at the expense of overheads that contribute to the critical path. Although it may seem counterintuitive to move overheads onto the critical path, this "work-first" principle has led to a portable Cilk-5 implementation in which the typical cost of spawning a parallel thread is only between 2 and 6 times the cost of a C function call on a variety of contemporary machines. Many Cilk programs run on one processor with virtually no degradation compared to equivalent C programs. This paper describes how the work-first principle was exploited in the design of Cilk-5's compiler and its runtime system. In particular, we present Cilk-5's novel "two-clone" compilation strategy and its Dijkstra-like mutual-exclusion protocol for implementing the ready deque in the work-stealing scheduler.

Introduction Cilk is a multithreaded language for parallel programming that generalizes the semantics of C by introducing linguistic constructs for parallel control. The original Cilk-1 release [3, 4, 18] featured a provably efficient, randomized, "work- stealing" scheduler [3, 5], but the language was clumsy, because parallelism was exposed "by hand" using explicit continuation passing. The Cilk language implemented by This research was supported in part by the Defense Advanced Research Projects Agency (DARPA) under Grant N00014-94-1-0985. Computing facilities were provided by the MIT Xolas Project, thanks to a generous equipment donation from Sun Microsystems. To appear in Proceedings of the 1998 ACM SIGPLAN Conference on Programming Language Design and Implementation (PLDI), Montreal, Canada, June 1998. our latest Cilk-5 release [8] still uses a theoretically efficient scheduler, but the language has been simplified considerably. It employs call/return semantics for parallelism and features a linguistically simple "inlet" mechanism for nondeterministic control. Cilk-5 is designed to run efficiently on contemporary symmetric multiprocessors (SMP's), which feature hardware support for shared memory. We have coded many applications in Cilk, including the ?Socrates and Cilkchess chess-playing programs which have won prizes in international competitions. The philosophy behind Cilk development has been to make the Cilk language a true parallel extension of C, both semantically and with respect to performance. On a parallel computer, Cilk control constructs allow the program to execute in parallel. If the Cilk keywords for parallel control are elided from a Cilk program, however, a syntactically and semantically correct C program results, which we call the C elision (or more generally, the serial elision) of the Cilk program. Cilk is a faithful extension of C, because the C elision of a Cilk program is a correct implementation of the semantics of the program. Moreover, on one processor, a parallel Cilk program "scales down" to run nearly as fast as its C elision. Unlike in Cilk-1, where the Cilk scheduler was an identifiable piece of code, in Cilk-5 both the compiler and runtime system bear the responsibility for scheduling. To obtain ef- ficiency, we have, of course, attempted to reduce scheduling overheads. Some overheads have a larger impact on execution time than others, however. A theoretical understanding of Cilk's scheduling algorithm [3, 5] has allowed us to identify and optimize the common cases. According to this abstract theory, the performance of a Cilk computation can be characterized by two quantities: its work , which is the total time needed to execute the computation serially, and its critical-path length , which is its execution time on an infinite number of processors. (Cilk provides instrumentation that allows a user to measure these two quantities.) Within Cilk's scheduler, we can identify a given cost as contributing to either work overhead or critical-path overhead. Much of the efficiency of Cilk derives from the following principle, which we shall justify in Section 3. The work-first principle: Minimize the scheduling overhead borne by the work of a computation. Specifically, move overheads out of the work and onto the critical path. The work-first principle played an important role during the design of earlier Cilk systems, but Cilk-5 exploits the principle more extensively. The work-first principle inspired a "two-clone" strategy for compiling Cilk programs. Our cilk2c compiler [23] is a source-to-source translator that transforms a Cilk source into a C postsource which makes calls to Cilk's runtime library. The C postsource is then run through the gcc compiler to produce object code. The cilk2c compiler produces two clones of every Cilk procedure-a "fast" clone and a "slow" clone. The fast clone, which is identical in most respects to the C elision of the Cilk program, executes in the common case where serial semantics suffice. The slow clone is executed in the infrequent case that parallel semantics and its concomitant bookkeeping are required. All communication due to scheduling occurs in the slow clone and contributes to critical-path overhead, but not to work overhead The work-first principle also inspired a Dijkstra-like [11], shared-memory, mutual-exclusion protocol as part of the runtime load-balancing scheduler. Cilk's scheduler uses a "work-stealing" algorithm in which idle processors, called thieves, "steal" threads from busy processors, called vic- tims. Cilk's scheduler guarantees that the cost of stealing contributes only to critical-path overhead, and not to work overhead. Nevertheless, it is hard to avoid the mutual-exclusion costs incurred by a potential victim, which contribute to work. To minimize work overhead, instead of using locking, Cilk's runtime system uses a Dijkstra-like protocol, which we call the THE protocol, to manage the runtime deque of ready threads in the work-stealing algorithm. An added advantage of the THE protocol is that it allows an exception to be signaled to a working processor with no additional work overhead, a feature used in Cilk's abort mechanism The remainder of this paper is organized as follows. Section 2 overviews the basic features of the Cilk language. Section 3 justifies the work-first principle. Section 4 describes how the two-clone strategy is implemented, and Section 5 presents the THE protocol. Section 6 gives empirical evidence that the Cilk-5 scheduler is efficient. Finally, Section 7 presents related work and offers some conclusions. 2 The Cilk language This section presents a brief overview of the Cilk extensions to C as supported by Cilk-5. (For a complete description, consult the Cilk-5 manual [8].) The key features of the language are the specification of parallelism and synchroniza- tion, through the spawn and sync keywords, and the specification of nondeterminism, using inlet and abort. #include !stdlib.h? #include !stdio.h? #include !cilk.h? cilk int fib (int n) if (n!2) return n; else - int x, return cilk int main (int argc, char *argv[]) int n, result; printf ("Result: %d"n", result); return 0; Figure 1: A simple Cilk program to compute the nth Fibonacci number in parallel (using a very bad algorithm). The basic Cilk language can be understood from an example Figure 1 shows a Cilk program that computes the nth Fibonacci number. 1 Observe that the program would be an ordinary C program if the three keywords cilk, spawn, and sync are elided. The keyword cilk identifies fib as a Cilk procedure, which is the parallel analog to a C function. Parallelism is created when the keyword spawn precedes the invocation of a procedure. The semantics of a spawn differs from a C function call only in that the parent can continue to execute in parallel with the child, instead of waiting for the child to complete as is done in C. Cilk's scheduler takes the responsibility of scheduling the spawned procedures on the processors of the parallel computer. A Cilk procedure cannot safely use the values returned by its children until it executes a sync statement. The sync statement is a local "barrier," not a global one as, for ex- ample, is used in message-passing programming. In the Fibonacci example, a sync statement is required before the statement return (x+y) to avoid the anomaly that would occur if x and y are summed before they are computed. In addition to explicit synchronization provided by the sync statement, every Cilk procedure syncs implicitly before it returns, thus ensuring that all of its children terminate before it does. Ordinarily, when a spawned procedure returns, the returned value is simply stored into a variable in its parent's frame: This program uses an inefficient algorithm which runs in exponential time. Although logarithmic-time methods are known [9, p. 850], this program nevertheless provides a good didactic example. cilk int fib (int n) int inlet void summer (int result) x += result; return; if (n!2) return n; else - summer(spawn fib (n-1)); summer(spawn fib (n-2)); return Figure 2: Using an inlet to compute the nth Fibonnaci number. Occasionally, one would like to incorporate the returned value into the parent's frame in a more complex way. Cilk provides an inlet feature for this purpose, which was inspired in part by the inlet feature of TAM [10]. An inlet is essentially a C function internal to a Cilk pro- cedure. In the normal syntax of Cilk, the spawning of a procedure must occur as a separate statement and not in an expression. An exception is made to this rule if the spawn is performed as an argument to an inlet call. In this case, the procedure is spawned, and when it returns, the inlet is invoked. In the meantime, control of the parent procedure proceeds to the statement following the inlet call. In princi- ple, inlets can take multiple spawned arguments, but Cilk-5 has the restriction that exactly one argument to an inlet may be spawned and that this argument must be the first argument. If necessary, this restriction is easy to program around. Figure illustrates how the fib() function might be coded using inlets. The inlet summer() is defined to take a returned value result and add it to the variable x in the frame of the procedure that does the spawning. All the variables of fib() are available within summer(), since it is an internal function of fib(). 2 No lock is required around the accesses to x by summer, because Cilk provides atomicity implicitly. The concern is that the two updates might occur in parallel, and if atomicity is not imposed, an update might be lost. Cilk provides implicit atomicity among the "threads" of a procedure in- stance, where a thread is a maximal sequence of instructions ending with a spawn, sync, or return (either explicit or implicit) statement. An inlet is precluded from containing spawn and sync statements, and thus it operates atomically as a single thread. Implicit atomicity simplifies reasoning 2 The C elision of a Cilk program with inlets is not ANSI C, because ANSI C does not support internal C functions. Cilk is based on Gnu C technology, however, which does provide this support. about concurrency and nondeterminism without requiring locking, declaration of critical regions, and the like. Cilk provides syntactic sugar to produce certain commonly used inlets implicitly. For example, the statement x += spawn fib(n-1) conceptually generates an inlet similar to the one in Figure 2. Sometimes, a procedure spawns off parallel work which it later discovers is unnecessary. This "speculative" work can be aborted in Cilk using the abort primitive inside an in- let. A common use of abort occurs during a parallel search, where many possibilities are searched in parallel. As soon as a solution is found by one of the searches, one wishes to abort any currently executing searches as soon as possible so as not to waste processor resources. The abort statement, when executed inside an inlet, causes all of the already-spawned children of the procedure to terminate. We considered using "futures" [19] with implicit synchro- nization, as well as synchronizing on specific variables, instead of using the simple spawn and sync statements. We realized from the work-first principle, however, that different synchronization mechanisms could have an impact only on the critical-path of a computation, and so this issue was of secondary concern. Consequently, we opted for implementation simplicity. Also, in systems that support relaxed memory-consistency models, the explicit sync statement can be used to ensure that all side-effects from previously spawned subprocedures have occurred. In addition to the control synchronization provided by sync, Cilk programmers can use explicit locking to synchronize accesses to data, providing mutual exclusion and atomicity. Data synchronization is an overhead borne on the work, however, and although we have striven to minimize these overheads, fine-grain locking on contemporary processors is expensive. We are currently investigating how to incorporate atomicity into the Cilk language so that protocol issues involved in locking can be avoided at the user level. To aid in the debugging of Cilk programs that use locks, we have been developing a tool called the "Nonde- [7, 13], which detects common synchronization bugs called data races. 3 The work-first principle This section justifies the work-first principle stated in Section 1 by showing that it follows from three assumptions. First, we assume that Cilk's scheduler operates in practice according to the theoretical analysis presented in [3, 5]. Sec- ond, we assume that in the common case, ample "parallel slackness" [28] exists, that is, the average parallelism of a Cilk program exceeds the number of processors on which we run it by a sufficient margin. Third, we assume (as is indeed the case) that every Cilk program has a C elision against which its one-processor performance can be measured. The theoretical analysis presented in [3, 5] cites two fundamental lower bounds as to how fast a Cilk program can run. Let us denote by TP the execution time of a given computation on P processors. Then, the work of the computation is and its critical-path length is . For a computation with work, the lower bound TP - T1=P must hold, because at most P units of work can be executed in a single step. In addition, the lower bound TP - must hold, since a finite number of processors cannot execute faster than an infinite Cilk's randomized work-stealing scheduler [3, 5] executes a Cilk computation on P processors in expected time assuming an ideal parallel computer. This equation resembles "Brent's theorem'' [6, 15] and is optimal to within a constant factor, since T1=P and are both lower bounds. We call the first term on the right-hand side of Equation (1) the work term and the second term the critical-path term. Importantly, all communication costs due to Cilk's scheduler are borne by the critical-path term, as are most of the other scheduling costs. To make these overheads explicit, we define the critical-path overhead to be the smallest constant c1 such that The second assumption needed to justify the work-first principle focuses on the "common-case" regime in which a parallel program operates. Define the average parallelism as which corresponds to the maximum possible speedup that the application can obtain. Define also the parallel slackness [28] to be the ratio P=P . The assumption of parallel slackness is that P=P AE c1 , which means that the number P of processors is much smaller than the average parallelism P . Under this assumption, it follows that T1=P AE c1T1 , and hence from Inequality (2) that and we obtain linear speedup. The critical-path overhead c1 has little effect on performance when sufficient slackness exists, although it does determines how much slackness must exist to ensure linear speedup. Whether substantial slackness exists in common applications is a matter of opinion and empiricism, but we suggest that slackness is the common case. The expressiveness of Cilk makes it easy to code applications with large amounts of parallelism. For modest-sized problems, many applications exhibit an average parallelism of over 200, yielding substantial slackness on contemporary SMP's. Even on Sandia National Laboratory's Intel Paragon, which contains 1824 nodes, the ?Socrates chess program (coded in Cilk-1) ran in its linear-speedup regime during the 1995 ICCA World Computer Chess Championship (where it placed second in a field of 24). Section 6 describes a dozen other diverse applications which were run on an 8-processor SMP with 3 This abstract model of execution time ignores real-life details, such as memory-hierarchy effects, but is nonetheless quite accurate [4]. considerable parallel slackness. The parallelisim of these applications increases with problem size, thereby ensuring they will run well on large machines. The third assumption behind the work-first principle is that every Cilk program has a C elision against which its one-processor performance can be measured. Let us denote by TS the running time of the C elision. Then, we define the work overhead by Incorporating critical-path and work overheads into Inequality (2) yields since we assume parallel slackness. We can now restate the work-first principle precisely. Minimize c1 , even at the expense of a larger c1 , because c1 has a more direct impact on performance. Adopting the work-first principle may adversely affect the ability of an application to scale up, however, if the critical-path overhead c1 is too large. But, as we shall see in Section 6, critical-path overhead is reasonably small in Cilk-5, and many applications can be coded with large amounts of parallelism. The work-first principle pervades the Cilk-5 implementa- tion. The work-stealing scheduler guarantees that with high probability, only O(PT1) steal (migration) attempts occur (that is, O(T1 ) on average per processor), all costs for which are borne on the critical path. Consequently, the scheduler for Cilk-5 postpones as much of the scheduling cost as possible to when work is being stolen, thereby removing it as a contributor to work overhead. This strategy of amortizing costs against steal attempts permeates virtually every decision made in the design of the scheduler. 4 Cilk's compilation strategy This section describes how our cilk2c compiler generates C postsource from a Cilk program. As dictated by the work- first principle, our compiler and scheduler are designed to reduce the work overhead as much as possible. Our strategy is to generate two clones of each procedure-a fast clone and a slow clone. The fast clone operates much as does the C elision and has little support for parallelism. The slow clone has full support for parallelism, along with its concomitant overhead. We first describe the Cilk scheduling algorithm. Then, we describe how the compiler translates the Cilk language constructs into code for the fast and slow clones of each procedure. Lastly, we describe how the runtime system links together the actions of the fast and slow clones to produce a complete Cilk implementation. As in lazy task creation [24], in Cilk-5 each proces- sor, called a worker , maintains a ready deque (doubly- ended queue) of ready procedures (technically, procedure instances). Each deque has two ends, a head and a tail , from which procedures can be added or removed. A worker operates locally on the tail of its own deque, treating it much int fib (int n) 3 fib-frame *f; frame pointer 8 return n; int x, live vars do C call return 0; frame stolen 22 return (x+y); Figure 3: The fast clone generated by cilk2c for the fib procedure from Figure 1. The code for the second spawn is omitted. The functions alloc and free are inlined calls to the runtime system's fast memory allocator. The signature fib sig contains a description of the fib procedure, including a pointer to the slow clone. The push and pop calls are operations on the scheduling deque and are described in detail in Section 5. as C treats its call stack, pushing and popping spawned activation frames. When a worker runs out of work, it becomes a thief and attempts to steal a procedure another worker, called its victim . The thief steals the procedure from the head of the victim's deque, the opposite end from which the victim is working. When a procedure is spawned, the fast clone runs. Whenever a thief steals a procedure, however, the procedure is converted to a slow clone. The Cilk scheduler guarantees that the number of steals is small when sufficient slackness exists, and so we expect the fast clones to be executed most of the time. Thus, the work-first principle reduces to minimizing costs in the fast clone, which contribute more heavily to work overhead. Minimizing costs in the slow clone, although a desirable goal, is less important, since these costs contribute less heavily to work overhead and more to critical-path overhead. We minimize the costs of the fast clone by exploiting the structure of the Cilk scheduler. Because we convert a procedure to its slow clone when it is stolen, we maintain the invariant that a fast clone has never been stolen. Further- more, none of the descendants of a fast clone have been stolen either, since the strategy of stealing from the heads of ready deques guarantees that parents are stolen before their children. As we shall see, this simple fact allows many optimizations to be performed in the fast clone. We now describe how our cilk2c compiler generates post- source C code for the fib procedure from Figure 1. An example of the postsource for the fast clone of fib is given in Figure 3. The generated C code has the same general structure as the C elision, with a few additional statements. In lines 4-5, an activation frame is allocated for fib and initialized. The Cilk runtime system uses activation frames to represent procedure instances. Using techniques similar to [16, 17], our inlined allocator typically takes only a few cycles. The frame is initialized in line 5 by storing a pointer to a static structure, called a signature, describing fib. The first spawn in fib is translated into lines 12-18. In lines 12-13, the state of the fib procedure is saved into the activation frame. The saved state includes the program counter, encoded as an entry number, and all live, dirty vari- ables. Then, the frame is pushed on the runtime deque in lines 14-15. 4 Next, we call the fib routine as we would in C. Because the spawn statement itself compiles directly to its C elision, the postsource can exploit the optimization capabilities of the C compiler, including its ability to pass arguments and receive return values in registers rather than in memory. After fib returns, lines 17-18 check to see whether the parent procedure has been stolen. If it has, we return immediately with a dummy value. Since all of the ancestors have been stolen as well, the C stack quickly unwinds and control is returned to the runtime system. 5 The protocol to check whether the parent procedure has been stolen is quite subtle-we postpone discussion of its implementation to Section 5. If the parent procedure has not been stolen, it continues to execute at line 19, performing the second spawn, which is not shown. In the fast clone, all sync statements compile to no-ops. Because a fast clone never has any children when it is exe- cuting, we know at compile time that all previously spawned procedures have completed. Thus, no operations are required for a sync statement, as it always succeeds. For exam- ple, line 20 in Figure 3, the translation of the sync statement is just the empty statement. Finally, in lines 21-22, fib deallocates the activation frame and returns the computed result to its parent procedure. The slow clone is similar to the fast clone except that it provides support for parallel execution. When a procedure is stolen, control has been suspended between two of the procedure's threads, that is, at a spawn or sync point. When the slow clone is resumed, it uses a goto statement to restore the program counter, and then it restores local variable state from the activation frame. A spawn statement is translated in the slow clone just as in the fast clone. For a sync statement, cilk2c inserts a call to the runtime system, which checks to see whether the procedure has any spawned children that have not returned. Although the parallel book- 4 If the shared memory is not sequentially consistent, a memory fence must be inserted between lines 14 and 15 to ensure that the surrounding writes are executed in the proper order. 5 The setjmp/longjmp facility of C could have been used as well, but our unwinding strategy is simpler. keeping in a slow clone is substantial, it contributes little to work overhead, since slow clones are rarely executed. The separation between fast clones and slow clones also allows us to compile inlets and abort statements efficiently in the fast clone. An inlet call compiles as efficiently as an ordinary spawn. For example, the code for the inlet call from Figure compiles similarly to the following Cilk code: Implicit inlet calls, such as x += spawn fib(n-1), compile directly to their C elisions. An abort statement compiles to a no-op just as a sync statement does, because while it is executing, a fast clone has no children to abort. The runtime system provides the glue between the fast and slow clones that makes the whole system work. It includes protocols for stealing procedures, returning values between processors, executing inlets, aborting computation subtrees, and the like. All of the costs of these protocols can be amortized against the critical path, so their overhead does not significantly affect the running time when sufficient parallel slackness exists. The portion of the stealing protocol executed by the worker contributes to work overhead, however, thereby warranting a careful implementation. We discuss this protocol in detail in Section 5. The work overhead of a spawn in Cilk-5 is only a few reads and writes in the fast clone-3 reads and 5 writes for the fib example. We will experimentally quantify the work overhead in Section 6. Some work overheads still remain in our im- plementation, however, including the allocation and freeing of activation frames, saving state before a spawn, pushing and popping of the frame on the deque, and checking if a procedure has been stolen. A portion of this work overhead is due to the fact that Cilk-5 is duplicating the work the C compiler performs, but as Section 6 shows, this overhead is small. Although a production Cilk compiler might be able eliminate this unnecessary work, it would likely compromise portability. In Cilk-4, the precursor to Cilk-5, we took the work-first principle to the extreme. Cilk-4 performed stack-based allocation of activation frames, since the work overhead of stack allocation is smaller than the overhead of heap alloca- tion. Because of the "cactus stack" [25] semantics of the Cilk stack, 6 however, Cilk-4 had to manage the virtual-memory map on each processor explicitly, as was done in [27]. The work overhead in Cilk-4 for frame allocation was little more than that of incrementing the stack pointer, but whenever the stack pointer overflowed a page, an expensive user-level ensued, during which Cilk-4 would modify the memory map. Unfortunately, the operating-system mechanisms supporting these operations were too slow and un- predictable, and the possibility of a page fault in critical sec- 6 Suppose a procedure A spawns two children B and C. The two children can reference objects in A's activation frame, but B and C do not see each other's frame. tions led to complicated protocols. Even though these overheads could be charged to the critical-path term, in practice, they became so large that the critical-path term contributed significantly to the running time, thereby violating the assumption of parallel slackness. A one-processor execution of a program was indeed fast, but insufficient slackness sometimes resulted in poor parallel performance. In Cilk-5, we simplified the allocation of activation frames by simply using a heap. In the common case, a frame is allocated by removing it from a free list. Deallocation is performed by inserting the frame into the management of virtual memory is required, except for the initial setup of shared memory. Heap allocation contributes only slightly more than stack allocation to the work overhead, but it saves substantially on the critical path term. On the downside, heap allocation can potentially waste more memory than stack allocation due to fragmentation. For a careful analysis of the relative merits of stack and heap based allocation that supports heap allocation, see the paper by Appel and Shao [1]. For an equally careful analysis that supports stack allocation, see [22]. Thus, although the work-first principle gives a general understanding of where overheads should be borne, our experience with Cilk-4 showed that large enough critical-path overheads can tip the scales to the point where the assumptions underlying the principle no longer hold. We believe that Cilk-5 work overhead is nearly as low as possible, given our goal of generating portable C output from our compiler. 7 Other researchers have been able to reduce overheads even more, however, at the expense of portability. For example, lazy threads [14] obtains efficiency at the expense of implementing its own calling conventions, stack layouts, etc. Although we could in principle incorporate such machine-dependent techniques into our compiler, we feel that Cilk-5 strikes a good balance between performance and portability. We also feel that the current overheads are sufficiently low that other problems, notably minimizing overheads for data synchronization, deserve more attention. 5 Implemention of work-stealing In this section, we describe Cilk-5's work-stealing mecha- nism, which is based on a Dijkstra-like [11], shared-memory, mutual-exclusion protocol called the "THE" protocol. In accordance with the work-first principle, this protocol has been designed to minimize work overhead. For example, on a 167-megahertz UltraSPARC I, the fib program with the THE protocol runs about 25% faster than with hardware locking primitives. We first present a simplified version of the protocol. Then, we discuss the actual implementation, which allows exceptions to be signaled with no additional overhead. 7 Although the runtime system requires some effort to port between architectures, the compiler requires no changes whatsoever for different platforms. Several straightforward mechanisms might be considered to implement a work-stealing protocol. For example, a thief might interrupt a worker and demand attention from this victim. This strategy presents problems for two reasons. First, the mechanisms for signaling interrupts are slow, and although an interrupt would be borne on the critical path, its large cost could threaten the assumption of parallel slack- ness. Second, the worker would necessarily incur some overhead on the work term to ensure that it could be safely interrupted in a critical section. As an alternative to sending interrupts, thieves could post steal requests, and workers could periodically poll for them. Once again, however, a cost accrues to the work overhead, this time for polling. Techniques are known that can limit the overhead of polling [12], but they require the support of a sophisticated compiler. The work-first principle suggests that it is reasonable to put substantial effort into minimizing work overhead in the work-stealing protocol. Since Cilk-5 is designed for shared-memory machines, we chose to implement work-stealing through shared-memory, rather than with message-passing, as might otherwise be appropriate for a distributed-memory implementation. In our implementation, both victim and operate directly through shared memory on the victim's ready deque. The crucial issue is how to resolve the race condition that arises when a thief tries to steal the same frame that its victim is attempting to pop. One simple solution is to add a lock to the deque using relatively heavyweight hardware primitives like Compare-And-Swap or Test-And- Set. Whenever a thief or worker wishes to remove a frame from the deque, it first grabs the lock. This solution has the same fundamental problem as the interrupt and polling mechanisms just described, however. Whenever a worker pops a frame, it pays the heavy price to grab a lock, which contributes to work overhead. Consequently, we adopted a solution that employs Di- jkstra's protocol for mutual exclusion [11], which assumes only that reads and writes are atomic. Because our protocol uses three atomic shared variables T, H, and E, we call it the THE protocol. The key idea is that actions by the worker on the tail of the queue contribute to work overhead, while actions by thieves on the head of the queue contribute only to critical-path overhead. Therefore, in accordance with the work-first principle, we attempt to move costs from the worker to the thief. To arbitrate among different thieves attempting to steal from the same victim, we use a hardware lock, since this overhead can be amortized against the critical path. To resolve conflicts between a worker and the sole thief holding the lock, however, we use a lightweight Dijkstra-like protocol which contributes minimally to work overhead. A worker resorts to a heavyweight hardware lock only when it encounters an actual conflict with a thief, in which case we can charge the overhead that the victim incurs to the critical path. In the rest of this section, we describe the THE protocol 9 T-; return FAILURE; return SUCCESS; Thief 7 return FAILURE; 9 unlock(L); return SUCCESS; Figure 4: Pseudocode of a simplified version of the THE protocol. The left part of the figure shows the actions performed by the victim, and the right part shows the actions of the thief. None of the actions besides reads and writes are assumed to be atomic. For example, T-; can be implemented as in detail. We first present a simplified protocol that uses only two shared variables T and H designating the tail and the head of the deque, respectively. Later, we extend the protocol with a third variable E that allows exceptions to be signaled to a worker. The exception mechanism is used to implement Cilk's abort statement. Interestingly, this extension does not introduce any additional work overhead. The pseudocode of the simplified THE protocol is shown in Figure 4. Assume that shared memory is sequentially consistent [20]. 8 The code assumes that the ready deque is implemented as an array of frames. The head and tail of the deque are determined by two indices T and H, which are stored in shared memory and are visible to all processors. The index T points to the first unused element in the array, and H points to the first frame on the deque. Indices grow from the head towards the tail so that under normal con- ditions, we have T - H. Moreover, each deque has a lock L implemented with atomic hardware primitives or with OS calls. The worker uses the deque as a stack. (See Section 4.) Before a spawn, it pushes a frame onto the tail of the deque. After a spawn, it pops the frame, unless the frame has been stolen. A thief attempts to steal the frame at the head of the deque. Only one thief at the time may steal from the deque, since a thief grabs L as its first action. As can be seen from the code, the worker alters T but not H, whereas the thief only increments H and does not alter T. The only possible interaction between a thief and its vic- 8 If the shared memory is not sequentially consistent, a memory fence must be inserted between lines 5 and 6 of the worker/victim code and between lines 3 and 4 of the thief code to ensure that these instructions are executed in the proper order. (b)000000000000000111111111111111000000000000111111111111111 H24(a) (c) Thief Figure 5: The three cases of the ready deque in the simplified THE protocol. A shaded entry indicates the presence of a frame at a certain position in the deque. The head and the tail are marked by T and H. occurs when the thief is incrementing H while the victim is decrementing T. Consequently, it is always safe for a worker to append a new frame at the end of the deque worrying about the actions of the thief. For a pop operations, there are three cases, which are shown in Figure 5. In case (a), the thief and the victim can both get a frame from the deque. In case (b), the deque contains only one frame. If the victim decrements T without interference from thieves, it gets the frame. Similarly, a thief can steal the frame as long as its victim is not trying to obtain it. If both the thief and the victim try to grab the frame, however, the protocol guarantees that at least one of them discovers that H ? T. If the thief discovers that H ? T, it restores H to its original value and retreats. If the victim discovers that H ? T, it restores T to its original value and restarts the protocol after having acquired L. With L acquired, no thief can steal from this deque so the victim can pop the frame without interference (if the frame is still there). Finally, in case (c) the deque is empty. If a thief tries to steal, it will always fail. If the victim tries to pop, the attempt fails and control returns to the Cilk runtime system. The protocol cannot deadlock, because each process holds only one lock at a time. We now argue that the THE protocol contributes little to the work overhead. Pushing a frame involves no overhead beyond updating T. In the common case where a worker can succesfully pop a frame, the pop protocol performs only 6 operations-2 memory loads, 1 memory store, 1 decre- ment, 1 comparison, and 1 (predictable) conditional branch. Moreover, in the common case where no thief operates on the deque, both H and T can be cached exclusively by the worker. The expensive operation of a worker grabbing the lock L occurs only when a thief is simultaneously trying to steal the frame being popped. Since the number of steal attempts depends on T1 , not on T1 , the relatively heavy cost of a victim grabbing L can be considered as part of the critical-path overhead c1 and does not influence the work overhead c1 . We ran some experiments to determine the relative performance of the THE protocol versus the straightforward protocol in which pop just locks the deque before accessing it. On a 167-megahertz UltraSPARC I, the THE protocol is about 25% faster than the simple locking protocol. This machine's memory model requires that a memory fence instruction (membar) be inserted between lines 5 and 6 of the pop pseudocode. We tried to quantify the performance impact of the membar instruction, but in all our experiments the execution times of the code with and without membar are about the same. On a 200-megahertz Pentium Pro running Linux and gcc 2.7.1, the THE protocol is only about 5% faster than the locking protocol. On this processor, the THE protocol spends about half of its time in the memory fence. Because it replaces locks with memory synchronization, the THE protocol is more "nonblocking" than a straightforward locking protocol. Consequently, the THE protocol is less prone to problems that arise when spin locks are used extensively. For example, even if a worker is suspended by the operating system during the execution of pop, the infrequency of locking in the THE protocol means that a can usually complete a steal operation on the worker's deque. Recent work by Arora et al. [2] has shown that a completely nonblocking work-stealing scheduler can be im- plemented. Using these ideas, Lisiecki and Medina [21] have modified the Cilk-5 scheduler to make it completely non- blocking. Their experience is that the THE protocol greatly simplifies a nonblocking implementation. The simplified THE protocol can be extended to support the signaling of exceptions to a worker. In Figure 4, the index H plays two roles: it marks the head of the deque, and it marks the point that the worker cannot cross when it pops. These places in the deque need not be the same. In the full THE protocol, we separate the two functions of H into two variables: H, which now only marks the head of the deque, and E, which marks the point that the victim cannot cross. exceptional condition has occurred, which includes the frame being stolen, but it can also be used for other exceptions. For example, setting causes the worker to discover the exception at its next pop. In the new protocol, E replaces H in line 6 of the worker/victim. Moreover, lines 7-15 of the worker/victim are replaced by a call to an exception handler to determine the type of exception (stolen frame or otherwise) and the proper action to perform. The thief code is also modified. Before trying to Program Size fib blockedmul 1024 29.9 0.0044 6730 1.05 4.3 7.0 6.6 notempmul 1024 29.7 0.015 1970 1.05 3.9 7.6 7.2 strassen 1024 20.2 0.58 35 1.01 3.54 5.7 5.6 *cilksort 4; 100; 000 5.4 0.0049 1108 1.21 0.90 6.0 5.0 yqueens 22 150. 0.0015 96898 0.99 18.8 8.0 8.0 yknapsack heat 4096 \Theta 512 62.3 0.16 384 1.08 9.4 6.6 6.1 4.3 0.0020 2145 0.93 0.77 5.6 6.0 Figure The performance of example Cilk programs. Times are in seconds and are accurate to within about 10%. The serial programs are C elisions of the Cilk programs, except for those programs that are starred (*), where the parallel program implements a different algorithm than the serial program. Programs labeled by a dagger (y) are nondeterministic, and thus, the running time on one processor is not the same as the work performed by the computation. For these programs, the value for T 1 indicates the actual work of the computation on 8 processors, and not the running time on one processor. steal, the thief increments E. If there is nothing to steal, the restores E to the original value. Otherwise, the thief steals frame H and increments H. From the point of view of a worker, the common case is the same as in the simplified protocol: it compares two pointers (E and T rather than H and T). The exception mechanism is used to implement abort. When a Cilk procedure executes an abort instruction, the runtime system serially walks the tree of outstanding descendants of that procedure. It marks the descendants as aborted and signals an abort exception on any processor working on a descendant. At its next pop, an aborted procedure will discover the exception, notice that it has been aborted, and return immediately. It is conceivable that a procedure could run for a long time without executing a pop and discovering that it has been aborted. We made the design decision to accept the possibility of this unlikely scenario, figuring that more cycles were likely to be lost in work overhead if we abandoned the THE protocol for a mechanism that solves this minor problem. 6 Benchmarks In this section, we evaluate the performance of Cilk-5. We show that on 12 applications, the work overhead c1 is close to 1, which indicates that the Cilk-5 implementation exploits the work-first principle effectively. We then present a break-down of Cilk's work overhead c1 on four machines. Finally, we present experiments showing that the critical-path overhead c1 is reasonably small as well. Figure 6 shows a table of performance measurements taken for 12 Cilk programs on a Sun Enterprise 5000 SMP with 8 167-megahertz UltraSPARC processors, each with 512 kilobytes of L2 cache, 16 kilobytes each of L1 data and instruction caches, running Solaris 2.5. We compiled our programs with gcc 2.7.2 at optimization level -O3. For a full description of these programs, see the Cilk 5.1 manual [8]. The table shows the work of each Cilk program T1 , the critical path and the two derived quantities P and c1 . The table also lists the running time T8 on 8 processors, and the speedup T1=T8 relative to the one-processor execution time, and speedup TS=T8 relative to the serial execution time. For the 12 programs, the average parallelism P is in most cases quite large relative to the number of processors on a typical SMP. These measurements validate our assumption of parallel slackness, which implies that the work term dominates in Inequality (4). For instance, on 1024 \Theta 1024 matri- ces, notempmul runs with an average parallelism of 1970- yielding adequate parallel slackness for up to several hundred processors. For even larger machines, one normally would not run such a small problem. For notempmul, as well as the other 11 applications, the average parallelism grows with problem size, and thus sufficient parallel slackness is likely to exist even for much larger machines, as long as the problem sizes are scaled appropriately. The work overhead c1 is only a few percent larger than 1 for most programs, which shows that our design of Cilk-5 faithfully implements the work-first principle. The two cases where the work overhead is larger (cilksort and cholesky) are due to the fact that we had to change the serial algorithm to obtain a parallel algorithm, and thus the comparison is not against the C elision. For example, the serial C algorithm for sorting is an in-place quicksort, but the parallel algorithm cilksort requires an additional temporary array which adds overhead beyond the overhead of Cilk it- self. Similarly, our parallel Cholesky factorization uses a quadtree representation of the sparse matrix, which induces more work than the linked-list representation used in the serial C algorithm. Finally, the work overhead for fib is large, because fib does essentially no work besides spawning procedures. Thus, the overhead good estimate of the cost of a Cilk spawn versus a traditional C function call. With such a small overhead for spawning, one can understand why for most of the other applications, which perform significant work for each spawn, the overhead of Cilk-5's scheduling is barely noticeable compared to the 10% "noise" in our measurements. 195 MHz MIPS R10000 Ultra SPARC I 200 MHz Pentium Pro Alpha 21164 overheads THE protocol frame allocation state saving 115ns 113ns 78ns 27ns Figure 7: Breakdown of overheads for fib running on one processor on various architectures. The overheads are normalized to the running time of the serial C elision. The three overheads are for saving the state of a procedure before a spawn, the allocation of activation frames for procedures, and the THE protocol. Absolute times are given for the per-spawn running time of the C elision. We now present a breakdown of Cilk's serial overhead c1 into its components. Because scheduling overheads are small for most programs, we perform our analysis with the fib program from Figure 1. This program is unusually sensitive to scheduling overheads, because it contains little actual computation. We give a breakdown of the serial overhead into three components: the overhead of saving state before spawning, the overhead of allocating activation frames, and the overhead of the THE protocol. Figure 7 shows the breakdown of Cilk's serial overhead for fib on four machines. Our methodology for obtaining these numbers is as follows. First, we take the serial C fib program and time its execution. Then, we individually add in the code that generates each of the overheads and time the execution of the resulting program. We attribute the additional time required by the modified program to the scheduling code we added. In order to verify our numbers, we timed the fib code with all of the Cilk overheads added (the code shown in Figure 3), and compared the resulting time to the sum of the individual overheads. In all cases, the two times differed by less than 10%. Overheads vary across architectures, but the overhead of Cilk is typically only a few times the C running time on this spawn-intensive program. Overheads on the Alpha machine are particularly large, because its native C function calls are fast compared to the other architectures. The state-saving costs are small for fib, because all four architectures have write buffers that can hide the latency of the writes required. We also attempted to measure the critical-path overhead c1 . We used the synthetic knary benchmark [4] to synthesize computations artificially with a wide range of work and critical-path lengths. Figure 8 shows the outcome from many such experiments. The figure plots the measured0.10.01 Normalized Normalized Machine Size Experimental data Model Work bound Critical path bound Figure 8: Normalized speedup curve for Cilk-5. The horizontal axis is the number P of processors and the vertical axis is the speedup T1=TP , but each data point has been normalized by dividing by T1=T1 . The graph also shows the speedup predicted by the formula speedup T1=TP for each run against the machine size P for that run. In order to plot different computations on the same graph, we normalized the machine size and the speedup by dividing these values by the average parallelism as was done in [4]. For each run, the horizontal position of the plotted datum is the inverse of the slackness P=P , and thus, the normalized machine size is 1:0 when the number of processors is equal to the average parallelism. The vertical position of the plotted datum is (T1=TP measures the fraction of maximum obtainable speedup. As can be seen in the chart, for almost all runs of this bench- mark, we observed TP - T1=P + 1:0T1 . (One exceptional data point satisfies TP - T1=P the work-first principle caused us to move overheads to the critical path, the ability of Cilk applications to scale up was not significantly compromised. 7 Conclusion We conclude this paper by examining some related work. Mohr et al. [24] introduced lazy task creation in their implementation of the Mul-T language. Lazy task creation is similar in many ways to our lazy scheduling techniques. Mohr et al. report a work overhead of around 2 when comparing with serial T, the Scheme dialect on which Mul-T is based. Our research confirms the intuition behind their methods and shows that work overheads of close to 1 are achievable. The Cid language [26] is like Cilk in that it uses C as a base language and has a simple preprocessing compiler to convert parallel Cid constructs to C. Cid is designed to work in a distributed memory environment, and so it employs latency-hiding mechanisms which Cilk-5 could avoid. (We are working on a distributed version of Cilk, however.) Both Cilk and Cid recognize the attractiveness of basing a parallel language on C so as to leverage C compiler technology for high-performance codes. Cilk is a faithful extension of C, however, supporting the simplifying notion of a C elision and allowing Cilk to exploit the C compiler technology more readily. TAM [10] and Lazy Threads [14] also analyze many of the same overhead issues in a more general, "nonstrict" language setting, where the individual performances of a whole host of mechanisms are required for applications to obtain good overall performance. In contrast, Cilk's multithreaded language provides an execution model based on work and critical-path length that allows us to focus our implementation efforts by using the work-first principle. Using this principle as a guide, we have concentrated our optimizing effort on the common-case protocol code to develop an efficient and portable implementation of the Cilk language. Acknowledgments We gratefully thank all those who have contributed to Cilk development, including Bobby Blumofe, Ien Cheng, Don Dailey, Mingdong Feng, Chris Joerg, Bradley Kuszmaul, Phil Lisiecki, Alberto Medina, Rob Miller, Aske Plaat, Bin Song, Andy Stark, Volker Strumpen, and Yuli Zhou. Many thanks to all our users who have provided us with feedback and suggestions for improvements. Martin Rinard suggested the term "work-first." --R Empirical and analytic study of stack versus heap cost for languages with closures. Thread scheduling for multiprogrammed multiprocessors. Executing Multithreaded Programs Ef- ficiently Scheduling multithreaded computations by work stealing. The parallel evaluation of general arithmetic expressions. Detecting data races in Cilk programs that use locks. Introduction to Algorithms. Solution of a problem in concurrent programming control. Polling efficiently on stock hardware. Efficient detection of determinacy races in Cilk programs. Lazy threads: Implementing a fast parallel call. Bounds on multiprocessing timing anoma- lies Heaps o' stacks: Time and space efficient threads without operating system support. The Cilk System for Parallel Multi-threaded Computing Jr. How to make a multiprocessor computer that correctly executes multiprocess programs. Personal communication Garbage collection is fast The function of FUNCTION in LISP or why the FUNARG problem should be called the environment prob- lem Parallel Symbolic Computing in Cid. VLSI support for a cactus stack oriented memory organization. A bridging model for parallel computation. --TR MULTILISP: a language for concurrent symbolic computation VLSI Support for a cactus stack oriented memory organization A bridging model for parallel computation Introduction to algorithms Fine-grain parallelism with minimal hardware support: a compiler-controlled threaded abstract machine Polling efficiently on stock hardware Whole-program optimization for time and space efficient threads The cilk system for parallel multithreaded computing Lazy threads Cilk Executing multithreaded programs efficiently Efficient detection of determinacy races in Cilk programs Thread scheduling for multiprogrammed multiprocessors Detecting data races in Cilk programs that use locks The Parallel Evaluation of General Arithmetic Expressions Solution of a problem in concurrent programming control Lazy Task Creation Parallel Symbolic Computing in Cid Garbage Collection is Fast, but a Stack is Faster The Function of FUNCTION in LISP, or Why the FUNARG Problem Should be Called the Environment Problem --CTR Liang Peng , Weng-Fai Wong , Chung-Kwong Yuen, SilkRoad II: mixed paradigm cluster computing with RC_dag consistency, Parallel Computing, v.29 n.8, p.1091-1115, 1 August Matteo Frigo, A fast Fourier transform compiler, ACM SIGPLAN Notices, v.39 n.4, April 2004 Kalyan S. Perumalla , Richard M. Fujimoto, Efficient large-scale process-oriented parallel simulations, Proceedings of the 30th conference on Winter simulation, p.459-466, December 13-16, 1998, Washington, D.C., United States Doug Lea, A Java fork/join framework, Proceedings of the ACM 2000 conference on Java Grande, p.36-43, June 03-04, 2000, San Francisco, California, United States Christopher J. Hughes , Radek Grzeszczuk , Eftychios Sifakis , Daehyun Kim , Sanjeev Kumar , Andrew P. Selle , Jatin Chhugani , Matthew Holliman , Yen-Kuang Chen, Physical simulation for animation and visual effects: parallelization and characterization for chip multiprocessors, ACM SIGARCH Computer Architecture News, v.35 n.2, May 2007 Yaron Shoham , Sivan Toledo, Parallel randomized best-first minimax search, Artificial Intelligence, v.137 n.1-2, p.165-196, May 2002 Philip Cox , Simon Gauvin , Andrew Rau-Chaplin, Adding parallelism to visual data flow programs, Proceedings of the 2005 ACM symposium on Software visualization, May 14-15, 2005, St. Louis, Missouri Voon-Yee Vee , Wen-Jing Hsu, Locality-preserving load-balancing mechanisms for synchronous simulations on shared-memory multiprocessors, Proceedings of the fourteenth workshop on Parallel and distributed simulation, p.131-138, May 28-31, 2000, Bologna, Italy Marcel van Lohuizen, A generic approach to parallel chart parsing with an application to LinGO, Proceedings of the 39th Annual Meeting on Association for Computational Linguistics, p.507-514, July 06-11, 2001, Toulouse, France Bryan Chan , Tarek S. Abdelrahman, Run-Time Support for the Automatic Parallelization of Java Programs, The Journal of Supercomputing, v.28 n.1, p.91-117, April 2004 Madanlal Musuvathi , Shaz Qadeer, Iterative context bounding for systematic testing of multithreaded programs, ACM SIGPLAN Notices, v.42 n.6, June 2007 Gregory W. Price , David K. Lowenthal, A comparative analysis of fine-grain threads packages, Journal of Parallel and Distributed Computing, v.63 n.11, p.1050-1063, November Kenjiro Taura , Kunio Tabata , Akinori Yonezawa, StackThreads/MP: integrating futures into calling standards, ACM SIGPLAN Notices, v.34 n.8, p.60-71, Aug. 1999 Dorit Naishlos , Joseph Nuzman , Chau-Wen Tseng , Uzi Vishkin, Towards a first vertical prototyping of an extremely fine-grained parallel programming approach, Proceedings of the thirteenth annual ACM symposium on Parallel algorithms and architectures, p.93-102, July 2001, Crete Island, Greece Robert Ennals , Simon Peyton Jones, Optimistic evaluation: an adaptive evaluation strategy for non-strict programs, ACM SIGPLAN Notices, v.38 n.9, p.287-298, September Radu Rugina , Martin Rinard, Pointer analysis for multithreaded programs, ACM SIGPLAN Notices, v.34 n.5, p.77-90, May 1999 John S. Danaher , I.-Ting Angelina Lee , Charles E. 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programming language;runtime system;multithreading;critical path;parallel computing

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Optimizing direct threaded code by selective inlining.

Achieving good performance in bytecoded language interpreters is difficult without sacrificing both simplicity and portability. This is due to the complexity of dynamic translation ("just-in-time compilation") of bytecodes into native code, which is the mechanism employed universally by high-performance interpreters.We demonstrate that a few simple techniques make it possible to create highly-portable dynamic translators that can attain as much as 70% the performance of optimized C for certain numerical computations. Translators based on such techniques can offer respectable performance without sacrificing either the simplicity or portability of much slower "pure" bytecode interpreters.

Introduction Bytecoded languages such as Smalltalk [Gol83], Caml [Ler97] and Java [Arn96, Lin97] offer significant engineering advantages over more conventional languages: higher levels of abstraction, dynamic execution environments with incremental debugging and code modification, compact representation of executable code, and (in most cases) platform independence. The success of Java is due largely to its promise of platform independence and compactness of code. The compactness of bytecodes has important advantages for net-work computing where code must downloaded "on-demand" for execution on an arbitrary platform and operating system while keeping bandwidth requirements to a minimum. The disadvantage is that bytecode interpreters typically offer lower performance than compiled code, and can consume significantly more resources. Most modern virtual machines perform some degree of dynamic translation to improve program performance [Deu84]. Such techniques significantly increase the complexity of the virtual machine, which must be tailored for Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. SIGPLAN '98 Montreal Canada c each hardware architecture in much the same way as a conventional compiler's back-end. This increases development costs (requiring specific knowledge about the target architecture and the time for writing specific code), and reduces reliability (by introducing more code to debug and support). Some of these languages (Caml for example) also have more traditional compilers that produce high-performance native code, but this defeats the advantages that come with platform independence and compactness. We propose a novel dynamic retranslation technique that can be applied to a certain class of virtual machines. This technique delivers high performance, up to 70% that of optimized C. It is easy to "retrofit" to existing virtual machines, and requires almost no effort to port to a new architecture. This paper continues as follows. The next section gives a brief survey of bytecode interpretation mechanisms, providing a context for the remainder of the paper. Our novel dynamic retranslation technique is explained in Section 3. Section 4 presents the results of applying the technique to two interpreters: the small RISC-like interpreter that inspired this work, and a "production" virtual machine for Objective Caml. The last two sections contrast our technique with related work and present some concluding remarks Background Interpreter performance can depend heavily on the representation chosen for executable code, and the mechanism used to dispatch opcodes. This section describes some of the common techniques. 2.1 Pure bytecode interpreters The inner loop of a pure bytecode interpreter is very simple: fetch the next bytecode and dispatch to the implementation using a switch statement. Figure 1 shows a typical pure bytecode interpreter loop, and an array of bytecodes that calculate will use this as a running example). The interpreter is an infinite loop containing a switch statement to dispatch successive bytecodes. Each case in the body of the switch implements one bytecode, and passes control to the next bytecode by breaking out of the switch to pass control back to the start of the infinite loop. Assuming the compiler optimizes the jump chains from the breaks through the implicit jump at the end of the for body back to its beginning, the overheads associated with this approach are as follows: compiled code: unsigned char bytecode-add, . -; bytecode implementations: unsigned char for unsigned char switch (bytecode) - case bytecode-push3: break; case bytecode-push4: break; case bytecode-add: *stackPointer += stackPointer[1]; break; Figure 1: Pure bytecode interpreter. ffl increment the instructionPointer; ffl fetch the next bytecode from memory; ffl a redundant range check on the argument to switch; ffl fetch the address of the destination case label from a table; ffl jump to that address; and then at the end of each bytecode: ffl jump back to the start of the for body to fetch the next bytecode. Eleven machine instructions must be executed on the PowerPC to perform the push3 bytecode. Nine of these instructions are dedicated to the dispatch mechanism, including two memory references and two jumps (among the most expensive instructions on modern architectures). Pure bytecoded interpreters are easy to write and under- stand, and are highly portable - but rather slow. In the case where most bytecodes perform simple operations (as in the push3 example) the majority of execution time is wasted in performing the dispatch. 2.2 Threaded code interpreters Threaded code [Bel73] was popularized by the Forth programming language [Moo70]. There are various kinds of threaded code, the most efficient of which is generally direct threading [Ert93]. Bytecodes are simply integers: dispatch involves fetching the next opcode (bytecode), looking up the address of the associated implementation (either in an explicit table, or implicitly using switch) and then transferring control to that address. Direct threaded code improves performance by eliminating this table lookup: executable code is represented as a sequence of opcode implementation addresses, and dispatch involves fetching the next opcode (implemen- tation address) and jumping directly to that address. An additional optimization eliminates the centralized dis- patch. Instead of returning to a central dispatch loop, each compiled code: void &&opcode-add, . -; opcode implementations: dispatch next instruction */ #define NEXT() goto *++instructionPointer void /* start execution: dispatch first opcode */ /* opcode implementations. */ opcode-push3: opcode-push4: opcode-add: *stackPointer += stackPointer[1]; Figure 2: Direct threaded code. direct threaded opcode's implementation ends with the code required to dispatch the next opcode. The direct threaded version of our '3 example is shown in Figure 2. 1 Execution begins by fetching the address of the first op- code's implementation from the compiled code and then jumping to that address. Each opcode performs its own work, and then dispatches to the next opcode implied by the compiled code. (Hence the name: control flow "threads" its way through the opcodes in the order implied by the compiled code, without ever returning to a central dispatch loop.) The overheads associated with threaded code are much lower than those associated with a pure bytecode inter- preter. For each opcode executed, the only additional overhead is dispatching to the next opcode: ffl increment the instructionPointer; ffl fetch the next opcode address from memory; ffl jump to that address. Five machine instructions are required to implement push3 on the PowerPC. Three of these are associated with opcode dispatch, with only one memory reference and one jump. We have saved six instructions over the "pure bytecode" approach. Most importantly we have saved one memory reference and one jump instruction (both of which are ex- pensive). 2.3 Dynamic translation to threaded code The benefits of direct threaded code can easily be obtained in a bytecoded language by translating the bytecodes into 1 The threaded code examples are written using the first-class labels provided by GNU C. The expression "void assigns the address (of type "void *") of the statement attached to the given label to addr. Control can be transferred to this location using a goto that dereferences the address: "goto *addr". Note that gcc's first-class labels are not required to implement these techniques: the same effects can be achieved with a couple of macros containing a few lines of asm. translation table: void *opcodes[]; dynamic translator: unsigned char *bytecodePointer = firstBytecode; void while (moreBytecodesToTranslate) Figure 3: Dynamic translation of bytecodes into threaded code. direct threaded code before execution. This is illustrated in Figure 3. The translation loop reads each bytecode, looks up the address of its implementation in a table, and then writes this address into the direct threaded code. The only complication is that most bytecode sets have extension bytes. These provide additional information that cannot be encoded within the bytecode itself: branch offsets, indices into literal tables or environments, and so on. These extension bytes are normally placed inline in the translated threaded code by the translator, immediately after the threaded opcode corresponding to the bytecode. Translation to threaded code permits other kinds of op- timization. For example, Smalltalk provides four bytecodes for pushing an implicit integer constant (between -1 and onto the stack. The translator loop could easily translate these as a single pushInteger opcode followed by the constant to be pushed as an inline operand. The same treatment can be applied to other kinds of literal quantity, relative branch offsets, and so on. Another possibility is "partial decoding", where the translator loop examines an "over- loaded" bytecode at translation time, and translates it into one of several threaded opcodes. The translator loop must be aware of the kind of operand that it is copying. A relative offset, for example, might require modification or scaling during the translation loop. It is possible to make an approximate evaluation of this approach in a realistic system. Squeak [Ing97] is a portable "pure bytecode" implementation of Smalltalk-80; it performs numerical computations at approximately 3.7% the speed of optimized C. BrouHaHa [Mir87] is a portable Smalltalk virtual machine that is very similar to the Squeak VM, except that it dynamically translates bytecodes into direct threaded code for execution [Mir91]. BrouHaHa performs the same numerical computations at about 15% the speed of optimized C. Both implementations have been carefully hand-tuned for performance; the essential difference between them is the use of dynamic translation to direct threaded code in BrouHaHa. 2.4 Optimizing common bytecode sequences Bytecodes can typically only represent Threaded opcodes can represent many more, since they are encoded as pointers. Translating bytecodes into threaded code therefore gives us the opportunity to make arbitrary transformations on the executable code. One such transformation is to detect common sequences of bytecodes and translate them as a single threaded "macro" opcode; this macro op-code performs the work of the entire sequence of original bytecodes. For example, the bytecodes "push literal, push variable, add, store variable" can be translated into a single "add-literal-to-variable" opcode in the threaded code. Such optimizations are effective because they avoid the overhead of the multiple dispatches that are implied by the original bytecodes (but elided within the macro opcode). A single macro opcode that is translated from a sequence of N original bytecodes avoids dispatches at execution time. This technique is particularly important in cases where the bytecodes are simple (as in the '3 the implementation of each bytecode can be as short as a single register-register machine instruction. The cost of threading can often be significantly larger than the cost of "useful" execution. If three instructions must be executed to dispatch to the next opcode then the overhead for this threading instructions executed and 12 instructions for dispatching the threaded opcodes). This overhead drops to 43% when the operation is optimized into a single macro opcode (four useful instructions and 3 instructions for threading). 2 Dispatching to opcode implementations at non-contiguous addresses also undermines code locality, causing un-necessary processor pipeline stalls and inefficient utilization of the instruction cache and TLBs. Combining common sequences of bytecodes into a single macro opcode considerably reduces these effects. The compiler will also have a chance to make inter-bytecode optimizations (within the implementation of the single macro opcode) that are impossible to make between the implementations of the individual bytecodes. Determining an appropriate set of common bytecode sequences is not difficult. The virtual machine can be instrumented to record execution traces, and a simple offline analysis will reveal the likely candidates. The corresponding pattern matching and macro opcode implementations can then be incorporated manually into the VM. For example, such analysis has been applied to an earlier version of the Objective Caml bytecode set, resulting in a new set of bytecodes that includes several "macro-style" operations. 2.5 Problems with static optimization The most significant problem with this static approach is that the number of possible permutations of even the shortest common sequences of consecutive bytecodes is pro- hibitive. For example, Smalltalk provides 4 bytecodes to push the most popular integer constants (minus one through two), and bytecodes to load and store 32 temporary and 256 "receiver" variables. Manually optimizing the possible permutations for incrementing and decrementing a variable by a small constant would require the translator to implement 2304 explicit special cases. This is clearly unreasonable. The problem is made more acute since different applications running on the same virtual machine will favor different sequences of bytecodes. Statically chosing a single "optimal" set of common sequences is therefore impossible. Our technique focuses on making this choice at runtime, which allows the set of common sequences to be nearly optimal for the particular application being run. "Instruction counting" is not a very accurate way to estimate the savings, since the instructions that we avoid are some of the most expensive to execute. dynamic-opcode-push3-push4-add: stackPointer-; *stackPointer += stackPointer[1]; goto Figure 4: Equivalent macro opcode for push3, push4, add. int nfibs(int n) return (n ! 2) Figure 5: Benchmark function in C. Dynamically rewriting opcode sequences We generate implementations for common bytecode sequences dynamically. These implementations are available as new macro opcodes, where a single such macro opcode replaces the several threaded opcodes generated from the original common bytecode sequence. These dynamically generated macro opcodes are executed in precisely the same manner as the interpreter's predefined opcodes; the original execution mechanism (direct threading) requires no modification at all. The transformation can be performed either during bytecode-to-threaded code translation, or as a separate pass over already threaded code. Figure 4 shows the equivalent C for a dynamically generated threaded opcode for the sequence of three bytecodes needed to evaluate the '3 + 4' example. The translator concatenates the compiled C implementations for several intrinsic threaded opcodes, each one corresponding to a bytecode in the sequence being optimized. Since this involves relocating code, it is only safe to perform this concatenation for threaded opcodes whose implementation is position independent. In general there are three cases to consider when concatenating opcode implementations: ffl A threaded opcode cannot be inlined if its implementation contains a call to a C function, where the destination address is relative to the processor's PC. Such destination addresses would be invalidated as they are copied to form the new macro opcode's implementation ffl Any threaded opcode that changes the flow of control through the threaded code must only appear at the end of a translated sequence. This is because different paths through the sequence might consume different numbers of inline arguments. ffl Any threaded opcode that is a branch destination can only appear at the beginning of a macro opcode, since incorporating it into the middle of a macro opcode would delete the branch destination in the final threaded code. The above can be simplified to the following rule: we only consider basic blocks for inlining, where a basic block begins with a jump destination and ends with either a jump nfibs: push r1 ; r1 saved during call move jge r0 r1 @ =cont pop r1 ; restore r1 return cont: move r0 r1 ; else arg -? r1 call @ =nfibs call @ =nfibs add add pop r1 ; restore r1 return start: move #32 r0 ; call nfibs(32) call @ =nfibs print Figure Threaded code for nfibs benchmark, before inlining. destination or a change of control flow. For inlining pur- poses, opcodes that contain a C function call are considered to be single-opcode basic blocks. (This restriction can be relaxed if the target architecture and/or the compiler used to build the VM uses absolute addresses for function call destinations.) Our technique was designed for (and works best with) fine-grained opcodes, where the implementations are short (typically a few machine instructions) and therefore the cost of opcode dispatch dominates. The next section presents an example in such a context. 3.1 Simple example We will illustrate our technique by applying it to a simple "RISC-like" virtual machine executing the "nfibs" func- tion, as shown in Figure 5. 3 Our example interpreter implements a register-based execution model. It has a handful of "registers" for performing arithmetic, and a stack that is used for saving return addresses and the contents of clobbered registers during subroutine calls. The direct threaded code has two kinds of in-line operand: instruction pointer-relative offsets for branch destinations, and absolute addresses for function call destinations The interpreter translates bytecodes into threaded code in two passes. It makes a first pass over the bytecodes, expanding them into threaded opcodes with no inlining, exactly as explained in Section 2.3. Figure 6 shows a symbolic listing of the nfibs function, implemented for our example interpreter's opcode set, after this initial translation into threaded code. Bytecode operands are placed inline in the threaded code during translation. For example, the offset for the jge op-code and the call destinations are placed directly in the opcode stream, immediately after the associated opcode. These are represented as the pseudo-operand '@' in the fig- 3 This doubly-recursive function has the interesting property that its result is the number of function calls required to calculate the result. nfibs: =cont =nfibs =nfibs Figure 7: Threaded code for nfibs benchmark, after inlining. The implementations of the new macro opcodes are shown on the right. ure, and appear on a separate line in the code prefixed with '='. After this initial translation to threaded code, a second pass performs inlining on the threaded code: basic blocks are identified, used to dynamically generate new threaded macro opcodes, and the corresponding original sequences of threaded opcodes are replaced with single macro opcodes. The rewriting of the threaded code can be performed in-situ, since optimizing an opcode sequence will always result in a shorter sequence of optimized code; there is no possibility of overwriting an opcode that has not yet been considered for inlining. Figure 7 shows the code for the nfibs function after in-lining has taken place. The function has been reduced to five threaded macro opcodes (shown as '%1' through `%5'), each replacing a basic block in the original code. The implementation of each new macro opcode is the concatenation of the implementations of the opcodes that it replaces. These new implementations are written in a separate area of memory called the macro cache. Five such implementations are required for nfibs, and are shown within curly braces in the figure. Each one ends with a copy of the implementation of the pseudo-opcode !thr?, which is the threading operation to dispatch the next opcode. Inline arguments are copied verbatim, except for cont (a jump offset) which is adjusted appropriately by the transla- tor. (These inline arguments are used by the macro opcode implementations at the points marked with '@' in the figure.) To help with the identification of basic blocks, we divide our threaded opcodes into four classes, as follows: INLINE - the opcode's implementation can be inlined into a macro opcode without restriction (the arithmetic opcodes belong to this class); PROTECT - the implementation contains a C function call and therefore cannot be inlined (the print opcode belongs to this class); FINAL - the opcode changes the flow of control and therefore defines the end of a basic block (e.g. the call RELATIVE - the opcode changes the flow of control and therefore defines the end of a basic block (e.g. the conditional branch jge). The only difference between FINAL and RELATIVE is the way in which the opcode's inline operand is treated. In the first case the operand is absolute, and can be copied directly into the final translated code. In the second case the operand is relative to the current threaded program counter, and so must be adjusted appropriately in the final translated code. Figure 8 shows the translator code that initializes the threaded opcode table, along with representative implementations of several of our threaded opcodes (each of the four classes of threaded opcode is represented). #define #define POP() (*sp-) #define GET() ((long)(*++ip)) /* read inline operand */ #define NEXT() goto *++ip /* dispatch next opcode */ #define PROTECT (0x00) /* never expanded */ #define INLINE (1!!0) /* expanded */ #define FINAL (1!!1) /* expanded, ends a basic block */ #define RELATIVE (1!!2) /* expanded, ends a basic block, offset follows */ #define OP(NAME, NARGS, FLAGS) " case if (!initialIP) break; " start-#NAME: /* opcode body */ #define /* initialize rather than execute (see macro 'OP') */ for (int switch (op) - OP(jge-r0-r1, 1, RELATIVE) - register long if (r0 ?= r1) ip += offset; OP(call, 1, FINAL) - register long dest = GET(); *)dest - default: fprintf(stderr, "panic: op %d is undefined!"n", op); abort(); Figure 8: Opcode table initialization. The translator's inlining loop is shown in Figure 9. It is not as complex as it might first appear. code is a pointer to the translated threaded code, which is rewritten in-situ. in and out are indices into code pointing to the next opcode to be copied (or inlined) and the location to which it will be copied, respectively (in ?= out at all times). The loop considers each in opcode for inlining: the inlining loop is entered only if both the current opcode and the opcode following it can be inlined. If this is not the case, the opcode at in is copied (along with any inline arguments) directly to out. nextMacro is a pointer to the next unused location in the macro cache. The inlining loop first writes this address to out (it represents the threaded opcode for the macro implementation that is about to be generated), and then copies the compiled implementations of opcodes from in into the macro cache. The inlined threaded opcodes are not copied, although any inline arguments that are encountered are copied directly to out. The inlining loop continues until it copies the implementation of an opcode that explicitly ends a basic block or RELATIVE), or until the next opcode is either non-inlinable int while int nextIn = in long if (info[thisOp].flags == INLINE && info[nextOp].flags != PROTECT && /* CAN INLINE: create new macro opcode at nextMacro */ void new macro opcode */ while (info[thisOp].flags != PROTECT) - icopy(info[thisOp].addr, ep, info[thisOp].size); if (info[thisOp].flags == RELATIVE) - locn of offset */ for (int if (info[thisOp].flags == FINAL - info[thisOp].flags == RELATIVE - destination[in]) break; /* end of basic block */ /* copy threading operation */ icopy(info[thr].addr, ep, info[thr].size); /* CAN'T INLINE: copy opcode and inline arguments */ if (info[thisOp].flags == RELATIVE) - /* copy literal arguments */ for (int Figure 9: Dynamic translator loop. (PROTECTED) or a branch destination (implicitly ending the current basic block). The translator then appends the implementation of the pseudo-opcode thr, which is the "thre- ading" operation itself. Finally, the nextMacro location is updated ready for the next inlining operation. The translator loop uses an array of flags "destination" to identify branch destinations within the threaded code. This array is easily constructed during the translator's first pass, when bytecodes are expanded into non-inlined threaded code. The loop also creates two arrays, relocations and patchList, that are used to recalculate relative branch offsets. 4 The inlining loop concatenates opcode implementations using the icopy function, shown in Figure 10. This function is similar to bcopy except that it also synchronizes the pro- cessor's instruction and data caches to ensure that the new macro opcode's implementation is executable. It contains the only line of platform-dependent code in our interpreter. 4 The branch destination identification and relative offset recalculation are not shown here. These can be seen in the full source code for the example interpreter (see the Appendix). static inline void icopy(void *source, void *dest, size-t size) bcopy(source, dest, size); while (size ? asm ("dcbst 0,%0; sync; icbi 0,%0; isync" :: "r"(p)); #elif defined(-sparc) asm ("flush %0; stbar" :: "r"(p)); /* no-op */ #elif defined(.) #endif dest += 4; size -= 4; Figure 10: The icopy function, containing the single line of platform-dependent code. 3.2 Saving space Translating multiple copies of the same opcode sequences would waste space. We therefore keep a cache of dynamically generated macro opcodes, keyed by a hash value computed from the incoming (unoptimized) opcodes during translation. In the case of a cache hit we reuse the existing macro opcode in the translated code, and immediately reclaim the macro cache space occupied by the newly translated version. In the case of a cache miss, the newly generated macro opcode is used in the translated code and the hash table updated to include the new opcode. This ensures that we never have more than one macro opcode corresponding to a given sequence of unoptimized opcodes. 4 Experimental results We are particularly interested in the performance benefits when dynamic inlining is applied to interpreters with fine-grain instruction sets. Nevertheless, we were also curious to see how the technique would perform when applied to an interpreter having a more coarse-grained bytecode set. We took measurements in both of these contexts, using our own RISC-like interpreter and the widely-used (but less suited) interpreter for the Objective Caml language. 4.1 Fine-grained opcodes Our RISC-like interpreter has an opcode set similar to that presented in Section 3.1. It can be configured (at compile time) to use bytecodes, direct threaded code, or direct threaded code with dynamically-generated macro opcodes. The performance of two benchmarks was measured using this in- terpreter: the function-call intensive Fibonacci benchmark presented earlier (nfibs), and a memory intensive, function call free, prime number generator (sieve). Table 1 shows the number of seconds required to execute these benchmarks on several architectures (133MHz Pentium, SparcStation 20, and 200MHz PowerPC 603ev). The figures shown are for a simple bytecode interpreter, the same interpreter performing translation into direct threaded code, direct threaded code with dynamic inlining of common opcode sequences, and the benchmark written in C and compiled with the same optimization options (-O2) as our interpreter. The final column shows the performance of the inlined threaded code compared to optimized C. nfibs machine bytecode threaded inlined C inlined/C Pentium 63.2 37.1 22.3 11.1 49.8% sieve machine bytecode threaded inlined C inlined/C Pentium 25.1 17.6 13.2 4.6 34.8% Table 1: nfibs and sieve benchmark results for the three architectures tested. The final column shows the speed of the inlined threaded code relative to optimized C. Pentium Pentium bytecode direct threaded inlined Figure 11: Benchmark performance relative to optimized C. nfibs spends much of its time performing arithmetic between registers. Memory (stack) operations are performed only during function call and return. Our interpreter allocates the first few VM registers in physical machine registers whenever possible. The opcodes that perform arithmetic are therefore typically compiled into a single machine instruction on the Sparc and PowerPC. These two architectures show a marked improvement in performance when common sequences are inlined into single macro opcodes, due to the significantly reduced ratio of op-code dispatch to "real" work. The effect is less pronounced on the Pentium, which has so few machine registers that all the VM registers must be kept in memory. Each arithmetic opcode compiles into several Pentium instructions, and therefore the ratio of dispatch overhead to real work is lower than for the RISC architectures. We observe a marked improvement (approximately a factor of two) between successive versions of the interpreter for nfibs. sieve shows a less pronounced improvement because it spends the majority of its time performing memory opera- tions. The contribution of opcode dispatch to the overall execution time is therefore smaller than with nfibs. It is also interesting to observe the performance of each version of the interpreter relative to that of optimized C. Figure 11 shows that nfibs gains approximately 14% the speed of optimized C when moving from a bytecoded representation to threaded code. The gain when moving from threaded to inlined threaded code is more dependent on the architecture: approximately 20% for the Pentium, and 38% for the Sparc. The gains for sieve are both smaller and less dependent on the architecture: approximately 9% at each step, for all three architectures. 4.2 Objective Caml We also applied our technique to the Objective Caml byte-code interpreter, in order to obtain realistic measurements of its performance and overheads in a less favorable environment Objective Caml was chosen because the design and implementation of the interpreter's core is clean and simple, and so understanding it before making the required modifications did not present a significant challenge. Furthermore it is a fully-fledged system that includes a bytecode com- piler, a benchmark suite, and some large applications. This made it easier to collect meaningful statistics. The interpreter is also equipped with a mechanism to bulk-translate the bytecodes into threaded code at startup (on those platforms that support it). 5 We needed only to extend this initial translation phase to perform the analysis of opcode sequences, generate macro opcode implementa- tions, and rewrite the threaded code in-situ to use these dynamically-generated macro opcodes. Implementing our technique for the Caml virtual machine took one day. There were only two small details that required careful attention. The first was the presence of the SWITCH opcode. This performs a multi-way branch, and is followed in the threaded code by an inline table mapping values onto branch offsets. We added a special case to our translator loop to handle this opcode. The second was the existence of a handful of opcodes that consume two inline arguments (a literal and a relative offset). We introduced a new opcode class RELATIVE2 for these, which differs from RELATIVE only by copying an additional inline literal argument before the offset in the translator loop. Our translation algorithm was identical in all other respects to the one presented in Section 3. We ran the standard Objective Caml benchmark suite 6 with our modified VM (see Table 2). The VM was instrumented to gather statistics relating to execution speed, 5 It uses gcc's first-class labels to do this portably.ftp://ftp.inria.fr/INRIA/Projects/cristal/Xavier.Leroy/ benchmarks/objcaml.tar.gz boyer fib genlex qsort qsort* sieve soli soli* takc taku speed (inlined/non-inlined) Pentium Sparc PowerPC Figure 12: Objective-Caml benchmark results for the three architectures tested. The vertical axis shows the performance relative to the original (non-inlining) interpreter. Asterisks indicate versions of the benchmarks compiled with array bounds checking disabled. boyer term processing, function calls fib integer arithmetic, function calls (1 arg) genlex lexing, parsing, symbolic processing kb term processing, function calls, functionals qsort integer arrays, loops sieve integer arithmetic, list processing, functionals soli puzzle solving, arrays, loops takc integer arithmetic, function calls (3 args, curried) taku integer arithmetic, function calls (3 args, tuplified) Table 2: Objective Caml benchmarks. memory usage, and the characteristics of dynamically generated macro opcodes. Figure 12 shows the performance of the benchmarks after inlining, relative to the original performance without inlining It is important to note that the Objective Caml byte-code set has already been optimized statically, as described in Section 2.4 [Ler98]. Any further improvements are therefore due mainly to the elimination of dispatch overhead in common sequences that are particular to each application. Virtual machines whose bytecode sets have not been "stat- ically" optimized in this way would benefit more from our technique. We can see from the figure that the majority of benchmarks benefit from a significant performance advantage after inlining. In most cases the inlined version runs more than 50% faster than the original, with two of the benchmarks running twice as fast as the original non-inlined version on the Sparc. It is clear that the improvements are related to the processor architecture. This is probably due to differences in the cost of the threading operation. On the Sparc, for ex- ample, avoiding the pipeline stalls associated with threading seems to make a significant difference. Figure 13 shows the final size of the macro cache for each benchmark on the Sparc, plotted as a factor of the size of the original (unoptimized) code. The final macro cache135 cache size original code size original code size (kbytes) Figure 13: Macro cache size (diamonds) and optimized threaded code size (crosses), plotted as a factor of the original code size. sizes vary slightly for each architecture, since they depend on the size of the bytecode implementations. However, the shape is the same in each case. The average ratios of original bytecode size to the macro cache size show that the cost is between three and four times the size of the original code on the Sparc. (The ratio is almost identical for the PowerPC, and slightly smaller for the Pentium.) We observe that this ratio decreases gradually as the original code size increases. This is to be expected, since larger bodies of code will tend to reuse macro opcodes rather than generating new ones. We tested this by translating the bytecoded version of the Objective Caml compiler: 421,532 bytes of original code generated 941,008 bytes of macro op-code implementation on the Sparc. This is approximately 2.2 times the size of the original code, and is shown as the rightmost point in the graph. Inlined threaded code is always smaller than the original code from which is generated. Figure 13 also shows the final optimized code size for each benchmark. We observe that the ratio is independent of the size of the benchmark. This is also to be expected, since the reduction in size is dependent on the average number of opcodes in a common sequence and the density of the corresponding macro opcodes in the final code. These depend mainly on the characteristics of the language and its opcode set. Some systems have a long-lived object memory, and generate new executable code at runtime. A realistic implementation for such systems would recycle the macro cache space, and possibly use profiling to optimize only popular areas of the program. For example, the 68040LC emulator found on Macintosh systems performs dynamic translation of 68040 into PowerPC code; it normally requires only 250Kb of cache in which the most commonly used translated code sequences are stored [Tho95]. A similar (fixed) cache size is effective in the BrouHaHa Smalltalk system [Mir97]. Translation speed is also an important factor. To measure this we ran the Object Caml bytecode compiler (a much larger program than any of the benchmarks) with our modified interpreter. The 105,383 opcodes of the Objective Caml compiler are translated in 0.22 seconds on the Sparc, a rate of 480,000 opcodes per second. The inlining interpreter executes the compiler at a rate of 2.4 million opcodes per sec- ond. Translation is therefore approximately five times slower than execution. 7 5 Related work BrouHaHa and Objective Caml have both demonstrated the benefits of creating specialized macro opcodes that perform the work of a sequence of common opcodes. In Objective Caml this led to a new bytecode set. In BrouHaHa the standard Smalltalk-80 bytecodes are translated into threaded code for execution; the detection of a limited number of pre-determined common bytecode sequences is performed during translation, and a specialized opcode is substituted in the executable code. Our contribution is the extension of this technique to dynamically analyze and generate implementations for new macro opcodes at runtime. Several systems use concatenation of pre-compiled sequences of code at runtime [Aus96, Noe98], but in a completely different context. Their precompiled code sequences are generic "templates" that can be parameterized at run-time with particular constant values. A template-based approach is also used in some commercial Smalltalk virtual machines that perform dynamic compilation to native code [Mir97]. However, this technique is complex and requires a significant effort to implement the templates for a new architecture. An interesting system for portable dynamic code generation is vcode [Eng96], an architecture-neutral runtime as- sembler. It generates code that approaches the performance of C on some architectures. Its main disadvantage is that retrofitting it to an existing virtual machine requires a significant amount of effort - certainly more than the single day that was required to implement our technique in a production virtual machine. (Our simple nfibs benchmark runs about 40% faster using vcode, compared to our RISC-like inlined threaded code virtual machine.) Superoperators [Pro95] are a technique for specializing a bytecoded C interpreter according to the program that it is to execute. This is possible because the specialized 7 Since translation is performed only once for each opcode, the "break-even" point is passed in any program that executes more than six times the number of opcodes that it contains. interpreter is generated at the same time as the compiled (bytecoded) representation of the program. A compile-time analysis of the program chooses likely candidates for super- operators, which are then implemented as new interpreter bytecodes. Superoperators are similar to our macro opcodes. One advantage is that their corresponding synthesized bytecodes can benefit from some of the inter-opcode optimizations that our simple concatenation of implementations fails to exploit. However, superoperators require bytecodes corresponding precisely with the nodes used to build parse trees - which might not always be the best choice of bytecode set. It would also be tricky to use superoperators in an incremental system such as Smalltalk, where new executable code is generated at runtime. Nevertheless, an investigation of merging some of the techniques of superoperators and dynamically-generated macro opcodes might be very worthwhile. 6 Conclusions This work was inspired by the need to create an interpreter with a very fine-grain RISC-like opcode set, that is both general (not tied to any particular high-level language) and amenable to traditional compiler optimizations. The cost of opcode dispatch is more significant in such a context, compared to more abstract interpreters whose bytecodes are carefully matched to the language semantics. The expected benefits of our technique are related to the average semantic content of a bytecode. We would expect languages such as Tcl and Perl, which have relatively high-level opcodes, to benefit less from macroization. Interpreters with a more RISC-like opcode set will benefit more - since the cost of dispatch is more significant when compared to the cost of executing the body of each bytecode. The Objective Caml bytecode set is positioned between these two extremes, containing both simple and complex opcodes. 8 Vcode has better performance than our technique because its instruction set matches very closely the underlying architecture. It can exert very fine control over the code that is generated, such as performing some degree of reordering for better instruction scheduling. We believe that similar results can be achieved with our RISC-like inlining threaded code interpreter, but in a more portable manner. The performance of macro opcodes is limited by the inability of the compiler to perform the inter-opcode optimizations that are possible when a static analysis is performed and new macro opcodes implemented manually in the in- terpreter. We believe that these limitations are less important when using a very fine-grain opcode set, corresponding more closely to a traditional RISC architecture. Most op-codes will be implemented as a single machine instruction, and new opportunities for inter-opcode optimization will be available to the translator's code generator. Our technique is portable, simple to implement, and orthogonal to the implementation of the virtual machine's op- codes. In reducing the overhead of opcode dispatch, it helps to bring the performance of fine-grained bytecodes to the same level as that of more abstract, language-dependent op-code sets. 8 Significant overheads are associated with the technique used to check for stack overflow and pending signals in Objective Caml, but a discussion of these is beyond the scope of this paper. speed (seconds) space (bytes) Pentium Sparc PowerPC Sparc benchmark original inlined original inlined original inlined original inlined cache boyer 2.0 1.81 (111%) 2.3 1.50 (154%) 1.4 1.19 (113%) 13800 8324 42012 fib 2.0 1.44 (140%) 4.0 2.47 (163%) 1.6 1.12 (139%) 5288 3320 20160 genlex 1.0 0.93 (110%) 1.1 0.84 (127%) 0.7 0.59 (118%) 45696 26856 156892 kb 10.3 8.15 (126%) 16.9 7.71 (219%) 6.3 5.36 (118%) 20968 13048 75868 qsort 5.8 3.95 (146%) 9.5 5.39 (175%) 4.1 2.98 (137%) 6676 3932 26416 qsort* 4.8 3.04 (158%) 8.0 4.26 (188%) 3.3 2.27 (147%) 6532 3884 25280 sieve 3.0 2.79 (107%) 2.5 2.22 (110%) 1.9 1.86 (100%) 5200 3312 20124 soli 3.1 2.18 (144%) 5.1 2.98 (170%) 2.1 1.50 (142%) 6644 3952 25516 soli* 2.4 1.38 (172%) 4.0 2.00 (202%) 1.6 0.93 (168%) 6544 3908 24548 takc 2.8 1.91 (144%) 5.0 3.26 (152%) 2.1 1.47 (142%) 4784 3012 18652 taku 4.9 3.20 (152%) 7.0 4.14 (170%) 3.2 2.33 (139%) 4812 3036 18296 Table 3: Raw results for the Objective-Caml benchmarks. 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Mathew Zaleski , Angela Demke Brown , Kevin Stoodley, YETI: a graduallY extensible trace interpreter, Proceedings of the 3rd international conference on Virtual execution environments, June 13-15, 2007, San Diego, California, USA Mourad Debbabi , Abdelouahed Gherbi , Azzam Mourad , Hamdi Yahyaoui, A selective dynamic compiler for embedded Java virtual machines targeting ARM processors, Science of Computer Programming, v.59 n.1-2, p.38-63, January 2006 Arun Kejariwal , Xinmin Tian , Milind Girkar , Wei Li , Sergey Kozhukhov , Utpal Banerjee , Alexander Nicolau , Alexander V. Veidenbaum , Constantine D. 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threaded code;bytecode interpretation;dynamic translation;inlining;just-in-time compilation

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Transient loss performance of a class of finite buffer queueing systems.

Performance-oriented studies typically rely on the assumption that the stochastic process modeling the phenomenon of interest is already in steady state. This assumption is, however, not valid if the life cycle of the phenomenon under study is not large enough, since usually a stochastic process cannot reach steady state unless time evolves towards infinity. Therefore, it is important to address performance issues in transient state.Previous work in transient analysis of queueing systems usually focuses on Markov models. This paper, in contrast, presents an analysis of transient loss performance for a class of finite buffer queueing systems that are not necessarily Markovian. We obtain closed-form transient loss performance measures. Based on the loss measures, we compare transient loss performance against steady-state loss performance and examine how different assumptions on the arrival process will affect transient loss behavior of the queueing system. We also discuss how to guarantee transient loss performance. The analysis is illustrated with numerical results.

Introduction For a queueing system with finite buffer, loss performance is usually of great interest. One may want to know the probability that loss of workload occurs, or the expected workload loss ratio, due to buffer overflow. In the existing literature, this issue is typically addressed under the assumption that the stochastic process modeling the phenomenon of interest is already in steady state. This assumption is, however, not valid if the life cycle of the This work was supported by Academia Sinica and National Natural Science Foundation of China. phenomenon under study is not large enough, since usually a stochastic process cannot reach steady state unless time evolves towards infinity. For example, let us consider performance guarantee for real-time communications in high-speed networks, where performance must be defined on a per-connection basis [10]. Since the duration of any connection in a real-world network is always finite, the performance of the connection is in fact dominated by transient behavior of the underlying traffic process carried by the connection. Therefore, it is important to address performance issues in transient state. In this work, we assume that arrival processes are stationary. For a non-stationary process, both steady state and transient state are meaningless. Previous work in transient analysis of queueing systems usually focuses on Markov models, for example, see [1, 17]. Due to complexity involved in analysis, one may have to re-sort to simulation. Nagarajan and Kurose have examined transient loss performance defined on an interval basis for packet voice connections in high-speed networks by simulation [13]. They observed that in transient state, voice connections experienced more serious performance degra- dations. Our work is different from the previous work in that we analyze transient loss performance of a class of finite buffer queueing systems that may not be necessarily Markovian. Now let us consider a finite buffer queueing system fed by a two-state fluid-type stochastic process, where the state of the arrival process represents the arrival rate of workload to the system, that is, the amount of workload arrived per time unit. When the arrival process is in a given state, the arrival rate is a constant. The server of the system processes workload in the queue at a constant service rate unless the queue is empty. This queueing system is not necessarily Markovian. Fig. 1 shows a typical scenario illustrating such a system. The notations in Fig. are explained as follows, which will also be used in the following sections. ffl R(t): the arrival process representing the amount of workload arrived per time unit at time t. ffl r1 and r0 : the arrival rates associated with the states of the arrival process where r1 ? r0 - 0. ffl ON and OFF: we use ON and OFF to refer to states corresponding to r1 and r0 , respectively, and assume OFF ON ON OFF time Figure 1: A finite buffer queueing system fed by a general two-state fluid-type stochastic process. that the initial state of the process is ON. Note that when R(t) is in the OFF state, r0 may not necessarily be zero. ffl Sn and 1: the lengths of the nth ON and OFF intervals, respectively. For the time being, we assume that both Sn and are i.i.d. random variables with arbitrarily given distributions. So we can also denote Sn and Tn by S and T , respectively. ffl B: the buffer size of the queueing system measured by the maximum amount of workload that can be accommodated by the buffer. ffl C: the service rate of the server measured by the amount of workload processed per time unit. In this paper, our aim is to investigate loss performance of such a class of queueing systems. Instead of loss performance defined in idealized steady state, we are interested in transient loss performance. For example, we want to know the probability that loss of workload occurs during the nth ON interval, or the expected workload loss ratio of the nth ON interval, for arbitrary n - 1. We have obtained closed-form loss performance measures defined in transient state. Based on the performance measures, we can therefore contrast transient loss performance with steady-state loss performance and compare transient loss behaviors of the queueing systems fed by different stochastic processes. For some applications, performance guarantee is important. So we will also discuss how to guarantee transient loss performance. The results obtained are useful, since many interesting phenomena in performance-oriented studies can be modeled by two-state fluid-type stochastic processes, and the application of our results can also be extended to multiple-state processes. In the following, we first present our analysis regarding how to define, compute and guarantee transient loss performance (Section 2). To compare transient loss against loss in steady state, we also describe how to compute steady-state limits of the loss measures (Section 3). Then we use numerical results to illustrate the analysis (Section 4). After a brief discussion on some implications and application of the results (Section 5), we conclude the paper and point out some issues to be explored further (Section 6). Transient Loss Performance Analysis For a finite buffer queueing system fed by a two-state fluid-type stochastic process with arrival rates r1 ? r0 , if the service rate C of the system is greater than r0 but less than r1 , then loss of workload due to buffer overflow can only occur when the arrival process is in the ON state In fact, this is the only non-trivial case that should be considered. If C - r0 , then loss performance will be out of control in transient state. On the other hand, if loss will never occur for certain. Therefore, in the following, we assume r0 ! C ! r1 , and consider only loss measures defined for ON intervals of an arrival pro- cess. Since transient performance of the queueing system depends on temporal behavior of the system state, before we derive transient loss measures, we shall first investigate dynamical evolution of system state variables. 2.1 Stochastic Dynamics of the Queueing System The state variables of interest regarding the queueing system are Wn and Qn , which represent the amounts of workload in the buffer at the beginning and the end of the nth ON interval of the arrival process, respectively. Suppose that initially where w1 is a given constant between 0 and B. According to flow balance, the evolution of Wn and Qn is as follows. ae ae where n - 1. Since S and T are random variables, Wn and Qn are also random variables. Denote by 'n(w) and /n(q) the probability density functions of Wn and Qn , re- spectively. Evidently, 'n(w) and /n(q) can be viewed as a solution of the above evolution equations in the probabilistic sense, which can be readily obtained by recurrence. and where for is the probability density function of Qn conditioned on is the probability density function of S, is the probability density function of Wn conditioned on is the probability density function of T , and ffi(\Delta) is the Dirac delta function. In general, the Dirac function is defined by ae An important property of the Dirac delta function is is a function defined for x 2 (\Gamma1; 1). It is easy to verify that the above probability density functions are nonnegative and normalizable. The process Wn has significant impact on transient loss performance of the queueing system. To reach steady state for realizing steady-state performance, the system needs time to forget its history characterized by the initial condition and the distributions of Wn for 2.2 Transient Loss Measures We consider two transient loss measures defined for ON intervals of an arrival process. In the following, we denote an arrival process R(t) by R(n) when t is in the nth ON interval where n - 1. Our first transient loss measure is the probability that loss of workload occurs during the nth ON interval due to buffer overflow, denoted by r1 g. The second transient loss measure is the expected workload loss ratio of the nth ON interval, denoted by We first compute the loss probability. In order to derive g, we consider of exposition, denote the nth ON interval by [0; S). We use w(t) to represent the amount of workload in the buffer at t where t 2 [0; S). Given the event that loss of workload due to buffer overflow occurs in the interval is equivalent to existing - 2 [0; S) such that depends on w, we express such dependence by -(w). It is easy to see Therefore, for For simply use as transient loss measure since w1 is a constant. For we have where 'n (w) can be readily obtained by recurrence as shown in the previous subsection. Now we consider the expected workload loss ratio. Denote by ' the fraction of workload lost due to buffer overflow in the nth ON interval. We assume first since in [0; S), loss of workload due to buffer overflow begins only after reaches -(w). If S ? -(w), then the amount of workload lost in [0; S) equals the amount of workload arrived in [0; S) minus the amount of workload accepted in [0; S), which is r1S \Gamma (CS where Therefore, ae \Theta ' depends on S, we can also denote ' by '(S). where FS (s) is the probability distribution function of S. Denote '(s) by u, we have s and When equivalent to Therefore, for Z u 0P ae oe du: Since w1 is a constant, for simply use Z u 0P ae oe du as transient loss measure. For n - 2, we have Note that for n - 2, the loss probability and the expected loss ratio depend on the distributions of both S and T since 'n(w) depends on the distributions of not only S but also 2. 2.3 Investigating Transient Loss Behavior We can investigate transient loss behavior of the queueing system by computing the loss measures for any given - 1. But it is not necessary for us to do so. In fact, we can focus only on a series of a small number of ON and OFF intervals. Since both ON and OFF intervals have impact on transient loss performance, such a series must contain at least two ON intervals and one OFF interval between them, as shown in Fig. 2. ON OFF ON Figure 2: An ON-OFF-ON series of the arrival process. Now let us consider such an ON-OFF-ON series. Without loss of generality, the two ON intervals in the series can be numbered by 2. Since loss behavior of the queueing system during the first ON interval in the series cannot capture the impact of the OFF interval, we examine loss behavior of the system in the second ON interval. Using the results in the previous subsections, we have dqdw dw dq and Z BZ u 0P ae oe dqdudw Z BZ u 0P ae oe Z u 0P ae oe du dq Z u 0P ae oe du: (2) Before the ON-OFF-ON series begins, the history of the system is summarized by As time evolves, the ON-OFF-ON series will probabilistically duplicate itself, that is, this pattern will appear repeatedly with lengths of ON and OFF intervals drawn from the same distribu- tions. At the beginning of each such duplication, there is an amount of workload w1 left in the buffer, reflecting the impact of the history of the system. Therefore, in order to investigate transient loss performance of the queueing system, it is sufficient to consider or only for examine the behavior of the loss measures as w1 varies between 0 and B. In fact, by focusing only on this simple pattern, we can observe any possible transient loss behavior of the queueing system. 2.4 Stochastically Bounding Transient Loss For some applications, users may require statistically guaranteed loss performance. For example, it may be required that the loss probability or the expected loss ratio should not exceed some given number. This is an important issue for real-time communications in high-speed networks and has been addressed typically under the assumption of steady state. Here we discuss how to guarantee loss performance in transient state for queueing systems with two-state fluid-type arrival processes. It may not be convenient to use the loss measures defined for individual ON intervals directly for transient loss performance guar- antee. Instead, we consider some upper bounds on the loss measures. For arbitrary n ? 1, if at the end of the (n \Gamma 1)th ON interval, the buffer has been full already, that is, if then for the nth ON interval, the loss measures will be relatively larger compared to the case of since a smaller Qn\Gamma1 will likely result in a smaller Wn and a smaller Wn implies a relatively less chance of buffer overflow and a relatively smaller expected loss ratio. There- fore, and B]. The probability density function of Wn under the condition 0; otherwise. Accordingly, and Now we show that the upper bounds also hold for the first ON interval if the buffer is initially empty, that is, In this case, we have of course ? 1. This is because to Wn - W1 . Therefore, during the nth ON interval, the queueing system is likely to suffer more severe loss than it does in the first ON interval. Based on the upper bounds on the transient loss performance measures, we can determine the buffer size B and the service rate C such that or is a given loss performance parameter. By doing so, the loss measures defined for individual ON intervals will not exceed the given loss performance parameter if initially the buffer is empty. In the above, we have assumed that both Sn and are i.i.d. random variables, so we can use S and T to represent Sn and respectively. Now we can allow Sn to have different distributions for different n. In this case, S can be viewed as a random variable with an arbitrarily given distribution and stochastically greater than all Sn , that is, Similarly, we can also allow Tn to have different distributions. In this case, T can be viewed as a random variable with an arbitrarily given distribution and all are stochastically greater than T , that is, PfTn ? tg - PfT ? tg for all t - 0. The intuition behind the above arguments is that a stochastically larger S or a stochastically smaller T will likely lead to a larger Wn and will therefore result in a larger loss probability and expected loss ratio. So the above bounds on transient loss measures are still valid. 3 Steady-State Limits of Transient Loss Measures To compare transient loss against loss in steady state, we need to compute the steady-state limits of the loss measures. The loss measures in steady state are defined by Pflossg For a given stationary arrival process, we have the following limiting conditional probability density functions and limiting probability density functions regarding the state variables of the queueing system. To compute the steady-state loss measures, we need to know the steady-state probability density function '(w). Now we show how to compute '(w). Since in steady state we have by definition Z B'(wjy)/(y)dy and Z B/(yjx)'(x)dx we see that '(w) satisfies a homogeneous Fredholm integral equation of the second kind Z BK(w;x)'(x)dx (7) with kernel R B'(wjy)/(yjx)dy. We can see readily where and H(w;x) can also be obtained readily. In general, the Fredholm integral equation can be solved numerically. However, due to the singularity of K(w;x) caused by the Dirac delta function ffi(w) at solve (7) directly. The singularity caused by the Dirac function ffi(w) at in '(w). Since this singularity in '(w) is intrinsic and cannot be removed from '(w), the functional form of '(w) will be where '0 (w) and v(w) are unknown functions. Due to the property of the Dirac delta function ffi(w), only v(0) rather than the whole v(w) will appear in the steady-state loss measures. In the following, we outline a procedure for computing '0(w) and v(0). Substituting (9) and (8) into (7), we have Z BK0(w;x)'0(x)dx Z BH(w;x)'0(x)dx Comparing both sides of the above equation, we see Z BK0(w;x)'0(x)dx: (10) Note that where j is a small positive variable. To avoid the trivial case of unknown, we let and divide both sides of (10) by v(0), so The above equation is a non-homogeneous Fredholm integral equation of the second kind in ' 0 (w) with kernel K0(w;x) and can be solved numerically with the standard method [4]. obtained by solving (12), now we can determine v(0) as follows. Since by definition integrating both sides of the above equation from 0 to B, we have R Bv(w)ffi(w)dw Therefore, Substituting (13) into (11), we obtain finally Based on the above results, the steady-state loss measures now can be computed as follows, and -Z u 0P ae oe du Z BZ u 0P ae oe 4 Numerical Studies In this section, we numerically investigate the following issues: 1) transient loss behaviors of the queueing system with different arrival processes, 2) transient loss versus steady-state loss, 3) the tightness of the upper bounds on the transient loss measures, and 4) the reliability of transient loss performance guarantee with system resources determined according to the upper bounds on the transient loss measures. For simplicity, we assume that S and T are identically distributed and all arrival processes considered have the same E[S] and E[T ]. We consider exponential distribution as well as two heavy-tailed distributions for S and T . The first heavy-tailed distribution is which is a variant of the conventional Pareto distribution. The reason for us to use this variant form of the Pareto distribution is that the random variable of interest in our study takes values from [0; 1) while for the conventional Pareto distribution, the random variable takes values from heavy-tailed distribution is defined by the following probability density function ae ffe \Gammaff A ff t \Gamma(ff+1) t ? A 2: The arrival processes determined by (16) and (17) are long-range dependent (LRD) processes [2, 5, 12], henceforth referred to as LRD process I and LRD process II, respec- tively. Clearly, the arrival process determined by exponentially distributed S and T is a Markov process. Therefore, the queueing system with such a Markov arrival process can also be modeled by a Markov process. However, the queueing system with an LRD arrival process is not Markovian anymore. For simplicity, we let Under the assumption that all arrival processes considered have the same E[S] and E[T ] and S and T are identically distributed, the expectations determined by (16) and (17) are the same. So the parameter A in (17) can be determined by the above parameters. As shown in Subsection 2.3, we can investigate transient loss behavior of the queueing system by considering only for 2. are functions of w1 , in the following, we simply use w to represent and denote by P (w) and by E(w) for convenience of exposition. 4.1 Transient Loss of Different Arrival Processes We consider three arrival processes determined by the distributions of S and T as described above. The assumption that all arrival processes considered have the same E[S] and E[T ] implies that the marginal state distributions of the arrival processes are the same. We assume first that the buffer size and the service rate all the arrival processes. Then, we compare transient loss behaviors of the arrival processes based on the functions P (w) and E(w) defined by (1) and (2), respectively. Note that w in P (w) and E(w) is previously denoted by w1 in (1) and (2). We observe that even though the marginal state distributions of the arrival processes are the same, the transient loss measures of the arrival processes can be significantly different (Fig. 3). The difference in transient loss behavior is only due to the distributions of S and T . We notice in Fig. 3 (b) that if w exceeds some critical value, then the expected loss ratio of the Markov process is less than that of LRD process I. This is due to the effect of the heavy-tailed distributions of S and T . Such effect becomes more significant if C (or B) is large. For example, suppose that we Fig. 3 (c) and (d). Note that a large C implies a stringent loss performance require- ment. In this case, we observe that for any w 2 [0; B], both the loss measures of the Markov process are less than those of LRD processes I and II. This is because for the Markov process, the transient loss measures decay exponentially fast as C or B increases, but for the LRD processes, the loss measures decay only hyperbolically. O O O O O O O O O O O E(w) (b) (d) O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O (a) (c)0.09580.0025810.034P(w) E(w) Markov Process LRD Process II O Process I Figure 3: Transient loss behaviors of different arrival pro- cesses. (a) The loss probability with (b) The expected loss ratio with The loss probability with 0:95. (d) The expected loss ratio with 4.2 Transient Loss versus Steady-State Loss Based on (1), (2), (14) and (15), we can compare transient and steady-state loss measures for different arrival processes with different buffer size B and service rate C. We first fix B and let C take several different values (Fig. 4). Then, we fix C and let B take different values (Fig. 5). The loss measures in steady state are computed according to the procedure outlined in Section 3. Unlike the steady-state loss measures, which are simply some constants, the transient loss measures are functions of the initial value w that summarizes the history of the system. For all the arrival processes considered, we observe that as the initial value varies, the transient loss measures will become greater than the corresponding steady-state loss measures. That is, steady-state loss measures can underestimate actual loss in transient state. As the buffer size B or the service rate C increases, we can still observe such underestimates. Although increasing B or C can decrease the probability for w to take larger values, due to the randomness of ON and OFF periods, it is still possible for w to take values greater than some critical value beyond which the steady-state loss measures will underestimate transient loss. We even observe that if C (or B) is large enough, then for all w 2 [0; B], the steady-state loss measures are strictly less than the corresponding transient loss measures, for example, see the case of C = 0:9 in Fig. 4.3 Upper Bounds on Transient Loss Performance We compute upper bounds (3) and (4) on transient (a)0 E(w) E(w) E(w)w w (d) (b)0.002 Markov Process LRD Process LRD Process II Steady-State Loss Performance Transient Loss Performance Figure 4: Transient loss versus steady-state loss for and 0:9. (a) The loss probability of the Markov process. (b) The expected loss ratio of the Markov process. (c) The loss probability of LRD process I. (d) The expected loss ratio of LRD process I. (e) The loss probability of LRD process II. (f) The expected loss ratio of LRD process II. loss measures for different arrival processes with the same buffer size 0:8. Then we simulate actual transient loss performance with 95% confidence intervals. The results are shown in Table 1. We observe that the upper bounds on transient loss performance for all the arrival processes are quite tight. Of course increasing will affect the tightness of the upper bounds, since for a large B, it is seldom for Qn\Gamma1 to reach B while a tight bound implies that the actual value of Qn\Gamma1 is close to B. The service rate C and the arrival process will also affect the tightness of the upper bounds. If we increase C, then the actual loss probability or expected loss ratio will decrease faster than the corresponding upper bound does. Accordingly, the bound will become loose. However, the decay of the actual loss measures also depends on the distributions of S and T that determine the behavior of the arrival process. In contrast, decreasing B or C will lead to tighter upper bounds. 4.4 Transient Loss Performance Guarantee Some applications need loss performance guarantee. As shown in Subsection 4.2 (Fig. 4 and Fig. 5), transient (a)0 E(w) E(w) E(w)(d) (b) Markov Process LRD Process I LRD Process II Steady-State Loss Performance Transient Loss Performance Figure 5: Transient loss versus steady-state loss for 0:8 and 4. The horizontal axis represents w scaled by B. (a) The loss probability of the Markov pro- cess. (b) The expected loss ratio of the Markov process. (c) The loss probability of LRD process I. (d) The expected loss ratio of LRD process I. (e) The loss probability of LRD process II. (f) The expected loss ratio of LRD process II. loss is significantly different from steady-state loss. Due to the randomness of ON and OFF periods, w can take any value between 0 and B and beyond some critical value of w, transient loss measures P (w) and E(w) will exceed the corresponding steady-state loss measures. Therefore, loss performance guarantee based on steady-state loss measures is not appropriate. Suppose that the buffer size 0:01. Note that for a given C, a small B implies a small delay. For a given loss performance parameter, we can determine the service rate C properly for guaranteeing transient loss performance from (5) and (6) based on the upper bounds. We consider both the loss probability and the expected loss ratio. To be specific, we let the loss performance parameter . The reliability of transient loss performance guarantee based on the upper bounds can be verified by simulation. Table 2 shows the service rate C determined according to the upper bounds and the transient loss performance measured from simulation with 95% confidence intervals for different arrival processes. We see that transient loss performance guarantee based on the upper bounds is indeed reliable. loss probability arrival process upper bound simulation result Markov process 0.323 0.264 \Sigma4:590 \Theta 10 \Gamma3 LRD process I 0.294 0.176 LRD process II 0.269 0.181 \Sigma4:593 \Theta 10 \Gamma3 expected loss ratio arrival process upper bound simulation result Markov process LRD process I 0.034 0.017 \Sigma5:477 \Theta 10 \Gamma4 LRD process II 0.028 Table 1. Upper bounds on the loss measures versus the corresponding simulation results. Markov process ffl C measured loss frequency ffl C measured loss ratio LRD process I ffl C measured loss frequency ffl C measured loss ratio LRD process II ffl C measured loss frequency ffl C measured loss ratio Table 2. Transient loss performance measured from simulation and service rate C determined based on the upper bounds. Discussions In this section we discuss briefly some implications of our results, regarding the choice of loss performance measures and the issue of network traffic modeling in the presence of long-range dependence (LRD) [2, 5, 7, 8, 12, 14, 16]. We also outline a scheme for guaranteeing transient loss performance for real-time communications. 5.1 Loss Performance and Modeling Issue Our analysis and numerical results have shown that transient loss is significantly different from steady-state loss. In idealized steady state, the system has forgotten its history. As a result, the steady-state loss measures are only some constants. In contrast, transient loss measures depend on the history of the system and therefore are vari- ables. If the convergence towards steady state is not fast enough or if the life cycle of the underlying physical process is not large enough for the system to reach steady state, then steady-state loss performance can fail to capture the actual loss behavior in transient state. To use steady-state loss measures to approximate transient loss performance, it may be necessary to verify whether the system can approach steady state fast enough within the life cycle of the underlying physical process. From the numerical results presented in Section 4, we see that for a given arrival process, the loss probability is greater than the expected loss ratio. This phenomenon can be explained as follows. Let us focus on the nth ON interval of an arrival process. According to the results in Subsection 2.2, given the loss probability equals but the expected loss ratio is since '(s) ! 1. Note that if the loss probability is equal to 1, then it only means that loss of workload due to buffer overflow will occur for certain during the ON interval and does not necessarily imply that all workload arrived in the interval is lost. As a consequence, for a given loss performance parameter ffl, requiring a loss probability less than ffl means a more stringent loss performance guarantee and hence needs more resources compared with the case of requiring an expected loss ratio less than the same ffl. Therefore, for loss-tolerant applications, it is more appropriate to use the expected loss ratio as the performance measure, which requires fewer resources. With the loss probability as the performance measure, resource allocation for loss-tolerant applications can be over-conservative and lead to poor utilization of resources. Within the framework of steady-state performance anal- ysis, it is important to study the marginal state distribution of the arrival process. However, the marginal distribution is not important to transient loss performance. What is important is the distributions of random variables such as S and T that determine the behavior of the arrival pro- cess. Our numerical results have shown that different two-state fluid-type arrival processes with the same marginal state distribution but different distributions of S and T can experience significantly different transient loss. The above conclusion may be helpful to resolve the controversy regarding the relevance of LRD in network traffic [5, 14, 18, 8, 7]. The issue is whether it is valid to use Markov processes such as the well-known Markov fluid models for traffic engineering in the presence of LRD. This issue has been extensively investigated under the assumption that the traffic process is already in steady state. Now let us consider a two-state fluid-type stochastic process in transient state. If the process is Markovian, then the distributions of S and T are exponential. However, if S and T obey some heavy-tailed distributions, then the two-state fluid is an LRD process. Although in steady state, the Markov model may still be used [7, 8, 18], transient loss behaviors of the Markov and LRD processes are significantly different due to different distributions of S and T . Our numerical results imply that LRD can have significant impact on loss performance in transient state. Therefore, Markov models are not valid for loss performance analysis in the presence of LRD in transient state. As shown in Fig. 3 (c) and (d), if a Markov process is used to model an LRD process, then the Markov model can indeed underestimate transient loss of the LRD process even though both processes have the same marginal state distribution. 5.2 Transient Loss Performance Guarantee for Real-Time Communications Now let us consider a real-time connection in a high-speed network. Real-time communications are typically delay-sensitive but tolerate some fraction of traffic loss specified by a given loss performance parameter. Suppose that the maximum allowed delay at a link is D (time units). A simple way to meet the delay requirement is to allocate a buffer DC to the connection where C is the bandwidth to be determined. As a result, excessive delay is turned into loss. Transient loss performance guarantee then can be achieved by characterizing bandwidth requirements properly such that the transient loss measure will not exceed a given performance parameter. Since the buffer size required by real-time communications is usually very small due to the delay constraint, we can assume that the buffer requirement can always be satisfied, so we consider only bandwidth allocation. For the traffic process carried by the connection, there are two cases. In the first case, static bandwidth allocation will still be relatively efficient in the sense that the waste of bandwidth is not serious. In this case, we can use a two-state fluid-type stochastic process to bound the bit rate of the underlying traffic source where the state of the two-state process represents the bit rate of the bounding process and r0 ! C ! r1 . Note that C ! r1 implies statistical multiplexing. By replacing B by DC in (5) or (6), we can determine the bandwidth C according to the performance parameter for guaranteeing transient loss performance In the second case, however, static bandwidth allocation will result in poor utilization of bandwidth due to multiple time scales in the traffic process [6]. Therefore, dynamic bandwidth allocation may be necessary to avoid the waste of bandwidth. Dynamic bandwidth allocation requires a mechanism for detecting transitions of the bit rate of the underlying traffic process. In fact, the problem of detecting the rate change has been extensively studied and there have been different methods proposed to solve the problem [9, 3]. In addition, dynamically increasing or decreasing bandwidth requires a signaling mechanism. Such signaling protocols exist. For example, the Dynamic Connection Management scheme in the Tenet Protocol Suite [15] or the ATM signaling protocol can be used for this purpose [11]. Therefore, we can assume that the rate transition can be effectively detected and dynamical band-width increase or decrease can be effectively accomplished, so in the following, we focus only on the issue of dynamic bandwidth allocation. The state space of a traffic process with multiple time scales exhibits typically the following structure. The whole state space consists of several subspaces. The transitions between states within each subspace are much more frequent than the transitions between states of any two sub- spaces. Within each subspace, the underlying traffic process can be modeled or bounded by a two-state fluid-type stochastic process with r1 and r0 representing bit rates . The two-state process depends on the sub- space, that is, for different subspaces, the two-state processes are different. For a given subspace, r0 bounds the bit rates represented by some states of the subspace, and r1 bounds the bit rates represented by all other states of the subspace that cannot be bounded by r0 . Now we can still apply our results regarding two-state fluid-type stochastic processes to allocate a bandwidth to the connection when the underlying traffic process is in an arbitrarily given sub- space. When the traffic process moves from a subspace to another subspace, the bandwidth allocated to the connection will also change accordingly. With such a scheme, for multiple time-scale traffic, bandwidth utilization will remain high while transient loss performance can still be reliably guaranteed. 6 Conclusions and Future Work Suppose that we model a physical process as an arrival process to a finite buffer queueing system. If the life cycle of the physical process is not large enough for the queueing system to reach steady state, then it is necessary to study loss performance of the system in transient state rather than in idealized steady state. However, previous work in transient analysis of queueing systems typically focuses on Markov models. In this paper, we have presented an analysis of transient loss performance for a class of finite buffer queueing systems fed by two-state fluid-type stochastic processes that may not be necessarily Marko- vian. We have discussed how to define, compute and guarantee transient loss performance and illustrated the analysis with numerical results. Our work is useful since many interesting phenomena in performance-oriented studies can be modeled by two-state fluid-type stochastic processes, and our results can also be used for multiple-state processes with multiple time scales. A lot of work remains to be done. For example, it may be interesting to implement and demonstrate the scheme of transient loss performance guarantee for real-time communications outlined in Subsection 5.2 in an experimental environment such as an ATM test bed. Another issue to be explored further is to extend the analytical results to more general stochastic processes. --R "Transient analysis of Markovian queueing systems and its application to congestion-control modeling" "Long range dependence in variable bit rate video traffic" "Predictive dynamic bandwidth allocation for efficient transport of real-time VBR video over ATM" Computational Methods for Integral Equations "Exper- imental queueing analysis with long-range dependent packet traffic" "RCBR: a simple and efficient service for multiple time-scale traffic" "On the relevance of long-range dependence in network traffic" "What are the implications of long-range dependence for VBR- video traffic engineering?" "Quick detection of changes in traffic statistics: application to variable rate com- pression" "On computing per-session performance bounds in high-speed multi-hop computer networks" "An empirical evaluation of adaptive QOS renegotiation in an ATM net- work" "On the self-similar nature of ethernet traffic (extended version)" "On defining, computing and guaranteeing quality-of-service in high-speed networks" "On the use of fractional Brownian motion in the theory of connectionless networks" "Dynamic management of guaranteed performance multimedia con- nections" "Wide-area traffic: the failure of Poisson modeling" "Transient analysis of cumulative measures of Markov models" "The importance of long-range dependence of VBR video traffic in ATM traffic engineering: myths and realities" --TR Computational methods for integral equations On defining, computing and guaranteeing quality-of-service in high-speed networks On computing per-session performance bounds in high-speed multi-hop computer networks On the self-similar nature of Ethernet traffic (extended version) Dynamic management of guaranteed-performance multimedia connections area traffic RCBR Experimental queueing analysis with long-range dependent packet traffic What are the implications of long-range dependence for VBR-video traffic engineering? The importance of long-range dependence of VBR video traffic in ATM traffic engineering On the relevance of long-range dependence in network traffic

stochastic modeling;transient loss performance;queueing systems

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Queueing-based analysis of broadcast optical networks.

We consider broadcast WDM networks operating with schedules that mask the transceiver tuning latency. We develop and analyze a queueing model of the network in order to obtain the queue-length distribution and the packet loss probability at the transmitting and receiving side of the nodes. The analysis is carried out assuming finite buffer sizes, non-uniform destination probabilities and two-state MMBP traffic sources; the latter naturally capture the notion of burstiness and correlation, two important characteristics of traffic in high-speed networks. We present results which establish that the performance of the network is a complex function of a number of system parameters, including the load balancing and scheduling algorithms, the number of available channels, and the buffer capacity. We also show that the behavior of the network in terms of packet loss probability as these parameters are varied cannot be predicted without an accurate analysis. Our work makes it possible to study the interactions among the system parameters, and to predict, explain and fine tune the performance of the network.

Introduction It has long been recognized that Wavelength Division Multiplexing (WDM) will be instrumental in bridging the gap between the speed of electronics and the virtually unlimited bandwidth available within the optical medium. The wave-length domain adds a significant new degree of freedom to network design, allowing new network concepts to be devel- oped. For a local area environment with a small number of users, the WDM broadcast-and-select architecture has emerged as a simple and cost-effective solution. In such a LAN, nodes are connected through a passive broadcast star coupler and communicate using transceivers tunable across the network bandwidth. This work was supported in part by NSF grant NCR-9701113. A significant amount of research effort has been devoted to the study of WDM architectures in recent years [4]. The performance analysis of these architectures has been typically carried out assuming uniform traffic and memoryless arrival processes [16, 3, 5]. However, it has been established that, in order to study correctly the performance of a net- work, one needs to use models that capture the notion of burstiness and correlation in the traffic stream, and which permit non-uniformly distributed destination probabilities [8, 9]. Two studies of optical networks that use non-Poisson traffic models appeared recently in [13, 14]. The work in [13] derives a stability condition for the HiPeR-' reservation protocol, while [14] studies the effects of wavelength conversion in wavelength routing networks. We are not aware of any queueing-based studies of broadcast WDM networks. In this paper we revisit the well known broadcast-and- select WDM architecture in an attempt to investigate the performance of broadcast optical networks under more realistic traffic assumptions and finite buffer capacity. Specif- ically, we develop a queueing-based decomposition algorithm to study the performance of a network operating under schedules that mask the transceiver tuning latency [6, 12, 1, 2, 11]. The analysis is carried out using Markov Modulated Bernoulli Process (MMBP) arrival models that naturally capture the important characteristics of traffic in high-speed networks. Additionally, our analysis allows for unequal traffic flows to exist between sets of nodes. Our work makes it possible to study the complex interaction among the various system parameters such as the arrival processes, the number of available channels, and the scheduling and load balancing algorithms. To the best of our knowledge, such a comprehensive performance analysis of a broadcast WDM architecture has not been done before. The next section presents the queueing and traffic model and provides some background information. The performance analysis of the network is presented in Sections 3 and 4, numerical results are given in Section 5, and we conclude the paper in Section 6. System Model In this section we introduce a model for the media access control (MAC) layer in a broadcast-and-select WDM LAN. The model consists of two parts, a queueing network and a transmission schedule. We also present a traffic model to characterize the arrival processes to the network. l l l l l l l l node N CCfixed optical filters l (1) l (N) passive star transmitting queues receiving queues to users node 1 node N node 1 tunable lasers from users Figure 1: Queueing model of a broadcast WDM architecture with N nodes and C wavelengths 2.1 The Queueing Model We consider an optical network architecture with N nodes communicating over a broadcast passive star that can support Figure 1). Each node is equipped with a laser that enables it to inject signals into the optical medium, and a filter capable of receiving optical signals. The laser at each node is tunable over all available wavelengths. The optical filters, on the other hand, are fixed to a given wavelength. Let -(j) denote the receiving wavelength of node j. Since C - N , a set Rc of nodes may be sharing a single wavelength -c : Each node consists of a transmitting side and a receiving side, as Figure 1 illustrates. New packets (from users) arrive at the transmitting side of a node i and are buffered at a finite capacity queue, if the queue is not full. Otherwise, they are dropped. As Figure 1 indicates, the buffer space at the transmitting side of each node is assumed to be partitioned into C independent queues. Each queue c; at the transmitting side of node i contains packets destined for the receivers which listen to wavelength -c . This arrangement eliminates the head-of-line problem, and permits a node to send several packets back-to-back when tuned to a certain wavelength. We let B (in) ic denote the capacity of the transmitting queue at node i corresponding to channel -c . Packets buffered at a transmitting queue are sent on a FIFO basis onto the optical medium by the node's laser. A schedule (discussed shortly) ensures that transmissions on a given channel will not collide, hence a transmitted packet will be correctly received by its destination node. Upon arriving at the receiving side of its destination node, a packet is placed in another finite capacity buffer before it is passed to the user for further processing. We let B (out) j denote the buffer capacity of the receiving queue at node j. Packets arriving to find a full receiving queue are lost. Packets in a receiving queue are also served on a FIFO basis. Packets in the network have a fixed size and the nodes operate in a slotted mode. Since there are N nodes but C - N channels, the passive star (i.e., each of the C channels) must run at a rate N C times faster than the rate at which users at each node can generate or receive packets ( N need not be an integer). In other words, the MAC-to-network interface runs faster than the user-to-MAC interface. Thus, we distinguish between arrival slots (which correspond to the packet transmission time at the user rate) and service slots (which are equal to the packet transmission time at the channel rate within the network). Obviously, the duration of a ga a ag g g1 N Nl c Frame (a) (b) arrival slot service slot c c Figure 2: (a) Schedule for channel -c , and (b) detail corresponding to node 2 a service slot is equal to C N times that of an arrival slot. All N nodes are synchronized at service slot boundaries. Using timing information about service slots and the relationship between service and arrival slots one can derive the timing of arrival slots. Hence, we assume that all users are also synchronized at arrival slot boundaries. 2.2 Transmission Schedules One of the potentially difficult issues that arises in a WDM environment, is that of coordinating the various transmit- ters/receivers. Some form of coordination is necessary because (a) a transmitter and a receiver must both be tuned to the same channel for the duration of a packet's transmis- sion, and (b) a simultaneous transmission by one or more nodes on the same channel will result in a collision. The issue of coordination is further complicated by the fact that tunable transceivers need a non-negligible amount of time to switch between wavelengths. Several scheduling algorithms have been proposed for the problem of scheduling packet transmissions in such an environment [6, 12, 1, 2, 11]. Although these algorithms differ in terms of their design and operation, surprisingly the resulting schedules are very similar. A model that captures the underlying structure of these schedules is shown in Figure 2. In such a schedule, node i is assigned a ic contiguous service slots for transmitting packets on channel -c . These a ic slots are followed by a gap of g ic - 0 slots during which no node can transmit on -c . This gap may be necessary to ensure that the laser at node i + 1 has sufficient time to tune from wavelength -c\Gamma1 to -c before it starts transmis- sion. Note that in Figure 2 we have assumed that an arrival slot is an integer multiple of service slots. This may not be true in general, and it is not a necessary assumption for our model. Observe also that, although a schedule begins and ends on arrival slot boundaries, the beginning or end of transmissions by a node does not necessarily coincide with the beginning or end of an arrival slot (although they are, obviously, synchronized with service slots). We assume that transmissions by the transmitting queues onto wavelength -c follow a schedule as shown in Figure 2. This schedule repeats over time. Each frame of the schedule consists of M arrival slots. Quantity a ic can be seen as the number of service slots per frame allocated to node i, so that the node can satisfy the required quality of service of its incoming traffic intended for wavelength -c . By fixing a ic , we indirectly allocate a certain amount of the bandwidth of wavelength -c to node i. This bandwidth could, for instance, be equal to the effective bandwidth [7] of the total traffic carried by node i on wavelength -c . In general, the estimation of the quantities a is part of the connection admission algorithm [7], and it is beyond the scope of this paper. We note that as the traffic varies, a ic may vary as well. In this paper, we assume that quantities a ic are fixed, since this variation will more likely take place over larger scales in time. 2.3 Traffic Model The arrival process to each transmitting queue of the net-work is characterized by a two-state Markov Modulated Bernoulli Process (MMBP), hereafter referred to as 2-MMBP. A 2-MMBP is a Bernoulli process whose arrival rate varies according to a two-state Markov chain. It captures the notion of burstiness and the correlation of successive interarrival times, two important characteristics of traffic in high-speed networks. For details on the properties of the 2-MMBP, the reader is referred to [10]. (We note that the algorithm for analyzing the network was developed so that it can be readily extended to MMBPs with more than two states.) We assume that the arrival process to transmitting queue given by a 2- MMBP characterized by the transition probability matrix by A ic as follows: ic q (01) ic ic q (11) ic and A ic (1) In (1), q (kl) is the probability that the 2- MMBP will make a transition to state l, given that it is currently at state k. Obviously, q (k0) Also, ff (0) ic (ff (1) ic ) is the probability that an arrival will occur in a slot at state 0 (1). Transitions between states of the occur only at the boundaries of arrival slots. We assume that the arrival process to each transmitting queue is given by a different 2-MMBP. From (1) and [10], the steady-state arrival probability for the arrival process to this queue is ic ff (0) ic ff (1) ic ic (2) the probability that a packet generated at node i will have j as its destination node. We will refer to as the routing probabilities; this description implies that the routing probabilities can be node-dependent and non-uniformly distributed. The destination probabilities of successive packets are not correlated. That is, in a node, the destination of one packet does not affect the destination of the packet behind it. Given these assumptions, the probability that a packet generated at node i will have to be transmitted on wavelength -c is: Obviously, the relationship between r ic and fl ic is given by Queueing Analysis In this section we analyze the queueing network shown in Figure 1, which represents the tunable-transmitter, fixed- receiver optical network under study. The arrival process to passive starN l filters optical fixed l c c corresponding to listening to l l c c l c l c transmitting queues receiving queues Figure 3: Queueing sub-network for wavelength -c each transmitting queue is assumed to be a 2-MMBP, and the access of the transmitting queues to the wavelengths is governed by a schedule similar to the one described in Section 2.2. We analyze this queueing network in order to obtain the queue-length distribution in a transmitting or receiving queue, from which performance measures such as the packet-loss probability can be obtained. 3.1 Transmitting Side Analysis We first note that the original queueing network can be decomposed into C sub-networks, one per wavelength, as in Figure 3. For each wavelength -c , the corresponding sub-network consists of N transmitting queues, and all the receiving queues that listen to wavelength -c . Each transmitting queue i of the sub-network is the one associated with wavelength -c in the i-th node. These transmitting queues will transmit to the receiving queues of the sub-network over wavelength -c . Note that, due to the independence among the C queues at each node, the transmission schedule (i.e., the fact that different nodes transmit on the same wave-length at different times), and the fact that each receiver listens to a specific wavelength, this decomposition is exact. In view of this decomposition, it suffices to analyze a single sub-network, since the same analysis can be applied to all other sub-networks. Consider now the sub-network for wavelength -c . We will analyze this sub-network by decomposing it into individual transmitting and receiving queues. As discussed in the previous section, each transmitting queue i of the sub-network is only served for a ic consecutive service slots per frame. During that time, no other transmitting queue is served. Transmitting queue i is not served in the remaining slots of the frame. In view of this, there is no dependence among the transmitting queues of the sub-network, and consequently each one can be analyzed in isolation in order to obtain its queue-length distribution. (Each receiving queue will also be considered in isolation in Section 3.2.) From the queueing point of view, the queueing network shown in Figure 3 can be seen as a polling system in discrete time. Despite the fact that polling systems have been extensively analyzed, we note that very little work has been done within the context of discrete time (see, for example, [18]). In addition, this particular problem differs from the typical polling system since we consider receiving queues, which are not typically analyzed in polling systems. a ic (a) Frame l c (b) observation instant transition instant service completion instant 2-MMBP state arrival instant Figure 4: (a) Service period of transmitting queue i on channel -c , and (b) detail showing the relationship among service completion, arrival, 2-MMBP state transition, and observation instants within a service and an arrival slot 3.1.1 The Queue-Length Distribution of a Transmitting Queue Consider transmitting queue i of the sub-network for -c in isolation. This queue receives exactly a ic service slots on wavelength -c , as shown in Figure 4(a). The block of a ic service slots may not be aligned with the boundaries of the arrival slots. For instance, in the example shown in Figure 4(a), the block of a ic service slots begins at the second service slot of arrival slot x \Gamma 1, and it ends at the end of the second service slot in arrival slot x number within a frame. For each arrival slot, define v ic (x) as the number of service slots allocated to transmitting queue i, that lie within arrival slot x 1 . Then, in the example in Figure 4(a), we 0 for all other x 0 . Obviously, we have We analyze transmitting queue i by constructing its underlying Markov chain embedded at arrival slot boundaries. The order of events is as follows. The service (i.e., trans- mission) completion of a packet occurs at an instant just before the end of a service slot. An arrival may occur at an instant just before the end of an arrival slot, but after the service completion instant of a service slot whose end is aligned with the end of an arrival slot. The 2-MMBP describing the arrival process to the queue makes a state transition immediately after the arrival instant. Finally, the Markov chain is observed at the boundary of each arrival slot, after the state transition by the 2-MMBP. The order of these events is shown in Figure 4(b). The state of the transmitting queue is described by the tuple (x; In Figure 4, we assume that each arrival slot contains an integral number of service slots. If this is not the case, v ic (x) is defined as the number of service slots that end within arrival slot x (i.e., if there is a service slot that lies partially within arrival slots x and x will be counted in v ic ffl x represents the arrival slot number within a frame ffl y indicates the number of packets in the transmitting queue ic ), and ffl z indicates the state of the 2-MMBP describing the arrival process to this queue, that is, It is straightforward to verify that, as the state of the queue evolves in time, it defines a Markov chain. Let \Phi denote modulo-M addition, where M is the number of arrival slots per frame. Then, the transition probabilities out of state (x; y; z) are given in Table 1. Note that, the next state after (x; always has an arrival slot number equal to x \Phi 1. In the first row of Table 1 we assume that the 2- MMBP makes a transition from state z to state z 0 (from (1), this event has a probability q (zz 0 ) ic of occurring), and that no packet arrives to this queue during the current slot (from (1) and (3), this occurs with probability at most v ic are serviced during arrival slot x \Phi 1, and since no packet arrives, the queue length at the end of the slot is equal to maxf0; y 1)g. In the second row of Table 1 we assume that the 2-MMBP makes a transition from state z to state z 0 and a packet arrives to the queue. This arriving packet cannot be serviced during this slot, and has to be added to the queue. Finally, the expression for the new queue length ensures that it will not exceed the capacity B (in) ic of the transmitting queue. The probability transition matrix of this Markov chain is straightforward to derive from Table 1. This matrix defines a p-cyclic Markov chain [15], and therefore it can be solved using any of the techniques for p-cyclic Markov chains in [15, ch. 7]. We have used the LU decomposition method in [15] to obtain the steady state probability - ic (x; z) that at the end of arrival slot x, the 2-MMBP is in state z and the transmitting queue has y packets. The steady-state probability that the queue has y packets at the end of slot x, independent of the state of the 2-MMBP is: Finally, we note that all of the results obtained in this subsection can be readily extended to MMBP-type arrival processes with more than two states. 3.2 Receiving Side Analysis Consider the sub-network for wavelength -c in Figure 3, and observe that the arrival process to the receiving queues sharing -c is the combination of the departure processes from the transmitting queues corresponding to -c . An interesting aspect of the departure process from the transmitting queues is that for each frame, during the sub-period a ic we only have departures from the i-th queue. This period is then followed by a gap g ic during which no departure occurs. This cycle repeats for the next transmitting queue. Thus, in order to characterize the overall departure process offered as the arrival process to these receiving queues, it suffices to characterize the departure process from each transmitting queue, and then combine them. (We note that this overall departure process is quite different from the typical superposition of a number of departure processes into a single stream, where, at each slot, more than one packet may be departing.) The overall departure process is completely defined given the queue-length distribution of all transmitting Table 1: Transition probabilities out of state (x; y; z) of the Markov chain Current State Next State Transition Probability ic ff (z) ic queues in the sub-network (which may be obtained using the analysis in Section 3.1), since then the probability that a packet will be transmitted on channel -c in any given service slot is known. However, the individual arrival processes to each of the receiving queues listening on -c are not independent. Specif- ically, if j and j 0 are two receivers on -c , and there is a transmission from transmitting queue i to receiving queue j in a given service slot, then there can be no arrival to receiving queue j 0 in the same service slot. We will nevertheless make the assumption that these arrival processes are indeed independent, and that each is an appropriately thinned (based on the routing probabilities) version of the departure process from the transmitting queues. Note that this is an approximation only when there are multiple nodes with receivers fixed on channel -c . This assumption allows us to decompose the sub-network of Figure 3 into individual receiving queues and to analyze each of them in isolation 2 . 3.2.1 The Queue-Length Distribution of a Receiving Queue As in the previous section, we obtain the queue-length distribution of receiving queue j at arrival slot boundaries. During an arrival slot x a packet may be transmitted to the user from the receiving queue. However, during slot x, there may be several arrivals to this receiving queue from the transmitting queues. Let (x; w) be the state associated with receiving queue j, where ffl x indicates the arrival slot number within the frame ffl w indicates the number of packets at the receiving queue We assume the following order of events. A packet will begin to depart from the receiving queue at an instant immediately after the beginning of an arrival slot and the departure will be completed just before the end of the slot. A packet from a transmitting queue arrives at an instant just before the end of a service slot, but before the end-of- departure instant of an arrival slot whose end is aligned with the end of the service slot. Finally, the state of the queue is observed just before the end of an arrival slot and after the arrival associated with the last service slot has occurred (see Figure 5(b)). We also note that the approach of analyzing each receiving queue in isolation gives correct results for the individual receiving queues; after all, in steady-state, the probability that a packet transmitted by node i on - c will have j as its destination will equal the routing This approach is an approximation only when one attempts to combine results from individual receiving queues to obtain the overall performance for the network. It is possible to apply techniques to adjust for this approximation when aggregating individual results [17]. We will not consider such techniques here, instead we will only concentrate on individual queues. (a) a i+1,c a ic Frame l c (b) instant at which departure starts observation instant arrival instant instant at which departure ends x x Figure 5: (a) Arrivals to receiving queue j from transmitting queues i and detail showing the relationship of departure, arrival, and observation instants Let u j (x) be the number of service slots of any transmitting queue on channel -c within arrival slot x. We have: where v ic (x) is as defined in (4). Quantity u j (x) represents the maximum number of packets that may arrive to receiving queue j within slot x. In the example of Figure 5(a) where we show the arrival slots during which packets from transmitting queues i and may arrive to receiving queue j, we have: u Observe now that (a) at each state transition x advances by one (modulo-M ), (b) exactly one packet departs from the queue as long as the queue is not empty, (c) a number packets may be transmitted from the transmitting queues to receiving queue j within arrival slot x \Phi 1, and that (d) the queue capacity is B (out) . Then, the transition probabilities out of state (x; w) for this Markov chain can be obtained from Table 2. In Table is the probability that transmitting queue packets to receiving queue j given that the system is at the end of arrival slot x (in other words, it is the probability that s i packets are transmitted within slot To obtain L ij as the conditional 3 Since in most cases only one or two transmitting queues will transmit to the same channel within an arrival slot (refer also to Figure 2), the summation and product in the expression in the last column of Table 2 do not necessarily run over all N values of i, only over one Table 2: Transition probabilities out of state (x; w) of the Markov chain Current State Next State Transition Probability probability that a packet is destined for node j, given that the packet is destined to be transmitted on -c , the receive wavelength of node j: r ic as the conditional probability of having y packets at the i-th transmitting queue given that the system is observed at the end of slot x: Then, for r 0 is given by ic Expression can be explained by noting that transmitting queue i will transmit s i packets to receiving queue j during arrival slot x \Phi 1 if (a) v ic packets in its transmitting queue for -c at the beginning of the slot (equivalently, at the end of slot x), and (c) exactly s i of minfy; v ic (x \Phi 1)g packets that will be transmitted by this queue in this arrival slot are for receiver j. Expression represents the "thinning" of the arrival processes to the various receiving queues of the sub-network using the r 0 routing probabilities, and discounts the correlation among arrival streams to the different queues. Expression (9) is the crux of our approximation for the receiving side of the network. If r 0 1, in which case j is the only node listening on wavelength -c , the expression for must be modified as follows (recall that there is no approximation in this case): ic Expressions and (10) are based on the assumption that v ic ic which we believe is a reasonable one. In the general case, quantity v ic (x \Phi 1) in both expressions must be replaced by minfv ic ic g. The transition matrix of the Markov chain defined by the evolution of the state (x; w) of receiving queue j also defines a p-cyclic Markov chain. We have used the LU decomposition method as prescribed in [15] to obtain - j (x; w), the steady-state probability that receiving queue j has w packets at the end of slot x. or two values of i. Thus, this expression can be computed very fast, not in exponential time as implied by the general form presented in the table. 4 Packet-Loss Probability We now use the queue-length distributions - ic (x; y) and derived in the previous section, to obtain the packet-loss probability at the transmitting and receiving queues. 4.1 The Packet-Loss Probability at a Transmitting Queue Let\Omega ic be the packet-loss probability at the c-th transmitting queue of node i, i.e., the probability that a packet arriving to that queue will be lost.\Omega ic can be expressed as: lost per frame at queue c; node i] E[# arrivals per frame at queue c; node i] The expectation in the denominator can be seen to be equal to M fl ic , where fl ic is the steady-state arrival probability of the arrival process to this queue from (2). To obtain the expectation in the numerator, let us refer to Figure which shows the service completion, arrival, and observation instants within slot x. We observe that, due to the fact that at most one packet may arrive in slot x, if the number v ic (x) of slots during which this queue is serviced within arrival slot x is not zero (i.e., v ic (x) ? 0), no arriving packet will be lost. Even if the c-th queue at node i is full at the beginning of slot x, v ic packets will be serviced during this slot, and the order of service completion and arrival instants in Figure 4(b) guarantees that an arriving packet will be accepted. On the other hand, if v ic for slot x, then an arriving packet will be discarded if and only if the queue is full at the beginning of x (equivalently, at the end of the slot before x). Since the 2-MMBP can be in one of two states, we have that the numerator of (11) is equal to x:v ic (x)=0 z=0 ff (z) \Psi denotes regular subtraction with the exception that, if and the summation runs over all x for which v ic Using these expressions and the fact that - ic M for all x, we obtain an expression for \Omega ic as follows: x:v ic (x)=0 z=0 ff (z) 4.2 The Packet-Loss Probability at a Receiving Queue The packet-loss probability at a receiving queue is more complicated to calculate, since we may have multiple packet arrivals to a given queue within a single arrival slot (re- fer to Figure 5(a)). Let us as the conditional probability that n packets will be lost at receiving queue j given that the current arrival slot is x. A receiving queue will lose n packets in slot x if (a) the queue had packets at the beginning of slot x, and (b) exactly B (out) arrived during slot x. We can then write: pkts arrive to j j x](13) similar to (8). The last probability in (13) can be easily obtained using (9) or (10), as in the last column of Table 2. Note that at most u j (x) packets may arrive (and get lost) in arrival slot x. Using (13), we can then compute the expected number of packets lost in slot x as: E[number of packets lost at The expected number of arrivals to receiving queue j in slot x can be computed as: E[# arrivals to j j sPr[s pkts arrive to j j x] (15) Finally, the probability\Omega j that an arriving packet to node j will be lost regardless of the arrival slot x can be found as follows: x=0 E[number of lost packets at j j x] x=0 E[number of arrivals to j j x] 5 Numerical Results We now apply our analysis to a network with nodes. The arrival process to each of the transmitting queues of the network is described by a different 2-MMBP. The 2-MMBPs selected exhibit a wide range of behavior in terms of two important parameters, the mean interarrival time and the squared coefficient of variation of the interarrival time. The routing probabilities we used are: ae That is, receiver 1 is a hot spot, receiving 10% of the total traffic, while the remaining traffic is evenly distributed to the other 15 nodes. The total rate at which packets are generated by users of the network is 1.98 packets per arrival slot. Most of the traffic is generated at node 1, as the rate of new packets generated at this node is 0.583 packets per arrival slot. The packet generation rate decreases monotonically for nodes 2 to 16. For load balancing purposes, we have allocated one of the C channels exclusively to node 1, since this node receives a considerable fraction of the total traffic. The remaining are shared by the other 15 receivers. The allocation of the receivers to the remaining wavelengths was performed in a round-robin fashion, and is given in Table 3 for The quantities a ic of the schedule, i.e., the number of packets to be transmitted by node i onto channel -c per frame (refer to Section 2.2 and Figure 2) were fixed to be as close to (but no less than) 0.5 arrival slots as possible. Recall that, while the length of an arrival slot is independent of C and is taken as our unit of time, the length of a service slot Table 3: Channel sharing for depends on the number of channels. In cases in which 0.5 arrival slots is not an integral number of service slots, the value a ic is rounded up to the next integer to ensure that every queue is granted at least 0.5 arrival slots of service during each frame 4 (i.e., a In constructing the schedules, we have assumed that the time it takes a laser to tune from one channel to another is equal to one arrival slot 5 . Finally, for all of the results we present in this section we have let all transmitting and receiving queues have the same buffer capacity B (i.e., B (in) to reduce the number of parameters that need to be controlled. In Figure 6 we show the part of the schedule corresponding to channel -1 for three different values of the number of channels and 8; the parts of the schedules for other channels are very similar. The schedules will help explain the performance results to be presented shortly. Since the number of nodes each arrival slot is exactly four service slots long. Each node is allocated arrival slots, or 2 service slots for transmissions on each channel, as Figure 6(a) illustrates. For the network is bandwidth limited [12], that is, the length of the schedule is determined by the bandwidth requirements on each channel arrival slots), not the transmission and tuning requirements of each node (= 4 \Theta 0:5 slots). The schedule for Figure 6(b) is an example where there is a non-integral number of service slots within each arrival slot. More precisely, one arrival slot contains 6 , or 2 2 service slots. Each node is assigned two service slots (a for transmissions on each channel, since one service slot is less than 0.5 arrival slots. For the network is again bandwidth limited, and the total schedule length becomes service slots, or 12 arrival slots. Finally, when slots, and the corresponding schedule is shown in Figure 6(c). However, in this case the network is tuning limited [12], i.e., the node transmission and tuning requirements determine the schedule length. Since each node has to transmit for 0.5 arrival slots on each channel, and to tune to each of the 8 channels (recall that the tuning time is one arrival 4 Other schemes for allocating a ic have been implemented, including setting a ic proportional to r ic , setting a ic proportional to ic g, and setting a ic to the effective bandwidth [7] of node i's total traffic carried on channel - c . Although the packet loss probability results do depend on the actual values of a ic , the overall conclusions drawn regarding our analysis are very similar. Thus, we have decided to include only the simplest case here. 5 Again, due to the synchronous nature of this network, if one arrival slot is not an integral number of service slots, the number of service slots for which a transmitter cannot transmit is rounded up to the next integer, thereby setting the required time for tuning to some value slightly greater than one arrival slot. As a result, the tuning time is always d N C e service slots. (a) (c) Transmitting queue number denotes unused slot) arrival slot arrival slot arrival slot service slot service slot service slot Figure Transmission schedules for -1 and slot), the total schedule length is 8 \Theta 0:5 slots. But the transmissions on each channel only take arrival slots; the remaining 4 arrival slots in Figure 6(c) are not used. Figures 7-10 show the packet loss probability (PLP) at four different transmitting queues as a function of the buffer size B for 8. We only show results for two nodes, namely, the node with the highest traffic intensity (node 1) in Figures 7 and 9, and a representative intermediate node (node Figures 8 and 10. We also consider only transmitting queues 1 and 2 (out of C) at each node. Queue 1 at each node is for traffic to be carried on wavelength -1 , which is dedicated to receiver 1 (the "hot spot"). Thus, the amount of traffic received by this queue does not change as we vary the number of channels, since the first channel is dedicated to receiver 1. Queue 2 at each node is for traffic to be carried on wavelength -2 . The amount of traffic received by this queue will decrease as the number of channels increases, since channel -2 will be shared by fewer receivers. The behavior of queue 2 is representative of the behavior of the other Figure 7 plots the PLP\Omega 1;1 (i.e., the PLP at transmitting queue 1 of node 1) as a function of the buffer size B for 8. As expected, the PLP decreases as the buffer size increases. For a given buffer size, however, the PLP changes dramatically and counter to intuition, as the number C of channels is varied. Specifically, the PLP increases with C; that is, adding more channels results in worse per- formance. When B is 10, there is roughly nine orders of magnitude difference between the PLP for and three orders of magnitude difference between As we discussed above, the traffic load of this queue does not change with C; the queue receives the traffic for destination 1, which is always 10% of the total traffic generated at node 1 (see (17)). What does change as C varies is the service rate of the queue, and this change can help explain the results in Figure 7. Referring to Figure 6, we note that when 4, each frame of the schedule is arrival slots long, and 2. Hence, at most 8 packets may arrive to this queue during a frame while as many as 2 packets will be serviced. When indicating a decrease in the service rate of the queue. Simi- larly, for further decrease in available service per frame for this queue. This decrease is the reason behind the sharp increase in PLP with C in Figure 7. Very similar behavior is observed in Figure 8 where we plot\Omega 8;1 , the PLP at transmitting queue 1 of node 8. The main difference between Figures 7 and 8 is in the absolute values values of PLP. The very small PLP numbers are due to the fact that the amount of traffic entering queue 1 of node 8 (0.004 packets per arrival slot) is significantly smaller than the traffic entering the same queue of node 1 (0.058 packets per arrival slot - recall that the traffic source were chosen so that the packet generation rate decreases as the node index increases). In fact, for buffer sizes our analysis gave PLP values that are essentially zero; these values are not plotted in Figure 8 because we believe that they are the result of round-off errors. Figures plot the PLP at transmitting queue 2 of nodes 1 and 8, respectively, against the buffer size. From Table 3 we note that the traffic received by this queue decreases from 30% of the overall network traffic when or 8; this decrease is due to the fact that 5 receivers share wavelength -2 when only 3 receivers share it when 8. Thus, the PLP behavior at this queue will depend not only on the change in the service rate as C varies, but also on the change in the amount of traffic received due to addition of new channels. In Figure 9, and for a given buffer size, the PLP decreases as C increases from 4 to 6 (compare to Figure 7). In this case, the decrease in the traffic arrival rate (from an average rate of 0.175 to 0.105 packets per arrival slot) more than offsets the decrease in the service rate that we discussed above. On the other hand, the PLP values for are less than those for (transmitting queue 2 of node 8) due to the fact that the decrease in the offered load (from 0.012 to 0.007 packets per arrival slot) is not substantial enough to offset the decrease in the service rate; still, this increase is less severe than the one in Figure 8 where there was no decrease in the arrival rate. As C increases to 8 there is no change in the offered traffic for either queue; as expected, the PLP rises with the decrease in the service rate. Finally, Figures 11 and 12 plot the PLP at receiving queues 1 and 8, respectively. Receiving queue 8 is representative of queues 2 through 16 in that it receives 6% of the total network traffic (see (17)). Again, the PLP decreases with increasing buffer size. Also, the lower values of PLP in Figure 12 compared to Figure 11 reflect the fact that only 6% of the total traffic is destined to receiving queue 8, as opposed to 10% for the hot spot queue 1. What is surprising in Figures 11 and 12, however, is that, for a given buffer size, the PLP decreases as the number C of channels increases. This behavior is in sharp contrast to the one we observed in the transmitting side case, and can be explained as follows. First, higher losses at the transmitting queues for larger values of C means that fewer packets will make it to the receiving queues, thus losses will be lower at the latter. But the dominant factor in the PLP behavior in Figures 11 and 12 is the change in the service rate of the receiving queues as C varies (refer to Figure 6). For 4, as many as packets may arrive to each receiving queue within a frame, and 8 packets may be served (i.e., transmitted to the users). When the number of potential arrivals in a frame remains at 32, but the frame is 12 arrival slots long, meaning that up to 12 packets may be served, leading to a drop in the PLP. Finally, for the number of packets served in a frame is the same as in but the maximum number of packets that may arrive becomes only 16, explaining the dramatic drop in the PLP. 6 Concluding Remarks In this paper we introduced a model for the media access control (MAC) layer of optical WDM broadcast-and-select LANs. The model consists of a queueing network of transmitting and receiving queues, and a schedule that masks the transceiver tuning latency. We developed a decomposition algorithm to obtain the queue-length distributions at the transmitting and receiving queues of the network. We also obtained analytic expressions for the packet-loss probability at the various queues. Finally, we presented a study case to illustrate the significance of our work in predicting and explaining the performance of the network in terms of the packet-loss probability. Overall, the results presented in this paper indicate that the performance of a WDM optical network can exhibit behavior that is counter to intuition, and which may not be predictable without an accurate analysis. The performance curves shown also establish that the packet-loss probability in such an environment depends strongly on the interaction among the scheduling and load balancing algorithms, the routing probabilities, and the number of available channels. Our work has made it possible to investigate the behavior of optical networks under more realistic assumptions regarding the traffic sources and the system parameters (e.g., finite buffer capacities) than was possible before, and it represents a first step towards a more thorough understanding of net-work performance in a WDM environment. Our analysis also suggests that simple slot allocation schemes similar to the ones used for our study case are not successful in utilizing the additional capacity provided by an increase in the number of channels. The specification and evaluation of more efficient slot allocation schemes should be explored in future research. --R Impact of tuning delay on the performance of bandwidth-limited optical broadcast networks with uniform traffic Efficient scheduling of nonuniform packet traffic in a WDM/TDM local lightwave network with arbitrary transceiver tuning laten- cies A media-access protocol for packet-switched wavelength division multiaccess metropolitan area networks Call admission control schemes: A review. area traffic: The failure of poisson modeling. Queueing systems for modelling ATM networks. Approximate analysis of discrete-time tandem queueing networks with bursty and correleated input traffic and customer loss Scheduling transmissions in WDM broadcast-and-select networks Packet scheduling in broadcast WDM networks with arbitrary transceiver tuning latencies. A performance model for wavelength conversion with non-poisson traffic Numerical Solutions of Markov Chains. The MaTPi protocol: Masking tuning times through pipelining in WDM optical networks. Stochastic Modeling and the Theory of Queues. Approximate analysis of a discrete-time polling system with bursty arrivals --TR Scheduling transmissions in WDM broadcast-and-select networks area traffic Packet scheduling in broadcast WDM networks with arbitrary transceiver tuning latencies Scheduling of multicast traffic in tunable-receiver WDM networks with non-negligible tuning latencies Queueing systems for modelling ATM networks Approximate Analysis of a Discrete-Time Polling System with Bursty Arrivals HiPeR-l A Performance Model for Wavelength Conversion with Non-Poisson Traffic

markov modulated bernoulli process;discrete-time queueing networks;optical networks;wavelength division multiplexing

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Modeling set associative caches behavior for irregular computations.

While much work has been devoted to the study of cache behavior during the execution of codes with regular access patterns, little attention has been paid to irregular codes. An important portion of these codes are scientific applications that handle compressed sparse matrices. In this work a probabilistic model for the prediction of the number of misses on a K-way associative cache memory considering sparse matrices with a uniform or banded distribution is presented. Two different irregular kernels are considered: the sparse matrix-vector product and the transposition of a sparse matrix. The model was validated with simulations on synthetic uniform matrices and banded matrices from the Harwell-Boeing collection.

Introduction Sparse matrices are in the kernel of many numerical appli- cations. Their compressed storage [?], which permits both operations and memory savings, generates irregular access patterns. This fact reduces and makes hard to predict the performance of the memory hierarchy. In this work a probabilistic model for the prediction of the number of misses on a K-way associative cache memory considering sparse matrices with a uniform or banded distribution is presented. We want to emphasize that an important body of the model is reusable in different algebra kernels. The most important approach to study cache behavior has traditionally been the use of trace-driven simulations [?], [?], [?] whose main drawback is the large amount of time needed to process the traces. Another possibility is nowadays provided by the performance monitoring tools of modern microprocessors (built-in hardware counters), that make This work was supported by the Ministry of Education and Science (CICYT) of Spain under project TIC96-1125-C03, Xunta de Galicia under Project XUGA20605B96, and E.U. Brite-Euram Project BE95-1564 available data such as the number of cache misses. These tools are obviously restricted to the evaluation of the specific cache architectures for which they are available. Finally, analytical models present the advantage that they reduce the times for obtaining the estimations and make the parametric analysis of the cache more flexible. Their weak point has traditionally been their limited precision. Although some models use parameters extracted from address traces [?], more general analytical models have been developed that require no input traces. While most of the previous works focuse on dense algebra codes [?], [?], little attention has been devoted to sparse kernels due to the irregular nature of their access patterns. For example, [?] studies the self interferences on the vector involved in the sparse matrix-vector product on a direct-mapped cache without considering interferences with other data structures and they do not derive a general framework to model this kind of codes. Nevertheless, as an example of the potential usability of such types of models, we have modeled the cache behavior of a common algebra kernel, the sparse matrix-vector product, and a more complex one, the transposition of a sparse-matrix. This last code includes different access patterns and presents an important degree of data reusability for certain vectors. The remainder of the paper is organized as follows: Section ?? presents the basic model parameters and concepts. Sparse matrix-vector product cache behavior is modeled in Section ??, while Section ?? is dedicated to the modeling of the transposition of a sparse matrix. Both models are extended to banded matrices in Section ??. In Section ?? the models are verified and the cache behavior they depict is studied. Finally, Section ?? concludes the paper. 2 Probabilistic model Our model considers three types of misses: intrinsic misses, and self and cross interferences. An intrinsic miss takes place the first time a memory block is accessed. Self interferences are misses on lines that have been previously replaced in the cache by another line that belongs to the same program vector. Cross interferences refer to misses on lines that have been replaced between two consecutive references by lines belonging to other vectors. In direct mapped caches each memory line is always mapped to the same cache line, so replacements take place whenever accesses to two or more memory lines mapped to the same cache line take place. However, in K-way assso- ciative caches lines are mapped to a set of K cache lines. In this case the line to be replaced is selected following a Cache size in words Ls Line size in words Nc Number of cache lines (Cs=Ls ) Nnz Number of non zero elements of the sparse matrix M Number of rows of the sparse matrix N Number of columns of the sparse matrix fi Average of non zeros elements per row (Nnz=M) K Associativity Nk Number of cache sets (Nc=K) pn Probability that a position in the sparse matrix contains a non zero element (fi=N) Probability that there is at least one entry in a group of Ls positions of the sparse matrix r size of an integer size of a real Table 1: Notation used. random or a LRU criterium. Our model is oriented to K-way associative caches with LRU replacement. In order to replace a line in a cache of this kind, K different new lines mapped to the same set must be accessed before reusing the line considered. When the behavior is that of direct mapped caches. The replacement probability for a given line grows with the number of lines affected by the accesses between two consecutive references to it, and the way these lines are distributed among the different sets. This depends on the memory location of the vectors to be accessed, as it determines the set corresponding to a given line. We handle the areas covered by the accesses to a given program vector V as an area vector, S is the ratio of sets that have received K or more lines of vector V, while is the ratio of sets that have received lines from V. This means that S Vi is the ratio of cache sets that require i accesses to new different lines to replace all the lines they contained when the access to V started. Besides K, the number of lines in a set, there is a number of additional parameters our model considers. They are depicted in Table ??. By word we mean the logical access unit, this is to say, the size of a real or an integer. We have chosen the size of a real, but the model is totally scalable. A uniform distribution in an M \Theta N sparse matrix with Nnz non zero elements (entries) allows us to state the following considerations: the number of entries in a group of Ls positions belongs to a binomial B(Ls ; pn) where pn is the probability of a given position of the matrix containing an entry, that is, This way, the probability of generating an access to a given line of the vector times which we are multiplying the sparse matrix is p, which is calculated as the table shows. 2.1 Area vectors union An essential issue is the way we obtain the total area vector. We add the area vectors corresponding to the accesses to the different program vectors. Given two area vectors S corresponding to the accesses to vectors U and V, we define the union area vector where is the ratio of sets that have received lines from these two vectors, and (S U [S V )0 U U V00110011 U U Figure 1: Area vectors corresponding to the accesses to a vector U and a vector V in a cache with N and resulting total area vector. l Figure 2: Area vectors corresponding to a sequential access and an access to lines with a uniform reference probability of a vector covering 11 lines in a cache with N 2. is the ratio of sets that have received K or more lines. Figure ?? illustrates the area vector union process. From now on the symbol [ will be used to denote the vector union op- eration. This method makes no assumptions on the relative positions of the program vectors in memory, as it is based in the addition as independent probabilities of the area ratios. 2.2 Sequential access The calculation of the area vector depends on the access pattern of the corresponding program vector, so we define an area vector function for each access pattern we find in the codes analyzed. For example, the area vector for the access to n consecutive memory positions is (2) is the average number of lines that fall into each set. If l - K then as all of the sets receive an average of K or more lines. The term Ls added to n stands for the average extra words brought to the cache in the first and last lines of the access. 2.3 Access to lines with a uniform reference probability The area vector for an access to an n word vector where each one of the cache lines into which it is divided has the same probability P l of being accessed may be calculated as li (n; P l li (n; P l being B(n,p) the binomial distribution 1 and )eg. An example for this access and the sequential access is presented We define the binomial distribution on a non integer number of elements Figure 3: Correspondence between the location of the non zeros in a banded matrix to be transposed and the groups where they are moved to in the vectors that define the output matrix. in Figure ??. In the case of k accesses of this type, the area vector is S k l (n; P l 2.4 Access to groups of elements with a uniform reference probability In the transposition of a sparse matrix we find the case of a vector of n words divided into groups of t words where the probability Pg of accessing each group is the same. It happens in the main loop of the algorithm, which accesses the non zeros of the input matrix per rows (sequentially) moving them to the group corresponding to the column they belong to in the vectors that define the output matrix. Each group has as many positions as non zeros has the associated column in the input matrix. Only one of the lines of the group is accessed during the processing of each row of the input matrix, and consecutive accesses to the same group address consecutive memory positions. The area covered by an access of this type is given by Sg (n; t; Pg ae where if t - Ls , each line of the vector has a uniform probability being accessed. Whereas if t ? Ls this probability is Ls=tPg . As in the case of S l , Sg may be extended in order to calculate the area covered by k accesses, S k (n; t; p). Due to space limitations we only show the main expressions of our model; all of them can be found in [?]. 2.5 Access to areas displaced with successive references The model may be extended to consider the case in which the area with some access probability is displaced with successive references. Let S gb (b; t; Pg ) be the area vector corresponding to an access to b consecutive groups of t elements with a uniform probability per group Pg of accessing one and only one of the elements of the group. Its value is given by Sg (b \Delta t; t; Pg ). We define S k gb (b; t; Pg ) as the area vector corresponding to k accesses of this type. In the banded sparse matrix transposition case each one of the k accesses corresponds to the processing of a new row of the sparse matrix, as in each row a column leaves the band (to the left), which makes the probability of accessing the group corresponding to it null, and a new one joins the band to the right, thus adding its group to the set of groups that may be accessed during the processing of the row. The situation is depicted in Figure ?? with 5: during the processing of the first row, groups 1 DO J=R(I), R(I+1)-1 ENDDO ENDDO Figure 4: Sparse matrix-vector product. to 5 may be accessed if there is a non zero in the associated column, while in the second row these sets are 2 to 6, and in the third only sets 3-7 may be accessed. The accesses following this pattern cover three adjacent areas: 1. A set of lines with a growing access probability (Area 1 in Figure ??). 2. A set of lines with a constant access probability (Area 2 in Figure ??). 3. A last set symmetric to the first one, of the same size, and with a decreasing access probability (Area 3 in Figure ??). Once the average access probabilities for the lines of this three consecutive areas are known (see [?]), it only remains to combine them to obtain the corresponding area vector. 2.6 Number of lines in a vector competing for the same cache set Finally, for the calculation of the self interference probability a function to compute the average number of lines with which a line of the vector competes for the same cache set is needed. This function is defined as ae is the average number of lines of the vector mapped to the same cache set. 3 Modeling the sparse matrix-vector product The code for this sparse matrix algebra kernel is shown in Figure ??. The format used to store the sparse matrix is the Compressed Row Storage (CRS) [?]: A contains the sparse matrix entries, C stores the column of each entry, and R indicates in which point of A and C a new row of the sparse matrix starts, which permits knowing the number of entries per row. These three vectors and D, the destination vector of the product, present a purely sequential access, thus most of the misses on them are intrinsic. There are no self interferences and very few misses due to cross interferences (specially taking into account that we are considering a K-way associate cache). Therefore the number of misses for these vectors is calculated as the number of cache lines they cover (intrinsic misses). Nevertheless vector X suffers a series of indirect accesses dependent on the location of the entries in the sparse ma- trix, as it is addressed from the value of C(J). The number of misses accessing this vector is calculated multiplying the number of lines referenced per dot product by the number of rows of the sparse matrix and the miss probability of one of these accesses. The first value is calculated as p \Delta N=Ls , as X covers N=Ls lines, and each one has a probability p of being accessed during the dot product times a given row of the sparse matrix, as we have stated in the previous section. The miss probability is calculated as the opposite of the hit probability. Hits take place whenever the accessed line has been referenced during a previous dot product, and it has suffered no self or cross interferences. Cross interferences are generated by the accesses to the remaining vectors, which present a sequential access. The cross interference area is given by the total area vector associated with the accesses to vectors A, C, R and D. This area vector is calculated in the way explained in Section ??, by adding the individual area vectors corresponding to the accesses to each program vector during the period consid- ered. In our case we are interested in calculating the cross interference area vector after i dot products: cross is the area vector covered by the accesses to a vector V during i dot products. The four vectors have a sequential access, so expression Ss derived in Section ?? is applied: These values are obtained considering that one element of D and R, and fi components of A and C are accessed per dot product. A scale factor r is applied to integer vectors in order to take into account that integer data are often stored using less bytes than real ones. This factor is the quotient of the number of bytes required by a real datum and the one required by an integer. As for the self interference probability, each line of X competes on average with C(N) lines of the same vector (see definition of C(n) in (??)) in the same cache set, and they all have the same probability p of being accessed during a dot product times a row of the sparse matrix. As a result, the number of different candidate lines of X to replace a given line that have been accessed after i dot products belongs to a binomial The hit probability in the first access to any line of X during the dot product of the j-'th row of the sparse matrix times vector X is: where p(1\Gammap) i\Gamma1 is the probability that the last access to the line has taken place i dot products ago and S int X (i) is the interference area vector generated by the accesses produced during these i dot products, which is calculated as where the cross interfence area vector obtained in (??) is added to the self interference area vector, which is given by an access with a uniform access probability per line of 1\Gamma(1\Gamma p) i to the lines of vector X that can generate self interferences with another line, which are calculated using (??). Finally, the average hit probability is calculated as: END DO J=C(I)+2 END DO END DO DO K=R(I), R(I+1)-1 CT(P)=I END DO END DO Figure 5: Transposition of a sparse matrix. We must point out that the model only takes into account the first access to each line of X in each dot product. The other Nnz \Gammap\DeltaM \Delta(N=L s) accesses have a very low probability of resulting in a miss, as they refer to lines that have been accessed in the previous iteration of the inner loop. 4 Modeling the transposition of a sparse matrix In this section the model is extended to a operation with a greater degree of complexity, the sparse matrix transposi- tion. As in the previous section, we assume that the sparse matrix and its transposed are stored in a CRS format. Figure ?? shows the transposition algorithm described in [?], where both matrices are represented by vectors(A, C, R) and (AT, CT, RT), respectively. Observe that in loop 4 there are multiple indirection levels both in the left and right side of the sentences. In what follows we employ a similar approximation to the one developed in the previous section to estimate the number of misses for vectors AT, CT and RT. The remaining vectors have sequential accesses, which have already been considered in the previous algorithm. 4.1 Vectors AT and CT These two vectors follow exactly the same access pattern, as may be observed in loop 4 of Figure ??. We will thus explain the estimation of the number of misses for AT, as the one for CT is identical but taking into account that it is an integer vector. The access pattern is the one modeled by Sg , described in Section ??, where and This pattern has similarities with the one explained in Section ?? for vector X, as the access probability is uniformly distributed along the vector, being the difference that for vector X the probability is constant for each line of the vec- tor, while for vectors AT and CT it corresponds to sets of as many elements as a column of the original sparse matrix holds. As a result, the general form of the hit probability during the process of the j-'th row of the sparse matrix is very similar to the one in (??), with the following differences: ffl The probability of accessing the considered line of the vector during the processing of a row is not p but ffl The probability of accessing another line mapped to the same cache set during the processing of the i previous rows is not 1\Gamma(1\Gammap) i but 1\Gamma(S i ffl Vector AT has Nnz elements, so the number of lines that compete in the set with the line considered is C(Nnz) in the equation that calculates the interference area vector. ffl When calculating the cross interference probability, the same scheme of adding the area vectors corresponding to the access to the remaining vectors is used. R (i), S C (i) and S A (i) are calculated according to (??) and the remaining area vectors are l (N \Delta (Nnz \Delta 4.2 Vector RT This vector is referenced in the four loops of the algorithm. In the first loop it has a totally sequential access that produces misses. In the second loop, the accesses to RT follow a similar pattern to those of vector X in the sparse matrix-vector product. The only differences consist in that there is only one possible source of cross interferences (vector C), and that RT is an integer vector and not a real value vector. The number of misses in loop 2 is first hit RT2 where P hit RT2 is calculated following expressions (??) and (??) introducing the modifications mentioned above. Vector RT has been completely accessed in a sequential manner in the previous loop. For this reason we must add the probability first hit RT2 of getting a hit due to the existence of portions of RT in the cache when initiating the loop. Again, a final expression can be found in [?]. In loop 3, vector RT is sequentially accessed without any accesses to other vectors. The estimated number of misses is where P first hit RT3 may be estimated as P first hit RT2 \Delta pr \Delta M . Finally, in loop 4 the access to RT is again similar to the sparse matrix-vector product described in Section ??. In this loop we must also consider a hit probability P first hit RT4 due to a previous load of RT generated by loop 3. 5 Extension to banded matrices A large number of real numerical problems in engineering lead to matrices with a sparse banded distribution of the entries [?]. In this section we present the modifications the model requires in order to describe the cache behavior for matrices of this type. The model parameter pn (see Table ??) is calculated as fi W , where W is the bandwidth. Consequently p takes the value 5.1 Sparse matrix-vector product The number of misses over vector X is the only one affected by the band distribution of the entries. The modeling of the behavior of this vector is identical to the one described in Section ??. The number of accesses to different lines in the dot products in which vector X is involved, multiplied by the miss probability calculated as For the calculation of P hit X in expression (??) M must be replaced by W , as one line of X may be accessed during the product of a maximum of W rows. In addition, the number of lines of X that compete with another in a cache set is C(W ) instead of C(N ), which influences P hit X (j) in expression (??), as the entries of each row are distributed among W positions instead of N . 5.2 Transposition of a sparse matrix The calculation of the number of misses on AT is modified in a similar way to that of vector X in the sparse matrix-vector product: each line of this vector spreads its access probability through the processing of W rows of the sparse matrix, instead of M , thus reducing to this limit the sum that calculates the hit probability. For the same reason, the number of lines with which a given line of this vector competes for the same cache set during its access period is not C(Nnz ), but C(Nnz=N \Delta W ). Besides, AT is accessed following the pattern described by area vector funcion S gb in Section ??, so the probability of accessing during the processing of the i previous rows a given line that is mapped to the same cache set as the line we are considering is 1\Gamma(S i Also the cross interference probability calculation needs to be modified according to the new access patterns. Similar changes are needed for the estimation of the number of misses on vector CT, taking into account that it is made up of integer values. The prediction of the number of misses on vector RT in loops 2 and 4 is based on the model for vector X in the sparse matrix-vector product, so the calculation of P hit RT2 and P hit RT4 requires the modifications explained in the previous section, and the number of different lines accesses in each dot product is no longer N \Delta r=Ls but W \Delta r=Ls . Some changes are needed also to calculate the probabilities of hit due to a reuse P first hit RT2 and P first hit RT4 . Finally, the value of P first hit RT3 is now P first hit RT2 \Delta pr \Delta W . 6 Results The model was validated with simulations on synthetic matrices with a uniform distribution of the non zero elements and banded matrices from the Harwell-Boeing collection [?]. Traces for the simulations were obtained by running the algorithms after replacing the original accesses by calls to functions that calculate the position to be accessed and write it to disk. These traces were fed to the dineroIII cache sim- ulator, belonging to the WARTS tools [?]. Tables ?? and ?? show the model accuracy for the sparse matrix-vector product for some combinations of the input parameters for uniform and banded sparse matrices respec- tively. Tables ?? and ?? display the results for the sparse matrix transposition model. Without any loss of generality we have considered square matrices M) in the analy- sis, and r = 1. Cs is expressed in Kwords and Ls in words. The maximum error obtained in the trial set was 5.15% for the synthetic matrices and 28.12% for the Harwell-Boeing matrices. The reason for this last result is that the entries distribution in the real matrices is not completely uniform, which produces high deviations for the sparse matrix transposition model (see Table ??). Nevertheless, we consider that such amount of deviation is still acceptable for our analysis purposes. Besides the small size of the matrices of the collection does not favor the convergence of our probabilistic model. We also want to point out that the average error predicted measured Table 2: Predicted and measured misses and deviation of model for sparse matrix-vector product with a uniform entries distribution. M , Nnz and numbers of misses in thousands. Name N W Nnz ff Cs Ls K predicted measured Table 3: Predicted and measured misses and deviation of model for sparse matrix-vector product of some Harwell-Boeing matrices. predicted measured Table 4: Predicted and measured misses and deviation of model for the transposition of a sparse matrix with a uniform entries distribution. M , Nnz and numbers of misses in thousands. Name N W Nnz ff Cs Ls K predicted measured Table 5: Predicted and measured misses and deviation of model for the transposition of a sparse matrix of some Harwell-Boeing matrices. Pn Log Ls Figure Number of misses in a 4-way associative cache with 2Kw during the sparse matrix-vector product of a 10K \Theta10K matrix as a function of Ls and pn . K=2 K=4 Log Ls Figure 7: Number of misses during the sparse matrix-vector product of a 10K \Theta 10K matrix with as a function of the line size of a 2Kw cache for different associativities. for the synthetic matrices was 0.65%, and 5% for the real matrices. In Figures ??-?? we present the relationship between the number of misses and the different parameters introduced in the models. In the case of Cs and Ls we display the base 2 logarithm of the number of cache Kwords and line words respectively. Figure ?? shows the relationship of the number of misses with Ls and pn in the sparse matrix-vector product. The evolution is approximately linear with respect to pn , as the number of accesses is directly proportional to Nnz and most of them follow a sequential pattern. It may be observed how the number of misses significantly decreases as Ls increases because the accesses to all of the vectors except X are sequential (see Figure ??) and the larger the lines, the more exploitation of the spatial locality we obtain. Nevertheless, when Ls is very large (? 64 words) and the matrix has a lower degree of sparsity (pn ? 0:1) the increase of the self and cross interferences probabilities over vector X begins to unbalance the advantages obtained from a more efficient use of the spatial locality exhibited by the remaining vectors for This effect, shown in Figure ??, increases with the value of pn . The relationship between W and Cs for the banded sparse matrix-vector product is shown in Figure ??. In this graph0.51.5x Figure 8: Number of misses during the sparse matrix-vector product for a sparse matrix 20K\Theta20K with tries, a function of W and Cs . K=2 K=4 Log Figure 9: Number of misses during the sparse matrix-vector product of a 20K \Theta 20K matrix with 2M entries and 8000 as a function of the cache size with Ls = 8 for different associativities. we have considered broad bands in order to illustrate the effect of self interferences in the access to vector X. The bandwidth reduction has a large influence on the number of misses because it reduces the self interference probability and increases the reuse probability for the lines of vector X, as the non zeros are spread on narrower rows (spatial locality improvement) and columns (temporal locality improve- ment). A number of misses near the minimum is reached when being the increases of Cs beyond that size of little use. It is intuitive that good miss rates can only be reached with Cs ? W , as with this size there is a line in the cache for each line in the band of the currently processed row. The extra room is needed to avoid the combined effect of self and cross interferences, as we shall now demostrate through an analysis for a fixed bandwidth for different degrees of associativity and cache sizes. In Figure ?? the number of misses for a matrix with displayed in relation to the cache size for different associativities. We can observe that for of the improvement is reached for the 8Kw cache, due to the elimination of the self interferences. The cache size increments from that size on help to gradually reduce the cross interferences. On the other hand, caches with K ? 1 have Log Figure 10: Number of misses during the transposition of a sparse matrix 20K\Theta20K with and a function of W and Cs . a different behavior: the misses reduction gradient is very high while Cs -16Kw. The reason is that in the 8Kw cache there are K different lines of X mapped to each cache set. As the lines are usually accessed in the same order and the cache uses a LRU replacement, any cross interference may generate misses for all of the lines of X that are mapped to the same cache set. The result is that the cross interferences affect as many lines as a set can hold, thus generating more misses. In fact we can see that the 4-way associate cache performs a little worse than the 2-way cache for this cache size. For caches with Cs - 16Kw, the cache sets have enough lines left to absorb the accesses to the vectors that comprise the sparse matrix and the destination vector without increasing the interferences on X due to their combination with the lines of this vector that reside in the same cache set. For small cache sizes all the associativities perform similarly due to the large number of interferences. The conclusion is that associative caches help reducing the interference effect when the number of lines that compete for a given cache line is smaller than or equal to K; otherwise the performance is quite similar to that of a direct mapped cache. Moreover, in this case, if the lines mapped to the same cache set are usually accessed in the same order, high associativities may perform worse. As for the sparse matrix transposition, Figures ?? and ?? represent the same data for this algorithm as Figures ?? and ?? for the sparse matrix-vector product respectively. The first one shows a decrease of the number of misses with the bandwidth reduction, being this more noticeable in the point where Cs becomes greater than W . The reasons are those explained for the previous algorithm. Anyway, this reduction is much softer than in the sparse matrix-vector product because the accesses to vector RT in loops 2 and 4, which are the most directly favoured by the bandwidth reduction, stand only for a small portion of the misses. On the other hand, the number of misses on vectors AT, and CT, which account for the vast majority of the misses, decreases slowly when Cs increases or W decreases, although they are also heavily dependent on the bandwidth. The reason is that in all of the cases shown in the figure the data belonging to these vectors during the process of all of the columns inside the band of the original matrix do not fit in the cache ((Nnz=N) \Delta W entries). Only for the case when does this set fit, and we can see that the number of misses becomes stable in this area K=2 K=4 Log Figure 11: Number of misses during the transposition of a sparse matrix 20K\Theta20K matrix with 2M entries and 8000 as a function of the cache size with Ls = 8 for different associativities. of the graph. The misses on C, R and A remain almost completely constant due to their sequential access, obtaining as only benefit from the bandwidth reduction a somewhat lower cross interference probability. Figure ?? shows that the general behavior of the algorithm with respect to K, the associativity degree, although having similarities with that of the sparse matrix-vector product, does not depend only on W to determine the cache sizes for which the hit rate obtains reasonable values. As explained before, the reason is that for none of the cache sizes considered contain a considerable portion of the data the algorithm accesses during the W iterations that process a whole band in loop 4, the one that causes most of the misses. Only accesses to RT, mainly in loop 2, benefit from the increase of Cs to values larger than W . This is specially noticeable for as in the sparse matrix-vector product. Another similarity is the worse behavior of caches with K ? 1 for a cache size very close to W due to the added effect of cross interferences with self interferences because of cache lines which are usually accessed in the same order. This penalizes the LRU replacement algorithm. Once the cache size is noticeable larger than the bandwidth, higher associativities perform better, as in Figure ??. In order to get hints about the values of the cache parameters for which the algorithm stabilizes its misses, Figures ?? and ?? show the same data for a smaller matrix using 4. During the process of a band of this matrix the working set for vectors AT and CT, those that account for most of the misses, comprise 25W elements, as there is an average of 25 elements per column. The hit rate obtains values near to its maximum in Figure ?? when the cache size exceeds this value. Increases of Cs beyond that limit provide little performance improvement. These improvements are only noticeable when the cache size increase is very high, due to the strong reduction of the cross interference. We must take into account that this graph is constructed for a 4-way associative cache. Somewhat greater cache sizes would be needed to obtain good cross interference reduction in a direct mapped cache (see Figure ??). associative caches help reducing the miss rate for the small cache sizes in Figure ?? because on average there are always less than 2 lines competing in any set, as during the process of each row there are only about 200 lines of vector Log Figure 12: Number of misses during the transposition of a sparse matrix 5K\Theta5K with 125K entries, as a function of W and Cs . K=2 K=4 Log Figure 13: Number of misses during the transposition of a sparse matrix 5K\Theta5K matrix with 125K entries and 200 as a function of the cache size with Ls = 4 for different associativities. AT with a non null access probability, 200 lines belonging to vector CT and a few more lines belonging to the other vectors, while the smallest of the caches considered has 512 lines. As expected, the increase of the cache size reduces the hit ratio difference between direct mapped caches and set associate caches. For Cs - 8Kw the cache size increase helps only by reducing the cross interferences. Finally, the relationship of the line size and the matrix density with the number of misses for this last algorithm has proved to be the same as in the sparse matrix-vector product, being the only difference that the gradient of the increase of misses with relation to pn is about three times larger, which is normal, as the number of accesses per non zero in the original matrix is eight, while in the sparse matrix-vector product algorithm it is three. 7 Conclusions and future work We have presented a fully-parametrizable model for the estimation of the number of misses on a generic K-way associative cache with LRU replacement and we have applied it to the sparse matrix-vector product and the transposition of a sparse matrix. The model deals with all the possible types dineroIII time model time 100 1000 1% 1000 100 1000 10% 100 100 1000 1% 1000 100 1000 10% 100 100 1000 1% 1000 100 1000 10% 100 Table Simulation and model user times to calculate the number of misses during the transposition of a banded sparse matrix on a 200 MHz Pentium. N and Nnz in thousands, time in seconds. of misses and has demonstrated a high level of accuracy in its predictions. It considers a uniform distribution of the entries on the whole matrix or on a given band. As Table ?? shows, besides providing more information, the modelization is much faster than the simulation, even when the time required to generate the trace, which takes almost as much time as the execution of the simulator itself, is not included in the table. We have illustrated how the model may be used to study the cache behavior for this code, and shown the importance of the bandwidth reduction in the case of the banded matri- ces, even for high degrees of associativity. The model can be also applied to study possible architectural cache parameter modifications in order to improve cache performance. Future work includes the extension of the model to consider prefetching and subblock placement. On the other hand, we are now working on the modeling for non uniform distributions of the entries in the sparse matrices focusing in the most common patterns that appear in real matrices suites. Finally, in order to obtain more accurate estima- tions, we are studying the inclusion of the data structures base addresses as a parameter of the model. --R "Analysis of Cache Performance for Operating Systems and Multiprogramming," Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. "User's Guide for the Harwell-Boeing Sparse Matrix Collection," "Cache Miss Prediction in Sparse Matrix Computations," "Cache Miss Equations: An Analytical Representation of Cache Misses," "Cache Profiling and the SPEC Benchmarks: A Case Study," Sparse Matrix Technology. "Sparse Matrix Computations: Implications for Cache Designs," "Characterizing the Behaviour of Sparse Algorithms on Caches," "Cache Interference Phenomena," "Trace-Driven Memory Simulation: A Survey," --TR Analysis of cache performance for operating systems and multiprogramming Characterizing the behavior of sparse algorithms on caches Sparse matrix computations Cache interference phenomena Block algorithms for sparse matrix computations on high performance workstations Trace-driven memory simulation Cache miss equations Cache Profiling and the SPEC Benchmarks --CTR Gerardo Bandera , Manuel Ujaldn , Emilio L. Zapata, Compile and Run-Time Support for the Parallelization of Sparse Matrix Updating Algorithms, The Journal of Supercomputing, v.17 n.3, p.263-276, Nov. 2000 Basilio B. Fraguela , Ramn Doallo , Emilio L. Zapata, Probabilistic Miss Equations: Evaluating Memory Hierarchy Performance, IEEE Transactions on Computers, v.52 n.3, p.321-336, March Chun-Yuan Lin , Jen-Shiuh Liu , Yeh-Ching Chung, Efficient Representation Scheme for Multidimensional Array Operations, IEEE Transactions on Computers, v.51 n.3, p.327-345, March 2002 B. B. Fraguela , R. Doallo , J. Tourio , E. L. Zapata, A compiler tool to predict memory hierarchy performance of scientific codes, Parallel Computing, v.30 n.2, p.225-248, February 2004 Jingling Xue , Xavier Vera, Efficient and Accurate Analytical Modeling of Whole-Program Data Cache Behavior, IEEE Transactions on Computers, v.53 n.5, p.547-566, May 2004 Chun-Yuan Lin , Yeh-Ching Chung , Jen-Shiuh Liu, Efficient Data Parallel Algorithms for Multidimensional Array Operations Based on the EKMR Scheme for Distributed Memory Multicomputers, IEEE Transactions on Parallel and Distributed Systems, v.14 n.7, p.625-639, July

probabilistic model;irregular computation;sparse matrix;cache performance

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Application and evaluation of large deviation techniques for traffic engineering in broadband networks.

Accurate yet simple methods for traffic engineering are important for efficient dimensioning of broadband networks. The goal of this paper is to apply and evaluate large deviation techniques for traffic engineering. In particular, we employ the recently developed theory of effective bandwidths, where the effective bandwidth depends not only on the statistical characteristics of the traffic stream, but also on a link's operating point through two parameters, the space and time parameters, which are computed using the many sources asymptotic. We show that this effective bandwidth definition can accurately quantify resource usage. Furthermore, we estimate and interpret values of the space and time parameters for various mixes of real traffic demonstrating how these values can be used to clarify the effects on the link performance of the time scales of burstiness of the traffic input, of the link parameters (capacity and buffer), and of traffic control mechanisms, such as traffic shaping. Our approach relies on off-line analysis of traffic traces, the granularity of which is determined by the time parameter of the link, and our experiments involve a large set of MPEG-1 compressed video and Internet Wide Area Network (WAN) traces, as well as modeled voice traffic.

Introduction The rapid progress and successful penetration of broadband communications in recent years has led to important new problems in traffic modeling and engineering. Among oth- ers, call admission control and network dimensioning for cases of guaranteed QoS (which is one of the most important features supported by the ATM technology) have attracted the attention of researchers. Successful approaches are closely related to the ability of quantifying the usage This work was supported in part by the European Commission under ACTS project CASHMAN (AC-039). y Also with the Dept. of Computer Science, University of Crete. In Proc. of ACM SIGMETRICS '98/ PERFORMANCE Joint International Conference on Measurement and Modeling of Computer Systems, Madison, Wisconsin, June 24-26, 1998. of resources, on the basis of traffic modeling and measurements For example, statistical analysis of traffic measurements [11, 14, 7] has shown a self-similar or fractal behavior; such traffic exhibits long range dependence or slowly decaying autocorrelation. Although the implications of such long range dependence is still an open issue (e.g., see [6, 8] and the references therein), recent work [16, 8] has shown that these implications can be of secondary importance to the buffer overflow probability when the buffer size is small, which applies to the case where real time communication is supported. This example motivates the need for a methodology to understand the impact of the various time scales of the burstiness of real broadband traffic on the performance of the network and on its resource sharing capabilities. In particular, some basic questions for which the network engineer must provide answers are the following: How much does the cell loss probability decrease when the link capacity or buffer size increases? How does traffic shaping 1 affect the multiplexing capability of a link and the amount of resources used by a bursty source? What is the necessary granularity of traces in order to capture the information that is important for performance analysis and network di- mensioning? What are the effects of the composition of the traffic mix on the multiplexing capability of a link? Traditional queueing theory, which requires elaborate traffic models, cannot be applied to answer such questions in the context of large multi-service networks; for such cases asymptotic methods are more appropriate. In this paper we answer such questions by applying and evaluating the many sources asymptotic and the effective bandwidth based on this asymptotic, for real broadband traffic. This traffic consists of MPEG-1 compressed video, Internet Wide Area Network (WAN) traffic, and traffic resulting from modeled voice. Problems related to resource sharing have often been analyzed using the notion of effective bandwidth, which is a scalar summarizing resource usage depending on the statistical properties and Quality of Service (QoS) requirements of a source. Effective bandwidths are usually derived by means of asymptotic analysis, which is concerned with how the buffer overflow probability decays as some quantity in- creases. If this quantity is the size of the buffer, we have the large buffer asymptotic [5, 10]. If the buffer per source and capacity per source are kept constant, and we are interested Related work on how traffic smoothing affects the multiplexing capability of a link can be found in [18] and the references therein. in how the overflow probability decays as the size of the system (the broadband link and the multiplexed sources) increases, then we have the many sources asymptotic; this asymptotic regime has been investigated in [4, 1, 17]. Effective bandwidth definitions based on the large buffer asymptotic were found, in some cases, to be overly conservative or too optimistic [2]. This occurs because the large buffer asymptotic does not take into account the gain when many independent sources are statistically multiplexed to- gether. On the other hand, the effective bandwidth defined according to the many sources asymptotic [9, 3] depends not only on the statistical properties and Quality of Service (QoS) requirements of a source, but also on the statistical properties of the other traffic it is multiplexed with and the parameters (capacity and buffer) of the multiplexing link. Only recently [9, 3] has it been understood how to incorporate such information into the definition of the effective bandwidth. This work has shown that the effective bandwidth of a source depends on the link's operating point through two parameters, called the space and time parame- ters, which in turn depend on the link parameters (capacity and buffer) and the statistical properties of the multiplexed traffic. These parameters can be computed using the many sources asymptotic and, as we will demonstrate with real broadband traffic, have important applications for traffic engineering. Since the effective bandwidth gives the amount of resources that must be reserved for that source in order to satisfy its QoS requirements, it helps simplify problems in traffic engineering and network dimensioning. The rest of this paper is structured as follows. In Section 2 we review basic results from the theory of effective bandwidths, as developed in [9], and the theory of many sources asymptotics [4, 1, 17]. In Section 2.2 we discuss the application of this framework to traffic engineering, giving emphasis on the interpretation of the space and time parameters. In Section 3 we present a detailed series of experiments which aim to evaluate the accuracy of the above framework for link capacities and buffer sizes anticipated for broadband networks and for real broadband traffic which consists of MPEG-1 compressed video and Internet WAN traces, in addition to modeled voice traffic. Finally, Section 4 summarizes the results of the paper and identifies areas for future research. 2 The Many Sources Asymptotic and its Implications In this section we summarize the key results of the many sources asymptotic and their implications for traffic engineering 2.1 Effective Bandwidths Suppose the arrival process at a broadband link is the superposition of independent sources of J types. be the number of sources of type j, and let (note that the n j s are not necessarily integers). This system can be viewed as having N sources of the same type, where a single source consists of a proportion of the J source types and can be characterized by the vector n. The broadband link has a shared buffer of size link capacity Parameter N is the scaling parameter (size of the system), and parameters b; c are the buffer and capacity per source, respectively. We suppose that after taking into account all economic factors (such as demand and com- petition) the proportions of traffic of each of the J types remains close to that given by the vector n, and we seek to understand the relative usage of network resources that should be attributed to each traffic type. Furthermore, let X j [0; t] be the total load produced by a source of type j in the time interval (0; t], which feeds the above link. We assume that X j [0; t] has stationary incre- ments. Then, the effective bandwidth of a source of type j is defined as [9] st log E \Theta e sX j [0;t] where s; t are system parameters which are defined by the context of the source, i.e., the characteristics of the multiplexed traffic, their QoS requirements, and the link resources (capacity and buffer). Specifically, the time parameter (measured in, e.g., milliseconds) corresponds to the most probable duration of the busy period of the buffer prior to overflow [4]. The space parameter s (measured in, e.g., kb \Gamma1 ) corresponds to the degree of multiplexing and depends, among others, on the size of the peak rate of the multiplexed sources relative to the link capacity. In particular, for link capacities much larger than the peak rate of the multiplexed sources, s tends to zero and ff j (s; t) approaches the mean rate of the source, while for link capacities not much larger than the peak rate of the sources, s is large and ff j (s; t) approaches the maximum value that the random variable X j [0; t]=t can attain. P(overflow) be the probability that in an infinite buffer which multiplexes sources and is served at rate C = Nc, the queue length is above the threshold Nb. The following holds for Q(Nc;Nb;Nn) [4]: lim log sup st where I is called the asymptotic rate function. The last equation is referred to as the many sources asymptotic, and has been proved for continuous time in [1] and for a special case in [17]. A similar asymptotic holds for the proportion of workload lost through the overflow of a finite buffer of size Nb. Due to equation (2), the overflow probability can be written as which leads to the following approximation when N is large: The accuracy of the above approximation and, more im- portantly, the achievable link utilization for MPEG-1 compressed video and Internet WAN traffic is investigated in Section 3.1. Consider the QoS constraint on the overflow probability to be P(overflow) - e \Gammafl , and assume be a subset of Z J such that vice versa), i.e., the QoS constraint on the overflow probability is met. Due to this property, A(N called the acceptance region. The region A(N to compute. However, for the scaled acceptance region, the following holds [9]: lim where s st Hence, the scaled acceptance region can be approximated by A. If is on the boundary of the region A and the boundary is differentiable at that point, then the tangent plane determines the half-space [9] s where (s; t) is an extremizing pair in equation (2) and c is the "effective capacity" per source. The case for two source types shown in Figure 1. the scaled acceptance region Approximation A of A 3 Figure 1: Approximation A of the scaled acceptance region three values of t in (4). To the extent that A(N can be approximated by NA, it follows from (5) that a point can be taken to satisfy s where, as in (5), (s; t) is an extremizing pair in equation (2), and C is the "effective capacity" of the system at the operating point (s; t). The effective bandwidth ff j (s; t) provides a relative measure of resource usage for a particular operating point of the link, expressed through parameters s; t. For example, if a source of type j1 has twice as much effective bandwidth as a source of type j2 , then, for this particular operating point of the link, one source of the first type can be substituted by two sources of the second type, while still satisfying the QoS constraint. The above measure of resource usage differs from the ordinary measure that is usually reported (i.e., the mean rate), which corresponds to Note that the QoS guarantees are encoded in the effective bandwidth definition through the value of fl, which appears on the right hand side of (6) and influences the form of the acceptance region. The asymptotics behind the above results assume only stationarity of sources. Illustrative examples discussed in [9] and [16] include periodic sources, fractional Brownian input, policed and shaped sources, and deterministic multiplexing Unlike the above definition of the effective bandwidth which takes into account the effects of statistical multi- plexing, the effective bandwidth based on the large buffer asymptotic depends solely on the source's characteristics and the QoS constraint. Specifically [5, 10], consider the QoS constraint P(overflow) - e \GammaffiB , where B is the total buffer. Then, the effective bandwidth based on the large buffer asymptotic of a source of type j, and the corresponding constraint is s lim \Theta e sX j [0;t] Observe that (7) is a special case of (1) for t ! 1. Indeed, equation (7) becomes accurate when the buffer of the system is large, in which case the time parameter t becomes large. However, for finite buffer sizes, equation (7) can lead to significant underutilization or even overutilization of link capacity [2]. The experiments in Section 3 include results which compare the performance of the large buffer asymptotic with that of the many sources asymptotic. Using the Bahadur-Rao theorem, the authors of [12] extended the proof of the many sources asymptotic in [4] to show that as N !1, the following holds: e \GammaN I where (s; t) is an extremizing pair of (2) and oe 2 is given by log \Theta e sX j [0;t] \Theta e sX j [0;t] is the moment generating func- tion. Based on (8), we have the following approximation: We will refer to the above equation as the many sources asymptotic approximation with the Bahadur-Rao improve- ment. The term 1log(2-N oe 2 s 2 ) can be approximated by2 log(4-NI) [13]. Hence, equation (9) does not require any additional computations compared to (3). It can be seen with some algebra that the many sources asymptotic with the Bahadur-Rao improvement (equation gives the following effective bandwidth constraint in the neighborhood of the extremizing pair (s; t) of (2) s where 2fl ). It is important to note that the same formula for the effective bandwidth, given by equation (1), is used in both (6) and (10). The Bahadur- Rao improvement only affects (increases) the amount of effective capacity C while the parameters s; t are computed using the same formula (2). 2.2 Implications to Traffic Engineering In this subsection we discuss the interpretation of the space s and time t parameters, and how they can be used for traffic engineering. For any traffic stream, the effective bandwidth ff j (s; t) in (1) is a template that must be filled with the system operating point parameters s; t in order to provide the correct measure of effective usage by the stream for that particular operating point. Although the operating point also depends on the individual stream, for a large system, due to heavy multiplexing, this dependence is ignored. Such an engineering approach simplifies considerably the analysis because there is no circle in the definitions of the effective bandwidth and the operating point. Furthermore, as we will see in Section 3.3, the values of s; t are, to a large ex- tent, insensitive to small variations of the traffic mix. It has been observed that in networks serving a large number of sources, the traffic mix exhibits a strong cyclic behav- ior. Hence, we can assign particular pairs (s; t) to periods of the day during which the traffic mix remains relatively constant. The values of s; t corresponding to a particular period of the day can be computed off-line using the supinf formula (2) and the effective bandwidth formula (1), where the expectation in (1) is replaced by the empirical mean; the latter can be computed from traffic traces taken during that period of the day. Recall that the time parameter t corresponds to the most probable duration of the busy period prior to buffer overflow. As we will discuss now, this parameter also identifies the time scales that are important for buffer over- flow. Assume that a link is operating at a particular operating point, expressed through parameters s; t. In the effective bandwidth formula (1) the statistical properties of the source appear in X j [0; t], which is the amount of work-load produced by the source in an interval of length t. If two sources have the same distribution of X j [0; t] for this value of t, then they will both have the same effective bandwidth. A case of practical interest where this result can be applied is traffic smoothing: To have an effect on the amount of resources used by a source, traffic smoothing must be performed on a time scale larger than t, since only then does it affect the distribution of X j [0; t]. We will investigate this with real broadband traffic in Section 3.3. The time parameter t also indicates the granularity that traffic traces must have 2 , since given a value for t it would be sufficient to choose the granularity to be a few times smaller than this value. Traditionally, the granularity of traces was chosen in a rather ad-hoc manner. By setting equal to the right hand side of (2) and taking the derivative with respect to B and C, we get the following expression for the space parameter s and the product st [3]: and st = @fl Thus, the space parameter s equals the rate at which the logarithm of the overflow probability decreases with the buffer size, for fixed capacity C, whereas the product st equals the rate at which the logarithm of the overflow probability decreases with the link capacity, for fixed buffer B. Here, by trace granularity we mean the time window in which we measure the amount of load produced by a source. 3 Experiments with Real Traffic In this section we apply and evaluate the traffic engineering framework discussed in the preceding sections for real broadband traffic. The specific issues we address are the ffl Compare the overflow probability and link utilization using the many sources asymptotic and its Bahadur- Rao improvement to the actual cell loss probability and the maximum utilization estimated using simula- tion. (Section 3.1) ffl Compare the values of parameters s; t computed by theory to the values estimated using simulation. (Sec- tion 3.2) ffl Estimate and interpret typical values of parameters s; t for real broadband traffic. (Section 3.2) ffl Investigate how the values of s; t, and subsequently the effective bandwidth, depend on the traffic mix. (Section 3.3) Our experiments involve MPEG-1 compressed video and Internet WAN traces, as well as modeled voice traffic. The sequences, made available 3 by O. Rose [15], have been encoded using the UC Berkeley MPEG-1 software encoder with the frame pattern IBBPBBPBBPBB. Each sequence consisted of 40,000 frames (approximately utes). For Internet Wide Area Network (WAN) traffic we used the Bellcore Ethernet trace BC-Oct89Ext which has been made available 4 by W. Leland and D. Wilson [11]. The trace had duration 122797.83 seconds. For voice traffic we use an on-off Markov modulated fluid model with peak rate 64 Kbps and average time spent in the "on" and "off " states 352 msec and 650 msec, respectively. Finally, we consider link capacities 34 Mbps, 155 Mbps, and 622 Mbps, and buffers introducing delay of up to 50 msec for MPEG-1 traffic, and up to 150 msec for Internet WAN traffic. 3.1 Overflow Probability and Link Utilization In this section we compare the overflow probability and link utilization using the many sources asymptotic and its Bahadur-Rao improvement with the actual cell loss probability and the maximum utilization estimated using simu- lation. We also derive a simple heuristic for computing the actual cell loss probability from the overflow probability. 3.1.1 Overflow probability Figure compares, for a fixed number of sources, the overflow probability estimated using the many sources asymptotic and its Bahadur-Rao improvement with the cell loss probability and frame overflow probability estimated using simulation; the latter is the probability that at least one cell of a frame is lost. Both the cell loss probability and the frame overflow probability are measured using a discrete time simulation model with an epoch equal to one frame In these and subsequent simulations, we report the average from a total of 80 independent simulation runs, each with a random selection of the starting frame for every source (we assume that frame boundaries are aligned). Each simulation run had duration 5 times the 3 Available at http://ftp-info3.informatik.uni-wuerzburg.de/pub/ 4 Available at The Internet Traffic Archive, size of the trace. Both the number of runs and the duration of each run were chosen empirically. For each method, the base-10 logarithm of the overflow probability is plotted against the buffer size (measured in milliseconds), while the link utilization remains the same. In Figure 2, first observe that for small buffer sizes there is a relatively fast decrease of the overflow probability as the buffer size increases. However, this stops to occur after some buffer size (e.g., 5 \Gamma 8 msec for a 155 Mbps link). Further increase of the buffer has a smaller effect on the overflow probability. Furthermore, the logarithm of the overflow probability in both of these regimes is almost linear with the buffer size. Second, observe that although the many sources asymptotic overestimates the Cell Loss Probability (CLP) by approximately 2-3 orders of magnitude, it qualitatively tracks its shape very well. The Bahadur-Rao improvement overestimates the CLP by 1-2 orders of magnitude. On the other hand, the large buffer asymptotic, in addition to overestimating the CLP by many orders of magnitude, also fails to track its shape. The actual cell loss probability differs from the overflow probability estimated using the many sources asymptotic and its Bahadur-Rao improvement because the latter is not a measure of the CLP, but a measure of the probability that in an infinite buffer the queue length becomes greater than B. This probability is closer in spirit to the frame overflow probability, i.e., the probability that at least one cell of a frame is lost. Indeed, as Figure 2 shows, the overflow probability estimated using the many sources asymptotic with the Bahadur-Rao improvement is very close to the frame overflow probability. 5 To further explain the above, we derive a simple expression for the cell loss probability in terms of the frame overflow probability L f . If one observes a large number of frames, say M , the average number of frames in which we have at least one lost cell is ML f . Let x be the average number of cells that are lost when a frame overflow occurs. Then the average number of cells that are lost in M frames is ML f x. Since there is a total of MF cells in these frames, where F is the average number of cells in a frame, we can approximate the cell loss probability with the percentage of lost cells, namely From the last equation we see that the cell loss probability differs from the frame overflow probability by a correction We assume that when an overflow occurs only a few cells are lost. This is expected for small loss probabilities, since the probability of loosing cells in a buffer of size B+ ffl is exponentially harder than loosing cells in a buffer of size B. In particular, we will assume that only one cell is lost, hence Lc - 1=F , and since the average number of cells in one frame is 25 we get Lc - This heuristic agrees with the difference between the frame overflow probability and the cell loss probability shown in Figure 2. Figure 3 shows the cell loss probability estimated using (12), where . Observe that the cell loss probability using this heuristic matches the cell loss probability estimated using simulation extremely well. 5 This is the case only when the simulation epoch equals the frame time. 3.1.2 Link utilization Nm=C be the link utilization, where N is the number of sources, m is the mean rate, and C is the link capacity. Figure 4 compares, for a range of buffer sizes and for overflow probability 10 \Gamma6 , the link utilization using the many sources asymptotic and its Bahadur-Rao improvement with the maximum achievable utilization (estimated using simulation). The utilization is computed by increasing the number of multiplexed sources in order to find the maximum number such that the overflow probability (3), computed using (1) and (2), is less than the target overflow Similar to our observations regarding the overflow prob- ability, there are significant gains in increasing the size of the buffer up to a certain value. Increasing the buffer size above this value has a very small effect on the link utilization Recall that the many sources asymptotic overestimated the CLP by 2-3 orders of magnitude. However, as Table 1 shows, it performs much better in estimating the maximum utilization. In particular, for 1 msec, the many sources asymptotic achieves a utilization which is approximately 79% of the maximum utilization. The Bahadur-Rao improvement increases this percentage to 88%. Furthermore, this percentage increases for larger link capacities; e.g., for the many sources asymptotic achieves a utilization which is 90% of the maximum (Table 1(b)). Of course, as Figure 5 shows, using the heuristic based on (12), we achieve a utilization which almost coincides with the maximum utilization Buffer Utilization % msec (cells) Simulation MSA MSA (a) Buffer Utilization % msec (cells) Simulation MSA MSA Table 1: Link utilization: theory vs. simulation for Star Wars traffic. The parenthesis contain the percentage of the maximum utilization (second column). MSA: Many Sources Asymptotic, B-R: Bahadur-Rao. [ P(overflow) - Finally, Figure 6 shows the link utilization in the case of Internet WAN traffic. Observe that while for Star Wars traffic the gains of increasing the buffer decrease abruptly, for Internet WAN traffic the gains of increasing the buffer decrease smoother as the buffer size increases. This indicates that there are more time scales in Internet traffic which, for different buffer sizes, become important for buffer overflow. 3.2 Space and Time Parameters The space s and time t parameters characterize a link's operating point and depend on the characteristics of the multiplexed traffic. In this section we compare the values of these parameters computed using the supinf formula (2) to the corresponding values estimated using simulation. Furthermore, we compute and interpret typical values of these parameters, demonstrating how they can be used for traffic engineering. 3.2.1 Space and time parameters: theory vs. simulation Recall from our discussion in Section 2.2 that the space parameter s equals the rate at which the logarithm of the overflow probability decreases with the buffer size, equation (11). Motivated by this equation, we can estimate s using the following ratio of differences: sim is the cell loss probability estimated using simulation. Figure 7(a) shows that the value of parameter s computed using the supinf formula (2) is in good agreement with the value computed using (13). Note that the "steps" in the value computed using the supinf formula are expected since the many sources asymptotic (and large deviations theory in general) captures only the most likely way overflow can occur. On the other hand, the curve labeled "simulation" in Figure 7(a) represents an average over all the events that contribute to overflow. Recall from our discussion in Section 2.1 that the time parameter t can be interpreted as the most probable duration of the busy period prior to buffer overflow. Figure 7(b) compares the value of parameter t computed using the supinf formula to the average value of the busy period prior to buffer overflow. As was the case for parameter s, the agreement between the two curves is good. 3.2.2 Typical values and interpretation of the space and time parameters Next we investigate how parameters s; t and the product st depend on the link capacity and buffer size. The values of s; t are computed using the supinf formula for a target Figure 8 shows the values of parameter s as a function of the buffer size, for various link capacities (Figure 8(a)) and for various video contents (Figure 8(b)). In Figure 8(a), the explanation of the steep decrease of the value of s is similar to the explanation of the knee of the curves in Figures 2 and 4. Specifically, according to equation (11), s equals the rate at which the logarithm of the cell loss probability decreases (i.e., the quality of service improves), when the buffer size increases. Up to some value, the buffer is used to smooth the fast time scales of the multiplexed traffic. Thus, increasing the buffer has a large affect on the overflow probability, and the value of s is high. Once the fast time scales have been smoothed, the slow time scales govern the buffer overflow. Thus, increasing the buffer has a very small effect on the overflow probability, and the value of s is small. Also, observe in Figure 8(a) that the steep decrease of the value of s occurs for smaller buffer sizes (measured in milliseconds) as the link capacity increases. Finally, see Figure 8(b), similar behavior is observed for MPEG-1 traffic with various contents. This indicates that the dependence of s on the link capacity and buffer size is related to the frame structure. The dependence of parameter t on the buffer size is shown in Figure 9(a). Observe that the steep increase of its value occurs for the same buffer size for which the increase of the value of s occurs (Figure 8(a)). The small values of t correspond to the regime where the fast time scales are important for buffer overflow, whereas the large values of t correspond to the regime where the slow time scales are important for buffer overflow. The product st, for different buffer sizes, is shown in Figure 9(b). Once again we observe a steep increase of its value, occurring at the same buffer sizes where the changes in the values of s and t occur. However, the explanation for the increase of the value of st is more subtle than the explanation for the behavior of s; t. Recall from our discussion in Section 2.2 that the product st equals the rate at which the logarithm of the overflow probability decreases with the link capacity, while the buffer size remains the same; see equation (11). Comparing Figure 9(a) with Figure 9(b), we observe that the larger values of st correspond to larger values of t. Larger values of t result in an averaging effect in the amount of load X j [0; t] that appears in the effective bandwidth formula (1). Hence, for the overflow phenomenon, the traffic appears smoother. But for a link which multiplexes smooth traffic and is operating with a cell loss probability greater than zero, a change of the link capacity has a greater effect on the overflow probability compared to a link which multiplexes more bursty traffic. This gives the intuition of why the value of st increases. Figure compares the values of s; t for Star Wars and voice traffic. Figure 10(a) shows that as the buffer size in- creases, the value of s for voice traffic decreases smoothly. Furthermore, the rate of decrease is smaller for larger buffer sizes. Comparing the value of s for MPEG-1 and voice traf- fic, we conclude that for buffer sizes up to 2 msec and above increasing the buffer has a larger effect for a net-work carrying voice traffic compared to a network carrying MPEG-1 traffic. This provides an example of how the space parameter s can be used for buffer dimensioning. Figure 10(b) shows that for voice traffic the time parameter increases almost linearly with the buffer size. This can be explained since for a high degree of multiplexing, voice sources (which are modeled as on-off Markov fluids) behave as Gaussian sources. For such sources, it can be shown [4] that the time parameter t increases linearly with the buffer size. Figure 11(a) compares the value of parameter s for Star Wars and Internet WAN traffic. For MPEG-1 traffic, the values of s form two distinct regimes, where s is almost constant, corresponding to the two distinct time scales that are important for buffer overflow. On the other hand, for Internet traffic, the values of s form more than two regimes, indicating that for such traffic there are more time scales which, for different buffer sizes, become important for buffer overflow. Recall that this is also the reason behind the smoother dependence of the link utilization on the buffer size for Internet WAN traffic compared to Star Wars traffic Figure 6). Finally, Figure 11(b) shows that parameter s can take different values for different Internet traffic segments, illustrating that different segments have different statistical properties. 3.3 Effects of the Traffic Mix As discussed in Section 2.1, periods of the day during which the traffic mix remains relatively constant can be characterized by corresponding pairs of (s; t). In this section we investigate the dependence of these parameters, hence of the effective bandwidth, on the traffic mix. The traffic mix we consider consists of traffic of different types (MPEG-1 compressed video and voice), traffic with the same structure but different contents (MPEG-1 compressed video with different contents), and smoothed/unsmoothed traffic of the same type and content. 3.3.1 Traffic mix containing Star Wars and voice traffic We first consider the traffic mix containing Star Wars and voice traffic. The horizontal axis in Figures 12(a) and 12(b) depicts the percentage of voice connections, each containing individual voice channels. The vertical axis of the above figures depicts the effective bandwidth of the Star Wars traffic stream. Observe that (1) the effective bandwidth, to a large extent, changes slowly with the traffic mix, (2) the dependence of the effective bandwidth on the traffic mix is smaller for larger capacities and larger buffer sizes, and (3) there are cases where increasing the percentage of voice connections leads to a sharp decrease of the value of the effective bandwidth. The first observation supports the argument that pairs of (s; t) can be assigned to periods of the day during which the traffic mix remains relatively constant. However, the third observation says that there are percentages of the traffic mix where the effective bandwidth exhibits sharp changes. If the link's operating point is close to such a percentage, then we can estimate the average amount of resources used by a source as a linear combination of the effective bandwidth to the left and to the right of the sharp change. The coefficients of the linear combination would be determined by the percentage of the time the network was operating on the left and on the right of the change. The sharp decrease in the value of the effective band-width identified above is due to the change of the time scale that is important for buffer overflow. In particular, as indicated above the curve for capacity 155 Mbps and buffer size 4 msec in Figure 12(a), the time parameter t increases finally 7 frames) for the same percentage of voice connection at which the sharp decrease in the value of the effective bandwidth occurs. The increase of t produces an averaging effect (also discussed in Section 3.2.2) in the amount of workload X j [0; t] that appears in the effective bandwidth formula (1); this averaging effect results in a smaller effective bandwidth. 3.3.2 Traffic mix containing MPEG-1 traffic with different contents Up to now we investigated the case where the traffic mix contains traffic of different nature. Next we investigate the case where the traffic mix contains MPEG-1 video traffic with the same encoding parameters but with different contents Figures 13(a) and 13(b) show the effective bandwidth of the Star Wars stream as a function of the percentage of news and talk show streams, respectively. These figures show that the content has a very small effect on the effective bandwidth; this also implies that parameters s; t are affected very little. 3.3.3 Traffic mix containing smoothed and unsmoothed Star Wars traffic Our final investigation deals with another important question in traffic engineering: How does traffic smoothing affect the network's operating point and the amount of resources used by a source? We will see that parameter t shows the time scale at which smoothing must be performed in order for it to affect resource usage. Figure 14 shows the effective bandwidth of the Star Wars stream for different percentages of a traffic mix which consists of unsmoothed and smoothed Star Wars traffic; the latter is created by evenly spacing the cells belonging to two consecutive frames. First, observe that the effects of the traffic mix on the effective bandwidth decreases when the link capacity and buffer size increases. Second, observe that there are cases where increasing the buffer size has a very small effect on the effective bandwidth, e.g., at the curves for practically co- incide. Third, observe that for some buffer sizes, smoothing has no effect on the effective bandwidth (and on the net- work's operating point), e.g., in Figure 14(a) the curve for and in Figure 14(b) the curves for are flat. We explain this behavior next. Figure 15 shows the effective bandwidth for both the smoothed and the unsmoothed Star Wars stream. When the percentage of smoothed traffic is small, the time parameter smaller than the time interval over which smoothing was performed (80 msec). For this reason, the amount of workload X j [0; t] that appears in the effective bandwidth formula (1) is smaller for the smoothed stream than it is for the unsmoothed stream. Hence, the effective bandwidth of the smoothed stream is smaller than the effective bandwidth of the unsmoothed stream. For some percentage of smoothed traffic (- 60%), the time parameter t (= 80 msec) is no longer smaller than the time interval over which smoothing is performed (80 msec). Because of this, the amount of workload X j [0; t] is the same for both the smoothed and the unsmoothed stream. Hence, the effective bandwidth of both streams is the same. Conclusions In this paper we employ the recently developed theory of effective bandwidths, and in particular the one based on the many sources asymptotic, whereby the effective bandwidth depends not only on the statistical characteristics of the traffic stream, but also on a link's operating point. The latter is summarized in two parameters: the space and time parameters. We have investigated the accuracy of the above frame- work, and how it can provide important insight to the complex phenomena that occur at a broadband link with a high degree of multiplexing. In particular, we estimate and interpret values of the space and time parameters for various mixes of real traffic, demonstrating how these can be used to clarify the effects on the link performance of the time scales of burstiness of the traffic input, of the link parameters (ca- pacity and buffer), and of traffic control mechanisms, such as traffic smoothing. Our approach is based on the off-line analysis of traffic traces, the granularity of which can be determined by the time parameter of the system. For the traffic mixes consid- ered, the space and time parameters are, to a large extent, insensitive to small variations of the traffic mix. This indicates that particular pairs of these parameters can characterize periods of the day during which the traffic mix remains relatively constant. The above result has important implications to charging, since simple charging schemes which are linear in time and volume and have important incentive properties can be created from tangents to bounds of the effective bandwidth [3]. Furthermore, the above result opens up some new possibilities for traffic modeling. Rather than developing general models that try to emulate real traffic in any operating environment, a new approach would be to develop models that emulate real traffic according to the particular operating point of the network. Such models would be parameterized with the pair (s; t), would be simple and efficient to implement, and can be the basis for fast and flexible traffic generators. The application of our approach for traffic engineering and network dimensioning in a real multi-service network environment that involves a large number of different source types constitutes a promising area for further research. A specific question is to investigate whether the number of discontinuities of the time parameter, that we have identified for a traffic mix containing two source types, increases with the number of source types. A second area for further work is to extend our investigations to links with priorities; some results in this area are presented in [9]. Acknowledgements The authors are particularly grateful to Frank P. Kelly for his helpful discussions and insights, and thank the anonymous reviewers for their constructive comments. --R Large deviations On the effectiveness of effective bandwidths for admission control in ATM networks. Buffer overflow asymptotics for a switch handling many traffic sources. Effective bandwidth of general Markovian traffic sources and admission control of high speed networks. Experimental queueing analysis with long-range dependent packet traf- fic On the relevance of long-range dependence in network traffic Notes on effective bandwidths. Effective bandwidths for multiclass Markov fluids and other ATM sources. On the self-similar nature of ethernet traffic Cell loss asymptotics for buffers fed with a large number of independent stationary sources. de Veciana. Statistical properties of MPEG video traffic and their impact on traffic modeling in ATM systems. The importance of the long-range dependence of VBR video traffic in ATM traffic engineering: Myths and realities Large deviations approximations for fluid queues fed by a large number of on/off sources. Smoothing, statistical multiplexing and call admission control for stored video. --TR Effective bandwidth of general Markovian traffic sources and admission control of high speed networks On the self-similar nature of Ethernet traffic Effective bandwidths for multiclass Markov fluids and other ATM sources Analysis, modeling and generation of self-similar VBR video traffic area traffic Experimental queueing analysis with long-range dependent packet traffic The importance of long-range dependence of VBR video traffic in ATM traffic engineering On the relevance of long-range dependence in network traffic --CTR A. Courcoubetis , Antonis Dimakis , George D. Stamoulis, Traffic equivalence and substitution in a multiplexer with applications to dynamic available capacity estimation, IEEE/ACM Transactions on Networking (TON), v.10 n.2, April 2002 C. Courcoubetis , V. A. Siris , G. D. Stamoulis, Network control and usage-based charging: is charging for volume adequate?, Proceedings of the first international conference on Information and computation economies, p.77-82, October 25-28, 1998, Charleston, South Carolina, United States Jun Jiang , Symeon Papavassiliou, Providing End-to-End Quality of Service with Optimal Least Weight Routing in Next-Generation Multiservice High-Speed Networks, Journal of Network and Systems Management, v.10 n.3, p.281-308, September 2002

broadband networks;large deviations;effective bandwidths;traffic engineering;ATM

277973

Fast Multigrid Solution of the Advection Problem with Closed Characteristics.

The numerical solution of the advection-diffusion problem in the inviscid limit with closed characteristics is studied as a prelude to an efficient high Reynolds-number flow solver. It is demonstrated by a heuristic analysis and numerical calculations that using upstream discretization with downstream relaxation ordering in a multigrid cycle with appropriate residual weighting leads to an efficient solution process. Upstream finite-difference approximations to the advection operator are derived whose truncation terms approximate "physical" (Laplacian) viscosity, thus avoiding spurious solutions to the homogeneous problem when the artificial diffusivity dominates the physical viscosity [A. Brandt and I. Yavneh, J. Comput. Phys., 93 (1991), pp. 128--143].

Introduction Efficient multigrid algorithms for the numerical solution of partial differential problems normally require good ellipticity measures on all scales of the problem, which implies that nonsmooth solution components can be re-solved by local processing [2]. But problems with small ellipticity measures are marked either by indefiniteness or by anisotropies. In the latter case, Technion-Israel Institute of Technology, Haifa, Israel y University of Twente, Enschede, The Netherlands z Weizmann Institute of Science, Rehovot, Israel there exist so-called characteristic directions of strong dependence. Some nonsmooth components of the solution are then advected along these char- acteristics, and hence they cannot be resolved locally [1]. A typical example is steady flow at high Reynolds numbers (small viscosity). When applied to such problems of small ellipticity the usual multigrid algorithms often exhibit a severe degradation of performance compared to that seen in elliptic problems. Indeed, most multigrid codes in use today for solving steady flows at high Reynolds numbers, although yielding a great improvement over previous single-grid solvers, fall far short of attaining the so-called textbook multigrid efficiency for general (even smooth) flows. To regain this efficiency the multigrid algorithm requires modifications that take into account the anisotropic properties of the operator. For example, it was shown in [4] and [9] that using upstream discretization and downstream relaxation-ordering yields a fully efficient multigrid solver for flows whose characteristics (streamlines) start at some part of the boundary and end at another without recirculating (entering flows). To obtain efficient multigrid solvers for flows with closed characteristics, however, different modifications were proposed, such as defect-correction cycles and residual overweighting [5]. The main drawbacks of the latter approaches are: (a) they are not likely to generalize efficiently to orders of accuracy higher than one; (b) they require W cycles, which may be substantially more expensive than simple V cycles in parallel computation; (c) they suggest different treatment for different types of flow, viz., recirculating versus entering flows. The upshot of the present work is to obtain a unified approach for both types of flow by employing upstream discretization and downstream relaxation ordering for recirculating flows as well. In Section 2 we formulate the simple model problem of advection-diffusion and present the First Differential Approximation to its discretized form. In Section 3 we present the two-level cycle and use the approximation of Section 2 in a heuristic analysis for a priori prediction of the performance of this algorithm. In Section 4 new first-order upstream discretizations for the advection operator are presented, whose first truncation terms approximate isotropic diffusivity. These schemes are shown to eliminate spurious solutions to the homogeneous (i.e., unforced) small-viscosity advection-diffusion equation, such as those reported in [3]. Section 5 presents numerical calculations testing the accuracy of the discretization and the efficiency of the multigrid algorithm and how it compares to the predictions of Section 3. Section 6 summarizes the main conclusions and further research plans. 2 The Scalar Advection-Diffusion Equation We study the scalar advection-diffusion equation with closed characteristics as a prelude to the study of flow problems. This equation serves well as a preliminary problem, since the advection part (i.e., momentum equations) is responsible for the degraded performance observed in the solution of the incompressible-flow equations by the usual multigrid algorithm [5]. Also, as is shown for entering flows in [4], the solution-process of the advection part of the system can be effectively decoupled from that of the elliptic part that is due to the continuity equation. Hence, efficient solution of the advection problem is a necessary stage in the development of a fully efficient flow-equations solver, and [4] suggests that the resulting advection-problem solver can indeed be used in designing the sought-after flow-solver. The advection-diffusion equation in two dimensions is where ffl is a positive constant and a, b, f , and g are given functions of x and y. Equation (1) is discretized on a uniform grid of meshsize h, whose gridlines lie parallel to the x and y coordinates. The characteristic direction of the advection operator in (1) is given (locally) by where OE is the (local) angle of nonalignment between the x coordinate and the characteristic direction. We will focus our attention on the particular case where the characteristics defined by a and b form closed loops (as in vortices), one of which may coincide with @\Omega (as in internal flows). Suppose that (1) is discretized by some stable finite-difference discretization of first-order accuracy. The main aspects of the problem can be analyzed by substituting for the discrete operator its First Differential Approximation [8] and also [1, 2, 9]. For the advection-diffusion equation with positive but vanishingly small ffl we need only consider the advection oper- ator, since the tiny diffusion will be dominated by the artificial diffusivity represented by the truncation terms (except at stagnation points). Let L h denote a first-order accurate discrete approximation to the advection opera- tor. Then, by a Taylor series expansion, we generally have where u h denotes the discretized function, h being the meshsize of a uniform grid. Here, ~ are functions of x and y, the specific details of which are determined by a and b and the discretization. The FDA is the approximation of L h by the differential operator that remains in (2) after the O(h 2 ) terms are neglected. (Hence, it applies only to sufficiently smooth u h , since the neglected terms are higher derivatives). We now assume for simplicity of the discussion that the equation is normalized such that a 2 introduce a (conformal) local coordinate system, (-; j), where - denotes the local "streamwise" coordinate parallel to the characteristic direction, while j denotes the "cross-stream" coordinate that is perpendicular to the characteristic. Thus, and The FDA of L h in the local coordinate system is therefore ~ where, by (2) and (4), We assume a consistent and stable discretization, which requires that the artificial viscosity operator represented by the first truncated term be elliptic, implying that the O(h) part of the operator in (5) is of positive type. Under special circumstances, such as consistent alignment of the characteristics with the grid, T h vanish, and this property is marginally violated. In this case the physical diffusion term becomes important, no matter how small ffl may be, and the analysis below does not apply. Accordingly, we will assume below that T h 1 is large compared to h, which is the usual case. 3 Two-Level Error-Reduction Analysis We now analyze the error reduction attainable with a two-level cycle using upstream discretization and downstream relaxation-ordering. 3.1 Two-level Cycle The proposed two-level cycle for a given discrete problem L h u defined as follows. ffl Starting with some approximation to u h perform - 1 (small integer) relaxation sweeps. ffl Calculate the residuals, r u h is the current approximation to the solution, and transfer them to a twice-coarser grid 2h, multiplied by a globally-uniform weight W . ffl Solve the coarse-grid problem, L 2h v for the correction. Here, r 2h is the restriction of r h to the coarse grid. ffl Interpolate and add the correction v 2h to the fine-grid approximation. ffl Perform - 2 (small integer) fine-grid relaxation sweeps. In studying the asymptotic performance of the two-level cycle, the number of pre-relaxation sweeps - 1 need not be distinguished from the number of post-relaxation sweeps - 2 . (Recall that we associate asymptotic performance with the spectral radius ae of the iteration matrix, and that ae(AB) = ae(BA) for any pair of square matrices A, B of the same dimension). We denote the total number of sweeps by In analyzing the two-level cycle we shall make many simplifying assump- tions. The degree to which these assumptions are justified needs to be judged by the degree to which numerical results match the predictions of the analyses 3.2 The Model Problem and Analysis We analyze the two-level convergence for the discrete approximation to (1) in the limit of vanishing ffl for problems with closed characteristics by considering the following model problem on grid h. where L h is, as above, the discretization of the advection operator. For the domain of solution and boundary conditions we require periodicity in - in order to simulate closed characteristics, and we choose for simplicity of the discussion - 2 [0; 1]. The boundary conditions in the cross-stream direction are not germane in the present context. For simplicity, we let The main point of our approach is to use discretization that is purely upstream, and to relax the equation in downstream ordering, starting at ordering means that we relax a variable only after relaxing all other unknowns which participate in the equation which corresponds to this variable (except, perforce, at Thus, a full relaxation sweep results in the elimination of all the residuals except at a narrow band (of O(h) width) that stretches from (which coincides with due to the periodicity). Neglecting the width of this band, we find that the residual function, r h , which remains after at least one full relaxation sweep has been carried out, can be modeled by r h (-; The fine-grid error v h satisfies the residual equation. Wherever the residual vanishes we now revert to FDA, obtaining ~ with ~ defined in (5). We now add a further simplifying assumption that in (5) are independent of j. Hence, we may expand (8) in a Fourier series in j. For an error component - (-) exp(i!j) of frequency !, Equations (5) and (8) then yield d- where all the O(h) terms but the first can be neglected in the homogeneous equation, since they multiply derivatives and are therefore small compared to d-v h d- . The solution to (9) in the interval (0; 1) is therefore given by Z -T h ds where A h ! is the amplitude of - v h ! at which we shall determine shortly. (Superscripts denote an infinitesimal positive (negative) incre- ment). In particular, at 1 is the average value of T h 1 over the entire domain, under the assumption that T his j \Gammaindependent (and recalling that the domain length in the - direction is one). It is important to note that D(h; !) is approximately the factor by which a single relaxation sweep amplifies (reduces) an error component that oscillates at frequency ! in the j direction. This is due to the fact that, given upstream differencing, the downstream relaxation ordering yields numerical integration; and D(h; !) is the factor by which this integration over the domain reduces the error. Equation (11) implies that relaxation reduces error components with large ! very efficiently, but components that are smooth along j need to be corrected on the coarse grid. the FDA is no longer useful. Instead, we have a jump in - v h ! that is proportional to the Fourier coefficient of R h (j) corresponding to frequency !. We denote this jump by ffi h . The periodicity in - now implies by (10) and (11) A h Now, following the two-level algorithm, we attempt to approximate the (weighted) residual equation on the coarse grid 2h. We assume that the same discretization stencil is used on the coarse grid as on the fine. (Note this important assumption on which the entire method hinges). We also assume that the restriction operator is such that the jump condition at is approximated correctly on the coarse grid. In practice this holds provided that a proper averaging is used, such as full-weighted residual transfers. Analogously to (10),(11), and (12), respectively, we obtain Z -T 2h(s) ds and A 2h where W is a constant weight to be chosen. Since the stencils of L h and L 2h are the same, we also assume - Also, since we assume that the restriction operator transfers the jump condition correctly, we have ffi 2h . Equations (12), (15), and (16) now yield A 2h A h Neglecting again the effects of intergrid transfers and aliasing, we assume that the remaining error after adding the coarse-grid correction is Hence, the fine-grid error is amplified by the factor (-v h ! . In addition, relaxation sweeps performed on the fine grid amplify the error by D(h; !) - , as noted above. The two-level error-amplification factor - tl is then given by the absolute value of the product of these terms. Note, however, that the determining values of - ! in the coarse-grid correction term depend on where one begins relaxing on the fine grid immediately following the coarse-grid correction (since all other fine-grid values at the end of one or more sweeps are determined solely by the values where relaxation begins, due to the upstream differencing and downstream relaxation ordering). If the fine- grid relaxation begins at that is, at or shortly after the point where the residual was nonzero, then we have A 2h A h However, if the fine-grid relaxation begins at there is some small overlap in the region being relaxed), then A h 3.3 Optimal Residual Weighting By (11) we have and in order to obtain an h-independent analysis we assume now that D can take on any value in the interval (0; 1). The optimal value of W is that which minimizes the supremum of - tl over 1). For any fixed W , the supremum is evidently obtained either for @D From (18) and (19) we can thus obtain Dm as a function of W and -, from which we can then calculate W opt (the value of W which yields the fastest convergence) for either of these cases. For for (18), yielding - For larger - the tedious calculations need to be carried out numerically. But W opt tends to 2 for both cases rather quickly. This is expected, since this value is the ratio of the Green's functions on the coarse and fine grids for components that are very smooth in the cross-characteristic direction; other components are reduced by relaxation. With we obtain for both (18) and (19) Equating the derivative of (20) with respect to D to zero, we get as the only relevant root, The two-level error-amplification factor is now obtained by substituting (21) into (20). For sufficiently small - \Gamma2 we may neglect this term, obtaining In fact, (22) gives an excellent approximation of the maximal - tl in (20) for any -, erring by less than 2% for - 2. By curious coincidence, the same asymptotic two-level error-amplification factor for large -, (2-e) \Gamma1 , is obtained for the Poisson equation on a rectangle using Gauss Seidel relaxation in Red-Black order [7]! Thus, this analysis leads us to expect efficiency that is comparable to that obtained for the Poisson problem. Example 1. We apply our algorithm to the advection-diffusion problem with closed characteristics used in [10] (originally in [6]): (0; 1), with u(x; on @ and For the advection term we use the same discretization as is used on the finest grid in [6] and [10]: standard upstream (SU), defined by where In We use this value only at the stagnation point, adding no viscosity elsewhere to maintain upstream discretization. Since the "physical" viscosity is dominated by the artificial viscosity elsewhere anyway, the difference is small. (Alternatively, we could use a much smaller ffl everywhere). Levels Grid W= 1 W= 2 MGD9V Table 1: The number of cycles necessary to reduce the L 2 residual norm by a factor of 10 8 in Example 1. The problem is taken from [10], and MGD9V is the automatic method of de Zeeuw which is used there. As in [10], the initial guess for u in\Omega is zero, and we cycle until the L 2 norm of the residuals is reduced to at most 10 \Gamma8 times its initial value. We performed this test with 4, 5, 6, 7, and 8 levels, with the coarsest grid always 5 \Theta 5, including boundary points (as in [10]). We used V(1,1) cycles throughout, with the usual full-weighted residual transfers and bilinear in- terpolation. (See Section 5 for details on implementation of the relaxation). In Table 1 we compare our results with W=1 and 2 to those reported for MGD9V, the automatic method of de Zeeuw (available up to six levels), using a so-called "sawtooth" cycle with one ILLU (Incomplete Line LU) relaxation sweep per level. It must be stressed that the efficiency and robustness of this method is convincingly demonstrated in [10], and all the results achieved on ten other problems (including non-recirculating advection diffusion) were far better than these. Evidently, the present method performs very well, with efficiency comparable to that of elliptic equations in this simple test problem. Clearly, the downstream relaxation ordering itself is not sufficient for the recirculation problem (nor is ILLU). Both MGD9V and the present method with no residual overweighting show clear deterioration as the grid is refined, while with proper overweighting the convergence rate remains excellent even for very fine grids. We reiterate that our residual overweighting approach may not apply to MGD9V, since there different stencils are used on the different grids, requiring different overweighting. 3.4 Several Bands of Residuals It may not always be easy to obtain a single band of residuals per vortex. This happens when the relaxation is carried out in piecewise-downstream ordering, as would be the case in a domain-decomposition setting for example. Our analysis can be extended to the case where several bands of nonzero residuals remain after relaxation. It is found that one then requires two coarse-grid corrections, with optimal weights 1 and 2 approximately. This seems to imply that a W cycle is required in this situation. Also, one must then use upstream intergrid transfers, so as to avoid averaging over interfaces between subdomains, which may actually cause divergence. 4 Discretization Flows in which the streamlines do not start and end at boundaries, but constitute closed curves, require special considerations in the discretization. In such cases, even a very small viscosity plays an important role in determining the main flow throughout the domain. The solution in the limit of vanishing viscosity depends very strongly on how these coefficients tend to zero. In effect, the advection terms determine the behavior of the solution along streamlines, whereas the viscous terms determine its cross-stream form. And since the boundary is often a streamline itself, the propagation of information from the boundary into the domain is governed by the viscous terms no matter how small they may be. This effect is discussed in detail in [3, 9], where it is shown for both the advection-diffusion problem and the incompressible Navier Stokes equations that solutions with schemes in which the numerical viscosity is anisotropic (having different viscosity co-efficients for the cross-stream and streamwise directions), such as standard upstream-difference schemes, may be spurious. In the most general case it can be shown that even isotropic viscosity is not sufficient for convergence of the solution, and one must actually specify a uniform viscosity. We do not know how to do this while retaining the purely- upstream structure (but see remarks in Section 6). However, for the homogeneous advection-diffusion problem there are several indications (though no proof) that isotropy suffices. This is shown below and also in [3, 9], where it is also shown (in a numerical example) to suffice for the incompressible Navier Stokes equations. This is consistent with the fact that the vorticity in the Navier Stokes equations satisfies a homogeneous advection-diffusion equation. To obtain a discretization scheme that exhibits the appropriate physical- like behavior for vanishing viscosity we must thus either add sufficient explicit isotropic viscosity that will dominate the artificial viscosity of the discrete advection operator, or else derive a discretization of the advection operator that satisfies the condition of isotropy in its lowest-order truncated terms. Since we want our scheme to remain purely upstream, we follow the latter approach. Consider the standard upstream scheme of (23), and assume for simplicity of discussion a - b - 0 . From (2) we have by a Taylor expansion ~ Hence, in order to obtain isotropic artificial viscosity we may either add some approximation of 0:5h(a \Gamma b)u xx , or else subtract some approximation of 0:5h(a \Gamma b)u yy . In order to retain an upstream scheme we define this additional viscosity at the point (i \Gamma For general a and b we obtain in the first case the Isotropic-Viscosity Upstream scheme IVU1, defined by if ja i;j j ? jb i;j j, and otherwise. In the second case we obtain scheme IVU2, defined by if ja i;j j ? jb i;j j, and otherwise. Here i1; j1 are defined as in (23), and similarly The first truncated term in scheme IVU1 is thus \Gamma0:5 min(ja i;j j; jb i;j j)\Deltau, while that of IVU2 is \Gamma0:5 max(ja i;j j; jb i;j j)\Deltau : Both schemes have isotropic artificial viscosity, but that of IVU1 is smaller, and in fact it vanishes upon alignment of the characteristic directions with the grid. Both discretizations are stable in downstream-ordered Gauss-Seidel relax- ation. The former is a nonnegative-weighted average of the standard first-order and second-order upstream schemes, both of which are stable in this relaxation. The latter produces an M-matrix. As expected, there were no stability problems in any of our many numerical calculations. 5 Numerical Experiments We first test numerically the discretizations derived in Section 4 on a model problem for which the standard upstream scheme has been shown to yield spurious solutions [3, 9]. Then, the asymptotic error reduction of two-level and multilevel cycles are investigated for several problems. 5.1 Accuracy Test The accuracy of the different discretizations is tested on the model problem: a with a and b given by (These coefficients are the same as those of Example 2 below, and a picture of the characteristics appears in Figure 1a). The domain of solution is the unit square, centered at the origin, with a square of diagonal 0.5, whose sides form a 45 degree angle with the axes, removed from its center. On the outer boundary prescribed and on the inner boundary solve this problem with the three upstream schemes-SU, IVU1, IVU2, and also the (non-upstream) isotropic-viscosity scheme used in [3], denoted ISO. viscosity is added: These solutions are compared to that obtained with a standard second-order upstream scheme with physical viscosity coefficient 0.001. The latter solution is obtained on a 257 \Theta 257 grid. Grid SU IVU1 IVU2 ISO 129 \Theta 129 0.0750 0.0057 0.0163 0.0098 Table 2: L 1 difference norms between the solutions obtained with several schemes at different resolutions, and a high-accuracy solution obtained on grid 257 \Theta 257 (see text for details). The three isotropic-viscosity schemes are seen to yield convergent solutions, but not the standard upstream (SU) scheme. In Table 2 we present L 1 norms of the differences between the test solutions at various resolutions, and the second-order accurate solution (restricted to the corresponding grid by injection). Since the solution is smooth, and the physical viscosity dominates the second-order truncation terms at this high resolution, the latter solution is assumed to be very accurate. Evidently, the three schemes with isotropic artificial viscosity produce convergent solutions but the SU scheme does not, despite the fact that its average viscosity is smaller than that of IVU2 and ISO. IVU1, which has the least artificial viscosity of the four schemes tested, produces the smallest error. 5.2 Efficiency Tests The remainder of our numerical calculations are aimed at testing the performance of the algorithm in various configurations and comparing to the analytical predictions of Section 3. We test the SU scheme (as this is the most widely used first-order scheme) and the IVU1 scheme (which is more accurate than IVU2 and also employs just a four-point stencil). In all these tests we use first-order upstream residual restriction and bilinear interpolation of the corrections. The restriction is performed as follows: for all even i and j on the fine grid we define i1, j1 as in (23), and restrict to the corresponding coarse-grid right-hand side at coarse-grid point (i=2; j=2) the average of the fine-grid residuals at points (i; j), (i1; j), (i; j1), and (i1; j1). This restriction gives slightly better results than standard full weighting in multi-vortex problems, since residuals are less likely to be transferred from one vortex to another. The finest grid in all the tests is 129 by 129, and six levels are employed except in the two-level tests. We include no "physical" viscosity except at stagnation points it is required for well-posedness. We calculate convergence factors as follows: the boundary conditions and right-hand sides are chosen to be zero; (the choice is immaterial for linear problems, but this allows us to normalize the solution by a constant factor every few cycles in order to avoid roundoff errors). The initial solution field is pseudo- random, and 100 cycles are performed. We calculate the convergence factor as the (geometric) average error-convergence per cycle over the last 80 cycles. The averaging is used because in some cases the convergence history is not smooth, and the value corresponding to any particular cycle may not carry much meaning. However, it should be noted that in all cases the convergence factors in the vicinity of the optimal W were not sensitive to the exact choice of W . Example 2. The first test for efficiency is the problem used above to test the discretization, but without the inner "island" (so as to have pure recirculation everywhere). The characteristics of this problem, which form a single clockwise-rotating vortex, are plotted in Figure 1a. A relaxation sweep is implemented by sweeping four times over the domain, and in each such sweep relaxing roughly one quarter of the variables as follows: in the first sweep only variables at locations where both a(x; y) and b(x; y) are nonnegative are relaxed (designated first quadrant); in the second sweep only variables corresponding to locations where a(x; y) is nonnegative and b(x; y) is nonpositive are relaxed (second quadrant); in the third sweep only variables at locations where both a(x; y) and b(x; y) are nonpositive are relaxed (third quadrant); in the fourth sweep only variables corresponding to locations where a(x; y) is nonpositive and b(x; y) is nonnegative are relaxed (fourth quadrant). Fi- nally, all stagnation points (in this case just one) are relaxed last. This entire process comprises a single clockwise sweep. Of course, each quarter-sweep is performed in downstream order i.e., x and y increasing in the first quarter- increasing and y decreasing in the second, etc. It is efficient to store the order of relaxation of the entire sweep during a setup-sweep (which costs very little), so that from then on each full clockwise sweep costs nearly the same as an ordinary lexicographic Gauss-Seidel sweep. Note that we specify nonnegative and nonpositive in the above descrip- tion. That is, it should be ensured that the boundaries of the quadrants are included in the quadrant. This was important in some of the tests. Such a clockwise sweep indeed eliminates all the residuals except along a narrow band that extends from the center of the vortex to the boundary. But if the vortex rotates counterclockwise, several such bands would remain. Hence, in Examples 3-5 below, where both clockwise and counterclockwise vortices exist (as would be the general case), we perform an analogous counterclockwise sweep following each clockwise one. The quadrant where this counterclockwise sweep starts has some bearing on performance. In our experiments we use exactly the reverse order. This allows us to save some work by performing the counterclockwise sweep over just three quadrants, since the fourth quadrant (a(x; y) nonpositive and b(x; y) nonnegative) has just been relaxed in the clockwise sweep. Thus, we begin with the third quadrant (a and b nonpositive), then the second, and finally the first quadrant. (Note, by the way, the difference from the usual symmetric Gauss-Seidel: here the sweeps are all performed in downstream ordering within each quadrant. But the quadrants are scanned symmetrically). In Problems (3-5) (especially there was also some sensitivity to which quadrant one chooses to start the relaxation sweep. That is, a somewhat different convergence rate and optimal overweighting were obtained if one performed the clockwise sweep as above, than if one relaxed, say, the second quadrant first, then the third, then the fourth, and finally the first. Hence, in these tests we shifted the relaxation starting-point by one quadrant after every cycle so as to obtain results that represent some average or "typical" case. We first performed two-level tests in order to compare the numerical results with the analysis. We used and the theoretically optimal W , calculated from (19). This is the relevant value for this example, since there is a one-line overlap between the quadrants. The results are given in Table 3. Evidently, the analysis captures the main features of this problem very well, despite the numerous simplifications, as seen especially in the SU results. Calculations were also carried out with V(1,0), V(1,1), and V(2,1) cycles, with and the optimal value, which was determined experimentally. The numerical results are summarized in Table 4. We find that the experimental V-cycle results also match the two-level predictions fairly well in the Table 3: Comparison of error-amplification factors obtained by analytical prediction and two-level numerical calculations of Example 2 for W opt . The optimal values of W were obtained from (19). vicinity of the optimal residual-weighting factors. But the optimal W 's are somewhat higher than predicted, although they do show the expected dependence on the number of relaxation sweeps performed. In other problems, reported below, the optimal value varied, but it was always fairly close to 2. The overall performance when the optimal W is used is very satisfactory. As noted above, this performance is not sensitive to moderate changes in W (see also below). Example 3. This example features flow with four vortices rather than just one. Here, a and b are given by and the characteristics are plotted in Figure 1b. The domain, as in all the examples except Example 5, is the unit square centered at the origin. The numerical results using V(1,1) and V(2,1) cycles appear in Tables 5 and 6, respectively. The convergence performance remains excellent, even though some nonvanishing residuals remain after the relaxation at parts of the borderlines between vortices (where the flow leads away from the borderline). Recall, however, that here and below each full relaxation sweep consists of one clockwise sweep followed by three quarters of a counterclockwise sweep (see description of implementation above), in order to allow for the opposite- sign vortices. Cycle W SU IVU1 V(1,0) 1. 0.795 0.898 V(1,1) 1. 0.676 0.831 V(2,1) 1. 0.536 0.724 2. 0.280 0.440 2. 0.143 0.302 2. 0.069 0.133 Table 4: Error-amplification factors obtained in numerical calculations of Example 2. The optimal values of W were obtained experimentally (see text). Example 4. In order to test the effect of grid-alignment of the borderlines between vortices we solve a problem in which this borderline is not aligned with the grid. In this problem a and b are given by a 1 respectively, where a with This represents a superposition of two opposite-sign vortices. The characteristics are depicted in Figure 1c, and the numerical performance is shown in Tables 5 and 6. Here we see some loss of efficiency, but the performance is still satisfactory and far better than is usually exhibited in such problems. Example 5. Here we test a mixed problem, where the flow enters and leaves through the boundary, but there also exists a large recirculation zone. This is obtained by redoing Example 2, but in an extended domain: Example SU IVU1 Table 5: Error-amplification factors obtained with V(1,1) cycles for and W opt . The latter were found experimentally. Example SU IVU1 Table obtained with V(2,1) cycles for and W opt . The latter were found experimentally. The mesh is still uniform, and the finest grid is now 193 by 129. The characteristics are shown in Figure 1d, and the numerical performance is given in Tables 5 and 6. As in the other examples, there was virtually no sensitivity to the order in which the relaxation sweeps were performed (that is, clockwise first and then counterclockwise or vice versa). Summary Often, one may not wish to search for optimal residual weighting factors for every problem. Instead, one can simply use the nominal value 2. In the "realistic" examples 3-5, one saves at most 25% of the time spent in relaxation by using the optimal value rather than 2, and usually much less. This would also be the case in Example 1 if we were to employ the "symmetric" relaxation. Even with this nominal value the convergence rates are comparable to those of elliptic problems. 6 Conclusions and Further Research An experimental approach that is hoped to eventually lead to a fully efficient solver for general high-Reynolds flows has been introduced, analyzed and tested on the advection-diffusion problem in the inviscid limit. The numerical tests mostly match the predictions very well, indicating that the main cause for slow convergence of the usual multigrid algorithms for recirculating flows has indeed been understood, and a way to eliminate it has been found. The multigrid V-cycle, using downstream relaxation and upstream dis- cretization, was shown to yield an efficient solver for the tested problem in several simple situations of closed characteristics and in a mixed en- tering/recirculation problem. The tests were performed with the classical standard first-order upstream discretization scheme and also with a novel first-order upstream discretization, that was shown to preclude the spurious solutions reported in [3]. The present approach is cheaper to implement than that developed in [5], and can straightforwardly be applied to mixed entering/recirculating flows. More important, there is potential of success with high-order discretization, for which the approach of [5] yields an inadequate compromise. However, the results obtained are still preliminary. The effect of the intergrid transfers on the small band of residuals and its consequences in terms of error- reduction efficiency should be investigated over a wide variety of cases, along with a study of how to deal with (or avoid) situations where there remain several bands of nonvanishing residuals per vortex. Then, further research should be directed towards higher-order discretization. For this case too, an effectively-upstream discretization needs to be developed, whose truncation error represents isotropic artificial diffusivity. One approach is to use a predictor-corrector type discretization, employing an upstream scheme as a (local) driver and a possibly higher-order (not necessarily upstream) scheme as a (local) corrector. Finally, the present approach has only been tested on the advection problem. Experiments with the incompressible Navier Stokes equations for flows with closed streamlines need to be performed, employing distributive Gauss-Seidel relaxation, as shown in [4]. These will no doubt raise further questions. It is anticipated that the techniques investigated here will carry over to three dimensions, although the implementation will be considerably more complicated. This is supported by the fact that simple experiments performed with the overweighting methods of [5] in three dimensions exhibited the expected performance. An obvious drawback of the entire approach is that it is inherently sequen- tial, and efficient parallel implementations are hard to envisage. Some parallelization might be achievable by performing a downstream line Gauss Seidel relaxation. Also conceivable is a domain decomposition approach which leaves several lines of residuals per vortex. Another drawback of the present approach is that it is not directly applicable for flows with significant additional viscosity, since this entails using discretizations that are not purely upstream. Methods that deal with such flows as well are presently being investigated. Acknowledgment This work was supported by The Royal Netherlands Academy of Arts and Sciences and The Feinberg Graduate School, by the United States-Israel Binational Science Foundation under grant no. 94-00250, and by The United States Air Force Grant F49620-92-J-0439 and by the Carl F. Gauss Minerva Center for Scientific Computation. --R "Multigrid Solvers for Non-Elliptic and Singular- Perturbation Steady State Problems," "1984 Multigrid Guide with Applications to Fluid Dynam- ics," Inadequacy of First-Order Upwind Difference Schemes for Some Recirculating Flows On Multigrid Solution of High-Reynolds Incompressible Entering Flows Accelerated Multigrid Convergence and High- Reynolds Recirculating Flows Efficient Solution of Finite Difference and Finite Element Equations by Algebraic Multigrid fundamental algo- rithms On the Correctness of First Differential Approximation of Difference Schemes Multigrid Techniques for Incompressible Flows --TR

recirculating flow;upstream discretization;advection-diffusion;multigrid

278097

An Integral Invariance Principle for Differential Inclusions with Applications in Adaptive Control.

The Byrnes--Martin integral invariance principle for ordinary differential equations is extended to differential inclusions on {Bbb R}N. The extended result is applied in demonstrating the existence of adaptive stabilizers and servomechanisms for a variety of nonlinear system classes.

Introduction . Suppose that - semidynamical system on R N with semiflow ' and so, for each x 0 2 R N , is the unique maximal forwards-time solution of the initial-value problem - In [2], Byrnes Martin prove the following integral invariance principle: if '(\Delta; x 0 ) is bounded and continuous function l : R N ! R+ := [0; 1), then 1, to the largest invariant (with respect to the differential set in l \Gamma1 (0), the zero level set of l. This result has ramifications in adaptive control, some of which are highlighted in the present paper. However, we wish to consider the (adaptive) control problem in a fairly general setting that allows time-variation in the underlying differential equations, possible non-uniqueness of solutions, and discontinuous feedback strategies: each of these features places the problem outside the scope of [2]. For this reason we develop, in Theorem 2.10, an integral invariance principle for initial-value problems of the form - the set-valued map X is defined on some open domain G ae R N and is assumed to be upper semicontinuous with non-empty, convex and compact values. In the case Theorem 2.10 contains the following generalization of the Byrnes-Martin result: if x(\Delta) : R+ ! R N is a bounded solution and R 1l(x(s))ds ! 1 for some lower semicontinuous l : R N ! R+ , then x(t) tends, as t !1, to the largest weakly- invariant (with respect to the differential inclusion) set in l \Gamma1 (0). One particular consequence of Theorem 2.10 is to facilitate the derivation of a nonsmooth extension, to differential inclusions, of LaSalle's invariance principle for differential equations: this extension may be of independent interest and is presented in Theorem 2.11. The remainder of the paper is devoted to the application (in a collection of five lemmas) of the generalized integral invariance principle to demonstrate, by construction and for a variety of nonlinear system classes, the existence of a single adaptive controller that achieves (without system identification, parameter estimation or injection of probing signals) some prescribed objective for every system in the underlying class. 2. Differential inclusions. Some known facts (tailored 1 to our immediate pur- pose) pertaining to differential inclusions are first assembled. Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom. 1 Variants of Propositions 2.2, 2.4 and 2.8 can be found in, for example, [8], [19]: for general treatments of differential inclusions and related topics in set-valued analysis, nonsmooth control and optimization see [1], [4], [6], [7], [8] and [12]. 2.1. Maximal solutions. Consider the non-autonomous initial-value problem G; (1) is an open subset of R N . The set-valued map (t; x) 7! X(t; x) ae R N in (1) is assumed to be upper semicontinuous 2 on R \Theta G, with non-empty, convex and compact values. This is sufficient (see, for example, [1, Chapter 2, Theorem 3]) to ensure that, for each admits a solution: an X-arc 3 x 2 Definition 2.1. A solution x of (1) is said to be maximal, if it does not have a proper right extension which is also a solution of (1). Proposition 2.2. Every solution of (1) can be extended to a maximal solution. Definition 2.3. A solution x of (1) is precompact if it is maximal and the closure cl(x([t 0 ; !))) of its trajectory is a compact subset of G. Proposition 2.4. If x 2 AC([t 0 ; !); G) is a precompact solution of (1), then 2.2. Limit sets. Here, we specialize to the autonomous case of (1), rewritten as G; (2) where, without loss of generality, t assumed. The map x 7! X(x) ae R N (with domain G) is upper semicontinuous with non-empty, convex and compact values. Definition 2.5. Let x 2 AC([0; !); G) be a maximal solution of (2). A point is an !-limit point of x if there exists an increasing sequence (t n ) ae [0; !) such that t x as n ! 1. The set\Omega\Gamma x) of all !-limit points of x is the !-limit set of x. Definition 2.6. Let C ae R N be non-empty. A function x 2 AC([0; !); G) is said to approach C if dC is the (Euclidean) distance function for C defined (on R N ) by dC (v) := Cg. Definition 2.7. Relative to (2), S ae R N is said to be a weakly-invariant set if, for each x 0 there exists at least one maximal solution x 2 AC([0; !); G) of Proposition 2.8. If x is a precompact solution of (2), is a non-empty, compact, connected subset of G. Moreover,\Omega\Gamma x) is the smallest closed set approached by x and is weakly invariant. 2.3. Invariance principles. For later use, the following fact (a specialization of a more general result [4, Theorem 3.1.7]) is first recorded. Proposition 2.9. Let I = [a; b], let non-empty K ae G be compact. If AC(I; K) is a sequence of X-arcs and there exists a scalar \Phi such that, for all n, xn (t)k - \Phi for almost all t 2 I, then subsequence converging uniformly to an X-arc x 2 AC(I; K). We now arrive at the main result, which generalizes [2, Theorem 1.2]. 2 The set-valued map X is upper semicontinuous if it is upper semicontinuous at every point - - of its domain in the sense that, for each " ? 0 there exists with denotes the open unit ball centred at 0 in R N . 3 For an interval I ae R and S ae R N , AC(I ; S) denotes the space of functions I ! S that are absolutely continuous on compact subintervals of I. For simplicity, we write AC(I) in place of the same notational convention applies to other function spaces. A function x 2 AC(I; G) is said to be an X-arc if it satisfies the differential inclusion in (1) almost everywhere. AN INTEGRAL INVARIANCE PRINCIPLE AND ADAPTIVE CONTROL 3 Theorem 2.10. Let l R be lower semicontinuous. Suppose that U ae G is non-empty and that l(z) - 0 for all z 2 U . If x is a precompact solution of (2) with trajectory in U and l x approaches the largest weakly-invariant set in \Sigma := fz 2 cl(U Proof. By Proposition 2.4, x has maximal interval of existence by Proposition 2.8, has non-empty !-limit G. Let and so there exists x as n !1. Define upper semicontinuity of X together with compactness of its values, X(K) is compact and so - xk1 . Write I = [0; 1] and define a sequence Evidently, x as n !1. By Proposition 2.9 (with subsequence - which we do not relabel - converging uniformly to an X-arc x 2 AC(I; K), with x and the sequence (- n -n lower semicontinuity of l, together with continuity of x, x and (uniform) convergence of (x n ) to x , it follows that - and -n , n 2 N, are lower semicontinuous with lim inf n!1 -n Z t- (s)ds - Z tlim inf -n ds - lim inf Z t-n ds 8 t 2 I: By the hypotheses, defines a monotone function W 2 AC(R+ ) with W (t) # 0 as t !1. Hence, Z t+tn l(x(s))ds Z t-n (s)ds - Z t- (s)ds 8 t 2 I: Seeking a contradiction, suppose ffl := - (0) ? 0. Then, by lower semicontinuity of - , there exists t 2 (0; 1] such that - the contradiction Therefore, is arbitrary, we have \Omega\Gamma x) ae \Sigma. By Proposition 2.8, x and the latter is a weakly- invariant set. Therefore, x approaches the largest weakly-invariant set in \Sigma. The next result is a nonsmooth extension, to differential inclusions, of LaSalle's theorem [11, Chapter 2, Theorem 6.4]: a smooth version (that is, restricted to smooth functions V ) is given in [19, Theorem 1] and a nonsmooth version is proved in [23, Theorem 3]; the alternative proof given below is considerably simpler, by virtue of its use of the integral invariance principle. First, we give Clarke's [4] definition of a generalized directional derivative V of a locally Lipschitz function at z in direction OE: The map (z; OE) 7! V semicontinuous (in the sense of real-valued func- tions) and, for each z, the map OE 7! V continuous. Theorem 2.11. R be locally Lipschitz. Define Suppose that U ae G is non-empty and that u(z) - 0 for all z 2 U . If x is a precompact solution of (2) with trajectory in U , then, for some constant approaches the largest weakly-invariant set in Proof. This result is essentially a corollary to Theorem 2.10 insofar as the essence of the proof is to show that the hypotheses of Theorem 2.10 hold with l := \Gammau. We first show that u is upper semicontinuous (and so l j \Gammau is lower semicon- tinuous). Let z 2 G be arbitrary and let (z n ) ae G be such that z n ! z as n ! 1. From (u(z n )) we extract a subsequence (u(z nk )) with u(z nk 1. For each k, let OE k be a maximizer of continuous V X(znk ), and so u(z nk upper semicontinuity of X, we have X(znk sufficiently large. Since OE k 2 X(znk ) and X(z) is compact, contains a subsequence converging to OE is arbitrary and X(z) is compact, OE 2 X(z). Thus, invoking upper semicontinuity of \Delta), we may conclude that lim sup n!1 u(z n semicontinuity of u. Observe that, for all z 2 U , By Proposition 2.4, x has interval of existence R+ . Let O ae R+ denote the set of measure zero on which the derivative - x(t) fails to exist. Since V is locally Lipschitz, for each t 2 R+nO there exists a constant L t such that, for all h ? 0 sufficiently small, x(t)j: Therefore, lim inf Next, we prove that V is nonincreasing on R+ . This we do by showing that V ffi x is nonincreasing on every compact subinterval. Let [ff; fi] ae R+ , and let K ae G be compact and such that x([ff; fi]) ae K. Since V is locally Lipschitz on G, it is Lipschitz on K. Thus, the restriction of V ffi x to [ff; fi] is a composition of a Lipschitz function and an absolutely continuous function and so is itself absolutely continuous. It now follows from (3) and (4) that u(x(s))ds AN INTEGRAL INVARIANCE PRINCIPLE AND ADAPTIVE CONTROL 5 Therefore, t 7! V (x(t)) is nonincreasing on [ff; fi]. Since [ff; fi] ae R+ is arbitrary, we conclude that V ffi x is nonincreasing on R+ with u(x(s))ds By continuity of V and precompactness of x, we conclude that V (x(\Delta)) is bounded. Therefore, It follows that An application of Theorem 2.10 completes the proof. 3. Adaptive control. Approaches to adaptive control may be classified into methods that - either implicitly or explicitly - exhibit some aspect of identification of the process to be controlled, and methods that seek only to control. The latter approach, to be adopted here and sometimes referred to as universal control, has its origins in the work of Byrnes and Willems [3, 27], M-artensson [13, 14, 15], Morse [16], Nussbaum [17] and others (see [9] for a survey and comprehensive bibliography). In common with its above-cited precursors, this section of the present paper is concerned with demonstrating the existence - under relatively weak assumptions - of a single controller that achieves some prescribed objective for every system in the underlying class. In contrast with its above-cited precursors (which deal mainly with classes of linear systems, possibly subject to "mild" nonlinear perturbations, in a context of adaptive linear feedback), the present paper considers strongly nonlinear systems and nonsmooth feedback (for an overview of adaptive control of nonlinear systems in the context of smooth feedback, see [18]). In essence, the ensuing two subsections provide a unified analysis - unified through its use of the integral invariance principle - of various problems in nonlinear adaptive control (some closely-related problems have been individually investigated, via alternative analyses, in [5], [10], [20] and [21]). 3.1. Scalar systems. First, consider scalar systems of the form where parameters b 2 R, P 2 N and functions f , p are unknown. The state y(t) is available for feedback. We will identify (6) with the quadruple (b; f; p; P ). For any function OE : R+ ! R+ that is both continuous and positive definite we denote, by N OE , the set of system quadruples (b; f; satisfying the following three assumptions. Assumption A. b 6= 0. Assumption B. (p; y) 7! f(p; y), R P \Theta R! R, is a continuous function and is OE-bounded uniformly with respect to p in compact sets: precisely, for every compact K ae R P , there exists scalar -K such that jf(p; y)j -K OE(jyj) for all (p; y) 2 K \Theta R. Assumption C. p(\Delta) 2 L 1 (R; R P ). By virtue of Assumption C, without loss of generality t may be assumed in this we will do, without further comment, throughout this subsection. Examples 3.1. (a) Let OE : jyj 7! exp(jyj). Then all polynomial systems, of arbitrary degree, of the form - coefficient functions are of class N OE . 6 E. P. RYAN (b) Suppose that Assumptions A, C hold and that the only a priori information on continuous f is its behaviour "at infinity", captured in the following manner: for some known continuous - OE(jyj)), as jyj !1, uniformly with respect to p in compact sets in the sense that, for every compact K ae R P , there exist scalars c K and CK such that, for all OE(jyj) for all jyj ? CK . Assumption B holds with OE OE, and so (b; f; 3.1.1. Adaptive stabilizer. Let OE : R+ ! R+ be a continuous, positive-definite function. Assuming only that the function OE and the instantaneous state y(t) are available for control purposes, we will show that the following adaptive feedback strategy (appropriately interpreted) is a N OE -universal stabilizer in the sense that it assures that the state of (6) approaches f0g for all quadruples (b; f; whilst maintaining boundedness of the controller function -(\Delta): where - is any continuous function R! R with the properties: (a) lim sup For example, suffices. In view of the discontinuous nature of the feedback (however, note that, if 0, then the feedback is continuous), we interpret the strategy (7) in the set-valued sense with y 7! oe(y) ae R given by Let (b; f; . By properties of f(\Delta; \Delta) and boundedness of p(\Delta), there exists a scalar - such that jf(p(t); y)j -OE(jyj) for all (t; y). We embed the feedback-controlled system in a differential inclusion on R where x 7! X(x) ae R 2 is given by X is upper semicontinuous on R 2 with non-empty, convex and compact values. There- fore, for each x 0 2 R 2 , the initial-value problem (11) has a solution and, by Proposition 2.2, every solution can be extended to a maximal solution. Lemma 3.2. Let x 0 2 R 2 be arbitrary and let be a maximal solution of (11), defined on its maximal interval of existence [0; !). Then exists and is finite; (iii) Proof. The essence of the proof is to establish boundedness of x(\Delta): whence, by Proposition 2.4, assertion (i) and, by monotonicity, assertion (ii): state convergence to zero (assertion (iii)) is then an immediate consequence of Theorem 2.10. AN INTEGRAL INVARIANCE PRINCIPLE AND ADAPTIVE CONTROL 7 For almost all t 2 [0; !), we have which, on integration, yields Z -(t) Seeking a contradiction, suppose that solution component -(\Delta) (monotone increasing) is unbounded. Fix - such that - 1. Dividing by -(t) - 1 in (13), gives Z -(t) Recalling that b 6= 0 and taking limit inferior as leads to a contradiction of one or the other of properties (8). Hence, -(\Delta) is bounded and so, by (13), y(\Delta) is also bounded. Therefore, precompact solution of (11) and so, by Proposition 2.4, by boundedness of -(\Delta), that l ffi x 2 L 1 (R+ ) and so, by Theorem 2.10 (with U approaches the set f0g \Theta R. In particular, y(t) ! 0 as t !1 and, by boundedness and monotonicity of -(\Delta), lim t!1 -(t) exists and is finite. 3.1.2. Adaptive servomechanism. We now turn attention to the servomechanism problem for scalar systems (6): that is, the construction of controls that cause the state to track, asymptotically, reference signals r(\Delta) of some given class in the sense that 1. For the class of reference signals (previously adopted in [20], [10], [21]) we take the (Sobolev) space R = W 1;1 (R) of functions essentially bounded derivative - equipped with the norm krk rk1 where k k1 denotes the norm on L 1 (R). We impose a stronger assumption on the function f by requiring that Assumption should hold for some known, continuous, positive-definite, nondecreasing function OE with the additional property that, for each R - 0, there exists a scalar ae R such that Note that, by positive definiteness of OE together with property (14), OE(0) ? 0. Example 3.3. Let OE : jyj 7! exp(jyj), which has the property (14), and so all polynomial systems (of arbitrary degree and with coefficients in L 1 (R)), as cited earlier in Example 3.1(a), remain admissible. \Sigma Let OE be a continuous, positive-definite, nondecreasing function with property (14). We claim that, in order to assure that the tracking error approaches f0g for all reference signals r 2 R and all quadruples (b; f; suffices to replace every occurrence of y(t) in (9) by e(t). Proof of this claim follows. Let (b; f; define the continuous function ~ s: Let ~ K ae R ~ P be compact and so there exist compact K ae R P and R ? 0 such that ~ K ae K \Theta [\GammaR; R] 2 . By properties of f and OE, there exist constants -K and ae R such that, for all (p; with ~ f is OE-bounded uniformly with respect to ~ in compact sets. Let r 2 R be arbitrary. Then ~ (b; ~ . Expressed in terms of the tracking error the system dynamics have the form We are now in precisely the same context, modulo notation, as in the case of an adaptive stabilizer and so, replacing all occurrences of y(t) in (9) by e(t) to yield then the same argument (as used to establish Lemma 3.2) applies mutatis mutandis to conclude that (15) is an (R; N OE )-universal servomechanism: for each r(\Delta) 2 R and (b; f; every solution (e(\Delta); -(\Delta)) of the controlled system has maximal interval of existence R+ with e(t) ! 0 as t !1, and lim t!1 -(t) exists and is finite. 3.1.3. Practical stabilization and tracking by continuous feedback. The adaptive strategies outlined above are (generically) of a discontinuous feedback nature. From a viewpoint of practical utility, this feature might be regarded as unpleasant. Here, we investigate the possibility of adopting smooth approximations to the discontinuous feedbacks. Of course, in so doing, one would expect to pay a price. It will be shown that, if the objective of attractivity of the zero state (in the stabilization case) or asymptotic tracking (in the case of a servomechanism) is weakened to requiring global attractivity of any (arbitrarily small) prescribed neighbourhood of zero or, for the servomechanism problem, tracking to within any prescribed (arbitrarily small but non-zero) error margin, then such approximations are feasible. We will present this result only in the context of the stabilization problem (imposing the additional property (14), the corresponding result for the servomechanism problem is readily inferred by analogy with Section 3.1.2). The na-ive idea, as developed in [10] and [21], is to inhibit the adaption whenever the state lies within the prescribed neighbourhood of zero. arbitrary and let d ffl denote the distance function for the set [\Gammaffl; ffl]: thus, d ffl (x) := continuous and positive-definite. Let R be any continuous function (arbitrarily smooth) such that (i) jsat ffl (x)j - 1 for all x and (ii) We will show that the following strategy assures that the state of (6) approaches the interval [\Gammaffl; ffl] for all quadruples (b; f; where, as before, - is any continuous function with properties (8). Let (b; f; and so there exists constant - such that jf(p(t); y)j -OE(jyj) for all (t; y). Define the set-valued map y 7! oe ffl (y) by Evidently, sat ffl is a continuous selection from oe ffl . We now embed the smooth-feedback- controlled system in the following differential inclusion on R AN INTEGRAL INVARIANCE PRINCIPLE AND ADAPTIVE CONTROL 9 where x 7! X ffl (x) ae R 2 is given by ffl is upper semicontinuous on R 2 with non-empty, convex and compact values. There- fore, for each x 0 2 R 2 , the initial-value problem (16) has a solution and every solution has a maximal extension. Lemma 3.4. Let x 0 2 R 2 be arbitrary and let be a maximal solution of (16), defined on its maximal interval of existence [0; !). Then exists and is finite; (iii) d ffl Proof. For almost all t 2 [0; !), which, on integration, yields Z -(t) valid for all t; - 2 [0; !), with t - . By precisely the same contradiction argument as employed previously in the case of the discontinuous stabilizer, we may deduce that precompact solution of (16) and so by boundedness of -(\Delta), that l ffi x 2 L 1 (R+ )). Therefore, by Theorem 2.10 (with approaches the set l 0g. In particular, d ffl by monotonicity of bounded -(\Delta), lim t!1 -(t) exists and is finite. 3.1.4. Dynamically perturbed scalar systems. Let OE : R+ ! R+ be continuous and positive definite. Let \Sigma We wish to consider the situation wherein \Sigma 1 is subject to perturbations generated through its interconnection with a dynamical system \Sigma 2 . Figure System \Sigma 2 is assumed to correspond to a differential equation (driven by the state of the scalar system \Sigma 1 ) on R N of the form with input y(t), and scalar output w(t) perturbing \Sigma 1 . Notationally, we identify the system \Sigma 2 with the triple (g; h; N ). The overall system has representation (on R\ThetaR N )! We will define, via Assumption D below, a class P/ of admissible systems \Sigma such that the N OE -universal stabilizer of Section 3.1.1 is readily modified to yield (N OE ; P/ )-universal stabilizer. Before stating Assumption D, we cite Sontag's concept of input-to-state stability [24, 25] (see also [26]) in the context of (18) with g assumed to be locally Lipschitz and with y(\Delta) regarded as an independent input of class loc (R+ ; R). System (18) is input-to-state stable (ISS) if there exist a continuous, strictly increasing function and a continuous function and having the properties that, for each t - 0, fi(\Delta; t) is strictly increasing and, for each s - 0, fi(s; t) # 0 as t !1, such that, for every i 0 2 R N and every y(\Delta) 2 L 1 loc (R+ ; R), the (unique) maximal solution i(\Delta) of the initial-value problem (18) satisfies ki(t)k - fi(ki 0 k; denotes the truncation of y at t, that is, y t t. If (18) is ISS, then it is forward complete and has the convergent- input, convergent-state property: for each loc (R+ ; R), the unique solution i(\Delta) of the initial-value problem has maximal interval of existence R+ and, if For continuous by P/ the set of system triples \Sigma satisfying the following: Assumption D. (i) g : R \Theta R N ! R N is locally Lipschitz; (ii) system (18) is input-to-state stable (ISS); (iii) h : R N ! R is continuous; (iv) there exist a function loc (R+ ; R)), the (unique) solution i(\Delta) of (18) satisfies Z t/(jy(s)j)jy(s)jds 8 t - 0: While Assumption D is rather restrictive, it is not difficult to identify non-trivial classes of systems for which the assumption holds. Three such examples follow, the first of which is easily seen, the second and third can be verified by arguments invoking [22, Theorem 2]. Examples 3.5. (a) Let / : jyj 7! jyj and suppose (g; h; N ) defines a linear system with If A has spectrum, spec(A), in the open left half complex plane (b) More generally, let locally Lipschitz and h : R N ! R is continuous. In addition, assume g and h are positively homogeneous of degree k. If f0g is an asymptotically stable equilibrium of the unforced system - . For example, with systems (with with unknown real parameters a i ) of the form - are of class P/ , provided that a 1 ! 0. (c) . Assume R \Theta R N ! R N is locally Lipschitz and that h : R N ! R is continuous and positively homogeneous of degree k h . If, in addition, g is positively homogeneous of degree k g , and f0g is an asymptotically stable equilibrium of the unforced system - . For example, systems (with unknown parameters ae, a i 2 R) of the form - linear output are of class P/ , provided that a 1 ! 0 and continuous and positive definite. Let / be continuous. We will show that, for (N OE ; P/ )-universal stabilization, it suffices to AN INTEGRAL INVARIANCE PRINCIPLE AND ADAPTIVE CONTROL 11 replace both occurrences of OE in (9) by OE + /, to yield Let (b; f; . Then there exists constant - such that y). The feedback-controlled system (19-20) can be embedded in a differential inclusion on R N+2 :! R N+2 is given by \Theta fg(y; i)g \Theta f(OE X is upper semicontinuous on R N+2 with non-empty, convex and compact values. Therefore, for each x 0 2 R N+2 , the initial-value problem (21) has a solution and every solution can be maximally extended. Lemma 3.6. Let x 0 2 R N+2 be arbitrary and let be a maximal solution of (21), defined on its maximal interval of existence [0; !). Then exists and is finite; (iii) (y(t); Proof. For almost all t 2 [0; !), we have which, on integration, yields h(i(s))y(s)ds Z -(t) valid for all t; - 2 [0; !), with t - . Invoking Assumption D, we have Z -(t) -: By the same contradiction argument as that employed in the proof of Lemma 3.2, we may deduce that -(\Delta) is bounded. Boundedness of y(\Delta) then follows from (22). That i(\Delta) is bounded is a consequence of the ISS property of \Sigma Therefore, precompact solution of (21) and so may conclude, by boundedness of -(\Delta), that l ffi x 2 L 1 (R+ ). Therefore, by Theorem 2.10 (with approaches the set f(y; In particular, so, by the convergent-input, convergent-state property of the ISS system \Sigma also conclude that i(t) ! 0 as t !1. Finally, by boundedness and monotonicity of -(\Delta), lim t!1 -(t) exists and is finite. Example 3.7. Linear, minimum-phase systems of relative degree one. This class has played a central r-ole in the development of universal adaptive control. In appropriate coordinates, such systems have state space representations of the form In the single-input, single-output case, we identify (23) and (19) with By the relative-degree-one assumption, and, by the minimum-phase assumption, 2 jyj, we see that every single- input, single-output, linear, minimum-phase system of relative degree one is of class and the control (20) reduces to the ubiquitous Byrnes-Willems strategy: Remarks 3.8. In considering the case of dynamically perturbed scalar systems, we treated only the problem of adaptive stabilization. The adaptive servomechanism of Section 3.1.2 can also be modified to incorporate dynamically perturbed sys- tems, when the dynamic perturbations are generated by linear systems - described in Example 3.5(a) above. For such per- turbations, the (modified) servomechanism assures convergence to zero of the tracking error, convergence to a finite limit of the adapting parameter, and boundedness of the evolution t 7! i(t) of the perturbing system. We omit full details here. 3.2. Planar systems. In all applications of the integral invariance principle in Section 3.1 above, the conclusion that x(t) tends, as t !1, to the zero level set l \Gamma1 (0) proved sufficient for our purposes: the additional property that x(\Delta) approaches the largest weakly-invariant subset of l \Gamma1 (0) was redundant. Here, we treat a class of systems for which the latter property can be fruitfully exploited. We consider planar systems (with scalar control u) described by a second-order differential equation:! where the parameters b 2 R, P 2 N and functions d, f , p are unknown. The variable but not its derivative - is available for feedback. We identify (24) with the quintuple (b; d; f; p; P ). For positive-definite, nondecreasing we denote, by N ffi;OE , the set of system quintuples (b; d; f; for which Assumption A (that is, b 6= together with the following three assumptions. Assumption t. Assumption F. (p; y) 7! f(p; y); R P \Theta R! R is continuous and is continuously differentiable in its first argument. Both f and D 1 f (j @f=@p) are OE-bounded uniformly with respect to p in compact sets: precisely, for every compact K ae R P , there exists scalar -K such that jf(p; y)j Assumption G. p(\Delta) 2 W 1;1 (R; R P ). By virtue of Assumptions E and G, without loss of generality t may be assumed in (24): this we will do, without further comment, throughout. In Section 3.2.1 below, we will show that (b; d; f; known continuous positive-definite nondecreasing function OE, is sufficient a priori information for adaptive stabilizability of (24) by feedback of the variable y(t) alone: in essence, Assumption E compensates for the inaccessibility of the velocity variable - y(t) by requiring that the system should exhibit dissipitive dependence (loosely quantifiable by the known constant ffi ) on that variable. Example 3.9. As motivation for (24), consider a single-degree-of-freedom mechanical system with position, but not velocity, available for feedback and with some AN INTEGRAL INVARIANCE PRINCIPLE AND ADAPTIVE CONTROL 13 constant (but unknown) natural damping d quantified by a known parameter ffi in the sense that d ! \Gammaffi ! 0. If we suppose that Assumption F holds with for example, the following particular realizations are admissible. Nonlinear pendulum with disturbed support (disturbance p(\Delta) 2 W 1;1 (R)):- Duffing equation with extraneous disturbance (p(\Delta) 2 W 1;1 (R)):- In the absence of control, such systems are potentially "chaotic". \Sigma 3.2.1. Adaptive stabilizer. Let R+ be a continuous, positive-definite, nondecreasing function. Assuming only that ffi , OE and the instantaneous state y(t) are available for control purposes, our goal is to demonstrate the existence of an adaptive feedback strategy that provides N ffi;OE -universal stabilization in the sense that it assures that the state of (24) approaches f0g for all quintuples (b; d; f; whilst maintaining boundedness of the controller function -(\Delta). Define the continuous function its indefinite fl. We claim that the following (formal) strategy is a N ffi;OE -universal stabilizer:! where - is any continuous function with properties (8). Let (b; d; f; . Introducing the coordinate transformation may express (24) in the form? ? The feedback (25) is interpreted in the set-valued sense:! with the map y 7! oe(y) defined as before in (10). Writing -(t)), the overall adaptive feedback controlled system may be embedded in the following differential inclusion on R 3 14 E. P. RYAN X is upper semicontinuous on R\ThetaR 3 and takes non-empty, convex and compact values in R 3 . Therefore, for each x 0 2 R 3 , the initial-value problem (28) has a solution and every solution can be extended into a maximal solution. Lemma 3.10. Let x 0 2 R 3 be arbitrary and let be a maximal solution of (28) defined on its maximal interval of existence [0; !). Then exists and is finite; (iii) (y(t); Proof. On R, define the locally Lipschitz function \Phi : r 7! \Gamma(jrj), with directional derivative at r in direction s given by fl(jrj)sgn(r)s; r 6= 0 denote the composition \Phi ffi y and let O 1 ae [0; !) be the set (of measure zero) of points t at which the derivative - fails to exist. A straightforward argument (analogous to that yielding (4)) gives By properties of f and p (Assumptions F and G), the function is of class AC([0; !); R). Let O 2 denote the set (of measure zero) of points t at which p(t) fails to exist. Again by properties of f and p, there exists - ? 0 such that parameterized by c ? 0, as follows F By Assumptions F, G and definition of \Gamma, F c is such that, for all c sufficiently large, Moreover, by (30) and (31), be the set of points t at which y(t) and - are not both zero. Observe that (i) every point t of the subset O 0 := is an isolated point implying that O 0 is countable and so has measure zero, and (ii) From these observations together with (29), (30) and writing O := O 0 [O 1 [ O 2 (of measure zero), we deduce that Invoking (33), (34) and Assumption AN INTEGRAL INVARIANCE PRINCIPLE AND ADAPTIVE CONTROL 15 for almost all t 2 [0; !), wherein we have used the fact that with sufficiently large so that \Delta 2 (4ffl) and (32) holds, in which case, V c (t) - 1\Theta cy t)] for all t and which, on integration, yields cy Z j(t) for all t; - 2 [0; !), with t - . We first show that the function j(\Delta) (and hence -(\Delta)) is bounded. By properties (8) of -, there exist increasing sequences (-j n ) n2N and (~j n ) n2N , with - ~ jn !1 as n !1, such that (a) 1 jn jn ~ jn Z ~ jn ~ as n !1. Without loss of generality, we may assume - Seeking a contradic- tion, suppose that j(\Delta) is unbounded. Now, j(\Delta) is bounded from below (in particular, so, by the supposition, j(\Delta) is unbounded from above. Therefore, there exist increasing sequences !) such that, for all n, jn and j( ~ t n combine to yield the contradiction ~ jn Z ~ jn ~ combine to yield the contradiction jn jn Therefore, j(\Delta) (and hence -(\Delta)) is bounded. Boundedness of j(\Delta) and -(\Delta), together with (37), imply boundedness of y(\Delta) and z(\Delta). This establishes assertion (i) and assertion (ii) follows by monotonicity of -(\Delta). It remains to prove assertion (iii). By boundedness of d(\Delta), p(\Delta) and x(\Delta), there exists ae ? 0 such that X 2 (t; x(t)) ae ae - is a precompact solution of the autonomous initial-value problem Moreover, by boundedness of -(\Delta), Z 1- Therefore, by Theorem 2.10, x(\Delta) approaches the largest weakly-invariant (relative to the autonomous differential inclusion (39)) set W in f(y; z; -)j . By definition of weak invariance, the initial-value problem has at least one solution maximal interval of existence R+ and with trajectory in W ae f(y; z; -)j Therefore, we conclude that the largest weakly-invariant set in W is contained in the set f(y; z; -)j and so the solution x(\Delta) approaches the set f(0; 0)g \Theta R. In particular, (y(t); (0; 3.2.2. Adaptive servomechanism. We now turn attention to the servomechanism problem for planar systems (26): that is, the construction of controls that cause the system to track, asymptotically, any reference signal r(\Delta) of some given class, in the sense that both the tracking error its derivative r(t) tend to zero as t ! 1. For the class of reference signals we take the (Sobolev) space R = W 3;1 (R) of functions r 2 (R) with (R) and - equipped with the norm For the servomechanism problem, we restrict the underlying class of systems by imposing a stronger assumption on the function f : Assumption F below should hold for some known, continuous, positive-definite, nondecreasing function OE having the additional property (14). Specifically, for real ffi ? 0 and continuous, positive-definite, nondecreasing function OE : R+ ! R+ with property (14), we denote, by N ffi;OE , the set of quintuples (b; d; f; Assumptions A, E and F following. Assumption F . (p; y) 7! f(p; y); R P \Theta R! R is continuously differentiable. Both f and its gradient function (j (@f=@p; @f=@y)) are OE-bounded uniformly with respect to p in compact sets: precisely, for every compact K ae R P , there exists scalar -K such that jf(p; y)j+kDf(p; y)k -K OE(jyj) for all (p; y) 2 K \ThetaR. Example 3.11. The function OE : jyj 7! 1+jyj 3 has property (14) and the systems described in Example 3.9 are admissible. \Sigma OE be a continuous, positive-definite, nondecreasing function with property (14). We claim that, in order to assure convergence to zero of both the tracking error its derivative - e(t) for all reference signals r 2 R and all quintuples (b; d; f; ffi;OE , it suffices to replace every occurrence of y(t) in (25) by e(t). Proof of this claim follows. Let (b; d; f; ffi;OE . Write ~ define the continuous function ~ By properties of f , ~ f is continuously differentiable with respect to its first argument (~p), with D 1 ~ f given by ~ Let ~ K ae R ~ P be compact and so there exist compact K ae R P and R ? 0 such that ~ K ae K \Theta [\GammaR; R] 3 . By properties of f and OE, there exist constants -K and ae R such AN INTEGRAL INVARIANCE PRINCIPLE AND ADAPTIVE CONTROL 17 that, for all (p; ~ with ~ Therefore, ~ f satisfies Assumption F. P is of class f; ~ . Expressed in terms of the tracking error adopting the coordinate transformation the underlying dynamics have the We are now in precisely the same context, modulo notation, as in the case of an adaptive stabilizer and so, replacing all occurrences of y(t) in (9) by e(t), viz.! then the same argument, as used to establish Lemma 3.10, applies mutatis mutandis to conclude that (41) is an (R; N )-universal servomechanism: for each r(\Delta) 2 R and (b; d; f; ffi;OE , every solution (e(\Delta); z(\Delta); -(\Delta)) of the feedback controlled system has maximal interval of existence R+ with (e(t); moreover, lim t!1 -(t) exists and is finite. 3.2.3. Dynamically perturbed planar systems. Let ffi ? 0 and let the function positive definite and nondecreasing. Let \Sigma (b; d; f; . Here, we consider the case where \Sigma 1 is subject to perturbations generated through its interconnection with a dynamical system \Sigma 2 (as depicted in Figure 1). The system \Sigma 2 is assumed to correspond to a differential equation (driven by the variable y(t) of system \Sigma 1 ) on R N of the form (18) with input y(t), and scalar output w(t) perturbing \Sigma 1 . As before, we identify the system \Sigma 2 with the triple (g; h; N ). Writing the overall system has representation (on R \Theta R \Theta R N We will define, via Assumption H below, a class P/ of admissible systems \Sigma such that the N ffi;OE -universal stabilizer of Section 3.2.1 is readily modified to yield a (N ffi;OE ; P/ )-universal stabilizer. For continuous by P/ the set of system triples \Sigma satisfying the following: Assumption H. (i) g : R \Theta R N ! R N is locally Lipschitz; (ii) system (18) is input-to-state stable there exist a function ff and a scalar ff 1 ? 0, such that, for each (i R N \Theta L 1 loc (R+ ), the (unique) solution i(\Delta) of (18) satisfies Z th 2 (i(s))ds - ff 0 (ki Z t/(jy(s)j)jy(s)jds: Examples 3.12. (a) Assumption H holds for the class of linear systems in Example 3.5(a). (b) More generally, assume g : R \Theta R N ! R N is locally Lipschitz and h : R N is continuous. Assume further that h is positively homogeneous of degree k - 1 and that g is positively homogeneous of degree k f0g is an asymptotically stable equilibrium of the unforced system - then it can be shown (by an argument invoking [22, Theorem 2]) that (g; h; N For example, if systems (with real parameters a i ) of the form - are of class P/ , provided that positive definite and nondecreas- ing. As before, let continous with indefinite integral R -/. We will show that, for (N ffi;OE ; P/ )-universal stabilization, it suffices to replace both occurrences of fl in (25) by fl +/ and to replace the single occurrence of \Gamma by to yield! Let (b; d; f; . The feedback-controlled system (42-43) can be embedded in a differential inclusion on R N+3 :! where the set-valued map (t; x) j (t; N+3 is given by X is upper semicontinuous on R \Theta R N+3 and takes non-empty, convex and compact values in R N+3 . Therefore, for each x 0 2 R N+3 , the initial-value problem (44) has a solution and every solution can be maximally extended. Lemma 3.13. Let x 0 2 R N+3 be arbitrary and let a maximal solution of (44) defined on its maximal interval of existence [0; !). Then exists and is finite; (iii) (y(t); z(t); AN INTEGRAL INVARIANCE PRINCIPLE AND ADAPTIVE CONTROL 19 Proof. Let F c and V c , parameterized by c ? 0, be defined as in the proof of Lemma 3.10. By an argument essentially the same as that adopted in the proof of Lemma 3.10 and choosing c sufficiently large, we arrive at a counterpart to (35): for almost all t 2 [0; !). Invoking the inequality jh(i)jjzj - 1 integrating and invoking Assumption H, we have (for c sufficiently large) Z j(t) for all t; - 2 [0; !), with t - . A straightforward modification of the contradiction argument previously used in the proof of Lemma 3.10, establishes boundedness of j(\Delta) (and hence of -(\Delta)). Boundedness of j(\Delta) and -(\Delta), together with (46), imply boundedness of y(\Delta) and z(\Delta). That i(\Delta) is bounded is a consequence of the ISS property of \Sigma This establishes assertion (i) and assertion (ii) follows by monotonicity of -(\Delta). It remains to prove assertion (iii). With minor modification, the argument used in the proof of Lemma 3.10 applies to conclude that x(\Delta) approaches the set f(y; z; zg. In particular, (y(t); by the convergent-input, convergent-state property of the ISS system \Sigma we may also conclude that i(t) ! 0 as t !1. Example 3.14. Linear Minimum-Phase Systems of Relative Degree Two. Let define a linear, single-input u, single-output y minimum-phase system on R N+2 of relative degree two. Denoting its Markov parameters by m k := CA and the system has a representation (on R 2 \Theta R N ) of the form If we assume that \Gammam 3 (that is, the system exhibits natural damping quantified by known ffi ), then we may identify (47) and (42) by setting By the relative-degree-two assumption, and, by the minimum-phase assumption, 2 jyj, we see that every relative- degree-two, minimum-phase system with m 3 \Gammaffi is of class (N ffi;OE ; P/ ) and we recover (modulo notation) the adaptive stabilizer proposed previously in [5]: Remarks 3.15. We conclude with some observations on the servomechanism problem for dynamically perturbed planar systems. Akin to Remarks 3.8, the adaptive servomechanism of Section 3.2.2 can also be modified to incorporate dynamically perturbed systems, when the dynamic perturbations are generated by linear systems . For such perturbations, the (modified) servomechanism assures convergence to zero of the tracking error and its derivative, convergence to a finite limit of the adapting parameter, and boundedness of the evolution t 7! i(t) of the perturbing system. For brevity, we omit full details here. --R An integral-invariance principle for nonlinear systems Adaptive stabilization of multivariable linear systems Optimization and Nonsmooth Analysis Adaptive control of a class of nonlinearly perturbed linear systems of relative degree two Nonlinear Functional Analysis New York Differential Equations with Discontinuous Righthand Sides The Stability of Dynamical Systems Optimal Control via Nonsmooth Analysis Adaptive stabilization of multivariable linear systems The order of a stabilizing regulator is sufficient a priori information for adaptive stabi- lization New directions in parameter adaptive control Some remarks on a conjecture in parameter adaptive control Adaptive stabilization of nonlinear systems in Control of Uncertain Systems Universal A nonlinear universal servomechanism Universal stabilization of a class of nonlinear systems with homogeneous vector fields Adaptive stabilization of multi-input nonlinear systems Smooth stabilization implies coprime factorization On characterizations of the input-to-state stability property Global adaptive stabilization in the absence of information on the sign of the high frequency gain --TR

differential inclusions;universal servomechanisms;invariance principles;nonlinear systems;adaptive control

278098

Optimal Boundary Control of the Stokes Fluids with Point Velocity Observations.

This paper studies constrained linear-quadratic regulator (LQR) problems in distributed boundary control systems governed by the Stokes equation with point velocity observations. Although the objective function is not well defined, we are able to use hydrostatic potential theory and a variational inequality in a Banach space setting to derive a first-order optimality condition and then a characterization formula of the optimal control. Since matrix-valued singularities appear in the optimal control, a singularity decomposition formula is also established, with which the nature of the singularities is clearly exhibited. It is found that in general, the optimal control is not defined at observation points. A necessary and sufficient condition that the optimal control is defined at observation points is then proved.

Introduction . In this paper, we are concerned with the problems in boundary control of fluid flows. We consider the following constrained optimal boundary control problems in the systems governed by the Stokes equation with point velocity observations. Let\Omega ae R 3 be a bounded domain with smooth boundary \Gamma, \Gamma 1 an open subset of min Z subject to div ~ where ~ w(x) is the velocity vector of the fluid at x 2 \Omega\Gamma p(x) is the pressure of the fluid at x 2 \Omega\Gamma w)(x) is the surface stress of the fluid along \Gamma defined by Received by the editors XX XX, 19XX; accepted by the editors XXXX XX, 19XX. y Department of Mathematics, Texas A&M University, College Station, 77843. Supported in part by NSF Grant DMS-9404380 and by an IRI Award of Texas A&M University. z Department of Aerospace Engineering, Texas A&M University, College Station, Current address: Department of Mathematical Sciences, University of Nevada-Las Vegas, Las Vegas, NV 89154-4020. P.You,Z.Ding and J.Zhou ~n(x) is the unit outnormal vector of \Gamma at x; ~g is a given (surface stress) Neumann boundary data (B.D.) on U is the (surface stress) Neumann boundary control on the surface U is the admissible control set to be defined later for well-posedness of the problem and for applications; are given weighting factors; are prescribed "observation points"; are prescribed "target values" at -, a positive quantity, is the kinematic viscosity of the fluid. For simplicity, throughout this paper we assume that and the density of the fluid is the constant one. Let which is the subspace of the rigid body motions in R 3 . Multiplying the Stokes equation by ~a integration by parts yield the compatibility condition of the Stokes system, i.e., Z or For q - 1, let A be a subspace of (L q (\Gamma)) 3 and denote The Stokes equation (1.1) describes the steady state of an incompressible viscous fluid with low velocity in R 3 . It is a frequently used model in fluid mechanics. It is also an interesting model in linear elastostatics due to its similarities. During the past years, considerable attention has been given to the problem of active control of fluid flows (see [1, 2, 7, 18, 19] and references therein). This interest is motivated by a number of potential applications such as control of separation, combustion, fluid-structure interaction, and super maneuverable aircraft. In the study of those control problems and Navier-Stokes equations, the Stokes equations, which describe the slow steady flow of a viscous fluid, play an important role because of the needs in stability analysis, iterative computation of numerical solutions, boundary control and etc. The theoretical and numerical discussion of the Stokes equations on smooth or Lipschitz domains can be found from [14, 16, 17, 22, 25, 26, 27]. Our objective of this paper is to find the optimal surface stress ~u(x) on \Gamma 1 , which yields a desired velocity distribution ~ w(x), s.t. at observation points the observation values ~ are as close as possible to the target values Z k with a least possible control cost Z which arise from the contemporary fluid control problems in the fluid mechanics. Notice that point observations are assumed in the problem setting, because they are much easier to be realized in applications than distributed observations. They can be used in modeling contemporary "smart sensors". OPTIMAL CONTROL OF STOKES FLUIDS WITH POINT OBSERVATIONS 3 Sensors can be used in boundary control systems (BCS) governed by partial differential equations (PDE) to provide information on the state as a feedback to the systems. According to the space-measure of the data that sensors can detect, sensors can be divided into two types, point sensors and distributed sensors. Point sensors are much more realistic and easier to design than distributed sensors. In contemporary "smart materials", piezoelectric or fiber-optic sensors (called smart sensors) can be embedded to measure deformation, temperature, strain, pressure,.,etc. Each smart sensor detects only the average of the data in between the sensor and its size can be less than in any sense, they should be treated as point sensors. As a matter of fact, so far distributed sensors have not been used in any real applications, to the best of our knowledge. However, once point observations on the boundary are used in a BCS, singularities will appear and very often the system becomes ill-posed. Mathematically and numerically, it becomes very tough to handle. On the other hand, when point observations are used in the problem setting, the state variable has to be continuous, so the regularity of the state variable stronger than the one in the case of distributed observations is required. The fact is that in the literature of related optimal control theory, starting from the classic book [23] by J.L. Lions until recent papers [3],[4] by E. Casas and others, distributed observations are always assumed and the optimal controls are characterized by an adjoint system. The system is then solved numerically by typically a finite-element method, which cannot efficiently tackle the singularity in the optimal control along the boundary. On the other hand, since it is important in the optimal control theory to obtain a state-feedback characterization of the optimal control, with the bound constraints in the system, the Lagrange-Kuhn-Tucher approach is not desirable because theoretically it cannot provide us with a state-feedback characterization of the optimal control which is important in our regularity/singularity analysis of the optimal control and numerically it leads to a numerical algorithm to solve an optimization problem with a huge number of inequality constraints. A refinement of the boundary will double the number of the inequality constraints, so the numerical algorithm will be sensitive to the partition number of the boundary. Since the BCS is governed by a PDE system in R 3 , the partition number of the boundary can be very large, any numerical algorithm sensitive to the partition number of the boundary may fail to carry out numerical computation or provide reliable numerical solutions. Recently in the study of a linear quadratic BCS governed by the Laplace equation with point observations, the potential theory and boundary integral equations (BIE) have been applied in [20],[10],[11], [12] to derive a characterization of the optimal control in terms of the optimal state directly and therefore bypass the adjoint system. This approach shows certain important advantages over others. It provides rather explicit information on the control and the state, and it is amenable to direct numerical computation through a boundary element method (BEM), which can efficiently tackle the singularities in the optimal control along the boundary. In [10],[11],[9] several regularity results are obtained. The optimal control is characterized directly in terms of the optimal state. The exact nature of the singularities in the optimal control is exhibited through a decomposition formula. Based on the characterization formula, numerical algorithms are also developed to approximate the optimal control. Their insensitivity to the discretization of the boundary and fast uniform convergences are mathematically verified in [12],[31]. The case with the Stokes system is much more complicate than the one with the Laplace equation due to the fact that the fundamental solution of the Stokes system 4 P.You,Z.Ding and J.Zhou is matrix-valued and has rougher singular behaviors. In this paper, we assume that the control is active on a part of the surface and the control variable is bounded by two vector-valued functions. A Banach space setting has been used in our approach, we first prove a necessary and sufficient condition for a variational inequality problem which leads to a first order optimality condition of our original optimization problem. A characterization of the optimal control and its singularity decomposition formula are then established. Our approach can be easily adopted to handle other cases and it shows the essence of the characterization of the optimal control, through which gradient related numerical algorithms can be designed to approximate the optimal control. The organization of this paper is as follows: In the rest of Section 1, we introduce some basic definitions and known regularity results that are required in the later development; In Section 2, we first prove an existence theorem for an orthogonal projection, next we derive a characterization result for a variational inequality which serves as a first order optimality condition to our LQR problem; then a state-feedback characterization of the optimal control is established. Section 3 will be devoted to study regularity/singularity of the optimal control. Since the optimal control contains a singular term, we first derive a singularity decomposition formula for the optimal control, with which we find that in general the optimal control is not defined at observation points. A necessary and sufficient condition that the optimal control is defined at observation points is then established. Some other regularities of the optimal control will also be studied in this section. Based upon our characterization formulas a numerical algorithm, in a subsequent paper, we design a Conditioned Gradient Projection Method (CGPM)) to approximated the optimal control. Numerical analysis for its (uniform) convergence and (uniform) convergence rate are presented there. We show that CGPM converges uniformly sub-exponentially, i.e., faster than any integer power of 1 n . Therefore CGPM is insensitive to discretization of the boundary. The insensitivity of our numerical algorithm to discretization of boundary is a significant advantage over other numerical algorithms. Since the fundamental solution of the Stokes system is matrix valued with a very rough singular behavior, numerical analysis is also much more complicated than the case with scalar-valued fundamental solution, e.g., the Laplacian equation. Let us now briefly recall some hydrostatic potential theory, BEM and some known regularity results. Throughout of this paper, for a sequence of elements in R n , we use superscript to denote sequential index and subscript to denote components, e.g., We may also use ~x k to emphasize that x k is a vector. We may write ~ w(x; ~u) to indicate that the velocity ~ w depends also on ~u. Unless stated otherwise, we assume p ? is the Euclidean norm in R n and k \Delta k is the norm in (L h (\Gamma)) n (h - 1). Let fE(x; -); ~e(x; be the fundamental solution of the Stokes systems, i.e. ae div x E(x; is the unit Dirac delta function at I 3 is the 3 \Theta 3 identity matrix. It is known [22] that OPTIMAL CONTROL OF STOKES FLUIDS WITH POINT OBSERVATIONS 5 where ffi i;j is the Kronecker symbol. Remark 1. The significant difference between the case with point observations and the case with distributed observations is as follows: for a given vector ~ the function has a singularity of order O( 1 however it may oscillate between so it is very tough to deal with. Whereas the function Z E(-; x) ~ is well-defined and continuous. On the other hand, if E(P k ; x) in (1.4) and (1.5) is replaced by the fundamental solution of the Laplace equation, in this case, E(P k ; x) becomes scalar-valued, then (1.4) has the same order O( 1 of singularity at but the limit as exists (including \Gamma1 or +1). So the singularity can be easily handled. It is then known that the solution ( ~ w; p) of the Stokes equation (1.1) has a simple- layer representation ~ Z Z for some constants ~a; ~ b 2 R 3 and a 2 R. ~j is called the layer density and ~a represents a rigid body motion. By the jump property of the layer potentials, we obtain the boundary integral equation Z Z where With a given Neumann B.D., the layer density ~j can be solved from the above BIE (1.8). Once the layer density is known, the solution ( ~ w(x); p(x)) can be computed from (1.6) and (1.7). The velocity solution ~ w(x) is unique only up to a rigid body motion and the pressure solution p(x) is unique up to a constant. 6 P.You,Z.Ding and J.Zhou In BEM, the boundary divided into N elements with nodal points Assume that the layer density ~j(x) is piecewise smooth, e.g. piecewise constant, then the BIE (1.8) becomes a linear algebraic system. This system can be solved for ~j(x i ) and then ( ~ w(x); p(x)) can be computed from a discretized version of (1.6) and (1.7) for any x 2 \Omega\Gamma For each ~ we define the simple layer potential of velocity f) by Z E(x; -) ~ For each ~ we define the boundary operators K and K by K( ~ Z Z Z Z where Next we collect some regularity results on S v ; K and K into a lemma. Let which represents the set of all layer densities corresponding to the zero Neumann B.D., with which the Stokes system has only a rigid body motion. Hence we have I Lemma 1.1. Let\Omega ae R 3 be a bounded simply connected domain with smooth boundary \Gamma. (a) (R 3 linear operator for p ? 2 and (b) For any 1 - is a bounded linear operator and K (K ) is the adjoint of K (K); (c) For is a bounded linear (d) For ?M0 is invertible, ?N is invertible. OPTIMAL CONTROL OF STOKES FLUIDS WITH POINT OBSERVATIONS 7 is a bounded linear operator. Therefore K ?N is invertible. Proof. (a)-(d) can be found from [5],[8], [13], [14] and [22]. To prove (e), since \Gamma ae R 3 is a compact set, it suffices to prove (e) for Then we have 1 q \Gamma1. There exists an " 2 (0; 1), s.t. 1 q \Gamma1. are the conjugates of It can be verified that s ff r s ;ff s s r Note and 'Z where M is a constant independent of x 2 \Gamma. Let f)(x). Applying H-older's inequality twice, we get 'Z 'Z 'Z ''Z ff doe - 'Z ''Z r 'Z 'Z Thus 'Z s s 'Z Z s s This proves the first part of (e). The second part follows from (c). To prove (f), by (1.10), Q ij (x; -) is weakly singular for 1 - 3. Thus K is an integral operator with weakly singular kernel. By Theorem 2.22 in [21], K is a compact operator from (C (\Gamma)) 3 to (C (\Gamma)) 3 . The rest follows from the Fredholm alternative (see [21], p.44). 8 P.You,Z.Ding and J.Zhou For a given Neumann B.D. ~g 2 (L p (\Gamma 0 our control bound constraints to the entire boundary \Gamma by ae and ae Bu(x) x with where ~ vector depending on ~g and will be specified later. Define the feasible control set stands for the compatibility condition of the Neumann B.D. in the Stokes system (1.1). It is clear that U is a closed bounded convex set in (L p (\Gamma)) 3 . According to Lemma 1.1 (a), for each given Neumann B.D. ~u 2 U , the Stokes system (1.1) has a solution ~ w in (C(\Omega to a vector ~a ~ where ~ That is, for each given ~u, the velocity state variable ~ w is multiple-valued, so the objective function J(~u) is not well-defined. However among all these velocity solutions, there is a unique solution ~ ~ h2M0 A direct calculation yields that ~ must satisfy Since such a ~ w is unique and continuous, the point observations ~ our LQR problem setting make sense and the LQR problem is well-posed. From (1.14) and Lemma 1.1, we know where C is a constant depending only on \Gamma. Let us observe (1.16). If we notice that ~ is linear in ~u, then we have Lemma 1.2. Let ~a 3 be the unique solution to OPTIMAL CONTROL OF STOKES FLUIDS WITH POINT OBSERVATIONS 9 Then for ~ and where C is a constant depending only on \Gamma. 2. Characterization of the Optimal Control. We establish an optimality condition of the LQR problem through a variational inequality problem (VIP). The characterization of the optimal control is then derived from the optimality condition. In optimal control theory it is important to obtain a state-feedback characterization of the optimal control, i.e., the optimal control is stated as an explicit function of the optimal state. So the optimal control can be determined by a physical measurement of the optimal state. Our efforts are devoted to derive such a result. For each ~ we define the vector-valued truncation function ~ f Bl Let h\Delta; \Deltai be the pairing on ((L q our feasible control set U defined in (1.11) is a convex closed bounded set in (L p (\Gamma)) 3 , it is known that ~u is an optimal control of the LQR problem if For any ff ? 0, (2.1) is equivalent to To derive an optimality condition, we need to find a characterization of a solution to the above variational inequality. Theorem 2.1. For each f 2 (L q (\Gamma)) 3 , u f is a solution to the variational inequal- ity if and only if Bl where z f 2 M 0 such that [f Theorem 2.2 for the existence of such a z f ). Moreover, (2.3) is well-defined in the sense that if z 1 and z 2 are two vectors in then Bl a.e. x 2 \Gamma: (2. P.You,Z.Ding and J.Zhou Proof. By Theorem 2.2, there exists z Bl . We have for each u 2 U , Z on doe x where the last inequality holds since each integrand, the product of two terms, is nonnegative. Next we assume that u f is a solution to the VIP, i.e., Take Bl , which is in U , we obtain By the first part, we have Taking Combining (2.5) with (2.7) gives us Thus The proof of the second part of the theorem follows directly from taking z Bl in (2.8). In a Hilbert space setting, the above theorem is called a characterization of pro- jection. When U is a convex closed subset of a Hilbert space H , for each f 2 H , u f is a solution to the VIP if and only if i.e., u f is the projection of f on U . This characterization is used to derive a first order optimality condition for convex inequality constrained optimal control prob- lems. However, this result is not valid in general Banach spaces. Instead we prove a characterization of truncation, which is a special case of a projection. Note that in a Hilbert space setting, a projection maps a point in the space into a subset of the same space. However our truncation is a projection that maps a point in (L q (\Gamma)) 3 into a subset of (L p 1). It crosses spaces. This characterization gives a connection between the truncation and the solution to VIP, in our case, an optimality OPTIMAL CONTROL OF STOKES FLUIDS WITH POINT OBSERVATIONS 11 condition in terms of the gradient. That is, by our characterization of truncation, ~u 2 U is a solution to the VIP (2.2) if and only if where ~z 2 M 0 is defined in Theorem 2.2 s.t. To prove the existence of a rigid body motion z f in (2.3), we establish the following existence theorem for an orthogonal projection, which is given in a very general case and plays a key role in establishing the optimality condition. It can be used to solve LQR problems governed by PDE's, e.g., the Laplacian, the Stokes, the linear elastostatics, .,etc. where the PDE has multiple solutions for a given a Neumann type boundary data satisfying certain orthogonality condition. Theorem 2.2. Let \Gamma be a bounded closed set in R n and \Gamma be a subset s.t. be given s.t. ~ where ~ is given by (2.17) and ~ Assume that M 0 is an m-dimensional subspace in (L q (\Gamma)) n (q - 2; 1 and g, then a necessary and sufficient condition that for each ~ Bl is that Moreover the set of all solutions ~z f in (2.10) is locally uniformly bounded in the sense that for each given ~ exist with k ~ Bl we have Proof. Case 1: dim be an orthonormal basis in M 1 (in M 0 as well). To prove the first part of the theorem, we have to show that for each ~ ~ Bl 12 P.You,Z.Ding and J.Zhou For each ~ by ~ Bl Then to prove the first part, it suffices to show that for each ~ exists It is easy to check that for any ~ exist two constants depending only on \Gamma and the basis y s.t. is a bounded (depends on Bl and Bu) Lipschitz continuous map. To show that T f has a zero, we prove that there exists a constant R ? 0 s.t. when Once (2.15) is verified, we have By Altman's fixed point theorem [15], the map C has a fixed point (BR is the ball of radius R at the origin), i.e., it remains to verify (2.15). Define It suffices to show that there exists R ? 0 s.t. for t ? R, In the following, we prove that for each given ~ exist have So the second part of the theorem also follows. For each C 2 D, we denote ~y C OPTIMAL CONTROL OF STOKES FLUIDS WITH POINT OBSERVATIONS 13 It is obvious that Z j~y C (x)jdoe x is continuous in C and positive on the compact set D, hence C2D f Z and we set R For any given " ? 0, we assume For each C 2 D; t ? 0, Bl Z Bl Z ~ Bl Z I C Z where for I C Z Let We have lim I C Z Z Z jy C Thus lim Z jy C Z Z jy C (x)jdoe x Z 14 P.You,Z.Ding and J.Zhou where m y given by (2.16) is independent of C. From (2.14), we see that T f (C) \Delta C is continuous in both ~ f and C, therefore there exist R C Since D is compact, there exist C Let So we only need to take ~ and ~ Bu; a.e. on Case 2: be an orthonormal basis in M 0 , where (~y By the proof in Case 1, for each ~ s.t. Bl Then for any c f by (2.18), we have ~ Bl ~ Bl On the other hand, when by (2.18), we have Bl Therefore ~ Bl OPTIMAL CONTROL OF STOKES FLUIDS WITH POINT OBSERVATIONS 15 if and only if i.e., (2.11) is satisfied. The proof is complete. Remark 2. In the above theorem, (1) when rigid body motion is considered, we have dim(M all the conditions in the theorem are satis- fied. So for each ~ there is ~a f such that Bl (2) if the conclusion still holds for each ~ an m-dimensional subspace of (L q (\Gamma)) n where q - 1, 1 (3) the vector C in (2.13) represents the rigid body motion in our case. From the above theorem, we can see that the solution C f such that T f (C f unique. The following error estimate contains an uniqueness result, which will also be used in proving the uniform convergence rate in a subsequent paper. Theorem 2.3. Let us maintain all the assumptions in Theorem 2.2. Let ~ given in (L 1 respectively two zeros of T f and T h defined by (2.13). If where meas c h then where the constant fl is independent of C f and C h . P.You,Z.Ding and J.Zhou Proof. We may assume that For T f (C), we denote where y C Write Since T f (C) is Lipschitz continuous in C, a direct calculation leads to the Frechet derivative hy k m\Thetam a.e. C a Gram-matrix, which is symmetric positive semi-definite, i.e., for any nonzero vector (b where "?" holds strictly if because f~y is linearly independent. On the other hand, we have hy k m\Thetam hy k m\Thetam hy k m\Thetam where the Gram-matrix hy k m\Thetam is also symmetric positive semi-definite. Therefore where "!" holds strictly in the first inequality if meas (\Gamma C holds strictly in the second inequality if meas and two respectively, we let OPTIMAL CONTROL OF STOKES FLUIDS WITH POINT OBSERVATIONS 17 Since T f (C) is Lipschitz continuous in C, once meas (\Gamma C f It follows that T 0 positive definite matrix with Therefore defines a symmetric positive definite matrix with For any 0 ! - ! 1, we have for some into account, we arrive at Consequently, we have and the proof is complete. As a direct consequence of Theorem 2.3, we obtain the following uniqueness result. Corollary 2.4. Let us maintain all the assumptions in Theorem 2.2. For given ~ is a zero of T f with then C f is the unique zero of T f . Now we present a state-feedback characterization of the optimal control. Theorem 2.5. Let\Omega ae R 3 be a bounded domain with smooth boundary \Gamma. The LQR problem has a unique optimal control ~u 2 U and a unique optimal velocity state ~ P.You,Z.Ding and J.Zhou and Bl \Theta ~x is defined in Theorem 2.2 s.t. ~u ? M 0 and M 0 is given in (1.2). Proof. Let ?M0 . Since our objective function J(~u) is strictly convex and differentiable, and the feasible control set U is a closed bounded convex subset in the reflexive Banach space X , the existence and uniqueness of the optimal control are well-established. Equation (2.20) is just a copy of (1.16). By our characterization of truncation, Theorem 2.1 with Bl is defined in Theorem 2.2 s.t. Bl To prove (2.21), we only need to show Applying (1.9), i.e., M and then Since rJ(~u) defines a bounded linear functional on X , for any ~ h 2 X , take (1.12) into account, we have Z I +K ) Z So (2.22) is verified and the proof is complete. OPTIMAL CONTROL OF STOKES FLUIDS WITH POINT OBSERVATIONS 19 3. Regularities of the Optimal Control. It is clear that (2.21) is a feedback characterization of the optimal control. To obtain such a characterization, in (2.9) is crucial. Later on we will see that is also crucial in proving the uniform convergence of our numerical algorithms in a subsequent paper. Observe that when corresponds to the LQR problem without constraints on the control variable, the optimal solution, if it exists, becomes \Theta ~x is defined in Theorem 2.2 s.t. ~u ? M 0 (see Remark 2). But according to Lemma 1.1(d) such a solution ~u is only in (L q (\Gamma)) 3 (q ! 2), since E(P k ; \Delta) is only in (L q (\Gamma)) 3 . So it is reasonable to apply bound constraints Bl and Bu on the control variable ~u. However we notice that the optimal control still contains a singular term which is not computable at . In order to carry out the truncation by Bl and Bu, we have to know the sign of this singular term. Hence we derive a singularity decomposition formula of (2.21), in which the singular term is expressed as continuous bounded terms plus a simple dominant singular term and a lower order singular term. With the simple dominant singular term, the nature of the singularity is clearly exposed. Theorem 3.1. For the optimal control ~u given in (2.21), let ~ f Then f where in the singular part, the second term 4K ~ f (x) is dominated by the first term f (x) whose nature of singularity can be determined at each P k and the regular term f (x) is continuous on \Gamma. Proof. For given ~g 2 (L q (\Gamma)) 3 ?N with q I +K) I \Theta ~x: Let ~ f By (2.23), ~ ?N for every q ! 2, thus (3.1) follows. The first part of Lemma 1.1 (e) states that the singularity in 2 ~ f dominates the one in 4K ~ f . While 20 P.You,Z.Ding and J.Zhou the second part of Lemma 1.1 (e) and (f) imply that ( 1I +K) \Gamma1 ffiK ffiK ~ f is continuous. The above singularity decomposition formula plays an important role in our singularity analysis and also in our numerical computation. It is used to prove the uniform convergence and to estimate the uniform convergence rate of our numerical algorithms in a subsequent paper. Note that the fundamental velocity solution E(-; is not defined when in the sense that when x some of the entries may oscillate between \Gamma1 and +1. So if we look at the simple dominant singular term in the singularity decomposition formula of the optimal control, we can see that in general, the optimal control ~u (x) is not defined at P k even with the truncation by Bl and Bu. This is a significant difference between systems with scalar valued fundamental solution and with matrix-valued fundamental solution. For the formal case, e.g., the Laplacian, the optimal control is continuous at every point where Bl and Bu are continuous. Of course, if Bl(P k which means the control is not active at P k , then trivially ~u prescribed value. This is the case when a sensor is placed at P k , then a control device can not be put at the same point P k . However, in general point observation case, the control may still be active at P k . The above analysis then states that the optimal control is not defined at P k unless some other conditions are posed. This is the nature of point observations. Notice that a distributed parameter control is assumed in our problem setting, theoretically the values of the control variable at finite points will not affect the system. But, in numerical computation we can only evaluate the optimal control ~u at finite number of points. The observation points P k 's usually are of the most interest. On the other hand, the optimal velocity state ~ w is well-defined and continuous at P k , no matter ~u (P k ) is defined or not. So if one does want the optimal control ~u to be defined at P k , when Bl(P k m, it is clear that is defined at each P k . When Bl(P m, then we have the following necessary and sufficient condition. Theorem 3.2. Let Bl(P m, then the optimal control ~u is well-defined at the observation points P k if and only if where for each fixed k and i, the equality holds for at most one j 6= i unless ~ When ~u is well-defined at P k , we have ae Bl i Proof. If we observe the fundamental velocity solution, we can see that the proof follows from the following argument. For lim OPTIMAL CONTROL OF STOKES FLUIDS WITH POINT OBSERVATIONS 21 exists (including \Sigma1) if and only if where at most one equality can hold unless c that when (3.5) holds, and two equalities hold in (3.5), then We can make the limit either equal to zero by taking x to sign (c i )1 by taking x i 6= \Upsilonx j or x i 6= \Upsilonx k and x ! 0. So the limit will not exist. When lim x!0 - e i (x) exists and lim With the above result and the singularity decomposition formula for the optimal control, the following continuous result can be easily verified. Theorem 3.3. Let Bu and Bl be continuous on \Gamma 1 . If for each either or the condition (3.3) holds strictly with ( ~ 0, then the optimal control ~u is continuous on \Gamma 1 . So the equality in (2.4) holds for every point on \Gamma. From the state-feedback characterization (2.20), the control can be determined by a physical measurement of the state at finite number of observation points m. The question is then asked, will a small error in the measurement of the state cause a large deviation in the control? Due to the appearance of the singular term in (2.20), in general the answer is yes, i.e., the state-feedback system is not stable. However under certain conditions, we can prove that the state-feedback system is uniformly stable. Theorem 3.4. Let ~ be the exact velocity state at observation points and ~u p be the control determined from (2.20) in terms of ~ w(P k ). If for each either Bl and Bu are continuous and equal at P k or Bu and Bl are locally bounded at P k , the condition (3.3) holds strictly with ( ~ then the state-feedback system (2.20) is uniformly stable in the sense that for any " ? 0, there is such that for any measurement ~ where ~u 0 is the control determined from (2.20) in terms of ~ Proof. For each " ? 0. For each fixed and Bu are continuous and equal at P k , there is d 0 Since the control variable is bounded by Bl and Bu, If instead the condition (3.3) holds strictly with ( ~ chosen so that when j ~ holds strictly with Due to the singular term in (2.20) and since Bu and Bl are 22 P.You,Z.Ding and J.Zhou locally bounded at P k , there is d k ? 0 such that when x some m, we have \Gammafl either ? Bu(x) i or After the truncation by Bu and Bl, it follows that So if we define then in either case we have Denote ~ I +K) \Gamma1 ~ and meas implies that there is nothing to prove. So we assume that meas (\Gamma CF Theorem 2.3 can be applied. For x compact set, by using (2.20) and triangle inequality, we obtain Bl Bl OPTIMAL CONTROL OF STOKES FLUIDS WITH POINT OBSERVATIONS 23 Since the operator linear and bounded, and the function E(P k ; \Delta) is continuous and bounded on the compact set As for I 2 (x), Theorem 2.3 yields where the constant fl depends only on \Gamma. Since there is constant C 0 independent of ~ there is Finally for The proof is complete. As a final comment, it is worth while indicating that though in the problem setting, the governing differential equation, the Stokes, is linear, the bound constraint on the control variable introduces a nontrivial nonlinearity into the system. This can be clearly seen in Theorem 2.2. Also our approach can be adopted to deal with certain nonlinear boundary control problems. --R Structural actuator control of fluid/structure interactions Feedback control of the driven cavity problem using LQR designs Control of an elliptic problem with pointwise state constraints Boundary control of semilinear elliptic equations with pointwise state constraints Lectures on Singular Integral Operators L'int'egral de Cauchy definit un operateur born'e sur L 2 pour les curbs Lipschitziennes "New Developments in Differential Equations" Boundary value problems for the systems of elastostatics in Lipschitz domains Topics on Potential Theory on Lipschitz Domains and Boundary Control Problems Constrained LQR problems in elliptic distributed control systems with point observations Constrained LQR problems governed by the potential equation on Lipschitz domain with point observations Constrained LQR problems in elliptic distributed control systems with point observations - convergence results The Dirichlet problems for the Stokes system on Lipschitz domains Fixed Point Theory Finite Element Methods for Finite Element Methods for Viscous Incompressible Flows: A Guide to Theory Boundary velocity control of incompressible flow with an application to viscous drag reduction A dissipative feedback control synthesis for systems arising in fluid dynamics The Mathematical Theory of Viscous Incompressible Flow Analysis IV: Linear and Boundary Integral Equations Layer potentials and boundary value problems for Laplace's equation on Lipschitz domains A fiber-optic combustion pressure sensor system for automotive engine control "Constrained LQR problems in elliptic distributed control systems with point observations- on convergence rates" --TR --CTR Zhonghai Ding, Optimal Boundary Controls of Linear Elastostatics on Lipschitz Domains with Point Observations, Journal of Dynamical and Control Systems, v.7 n.2, p.181-207, April 2001

hydrostatic potential;boundary integral equation;stokes fluid;singularity decomposition;distributed boundary control;point observation;LQR

278124

Lipschitzian Stability for State Constrained Nonlinear Optimal Control.

For a nonlinear optimal control problem with state constraints, we give conditions under which the optimal control depends Lipschitz continuously in the L2 norm on a parameter. These conditions involve smoothness of the problem data, uniform independence of active constraint gradients, and a coercivity condition for the integral functional. Under these same conditions, we obtain a new nonoptimal stability result for the optimal control in the $L^\infty$ norm. And under an additional assumption concerning the regularity of the state constraints, a new tight $L^\infty$ estimate is obtained. Our approach is based on an abstract implicit function theorem in nonlinear spaces.

Introduction We consider the following optimal control problem involving a parameter: minimize subject to where the state x(t) 2 R dt x, the control u(t) 2 R m , the parameter p lies in a metric space, the functions h Throughout the paper, L ff (J denotes the usual Lebesgue space of functions equipped with its standard norm Z is the Euclidean norm. Of course, corresponds to the space of essentially bounded functions. Let W m;ff (J ; R n ) be the usual Sobolev space consisting of vector-valued functions whose j-th derivative lies in L ff for all its norm is When either the domain J or the range R n is clear from context, it is omitted. We let denote the space W m;2 , and Lip denote W 1;1 , the space of Lipschitz continuous functions. Subscripts on spaces are used to indicate bounds on norms; in particular, - denotes the set of functions in W m;ff with the property that the L ff norm of the m-th derivative is bounded by -, and Lip - denotes the space of Lipschitz continuous functions with Lipschitz constant -. Throughout, c is a generic constant, independent of the parameter p and time t, and B a (x) is the closed ball centered at x with radius a. The L 2 inner product is denoted h\Delta; \Deltai, the complement of a set A is A c , and the transpose of a matrix B is B T . Given a vector y yA denotes the subvector consisting of components associated with indices in A. And then YA is the submatrix consisting of rows associated with indices in A. We wish to study how a solution to either (1), or to the associated variational system representing the first-order necessary condition, depends on the parameter p. We assume that the problem (1) has a local minimizer (x; corresponding to a reference value of the parameter, and the following smoothness condition holds: Smoothness. The local minimizer (x ; u ) of (1) lies in W 2;1 \Theta Lip. There exists a closed set \Delta ae R n \Theta R m and a The function values and first two derivatives of f p (x; u), g p (x; u), and h p (x; u), and the third derivatives of g p (x), with respect to x and u, are uniformly continuous relative to p near p and (x; u) 2 \Delta. And when either the first two derivatives of f p (x; u) and or the first three derivatives of g p (x), with respect to x and u, are evaluated at resulting expression is differentiable in t and the L 1 norm of the time derivative is uniformly bounded relative to p near p . Let A, B, and K be the matrices defined by Here and elsewhere the * subscript is always associated with p . Let A(t) be the set of indices of the active constraints at (x (t); p ); that is, We introduce the following assumption: Uniform Independence at A. The set A(0) is empty and there exists a scalar ff ? 0 such that for each t 2 [0; 1] where A(t) 6= ; and for each choice of v. Uniform Independence implies that the state constraints are first-order (see [12] for the definition of the order of a state constraint). This condition can be generalized to higher order state constraints (see Maurer [17]), however, the generalization of the stability results in this paper to higher order state constraints is not immediate. It is known (see, for instance, Theorem 7.1 of the recent survey [12] and the regularity analysis in [8]), that under appropriate assumptions, the first-order necessary conditions (Pontryagin's minimum principle) associated with a solution (x ; u ) of (1) can be written in the following way: There exist / 2 W 2;1 and - 2 Lip such that are a solution at of the variational system: Here H p is the Hamiltonian defined by and the set-valued map N is defined in the following way: Given a nondecreasing Lipschitz continuous function -, a continuous function y lies in N(-) if and only if Defining where w be the quadratic form and let L be the linear and continuous operator from H 1 \Theta L 2 to L 2 defined by We introduce the following growth assumption for the quadratic form: Coercivity. There exists a constant ff ? 0 such that where In the terminology of [12], the form of the minimum principle we employ is the "indirect adjoining approach with continuous adjoint function." A different approach, found in [13] for example, involves a different choice for the multipliers and for the Hamiltonian. The multipliers in these two approaches are related in a linear fashion as shown in [11]. Normally, the multiplier -, associated with the state constraint, and the derivative of / have bounded variation. In our statement of the minimum principle above, we are implicitly assuming some additional regularity so that - and / are not only of bounded variation, but Lipschitz continuous. This regularity can be proved under the Uniform Independence and Coercivity conditions (see [8]). In Section 3 we establish the following result: Theorem 1.1. Suppose that the problem (1) with has a local minimizer and that the Smoothness and the Uniform Independence conditions hold. Let / and - be the associated multipliers satisfying the variational system (2)-(5). If the Coercivity condition holds, then there exist a constant - and neighborhoods V of p and U of w such that for every is a unique solution U to the first-order necessary conditions (2)-(5) with the property that ( - u) is a local minimizer of the problem (1) associated with p. Moreover, for every is the corresponding solution of (2)-(5), the following estimate holds: where (w kr (w In addition, we have The proof of Theorem 1.1 is based on an abstract implicit function theorem appearing in Section 2. In Section 4 we show that the L 1 estimate of Theorem 1.1 can be sharpened if the points where the state constraints change between active and inactive are separated. In Section 5 we comment briefly on related work. 2. An implicit function theorem in nonlinear spaces The following lemma provides a generalization of the implicit function theorem that can be applied to nonlinear spaces. To simplify the notation, we let denote the distance between the elements x and y of the metric space X. Lemma 2.1. Let X and \Pi be metric spaces with X complete, let Y be a subset of \Pi, and let P be a set. Given w 2 X and r ? 0, let W denote the ball B r (w ) in X and suppose that T (the subsets of \Pi) have the following properties: restricted to Y is single-valued and Lipschitz continuous, with Lipschitz constant -. there exists a unique w 2 W such that T (w; p) 2 F (w). Moreover, for every denotes the w associated with p i , then we have Proof. Fix . Observe that for each w a contraction on W with contraction constant -ffl. Let w 2 W . Since w Thus \Phi maps W into itself. By the Banach contraction mapping principle, there exists a unique w 2 W such that is equivalent to we conclude that for each there is a unique have Rearranging this inequality, the proof is complete. Let X, Y , and P be metric spaces and let w 2 X. Using the terminology of [3], strictly stationary at uniformly in p near p , if for each with the property that for all w Theorem 2.2. Let X be a complete metric space, let \Pi be a linear metric space, let Y be a subset of \Pi, and let P be a metric space. Suppose that F : continuous, and that for some w 2 X and is continuous at p . strictly stationary at uniformly in p near p . restricted to Y is single-valued and Lipschitz continuous, with Lipschitz constant -. maps a neighborhood of (w ; p ) into Y . Then for each -, there exist neighborhoods W of w and P of p such that for each moreover, for every denotes the w 2 W associated with p i , then we have Proof. By (Q5) there exist neighborhoods U 0 of w and P 0 of p such that We apply Lemma 2.1 with the following identifications: X, Y , and \Pi are as defined in the statement of the follow immediately from (Q1) and (Q4), respectively. Choose ffl ? 0 such that ffl ! that for this choice of ffl, we have ffl- and By (Q3) and the identity T (w exist neighborhoods of w such that (P3) of Lemma 2.1 holds. Let fi satisfy -fi=(1 \Gamma ffl- r and by (Q2), choose P smaller if necessary so that (P2) holds. By Lemma 2.1, for each exists a unique w 2 W such that T (w; p) 2 F (w), and the estimate (8) holds. Since T (w; p) 2 F (w) if and only if T (w; p) 2 F(w), the proof is complete. A particular case of Theorem 2.2 corresponds to the well-known Robinson implicit function theorem [20] in which X is a Banach space, Y is its dual X , N\Omega (w), \Omega is a closed, convex set in X, N\Omega (w) is the normal cone to the set\Omega at the point differentiable with respect to w, both T and its derivative rwT are continuous in a neighborhood of (w ; p ), and is the linearization of T . The Robinson framework is applicable to control problems with control constraints after the range space X is replaced by a general Banach space Y (see the discussion in Section 5). However, for problems with state con- straints, there are difficulties in applying Robinson's theory since stability results for state constrained quadratic problems, analogous to the results for control constrained problems, have not been established. In our previous paper [3], we extend Robinson's work in several different directions. For the solution map of a generalized equation in a linear metric space, we showed that Aubin's pseudo-Lipschitz property, that the existence of a Lipschitzian selection, and that local Lipschitzian invertibility are "robust" under nonlinear perturbations that are strictly stationary at the reference point. In Theorem 2.2, we focus on the latter property, giving an extension of our earlier result to nonlinear spaces. In this nonlinear setting, we are able to analyze the state constrained problem, obtaining a Lipschitzian stability result for the solution. 3. Lipschitzian stability in L 2 To prove Theorem 1.1, we apply Theorem 2.2 using the following identifications. First, we define where - (with the H 1 norm), (with the L 2 norm), -(1) - 0 and - An appropriate value for - is chosen later in the analysis. The space X consists of the collection of functions x, /, u, and - satisfying (10) and (11) with the norm defined in (10) and (11). Observe that the norms we use are not the natural norms. For example, the u and - components of elements in X lie in W 1;1 , but we use the L 2 norm to measure distance. Despite the apparent mismatch of space and norm, X is complete by Lemma 3.2 below. The functions T and F of Theorem 2.2 are selected in the following way: r The continuous operator L is obtained by linearizing the map T (\Delta; p ) in L 1 at the reference point w In particular, denote the components of - : a r The space \Pi is the product L 2 \Theta L 2 \Theta L 2 \Theta H 1 while the elements - in Y have the (with the L 2 norm), b 2 W 2;1 (with the H 1 norm), ks where - is a small positive constant chosen so that two related quadratic programs, (37) and (41), introduced later have the same solution. As we will see, the constant - associated with the space X must be chosen sufficiently large relative to -. Note that the inverse is the solution (x; /; u; -) of the linear variational system: Referring to the assumptions of Theorem 2.2, (Q1) holds by the definition of X and by the minimum principle, (Q2) follows immediately from the Smoothness condition. In Lemma 3.3, we deduce (Q3) from the Smoothness condition and a Taylor expansion. In Lemma 3.6, (Q5) is obtained by showing that for w near w and p near its associated derivatives are near those of - L(w ). Finally, in a series of lemmas, (Q4) is established through manipulations of quadratic programs associated with (15)-(18). To start the analysis, we show that X is complete using the following lemma: Lemma 3.1. If u 2 Lip - ([0; 1]; R 1 ), then we have Proof. Since u is continuous, its maximum absolute value is achieved at some on the interval [0; 1]. Let um the associated value of u. We consider two cases. Case 1: um ? -. Let us examine the maximum ratio between 1-norm and the maximize fkukL 1=kuk um ? -, the maximum is attained by the linear function v satisfying um and - \Gamma-. The 2-norm of this function is readily evaluated: this interval, we have kvk 2 Taking square roots gives which establishes the lemma in Case 1. Case 2: um -. In this case, let us examine the maximum ratio between 1-norm and the 2-norm to the 2/3-power: maximize fkukL 1=kuk The maximum is attained by the piecewise linear function v satisfying it follows that which completes the proof of case 2. Lemma 3.2. The space X of functions w satisfying (9), (10), and (11), is complete. Proof. Suppose that w sequence in X. We analyze the -component of w k . The sequence - k is a Cauchy sequence in L 1 by Lemma 3.1. Since L 1 is complete, there exists a limit point - 2 L 1 . Since the - k converge pointwise to - - and since each of the - k is Lipschitz continuous with Lipschitz constant - is Lipschitz continuous with Lipschitz constant -. Since each of the - k is non- decreasing, it follows from the pointwise convergence that - is nondecreasing; hence, for each k, the pointwise convergence implies that - This shows that the -component of X is complete. The other components can be analyzed in a similar fashion. Lemma 3.3. If the Smoothness condition holds, then for T and L defined in (12) and strictly stationary at w , uniformly in p near p . Proof. Only the first component of T (w; p) \Gamma L(w) is analyzed since the other components are treated in a similar manner. To establish strict stationarity for the first component, we need to show that for any given ffl ? 0, for p near p and for (x; u) and (y; v) 2 W 2;1 - \Theta Lip - near (x ; u ) in the norm of (y; v) are also near (x ; u ) in L 1 . After writing the difference f p (x; an integral over the line segment connecting (x; u) and (y; v), we have where is the average of the gradient of f p along the line segment connecting (x; u) and (y; v). By the Smoothness condition, as p approaches p and as both (x; u) and (y; v) approach (x ; u ) in L 1 . This completes the proof. Lemma 3.4. If the Smoothness condition holds, then for T and L defined in (12) and (13) respectively, and for any choice of the parameter - ? 0 in (14), there exists Proof. Again, we focus on the first component of T \Gamma L since the other components are treated in a similar manner. Referring to the definition of Y , we should show that for p near p and for (x; u) 2 W 2;1 - \Theta Lip - near (x ; u ) in the norm of H 1 \Theta L 2 . The W 1;1 norm in (20) is composed of two norms, the L 1 norm of the function values, and the L 1 norm of the time derivative. By the same expansion used in Lemma 3.3, we obtain the bound for p near p and for (x; u) near Differentiating the expression within the norm of (20) gives d By the Smoothness condition, - A and - lie in L 1 , and by the definition of X, we have By the triangle inequality and by Lemma 3.1, for x near x . Moreover, by Lemma 3.1 and by the Smoothness condition, r x f p (x; u) approaches A and r u f p (x; u) approaches B in L 1 as p approaches p and (x; u) approaches dt Analyzing each of the components of T \Gamma L in this same way, the proof is complete. We now begin a series of lemmas aimed at verifying (Q4). After a technical result (Lemma 3.5) related to the constraints, a surjectivity property (Lemma 3.6) is established for the linearized constraint mapping. Then we study a quadratic program corresponding to the linear variational system (15)-(18). We show that the solution (Lemma 3.9) and the multipliers (Lemma 3.10) depend Lipschitz continuously on the parameters. And utilizing the solution regularity derived in [8], the solution and the multipliers lie in X for - sufficiently large. To begin, let I be any map from [0; 1] to the subsets of f1; 2; with the property that the following sets I i are closed for every i: I We establish the following decomposition property for the interval [0; 1]: Lemma 3.5. If Uniform Independence at I holds, then for every ff there exists sets J 1 , J 2 , \Delta \Delta \Delta, J l , corresponding points a positive constant ae ! min i (- such that for each t 2 [- we have I(t) ae J i , and if J i is nonempty, then for every choice of v. The set J 1 can always be chosen empty. Proof. For each t 2 (0; 1) with I(t) c 6= ;, there exists an open interval O centered at t with O ae " i2I(t) cI c then we can choose a half-open interval O, with t the closed end of the interval, such that O ae " i2I(t) cI c . If I(t) c is empty, take fixed t 2 [0; 1] with I(t) 6= ;, choose O smaller if necessary so that for each s 2 O and for each choice of v. Since B and K are continuous, it is possible to choose O in this way. Observe that by the construction of O, we have I(s) ae I(t) for each s 2 O and (22) holds if I(t) is nonempty. Given any interval O on (0; 1), let O 1=2 denote the open interval with the same center, but with half the length; for the open intervals associated with denote the half-open interval with the same endpoint, 0 or 1, but with half the length. The sets O 1=2 form an open cover of [0; 1]. Let O 1 , O 2 , \Delta \Delta \Delta, O l be a finite subcover of [0; 1] and let t 1 , the associated centers of interior intervals, and the closed endpoint of the intervals associated with or 1. It can be arranged so that no O i is contained in the union of other elements of the subcover (by discarding these extra sets if necessary). Arrange the indices of the O i so that the left side of O i is to the left of the left side of O i+1 for each i. Let - 1 denote the successive left sides of the O i , and let ae be 1/4 of the length of the smallest O i . Defining J from the construction of the O i that I(t) ae J i and (22) holds for each t in an interval associated with t i and with length twice that of O i . Since (- By taking ae smaller if necessary, we can enforce the condition ae ! min i (- Lemma 3.6. If Uniform Independence at I holds, then for each a 2 L 1 and there exist x 2 W 1;1 and u 2 L 1 such that L(x; u) and This (x; u) pair is an affine function of (a; b), and for each ff - 1, there exists a constant c ? 0 such that for every (a is the pair associated with Proof. We use the decomposition provided by Lemma 3.5 to enforce the equations holds trivially on [- that i ? 1, and let us consider (23) on the interval [- we conclude that any j 2 I(t) is contained in either J then by (27), (23) holds. If j 2 J i n J then by the construction of the implies that (23) holds. Suppose that j 2 J i and let oe j be any given Lipschitz continuous function. Observe that if d dt then K Carrying out the differentiation in the second relation of (28) and substituting for - x using the state equation (25), we obtain a linear equation for u. By Lemma 3.5, this equation has a solution, and for fixed t and x, the minimum norm solution can be written: where In the special case where J i is empty, we simply set u(t; These observations show how to construct x and u in order to satisfy (26) and (27). On the initial interval [0; - 2 ], u is simply 0 and x is obtained from (25). Assuming x and u have been determined on the interval [0; - i ], their values on [- are obtained in the following way: The control is given in feedback form by (29), where For is linear on [- With this choice for oe, the first equation in (28) is satisfied, and with x and u given by (25) and (29) respectively, the second equation in (28) is satisfied. Also, by the choice of oe, for each Hence, (26) and (27) hold, which yields (23). For it follows from the definition of oe that When u in (29) is inserted in (25) and this bound on j - oe j (t)j is taken into account, we obtain by induction that x 2 W 1;1 and u 2 L 1 . By the equations (25) for the state, (29) for the control, and (31)-(32) for oe, (x; u) is an affine function of (a; b). Moreover, the change (ffix; ffiu) in the state and control associated with the change (ffia; ffib) in the parameters satisfies: for each i where oe is specified in (31)-(32). To complete the proof, we need to relate the oe term of (33) to the b term of (24). For Consequently, for almost every t 2 [- us proceed by induction and assume that Combining this with (34) and (33) for Since jffix(- j+1 )j - kffixk W 1;ff [0;- j+1 ] , the induction step is complete. In the following lemma, we prove a pointwise coercivity result for the quadratic form B. See [4] and [7] for more general results of this nature. Lemma 3.7. If Coercivity holds, then there exists a scalar ff ? 0 such that xi] for all (x; u) 2 M; (35) and Proof. If Hence, the L 2 norm of x and - x are bounded in terms of the L 2 norm of u, and (35) follows directly from the coercivity condition. To establish (36), we consider the control u ffl defined by Let the state x ffl be the solution to have lim ffl!0 Combining this with the coercivity condition gives (36). Consider the following linear-quadratic problem involving the parameters a, s, minimize subject to If the feasible set for (37) is nonempty, then Coercivity implies the existence of a unique minimizer over H 1 \Theta L 2 . Using the following lemma, we show that this minimizer lies in W 1;1 \Theta L 1 , and that it exhibits stability relative to the L 2 norm. Lemma 3.8. If Coercivity and Uniform Independence at I hold, then (37) has a unique solution for every a; Moreover, the change (ffix; ffiu) in the solution to (37) corresponding to a change (ffia; ffib; ffis; ffir) in the parameters satisfies the estimate Proof. By Lemma 3.6, Uniform Independence at I implies that the feasible set for (37) is nonempty while the Coercivity condition implies the existence of a unique solution From duality theory (for example, see [10]), there exists with the property that is the minimum with respect to u of the expression h- over all u 2 L 1 . It follows that and by (36), u (t) is uniformly bounded in t. From the equations L(x ; u x The estimate (38) can be obtained, as in Lemma 5 in [2], by eliminating the perturbation in the constraints. Let be the affine map in Lemma 3.6 relating the feasible pair (x; u) to the parameters (a; b). By making the substitution (x; to an equivalent problem of the form minimize subject to Here oe and ae are affine functions of a; b; s and r. Utilizing the Coercivity condition and the analysis of [9, Sect. 2], we obtain the following estimate for the change corresponding to the change (ffioe; ffiae): 2: Hence, Taking into account the relations between (x; u), (y; v), (oe; ae), and (a; b; s; r), the proof is complete. Now let us consider the full linear-quadratic problem where the subscript I on the state constraint has been removed: minimize subject to The first-order necessary conditions for this problem are precisely (15)-(18). Observe that x , u , / , and - satisfy (15)-(18) when - . Since the first-order necessary conditions are sufficient for optimality when Coercivity holds, (x ; u ) is the unique solution to (41) at - . In addition, if Uniform Independence holds, we now show that the multipliers / and - satisfying (16)-(18) are unique; hence, x , u , / , and - are the unique solution to (15)-(18) for - . To establish this uniqueness property for the multipliers, we apply Lemma 3.5 to the active constraint map A of Section 1. Let J i be the index sets associated with the complementary slackness condition - (1) T g associated with the condition (5) of the minimum principle, implies that (- ) J c l along with (16) and (17) imply that (- ) J l and / are uniquely determined on [- l ; 1]. Proceeding by induction, suppose that / and - are uniquely determined on the interval [- is constant on [- it is uniquely determined by the continuity of - , while (- and / on [- are uniquely determined by (21), (16), and (17). This completes the induction step. We now use Lemma 3.8 to show that the solution to (41) depends Lipschitz continuously on the parameters when Coercivity and Uniform Independence at A hold. We do this by making a special choice for the map I. Again, let J i be the index sets associated with I = A by Lemma 3.5. Since A(t) ae J i for each t 2 [- the parameter is strictly positive for each i. Setting in the case I = A ffl where A ffl (t) is the index set associated with the ffl-active constraints for the linearized problem: Since A ffl (t) ae J i for each t 2 [- implies that Uniform Independence at A ffl holds. We now observe that the solution (x ; u ) of (41) at - is the solution of (37) for I = A ffl and - . First, (x ; u ) is feasible in (37) since there are fewer constraints than in (41). By the choice I = A ffl , all feasible pairs for (37) near are also feasible in (41). Since (x ; u ) is optimal in (41), it is locally optimal in (37) as well, and by the Coercivity condition and Lemma 3.7, (x ; u ) is the unique minimizer of (37) for - . By Lemma 3.8, we have an estimate for the change in the solution to (37) corresponding to a change in the parameters. Since kffixk L 1 - kffixk H 1, it follows that for small perturbations in the data, the solution to (37) is feasible, and hence optimal, for (41). Hence, our previous stability analysis for (37) provides us with a local stability analysis for (41). We summarize this result in the following way: Lemma 3.9. If Coercivity and Uniform Independence at A hold, then for s, r, and a in an L 1 neighborhood of s , r , and a respectively, and for b in a W 1;1 neighborhood of b , there exists a unique minimizer of (41), and the estimate (38) holds. Moreover, taking I = A ffl with defined in (42), the solutions to (37) and (41) are identical in these neighborhoods. Now let us consider the multipliers associated with (41): Lemma 3.10 If Coercivity and Uniform Independence at A hold, then for s, r, and a in an L 1 neighborhood of s , r , and a respectively, and for b in a W 1;1 neighborhood of b , there exists a unique minimizer of (41) and associated unique multipliers satisfying the estimate: Proof. Let A ffl be the ffl-active constraints defined by (43), where Let J i be the index sets and let ae be the positive number associated with by Lemma 3.5. Consider - small enough that the active constraint set for (41) is a subset of A ffl (t) for each t. By the same analysis used to establish uniqueness of (/ ; - ), there exists unique Lagrange multipliers (/; corresponding to - + ffi-. We will show that Combining this with Lemma 3.9 yields Lemma 3.10. We prove (45) by induction. Let us start with the interval [- l \Gamma ae; 1]. If i 2 J c l , l Multiplying (17) by KB, we can solve for ffi- J l and substitute in (16) to eliminate -. Since it follows that for Proceeding by induction, suppose that (46) holds for we wish to show that it holds for is constant on [- and we have ae Combining this with (46) for for multiplying (17) by KB, we solve for ffi- J j and substitute in (16). the induction bound (46) for coupled with the bound already established for ffi- i , This completes the induction. Lemma 3.11. Suppose that Smoothness, Coercivity, and Uniform Independence at A hold and let - be small enough that Y is contained in the neighborhoods of Lemmas 3.9 and 3.10. Then for some - ? 0 and for each - 2 Y , there exists a unique solution (x; u) to (41) and associated multipliers (/; -) satisfying the estimates (38) and (44), (x; /; u; Proof. If the first-order necessary conditions (15)-(18) associated with (41). Lemmas 3.9 and 3.10 tell us that the unique solution and multipliers for (41) satisfy the estimates (38) and (44) for - near - . Since the first-order necessary conditions are sufficient for optimality when Coercivity holds, the variational system (15)-(18) has a unique solution, for - near - , that is identical to the solution and multipliers for (41), and the estimates (38) and (44) are satisfied. To complete the proof, we need to show that - This follows from the regularity results of [8], where it is shown that the solution to a constant coefficient, linear-quadratic problem satisfying the Uniform Independence condition and with R positive definite, Q positive semidefinite, and has the property that the optimal u and associated - are Lipschitz continuous in time while the derivatives of x and / are Lipschitz continuous in time. Moreover, the Lipschitz constant in time is bounded in terms of the constant ff in the Uniform Independence condition and the smallest eigenvalue of R. Exactly the same analysis applies to a linear-quadratic problem with time-varying coefficients, however, the bound for the Lipschitz constant of the solution depends on the Lipschitz constant of the matrices of the problem and of the parameters a, r, s, and - b, as well as on a uniform bound for the smallest eigenvalue of R(t) on [0; 1] and for the parameter ff in the Uniform Independence condition. By Lemma 3.9, and with the choice for I given in the statement of the lemma, the quadratic programs (37) and (41) have the same solution for s, r, and a in an L 1 neighborhood of s , r , and a , and for b in a W 1;1 neighborhood of b . Hence, for parameters in this neighborhood of - , the indices of the active constraints are contained in I(t) for each t, and the independence condition (21) holds. Lemma 3.7 provides a lower bound for the eigenvalues of R(t). If (a; s; then the Lipschitz constants for a, s, r, and - b are bounded by those for a , s , r , and - b plus -. Hence, taking - sufficiently large, the proof is complete. Proof of Theorem 1.1. We apply Theorem 2.2 with the identifications given at the beginning of this section, and with - chosen sufficiently large in accordance with Lemma 3.11. The completeness of X is established in Lemma 3.2, (Q1) is immediate, (Q2) follows from Smoothness, (Q3) is proved in Lemma 3.3, (Q4) follows from Lemma 3.11, and (Q5) is established in Lemma 3.4. Applying Theorem 2.2, the estimate (7) is established. Under the Uniform Independence condition, Coercivity is a second-order sufficient condition for local optimality (see [4], Theorem 1) which is stable under small changes in either the parameters or the solution of the first-order optimality conditions. Finally, we apply Lemma 3.1 to obtain the L 1 estimate of Theorem 1.1. We note that the Coercivity condition we use here is a strong form of a second-order sufficient optimality condition; it not only provides optimality, but also guarantees Lipschitz continuity of the optimal solution and multipliers when Uniform Independence holds. As recently proved in [6] for finite-dimensional optimization problems, Lipschitzian stability of the solution and multipliers necessarily requires a coercivity condition stronger than the usual second-order condition. For the treatment of second-order sufficient optimality under conditions equivalent to Coercivity, see [18] and [21]. These sufficient conditions can be applied to state constraints of arbitrary order. For recent work concerning the treatment of second-order sufficient optimality in state constrained optimal control, see [16], [19], and [22]. 4. Lipschitzian stability in L 1 One way to sharpen the L 1 estimate of Theorem 1.1 involves an assumption concerning the regularity of the solution to the linear-quadratic problem (41). The time t is a contact point for the i-th constraint of Kx+ b - 0 if (K(t)x(t) and there exists a sequence ft k g converging to t with (K(t k )x(t k each k. Contact Separation: There exists a finite set I 1 I N of disjoint, closed intervals contained in (0; 1) and neighborhoods of (a ; r ; s ) in W 1;1 and of b in W 2;1 with the property that for each a, r, s, and b in these neighborhoods, and for each solution to (41), all contact points are contained in the union of the intervals I i with exactly one contact point in each interval and with exactly one constraint changing between active and inactive at this point. Observe that if for (1) with there are a finite number of contact points, at each contact point exactly one constraint changes between active and inactive, and each contact point in the linear-quadratic problem (41) depends continuously on the parameters, then Contact Separation holds. The finiteness of the contact set is a natural condition in optimal control; for example, in [5] it is proved that for a linear-quadratic problem with time invariant matrices and one state constraint, the contact set is finite when Uniform Independence and Coercivity hold. Theorem 4.1. Suppose that the problem (1) with has a local minimizer that Smoothness, Contact Separation, and Uniform Independence at A hold. Let / and - be the associated multipliers satisfying the first-order necessary conditions (2)-(5). If the Coercivity condition holds, then there exist neighborhoods V of p and U of w such that for every there exists a unique solution U to the first-order necessary conditions (2)-(5) and (x; u) is a local minimizer of the problem (1) associated with p. Moreover, for every is the corresponding solution of (2)-(5), the following estimate holds: To prove this result, we need to supplement the 2-norm perturbation estimates provided by Lemmas 3.9 and 3.10 with analogous 1-norm estimates. Lemma 4.2. If Coercivity, Uniform Independence at A, and Contact Separation hold, then there exist neighborhoods of (a ; r ; s ) in W 1;1 and of b in W 2;1 such that for each a in these neighborhoods, the associated solutions c(kffiak Proof. Letting A ffl denote the ffl-active set defined in (43), we again choose defined in (42). We consider parameters a, r, s, and b chosen within the neighborhoods of the Contact Separation condition, and sufficiently close to a , r , s , and b that the active constraint set for the solution of the perturbed linear-quadratic problem (41) is contained in A ffl (t) for each t. By eliminating the perturbations in the constraints, as we did in the proof of Lemma 3.8, there is no loss of generality in assuming that a We refer to the quadratic program corresponding to the parameters (r Problem 2. Let (x; u) be either is a time for which K i for some i, then d Substituting for - x using the state equation for u using the necessary condition (17) yields: This equation has the form for suitable choices of the row vectors N i , S i , T i , and U i . Hence, at any time t where the change in solution and multipliers corresponding to a change in parameters satisfies the equation By the Contact Separation condition, Problems 1 and 2 have the same active set near 1. Since the components of - corresponding to inactive constraints are constant and since - i The relation (49) combined with Uniform Independence, with the L 2 estimates provided in Lemmas 3.9 and 3.10, and with a bound for the L 1 norm in terms of the H 1 norm, gives Using the bound (36) of Lemma 3.7 in (17) and applying Gronwall's lemma to (16), we have for all t ! 1 in some neighborhood of As t decreases, this estimate is valid until the first contact point is reached for either Problem 1 or Problem 2. Proceeding by induction, suppose that we have established (51) up to some contact point; we now wish to show that (51) holds up to the next contact point. Again, by the Contact Separation condition, there is precisely one constraint, say constraint j, that makes a transition between active and inactive at the current contact point. Suppose that on the interval (ff; fi), the active sets for Problems 1 and differ by the element j, and let - for the first contact point to the left of ff for either Problem 1 or Problem 2. If there is no such point, we take By the Contact Separation condition, the difference ff \Gamma - is uniformly bounded away from zero for all choices of the parameters s and r near s and r . There are essentially two cases to consider. Case 1: Constraint j is active in Problem 2 to the left of active in Problem 1 to the left of Case 2: Constraint j is active in Problem 2 to the right of is active in Problem 1 to the right of Case 1. Since constraint j is active in both Problem 1 and 2 at from (49) and from the Uniform Independence condition that is the set of indices of active constraints at on (ff; fi), the induction hypothesis yields Hence, we have is constant in Problem 1 on (ff; fi), and since it is monotone in Problem 2, the bound (53) coupled with the bound (51) at implies that Since ffi- i is constant on (ff; fi) for it follow from (51) that Relation (49), for along with (54) and (55) yield Combining (54)-(56) gives On the interval from to the next contact point - , precisely the same constraints are active in both Problems 1 and 2. Again, the relation (49) combined with Uniform Independence, with the L 2 estimates provided in Lemmas 3.9 and 3.10, and with a bound for the L 1 norm in terms of the H 1 norm gives Relation (50) for along with (57) and (58), give And combining this with (15)-(17) gives (51) for This completes the induction step in Case 1. Case 2. The mean value theorem implies that for some fl 2 (-; ff), we have d dt Hence, even though the derivative of K j x i may not vanish on (-; ff), the derivative of the change K j ffix is still bound by the perturbation in the parameters at some d dt Since ff and - lie in disjoint closed sets I k associated with the Contact Separation bounded away from zero by the distance between the closest pair of sets. Focusing on the left side of (59), we substitute ffi - substitute for ffiu using (17) to obtain the relation where denote the set of indices of the active constraints at Combining (60) with (49) for The analysis for Case 1 can now be applied, starting with (52), but with ff replaced by fl. Remark 4.3. In the proof of Lemma 4.2, we needed to ensure that the difference appearing in case 2, was bounded away from zero. The Contact Separation condition ensures that this difference is bounded away from zero since ff and - lie in disjoint closed intervals I k . On the other hand, any condition that ensures a positive separation for the contact points ff and - in case 2 can be used in place of the Contact Separation assumption of Theorem 4.1 and Lemma 4.2. Proof of Theorem 4.1. The functions T , F , and L and the sets X, \Pi, and Y are the same as in the proof of Theorem 1.1 except that L 2 is replaced by L 1 and H 1 is replaced by W 1;1 everywhere. Except for this change in norms, and the replacement of the L 2 estimates (38) and (44) referred to in Lemma 3.11 by the corresponding estimate (47) of Lemma 4.2, the same proof used for Theorem 1.1 can be used to establish Theorem 4.1. 5. Remarks As mentioned in Section 2, Theorem 2.2 is a generalization of Robinson's implicit function theorem [20] to nonlinear spaces. His theorem assumes that the nonlinear term is strictly differentiable and that the inverse of the linearized map is Lipschitz continuous. In optimal control, the latter condition amounts to Lipschitz continuity in 1 of the solution-multiplier vector associated with the linear-quadratic approxima- tion. For problems with control constraints, this property for the solution is obtained, for example, in [1] or [4]. In this paper, we obtain Lipschitzian stability results for state constrained problems utilizing a new form of the implicit function theorem applicable to nonlinear spaces. We obtain optimal Lipschitzian stability results in L 2 and nonoptimal stability results in L 1 under the Uniform Independence and the Coercivity conditions. And with an additional Contact Separation condition, we obtain a tight L 1 stability result. These are the first L 1 stability results that have been established for state constrained control problems. The Uniform Independence condition was introduced in [8] where it was shown that this condition together with the Coercivity condition yield Lipschitz continuity in time of the solution and the Lagrange multipliers of a convex state and control constrained optimal control problem. Using Hager's regularity result, Dontchev [1] proved that the solution of this problem has a Lipschitz-type property with respect to perturbations. Various extensions of these results have been proposed by several authors. A survey of earlier results is given in [2]. In a series of papers (see [14], [15], and the references therein), Malanowski studied the stability of optimal control problems with constraints. In [15] he considers an optimal control problem with state and control constraints. His approach differs from ours in the following ways: He uses an implicit function theorem in linear spaces and a compactness argument, and the second-order sufficient condition he uses is different from our coercivity condition. Although there are some similar steps in the analysis of L 2 stability, the two approaches mainly differ in their abstract framework. A prototype of Lemma 3.5 is given in [1], Lemma 2.5. Lemma 3.6 is related to Lemma 3 in [2], although the analysis in Lemma 3.6 is much simpler since we ignore indices outside of A(t). In the analysis of the linear-quadratic problem (37), we follow the approach in [4]. Acknowledgement . The authors wish to thank both Kazimierz Malanowski for his comments on an earlier version of this paper, and the reviewers for their constructive suggestions. --R Lipschitzian stability in nonlinear control and optimization An inverse function theorem for set-valued maps On regularity of optimal control Characterizations of strong regularity for variational inequalities over polyhedral convex sets Variants of the Kuhn-Tucker sufficient conditions in cones of nonnegative functions Lipschitz continuity for constrained processes Multiplier methods for nonlinear optimal control Dual approximations in optimal control Lagrange duality theory for convex control problems A survey of the maximum principles for optimal control problems with state constraints Theory of Extremal Problems Stability and sensitivity of solutions to nonlinear optimal control problems Sufficient optimality conditions in optimal control On the minimum principle for optimal control problems with state constraints First and second order sufficient optimality conditions in mathematical programming and optimal control Second order sufficient conditions for optimal control problems with control-state constraints Strongly regular generalized equations Sufficient conditions for nonconvex control problems with state constraints The Riccati equation for optimal control problems with mixed state- control constraints --TR --CTR Stephen J. Wright, Superlinear Convergence of a Stabilized SQP Method to a Degenerate Solution, Computational Optimization and Applications, v.11 n.3, p.253-275, Dec. 1998 Olga Kostyukova , Ekaterina Kostina, Analysis of Properties of the Solutions to Parametric Time-Optimal Problems, Computational Optimization and Applications, v.26 n.3, p.285-326, December W. Hager, Stabilized Sequential Quadratic Programming, Computational Optimization and Applications, v.12 n.1-3, p.253-273, Jan. 1999 D. Goldfarb , R. Polyak , K. Scheinberg , I. Yuzefovich, A Modified Barrier-Augmented Lagrangian Method for Constrained Minimization, Computational Optimization and Applications, v.14 n.1, p.55-74, July 1999

optimal control;state constraints;lipschitzian stability;implicit function theorem

278885

Connectors for Mobile Programs.

AbstractSoftware Architecture has put forward the concept of connector to express complex relationships between system components, thus facilitating the separation of coordination from computation. This separation is especially important in mobile computing due to the dynamic nature of the interactions among participating processes. In this paper, we present connector patterns, inspired in Mobile UNITY, that describe three basic kinds of transient interactions: action inhibition, action synchronization, and message passing. The connectors are given in COMMUNITY, a UNITY-like program design language which has a semantics in Category Theory. We show how the categorical framework can be used for applying the proposed connectors to specific components and how the resulting architecture can be visualized by a diagram showing the components and the connectors.

Introduction As the complexity of software systems grows, the role of Software Architecture is increasingly seen as the unifying infrastructural concept/model on which to analyse and validate the overall system structure in various phases of the software life cycle. In consequence, the study of Software Architecture has emerged, in recent years, as an autonomous discipline which requires its own concepts, formalisms, methods, and tools [1], [2]. The concept of connector has been put forward to express complex relationships between system components, thus facilitating the separation of coordination from computation. This is especially important in mobile computing due to the transient nature of the interconnections that may exist between system components. In this paper we propose an architectural approach to mobility that encapsulates this dynamic nature of interaction in well-defined connectors. More precisely, we present connector patterns for three fundamental kinds of transient interaction: action inhibition, action synchronization, and message passing. Each pattern is parameterized by the condition that expresses the transient nature of the interaction. The overall architecture is then obtained by applying the instantiated connectors to the mobile system components. To illustrate our proposal, components and connectors will be written in COMMUNITY [3], [4], a program design language based on UNITY [5] and IP [6]. The nature of the connectors proposed in the paper was motivated and inspired by Mobile UNITY [9], [10], an extension of UNITY that allows transient interactions among programs. However, our approaches are somewhat different. Mobile UNITY suggests the use of an interaction section to define coordination within a system of components. We advocate an approach based on explicitly identified connectors, in order to make the architecture of the system more explicit and promote interactions to first-class entities (like programs). Moreover, while we base our approach on the modification of the superposition relation between programs, Mobile UNITY introduces new special programming constructs, leading to profound changes in UNITY's syntax and computational model. However, we should point out that some of these syntactic and semantic modifications (like naming of program actions and locality of variables) were already included in COMMUNITY. To make it easier for interested readers to compare our approach with Mobile UNITY we use the same example as in [9]: a luggage distribution system. It consists of carts moving on a closed track transporting bags from loaders to unloaders that are along the track. Due to space limitations we have omitted many details which, while making the example more realistic, are not necessary to illustrate the main ideas. In this paper we follow the approach proposed in [7] and give the semantics of connectors in a categorical framework. In this approach, programs are objects of a category in which the morphisms show how programs can be superposed. Because in Category Theory [8] objects are not characterized by their internal structure but by their morphisms (i.e., relationships) to other objects, by changing the definition of the morphisms we can obtain different kinds of relationships between the programs, without having to change the syntax or semantics of the programming language. In fact, the core of the work to be presented in the remainder of this paper is an illustration of that principle: by changing program morphisms in a small way such that actions can be "ramified", transient action synchronization becomes possible. This work was partially supported by JNICT through contract PRAXIS XXI 2/2.1/MAT/46/94 (ESCOLA) and by project ARTS under contract to EQUITEL SA. Michel Wermelinger is with the Departamento de Inform'atica, Universidade Nova de Lisboa, 2825 Monte da Caparica, Portugal. E-mail: mw@di.fct.unl.pt. Jos'e Luiz Fiadeiro is with the Departamento de Inform'atica, Faculdade de Ci-encias, Universidade de Lisboa, Campo Grande, 1700 Lisboa, II. Mobile Community The framework to be used consists of programs and their morphisms. This section introduces just the necessary definitions. For a more thorough formal treatment, the interested reader should consult [4]. A COMMUNITY program is basically a set of named, guarded actions. Action names act as rendez-vous points for program synchronization. At each step, one or more actions whose guards are true execute in parallel. Each action consists of one or more assignments to execute simultaneously. Each attribute used by a program is either external-its value is provided by the environment and may change at any time-or local-its value is initialized by the program and modified only by its actions. Attributes are typed by a fixed algebraic data type specification a set of sort symbols,\Omega is an S \Theta S-indexed family of function symbols, and \Phi is a set of first-order axioms defining the properties of the operations. We do not present the specification of the sorts and predefined functions used in this paper. A COMMUNITY program has the following structure program P is read R init I do [] a := F (g; a)] where ffl V is the set of local attributes, i.e., the program "variables"; ffl R is the set of external attributes used by the program, i.e., read-only attributes that are to be instantiated with local attributes of other components in the environment; ffl each attribute is typed by a data sort in ffl I is the initialisation condition, a proposition on the local attributes; is the set of action names, each one having an associated statement (see below); ffl for every action g 2 \Gamma, the guard B(g) is a proposition on the attributes stating the necessary conditions to execute ffl for every action g 2 \Gamma, its domain D(g) is the set of local attributes that g can change; ffl for every action local attribute a 2 D(g), F (g; a) is a term denoting the value to be assigned to a each time g is executed. Formally, the signature of a program defines its vocabulary (i.e., its attributes and action names). program signature is a tuple hV; R; \Gammai where is a set of local attributes ; R s is a set of external attributes ; is a set of actions. The sets V s , R s and \Gamma d are finite and mutually disjoint. The domain of an action is the set d ' V such that Notation. The program attributes are A = S The sort of attribute a will be denoted by s a . The domain of action g is denoted by D(g). Inversely, for each a 2 V the set of actions that can change a is A program's body defines the initial values of its local attributes and also when and how the actions modify them. For that purpose the body uses propositions and terms built from the program's attributes and the predefined function symbols. program is a pair h'; \Deltai where is a program signature and is a program body where ffl I is a proposition over F assigns to every and to every a 2 D(g) a term of sort s a ; assigns to every proposition over A. Notation. If D(g) is empty, then F will be denoted by skip. 2 Locations are an important aspect of mobility [11]. We take the same approach as Mobile UNITY and represent location by a distinguished attribute. However, our framework allows us to handle locations in a more flexible way. We can distinguish whether the program controls its own motion or if it is moved by the environment by declaring the location attribute as local or external, respectively. The formal treatment of locations is the same as for any attribute because they have no special properties at the abstract level we are working at. However, any implementation of COMMUNITY will have to handle them in a special way, because a change in the system's location implies a change in the value of the location attribute and vice-versa. We assume therefore some special syntactic convention for location attributes such that a compiler can distinguish them from other attributes. Following the notation proposed by Mobile UNITY, in this paper location attributes start with -. To give an example of a COMMUNITY program, we present the specification of a cart. Like bags and (un)loaders, carts have unique identifiers, which are represented by external integer attributes, so that a cart cannot change its own identity. A cart can transport at most one bag at a time from a source loader to a destination unloader. Initially, the cart's destination is the loader from which it should fetch its first bag. The unloader at which a bag must be delivered depends on the bag's identifier. After delivering a bag, or if a loader is empty, the cart proceeds to the next loader. Absence of a bag will be denoted by the identifier zero. The track is divided into segments, each further divided into ten units. The location of a cart is therefore given by an integer. Carts can move at two different speeds: slow (one length unit per time unit) and fast (two length units). A cart stops when it reaches its destination. The action to be performed at the destination depends on whether the cart is empty or full. program Cart is dest : int; read id, nbag : int; do slow: [-6= dest ! -+ 1] [] fast: [-6= dest ! -+ 2] [] load: [-= dest nbag k dest := Dest(nbag, dest)] [] unload: [-= dest - bag dest := Next(dest)] We now turn to program morphisms, the categorical notion that expresses relationships between (certain) pairs of programs. In the previous definitions of COMMUNITY [4], [7], a morphism between two programs P and P 0 is just a mapping from P 's attributes and actions to those of P 0 , stating in which way P is a component of P 0 . It is therefore called a superposition morphism, since it captures the notion of superposition of [5], P being the underlying program and P 0 the transformed one. In this paper we keep the basic intuition but introduce a small although fundamental change. In a mobile setting, a program may synchronize each of its actions with different actions from different programs at different times. To allow this, a program morphism may associate an action g of the base program P with a set of actions fg of the superposed program P 0 . The intuition is that those actions correspond to the behaviour of g when synchronizing with other actions of other components of P 0 . Each action g i must preserve the basic functionality of g, adding the functionality of the action that has been synchronized with g. The morphism is quite general: the set fg may be empty. In that case, action g has been effectively removed from P 0 . Put in other words, it has been permanently inhibited, as if the guard had been made false. Due to technical reasons the mapping between actions of P and sets of actions of P 0 is formalised as a partial function from P 0 to P . However, in examples and informal discussions we use the "set version" of the action mapping. Morphisms must preserve the types, the locality, and the domain of attributes. Preserving locality means that local attributes are mapped to local attributes, and preserving domains means that new actions of the system are not allowed to change local attributes of the components. Definition 3 Given program signatures consists of a total function oe ff : A ! A 0 and a partial function oe Notation. In the following, the indices ff and fl are omitted. We denote the pre-image of oe fl by oe / . Also, if x is a term (or proposition) of ', then oe(x) is the term (resp. proposition) of ' 0 obtained by replacing each attribute a of x by Notice that through the choice of an appropriate morphism, it is possible to state whether a given component and a given system are co-located (i.e., whenever one moves, so does the other) or if the component can move independently within the system. This can be modeled by a morphism that maps (or not) the location attribute of the component to the location attribute of the system. Our first result is that signatures and their morphisms constitute a category. This basically asserts that morphisms can be composed. In other words, the "component-of" relation is transitive (and reflexive, of course). Proposition 1 Program signatures and signature morphisms constitute a category SIG. Superposition of a program P 0 on a base program P is captured by a morphism between their signatures that obeys the following conditions: ffl the initialization condition is not weakened; ffl the assignments are equivalent; ffl the guards are not weakened. where means validity in first-order sense. The category of signatures extends to programs. Proposition 2 Programs and superposition morphisms constitute a category PROG. To give an example of a program morphism, consider the need to prevent carts from colliding at intersections. We achieve that goal in two steps, the second of which to be presented in subsection IV-B. When two carts enter two segments that intersect, due to the semantics of COMMUNITY allowing only one cart to move at each step, one of the carts will be further away from the intersection. The first step to avoid collisions is to force that cart to move slowly. In other words, its fast action is inhibited. Notice that in this case the inhibition depends on the presence of another cart, and therefore a second (external) location attribute - 2 is needed. The Cart program is thus transformed into an InhibitedCart as given by the diagram program Cart is dest : int; read id, nbag : int; do slow: [-6= dest ! -+1] [] fast: [-6= dest ! -+2] [] load: [-=dest - bag=0 [] unload: [-=dest - bag 6= 0 program InhibitedCart is dest : int; read id, nbag, do slow: [-6= dest ! -+1] [] fast: [-6= dest - :I ! -+2] [] load: [-=dest - bag=0 [-=dest - bag6= 0 where the inhibition condition is I DistanceToCrossing(-). The morphism is an injection: - 7! -, fast 7! fast, etc. The next section shows how the InhibitedCart program can be obtained by composition of two components. III. The Architecture The configuration of a system is described by a diagram of components and channels. The components are programs, and the channels are given by signatures that specify how the programs are interconnected. Given programs P and P 0 , the signature S is constructed as follows: for each pair of attributes (or actions) a 2 P and a that are to be shared (resp. synchronized), the signature contains one attribute (resp. action) b; the morphism from S to P maps b to a and the morphism from S to P 0 maps b to a 0 . We have morphisms only between signatures or only between programs, but a signature can be seen as a program F(') with an "empty" body [7]. In categorical terms, the operator F is a functor (i.e., a morphism between objects of different categories). As a simple example consider the following diagram, which connects (through a channel that represents attribute sharing) the generic cart program with a program that initializes an integer attribute with the value 2. program Init 2 is signature Share is program Cart is var . init . do . The program that describes the whole system is given by the colimit of the diagram, which can be obtained by computing the pushouts of pairs of components with a common channel. The program P resulting from the pushout of obtained as follows. The initialization condition is the conjunction of the initialization conditions of the components, and the attributes of P are the union of the attributes of P 1 and P 2 , renaming them such that only those that are to be shared will have the same name. An attribute of P is local only if it is local in at least one component. For the above example, the resulting pushout will represent the cart with identifier 2. program Cart 2 is var bag, -, dest, id : int read nbag : int do . As for the actions of P , they are basically a subset of all pairs of actions Only those pairs such that g 1 and g 2 are mapped to the same action of the channel may appear in P . If an action of P 1 (or not mapped to any action of the channel-i.e., it is not synchronized with any action of P 2 (resp. P 1 )-then it appears "unpaired" in P . Synchronizing two actions g 1 and g 2 (i.e., joining them into a single one g 1 taking the union of their domains, the conjunction of their guards, and the parallel composition of their assignments. If the actions have a common attribute a then the resulting assignment is a := F (g 1 ; a) and the guard is strengthened by a). If the actions are "incompatible" (i.e., the terms denote different values for a) then the equality is false and therefore the synchronized action will never execute, as expected. As an illustration, the pushout of the diagram program Cart is var bag, -, dest : int read id, nbag : int do slow: [-6= dest ! -+1] [] fast: [-6= dest ! -+2] [] load: [-=dest - bag=0 [] unload: [-=dest - bag 6= 0 signature S is do i i7!fast program Inhibitor is read -: int do i: [:I ! skip] is program InhibitedCart shown in the previous section: actions fast and i were paired together, joining their guards and assignments. Notice that attribute - of the Inhibitor program has been renamed to - 0 because names are local. The next result states that every finite diagram has a colimit. Proposition 3 Category PROG is finitely cocomplete. Channels (i.e., signatures) only allow us to express simple static connections between programs. To express more complex or transient interactions, we use connectors, a basic concept of Software Architecture [2]. A connector consists of a glue linked to one or more roles through channels. The roles constrain what objects the connector can be applied to. In a categorical framework, the connectors (and therefore the architectures) that can be built depend on the categories used to represent glues, roles, and channels, and on the relationships between those categories. It is possible to use three different categories for the three parts of a connector (e.g., [7] proposes roles to be specifications written in temporal logic) but for simplicity we assume that roles and glues are members of the same category. We therefore adopt only the basic definitions of [7]. connection is a tuple hC; G; R; fl; aei where is the channel ; is the glue; is the role; are morphisms in PROG. A connector is a finite set of connections with the same glue. The semantics of a connector is given by the colimit of the connections diagram. By definition, there are superposition morphisms from each object in the diagram to the colimit. Therefore superposition becomes in a sense "symmetric", a necessary property to capture interaction [10]. A connector can be applied only to programs which are instantiations of the roles. In categorical terms, there must exist morphisms from the roles to the programs. Definition 6 A correct instantiation of a connector fhC i ; G; R is a set of morphisms PROG. The resulting system is the colimit of the diagram formed by morphisms As an illustration, an instantiated connector with two roles has the diagram IV. Interactions An interaction between two programs involves conditions and computations. Therefore it cannot be specified just by a signature; we must use a connector, where the programs are instances of the roles, the interaction is the glue, and each channel states exactly what is the part of each program in the interaction. A distributed system may consist of many components, but usually it can be classified into a relatively small set of different types. Since interaction patterns normally do not depend on the individual components but on their types, it is only necessary to define connectors for the existing component types. To obtain the resulting system, the connectors will be instantiated with the actual components. Therefore, in the following we only consider the programs that correspond to component types. In the luggage distribution example there are only three different program types: carts, loaders, and unloaders. The programs for the individual components only differ in the initialization condition for the identifier attribute. In a mobile setting one of the important aspects of interactions is their temporary nature. This is represented by conditions: an interaction takes place only while some proposition is true. Usually that proposition is based on the location of the interacting parties. We consider three kinds of interactions: inhibition An action may not execute. 1 synchronization Two actions are executed simultaneously. communication The values of some local attributes of one program are passed to corresponding external attributes of the other program. For each kind of interaction we develop a connector template which is parameterized by the interaction conditions. This means that, given the interacting programs (i.e., the roles) and the conditions under which they interact, the appropriate connector can be instantiated. Given the set of components that will form the overall system, the possible interactions are specified as follows: ffl An inhibition interaction states that an action g of some program P will not be executed whenever the interaction condition I is true. ffl A synchronization interaction states that action g of program P will execute simultaneously with action g 0 of program I is true. ffl A communication interaction states that the value of the local attributes M (the "message") of program P can be written into the external attributes M 0 of program P 0 if I is true. The sets M and M 0 must be compatible. Moreover, each program must indicate which action is immediately executed after sending (resp. receiving) the message. Definition 7 Given a set P of programs, a transient interaction is either one of the following: ffl a transient inhibition hg; ffl a transient synchronization hg; ffl a transient communication hg; M;P; where there is a bijection ffl I is a proposition over attributes of P . The following subsections present the connector patterns corresponding to the above interactions. The glue of a connector only needs to include the attributes that occur in the interaction condition. However, to make the formal definitions easier, the glue patterns will include all the attributes of all the roles. Due to the locality of names, attributes from different roles must be put together with the disjoint union operator (written ]) to avoid name clashes. For further simplication, we assume that the interaction condition only uses attributes from the interacting programs thus only those roles are presented in the patterns. If this is not the case, the instantiated connector must have further roles that provide the remaining attributes. The next subsection provides an example. A. Inhibition Inhibition is easy and elegant to express: if an action is not to be executed while I is true, then it can be executed only while :I is true. Definition 8 The inhibition connector pattern corresponding to inhibition interaction hg; 1 In this case the interaction is between the program and its environment. program P is read R init . do g: [B(g) ! . ] [] . signature Target is do g program Inhibitor is init true do g: [:I ! skip] For illustration, the action inhibition example of Sections II and III can be achieved through the following connector. signature Context is read -: int program InhibitCrossing is read init true do fast: [:I ! skip] signature Target is read -: int do fast fast7!fast program Cart is. program Cart is. Again, the inhibition condition is I Notice that the connector has two roles, one for the cart whose action is to be temporarily inhibited, the other for the cart that provides the context for the inhibition to occur. An application of this connector and the resulting colimit will be presented in the next subsection. B. Synchronization Synchronizing two actions g and g 0 of two different components can be seen as merging them into a single action gg 0 of the system, the only difference between the static and the mobile case being that in the latter the merging is only done while some condition is true. When gg 0 executes, it corresponds to the simultaneous execution of g and g 0 . Therefore, if g would be executed by a component, the system will in fact execute gg 0 which means that it is also executing g 0 , and vice-versa. To sum it up, when two actions synchronize either both execute simultaneously or none is executed. This contrasts with the approach taken by Mobile UNITY which allows two kinds of synchronization: coexecution and coselection [10]. The former corresponds to the notion exposed above, while the latter forces the two actions to be selected simultaneously but if one of them is inhibited or its guard is false then only the other action executes. This extends the basic semantics of UNITY where only one action can be selected at a time. Since COMMUNITY already allows (but does not impose) simultaneous selection of multiple actions, and because we believe that the intuitive notion of synchronization corresponds to coexecution, we will not handle coselection. The key to represent synchronization of two actions subject to condition I is to ramify each action in two, one corresponding to its execution when I is false and the other one when I is true. Put in other words, each action has two "sub-actions", one for the normal execution and the other for synchronized execution. As the normal sub-action can only execute when the condition is false, it is inhibited when I is true, and the opposite happens with the synchronization sub-action. Therefore we can use the same technique as for inhibition. Since there are two actions to be synchronized, and the synchronization sub-action must be shared by both, there will we three (instead of four) sub-actions. To facilitate understanding, the name of a sub-action will be the set of the names of the actions it is part of. Definition 9 The synchronization connector pattern corresponding to synchronization interaction hg; signature C is do g program Synchroniser is read init true do g: [:I ! skip] signature C 0 is read do program P is read R init . do g: [B(g) ! . ] [] . program P 0 is read R 0 init . do [] . In the colimit, the action gg 0 will have the guards and the assignments of g and g 0 . Therefore, if either B(g) or B(g 0 ) is false, or if the assignments are incompatible, then gg 0 will not get executed. This connector describes what is called "non-exclusive coexecution" in [10]: outside the interaction period the actions execute as normal. It is also possible to simulate exclusive coexecution which means that the actions are only executed (synchronously) when the interaction condition is true. To that end, simply eliminate actions g and g 0 from the inhibition connector shown above, just keeping the synchronized action gg 0 . Continuing with the example, the second step to avoid collisions at crossings is to force the nearest cart to move fast whenever the most distant one moves. Since the latter can only move slowly, the nearest cart is guaranteed to pass the crossing first. Using the same interaction condition as in the previous section one gets the diagram signature C 1 is read -: int do fast program SynchCrossing is read init true do fast: [:I ! skip] signature C 2 is read -: int do slow slow7!fslow;fastslowg program Cart is. program Cart is. To prevent collisions between Cart 1 and Cart 2 (obtained as shown in Section III) one must consider two symmetrical cases, depending on which cart is nearer to the intersection. Let us assume that Cart 1 is nearer. Thus we must block the fast action of Cart 2 with the inhibitor shown in the previous section and synchronize its slow action with the fast action of Cart 1 using the connector above. The diagram is Cart Context InhibitCrossing Target // Cart Cart OO SynchCrossing C 2 // Cart with the following colimit (where i ranges over 1 and 2 to abbreviate code duplication) program System is read nbag dest do slow dest dest dest fast 1 slow dest dest dest dest dest i := Dest(nbag i , dest i )] dest dest i := Next(dest i )] To see that synchronization is transitive, consider the following example where action g 0 is synchronized with two other actions g and g 00 whenever I 1 and I 2 are true, respectively. The resulting system must provide actions for all four combinations of the truth values of the interaction conditions. For example, if I 1 - I 2 is true then all actions must occur simultaneously, but if I 1 -I 2 is false, then any subset of the actions can occur. This happens indeed because the pushout of two morphisms m g is basically given by the pairs fg 1 g 0 with morphism oe(g i g. Putting into words: if an action g "ramifies" into actions g, it means that whenever g would be executed, any subset of oe(g) executes in the superposed program, and vice-versa, the execution of any g i implies that g is executed in the base program. Therefore, if g can be ramified in two distinct ways, in the pushout any combination of the sub-actions can occur whenever g executes. The pushout morphisms just state to which combinations each sub-action belongs. do do ssh do do do ssg gggggggggggggggggggg do As one can see, for all combinations of I 1 and I 2 the correct actions are executed. The colimit includes the combination of all actions that share the name actions g 0 and gg 0 of the left middle pushout are synchronized with g 0 and g 0 g 00 on the right in the four possible ways. C. Communication In Mobile UNITY communication is achieved through variable sharing. The interaction x - y when C engage I disengage F x k F y states the sharing condition C, the (shared) initial value I of both variables, and the final value F x and F y of each variable. The operational semantics states that whenever a program changes x, y gets the same value, and vice-versa. This approach violates the locality principle. Furthermore, as pointed out in [10], several restrictions have to be imposed in order to avoid problems like, e.g., simultaneous assignments of different values to shared variables. We also feel that communication is a more appropriate concept than sharing for the setting we are considering, namely mobile agents that engage into transient interactions over some kind of network. In the framework of COMMUNITY programs, communication can be seen as some kind of sharing of local and external attributes, which keeps the locality principle. We say "some kind" because we cannot use the same mechanism as in the static case, in which sharing meant to map two different attributes of the components into a single one of the system obtained by the colimit. In the mobile case the same local attribute may be shared with different external attributes at different times, and vice-versa. If we were to apply the usual construction, all those attributes would become a single one in the resulting system, which is clearly unintended. We therefore will obtain the same effect as transient sharing using a communication perspective. To be more precise, we assume program P wants to send a message M , which is a set of local attributes. If P 0 wants to receive the message, it must provide external attributes M 0 which correspond in number and type to those of M . Program P produces the values, stores them in M , and waits for the message to be read by P 0 . Since COMMUNITY programs are not sequential, "waiting" has to be understood in a restricted sense. We only assume that P will not produce another message before the previous one has been read (i.e., messages are not lost); it may however be executing other unrelated actions. To put it in another way, after producing M , program P is expecting an acknowledge to produce the new values for the attributes in M . For that purpose, we assume P has an action g which must be executed before the new message is produced. Similarly, program P 0 must be informed when a new message has arrived, so that it may start processing it. For that purpose we assume that P 0 has a single action g 0 which is the first action to be executed upon the receipt of a new message 2 . That action may simply start using M 0 directly or it may copy it to local attributes of P 0 . To sum up, communication is established via one single action for each program 3 : the action g of P is waiting for M to be read, the action g 0 of P 0 reads M (i.e., starts using the values in M 0 ). As expected, it is up for the glue of the interaction connector to transfer the values from M to M 0 and to notify the programs. The solution is to explicitly model the message transmission as the parallel assignment of the message attributes, which we abbreviate as M 0 := M . For this to be possible, the local attributes M of P must be external attributes of the glue, and the external attributes M 0 of P 0 must be local attributes of the glue. The assignment can be done in parallel with the notification of P . Moreover, the programs may only communicate when proposition I is true. Therefore the glue contains an action wait : [I !M 0 := M ] to be synchronized with the "waiting" action g of P . The "reading" action g 0 of P 0 can only be executed after the message has been transmitted. The solution is to have another action read in the glue that is synchronized with g 0 . To make sure that read is executed after wait we use a boolean attribute. Thus 0 is inhibited while no new values have been transferred to M 0 . Again, this is like a blocking read primitive, except 2 It is always possible to write P 0 in such a way. 3 This is similar to pointed processes in the -calculus, or to ports in distributed systems. that P 0 may execute actions unrelated to M 0 . Since a receiver may get messages from different senders different times or not), there will be several possible assignments M 0 := M i . Due to the locality principle, all assignments to an attribute must be in a single program. Therefore for each message type a receiver might get, there will be a single glue connecting it to all possible senders. On the other hand, a message might be sent to different receivers m. Therefore there will be several possible assignments M 0 associated with the same wait action of the sender of message M . So there must be a single glue to connect a sender with all its possible recipients. To sum up, for each message type there will be a single glue acting like a "demultiplexer": it synchronizes sender i with receiver j when interaction condition I ij is true. This assumes that the possible communication patterns are known in advance. The communication connector pattern corresponding to communication interactions m) is signature Sender i is read do wait i program Communicator is read init :new j do signature Receiver j is read M 0 do read j program P i is read R i init . do [] . program j is read M 0 init . do read [] . Notice that several actions wait ij may occur simultaneously, in particular for the same receiver j if the messages sent have the same value. To distinguish messages sent by different senders, even if their content is the same, one can add a local integer attribute s to the glue and add the assignment s := i to each action wait ij . This prevents two different senders from sending their messages simultaneously. In the luggage delivery example, communication takes place when a cart arrives at a station (i.e., a loader or an unloader), the bag being the exchanged message. Loaders are senders, unloaders are receivers, and carts have both roles. The bags held by a station will be stored in an attribute of type queue of integers. Although the locations of stations are fixed they must be represented explicitly in order to represent the communication condition, namely that cart and station are co-located. Since it is up for the connector to describe the interaction, the programs for the stations just describe the basic computations: loaders remove bags from their queues, unloaders put bags on their queues. The loader program must have separate actions to produce the message (i.e., the computation of the value of the bag attribute) and to send the message (i.e., the bag has been loaded onto the cart). The c carts are connected to the l loaders through a connector with c identical roles (each one being the Cart program of Section I) and l identical roles, each being the Loader program. We only show the roles and respective morphisms for the i-the loader (sender) and the j-th cart (receiver). signature Sender is read do load load7!fwait i1 ;:::;wait ic g program Load is init :new j do nbag new j :=true] new j :=false] signature Receiver is read -, nbag : int do load load7!read j program Loader is loaded init loaded -= InitLoc(id) do newbag: [q loaded loaded:=false] loaded [] load: [:loaded loaded:=true] program Cart is var -, dest, bag : int read id, nbag : int -=InitLoc(id) do slow: [-6= dest ! -+1] [] fast: [-6= dest ! -+2] [] load: [-=dest - bag=0 bag:=nbag [-=dest - bag6=0 Similarly, there is a connector with u roles for the unloaders and c roles for the carts. The i-the cart (sender) is connected to the j-th unloader (receiver) as follows. signature Sender is read do load load7!fwait i1 ;:::;wait iu g program Unload is read init :new j do signature Receiver j is read do unload program Cart is var -, dest, bag : int read id, nbag : int -=InitLoc(id) do slow: [-6= dest ! -+1] [] fast: [-6= dest ! -+2] [] load: [-=dest - bag=0 bag:=nbag [-=dest - bag6=0 program Unloader is read do unload: [true be the program obtained by the pushout of programs Init i (of Section III) and X . Then the program corresponding to a system consisting of two carts, one loader, and one unloader is obtained by computing the colimit of the following diagram, which only shows the role instantiation morphisms between the connectors (which have the same name as their glues) and the components. SynchCrossing slow fast InhibitCrossing fast ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Loader 3 Load O O O O O O O O O O O O Unload ggO O O O O O O O O O O O Unloader 4 SynchCrossing fast slow InhibitCrossing fast __ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ Notice that the binary connectors dealing with crossings are not symmetric; they distinguish which cart is supposed to be nearer to the crossing. Therefore one must apply those connectors twice to each pair of carts. V. Concluding Remarks We have shown how some fundamental kinds of transient interactions, inspired by Mobile UNITY [9], [10], can be represented using architectural connectors. The semantics has been given within a categorical framework, and the approach has been illustrated with a UNITY-like program design language [3], [4]. As argued in [3], [12], the general benefits of working within a categorical framework are: ffl mechanisms for interconnecting components into complex systems can be formalized through universal constructs (e.g., colimits); ffl extra-logical design principles are internalized through properties of universal constructs (e.g., the locality of names); ffl different levels of design (e.g., signatures and programs) can be related through functors. For this work in particular, the synergy between Software Architecture and Category Theory resulted in several conceptual and practical advantages. First, systems are constructed in a principled way: for each interaction kind there is a connector template to be instantiated with the actual interaction conditions; the instantiated connectors are applied to the interacting programs thus forming the system architecture, which can be visualized by a diagram; the program corresponding to the overall system is obtained by "compiling" (i.e., computing the colimit of) the diagram. Second, separation between computation and coordination, which is already supported by Software Architecture, has been reinforced by two facts. On the one hand, the glue of a connector uses only the signatures of the interacting programs, not their bodies. On the other hand, the superposition morphisms impose the locality principle. Third, to capture transient interactions, only the morphism between program actions had to be changed; the syntax and semantics of the language remained the same. There are two ways of dealing with architectures of mobile components. In a system with limited mobility or with a limited number of different component types, all possible interaction patterns can be foreseen, and thus a static architecture with all possible interconnections can represent such a system. To cope with systems having a greater degree of mobility, one must have evolving architectures, where components and connectors can be added and removed unpredictably. This paper, being inspired by Mobile UNITY, follows the first approach. Our future work will address the second approach. One of the ideas we wish to explore is to remove the interaction condition from the glue's actions and instead associate it to the application of the whole connector. The diagram of the system architecture thus becomes dynamic, at each moment including only the connectors whose conditions are true. Another possibility is to apply graph rewriting techniques to the system diagrams. A third venue is to change (again) the definition of morphism to represent the notion of "changes-to" instead of "component-of". In other words, a morphism form P to P 0 indicates that P may become . For the moment, these are just some of our ideas to capture software architecture evolution in a categorical setting. Their suitability and validity must be investigated. Acknowledgements We would like to thank Ant'onia Lopes for many fruitful discussions and the anonymous referees for suggestions on how to improve the presentation. --R "Special issue on software architecture," Perspectives on an Emerging Discipline Parallel Program Design-A Foundation "Semantics of architectural connectors," Basic Category Theory for Computer Scientists "Mobile UNITY: Reasoning and specification in mobile computing," "Mobile UNITY: A language and logic for concurrent mobile systems," "Towards a general location service for mobile environments," --TR --CTR Michel Wermelinger , Cristvo Oliveira, The community workbench, Proceedings of the 24th International Conference on Software Engineering, May 19-25, 2002, Orlando, Florida Performance evaluation of mobility-based software architectures, Proceedings of the 2nd international workshop on Software and performance, p.44-46, September 2000, Ottawa, Ontario, Canada Michel Wermelinger , Antnia Lopes , Jos Luiz Fiadeiro, Superposing Connectors, Proceedings of the 10th International Workshop on Software Specification and Design, p.87, November 05-07, 2000 Antonio Brogi , Carlos Canal , Ernesto Pimentel, On the semantics of software adaptation, Science of Computer Programming, v.61 n.2, p.136-151, July 2006 Antnia Lopes , Jos Luiz Fiadeiro , Michel Wermelinger, Architectural primitives for distribution and mobility, Proceedings of the 10th ACM SIGSOFT symposium on Foundations of software engineering, November 18-22, 2002, Charleston, South Carolina, USA Antnia Lopes , Jos Luiz Fiadeiro , Michel Wermelinger, Architectural primitives for distribution and mobility, ACM SIGSOFT Software Engineering Notes, v.27 n.6, November 2002 Andrea Bracciali , Antonio Brogi , Carlos Canal, A formal approach to component adaptation, Journal of Systems and Software, v.74 n.1, p.45-54, January 2005 Lus Filipe Andrade , Jos Luiz Fiadeiro, Agility through coordination, Information Systems, v.27 n.6, p.411-424, September 2002 Marco Antonio Barbosa , Lus Soares Barbosa, An Orchestrator for Dynamic Interconnection of Software Components, Electronic Notes in Theoretical Computer Science (ENTCS), 181, p.49-61, June, 2007 Michel Wermelinger , Jos Luiz Fiadeiro, A graph transformation approach to software architecture reconfiguration, Science of Computer Programming, v.44 n.2, p.133-155, August 2002 Michel Wermelinger , Jos Luiz Fiadeiro, Algebraic software architecture reconfiguration, ACM SIGSOFT Software Engineering Notes, v.24 n.6, p.393-409, Nov. 1999 Antnia Lopes , Jos Luiz Fiadeiro, Adding mobility to software architectures, Science of Computer Programming, v.61 n.2, p.114-135, July 2006 Dianxiang Xu , Jianwen Yin , Yi Deng , Junhua Ding, A Formal Architectural Model for Logical Agent Mobility, IEEE Transactions on Software Engineering, v.29 n.1, p.31-45, January

transient interactions;connectors;UNITY;software architecture

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A Framework-Based Approach to the Development of Network-Aware Applications.

AbstractModern networks provide a QoS (quality of service) model to go beyond best-effort services, but current QoS models are oriented towards low-level network parameters (e.g., bandwidth, latency, jitter). Application developers, on the other hand, are interested in quality models that are meaningful to the end-user and, therefore, struggle to bridge the gap between network and application QoS models. Examples of application quality models are response time, predictability, or a budget (for transmission costs). Applications that can deal with changes in the network environment are called network-aware. A network-aware application attempts to adjust its resource demands in response to network performance variations. This paper presents a framework-based approach to the construction of network-aware programs. At the core of the framework is a feedback loop that controls the adjustment of the application to network properties. The framework provides the skeleton to address two fundamental challenges for the construction of network-aware applications: 1) how to find out about dynamic changes in network service quality and 2) how to map application-centric quality measures (e.g., predictability) to network-centric quality measures (e.g., QoS models that focus on bandwidth or latency). Our preliminary experience with a prototype network-aware image retrieval system demonstrates the feasibility of our approach. The prototype illustrates that there is more to network-awareness than just taking network resources and protocols into account and raises questions that need to be addressed (from a software engineering point of view) to make a general approach to network-aware applications useful.

INTRODUCTION applications use networks to provide access to remote services and resources. However, in today's net- works, users experience large variations in performance; e.g., bandwidth or latency may change by several orders of magnitude during a session. Such dramatic changes are observed in mobile environments (where a user moves from one location to another) as well as in stationary environments (where other network users cause con- gestion). Variations in network performance are a problem for applications since they result in unpredictable application be- havior. Such unpredictability is annoying-e.g., if a user looks through an on-line catalogue, a certain bandwidth must be continuously available if the system wants to display images at the J. Bolliger is with the Department of Computer Science, Swiss Federal Institute of Technology (ETH), Z-urich, Switzerland. E-mail: bolliger@inf.ethz.ch. Effort sponsored in part by ETH Polyprojekt 41-2641.5. T. Gross is with the Department of Computer Science, ETH, Z-urich, Switzer- land, and with the School of Computer Science, Carnegie Mellon University, Pittsburgh, PA 15213. E-mail: thomas.gross@cs.cmu.edu. Effort sponsored in part by the AdvancedResearch Projects AgencyandRome Laboratory, Air Force Materiel Command, USAF, under agreement number F30602-96-1-0287. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or en- dorsements, either expressed or implied, of the Advanced Research Projects Agency, Rome Laboratory, or the U.S. Government. speed expected by the user, or when congestion frustrates the user to the point that the software becomes unusable. To bridge the gap between network reality and application expectation, i.e. to cope with the performance variations and to provide for a certain predictability of the application behav- ior, a number of researchers have proposed the development of network-aware applications. The basic idea is to allow an application to adapt to its network environment, e.g., by trading off the volume (and with it the quality) of the data to be transferred and the time needed for the transfer. That is, the application responds to a drop in bandwidth by reducing its demands on the networks, and increases its demands when there are additional resources. To develop a meaningful approach to such adaptation, we must understand the realities of today's network architectures and the dynamics of the services provided. There can be many reasons for the variation in network performance. Some of the reasons are inherent (e.g., for mobile wireless communication), others are caused by the tremendous demand that always seems to outgrow any capacity improvement. In response to this sit- uation, modern networks are beginning to move away from the best-effort service model to QoS models that allow the definition of quality metrics based on a variety of parameters. Un- fortunately, current QoS models are oriented towards low-level network parameters (e.g., bandwidth, latency, jitter). Application developers, on the other hand, are interested in quality models that are meaningful to the end-user, such as response time. Thus, network awareness includes mapping application-centric quality measures (e.g., predictability) to network-centric quality measures and vice versa. Another motivation for network awareness is to avoid the distinction between different application modes. For example, some image retrieval systems distinguish between a preview (or browse) mode, where only thumbnails are provided, and a mode of higher quality image delivery. Avoiding the concept of a mode simplifies implementation of the application components and allows the system to dynamically take advantage of available resources. A user on a high-bandwidth local area net-work does not have to live with a thumbnail-sized view that is statically defined and optimized for users accessing the image server across a (slow) wide-area network. A time limit parameter that controls how long a client is willing to wait provides enough flexibility to toggle implicitly between the browsing and the high-quality mode. Applications may need to adapt either at start-up time or dynamically during the course of a session or both. There exist a number of network-aware applications, in particular from the realm of multimedia [22], [2]. However the solutions to the problem of network variability adopted by this class of applications are often tailored to the specific needs of IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. 24, NO. 5, MAY 1998, 376-390 an individual application or a specific programming model [41], and there exists no general approach to develop network-aware applications for other application domains. As network awareness continues to be an important aspect of application develop- ment, the need arises to identify and provide a general approach to build network-aware systems. We propose to use frameworks as an approach that encapsulates (and integrates solutions to) the problems of adapting an application's behavior to the availability of network resources. A framework provides a basic solution to a class of problems; clients of the framework employ the basic structure by exten- sion, i.e. they provide concrete methods where the framework relies on abstract methods [39]. So to build network-aware applications by extending a framework, we must develop the over-all structure, which is the foundation for a framework, as well as the specific extensions that result in a real system, as has been done in other application domains where frameworks have proven useful. The paper is organized as follows: Section II discusses issues related to the problem of network-awareness. Section III introduces the basic structure of our framework and the service model supported; Sections IV and V provide a detailed description of the methods employed to obtained information about net-work resource availability and the strategies used to adapt to changes in service quality respectively. After presenting performance measurements in Section VI we summarize related work and present our conclusions. II. NETWORK- AND SYSTEM-AWARENESS Networks are just one of the many resources employed by an application. The model of a network-aware application emphasizes the crucial role of the network connection: in many cases, the network is on the critical path, and performance problems in the network are the cause of the degradation of application per- formance. However in other cases, a system is bottlenecked by other components, e.g., the transfers across a local bus or from the disks, or the amount of computation. (Some experimental systems support a QoS model for internal transfers [15], [10], [9].) If application performance is limited by parts other than the network, then such an application should not be network-aware but system-aware, i.e. it should be able to adjust its behavior in response to other aspects of the system (response time, disk I/O latency, bus bandwidth, etc. In the context of this paper we focus on the concept of network-awareness implying that an ap- plication's behavior is primarily controlled by the availability of network resources, but we point out where an application must go beyond network issues. System-awareness is especially important if an application wants to trade off communication and computation, i.e. an application may adjust to network changes by computing, e.g., compression. In such cases it is important to make sure that the computation overhead is not worse than the network overload. For our discussion of network service quality awareness we concentrate on unicast request-response type communication between clients and servers, where the traffic in at least one direction can be described as bulk-transfer type network traffic. This traffic pattern makes up a large fraction of application traffic patterns observable in today's networks [5], [29]. A. Reservation vs. adaptation Another approach to couple a service quality-aware application to a network is to allow the application to reserve network services in advance [40]. We do not discuss the relative benefits of either approach since in practice, both of them coexist (and continue to do so for a long time). Some network architectures (or their implementations) may not support reservations at all or may support them only to a limited degree (either by choice or due to implementation faults), and although future versions of popular protocol suites may support reservations, not all sites will run the most recent software. Furthermore, as network providers attempt to develop usage-based charging schemes, there will be financial incentives to restrain applications from uncontrolled use of network resources. (Today's networks have really two aspects that make adaptivity unattractive: there is almost no usage-based charging, and what is worse, the most aggressive applications are often rewarded with the largest share of the bandwidth pie [12].) In a reservation-based approach an application must address the two issues of (i) how to find out what and how much to reserve (e.g., given some limit on the costs) and (ii) how to adjust to meet the confirmed reservation, which may be less than the application has asked for. From a software engineering point of view, however, both techniques require the same software technology: an application must be able to adjust its resource demands, either to meet a limit imposed by a reservation or to meet some constraints imposed by the network. In either case the application must be adaptive. B. Quality The objective of network-awareness is to allow an application to be sensitive to changes in the network environment with the goal of maximizing user-perceived quality. In our context, quality means "conformance to a standard or a specification". Our focus on system-awareness means that we are interested in "the totality of features and characteristics of a product or service that bears on its ability to satisfy given needs" [17]. Only the application (developer) knows what "quality" is. So we build an infrastructure for those applications that are interested in a quality-time tradeoff, i.e. applications that are willing to sacrifice some degree of quality in return for faster response time (or are willing to wait a little longer to get better results). So the central issue is that we must find a software structure that allows the application developer to specify what "quality" means in the context of a specific application. III. FRAMEWORK FOR NETWORK-AWARE APPLICATIONS Before we can discuss a specific framework, we first want to lay out a roadmap for the kind of interaction that is possible or profitable between an application and the network. The framework then provides, for some class of applications, a way to structure their interaction with the network through extending the framework. We start with principles of application-network these principles stem from our experience with various application projects and reflect study and rework involved while factoring possible framework structures. We illustrate BOLLIGER AND GROSS: A FRAMEWORK-BASED APPROACH TO THE DEVELOPMENT OF NETWORK-AWARE APPLICATIONS 3 the general principles with examples from a specific project, the Chariot (Swiss Analysis and Retrieval of Image ObjecTs) project [1], which is described in more detail in Section V-A. The objective of the Chariot project is to allow networked clients to search a remote image database. The Chariot system contains an adaptive image server that serves as proof-of-concept for the general ideas presented in the remainder of this paper. A. Service model Many networked applications using request-response type communication include a user (client) that requests a set of objects (images, texts, videos, byte code, etc.) from a remote site (server), which is responsible for retrieving the requested objects (from secondary storage) and delivering them to the client. The response usually has a larger volume than the request and dominates the transmission costs. In the following, we sometimes refer to such servers and clients as sender and receiver of the bulk-transfer, respectively. A server accepts and acts upon request messages containing a list of objects to be retrieved (or computed) and some QoS- restrictions, where QoS-restrictions characterize the minimum quality tolerable for the objects delivered, the maximum quality that is beneficial for the user, and a limit T on the time allowed for processing the request and transmitting the response. The bounds on the quality may be (implicitly) imposed by the re- quester's processing or display capabilities. The application decides what kind of objects can be requested; quality is a property of a requestable object and must also be defined by the application Example (Chariot): Requestable objects are images or image sequences. The quality of an image is defined by the resolution, color depth, the image format (e.g., JPEG, GIF), a format-specific parameter, such as JPEG's compression factor [18], and a user-defined weighting of these image characteristics. The server's task is to deliver all the requested objects to the client within time T , attempting to maximize the overall quality of the objects transmitted while respecting the QoS-restrictions. The range for dynamic adaptation to bandwidth availability is bounded by the minimal and maximal quality specified by the client. To quantify the task of the server, a quality metric must be defined by the application, e.g., as a weighted sum of the individual object qualities to be delivered. Weights for the quality metric may include the relative importance of an object in comparison to the other objects in the request list. Example (Chariot): The weight of an image in the image request list is determined by a value for the similarity of the image with respect to a query image. Such a network-aware server need not only dynamically adapt due to network service degradation (e.g., a drop in bandwidth), but should also try to opportunistically exploit extra bandwidth to deliver as many high quality objects as possible within time T . Network-aware applications adhering to the service model above must address the following questions: (i) how to find out about dynamic changes in network service quality on the path from the sender to the receiver, and (ii) how to adapt the delivery process to such dynamic changes such that the objectives of the service model are met. Before we turn to each question in detail in Sections IV and V, we present a general structure for the type of network-aware application under consideration. B. Application structure: software feedback control loop A useful structure for network-aware applications using request-response type communication is a software feedback control loop, where the time left for the response-initially set to T -constitutes the command variable of the closed-loop con- trol. The feedback driving the sender adaptation comprises information about the currently available bandwidth as obtained by mechanisms described in Section IV. We focus on closed-loop control systems because they are in a position to deal with bursty applications. Other applications, e.g., those that deal with continuous media streams, may use a different control structure [2], [22]. We model sender-initiated adaptation in a closed-loop control system with the three phases monitor and react (P mr ), prepare (P prep ), and transmit (P trans ), as depicted in Fig. 1. The three phases work independently and share the list L of requested but not yet transmitted objects. P mr is responsible for obtaining information (or feedback) about the available bandwidth, determining whether the amount of data to transmit must be reduced or whether it may be increased. In case adaptation is needed, the P mr phase must decide which objects to adapt, which transformations to apply, and must then set the quality state of the objects according to these decisions. The term "transformation" refers here to any activity, including transfers, conversions, or computation. Once a (final) decision on the quality of an object to be delivered has been made, P prep must transform the object to the quality assigned by P mr . P trans delivers completely prepared objects to the client. Note that P mr does not invoke transformations directly, but defers their execution to forthcoming phases P prep to allow for "last-minute" adaptation. Furthermore, note that while P mr may need to change the quality state of several objects at the same time, P prep makes only one object ready for transmission at a time (on a uniprocessor). IV. FEEDBACK FROM THE NETWORK A central issue that determines the effectiveness of the control loop (and the frameworks built on this loop) is how it obtains information about the state of the network. A. What does a network-aware application want to know? With the aim to provide predictable service, i.e. response delivery within a specified amount of time T , an application ideally wants to know the network service quality, and in particular the bandwidth available for the time T . With a best-effort network service model, such as IP's and ATM ABR's [7], there is no way of getting such information in advance. Thus, all we can do is gather as much QoS-information about the past behavior as possible (and useful) and extrapolate future network behavior from the observed QoS-values. We can distinguish two different application-relevant characteristics as far as bandwidth feedback is concerned: bottleneck bandwidth and available bandwidth [30]. The former gives an upper bound on how fast and how much an application may possibly transmit, while the latter gives an estimate on how fast the connection should transmit to preserve network stability, which is an issue of primary concern to congestion control mechanisms 4 IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. 24, NO. 5, MAY 1998, 376-390 mon t or & reac - mon tor/po bandw i dth - nu l y | t d | by object quality reduction or expansion deliver object to client prepare transform available version of object to desired object quality reques t recep t i on connection handling request processing ng list of requested objects feedback from network ob ject de l very across network app l ca t i on ayer adap t a t i on aye r ( con t r owe r aye r s Fig. 1. Control-loop consisting of three phases monitor and react (P mr ), prepare (P prep ), and transmit (P trans ) While knowledge about the bottleneck bandwidth is useful in bounding the approximations for the bandwidth estimates used by a network-aware sender, information about the dynamics of the available bandwidth on the end-to-end network path is indispensable to enable timely adaptation of the volume of data to be transmitted. B. Three approaches to obtaining feedback This section discusses three approaches to obtain feedback about the characteristics and the dynamic behavior of an end- to-end network path. The distinction is based on the layering of the ISO/OSI-protocol stacks. The higher the layer providing the feedback, the less cooperation is required from network protocols on one side, but the less accurate and frequent will the feedback information be on the other side. Feedback about network service quality may be provided by: Application-level QoS monitoring: A monitor assesses the dynamics of network service quality by measuring sender and receiver network quality parameters (e.g., packet inter-arrival times, bandwidth, etc.) and repeatedly exchanges the QoS-state between the peers, similar to the model proposed in RTP [35]. The timeliness and accuracy of the information depends on the averaging interval used for the computation of the QoS-values and the frequency of the QoS-state exchange. The monitoring approach provides only a black box view of the network and transport ser- vices. Therefore the sender has difficulties in distinguishing between service degradation caused by the network and degradation caused by the application or the end-system. E.g., a (temporarily) slow receiver may lead the sender to wrongly assume a network service degradation. End-to-end transport-level congestion control: The goal of a congestion control algorithm is to operate at a connection's fair share of the bandwidth. To do so it must deploy mechanisms to find the bottleneck bandwidth and detect incipient congestion or network under-utilization. The implicit feed-back that drives the adaptation of the sending rate may include the fraction of packets lost or measurements of delay variations, interarrival times of packet-pairs, etc. Several benefits can be gained from making such transport-level feedback information transparent to a network-aware appli- cation: the feedback-loop is shortened and queuing unnecessary data for transmission can be avoided in times of con- gestion. Such information may help in bringing the appli- cation's behavior in line with the protocol's behavior, since the application has the same view of the network resources as the protocol. Furthermore, if the congestion control algorithm can make transparent its conclusions about the available bandwidth, an even tighter coupling between application and network can be achieved. Network-level traffic management: Routers are most suited to fairly allocate resources among competing connections. Routers are the only authority capable of identifying and isolating misbehaving senders. Furthermore, routers are able to provide explicit feedback about their congestion state to the end-systems. Each router on an end-to-end path may generate feedback messages (either in binary form [31] or as an explicit rate information [7]). The feed-back must be processed in the end-systems to find the available bandwidth used to control the sending rate. It is important to note that the different layers may have different perceptions of the current network status since they employ different mechanisms to deal with exceptions such as loss events. However, as far as the estimation of available resources is concerned they all strive for a view as accurate as possible as it helps them avoid exception situations. Therefore, each layer may provide the information needed by a network-aware appli- cation, however, the lower the layer the more timely and accurate the information will be. BOLLIGER AND GROSS: A FRAMEWORK-BASED APPROACH TO THE DEVELOPMENT OF NETWORK-AWARE APPLICATIONS 5 ca t on mon ne t wo r k adapta t i on app l ca on mon ne wo r k adap t at on response (bulk data stream) reques response QoS-state exchange error ow con t ro exchange rate contro exchange send() ge _bw() r ecv Fig. 2. Layered architecture of network-aware applications with the adaptation layer implementing the closed-loop control of Fig. 1. C. Unified API Although all three layers employ different feedback mecha- nisms, they all aim at finding the available network service quality to control an application's sending rate. Therefore, we devise a unified API for network service quality feedback in general and bandwidth feedback in particular that provides a network-aware application with the required information. As the application is interested in obtaining predictions about the band-width to be expected (or any other QoS-value) and an estimation for the reliability of the prediction, we extend a common transport protocol API, the socket API [37], by a function get bw() that returns bw(t) predicting the bandwidth for t ? now, and prob bw (t), an estimate for the stability of the prediction 1 . Note that to ease framework development we provide the same QoS- interface at each layer in Fig. 2. Note also that both the monitor- and the adaptation-layer are logically part of the application. As provision of dynamic network QoS-information is not the main topic of this paper we restrict our discussion to exemplifying how end-to-end congestion control information can be made transparent to an application through the API described. Our implementation of a (TCP-based) user-level transport protocol [4] distinguishes three high-level sender (congestion) states: start-up (slow start), congestion avoidance, and congestion recovery [19]. Each of the three state-classes (see State pattern in [14]) provides the function get bw(): (i) The slow-start phase uses packet-pair probing to estimate the bottleneck bandwidth 2 bw max and returns the function RTT ), where cwnd denotes the current congestion window and RTT stands for the (mea- sured) round-trip time. In this phase, bw(t) reflects slow- start's doubling of the bandwidth occupied every round-trip time, which is represented by the ratio of cwnd and RTT . The exponential increase continues up to (at most) the net-work path's bottleneck capacity bw max . (ii) When the protocol is in the congestion avoidance state, i.e. when operating at the bandwidth effectively available, we deploy TCP Vegas-style network-path adaptation [3], and can therefore approximate bw(t) - cwnd RTT , as changes to cwnd are supposed to happen on a fairly large time-scale (multiples of the round-trip time). (iii) In the congestion recovery state, which effects a rate 1 For the sake of brevity we only discuss bandwidth-related functions. Similar API-extensions exist for other QoS-parameters, such as delay or loss. This process is known as initial slow-start threshold (ssthresh) estimation [3], [16]. Note that standard TCP uses a statically defined ssthresh of 64 KBytes. halving, bw(t) is modeled according to [21]. Congestion control's use-it-or-lose-it property [11] requires the sender to be almost constantly sending, otherwise the feed-back may not be useful. Moreover, the issue of dynamically assessing the stability of end-to-end network-path characteristics is an open research question, which is why we refrain from discussing how to compute prob bw (t) here and refer to off-line studies on this topic [30]. V. FEEDBACK LOOP AND ADAPTATION As stated in the previous sections, the goal of a network-aware sender is to meet a user-specified bound on the delivery time by adapting the quality of the objects delivered to the mea- sured/available network capacity. The adaptation process' objective must be to utilize the available resources as efficiently as possible and therefore to maximize the user-perceived quality within the bounds (time, bandwidth, and boundary conditions on quality) given. The following sections discuss in more detail the mechanisms deployed in our prototype network-aware system and elaborate on where and how application-specific information can/must be factored out of the software control system described to provide a reusable framework. However, before we turn to the framework structure and its interaction with an appli- cation, we briefly introduce the Chariot system as an example of the type of application that can be based upon this framework. A. Chariot: sample framework instantiation The objective of the Chariot project is to allow networked clients to search a remote image database. The Chariot system uses query-by-example to let a user formulate a query for similar images [1]. The low-level content (e.g., color and texture) of each image in the repository is extracted to define feature vectors, which are organized in a database index at the search engine. The core of the system (as depicted in Fig. of a client (to handle user access to the image library), a search engine to identify matching images, and one or more network-aware servers, which deliver the images in the best possible quality, considering network performance, server load, and a client-specified delivery time. Physical separation of the image library index (in the search engine) from the image repository (in the server) facilitates distribution and mirroring of the library. The core components are connected by a coordination layer that isolates the details of network access and adaptation and gives each component a maximum of flexibility to take advantage of future developments. 6 IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. 24, NO. 5, MAY 1998, 376-390 GUI Client Server Search Engine request delivery query reply Coordinator: connection handling Coordinator: connection handling and message convertion image retrieval and delivery Coordination Layer feature vector extraction and indexing Coordinator: connection handling Fig. 3. Chariot architecture It is the subsystem comprising client and the adaptive image repository that is relevant to our discussion of network- awareness and which serves as proof-of-concept for the ideas presented in this paper. B. Monitor- and react-phase Table I summarizes the terms and abbreviations introduced in the next sections. The discussion of application-specific information that is factored out of the control loop framework and which must be provided by the application (developer) always refers to the OMT-style [33] class hierarchy depicted in Fig. 4. The name of abstract classes, which are part of the frame- work, and abstract methods is shown in italics. Concrete classes provided by the application that instantiates the framework- Chariot in our example-are shaded. In the text we use "func- tional" notation. E.g., foo(ob j) or bar(class) indicate that the method foo is invoked on object ob j (ob j: foo() in our C++ im- plementation) or that the method bar is invoked from class class (class :: bar()), respectively. The monitor- and react-phase (P mr ) is the key phase in our framework. It is responsible for repeatedly obtaining feedback from lower protocol layers and deciding whether adaptation is required or not. The software control loop is part of the application and may be layered on top of a network monitor (see Fig. 2), from which it extracts feedback information about the available network service quality (e.g., bandwidth). As P mr is primarily interested in feedback about "relevant" changes in service quality, it either must deploy a polling policy to obtain feedback about the available bandwidth and assess the significance of a QoS change on its own, or it has to register with the monitor layer for asynchronous notification of QoS change events. Whether a change in network service quality is relevant is application-specific and depends on the granularity of the adaptation possible, the cost incurred by the adaptation mechanisms as well as on the bandwidth and processing power available (Section V-C). In both cases, P mr is executed repeatedly to establish whether adaptation (e.g., data reduction) is required to account for a net-work service degradation or whether adaptation is beneficial to prevent network under-utilization. To do so the application-level quality must be mapped down to network-level quality parameters such as the bandwidth required or the amount of data remaining to be shipped (d left ). d left , together with the feedback on the available bandwidth, can be used to compute the time needed needed ) for the transfer. Corrective action must be taken if t needed and the time left (t left ) differ "significantly". (Signifi- cance depends also on the size of the objects as well as network and application properties.) B.1 Application-to-network QoS-mapping The kind of QoS-mapping that enables the comparison between t needed and t left requires the application to provide a function data(quality) that computes the amount of data necessary for a given object quality (see member function Quality :: data() in Fig. 4). d left is then determined by the sum of data(quality(ob j)) of the objects ob j not yet delivered. Given d left and get bw(), which estimates the band-width available at time t in the future, we can compute t needed by integrating (i.e. by summing up piecewise continuous parts of) the function bw(t) over time t until an area (i.e. data vol- ume) is covered which exceeds d left . Thus, t needed represents the time needed to transfer d left given bandwidth bw(t). This fairly general statement must be qualified to avoid misinterpretations: with a best-effort network service model bw(t) can hardly be predicted for more than a few round-trip times with a reasonably high probability at the transport level (Section IV). Therefore, approximates the available bandwidth after these first few round-trip times with simple constant or linear functions based on past measurements. This approximation simplifies the computation of t needed ; knowledge about the bottleneck bandwidth is used to bound the approximation. Note that it is only for the time needed to prepare and transmit the next object that P mr needs to estimate future network behavior to be able to satisfy the user's request within the time limit- the reason is that the control loop gets an opportunity to take corrective action during the next iteration of P mr , if required. In case we do not have such estimates, or if the conditions above cannot be met, e.g., because we are dealing with large objects, for which transmission takes longer than the system can reliably predict bw(t), the situation is more complicated. Either the control loop gets a chance to take corrective action (because the time limit did not expire), or the data cannot be sent in the allot- BOLLIGER AND GROSS: A FRAMEWORK-BASED APPROACH TO THE DEVELOPMENT OF NETWORK-AWARE APPLICATIONS 7 I ABBREVIATIONS USED IN THE PAPER react-phase prep prepare phase trans transmit phase user specified time limit for response delivery t left time left to deliver response, initialized to T c prep CPU resources used to prepare objects for transmission t prep time needed to prepare objects (given c prep , load(t)) t trans time needed to transmit objects (given bw(t)) needed time needed to deliver response (given t trans , t prep ) error variable of control loop (t needed \Gamma t left ) d left data remaining to be transmitted d reduction reduction potential of an object bw(t) bandwidth estimation/prediction, t ? now load(t) system load estimation/prediction, t ? now Chariot Class Object Quality data () Algorithm prepare_costs() Request original, current transforms requested_objects ImageObject ImageQuality data () ImageScaling quality(obj, p) prepare_costs() prepare() algorithm_iter() ImageCompr prepare_costs() return w h - depth compr_ratio weight time_limit h, depth, compr_ratio new ImageQuality (-p obj.orig.w, -p - obj.orig.h, obj.orig.depth, obj.orig.compr_ratio) cur_trans cjpeg -quality param obj.orig cur_trans.transform(this, param) Framework Class abstract_method() Fig. 4. Application-specific part of the class hierarchy (OMT-notation [33]) ted time. In the latter case, the application must be able to deal with the breakdown of the service model (Section V-E). B.2 Network-to-application QoS-mapping The goal of P mr is to bring t needed in line with t left by either reducing or increasing the (overall) quality of the objects remaining to be delivered; these actions thereby reduce or increase d left . The following questions must be considered while the sender tries to compensate for the difference t dif needed by adapting the quality of the data awaiting delivery: (i) Which object(s) should be chosen for adaptation (victim choice)? (ii) How should the amount of quality adaptation be distributed among the chosen objects? Most importantly, how does the sender find the amount of quality adaptation needed given the volume of data adaptation required (d dif f ) (quality distribution)? (iii) Which algorithms should be used to accomplish a desired adaptation (algorithm selection)? To a certain extent most of the questions above are application-specific and therefore cannot be answered in gen- eral. Thus, a framework for network-aware applications must provide flexibility in replacing, refining, or extending strategies as described in the following paragraphs. (i) Victim choice: One strategy to chose objects for adaptation (i.e. victims) proceeds along the following idea: if quality reduction is required, choose the objects with the lowest weight- quality product, because they influence the overall quality the least. In case expansion is needed, the objects with the highest weight-quality product should be chosen for analogous reasons. Note that the metric used for the victim choice depends on the application, e.g., in an image retrieval system it may be better to only decide according to the weights, which are based on similarity measures, because they reflect which images the user is really interested in. (ii) Quality distribution: Given a set of victims to be reduced (or expanded), ideally the individual objects are reduced in quality inversely proportional to their weight-quality product (or their weight respectively). However, a problem arises 8 IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. 24, NO. 5, MAY 1998, 376-390 here because the system must satisfy two objectives at two different levels: On one hand, it aims to balance the application-level quality reduction according to the relative importance of the objects (i.e. their weight) and on the other hand, it needs to achieve a data reduction of a certain amount (d dif f ) at the net-work level. The problem is that the mapping from network to application quality measures is generally ambiguous (in contrast to application-to-network mapping). Example (Chariot): to effect an image size reduction of a factor N the server may either scale the image down by N, reduce the color depth by a factor of N, find a JPEG quality factor that achieves such a compression ratio or use any combination of the image transformation algorithms mentioned. Although we find transformations that achieve a certain data reduction, the direct effect on the quality reduction is not known since the application-level quality depends on the user-specified weighting of the individual quality attributes (e.g., resolution, color depth). This ambiguity makes it hard to guarantee balanced quality reduction and to find the required quality reduction efficiently in general. A straightforward but inefficient solution simply computes and compares the data and quality reduction of all the algorithms. Unless the application provides additional hints such as the continuity of the data(quality) function, there is not much chance to improve upon such an approach. Finding efficient and generally applicable approaches to this aspect of application-controlled QoS-mapping is still an area of ongoing research. For our prototype system we make the simplification that data(quality) is a linear function of quality; this assumption implies that the system must find only a "fair" distribution of d dif f that respects the weights of the individual objects (see Section V-D.2). The problem of network-to-application QoS-mapping is further complicated since (a) the adaptation potential of an object (limited by the boundary conditions on min/max quality) must be taken into account, and (b) the transformations applied on the objects consume host resources and time. Therefore, the transformations indirectly impact t needed . We address the issues related to (b) in Section V-C. (iii) Algorithm selection: The choice of the (transformation) algorithm to accomplish a given quality adaptation is closely related to the issue of how much quality adaptation is required for each victim. There is usually an application-dependent choice as indicated in the example above. In our prototype framework we require the application to specify a list of transformation algorithms for each class of objects that can be part of a request. Each algorithm must provide a list of parameter values appli- cable. In addition, the application must provide functions that help the adaptation process estimate the data and quality reduction potential of an algorithm on a per-object basis. C. System-awareness Quality adaptations (e.g., by means of transformations, such as compression) cost CPU-resources and take a non-negligible amount of time to be completed. On one hand, a reduction in object quality may result in the desired reduction of transmission time, on the other hand, the transformations necessarily imply higher CPU-costs than simply retrieving an object (or image) from disk. Obviously, we want to avoid situations where a reduction of object quality in an attempt to reduce the error variable incurs prepare costs (t prep ) that are higher than the gain in transmission time (i.e. t prep ? t dif f ). Therefore, our resource model also includes t prep , the time needed for the phases P prep . The adaptation process is still driven by network resource availability but additionally controlled by host resource consumption and availability. Each transformation algorithm registered for the requested objects must provide a function prepare costs(ob j; param;cpu) returning an estimate for the costs (c prep ) to transform ob j from its original quality state to the one currently assigned on a given cpu. c prep denotes the costs in terms of resources used, e.g., as given by system and user CPU time on Unix systems. c prep is used to compute an estimate of the effective t prep needed for a transformation by using an operating system dependent function prepare time(c prep ; load), where load denotes the average length of the process run-queue for example. (For most Unix systems the time needed for a given task using c prep CPU time at a system load of load can be approximated by c prep \Delta load, up to a certain maximum load-level). The effectiveness of the adaptation process and the reliability of the server to meet the QoS-constraints depend on the accuracy of all the models and estimates introduced in the last sec- tions: bw(t), data(quality), prepare costs(ob j; param;cpu), prepare time(cost; load), etc. The more accurate the estimates used in the decision-making, the higher the probability that the sender is able to meet the time constraints. Example (Chariot): The server computes c prep as a function of image size and param used for the transformation algorithm. In contrast to approaches typically found in real-time systems, which rely on worst-case predictions for c prep , our server bases its estimates on statistical data gained during past measurements of request processing. We derived regression models for both an al- gorithm's cost and its reduction potential. The regression models are regularly updated with new measurements. C.1 Practical considerations: communication latency hiding In a simple implementation of the software control loop, the phases of the framework execute sequentially. The adaptation produces stable results if t prep trans for the adapted object is smaller than t trans of the original object. However, sequential operation wastes bandwidth while the host is busy preparing the next object for transmission and wastes CPU resources while transmitting objects over a slow end-to-end path. With a slow connection, the sender is almost constantly congestion- controlled, and there are ample CPU cycles. An improved control loop tries to keep P trans constantly sending and uses threaded prepare and transmit phases to hide the latency of the object de- livery. Communication latency hiding calls for a different cost model: t needed is no longer computed as t prep trans , but is approximated denotes the fraction of t prep that is not available for latency hiding [38]. Although the intrinsics of the various resource models are outside the scope of this paper, the discussion above emphasizes the need for suitable abstractions. To allow for future refinements and extensions we encapsulate the computation and communication model deployed by a function overall time(t prep ; t trans ) that can be used to compute t needed . BOLLIGER AND GROSS: A FRAMEWORK-BASED APPROACH TO THE DEVELOPMENT OF NETWORK-AWARE APPLICATIONS 9 D. Using the framework to implement an adaptive system Fig. 5 summarizes the steps involved in computing the error variable t dif f that drives the adaptation process. The function compute t dif f () takes the request, i.e. the list of objects not yet transmitted, and the functions bw(t), load(t) as argu- ments. In addition, it uses the global variables t left and cpu. compute t dif f () is then used by the function adapt(), which is sketched in Fig. 6, and is invoked repeatedly by P mr after obtaining new bandwidth feedback bw(t). If t dif f exceeds an application-specific threshold e limiting oscillation, the remaining objects in the request are subject to the adaptation process described in the next sections. To accomplish the adaptation, the sender must find objects to transform. Given the list of objects that must be transmitted, there are several possible approaches to identify the victims, distribute the quality reduction, and select the transformation algo- rithms. We discuss here two such approaches. D.1 General exhaustive search To avoid congestion and network under-utilization the adaptation process should aim to find a combination of objects to adapt and transformations to apply such that j t dif f j is minimized and the overall quality metric is maximized. Unfortunately, an exhaustive search for the global minimum of j t dif f j in the whole solution space is not attractive, as we illustrate in the next paragraphs Given a request consisting of N objects and given M transformation algorithms each taking m different parameter values on average, there are n - M possible transformations applicable to each of the objects. If we assume that all the possible combinations fulfill the QoS-restrictions, there are approximately possibilities to adapt the request to the currently available bandwidth. In each iteration of P mr , the sender must compute t dif f for each of the N n points in the solution space and, e.g., find the combination with the smallest j t dif f j. As an alter- native, the sender can try to find the combinations with jt dif f and choose the one with maximal -weight \Delta quality. As long as there is no additional information about the functions used to compute t dif f (e.g., gradients), or as long as the quality boundaries are not very restrictive, the size of the solution space cannot be reduced, and hence the complexity is too high to make this approach feasible in the general case. There- fore, we cannot include a generic method to perform exhaustive search in the framework, since we expect the methods of the framework to provide a solution for all possible extensions. However, we can provide the application with several strategies [14] for the adaptation process (one being exhaustive search for example) and leave it to the application developer to decide on the most appropriate strategy to use in the context of the application D.2 A practical approximative search If N or n are large, the sender must either employ some approximations or introduce simplifications in the adaptation process to reduce the complexity of the adaptation process, otherwise the search is so expensive that the resource consumption of P mr must be included in the cost models. For the sake of simplicity we restrict our discussion to the former case. The idea that forms the basis of the currently implemented adaptation process is to approximate the search for a minimal iteratively trying to apply the possible transformation algorithms with their respective parameters with the objective to find a local minimum that is within the tolerance. If one algorithm does not achieve the desired result, the next algorithm is chosen [24]. The adaptation phase, i.e. the reduce() function in Fig. 6, then proceeds along the following steps (see Section V- (i) the victims are chosen as the first n objects from the request list, which is ordered by increasing weights, such that reduction (ob represents the amount of data reduction that is required to compensate for t dif f . d reduction (ob j) denotes the reduction potential of the current quality state of object ob j, which is bounded by the minimal quality tolerated by the user. If no such set of n objects exists with which the necessary data reduction can be achieved an exception is thrown, which is caught and handled in adapt() (Fig. 6). (ii) With the simplifying assumption data - quality, d dif f is distributed among the victims by assigning the reduction needed for each object to a fraction of d dif f inversely proportional to the object's relative weight (in the request list), unless d reduction poses a limit on the reduction attainable. In such a case, the distribution step is repeated, as long as there are objects whose reduction is limited by d reduction and as long as t dif f has not been fully compensated for. (iii) The transformation algorithm selection is done by iterating over the algorithms, the objects, and the parameters. In each step t dif f is computed and the iteration terminates Note that the adaptation process outlined makes heavy use of the iterators shown in Fig. 4. Use of iterators facilitates experimentation with different priorities of the transformation algorithms used. Based on our experience with Chariot, we found that being able to cleverly apply application-knowledge to set priorities is essential for the effectiveness of the approximative search. E. Problems with feedback control This paper describes the overall structure of a framework for network-aware applications. Several practical issues have not been mentioned or discussed in detail: Start-up behavior: Special care must be applied to find the optimal operating point of the control loop as soon as possible while avoiding overshooting and an excessively conservative (i.e. slow) start-up. For a network-aware sender this requirement means that the server ought to start delivering objects as soon as possible to get early feedback. Furthermore, the sender should refrain from sending too large an object at the start, in case bandwidth turns out to be unexpectedly low. These requirements impact application design as follows: an application should either (i) allow the list of requested objects to be reordered, such that objects ob j with small data(ob j) that need not or cannot be adapted are sent first; (ii) be able to cope with an interrupted object delivery that may be restarted in lower ob j2request data(quality(ob j)) ob j2request prepare costs(ob j; algorithm(ob j); param(ob j);cpu) prepare time(c prep ; load(t)) Fig. 5. Function compute t dif f (request;bw(t); load(t)) returning t dif f try f prevent congestion else prevent under-utilization catch (NoAdaptationPossible exception) f handle (exception); // application specific handler Fig. 6. Function adapt(request;bw(t); load(t)) quality; or (iii) support hierarchical encoding and progressive delivery of objects, such that the transmission can be stopped at any time. Bandwidth probing by the lower layers of the communication system allows to estimate the expected bandwidth after just a few RTTs (e.g., packet-pair probing [20], [30]) and can also help to alleviate the problems with start-up behavior. Communication idle time: Gaps in the sequence of object transmissions should not only be avoided because of the transmission opportunities lost at the application level, but also because many congestion control mechanisms exhibit a use-it-or-lose-it property [11]. That is, communication idle time results in loss of the fair share of the bottleneck bandwidth previously held by the connection and consequently results in repeated start-up behavior. Latency of prepare and transmit activities: With our model of dynamic adaptation to network service quality, a network-aware sender must rely on either good bandwidth estimates or on the expectation that network service does not degrade more during t prep trans of the next object than there is data reduction potential inherent to the remaining objects in the request list. Due to the nature of best-effort network service these assumptions may not be fulfilled. Such a situation results in the breakdown of the service model. Ill-specified boundary conditions are another cause of failure that requires application-specific reaction. No application should set T and then require a high minimal quality such that even sending at minimal quality exceeds the time limit. However, the appropriate settings of the boundary conditions cannot always be anticipated. Therefore, an application must be able to deal with such situations. Possible reactions include delivery of objects at minimal quality (de- sirable in an image retrieval system), a user-application dialogue to renegotiate the boundary parameters, or termination of transfers altogether. This last option is attractive if it allows an overloaded server to catch up. The application-provided exception handler in Fig. 6 deals with such situations VI. EVALUATION This section presents results from the Chariot system, which is an extension of the framework presented here. We concentrate here on assessing the ability of the (adaptive) server to respond to bandwidth fluctuations, i.e. its network-awareness. Note that the examples presented here serve the purposes of validating the approach as well as pointing out areas of further research. The restricted nature of selected examples can by no means replace an extensive evaluation and quantification of the adaptation potential in practice. However, such a study is beyond the scope of this paper. A. Evaluation Methodology Our approach to evaluate the system's network-awareness proceeds in two steps: First, we subject the system to synthetic reference bandwidth waveforms (the example presented here is the Step-Down waveform shown in Fig. 7a) to characterize its ability to adapt in general and in accordance with the (well- established) principles for measuring dynamic response from the field of control systems [32]. Second, field tests in the Internet with its high bandwidth dynamics enable us to assess the BOLLIGER AND GROSS: A FRAMEWORK-BASED APPROACH TO THE DEVELOPMENT OF NETWORK-AWARE APPLICATIONS 11 system's agility with respect to real-world network traffic. Since ensuring reliable and reproducible experiments on real networks is extremely difficult, we follow the approach of other researchers and resort to a technique called trace modulation [28]. Trace modulation performs an application-transparent emulation of a slower target network on a faster, wired LAN. Each application's network traffic is delayed according to delay and bandwidth parameters read from a so-called replay trace, which is gathered from monitored transfers. B. Experimental Setup In our experiments the Chariot server runs on a 150 MHz MIPS R4400 SGI Challenge S with 128 MB of memory. A 134 MHz MIPS R4600 SGI Indy with 64MB of memory serves as the platform for the client. For both of the experiments shown below, the client requests transmission of 90 JPEG images stored at the server in a resolution of 380 \Theta 250 pixels and a JPEG quality factor of 95-97. The 90 images total 5.2 MB of data to be transmitted. The images are assumed to be equally relevant, which means that equal weights are assigned to the 90 images. The user-imposed time limit for request processing is arbitrarily chosen to be 60 seconds with a tolerance interval of [-2, 2] seconds. The bandwidth replay traces used for the two experiments conducted are depicted in Fig. 7. The Step-Down waveform of Fig. 7a is an idealization of real network scenarios; it approximates possible situations in an overlay network for instance, where a mobile client may seamlessly switch between different network interfaces. Fig. 7b shows the monitor layer's perception of the available bandwidth during a transfer between the ETH Z-urich (Switzerland) and the University of Linz (Austria). This bandwidth curve has been smoothed using a two second averaging interval. Hence the system under test does not deal with the problems of start-up behavior. The Chariot server operates using the "approximative search" adaptation process described in Section V-D.2. Chariot's reduction algorithms registered with the framework are image scaling (with factors 1/2 and 1/4) and image compression (with quality factors 75, 50, and 25 [18]). The server performs communication latency hiding by means of a separate thread for P prep . As a consequence, P trans for image i of the sequentially processed request list operates concurrently to P prep for image i + 1. C. Experimental Results C.1 Step-Down waveform Fig. 8-a data vs. time plot as introduced in [19]-shows that Chariot is able to both adapt the amount of data transmitted (curve named "actual") to the amount of data transmittable ("possible") and deliver the 90 images within the 60-second time limit. The Step-Down waveform of the available band-width in Fig. 7a represents the derivative of the curve named "possible". The sharp drop in bandwidth at seconds is absorbed almost without loss of transmission possibilities. Loss of transmission possibilities, which is characterized by the vertical difference between the curve showing the data theoretically transmittable ("possible") and the data actually transmitted ("ac- tual"), can be caused by prepare or control loop overhead. The curve depicting the control loop's estimate of the total amount of data transmittable within the time limit ("estimated") shows that the adaptation at place swiftly (within a small fraction of a second). The estimate is based on the amount of data already transmitted, the monitor's estimate of the available bandwidth bw(t) and t left . Fig. 9 plots the control loop's error variable t dif f that drives Chariot's adaptation. The two horizontal lines at t dif tolerance interval specified. The "time difference" plot shows that in fact three different (major) adaptation events occurred (adaptation is necessary when j t dif f j? 2 s). First, around adaptation steps are necessary to reduce the 5.2 MB to the 4.7 MB estimated to be transferable. Second, due to the sharp bandwidth drop at needed and hence t dif f increase by approx. 33 seconds; this drop is compensated in subsequent reduction steps. Third, t dif f exceeds the 2 second-tolerance twice at t - 33 s although no change in band-width could be observed. This fact may be attributed to inaccuracies in the estimates of c prep and the reduction potential of images. Although provision of inaccurate estimates by the application can have a detrimental impact on the overall performance (i.e. the quality deliverable), the example shows that our control loop mechanism is flexible enough to even cope with such situations. C.2 Internet traffic Fig. 10 shows that Chariot is even capable of dealing with frequent oscillations in the available bandwidth as present on to- day's wide-area network paths. Note, however, that the penalty in terms of transmission possibilities lost is higher than in the previous case. The curve depicting the data volume transmittable ("possible") relates to the bandwidth waveform shown in Fig. 7b. Careful examination of the curve plotting the data effectively transmitted reveals two cases (at t - 3 s and t - 20 s) where transmission lulls had to be accepted. The reason is that in these cases P trans for image i finished before the concurrently executed phase P prep for image i +1 and thus had to wait before starting transmission of image i + 1. The causes for this behavior can be twofold: Either c prep (img case the adaptation process could try to reorder the images in the request list to avoid communication idle time, or the server's load is too high, such that t prep (img trans (img i ). The latter problem calls for host resource reservation by the operating system as other researchers have suggested [25], [26]. Keep in mind, that although the examples presented show that adaptation to meet the given time limit works, the whole process of adaptation is quite sensitive to the choice of the "boundary conditions", such as the time limit. Since the adaptation potential is limited by the reduction potential of the objects/images to be transmitted and the cost incurred for their transformation, unrealistic expectations from the user may simply result in the break-down of the service model. VII. RELATED WORK We can divide approaches to provide predictability of service quality to the application/user into two categories: those that are bandwidth time [sec] bandwidth (a) Step-Down waveform0.10.30.50.70.90 bandwidth time [sec] bandwidth (b) Bandwidth of Internet image transfer Fig. 7. Bandwidth replay traces used DDDDDDDDDDD D DD D D D D D D D D D D D D D D D D D D D D D DD D D D D D DD D D D D D D DDD D D DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD D DDD DDDDDDDDDDDDDDDDDDDD DD DD D D D D D D D D D D D D D D D D DD D D D D D D D D D D D D D D D D D D D D data time [sec] Possible . Actual Estimated Fig. 8. Data volume transmitted in Step-Down scenario t_diff time [sec] . t_diff lower bound upper bound Fig. 9. Time difference (t dif ) plot for Step-Down example BOLLIGER AND GROSS: A FRAMEWORK-BASED APPROACH TO THE DEVELOPMENT OF NETWORK-AWARE APPLICATIONS 13 DDDDDDDDDDDDDDDDD D D D D D D D D D D D DDDDDDD DD D DDDDDDDDD D DDDDDDDDDDDD D D DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD D D DDDDDDDDD D DDDDDDDDDDDDDDDDDDDDDDDDDD D D D DDDDDDDD DDDDDDDDDDDDDDDD D D D DDDDDD DDDDDDDDDDDDDDDDDDDDDDD DD DDDDDDDDDDDDD D D DDDDDDDDDDDDDDDDDDDDDDDDDDD DDDDDD D DDDDDD DDDDDDD DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD D D D DDDDD DDDDDDDDDDDDDDDDDDDDDDDDDDDDD D DDDDDDDDDDD DDDDD D DDDDDD DDDDDDDDDD DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD DDDDDDDDDDDD D D D D DD DDDDD D D D DD DD D D DD D D D D D D D D D D D D DD DD data time [sec] Possible . Actual Estimated Fig. 10. Data volume transmitted from Z-urich to Linz based on reservations and those that are based on adaptation (see Section II-A). A. Reservation There exists a long tradition of research into reservation of network resources, with a trend towards integrating multiple service models in a single cell- or packet-switched network [8], [40]. It has been recognized that to support end-to-end QoS guarantees not only network aspects must be considered, but the end-system and OS-resources must also be taken into account [26]. This requirement holds especially for continuous media applications as they have the most stringent resource requirements [36], [34]. In step with advances in resource guarantee provision in both fields, researchers identified the need for resource orchestration and developed methods that allow for meeting the user's QoS requirements on an end-to-end basis [25], [6]. Most methods involve QoS-negotiation procedures mainly based on application-to-network QoS-mapping. B. Adaptation Adaptation is an effective way of enhancing the user's perception of service quality in environments where resource reservation is not possible, or in situations where it is impossible for an application to specify its resource requirements in advance. Recent adaptive system's such as RLM [22], [23] or IVS [2] have shown that even continuous media applications can benefit from adaptation in environments lacking reservation capabili- ties. Their feedback-driven adaptation scales back quality and hence resource consumption when application performance is poor, and they attempt to discover additional resources by optimistically scaling up usage from time to time. While IVS employs sender-based bandwidth adaptation, RLM pioneered receiver-based adaptation in a multicast environment. Also, both systems continuously adapt their play-out point to account for variations in the transmission latency. In contrast to these systems, Odysee [27] seeks to provide a more general approach to the construction of resource-aware applications by modifying the interface between applications and the operating system. Their measurement-based approach employs receiver-driven adaptation and concentrates on orchestrating multiple concurrent resource-aware applications on the client rather than on the server. In contrast, our framework uses sender-based adaptation and identifies a wide range of methods that can be customized by the user. Fox et al. [13] propose a proxy-based architecture employing so-called distillation services to adapt the quality of the service for the client to the variations in network resource availabil- ity. Their system-in addition to being network-aware-also accounts for variability in client software and hardware sophistication VIII. CONCLUDING REMARKS This paper presents a simple framework for the construction of network-aware applications. Given the framework, the application developer must specify functions to determine the relationships between quality and size as well as provide estimates on the effectiveness of various transformations to reduce size. Fig. 5 summarizes the functions required. Undoubtedly, further work is required to find more elaborate solutions to the problems discussed in this paper. However, the abstractions identified in the adaptation process allow for experimentation with various methods for information collection and with methods providing better estimates, such that tradeoffs can be found between the accuracy achieved, the efforts involved in providing the estimates, and their effect on the bandwidth adap- tation. As it is not always possible to provide good estimates for network behavior (or for the application's resource demands) it is important that systems are designed for adaptivity. Such systems can observe the actions involved with a decision and can take corrective action if necessary. A framework provides the context for such experimentation by application developers, and frees the developer from the need to acquire a detailed understanding of the monitoring system, network protocols, the net-work interface or router capabilities. Our experience with the development of an adaptive image server has demonstrated the practicability and benefits of this approach. The development of network-aware application requires considerable effort, and no amount of adaptation can accomplish the impossible-satisfy unrealistic expectations by an application or a user. However, with adaptation, applications can push the envelope of acceptable network performance, and we expect increased use of adaptation techniques both in stationary and 14 IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. 24, NO. 5, MAY 1998, 376-390 mobile network applications. The framework outlined here provides an approach that shelters the application developers from many details of adaptivity and thus helps to reduce the effort involved in the development of network-aware applications. ACKNOWLEDGEMENTS We thank S. Blott, P. Brandt, A. Dimai, R. Karrer, M. N-af, M. Stricker, P. Walther and R. Weber for their contributions to the design and implementation of the Chariot system. We appreciate the feedback of the referees which improved the paper considerably. Finally, we acknowledge the discussions during the workshop on network-aware and mobile applications held in conjunction with ESEC/FSE '97 in Z-urich. --R Architecture of a networked image search and retrieval system. Scalable feedback control for multicast video distribution in the Internet. Vegas: New techniques for congestion detection and avoidance. Adaptives Transportprotokoll (in German). Characteristics of wide-area TCP/IP conversations A continuous media transport and orchestration service. 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Unsupervised Segmentation of Markov Random Field Modeled Textured Images Using Selectionist Relaxation.

AbstractAmong the existing texture segmentation methods, those relying on Markov random fields have retained substantial interest and have proved to be very efficient in supervised mode. The use of Markov random fields in unsupervised mode is, however, hampered by the parameter estimation problem. The recent solutions proposed to overcome this difficulty rely on assumptions about the shapes of the textured regions or about the number of textures in the input image that may not be satisfied in practice. In this paper, an evolutionary approach, selectionist relaxation, is proposed as a solution to the problem of segmenting Markov random field modeled textures in unsupervised mode. In selectionist relaxation, the computation is distributed among a population of units that iteratively evolves according to simple and local evolutionary rules. A unit is an association between a label and a texture parameter vector. The units whose likelihood is high are allowed to spread over the image and to replace the units that receive lower support from the data. Consequently, some labels are growing while others are eliminated. Starting with an initial random population, this evolutionary process eventually results in a stable labelization of the image, which is taken as the segmentation. In this work, the generalized Ising model is used to represent textured data. Because of the awkward nature of the partition function in this model, a high-temperature approximation is introduced to allow the evaluation of unit likelihoods. Experimental results on images containing various synthetic and natural textures are reported.

INTRODUCTION Textured image segmentation consists in partitioning an image into regions that are homogeneous with regards to some texture measure. Texture description is an important issue with respect to this task. Existing texture segmentation methods are commonly classified according to the texture description they rely on. In structural methods, textures are assumed to consist of structural elements obeying placement rules. In feature based methods, a vector of texture features is computed for each pixel. In stochastic model-based methods, textures are assumed to be realizations of two-dimensional stochastic processes such as, for example, Markov random fields. Among the existing texture segmentation methods [30], those based on Markov random fields [23, 15] have retained substantial attention. Markov random fields are attractive because they yield local and parsimonious texture descriptions. Past studies have also shown the efficiency of Markov random fields in texture modeling, com- pression, and classification [12, 6, 5]. Besides, the use of texture models presents a methodological advantage. Textured images can be generated according to the specified model so that the segmentation method can be evaluated independently of the adequacy of the underlying texture characterization. Texture segmentation using Markov random fields can be achieved through maximum likelihood labelization [8]. Besides their ability to model textured data, Markov random fields can also be used to incorporate a priori knowledge concerning the properties of the labels themselves [15, 13, 14, 8, 27]. Segmentation is then achieved by searching the labelization that maximizes the posterior probability of the labeling conditioned on the data. This optimization problem can be solved using the Gibbs Sampler combined with simulated annealing [16]. However, the parameter estimation problem is a crucial issue for methods based on Markov random fields, and their performance depends on the availability of correct parameter estimates [15]. These methods work well in supervised mode, wherein the number of textures and their associated parameters are known, or can be esti- mated, beforehand. In unsupervised mode, when such knowledge is not available, a circular problem arises [26, 22]: parameter estimates are needed to segment the image, while homogeneous texture samples, which can be provided in the form of an already segmented image, are needed to compute these estimates. Different solutions to this problem have been proposed. A first approach consists in assuming that the shapes of the textured regions are such that the image can be divided into a number of homogeneously textured blocks. Each such block provides a parameter estimate. The number of textures and their associated parameters are determined by applying a clustering algorithm on the parameter estimate set. These final estimates are used to compute the segmentation that optimizes a criterion such as the likelihood criterion [9, 29], the a posteriori probability criterion [26, 22, 29], or the classification error criterion [26]. Conversely, a second approach consists in iterating an estimation/segmentation cycle [24, 33]. Given a candidate number of textures and an initial random set of texture parameters, a first segmentation is computed. Texture parameters are then recomputed using the current segmentation. This cycle is repeated several times until convergence. The whole procedure is repeated with different candidate numbers of textures, and the number that optimizes a model fitting criterion is retained as the true number of regions [33]. These solutions are feasible for images that contain large textured regions or that contain only a limited number of textured regions. In practice, these conditions may not be satisfied. Moreover, besides the parameter estimation problem, relaxation methods based on Markov random fields are often computationally expensive. The problem of segmenting textures modeled using Markov random fields indeed represents a large combinatorial search problem. In this work, a genetic algorithm based approach is adopted to overcome some of the aforementioned difficulties of existing methods. Genetic algorithms [21, 18] are stochastic search methods inspired by the conception that natural evolution is an optimization procedure, which, if simulated, can be applied to solve artificial optimization problems. In a genetic algorithm, a population of candidate solutions, initially generated at random, undergoes a simulated evolutionary process, whereby solutions of increasing quality progressively appear. Each generation, a new population is computed from the previous one through a two-step cycle. During the first step, good solutions are selected and duplicated to replace bad ones. During the second step, new solutions are generated by recombining and mutating the solutions that have been selected. Consequently, good solutions are progressively spreading within the popu- lation, while being permanently exploited to build possibly better solutions. Genetic algorithms are attractive in combinatorial problems because they achieve an efficient, parallel exploration of the search space (which in particular may avoid being stuck in local optima), while requiring minimum information on the function to be optimized (in particular, the derivatives are not required) [18]. In the standard genetic algorithm [18], the population is panmictic, i.e., each individual can compete or recombine with any other one in the population. Alternatively, in distributed genetic algorithms, each individual is constrained to interact with a limited number of other individuals. In coarse-grained distributed genetic algorithms [32, 10], the population is divided into several subpopulations submitted to their own genetic algorithm, and periodically exchanging some of their individuals. In fine-grained distributed genetic algorithms [28, 25, 31, 11], the population is mapped onto a grid whereupon a neighborhood system is defined to constrain interactions among individ- uals. The purpose of distributed genetic algorithms is to increase the quality of the obtained solutions, in particular by avoiding premature convergence to non-optimal solutions, and to reduce the time needed to obtain the solutions. The method presented herein is an unsupervised segmentation method whereby the transformation of an input image into an output segmented image is computed by a population of units that are mapped onto the image. Initially generated at random, the population is iteratively updated and reorganizes through a fine-grained distributed genetic algorithm. Consequently, a sequence of segmented images is generated. This sequence progressively converges to a stable labelization, which is taken as the resulting segmented image. This method, called selectionist relaxation to emphasize the role of selection, has been previously applied to the unsupervised gray-level image segmentation problem [1]. It is shown here how selectionist relaxation can be generalized to the unsupervised textured image segmentation problem. In this work, textures are represented using the generalized Ising model [16], also known as the Derin-Elliott model [14]. With this model, the likelihood of a texture window, the evaluation of which is required in selectionist relaxation, is not computable in practice because of the intractability of the partition function. An approximation of the partition function is therefore introduced to overcome this problem. The organization of the paper is as follows. Background on the Markov random field approach to texture modeling is given in Section 2. The approximation of the partition function is set out in Section 3. Selectionist relaxation is presented in Section 4. Results on synthesized texture patches are reported in Section 5. Section 6 is devoted to a final discussion and conclusion. s Figure 1: Neighborhoods and cliques. A. The spatial extent of the neighborhood of a site s depends on the order of the model. At order n, the neighborhood contains all the sites that are assigned a digit less than or equal to n. B. The 10 clique types associated to a second-order model. Cliques with non-zero potential in the model used in this paper are shown in gray. The input image is assumed to contain several textures, each of which is considered as a realization of a Markov random field [23]. Further, the same model is used for all textures, because it is assumed that these textures are different instances of the same texture model. 2.1 Markov/Gibbs random fields Consider the two-dimensional set of sites NC g, wherein NR and NC are the numbers of rows and columns of the texture image, respectively. A collection of subsets of sites defines a neighborhood system if it satisfies the following two conditions: (1) 8s 2 clique is either a single site or a set of mutually neighboring sites. The set of cliques is being the number of different clique types. Neighborhood structures and clique types are illustrated in Fig. 1. A texture sample is considered herein as a realization of a random field s is a random variable taking values in a discrete set being the number of gray levels in the image. A realization x of X is called a configuration. The state space of X . The restriction of a configuration to a subset R of S is noted xR . A collection of real-valued functions defined on\Omega and such that VR (x) only depends on xR is called a potential. Further, V is a neighbor potential if 8x According to the Hammersley-Clifford theorem [2], the random field X is a Markov random field on S with respect to a neighborhood system N if and only if its distribution on\Omega is a Gibbs distribution induced by a neighbor potential, that is to say 8x is the energy of configuration x and the normalizing constant y2\Omega expf\GammaE(y)g is the partition function. 2.2 Generalized Ising model Various Markov random field texture models have been proposed, each of which being defined by its associated potential: the autologistic model [20], the autobinomial model [12], the autonormal (Gaussian Markov random field) model [4] and the generalized Ising model [16, 14, 17]. Despite its simplicity, we have retained this last model to work with in this work. Indeed, the first step in our work is to test the feasibility of applying selectionist relaxation to segment images that contain Markov random field samples, regardless of whether the model is able or not to capture the complexity of natural textures. The generalized Ising model is a pairwise interaction model [15]: only those cliques that contain no more than 2 sites have non zero potentials. Singleton potentials are set to zero so that the first-order histogram is uniform. Because we use a second-order model, the effective number of clique types is clique types are shown in gray in Fig. 1-B). For a pair clique the potential function is given by is the parameter associated to clique type i, and which case ffi xsxr = 1. Letting the vector of model parameters, the energy of any configuration x can be written as wherein the vector defined by 3 PARTITION FUNCTION APPROXIMATION As described in the next section, selectionist relaxation requires that, given any w \Theta w window W and any candidate vector model parameters, the likelihood of a configuration xW is practically computable. Letting \Omega W denote the state space of XW , this likelihood is given by: Considering the generalized Ising model, the exact computation of the likelihood cannot be achieved in practice: the partition function ZW (B) neither has a simple analytical form, nor can be computed. This would involve calculating the energy of all possible configurations over W , which is computationally intractable because of the huge number of such configurations. We therefore propose an approximation of the partition function. It consists in approximating each of the terms in the expression of ZW (B) using its second-order expansion: The approximated terms are then summed up over y 2\Omega W . It is shown in the Appendix how, assuming the window W has a toroidal structure, the resulting expression Figure 2: Plot of the relative approximation error as a function of temperature. The error was numerically determined with the following conditions: W is a 3 \Theta 3 window, the number of gray levels is and the parameter set is B of the partition function can be rearranged and simplified. This eventually leads to the following approximated partition function: ~ wherein g is the number of gray levels and w is the number of sites in the window W . It should be noted that the approximation is not only valid in the second-order model case, but stands for any order of the model. This approximation can be interpreted as a high-temperature approximation of the partition function. Up to now, the temperature T of the Gibbs distribution (1) has indeed been considered as incorporated in the energy, but if we define then, from (2), the energy can be rewritten as The error due to the approximation (3) vanishes as E(y; B) ! 0. From (6), this clearly happens when T ! 1. The dependency on T of the resulting approximation error is illustrated in Fig. 2, which plots the relative error defined by T has been made explicit here only for the sake of the demonstration. In the remainder of the paper, we return to the use of B (instead of B and T ), considering T as an implicit scaling factor: the condition fi i - 1, which will be imposed to keep the approximation error small enough, will be interpreted as an absorption of T within the parameters themselves, according to (5). input image image output algorithm s s s Figure 3: Selectionist relaxation. The unit U s assigned to site s consists of a feature vector B s and of a label L s . The fitness of U s depends on how well B s matches the data in the input window W s . The genetic algorithm applied to the population of units results in a relaxation process, whereby highly fitted units spread over the image, replacing badly fitted ones. In this process, unit U s primarily interacts with the units located within its neighborhood N s . At the end of the process, the resulting segmented image is build by attributing to each site s the corresponding label L s . 4.1 Outline of the method Selectionist relaxation is an unsupervised segmentation method whereby the transformation of an input image into an output image is computed by a population of units that iteratively evolves through a distributed genetic algorithm (Fig. 3). Each unit is an association between a candidate feature vector and a label. The latter is used to label the unit pixel. The former is used to assign a fitness value to the unit. The fitness of a unit is a measure of the matching between its feature vector and the data in the image window whereupon the unit is centered. The features that compose unit feature vectors are arbitrarily chosen on the basis of the desired segmentation type. For example, pixel matrices were used as feature vectors for grey-level image segmentation [1]. Here, it is proposed that texture segmentation can be achieved using texture model parameters as feature vectors. The population of units iteratively evolves through the application of genetic operators [18]: the units whose feature vectors find good support from image data are selected, recombined and mutated. These mechanisms allow units with high fitness values to spread over their neighborhood by replacing the neighboring units that do not fit the local image data. Additionally, some units can jump over large distances to invade distant regions with similar characteristics. This results in a mixed local/distant parameters label Figure 4: Unit U s is made of a vector B of texture model parameters and of a label L s . label spreading process that eventually leads to a stable label configuration, which is taken as the resulting segmentation. 4.2 Units As illustrated in Fig. 3, each site s of the input image has an associated unit U s . U s is a couple U candidate vector of texture model parameters and L s is a label (Fig. 4). A collection of units is called a population (because selectionist relaxation is an iterative method, units or population of units will be indexed with time whenever this is necessary). L s is the label assigned to site s. The output of the algorithm, the segmented image, is stands for the stopping time step. Each unit U s is assigned a fitness value f(U s ), which quantifies how well the unit matches, according to the texture model, the w \Theta w texture window W s centered on site s. The likelihood P measure of this match. However, with the generalized Ising model, this criterion cannot be retained because of the aforementioned awkwardness of the partition function. Instead, f(U s ) is defined as the approximated likelihood: ZWs (B s ) is the approximated partition function given in (4). Using this approximation constrains the domain wherein unit parameters may be reliably searched for by the genetic algorithm. Unsatisfactory segmentation results are indeed expected if the fitness function is unreliable, due to a large approximation error. As explained in Section 3, this error vanishes as the parameters go to zero. Thus unit parameters must be initially close to zero (constraint on initialization), and they must stay close to zero during the whole run (constraint on mutation). How close to zero the parameters must be to yield good segmentation results is determined experimentally. How these constraints are taken into account in the algorithm is explained in the following subsection. Selectionist relaxation implements a fine-grained distributed genetic algorithm. This means that the population is spatially organized, each unit primarily interacting with its neighboring units. For the unit at site s, these are the units located within the j \Theta j window N s centered on site s (Fig. 3). Units located on the borders of the image have fewer neighbors than interior units. selection crossover/mutation states Figure 5: Selection, crossover and mutation are state-dependent. The population is here in a state with three coexisting labels, corresponding to three subpopulations of units. Their boundaries are shown as thin broken lines. Left. The sites whose neighborhood (dark gray) crosses a label boundary have state 1 (light gray); the others have state 0 (white). Middle. Selection at sites with state 0 only involves neighboring units, while it additionally involves a randomly picked unit at sites with state 1. Right. Only units with state 0 undergo crossover with a neighbor and mutation. Each unit is attributed a binary value, called its state, which depends on the labels of its neighbors. The state S s of unit U s is defined as follows: This variable allows to discriminate units according to their distance from units with different labels. As relaxation proceeds, some units spread over the image by being copied from site to site, and so do their associated labels. Consequently, homogeneously labeled regions are growing. As illustrated in Fig. 5 (Left ), units located in the neighborhood of a boundary between two or more such regions have state 1, while units located inwards these regions have state 0. As explained in the next subsec- tion, in selectionist relaxation, genetic operators (selection, crossover and mutation) are state-dependent. 4.3 Algorithm Initialization. The first step in selectionist relaxation consists in creating the initial population U(0) as follows. For each unit U s , each component fi s;i of its parameter vector B s is assigned a value sampled from the uniform distribution over the interval As explained in the previous subsection, the parameter initialization domain is constrained because the fitness function relies on the approximation of the partition Accordingly, ffi must be chosen small enough so that the approximation error is acceptable. In the experiments reported in the next section, the simple rule has been used with success. Unit label L s is chosen as the raster scan index of site s. Consequently, there are initially as many labels as there are pixels in the image, and all sites are in state 1. Relaxation cycle. After the population of units has been initialized, selectionist relaxation consists in repeating a two-step relaxation cycle until the stopping criterion crossover mutation Figure Crossover and mutation. Top, Crossover between unit U s and a neighboring unit U r . The crossover position k is chosen at random. Bottom, Mutation of unit U s . Mutation position l and mutation amount m are chosen at random. Unit labels are not shown because they are not affected by crossover nor by mutation. is met. Each time step t, the population U computed by, first, applying selection to the population U (t) and, second, by applying crossover and mutation to the selected population. The operators are state-dependent (Fig. 5). During selection at a site with state 0, competition only involves the neighboring units. At a site with state 1, it additionally involves a remote unit. This mechanism allows spatially distant units to interact, and was introduced so that a same texture can be assigned a unique label even though it appears in different disconnected regions. Without this mechanism, such a texture would be assigned as many labels as it forms separate regions, because labels would exclusively be propagated from one site to the next. During the second step of the cycle, crossover and mutation are applied only to units with state 0. This prevents the label boundaries that are formed as relaxation proceeds from being perturbed and disrupted by sudden fitness changes. Each of the three operators synchronously affects all image sites. They are described in detail below for a generic site s. ffl Selection. The selection scheme implemented in selectionist relaxation is local tournament selection [31], a variant of tournament selection [19]: the unit whose fitness is the highest in a subpopulation U s of population U is selected to replace the unit at site s. As said before, selection is state-dependent (Fig. 5, Middle): where r is a randomly picked site. Once U s is build, the unit with the greatest fitness value in U s is selected to replace the unit at site s: ffl Crossover. If S does not undergo crossover (Fig. 5, Right ). Otherwise, a neighboring unit U r ; r 2 N s , is randomly picked. Then, one component in the parameter vector B s is chosen and is assigned the corresponding value of the parameter vector B r (Fig. 6, Top). ffl Mutation. As for crossover, unit U s does not undergo mutation whenever S 1. Otherwise, a parameter index l 2 randomly chosen and a value m sampled from the uniform distribution in the interval [\Gamma- l s ] is added to the corresponding parameter fi s;l of U s (Fig. 6, Bottom). Two constraints are imposed on mutation through - l s . First, mutation amplitude must be small compared to the initial parameter range ffi. Second, preliminary experiments have shown that a texture dependent mutation scheme leads to better results than a texture independent one. These constraints are taken into account by letting The first term, ffl ffi, enforces the first constraint, provided ffl is small. In the experiments reported in the next section, ffl is set to 0:02. The second term makes mutation amplitude depend on the local texture configuration by allowing greater mutation amplitude when j- l j is small. This occurs when there is some ambiguity in the texture along clique type l, since by definition, j- l j small means that about half of the cliques of type l contribute by +1 while the other half contribute by \Gamma1. It experimentally appeared that a greater mutation amplitude was beneficial to overcome such ambiguities. 5.1 Experimental setup The results reported in this section illustrate selectionist relaxation segmentation of images containing textures that are realizations of the generalized Ising model presented in Section 2. The images contain 8 gray levels and are 256 \Theta 256 pixels wide. Texture samples were synthesized using the Gibbs Sampler [16] for 100 steps. For each test image, selectionist relaxation was run for 300 time steps. Each unit has 8 neighbors (j = 3). The only externally tuned parameter is w, which defines the size of the texture window that is used to compute each unit fitness. As explained in Section 4, w automatically determines the initial parameter range as well as the mutation range. The number of textures and their associated parameters are automatically determined by the algorithm through simulated evolution among the population of units. Segmentation results are evaluated by visual examination and by computing the error rate. Misclassified pixels are determined as follows: for each region of the true segmentation, the label which is the most represented over that region in the segmented image is determined. The pixels that are assigned another label are considered as misclassified. The error rate is the percentage of misclassified pixels over the whole image. 5.2 Segmentation results Fig. 7 displays the segmentation result for an image that contains two textures spatially arranged according to a simple geometry. Texture windows used to evaluate the fitness of the units are w wide. The segmentation of an image containing the same two textures arranged in a more complex fashion is illustrated on Fig. 8. Though more complex, the two textures still form connected regions. Fig. 9 shows that the same textures can also be correctly segmented when they form spatially disconnected regions. This example proves that, though selectionist relaxation mainly proceeds by propagating labels over nearest neighboring sites, spatially separated blobs of the same Figure 7: Segmentation of Wave, a 2-textures image. A. True segmentation. B. Input image. C. Selectionist relaxation segmentation. D. Misclassified pixels. texture can be assigned the same label. This is because a randomly chosen unit is systematically involved in the selection process at those sites s such that S s = 1. Consequently, some units can literally jump over large spatial distances. As illustrated in Fig. 10, images that contain a larger number of textures can be segmented as well. In this last case, it was necessary to compute the fitness of the units over larger (w 11 \Theta 11) texture windows, to take into account the coarseness of the different textures. rates are given in Table 1 (Middle). These are reasonably low, and, as can be seen in Fig. 7, 8, 9 and 10, errors exclusively occur at the boundaries between the textured regions. This suggests that comparing error rates among the four cases is misleading, because the total length of texture boundaries differ among the four cases. A relative error rate was defined as the number of misclassified pixels divided by the total length of region boundaries in the true segmentation. According to the relative error rate (last column in Table 1), it appears that, in spite of the varying region shape, connectivity, and number, the performance of selectionist relaxation is relatively constant among the four cases, and, in particular, the relative error rate is always less than 1. However, using a larger window size (fourth case) seems to result in an increased number of errors at region boundaries, which is not unexpected. Figure 8: Segmentation of Spiral, a 2-textures image. A. True segmentation. B. Input image. C. Selectionist relaxation segmentation. D. Misclassified pixels. 5.3 Estimated parameters The issue naturally arises of the extent to which the parameters of the units that are found through selectionist relaxation on a given texture match the true parameters of that texture (i.e., the parameters that were used to synthesize the original tex- ture). These parameter sets will be respectively referred to as B units and B true . Any attempt to assess the correctness of B units with regards to B true is however hampered by the constraints imposed on B units because of the partition function approximation. As previously mentioned, B units must be considered as containing both the estimated parameters of the texture (B estim ) and the temperature B estim can thus be determined from B units (and subsequently compared to B true T is known. The problem is that T is only implicit, and, consequently, unknown. However, under the assumption that the estimated parameters are correct (i.e. assuming the criterion Figure 9: Segmentation of Blobs, a 2-textures image. A. True segmentation. B. Input image. C. Selectionist relaxation segmentation. D. Misclassified pixels. Using this criterion, a value of T can be computed and B estim can be determined from B units . The textures can also be resynthesized using B estim and compared to the originals. This has been done for the Wave experiment reported in Fig. 7. For each texture, the vector B units is computed by averaging unit parameters over all the units whose label is the most represented label on that texture. Table 2 gives theoretical, unit, and estimated parameters for each texture. Comparing B estim with B true shows that the relative parameter values are acceptable for texture L, but are far from the original for texture U. The textures resynthesized using estimated parameters are shown in Fig. 11. In this work, selectionist relaxation is proposed as a new method for segmenting images that contain textures modeled using Markov random fields. Using a high temperature approximation of the partition function, the ability of selectionist relaxation to segment samples of the generalized Ising model has been demonstrated. Selectionist relaxation is unsupervised in so far as knowledge concerning the number of textures and their associated parameters is not required beforehand. Estimation of these unknowns and Figure 10: Segmentation of Rose, a 8-textures image. A. True segmentation. B. Input image. C. Selectionist relaxation segmentation. D. Misclassified pixels. segmentation are achieved simultaneously. The algorithmic complexity of the method neither depends on the number of textures nor on the number of gray levels. Several solutions to the problem of segmenting Markov modeled textured images in unsupervised mode have been proposed [26, 22, 9, 33, 29]. These methods rest upon the assumptions that the image contains only a limited number of textured regions or that the shapes of the textured regions are such that the image can be divided into non-overlapping blocks, the majority of which is homogeneous so that texture parameters can be estimated on each such block. Selectionist relaxation loosens these constraints since no assumption is made neither on the number of regions nor on the shapes of these regions. Relaxing these assumptions results in an increased difficulty. The problem can be decomposed into three subproblems to be solved simultaneously. They respectively consist in the determination of: (1) the number of different textured regions (which is naturally bounded by the number of pixels in the input image); (2) the corresponding set of model parameter vectors; (3) the optimal labelization of the data. It is noted that the trivial solution, which consists in assigning a different label to each pixel, is not observed. Partial suboptimal solutions, in which several labels are assigned to a same region, are not observed either. On the contrary, the number of regions Image Wave 0.26 0.47 Spiral 3.40 0.52 Blobs 3.07 0.49 Rose 2.09 0.87 Table 1: Segmentation errors. percentage of misclassified pixels. number of misclassified pixels divided by the total length of region boundaries in the true segmentation. Texture Parameters Temperature true 1.000 -1.000 -1.400 -1.400 true -1.000 -1.000 -1.400 -1.400 Table 2: Comparison between actual texture parameters and unit parameters found by selectionist relaxation (case of the Wave experiment reported in Fig. 7). Left : U and L refer to the upper and to the lower textures in Fig. 7, respectively. Middle: parameters used to synthesize the original textures (B true ), unit parameters (B units ), and estimated parameters (B estim determined using criterion (7) and used to compute B estim from B units . has been correctly identified in all our experiments. This suggests that, though no priors are imposed on the labels, regularizing constraints are implicitly incorporated into the algorithm. The issue of identifying these constraints is the matter for further investigations. Though it is unsupervised, the method is not, however, fully data-driven, since the size w of the texture window used to compute the fitness of the units has to be specified by the user. Tuning this parameter is easy because it is in natural correspondence with the coarseness of the textures. The coarser the textures are, the greater the size of the window should be to insure that it contains enough information to yield reliable fitness values. As a counterpart, this may affect segmentation accuracy, since large errors are expected to occur at region boundaries when using large windows. The results reported here show that in most cases, the spatial extent of boundary misclassifications is unexpectedly small compared to the size of the window. A comparison between actual texture parameters and parameters estimated through selectionist relaxation has been done. For the first texture (texture L in Table 2), estimated parameters agree with actual parameters. For the second texture (texture U), the estimation is less satisfactory. Though correct signs and roughly correct absolute values are obtained, pairwise parameter ratios within the estimated parameter set and within the true parameter set largely differ. We propose that this may result from several, possibly non-exclusive, causes. First, it can be argued that error in parameter estimation results from the high temperature approximation of the partition function. However, if this were systematically true, then correct parameters would not Figure 11: Textures resynthesized using estimated parameters. A. Original patch (same as Fig. 7-A). B. Reconstructed patch: the textures have been resynthesized using the estimated parameters (given in Table 2). be obtained for texture L. Second, the size w of window W may be too small, and texture U may be more sensitive to this parameter because it is coarser than texture L. Third, the computation of B estim relies on the assumption that all the parameters are at the same temperature. This is certainly far from being a correct assumption, because the mutation range is texture- and parameter-dependent, so that the parameters do not evolve at the same rate. Texture U is such that the mutation range on fi 1 is, on average, larger than the mutation range on the other parameters, while texture L is such that the mutation range is roughly parameter-independent, because this texture is isotropic. Fourth, it is also likely that sampling bias in the procedure used to synthesize the textures is stronger for texture U than for texture L. This is confirmed by experiments in which parameters were estimated (using maximum pseudo-likelihood estimation [3]) on 100 \Theta 100 homogeneous texture samples (data not shown). Pairwise parameter ratios were in good agreement between estimated and actual parameter sets for texture L, but not for texture U. It has been suggested that the performance of the generalized Ising model on a texture classification task was poorer than the performance of the autobinomial and autonormal models [7]. It has also been argued that using either of these two other models was better for natural texture segmentation [22]. We are currently trying to apply selectionist relaxation to natural texture segmentation using more appropriate models than the generalized Ising model. If the assumption that was made here is valid, i.e., if the ability of the method to segment Markov random field texture samples is independent of any particular model, then selectionist relaxation will appear as a promising approach towards unsupervised texture image segmentation. A Given a window W and a set of texture parameters the problem is to obtain an approximated, manageable expression of the partition function ZW (B). For simplicity, and without loss of generality, the case considered here. The partition function to be approximated is thus x2\Omega expf\GammaE(x; B)g is the set of all possible configurations over S, n being the number of sites in S and being the set of gray level values. Remember that is the set of all cliques in S, C i being the set of type i cliques. Under the assumption that the grid S is toroidal, the number of cliques of each type equals the number of sites: jC For the generalized Ising model, the energy of a configuration can be written as , the vector being defined by with, for any clique Approximating each term in the sum Z(B) by its second-order expansion (3) yields the approximated partition function: ~ with x2\Omega E(x; B); The problem is now to rearrange Z 1 and Z 2 as functions of the fi i s (Z 0 is a constant equals to g n ). These calculations rely on some preliminary results that are given in the next subsection. A.1 Preliminary results For any clique c 2 C, and for any two cliques c 1 6= c 2 , one can show that A.2 Calculation of Using the definitions of Z 1 (B), E(x; B) and - i (x) leads to which, together with (9), yields A.3 Calculation of Proceeding as for Z 1 (B) leads to The problem is now to compute the coefficients These coefficients are calculated by distinguishing, for each clique c 1 , those cliques c 2 that are equal to c 1 from those that differ from c 1 , and then using (10) and (11). In the expression of w ii , there will be only one clique c In the expression of cliques c 2 necessarily differ from c 1 , since they belong to different clique types. This gives ng n\Gamma2 With these coefficients, the following expression of finally obtained Collecting (8), (12) and (13) leads to the expression of the approximated partition function given in (4). ACKNOWLEDGMENTS The authors would like to thanks Evelyne Lutton; her comments on an earlier version of this work greatly contributed to improve the presentation of the paper and were very much appreciated. --R Unsupervised image segmentation using a distributed genetic algorithm. Spatial interaction and the statistical analysis of lattice systems. On the statistical analysis of dirty pictures. Classification of textures using Gaussian Markov random fields. Texture synthesis and compression using Gaussian-Markov random field models Markov random fields for texture classification. Simple parallel hierarchical and relaxation algorithms for segmenting noncausal Markovian random fields. Maximum likelihood unsupervised textured image segmentation. Distributed genetic algorithms for the floorplan design problem. Studies in Artificial Evolution. Markov random field texture models. Segmentation of textured images using Gibbs random fields. Modeling and segmentation of noisy and textured images using Gibbs random fields. Random field models in image analysis. Stochastic relaxation Probabilistic models of digital region maps based on Markov random fields with short- and long-range interaction Genetic Algorithms in Search A comparative analysis of selection schemes used in genetic algorithms. The use of Markov random fields as models of texture. Adaptation in Natural and Artifical Systems: An Introductory Analysis with Applications to Biology Texture segmentation based on a hierarchical Markov random field model. Markov Random Fields and Their Applications Simultaneous parameter estimation and segmentation of Gibbs random fields using simulated annealing. Unsupervised texture segmentation using Markov random field models. Stochastic and deterministic networks for texture segmentation. Parallel genetic algorithms Gibbs random fields du Buf. A massively parallel genetic algorithm. Distributed genetic algorithms. Unsupervised segmentation of noisy and textured images using Markov random fields. --TR --CTR C.-T. Li, Multiresolution image segmentation integrating Gibbs sampler and region merging algorithm, Signal Processing, v.83 n.1, p.67-78, January Eun Yi Kim , Se Hyun Park , Sang Won Hwang , Hang Joon Kim, Video sequence segmentation using genetic algorithms, Pattern Recognition Letters, v.23 n.7, p.843-863, May 2002 J. Veenman , Marcel J. T. Reinders , Eric Backer, A Maximum Variance Cluster Algorithm, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.24 n.9, p.1273-1280, September 2002

genetic algorithms;unsupervised texture segmentation;selectionist relaxation;partition function approximation;markov/gibbs random fields

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A Hierarchical Latent Variable Model for Data Visualization.

AbstractVisualization has proven to be a powerful and widely-applicable tool for the analysis and interpretation of multivariate data. Most visualization algorithms aim to find a projection from the data space down to a two-dimensional visualization space. However, for complex data sets living in a high-dimensional space, it is unlikely that a single two-dimensional projection can reveal all of the interesting structure. We therefore introduce a hierarchical visualization algorithm which allows the complete data set to be visualized at the top level, with clusters and subclusters of data points visualized at deeper levels. The algorithm is based on a hierarchical mixture of latent variable models, whose parameters are estimated using the expectation-maximization algorithm. We demonstrate the principle of the approach on a toy data set, and we then apply the algorithm to the visualization of a synthetic data set in 12 dimensions obtained from a simulation of multiphase flows in oil pipelines, and to data in 36 dimensions derived from satellite images. A Matlab software implementation of the algorithm is publicly available from the World Wide Web.

Introduction Many algorithms for data visualization have been proposed by both the neural computing and statistics communities, most of which are based on a projection of the data onto a two-dimensional visualization space. While such algorithms can usefully display the structure of simple data sets, they often prove inadequate in the face of data sets which are more complex. A single two-dimensional projection, even if it is non-linear, may be insufficient to capture all of the interesting aspects of the data set. For example, the projection which best separates two clusters may not be the best for revealing internal structure within one of the clusters. This motivates the consideration of a hierarchical model involving multiple two-dimensional visualization spaces. The goal is that the top-level projection should display the entire data set, perhaps revealing the presence of clusters, while lower-level projections display internal structure within individual clusters, such as the presence of sub-clusters, which might not be apparent in the higher-level projections. Once we allow the possibility of many complementary visualization projections, we can consider each projection model to be relatively simple, for example based on a linear projection, and compensate for the lack of flexibility of individual models by the overall flexibility of the complete hierarchy. The use of a hierarchy of relatively simple models offers greater ease of interpretation as well as the benefits of analytical and computational simplification. This philosophy for modelling complexity is similar in spirit to the "mixture of experts" approach for solving regression problems [1]. The algorithm discussed in this paper is based on a form of latent variable model which is closely related to both principal component analysis (PCA) and factor analysis. At the top level of the hierarchy we have a single visualization plot corresponding to one such model. By considering a probabilistic mixture of latent variable models we obtain a soft partitioning of the data set into 'clusters', corresponding to the second level of the hierarchy. Subsequent levels, obtained using nested mixture representations, provide successively refined models of the data set. The construction of the hierarchical tree proceeds top down, and can be driven interactively by the user. At each stage of the algorithm the relevant model parameters are determined using the expectation-maximization (EM) algorithm. In the next section we review the latent-variable model, and in Section 3 we discuss the extension to mixtures of such models. This is further extended to hierarchical mixtures in Section 4, and is then used to formulate an interactive visualization algorithm in Section 5. We illustrate the operation of the algorithm in Section 6 using a simple toy data set. Then we apply the algorithm to a problem involving the monitoring of multi-phase flows along oil pipes in Section 7 and to the interpretation of satellite image data in Section 8. Finally, extensions to the algorithm, and the relationships to other approaches, are discussed in Section 9. Latent Variables We begin by introducing a simple form of linear latent variable model and discuss its application to data analysis. Here we give an overview of the key concepts, and leave the detailed mathematical discussion to Appendix A. The aim is to find a representation of a multi-dimensional data set in terms of two latent (or 'hidden') variables. Suppose the data space is d-dimensional with coordinates y and that the data set consists of a set of d-dimensional vectors ft n g where . Now consider a two-dimensional latent space together with a linear function which maps the latent space into the data space where W is a d \Theta 2 matrix and - is a d-dimensional vector. The mapping (1) defines a two-dimensional planar surface in the data space. If we introduce a prior probability distribution p(x) over the latent space given by a zero-mean Gaussian with a unit covariance matrix, then (1) defines a singular Gaussian distribution in data space with mean - and covariance matrix h(y . Finally, since we do not expect the data to be confined exactly to a two-dimensional sheet, we convolve this distribution with an isotropic Gaussian distribution p(tjx; oe 2 ) in data space having a mean of zero and covariance oe 2 I where I is the unit matrix. Using the rules of probability, the final density model is obtained from the convolution of the noise model with the prior distribution over latent space in the form Z Since this represents the convolution of two Gaussians, the integral can be evaluated analytically, resulting in a distribution p(t) which corresponds to a d-dimensional Gaussian with mean - and covariance matrix WW T If we had considered a more general model in which conditional distribution p(tjx) is given by a Gaussian with a general covariance matrix (having d independent parameters) then we would obtain standard linear factor analysis [2, 3]. In fact our model is more closely related to principal component analysis, as we now discuss. The log likelihood function for this model is given by likelihood can be used to fit the model to the data and hence determine values for the parameters -, W and oe 2 . The solution for - is just given by the sample mean. In the case of the factor analysis model, the determination of W and oe 2 corresponds to a non-linear optimization which must be performed iteratively. For the isotropic noise covariance matrix, however, it was shown by Tipping and Bishop [4, 5] that there is an exact closed form solution as follows. If we introduce the sample covariance matrix given by then the only non-zero stationary points of the likelihood occur for: where the two columns of the matrix U are eigenvectors of S, with corresponding eigen-values in the diagonal matrix , and R is an arbitrary 2 \Theta 2 orthogonal rotation matrix. Furthermore, it was shown that the stationary point corresponding to the global maximum of the likelihood occurs when the columns of U comprise the two principal eigenvectors of S (i.e. the eigenvectors corresponding to the two largest eigenvalues) and that all other combinations of eigenvectors represent saddle-points of the likelihood surface. It was also shown that the maximum-likelihood estimator of oe 2 is given by d which has a clear interpretation as the variance 'lost' in the projection, averaged over the lost dimensions. Unlike conventional PCA, however, our model defines a probability density in data space, and this is important for the subsequent hierarchical development of the model. The choice of a radially symmetric rather than a more general diagonal covariance matrix for p(tjx) is motivated by the desire for greater ease of interpretability of the visualization results, since the projections of the data points onto the latent plane in data space correspond (for small values of oe 2 ) to an orthogonal projection as discussed in Appendix A. Although we have an explicit solution for the maximum-likelihood parameter values, it was shown by Tipping and Bishop [4, 5] that significant computational savings can sometimes be achieved by using the following EM (expectation-maximization) algorithm [6, 7, 8]. Using (2) we can write the log likelihood function in the form Z in which we can regard the quantities x n as missing variables. The posterior distribution of the x n , given the observed t n and the model parameters, is obtained using Bayes' theorem and again consists of a Gaussian distribution. The E-step then involves the use of 'old' parameter values to evaluate the sufficient statistics of this distribution in the form I is a 2 \Theta 2 matrix, and h i denotes the expectation computed with respect to the posterior distribution of x. The M-step then maximizes the expectation of the complete-data log likelihood to give f e Nd Whx in which e denotes 'new' quantities. Note that the new value for f W obtained from (9) is used in the evaluation of oe 2 in (10). The model is trained by alternately evaluating the sufficient statistics of the latent-space posterior distribution using (7) and (8) for given oe 2 and W (the E-step), and re-evaluating oe 2 and W using (9) and (10) for given hx n i and hx n x T (the M-step). It can be shown that, at each stage of the EM algorithm, the likelihood is increased unless it is already at a local maximum, as demonstrated in Appendix For N data points in d dimensions, evaluation of the sample covariance matrix requires any approach to finding the principal eigenvectors based on an explicit evaluation of the covariance matrix must have at least this order of computational complexity. By contrast, the EM algorithm involves steps which are only O(Nd). This saving of computational cost is a consequence of having a latent space whose dimensionality (which, for the purposes of our visualization algorithm, is fixed at 2) does not scale with d. If we substitute the expressions for the expectations given by the E-step equations (7) and (8) into the M-step equations we obtain the following re-estimation formulae f e d which shows that all of the dependence on the data occurs through the sample covariance matrix S. Thus the EM algorithm can be expressed as alternate evaluations of (11) and (12). (Note that (12) involves a combination of 'old' and `new' quantities.) This form of the EM algorithm has been introduced for illustrative purposes only, and would involve cost due to the evaluation of the covariance matrix. We have seen that each data point t n induces a Gaussian posterior p(x n jt n ) distribution in the latent space. For the purposes of visualization, however, it is convenient to summarize each such distribution by its mean, given by hx n i, as illustrated in Figure 1. Note that x prior posterior Figure 1: Illustration of the projection of a data point onto the mean of the posterior distribution in latent space. these quantities are obtained directly from the output of the E-step (7). Thus a set of data points projected onto a corresponding set of points fhx n ig in the 2-dimensional latent space. 3 Mixtures of Latent Variable Models We can perform an automatic soft clustering of the data set, and at the same time obtain multiple visualization plots corresponding to the clusters, by modelling the data with a mixture of latent variable models of the kind described in Section 2. The corresponding density model takes the form where M 0 is the number of components in the mixture, and the parameters - i are the mixing coefficients, or prior probabilities, corresponding to the mixture components p(tji). Each component is an independent latent variable model with parameters - i , W i and oe 2 This mixture distribution will form the second level in our hierarchical model. The EM algorithm can be extended to allow a mixture of the form (13) to be fitted to the data (see Appendix B for details). To derive the EM algorithm we note that, in addition to the fx n g, the missing data now also includes labels which specify which component is responsible for each data point. It is convenient to denote this missing data by a set of variables z ni where z generated by model i (and zero otherwise). The prior expectations for these variables are given by the - i and the corresponding posterior probabilities, or responsibilities, are evaluated in the extended E-step using Bayes' theorem in the form Although a standard EM algorithm can be derived by treating the fx n g and the z ni jointly as missing data, a more efficient algorithm can be obtained by considering a two-stage form of EM. At each complete cycle of the algorithm we commence with an 'old' set of parameter values - i , - i , W i and oe 2 i . We first use these parameters to evaluate the posterior probabilities R ni using (14). These posterior probabilities are then used to obtain 'new' values e using the following re-estimation formulae e R ni (15) e The new values e are then used in evaluation of the sufficient statistics for the posterior distribution for x ni Finally, these statistics are used to evaluate 'new' values f and e oe 2 using f R ni hx ni x T e d R ni kt \Gamma2 R ni hx T R ni f which are derived in Appendix B. As for the single latent variable model, we can substitute the expressions for hx ni i and ni i, given by (17) and (18) respectively, into (19) and (20). We then see that the re-estimation formulae for f i take the form f d f where all of the data dependence been expressed in terms of the quantities and we have defined R ni . The matrix S i can clearly be interpreted as a responsibility-weighted covariance matrix. Again, for reasons of computational efficiency, the form of EM algorithm given by (17) to (20) is to be preferred if d is large. Hierarchical Mixture Models We now extend the mixture representation of Section 3 to give a hierarchical mixture model. Our formulation will be quite general and can be applied to mixtures of any parametric density model. So far we have considered a two-level system consisting of a single latent variable model at the top level and a mixture of M 0 such models at the second level. We can now extend the hierarchy to a third level by associating a group G i of latent variable models with each model i in the second level. The corresponding probability density can be written in the where p(tji; latent variable models, and - jji correspond to sets of mixing coefficients, one for each i, which satisfy 1. Thus each level of the hierarchy corresponds to a generative model, with lower levels giving more refined and detailed representations. This model is illustrated in Figure 2. Determination of the parameters of the models at the third level can again be viewed as a missing data problem in which the missing information corresponds to labels specifying which model generated each data point. When no information about the labels is provided COPY Second Level Mixture Model en t Mode Third Level Mixture Model Figure 2: The structure of the hierarchical model. the log likelihood for the model (24) would take the form If, however, we were given a set of indicator variables z ni specifying which model i at the second level generated each data point t n then the log likelihood would become z ni ln In fact we only have partial, probabilistic, information in the form of the posterior responsibilities R ni for each model i having generated the data points t n , obtained from the second level of the hierarchy. Taking the expectation of (26) we then obtain the log likelihood for the third level of the hierarchy in the form R ni in which the R ni are constants. In the particular case in which the R ni are all 0 or 1, corresponding to complete certainty about which model in the second level is responsible for each data point, the log likelihood (27) reduces to the form (26). Maximization of (27) can again be performed using the EM algorithm, as discussed in Appendix C. This has the same form as the EM algorithm for a simple mixture, discussed in Section 3, except that in the E-step, the posterior probability that model (i; generated data point t n is given by in which Note that R ni are constants determined from the second level of the hierarchy, and R njji are functions of the 'old' parameter values in the EM algorithm. The expression (29) automatically satisfies the relation so that the responsibility of each model at the second level for a given data point n is shared by a partition of unity between the corresponding group of offspring models at the third level. The corresponding EM algorithm can be derived by a straightforward extension of the discussion given in Section 3 and Appendix B, and is outlined in Appendix C. This shows that the M-step equations for the mixing coefficients and the means are given by e e The posterior expectations for the missing variables z ni;j are then given by Finally, the W i;j and oe 2 i;j are updated using the M-step equations f R ni;j hx ni;j x T e d R ni;j kt \Gamma2 R ni;j hx T R ni;j f Again, we can substitute the E-step equations into the M-step equations to obtain a set of update formulae of the form f e d f where all of the summations over n have been expressed in terms of the quantities in which we have defined . The S i;j can again be interpreted as responsibility- weighted covariance matrices. It is straightforward to extend this hierarchical modelling technique to any desired number of levels, for any parametric family of component distributions. 5 The Visualization Algorithm So far we have described the theory behind hierarchical mixtures of latent variable models, and have illustrated the overall form of the visualization hierarchy in Figure 2. We now complete the description of our algorithm by considering the construction of the hierarchy, and its application to data visualization. Although the tree structure of the hierarchy can be pre-defined, a more interesting possi- bility, with greater practical applicability, is to build the tree interactively. Our multi-level visualization algorithm begins by fitting a single latent variable model to the data set, in which the value of - is given by the sample mean. For low values of the data space dimensionality d we can find W and oe 2 directly by evaluating the covariance matrix and applying (4) and (5). However, for larger values of d it may be computationally more efficient to apply the EM algorithm, and a scheme for initializing W and oe 2 is given in Appendix D. Once the EM algorithm has converged, the visualization plot is generated by plotting each data point t n at the corresponding posterior mean hx n i in latent space. On the basis of this plot the user then decides on a suitable number of models to fit at the next level down, and selects points x (i) on the plot corresponding, for example, to the centres of apparent clusters. The resulting points y (i) in data space, obtained from (1), are then used to initialize the means - i of the respective sub-models. To initialize the remaining parameters of the mixture model we first assign the data points to their nearest mean vector - i and then either compute the corresponding sample covariance matrices and apply a direct eigenvector decomposition, or use the initialization scheme of Appendix D followed by the EM algorithm. Having determined the parameters of the mixture model at the second level we can then obtain the corresponding set of visualization plots, in which the posterior means hx ni i are again used to plot the data points. For these it is useful to plot all of the data points on every plot, but to modify the density of 'ink' in proportion to the responsibility which each plot has for that particular data point. Thus, if one particular component takes most of the responsibility for a particular point, then that point will effectively be visible only on the corresponding plot. The projection of a data point onto the latent spaces for a mixture of two latent variable models is illustrated schematically in Figure 3. The resulting visualization plots are then used to select further sub-models, if desired, Figure 3: Illustration of the projection of a data point onto the latent spaces of a mixture of two latent variable models. with the responsibility weighting of (28) being incorporated at this stage. If it is decided not to partition a particular model at some level, then it is easily seen from (30) that the result of training is equivalent to copying the model down unchanged to the next level. Equation (30) further ensures that the combination of such copied models with those generated through further sub-modelling defines a consistent probability model, such as that represented by the lower three models in Figure 2. The initialization of the model parameters is by direct analogy with the second-level scheme, with the covariance matrices now also involving the responsibilities R ni as weighting coefficients, as in (23). Again, each data point is in principle plotted on every model at a given level, with a density of 'ink' proportional to the corresponding posterior probability, given for example by (28) in the case of the third level of the hierarchy. Deeper levels of the hierarchy involve greater numbers of parameters and it is therefore important to avoid over-fitting and to ensure that the parameter values are well-determined by the data. If we consider principal component analysis then we see that three (non- data points are sufficient to ensure that the covariance matrix has rank two and hence that the first two principal components are defined, irrespective of the dimensionality of the data set. In the case of our latent variable model, four data points are sufficient to determine both W and oe 2 . From this we see that we do not need excessive numbers of data points in each leaf of the tree, and that the dimensionality of the space is largely irrelevant. Finally, it is often also useful to be able to visualize the spatial relationship between a group of models at one level and their parent at the previous level. This can be done by considering the orthogonal projection of the latent plane in data space onto the corresponding plane of the parent model, as illustrated in Figure 4. For each model in the hierarchy (except those at the lowest level) we can plot the projections of the associated models from the level below. In the next section, we illustrate the operation of this algorithm when applied to a simple toy data set, before presenting results from the study of more realistic data in Sections 7 and 8. Figure 4: Illustration of the projection of one of the latent planes onto its parent plane. 6 Illustration using Toy Data We first consider a toy data set consisting of 450 data points generated from a mixture of three Gaussians in a three-dimensional space. Each Gaussian is relatively flat (has small variance) in one dimension, and all have the same covariance but differ in their means. Two of these pancake-like clusters are closely spaced, while the third is well separated from the first two. The structure of this data set has been chosen order to illustrate the interactive construction of the hierarchical model. To visualize the data, we first generate a single top-level latent variable model, and plot the posterior mean of each data point in the latent space. This plot is shown at the top of Figure 5, and clearly suggests the presence of two distinct clusters within the data. The user then selects two initial cluster centres within the plot, which initialize the second- level. This leads to a mixture of two latent variable models, the latent spaces of which are plotted at the second level in Figure 5. Of these two plots, that on the right shows evidence of further structure, and so a sub-model is generated, again based on a mixture of two latent variable models, which illustrates that there are indeed two further distinct clusters. At this third step of the data exploration, the hierarchical nature of the approach is evident as the latter two models only attempt to account for the data points which have already been modelled by their immediate ancestor. Indeed, a group of offspring models may be combined with the siblings of the parent and still define a consistent density model. This is illustrated in Figure 5, in which one of the second level plots has been 'copied down' (shown by the dotted line) and combined with the other third-level models. When offspring plots are generated from a parent, the extent of each offspring latent space (i.e. the axis limits shown on the plot) is indicated by a projected rectangle within the parent space, using the approach illustrated in Figure 4, and these rectangles are numbered sequentially such that the leftmost sub-model is '1'. In order to display the relative orientations of the latent planes, this number is plotted on the side of the rectangle which corresponds to the top of the corresponding offspring plot. The original three clusters have been individually coloured and it can be seen that the red, yellow and blue data points have been almost perfectly separated in the third level. Figure 5: A summary of the final results from the toy data set. Each data point is plotted on every model at a given level, but with a density of ink which is proportional to the posterior probability of that model for the given data point. 7 Oil Flow Data As an example of a more complex problem we consider a data set arising from a non-invasive monitoring system used to determine the quantity of oil in a multi-phase pipeline containing a mixture of oil, water and gas [9]. The diagnostic data is collected from a set of three horizontal and three vertical beam-lines along which gamma rays at two different energies are passed. By measuring the degree of attenuation of the gammas, the fractional path length through oil and water (and hence gas) can readily be determined, giving 12 diagnostic measurements in total. In practice the aim is to solve the inverse problem of determining the fraction of oil in the pipe. The complexity of the problem arises from the possibility of the multi-phase mixture adopting one of a number of different geometrical configurations. Our goal is to visualize the structure of the data in the original 12-dimensional space. A data set consisting of 1000 points is obtained synthetically by simulating the physical processes in the pipe, including the presence of noise dominated by photon statistics. Locally, the data is expected to have an intrinsic dimensionality of 2 corresponding to the two degrees of freedom given by the fraction of oil and the fraction of water (the fraction of gas being redundant). However, the presence of different flow configurations, as well as the geometrical interaction between phase boundaries and the beam paths, leads to numerous distinct clusters. It would appear that a hierarchical approach of the kind discussed here should be capable of discovering this structure. Results from fitting the oil flow data using a 3-level hierarchical model are shown in Figure 6. Homogeneous Annular Laminar Figure Results of fitting the oil data. Colours denote different multi-phase flow configurations corresponding to homogeneous (red), annular (blue) and laminar (yellow). In the case of the toy data discussed in Section 6, the optimal choice of clusters and sub-clusters is relatively unambiguous and a single application of the algorithm is sufficient to reveal all of the interesting structure within the data. For more complex data sets, it is appropriate to adopt an exploratory perspective and investigate alternative hierarchies, through the selection of differing numbers of clusters and their respective locations. The example shown in Figure 6 has clearly been highly successful. Note how the apparently single cluster, number 2, in the top level plot is revealed to be two quite distinct clusters at the second level, and how data points from the 'homogeneous' configuration have been isolated and can be seen to lie on a two-dimensional triangular structure in the third level. 8 Satellite Image Data As a final example, we consider the visualization of a data set obtained from remote-sensing satellite images. Each data point represents a 3x3 pixel region of a satellite land image, and for each pixel there are four measurements of intensity taken at different wavelengths (approximately red and green in the visible spectrum, and two in the near infra-red). This gives a total of 36 variables for each data point. There is also a label indicating the type of land represented by the central pixel. This data set has previously been the subject of a classification study within the Statlog project [10]. We applied the hierarchical visualization algorithm to 600 data points, with 100 drawn at random of each of six classes in the 4435-point data set. The result of fitting a 3-level hierarchy is shown in Figure 7. Note that the class labels are used only to colour the data red soil cotton crop grey soil damp grey soil soil with vegetation stubble very damp grey soil Figure 7: Results of fitting the satellite image data. points and play no role in the maximum likelihood determination of the model parameters. Figure 7 illustrates that the data can be approximately separated into classes, and the 'very damp grey soil' continuum is clearly evident in component 3 at the second level. One particularly interesting additional feature is that there appear to be two distinct and separated clusters of 'cotton crop' pixels, in mixtures 1 and 2 at the second level, which are not evident in the single top-level projection. Study of the original image [10] indeed indicates that there are two separate areas of 'cotton crop'. 9 Discussion We have presented a novel approach to data visualization which is both statistically principled and which, as illustrated by real examples, can be very effective at revealing structure within data. The hierarchical summaries of Figures 5, 6 and 7 are relatively simple to in- terpret, yet still convey considerable structural information. It is important to emphasize that in data visualization there is no objective measure of quality, and so it is difficult to quantify the merit of a particular data visualization tech- nique. This is one reason, no doubt, why there is a multitude of visualization algorithms and associated software available. While the effectiveness of many of these techniques is often highly data-dependent, we would expect the hierarchical visualization model to be a very useful tool for the visualization and exploratory analysis of data in many applications. In relation to previous work, the concept of sub-setting, or isolating, data points for further investigation can be traced back to Maltson and Dammann [11], and was further developed by Friedman and Tukey [12] for exploratory data analysis in conjunction with projection pursuit. Such sub-setting operations are also possible in current dynamic visualization software, such as 'XGobi' [13]. However, in these approaches there are two limitations. First, the partitioning of the data is performed in a hard fashion, while the mixture of latent variable models approach discussed in this paper permits a soft partitioning in which data points can effectively belong to more than one cluster at any given level. Second, the mechanism for the partitioning of the data is prone to sub-optimality as the clusters must be fixed by the user based on a single two-dimensional projection. In the hierarchical approach advocated in this paper, the user selects only a 'first guess' for the cluster centres in the mixture model. The EM algorithm is then utilized to determine the parameters which maximize the likelihood of the model, thus allowing both the centres and the widths of the clusters to adapt to the data in the full multi-dimensional data space. There is also some similarity between our method and earlier hierarchical methods in script recognition [14] and motion planning [15] which incorporate the Kohonen Self-Organizing Feature Map [16] and so offer the potential for visualization. As well as again performing a hard clustering, a key distinction in both of these approaches is that different levels in the hierarchies operate on different subsets of input variables and their operation is thus quite different from the hierarchical algorithm described in this paper. Our model is based on a hierarchical combination of linear latent variable models. A related latent variable technique called the generative topographic mapping (GTM) [17] uses a non-linear transformation from latent space to data space and is again optimized using an EM algorithm. It is straightforward to incorporate GTM in place of the linear latent variable models in the current hierarchical framework. As described, our model applies to continuous data variables. We can easily extend the model to handle discrete data as well as combinations of discrete and continuous vari- ables. In case of a set of binary data variables y k 2 f0; 1g we can express the conditional distribution of a binary variable, given x, using a binomial distribution of the form is the logistic sigmoid function, and w k is the k th column of W. For data having a 1-of-D coding scheme we can represent the distribution of data variables using a multi-nomial distribution of the form are defined by a softmax, or normalized exponential, transformation of the form If we have a data set consisting of a combination of continuous, binary and categorical variables, we can formulate the appropriate model by writing the conditional distribution p(tjx) as a product of Gaussian, binomial and multi-nomial distributions as appropriate. The E-step of the EM algorithm now becomes more complex since the marginalization over the latent variables, needed to normalize the posterior distribution in latent space, will in general be analytically intractable. One approach is to approximate the integration using a finite sample of points drawn from the prior [17]. Similarly, the M-step is more complex, although it can be tackled efficiently using the iterative re-weighted least squares One important consideration with the present model is that the parameters are determined by maximum likelihood, and this criterion need not always lead to the most interesting visualization plots. We are currently investigating alternative models which optimize other criteria such as the separation of clusters. Other possible refinements include algorithms which allow a self-consistent fitting of the whole tree, so that lower levels have the opportunity to influence the parameters at higher levels. While the user-driven nature of the current algorithm is highly appropriate for the visualization context, the development of an automated procedure for generating the hierarchy would clearly also be of interest. A software implementation of the probabilistic hierarchical visualization algorithm in Matlab is available from: http://www.ncrg.aston.ac.uk/PhiVis Acknowledgements This work was supported by EPSRC grant GR/K51808: Neural Networks for Visualization of High-Dimensional Data. We are grateful to Michael Jordan for useful discussions, and we would like to thank the Isaac Newton Institute in Cambridge for their hostpitality. --R "Hierarchical mixtures of experts and the EM algo- rithm.," An Introduction to Latent Variable Models. Multivariate Analysis Part 2: Classification "Mixtures of principal component analysers," "Mixtures of probabilistic principal component analysers," "Maximum likelihood from incomplete data via the EM algorithm," "EM algorithms for ML factor analysis," Neural Networks for Pattern Recognition. "Analysis of multiphase flows using dual-energy gamma densitometry and neural networks," Neural and Statistical Classification. "A technique for determining and coding sub-classes in pattern recognition problems," "A projection pursuit algorithm for exploratory data analysis," "Interactive high-dimensional data visualiza- tion," "Script recognition with hierarchical feature maps," "Learning fine motion by using the hierarchical extended Kohonen map," "GTM: the generative topographic mapping," Chapman and Hall --TR --CTR Peter Tino , Ian Nabney, Hierarchical GTM: Constructing Localized Nonlinear Projection Manifolds in a Principled Way, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.24 n.5, p.639-656, May 2002 Michalis K. Titsias , Aristidis Likas, Mixture of experts classification using a hierarchical mixture model, Neural Computation, v.14 n.9, p.2221-2244, September 2002 Tien-Lung Sun , Wen-Lin Kuo, Visual exploration of production data using small multiples design with non-uniform color mapping, Computers and Industrial Engineering, v.43 n.4, p.751-764, September 2002 Neil D. Lawrence , Andrew J. Moore, Hierarchical Gaussian process latent variable models, Proceedings of the 24th international conference on Machine learning, p.481-488, June 20-24, 2007, Corvalis, Oregon Alexei Vinokourov , Mark Girolami, A Probabilistic Framework for the Hierarchic Organisation and Classification of Document Collections, Journal of Intelligent Information Systems, v.18 n.2-3, p.153-172, March-May 2002 Daniel Boley, Principal Direction Divisive Partitioning, Data Mining and Knowledge Discovery, v.2 n.4, p.325-344, December 1998 Hiroshi Mamitsuka, Essential Latent Knowledge for Protein-Protein Interactions: Analysis by an Unsupervised Learning Approach, IEEE/ACM Transactions on Computational Biology and Bioinformatics (TCBB), v.2 n.2, p.119-130, April 2005 Ting Su , Jennifer G. Dy, Automated hierarchical mixtures of probabilistic principal component analyzers, Proceedings of the twenty-first international conference on Machine learning, p.98, July 04-08, 2004, Banff, Alberta, Canada Ian T. Nabney , Yi Sun , Peter Tino , Ata Kaban, Semisupervised Learning of Hierarchical Latent Trait Models for Data Visualization, IEEE Transactions on Knowledge and Data Engineering, v.17 n.3, p.384-400, March 2005 Michael E. Tipping , Christopher M. Bishop, Mixtures of probabilistic principal component analyzers, Neural Computation, v.11 n.2, p.443-482, Feb. 15, 1999 Unsolved Information Visualization Problems, IEEE Computer Graphics and Applications, v.25 n.4, p.12-16, July 2005 Kui-Yu Chang , J. Ghosh, A Unified Model for Probabilistic Principal Surfaces, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.23 n.1, p.22-41, January 2001 Wang , Yue Wang , Jianping Lu , Sun-Yuan Kung , Junying Zhang , Richard Lee , Jianhua Xuan , Javed Khan , Robert Clarke, Discriminatory mining of gene expression microarray data, Journal of VLSI Signal Processing Systems, v.35 n.3, p.255-272, November Pradeep Kumar Shetty , R. Srikanth , T. S. Ramu, Modeling and on-line recognition of PD signal buried in excessive noise, Signal Processing, v.84 n.12, p.2389-2401, December 2004

maximum likelihood;latent variables;principal component analysis;factor analysis;statistics;density estimation;hierarchical mixture model;EM algorithm;data visualization;clustering

279092

Interpolating Arithmetic Read-Once Formulas in Parallel.

A formula is read-once if each variable appears in it at most once. An arithmetic formula is one in which the operations are addition, subtraction, multiplication, and division (and constants are allowed). We present a randomized (Las Vegas) parallel algorithm for the exact interpolation of arithmetic read-once formulas over sufficiently large fields. More specifically, for $n$-variable read-once formulas and fields of size at least 3(n2+3n-2), our algorithm runs in $O(\log^2 n)$ parallel steps using O(n4) processors (where the field operations are charged unit cost). This complements some results from [N. H. Bshouty and R. Cleve, Proc. 33rd Annual Symposium on the Foundations of Computer Science, IEEE Computer Science Press, Los Alamitos, CA, 1992, pp. 24--27] which imply that other classes of read-once formulas cannot be interpolated---or even learned with membership and equivalence queries---in polylogarithmic time with polynomially many processors (even though they can be learned sequentially in polynomial time). These classes include boolean read-once formulas and arithmetic read-once formulas over fields of size $o(n / \log n)$ (for n variable read-once formulas).

Introduction The problem of interpolating a formula (from some class C) is the problem of exactly identifying the formula from queries to the assignment (membership) oracle. The interpolation algorithm queries the oracle with an assignment a and the oracle returns the value of the function at a. There are a number of classes of arithmetic formulas that can be interpolated sequentially in polynomial-time as well as in parallel in poly-logarithmic-time (with polynomially many processors). These include sparse polynomials and sparse rational functions ([BT88,BT90,GKS90,GrKS88,RB89,M91]). Research supported in part by NSERC of Canada. Author's E-mail addresses: bshouty@cpsc.ucalgary.ca and cleve@cpsc.ucalgary.ca. A formula over a variable set V is read-once if each variable appears at most once in it. An arithmetic read-once formula over a field K is a read-once formula over the basic operations of the field K: addition, subtraction, multiplication, division, and constants are also permitted in the formula. The size of an arithmetic formula is the number of instances of variables (i.e. leaves) in it. Bshouty, Hancock and Hellerstein [BHH92] present a randomized sequential polynomial-time algorithm for interpolating arithmetic read-once formulas (AROFs) over sufficiently large fields. Moreover, they show that, for arbitrarily-sized fields, read-once formulas can be learned using equivalence queries in addition to membership queries. The question of whether arithmetic read-once formulas can be interpolated (or learned) quickly in parallel depends on the size of the underlying field. It is shown in [BC92] that for arithmetic read-once formulas over fields with o(n= log n) elements there is no poly-logarithmic-time algorithm that uses polynomially many processors (for interpolating, as well as learning with membership and equivalence queries). Also, a similar negative result holds for boolean read-once formulas. We present a (Las Vegas) parallel algorithm for the exact interpolation of arithmetic read-once formulas over sufficiently large fields. For fields of size at least 2), the algorithm runs in O(log 2 n) parallel steps using O(n 4 ) processors (where the field operations are charged unit cost). If the "obvious" parallelizations are made to the interpolating algorithm in (i.e., parallelizations of independent parts of the computation) one obtains a parallel running time that is \Theta(d), where d is the depth of the target formula. Since, in general, d can be as large as \Theta(n), this does not result in significant speedup. Our parallel algorithm uses some techniques from the sequential algorithm of [BHH92] as well as some new techniques that enable nonlocal features of the AROF to be determined in poly-logarithmic-time. The parallel algorithm can be implemented on an oracle parallel random access machine (PRAM). More specifically, it is an exclusive-read exclusive-write (EREW) PRAM-which means that processor's accesses to their communal registers are constrained so that no two processors can read from or write to the same register simultaneously. The EREW PRAM initially selects some random input values (uniformly and independently distributed) and then performs O(n 3 ) membership queries (via its oracle). queries The learning criterion we consider is exact identification. There is a formula f called the target formula, which is a member of a class of formulas C defined over the variable set V . The goal of the learning algorithm is to halt and output a formula h from C that is equivalent to f . In a membership query, the learning algorithm supplies values (x (0) the variables in V , as input to a membership oracle, and receives in return the value of f(x (0) the projection of f obtained by hard-wiring x to the value x (0) . An assignment of values to some subset of a read-once formula's variables defines a projection, which is the formula obtained by hard-wiring those assigned variables to their values in the formula and then rewriting the formula to eliminate constants from the leaves. Note that if f 0 is a projection of f , it is possible to simulate a membership oracle for f 0 using a membership oracle for f . We say that the class C is learnable in polynomial time if there is an algorithm that uses the membership oracle and interpolates any f 2 C in polynomial time in the number of variables n and the size of f . We say that C is efficiently learnable in parallel if there is a parallel algorithm that uses the membership oracle and interpolates any f 2 C in polylogarithmic time with polynomial number of processors. In the parallel computation p processors can ask p membership queries in one step. 3 Preliminaries A formula is a rooted tree whose leaves are labeled with variables or constants from some domain, and whose internal nodes, or gates, are labeled with elements from a set of basis functions over that domain. A read-once formula is a formula for which no variable appears on two different leaves. An arithmetic read-once formula over a field K is a read-once formula over the basis of addition, subtraction, multiplication, and division of field elements, whose leaves are labeled with variables or constants from K. In [BHH92] it is shown that a modified basis can be used to represent any read-once formula. Let K be an arbitrary field. The modified basis for arithmetic read-once formulas over K includes only two non-unary functions, addition (+) and multiplication (\Theta). The unary functions in the basis are (ax d) for every a; b; c; d 2 K such that ad \Gamma bc 6= 0. This requirement is to prevent being identically 0 or differing by just a constant factor. We can also assume that non-constant formulas over this modified basis do not contain constants in their leaves. We represent such a unary function as f A , where a b c d The restriction on a, b, c, and d is equivalent to saying the determinant of A (denoted det(A)) is non-zero. The value of a read-once formula on an assignment to its variables is determined by evaluating the formula bottom up. This raises the issue of division by zero. In [BHH92] this problem is handled by defining basis functions over the extended domain K [ f1;ERRORg, where 1 represents 1=0 and ERROR represents 0=0. For the special values we define our basis function as follows (assume x and A is as above). f A ( a c It is shown in [BHH92] that these definitions are designed so that the output of the read-once formula is the same as it would be if the formula were first expanded and simplified to be in the form p(x p and q where gcd(p; We say that a formula f is defined on the variable set V if all variables appearing in f are members of V . Let g. We say a formula f depends on variable x i if there are values x (0) n and x (1) i in K for which and for which both those values of f are not ERROR. We call such an input vector justifying assignment for x i . Between any two gates or leaves ff and fi in an AROF, the relationships ancestor, descendant, parent, and child refer to their relative position in the rooted tree. Let that ff is a descendant of fi (or, equivalently, that fi is an ancestor of ff). Let ff ! fi denote that ff is a proper descendant of fi (i.e., ff - fi but ff 6= fi). For any pair of variables x i and x j that appear in a read-once formula, there is a unique node farthest from the root that is an ancestor of both x i and x j , called their lowest common ancestor, which we write as lca(x i ; x j ). We shall refer to the type of lca(x to mean the basis function computed at that gate. We say that a set W of variables has a common lca if there is a single node that is the lca of every pair of variables in W . We define the skeleton of a formula f to be the tree obtained by deleting any unary gates in f (i.e. the skeleton describes the parenthesization of an expression with the binary operations, but not the actual unary operations or embedded constants). We now list a basic property of unary functions f A that is proved in [BHH92]. 1. The function f A is a bijection from K [ f1g to K [ f1g if and only if is either a constant value from K[f1;ERRORg or else is a constant value from K[f1g, except on one input value on which it is ERROR. 2. The functions f A and f -A are equivalent for any - 6= 0. 3. Given any three distinct points (a) If are on a line then there exists a unique function f A with f A (b) If are not on a line then there exists a unique function f A with 4. If functions f A and f B are equivalent and det (A); det (B) 6= 0, then there is a constant - for which -A = B. 5. The functions (f A are equivalent. 6. If det(A) 6= 0, functions f A and f A \Gamma1 are equivalent. 7. f A A A A A (x). 4 Collapsibility of Operations Whenever two non-unary gates of the same type in an AROF are separated by only a unary gate it may be possible to collapse them together to a single non-unary gate of the same type with higher arity. For ? 2 f+; \Thetag, a unary operation f A is called ?-collapsible if for some unary operations f B and f C . Intuitively, the above property means that if the f A gate occurs between two non-unary ? gates then the two ? gates can be "collapsed" into a single ? gate of higher arity, provided that new unary gates can be applied to the inputs. In [BHH92] it is explained that a unary gate f A is \Theta-collapsible if and only if A is of the form ' a 0 or and +-collapsible if and only if A is of the form ' a b The following are equivalent definitions of ?-collapsible that will be used in this paper. Property 2 The following are equivalent 1. f A is +-collapsible. 2. f A 3. f A (1) = 1. The following are equivalent 1. f A is \Theta-collapsible. 2. f A 3. ff A (1); f A (0)g = f0; 1g. Proof: We prove the property by showing that 1 , 2 , 3. If f A is +-collapsible then ' a b and therefore (b=c). Since A is nonsingular a 6= 0 and c 6= 0 and a=c f A d c must have The result for \Theta-collapsible is left for the reader.2 In [BHH92], a three-way justifying assignment is defined as an assignment of constant values to all but three variables in an AROF such that the resulting formula depends on all of the three remaining variables. For the present results, we require assignments that meet additional requirements, which are defined below. For any two gates, ff and fi, with ff ! fi, define the ff-fi path as the sequence of gate operations along the path in the tree from ff to fi (including the operations of ff to fi at the endpoints of the path). Define a non-collapsing three-way justifying assignment as a three-way justifying assignment with the following additional property. For the unassigned variables x, y, and z, if lca(x; y) ! lca(x; z) and all non-unary operations in the lca(x; y)-lca(x; z) path are of the same type ? (for some then the function that results from the justifying assignment is of the form for some unary operations f A , f B , f C , f D and f E , where f C is not ?-collapsible. Intuitively, this means that, after the justifying assignment, the two gates, lca(x; y) and lca(x; z), cannot be collapsed-and thus the relationship lca(x; y) ! lca(x; z) can still be detected in the resulting function. Now, define a total non-collapsing three-way justifying assignment as a single assignment of constant values to all variables in an AROF such that, for any three variables, if all but those three are assigned to their respective constants then the resulting assignment is non-collapsing three-way justifying. 5 Parallel Learning Algorithm In this section, we present a parallel algorithm for learning AROFs. The algorithm has three principal components: finding a total non-collapsing three-way justifying assignment; determining the skeleton of the AROF; and, determining the unary gates of the AROF. The basic idea is to first construct a graph (that will later be referred to as the LCAH graph) that contains information about the relative positions of the lcas of all pairs of variables. This cannot be obtained quickly in parallel from justifying assignments, because of the possibility that some of the important structure of an AROF "collapses" under any given justifying assignment. However, we shall see that any total non-collapsing justifying assignment is sufficient to determine the entire structure of the AROF at once (modulo some polylog processing). Once the LCAH graph has been constructed, the skeleton of the AROF can be constructed by discarding some of the structure of the LCAH graph (a "garbage collection" step). This is accomplished using some simple graph algorithms, as well as a parallel prefix sum computation (which is NC 1 computable [LF80]). Finally, once that skeleton is determined, the unary gates can be determined by a recursive tree contraction method (using results from [B74]). 5.1 Finding a Total Non-Collapsing Three-Way Justifying Assignment In [BHH92], it is proven that, for any triple of variables x, y and z, by drawing random values (independently) from a sufficiently large field, and assigning them to the other variables in an AROF, a three-way justifying assignment for those variables is obtained with high probability. In the parallel algorithm, a three-way justifying assignment that is total non-collapsing is required. We show that, if the size of the field K is at least O(n 2 ) then the same randomized procedure also yields a total non-collapsing three-way justifying assignment with probability at least 1Therefore in time O(1) this step can be implemented. We shall begin with some preliminary lemmas and then the precise statement that we require will appear in Corollary 4. Lemma 1: If A is not ?-collapsing then there exists at most one value z (0) for z such that f C (y) j g(y; z (0) ) is ?-collapsing. then by property 2 we have where ff 2 Knf0g and fi 2 K. We substitute f A is not +-collapsible, by property 2, we have f A Solving the above system using property 1 we get This shows that there is at most one value of z that makes f B (f A (y) collapsible. then by property 2 we have where ff 2 Knf0g and fi 2 f+1; \Gamma1g. We substitute f A is not \Theta-collapsible, by property 2, we have either f A (0) or f A (1) is not in f0; 1g. Suppose f A (0) 62 f0; 1g and suppose f B (f A (0)z 0 are similar). Solving this gives This shows that there is at most one value of z that makes f B (f A (y)z) \Theta-collapsible.2 Lemma 2: Let F be an AROF with suppose that all non-unary operations in the lca(x 1 are of the same type ? 2 f+; \Thetag. Let x (0) n be independently uniformly randomly chosen from S ' K, where m. Then the probability that x (0) n is a non-collapsing three-way justifying assignment is at least 1 \Gamma Proof: Note that x (0) n is not a non-collapsing three-way justifying assignment if and only if it is not a justifying assignment or there exists a path between the lcas of x 1 , x 2 and x 3 such that all non-unary operations are of the same type and the path collapses under the assignment. From [BHH92], the probability of the former condition is at most 2n+4 . We need to bound the probability of the latter condition. We have that F is of the form E(fH k may depend on variables from x in addition to their marked arguments. Let - E(y) denote the above formulas (respectively) with x (0) substituted for the variables denote the degrees of C as functions of x . By the assumption that F is in normal form, f H0 is not ?-collapsing. Therefore, by Lemma 1, there exists at most one value of C 1 for which f H 1 ?-collapsing. We can bound the probability of this value occurring for C 1 . Since the degree of C 1 is d 1 , an application of Schwartz's result in [Sch80] implies that probability that this value occurs for C 1 is at most d 1 =m. Similarly, if f H 1 ?-collapsing then Lemma 1 implies that there exists at most one value of C 2 for which f H 2 collapsing, which occurs with probability at most d 2 =m, and so on. It follows that the probability that is ?-collapsing is at most (d The result now follows by summing the two bounds. 2 Theorem 3: Let F be an AROF over K, and x (0) n be chosen uniformly from a set S ' K with m. Then the probability that x (0) is a total non-collapsing three-way justifying assignment is at least 1 \Gamma 6n 2 Proof: First, note that, from Lemma 2, we can immediately infer that if are drawn independently uniformly randomly from S ' K, where then the probability that x (0) n is a non-collapsing three-way justifying assignment is at most To obtain a better bound, consider each subformula C i that is an input to some non-unary gate in the AROF. By results in [BHH92], there are at most two possible values of C i that will result in some triple of variables with respect to which the the assignment is not three-way justifying (the values are 0 and 1). Thus, as in the proof of Lemma 2, the probability of one of these values arising for C i is at most 2d , where d is the degree of C i . Also, from Lemma 2, there is at most one value of C i that will result in a collapsing assignment, and the probability of this arising is at most d . Thus, the probability of one of the two events above arising is at most 3d , and, since d - n, this is at most 3n Since there are at most 2n such subformulas C i , the probability of any one of them attaining one of the above values is at most 6n 2 . 2 The constant in the proof of theorem 3 can be improved to obtain probability of by using the following observation. Notice that we upper bounded the degree of each subtree by n. In fact we can upper bound the degree of the leaves (there are n leaves) by degree 1 since they are variables. Then we have another subformulas of degrees . It is easy to show that d i - i +1 (simple induction on the number of nodes). Taking all this into account we obtain the above bound. By setting m - we obtain the following. Corollary 4: Let F be an AROF over K, and x (0) n be chosen uniformly from a set S ' K with 2). Then the probability that n is a total non-collapsing three-way justifying assignment is least 1 This Corollary implies that the expected time complexity of finding a total non- collapsing three-way justifying assignment is O(1). 5.2 Determining the Skeleton of a Read-Once Formula in Parallel In this section, we assume that a total non-collapsing three-way justifying assignment is given and show how to construct the skeleton with O(n 3 ) membership queries in one parallel step followed by O(log n) steps of computation. Firstly, suppose that, for a triple of variables x, y, and z, we wish to test whether or not lca(x; y) ! lca(x; z). If op(x; y) 6= op(x; z) then this can be accomplished by a direct application of the techniques in [BHH92], using the fact that we have an assignment that is justifying with respect to variables x, y, and z. On the other hand, if could be difficult to detect with a mere justifying assignment because the justifying assignment might collapse the relative structure between these three variables. If all the non-unary operations in the lca(x; y)-lca(x; z) path are identical then, due to the fact that we have a non-collapsing justifying assignment, we are guaranteed that the sub-structure between the three variables does not collapse, and we can determine that lca(x; y) ! lca(x; z) in O(1) time (again by directly applying techniques in [BHH92]). This leaves the case where op(x; but the non-unary operations in the lca(x; y)-lca(x; z) path are not all of the same type. In this case, the techniques of [BHH92] might fail to determine that lca(x; y) ! lca(x; z) and report them as equal. We shall overcome this problem at a later stage in our learning algorithm, by making inferences based on hierarcical relationships with other vari- ables. For the time being, we can, in time O(1) with one processor, compute the following. YES if lca(x; y) ! lca(x; z) and all non-unary operations in the lca(x; y)-lca(x; z) path are of the same type; YES or MAYBE if lca(x; y) ! lca(x; z) and op(x; but not all non-unary operations in the lca(x; y)-lca(x; z) path are of the same type; MAYBE otherwise. Note that if DESCENDANT(x; must be that lca(x; y) ! then it is possible that and the non-unary operations on the are not of the same type, or that lca(x; y) 6! lca(x; z). To construct the extended skeleton of an AROF, we first construct its least common ancestor hierarchy (LCAH) graph, which is defined as follows. Definition: The least common ancestor hierarchy (LCAH) graph of an AROF with n variables consists of vertices, one corresponding to each (unordered) pair of variables. For the distinct variables, x and y, denote the corresponding vertex by xy or, equivalently, yx. Then, for distinct vertices xy and zw, the directed edge present in the LCAH graph if and only if lca(x; y) - lca(z; w). We shall prove that the following algorithm constructs the LCAH graph of an AROF. Algorithm CONSTRUCT-LCAH-GRAPH 1. in parallel for all distinct variables x; y; z do if DESCENDANT(x; insert edges xy ! xz and xy ! yz and xz ! yz and yz ! xz 2. in parallel for all distinct variables x; if edges xy ! xw ! xz are present then insert edge xy ! xz 3. in parallel for all distinct variables x; y; z do if no edges between any of xy; xz; yz are present then insert edges in each direction between every pair of xy; xz; yz 4. in parallel for all distinct variables x; if edges xy ! xw ! zw present or edges xy ! yw ! zw present then insert edge xy ! zw Theorem 5: Algorithm CONSTRUCT-LCAH-GRAPH constructs the LCAH graph of an AROF. Proof: The proof follows from the following sequence of observations: (i) For all distinct variables x, y and z for which lca(x; y) ! lca(x; after executing steps 1 and 2 of the algorithm, the appropriate edges pertaining to vertices xy, xz and yz (namely, xy ! xz, xy ! yz, xz ! yz and yz ! xz) are present. (ii) For all distinct variables x, y and z for which lca(x; after executing step 3 of the algorithm, the appropriate edges pertaining to vertices xy, xz and yz (namely, edges in both directions between every pair) are present. (iii) For all distinct variables x, y, z and w, after executing step 4 of the algo- rithm, the edge xy ! zw is present if and only if lca(x; y) - lca(z; w).2 It is straightforward to verify that algorithm CONSTRUCT-LCAH-GRAPH can be implemented to run in O(log n) time on an EREW PRAM with O(n 4 ) processors. Moreover, the O(n 3 ) membership queries can be made initially in one parallel step. In an AROF, each non-unary gate corresponds to a biconnected component (which is a clique) of its LCAH graph. Thus, to transform the LCAH graph into the extended skeleton of the AROF, we simply "compress" each of its biconnected components into a single vertex and then extract the underlying tree structure of this graph (where the underlying tree structure of a graph is the tree whose transitive closure is the graph 1 ). This is accomplished using standard graph algorithm techniques, including a parallel prefix sum computation ([LF80]). The details follow. We first designate a "leader" vertex for each biconnected component. We then record the individual variables that are descendants of each non-unary gate, and then discard the other nodes in each biconnected component. The algorithm below selects a leader from each connected component in an LCAH graph. We assume that there is a total ordering OE on the vertices of the LCAH graph (for example, the lexicographic ordering on the pair of indices of the two variables corresponding to each vertex). Algorithm LEADER in parallel for all vertices xy OE zw do if edges xy ! zw and zw ! xy are present then mark xy with X It is easy to prove the following. Lemma executing algorithm LEADER, there is precisely one unmarked node (namely, the largest in the OE ordering) in each biconnected component of the LCAH graph. After selecting a leader from each biconnected component of the LCAH graph, we add n new nodes to this graph that correspond to the n variables. The edge inserted if and only if the variable x is a descendant of lca(y; z). This is accomplished by the following algorithm. Algorithm LEAVES in parallel for all distinct variables x; insert edge x ! xy if edge xy ! zw is present then insert edge x ! zw Lemma 7: After executing algorithm LEAVES, the edge x ! yz is present if and only if variable x is a descendant of lca(y; z). Both algorithms LEADER and LEAVES can be implemented in O(log n) time with O(n 4 ) processors. After these steps, the marked nodes are discarded from the augmented LCAH graph (that contains resulting in a graph with at most 2n \Gamma 1 vertices that is isomorphic to the extended skeleton of the AROF. This discarding is accomplished by a standard technique involving the computation of prefix sums. We first adopt the convention that the order OE extends to the augmented LCAH graph as x 1 OE \Delta \Delta \Delta OE x n and x OE yz for any variables x, y and z. Then, for each All edges are directed towards the root. node v, set ae 1 if v is unmarked and compute the prefix sums u-v With algorithms for parallel prefix sum computation ([LF80]) this can be accomplished in O(log( processors. The function oe is a bijection between the unmarked nodes of the augmented LCAH graph and some S ' ng. The following algorithm uses the values of this function to produce the extended skeleton of the AROF. Algorithm COMPRESS-AND-PRUNE in parallel for all distinct vertices u; v do if vertices u; v are both unmarked and edge is in augmented LCAH graph then insert edge oe(u) ! oe(v) in skeleton graph in parallel for all distinct do if edges are in skeleton graph then remove edge from skeleton graph The following is straightforward to prove. Lemma 5: The "skeleton" graph that COMPRESS-AND-PRUNE produces is isomorphic to the extended skeleton of the AROF, where the inputs x correspond to the vertices of the graph. 5.3 Determining a Read-Once Formula from its Skeleton Once the skeleton of an AROF is determined, what remains is to determine the constants in its unary gates (note that the non-unary operations are easy to determine using the techniques in [BHH92]). We show how to do this in O(log 2 n) steps with O(n log n) processors. The main idea is to find a node that partitions the skeleton into three parts whose sizes are all bounded by half of the size of the skeleton. Then the unary gates are determined on each of the parts (in a recursive manner), and the unary gates required to "assemble" the parts are computed. The following lemma is an immediate consequence from a result in [B74]. Lemma 9 [B74]: For any formula F exists a non-unary gate of type ? that "evenly" partitions it in the following sense. With a possible relabelling of the indices of the variables, and the number of variables in G(y; x are all bounded above by d ne. A minor technicality in the above lemma is that, since the skeleton is not necessarily a binary tree, it may be necessary to "split" a non-binary gate into two smaller gates. It is straightforward to obtain the above decomposition of a skeleton in NC 1 . Once this decomposition is obtained, the recursive algorithm for computing the unary gates of the ROF follows from the following lemma. Lemma 10: Let x (0) n be a total non-collapsing justifying assignment for the AROF F (i) Given the skeleton of F and the subformulas G(y; x possible to determine A, B and C, and, thus, the entire structure of F steps with O(n log n) processors (ii) Given the skeleton of F the problem of determining G(y; x reducible to the problem of determining a ROF given its skeleton. Proof: For part (i), assume that the subformulas G(y; x and I(x are given. Since x (0) n is a justifying assignment, G(y; x (0) l ) are all nonconstant unary functions, so there exist nonsingular matrices A 0 , (which are easy to determine in O(log n) parallel steps) such that l Also, so the matrices A 0 can be determined in O(1) steps, [BHH92]. From this, the matrices A, B, C can be determined. For part (ii), consider the problem of determining G(y; x for some nonsingular A 00 . Therefore, if we fix x l to x (0) l then we have a reduction from the problem of determining G(fA 00 (y); x Similarly, we have reductions from the problem of determining f B 00 (H(x and f C 00 (I(x Since the matrices can be absorbed into the processing of part (i) this is sufficient.2 By recursively applying Lemmas 9 and 10, we obtain a parallel algorithm to determine an AROF given its skeleton and a total noncollapsing three-way justifying assignment in O(log 2 n) steps. The processor count for this can be bounded by O(n log n). --R Machine Learning Learning Read-Once Formulas with Queries When Won't Membership Queries Help? The parallel evaluation of general arithmetic expressions. Learning arithmetic read-once formulas On the exact learning of formulas in par- allel Learning boolean read-once formulas with arbitrary symmetric and constant fan-in gates Asking Questions to Minimize Errors. An Algorithm to Learn Read-Once Threshold Formulas A deterministic algorithm for sparse multivariate polynomial interpolation. On the Decidability of Sparse Univariate Polynomial Interpolation. Exact Identification of Read-Once Formulas Using Fixed Points of Amplification Functions Fast parallel algorithms for sparse multivariate polynomial interpolation over finite fields. Interpolation of sparse rational functions without knowing bounds on the exponent. Learning read-once formulas over fields and extended bases Testing polynomials that are easy to com- pute Parallel prefix computation. Learning Quickly When Irrelevant Attributes Abound: A New Linear Threshold Algorithm Randomized approximation and interpolation of sparse polynomials. On the complexity of learning from counterexamples and membership queries. Interpolation and approximation of sparse multivariate polynomials over GF(2). Learning sparse multivariate polynomials over a field with queries and counterexamples. Fast polynomial algorithms for verification of polynomial identities. A theory of the learnable. Learning in parallel --TR --CTR Amir Shpilka, Interpolation of depth-3 arithmetic circuits with two multiplication gates, Proceedings of the thirty-ninth annual ACM symposium on Theory of computing, June 11-13, 2007, San Diego, California, USA

read-once formula;learning theory;parallel algorithm

279128

Localizing a Robot with Minimum Travel.

We consider the problem of localizing a robot in a known environment modeled by a simple polygon P. We assume that the robot has a map of P but is placed at an unknown location inside P. From its initial location, the robot sees a set of points called the visibility polygon V of its location. In general, sensing at a single point will not suffice to uniquely localize the robot, since the set H of points in P with visibility polygon V may have more than one element. Hence, the robot must move around and use range sensing and a compass to determine its position (i.e., localize itself). We seek a strategy that minimizes the distance the robot travels to determine its exact location.We show that the problem of localizing a robot with minimum travel is NP-hard. We then give a polynomial time approximation scheme that causes the robot to travel a distance of at most (k - 1)d, where which is no greater than the number of reflex vertices of P, and d is the length of a minimum length tour that would allow the robot to verify its true initial location by sensing. We also show that this bound is the best possible.

Introduction Numerous tasks for a mobile robot require it to have a map of its environment and knowledge of where it is located in the map. Determining the position of the robot in the environment is known as the robot localization problem. To date, mobile robot research that supposes the use of a map generally assumes either that the position of the robot is always known, or that it can be estimated using sensor data acquired by displacing the robot only small amounts [BR93, KMK93, TA92]. However, self-similarities between separate portions of the environment prevent a robot that has been dropped into or activated at some unknown place from uniquely determining its exact location without moving around. This motivates a search for strategies that direct the robot to travel around its environment and to collect additional sensory data [BD90, KB87, DJMW93] to deduce its exact position. In this paper, we view the general robot localization problem as consisting of two phases: hypothesis generation and hypothesis elimination. The first phase is to determine the set H of hypothetical locations that are consistent with the sensing data obtained by the robot at its initial location. The second phase is to determine, in the case that H contains two or more locations (see Figure 1), which location is the true initial position of the robot; i.e. to eliminate the incorrect hypotheses. Ideally, the robot should travel the minimum distance necessary to determine its exact location. This is because the time the robot takes to localize itself is proportional to the distance it must travel (assuming sensing and computation time are negligible in comparison). Also, the most common devices for measuring distance, and hence position, on actual mobile robots are relative measurement tools such as odometers. Therefore, they yield imperfect estimates of orientation, distance and velocity, and the errors in these estimates accumulate disastrously with successive motions [Dav86]. Our strategy is well-suited to handling the accumulation of error problem via simple recalibration, as we will point out later. A solution to the hypothesis generation phase of robot localization has been given by Guibas, Motwani and Raghavan in [GMR92]. We describe this further in the next section, after making more precise the definitions of the two phases of robot localization. Our paper is concerned with minimizing the distance traveled in the hypothesis elimination phase of robot localization. It begins where [GMR92] left off. Together, the two papers give a solution to the general robot localization problem. In this paper, we define a natural algorithmic variant of the problem of localizing a robot with minimum travel and show this variant is NP-hard. We then solve the hypothesis elimination phase with what we call a greedy localization strategy. To measure the performance of our strategy, we employ the framework of competitive analysis for on-line algorithms introduced by Sleator and Tarjan [ST85]. That is, we examine the ratio of the distance traveled by a robot using our strategy to the length d of a minimum length tour that allows the robot to verify its true initial position. The worst-case value of this ratio over all maps and all starting points is called the competitive ratio of the strategy. If this ratio is no more than k, then the strategy is called k-competitive. Since our strategy causes the robot to travel a distance no more than (k 0 1)d, where no greater than the number of reflex vertices of P ), our strategy is (k 0 1)-competitive. We also show that no on-line localization strategy has a competitive ratio better than k 0 1, and thus our strategy is optimal. The rest of this paper is organized as follows. In Section 2 we give a formal definition of the robot localization problem, we define some of the terms used in the paper, and we comment on previous work. In Section 3 we prove that given a solution set H to the hypothesis generation phase of the localization problem that contains more than one hypothetical location, the hypothesis elimination phase, which localizes the robot by using minimum travel distance, is NP-hard. In Section 4 we define the geometric structures that we use to set-up our greedy localization strategy. In Section 5 we give our greedy localization strategy and prove the previously mentioned performance guarantee of k01 times optimum. We also give an example of a map polygon for which no on-line localization strategy is better than (k01)-competitive. Section 6 summarizes and comments on open problems. Localization Through Traveling and Probing In this section, we describe our robot abstraction and give some key definitions. The most common application domain for mobile robots is indoor "structured" environments. In such environments it is often possible to construct a map of the environment, and it is acceptable to use a polygonal approximation P of the free space [Lat91] as a map. A common sensing method used by mobile robots is range sensing (for example, sonar sensing or laser range sensing). 2.1 Assumptions about the robot We assume the following throughout this paper. ffl The robot moves in a static 2-dimensional obstacle-free environment for which it has a map. The robot has the ability to make error-free motions between arbitrary locations in the environment 1 . We model the movement of the robot in the environment by a point p moving inside and along the boundary of an n-vertex simple polygon P positioned somewhere in the plane. ffl The robot has a compass and a range sensing device. It is essential that the robot be able to determine its orientation (with the compass); otherwise it can never uniquely determine its exact location in an environment with non-trivial symmetry (such as a square). ffl The robot's sensor can detect the distances to those points on walls for which the robot has an unobstructed straight line of sight, and the robot's observations at a particular location determine a polygon V of points that it can see (see the next subsection for a definition of This is analogous to what can be extracted by various real sensors such as laser range finders. The robot also knows its location in V . In practice, position estimation errors accrue in the execution of such motions; however the strategy we present here is exceptionally well suited to various methods for limiting these errors using sensor feedback (see Section 5.1). 2.2 Some definitions and an example Two points in P are visible to each other or see each other if the straight line segment joining them does not intersect the exterior of P . The visibility polygon V (p) for a point is the polygon consisting of all points in P that are visible from p. The data received from a range sensing device is modeled as a visibility polygon. The visibility polygon of the initial location of the robot is denoted by V , and the number of its vertices is denoted by m. Since the robot has a compass, we assume that P and V have a common reference direction. We break the general problem of localizing a robot into two phases as follows. The Robot Localization Problem Hypothesis Generation: Given P and V , determine the set H of all points p i 2 P such that the visibility polygon of p i is congruent under translation to V (denoted by V (p Hypothesis Elimination: Devise a strategy by which the robot can correctly eliminate all but one hypothesis from H , thereby determining its exact initial location. Ideally, the robot should travel a distance as small as possible to achieve this. As previously mentioned, the hypothesis generation phase has been solved by Guibas, Motwani and Raghavan. We describe their results in the next subsection. This paper is concerned with the hypothesis elimination phase. Consider the example illustrated in Figure 1. The robot knows the map polygon P and the visibility polygon V representing what it can "see" in the environment from its present location. Suppose also that it knows that P and V should be oriented as shown. The black dot represents the robot's position in the visibility polygon. By examining P and V , the robot can determine that it is at either point p 1 or point p 2 in P , i.e. g. It cannot distinguish between these two locations because V (p 1 However, by traveling out into the "hallway" and taking another probe, the robot can determine its location precisely. Figure 1: Given a map polygon P (left) and a visibility polygon V (center), the robot must determine which of the 2 possible initial locations p 1 and p 2 (right) is its actual location in P . An optimal strategy for the hypothesis elimination phase would direct the robot to follow an optimal verification tour, defined as follows. verification tour is a tour along which a robot that knows its initial position a priori can travel to verify this information by probing and then return to its starting position. An optimal verification tour is a verification tour of minimum length d. Since we do not assume a priori knowledge of which hypothetical location in H is correct, an optimal verification tour for the hypothesis elimination phase cannot be pre-computed. Even if we did have this knowledge, computing an optimal verification tour would be NP-hard. This can be proven using a construction similar to that in Section 3 and a reduction to hitting set [GJ79]. For these reasons, we seek an interactive probing strategy to localize the robot. In each step of such a strategy, the robot uses its range sensors to compute the visibility polygon of its present position, and from this information decides where to move next to make another probe. To be precise, the type of strategy we seek can be represented by a localizing decision tree, defined as follows. localizing decision tree is a tree consisting of two kinds of nodes and two kinds of weighted edges. The nodes are either sensing nodes (S-nodes) or reducing nodes (R-nodes), and the node types alternate along any path from the root to a leaf. Thus tree edges directed down the tree either join an S-node to an R-node (SR-edges), or join an R-node to an S-node (RS-edges). ffl Each S-node is associated with a position defined relative to the initial position of the robot. The robot may be instructed to probe the environment from this position. ffl Each R-node is associated with a set H 0 ' H of hypothetical initial locations that have not yet been ruled out. The root is an R-node associated with H , and each leaf is an R-node associated with a singleton hypothesis set. ffl Each SR-edge represents the computation that the robot does to rule out hypotheses in light of the information gathered at the S-node end of the edge. An SR-edge does not represent physical travel by the robot and hence has weight 0. ffl Each RS-edge has an associated path defined relative to the initial location of the robot. This is the path along which the robot is directed to travel to reach its next sensing point. The weight of an RS-edge is the length of its associated path. Since we want to minimize the distance traveled by the robot, we define the weighted height of a localizing decision tree as follows. Definition. The weight of a root-to-leaf path in a localizing decision tree is the sum of the weights on the edges in the path. The weighted height of a localizing decision tree is the weight of a maximum-weight root-to-leaf path. An optimal localizing decision tree is a localizing decision tree of minimum weighted height. In the next section, we show that the problem of finding an optimal localizing decision tree is NP-hard. We call a localization strategy that can be associated with a localizing decision tree a localizing decision tree strategy. As an example of such a strategy, consider the map polygon P shown on the left in Figure 2. Imagine that from the visibility polygon sensed by the robot at its initial location it is determined R R R Go west d 1 Go south d 2 Go south d 3 Figure 2: A map polygon and 4 hypothetical locations fp (left) with a localizing decision tree for determining the true initial position of the robot (right). that the set of hypothetical locations is g. Hence the root of the localizing decision tree (shown on the right in Figure 2) is associated with H . In the figure, the SR-edges are labeled with the visibility polygons seen by the robot at the S-node endpoints of these edges. Assuming that north points straight up, the strategy given by the tree directs the robot first to travel west a distance d 1 , which is the distance between p i and p 0 and then to take another probe at its new location. Depending on the outcome of the probe, the robot knows it is located either at one of fp 0 2 g or at one of fp 0 g. If it is located at p 0 1 or p 0 , then the strategy directs it to travel south a distance d 2 , which is the distance between p 0 2, to a position just past the dotted line segment shown in P . By taking a probe from below this line segment, it will be able to see the vertex at the end of the segment if it is at location p 00 1 , and it will not see this vertex if it is at location p 00 2 . Thus after this probe it will be able to determine its unique location in P . Similarly, if the robot is located at p 0 3 or p 0 4 , then the strategy directs it to travel south a distance d 3 and take another probe to determine its initial position. The farthest that the robot must travel to determine its location is so the weighted height of this decision tree is d 1 2.3 Previous work Previous work on robot localization by Guibas, Motwani, and Raghavan [GMR92] showed how to preprocess a map polygon P so that given the visibility polygon V that a robot sees, the set of points in P whose visibility polygon is congruent to V , and oriented the same way, can be returned quickly. Their algorithm preprocesses P in O(n 5 log n) time and O(n 5 ) space, and it answers queries in O(m is the number of vertices of P , m is the number of vertices of V , and k is the size of the output (the number of places in P at which the visibility polygon is V ). They also showed how to answer a single localization query in O(mn) time with no preprocessing. Kleinberg [Kle94b] has independently given an interactive strategy for localizing a robot in a known environment. As in our work, he seeks to minimize the ratio of the distance traveled by a robot using his strategy to the length of an optimal verification path (i.e. the competitive ratio). Kleinberg's model differs from ours in several ways. First of all, he models the robot's environment as a geometric tree rather than a simple polygon. A geometric tree is a pair (V; E), where V is a finite point set in R d and E is a set of line segments whose endpoints all lie in V . The edges do not intersect except at points of V and do not form cycles. Kleinberg only considers geometric trees with bounded degree 1. Also, his robot can make no use of vision other than to know the orientation of all edges incident to its current location. Using this model, Kleinberg gives an O(n 2=3 )-competitive algorithm for localizing a robot in a geometric tree with bounded degree 1, where n is the number of branch vertices (vertices of degree greater than two) of the tree. The competitive ratio of Kleinberg's algorithm appears to be better than the lower bound illustrated by Figure 10 in Section 5.3. However, if this map polygon were modeled as a geometric tree it would have degree n, where n is the number of branch vertices, rather than a constant degree, and the distance traveled by a robot using Kleinberg's algorithm can be linear in the degree of the tree. If Kleinberg's algorithm ran on this example, it would only execute Step 1, which performs a spiral search, and it would cause the robot to travel a distance almost 4n times the length of an optimal verification path. Our algorithm causes the robot to travel a distance less than 2n times the length of an optimal verification path on this example. Our algorithm is similar to Step 3 of Kleinberg's algorithm, and he gives a lower bound example (Figure 3 of [Kle94b]) illustrating that an algorithm using only Steps 1 and 3 of his algorithm is no better than O(n)-competitive. Although this example does not directly apply to our model since the robot in our model has the ability to see to the end of the hallway, by adding small jogs in the hallway a similar example can be constructed where our strategy is no better than O(n)-competitive. In this example, the number of branch vertices of the geometric tree represented by P would be n and the number of vertices of P would be O(n). However, in this example jH does not contradict our results. Other theoretical work on localizing a robot in a known environment has also been done. Betke and Gurvits [BG94] gave an algorithm that uses the angles subtended by landmarks in the robot's environment to localize a robot. Their algorithm runs in time linear in the number of landmarks, and it assumes that a correspondence is given between each landmark seen by the robot and a point in the map of the environment. Avis and Imai [AI90] also investigated the problem of localizing a robot using angle measurements, but they did not assume any correspondence between the landmarks seen by the robot and points in the environment. Instead they assumed that the environment contains n identical markers, and the robot takes k angle measurements between an unknown subset of these markers. They gave polynomial time algorithms to determine all valid placements of the robot, both in the case where the robot has a compass and where it does not. In addition they showed that with polynomial-time preprocessing location queries can be answered in O(log n) time. Theoretical work with a similar flavor to ours has also been done on navigating a robot through an unknown environment. In this work a point robot must navigate from a point s to a target t, which is either a point or an infinite wall, where the Euclidean distance from s to t is n. There are obstacles in the scene, which are not known a priori, but which the robot learns about only as it encounters them. The goal is to optimize (i.e. minimize) the ratio of the distance traveled by the robot to the length of a shortest obstacle-free path from s to t. As with localization strategies, the worst-case ratio over all environments where s and t are distance n apart is called the competitive ratio of the strategy. Papadimitriou and Yannakakis [PY91] gave a deterministic strategy for navigating between two points, where all obstacles are unit squares, that achieves a competitive ratio of 1.5, which they show is optimal. For squares of arbitrary size they gave a strategy achieving a ratio of 26=3. They also showed, along with Eades, Lin and Wormald [ELW93], that when t is an infinite wall and the obstacles are oriented rectangles, there is a lower bound of \Omega# p n) on the ratio achievable by any deterministic strategy. Blum, Raghavan and Schieber [BRS91] gave a deterministic strategy that matched n) lower bound for navigating between two points with oriented, rectangular obstacles. Their strategy combines strategies for navigating from a point to an infinite wall and from a point on the wall of a room to the center of the room, with competitive ratios of O( n) and O(2 3 log n ) respectively. The competitive ratio for the problem of navigating from a corner to the center of a room was improved to O(ln n) by a strategy of Bar-Eli, Berman, Fiat and Yan [BEBFY92], who also showed that this ratio is a lower bound for any deterministic strategy. Berman et al. [BBF gave a randomized algorithm for the problem of navigating between two points with oriented, rectangular obstacles with a competitive ratio of O(n 4=9 log n). Several people have studied the problem of navigating from a vertex s to a vertex t inside an unknown simple polygon. They assume that at every point on its path the robot can get the visibility polygon of that point. Klein [Kle92] proved a lower bound of 2 on the competitive ratio and gave a strategy achieving a ratio of 5:72 for the class of street polygons. A street is a simple polygon such that the clockwise chain L and the counterclockwise chain R from s to t are mutually weakly visible. That is, every point on L is visible to some point on R and visa versa. Kleinberg [Kle94a] gave a strategy that improved Klein's ratio to 2 2, and Datta and Icking [DI94] gave a strategy with a ratio of 9.06 for a more general class of polygons called generalized streets, where every point on the boundary is visible from a point on a horizontal line segment joining L and R. They also showed a lower bound of 9 for this class of polygons. Previous work in the area of geometric probing has examined the complexity of constructing minimum height decision trees to uniquely identify one of a library of polygons in the plane using point probes. Such probes examine a single point in the plane to determine if an object is located at that point. If each polygon in the library is given a fixed position, orientation and scale, then it has been shown that both the problem of finding a minimum cardinality probe set (for a noninteractive probing strategy) [BS93] and the problem of constructing a minimum height decision tree for probing (for an interactive strategy) [AMM + 93] are NP-Complete. Arkin et al.[AMM a greedy strategy that builds a decision tree of height at most dlog ke times that of an optimal decision tree, where k is the number of polygons in the library. The minimum height decision tree used for probing in [AMM + 93] is different than our localizing decision tree. It is a binary decision tree whose internal nodes represent point probes whose outcome is either positive or negative and whose edges are unweighted. The height of such a decision tree is the number of levels of the tree, and it represents the maximum number of probes necessary to identify any polygon in the library. 3 Hardness of Localization In this section we show that the problem of constructing an optimal localizing decision tree, as defined in the previous section, is NP-hard. To do this, we first formulate the problem as a decision problem. Robot-Localizing Decision Tree (RLDT) INSTANCE: A simple polygon P and a star-shaped polygon V , both with a common reference direction, the set H of all locations positive integer h. QUESTION: Does there exist a localizing decision tree of weighted height less than or equal to h that localizes a robot with initial visibility polygon V in the map polygon P? We show that this problem is NP-hard by giving a reduction from the Abstract Decision Tree problem, proven NP-complete by Hyafil and Rivest in [HR76]. The Abstract Decision Tree problem is stated as follows: Abstract Decision Tree (ADT) INSTANCE: A set of objects, a set of subsets of X representing binary tests, where test T j is positive on object x i if x and is negative otherwise, and a positive QUESTION: Does there exist an abstract decision tree of height less than or equal to h 0 , where the height of a tree is the maximum number of edges on a path from the root to a leaf, that can be constructed to identify the objects in X? An abstract decision tree has a binary test at all internal nodes and an object at every leaf. To identify an unknown object, the test at the root is performed on the object, and if it is positive the right branch is taken, otherwise the left branch is taken. This procedure is repeated until a leaf is reached, which identifies the unknown object. Theorem 1 RLDT is NP-hard. Proof: Given an instance of ADT, we create an instance of RLDT as follows. We construct P to be a staircase polygon, with a stairstep for each object x Figure 3). For each stairstep we construct protrusions, one for each test in T (see Figure 4). If test T j is a positive test for object x i , then protrusion T j on stairstep x i has an extra hook on its end (such as T 3 , T 4 , and T n in Figure 4). The length of a protrusion is denoted by l and the distance between protrusions denoted by d, where d and l are chosen so that dh 0 ! l. The vertical piece between adjacent stairsteps is longer than (2l +d)h 0 , and the width w of each stairstep is much smaller than the other measurements. The polygon P has O(nk) vertices, where Consider a robot that is initially located at the shaded circle shown in Figure 4 on one of the k stairsteps. The visibility polygon V at this point has O(n) vertices and is the same at an analogous point on any internal stairstep x i . We output the polygons P and V , which can be constructed in polynomial time, the k locations weighted height as an instance of RLDT. In order for the robot to localize itself, it must either travel to one of the "ends" of P (either the top or the bottom stairstep) to discover on which stairstep it was located initially, or it must examine Figure 3: Construction showing localization is NP-hard. a sufficient number of the n protrusions on the stairstep where it is located to distinguish that stairstep from all the others. Since the vertical piece of each stairstep is longer than only a strategy that directs the robot to remain on the same stairstep can lead to a decision tree of weighted height less than or equal to h. Any decision tree that localizes the robot by examining protrusions on the stairstep corresponds to an equivalent abstract decision tree to identify the objects of X using tests in T , and visa versa. Each time the robot travels to the end of protrusion T j to see if it has an extra hook on its end, it corresponds to performing binary test T j on an unknown object to observe the outcome. The robot must travel 2l to perform this test, and it travels at most d in between tests. Therefore, if the robot can always localize itself by examining no more than h 0 protrusions, then it has a decision tree of weighted height no more than which corresponds to an abstract decision tree l d Figure 4: Close-up of a stairstep x i in NP-hard construction. Not to scale: l ?? d ?? w. of height h 0 for the ADT problem. Since dh 0 ! l, in a localizing decision tree of weighted height - h the robot cannot examine more than h 0 protrusions on any root-to-leaf path. ut 4 Using a Visibility Cell Decomposition for Localization In this section we discuss the geometric issues involved in building a data structure for our greedy localization strategy. 4.1 Visibility cells and the overlay arrangement When we consider positions where the robot can move to localize itself, we reduce the infinite number of locations in P to a finite number by first creating a visibility cell decomposition of P [Bos91, BLM92, GMR92]. A visibility cell (or visibility region) C of P is a maximally connected subset of P with the property that any two points in C see the same subset of vertices of P ([Bos91, BLM92]). A visibility cell decomposition of P is simply a subdivision of P into visibility cells. This decomposition can be computed in O(n 3 log n) using techniques in [Bos91, BLM92]. It is created by introducing O(nr) line segments, called visibility edges, into the interior of P , where r is the number of reflex vertices 2 of P . Each line segment starts at a reflex vertex u, ends at the boundary of P , and is collinear with a vertex v that is either visible from u or is adjacent to it. The number of cells in this decomposition, as well as their total complexity, is O(n 2 r) (see [GMR92] for a proof). Although two points p and q in the same visibility cell C see the same subset of vertices of P , they may not have the same visibility polygon (i.e. it may be that V (p) 6= V (q)). This is because some edges of V (p) may not actually lie on the boundary of P (these edges are collinear with p and are produced by visibility lines), so these edges may be different in V (q). Therefore, in order to represent the portion of P visible to a point p in a visibility cell C in such a way that all points in C are equivalent, we need a different structure than the visibility polygon. The structure that we use is the visibility skeleton of p. Definition. The visibility skeleton V 3 (p) of a location is the skeleton of the visibility polygon V (p). That is, it is the polygon induced by the non-spurious vertices of V (p), where a spurious vertex of V (p) is one that lies on an edge of V (p) that is collinear with p, and the other endpoint of this edge is closer to p. The non-spurious vertices of V (p) are connected to form V 3 (p) in the same cyclical order that they appear in V (p). The edges of the skeleton are labeled to indicate which ones correspond to real edges from P and which ones are artificial edges induced by the spurious vertices. If p is outside P , then V 3 (p) is equal to the special symbol ;. For a complete discussion of visibility skeletons and a proof that V 3 any two points p and q in the same visibility cell, see [Bos91, BLM92, GMR92]. As stated in Section 2, the hypothesis generation phase of the robot localization problem generates a set ae P of hypothetical locations at which the robot might be located reflex vertex of P is a vertex that subtends an angle greater than 180 ffi . initially. The number k of such locations is bounded above by r (see [GMR92] for a proof). From this set H , we can select the first location p 1 (or any arbitrary location) to serve as an origin for a local coordinate system. For each location we define the translation vector that translates location p j to location p 1 , and we define P j to be the translate of P by vector t j . We thus have a set fP of translates of P corresponding to the set H of hypothetical locations. The point in each P j corresponding to the hypothetical location p j is located at the origin. In order to determine the hypothetical location corresponding to the true initial location of the robot, we construct an overlay arrangement A that combines the k translates P j that correspond to the hypothetical locations, together with their visibility cell decompositions. More formally, we define A as follows. Definition. The overlay arrangement A for the map polygon P corresponding to the set of hypothetical locations H is obtained by taking the union of the edges of each translate P j as well as the visibility edges in the visibility cell decomposition of P j . Figure 5 for an example of an overlay arrangement. Since each visibility cell decomposition is created from O(nr) line segments introduced into the interior of P j , a bound on the total number of cells in the overlay arrangement as well as their total complexity is O(k which may be O(n 6 ). Figure 5: A visibility polygon, a map polygon and the corresponding overlay arrangement. 4.2 Lower bound on the size of the overlay arrangement Figure 6 shows a map polygon P whose corresponding overlay arrangement for the visibility polygon shown in Figure 7(a) cells. This polygon has a long horizontal "hallway" with k identical, equally spaced "rooms" on the bottom side of it Figure 6). Each room has width 1 unit, and the distance between rooms is 2k 01 units. If the robot is far enough inside one of these rooms so that it cannot see any of the rooms on the top of the hallway, then its visibility polygon is the same no matter which room it is in. The k 0 1 rooms on the top side of the hallway are identical, have width 1 unit, and are spaced 2k+1 units apart. Each top room is between two bottom rooms. The i th top room from the left has its left edge a distance 2i 0 1 to the right of the right edge of the bottom room to its left, and it has its right edge a distance 2(k 0 i) 0 1 to the left of the left edge of the bottom room to its right (see Figure 6). Figure A map polygon whose overlay arrangement Figure 7: (a) A visibility polygon (b) Visibility cells in a bottom room Consider the visibility edges starting from the reflex vertices of the bottom rooms that are generated by (i.e. collinear with) the reflex vertices of the top rooms. The i th bottom room from the left will have 2(k0i) such visibility edges starting from its right reflex vertex and 2(i01) starting from its left reflex vertex. Due to the spacing of the top rooms, the visibility edges starting from the reflex vertices of one bottom room will be at different angles than those in any other bottom room. See the picture in Figure 7(b) for an illustration of the visibility cells inside a bottom room. When the overlay arrangement A for the visibility polygon shown in Figure 7(a) is constructed, it will consist of k translates, one for each of the bottom rooms of P . Since these rooms are identical and equally spaced, A will have 2k 0 1 rooms on its bottom side. Since the visibility edges inside each bottom room are at different angles, these edges will not coincide when bottom rooms from two different translates overlap in A. This means that A will have \Omega# visibility edges starting from the left reflex vertex, edges starting from the right reflex vertex, resulting in \Omega# k 4 ) cells inside each of these bottom rooms of A. Therefore, A will have \Omega# k 5 ) cells in total. Since the number of vertices of P is 8k, A Closing the gap between the upper and lower bounds on the size of the arrangement is an open problem. 4.3 The reference point set Q Each cell in the overlay arrangement A represents a potential probe position, which can be used to distinguish between different hypothetical locations of the robot. For each cell C of A and for each translate P j that contains C, there is an associated visibility skeleton V 3 (C). If two translates and P j have different skeletons for cell C, or if C is outside of exactly one of P i and P j , then C distinguishes p i from p j . For our localization strategy we choose a set Q of reference points in A that will be used to distinguish between different hypothetical locations. For each cell C in A that lies in at least one translate of P , and for each translate P j that contains C, let q C;j denote the point on the boundary of C that is closest to the origin. Here, the distance d j (q C;j ) from the origin to the closest point in C is measured inside P j . We choose g. In the remainder of this paper we drop the subscripts from q C;j when they are not necessary. Computing the reference points in Q involves computing Euclidean shortest paths in P j from the origin to each cell C. To compute these paths we can use existing algorithms in the literature for shortest paths in simple polygons. We first compute for each hypothetical initial location p j the shortest path tree from the origin to all of the vertices of P j in linear time using the algorithm given in [GHL + 87]. This algorithm also gives a data-structure for storing the shortest path tree so that the length of the shortest path from the origin to any point x 2 P j can be found in time O(log n) and the path from the origin to x can be found in time O(log n+ l), where l is the number of segments along this path. We can use this data-structure later to extract the shortest path to any cell C in A within any translate P j . We use -(p to denote the shortest path from the origin to x in P j . To find the shortest path from the origin to a segment xy contained in P j we use the following theorem. Theorem 2 If P is a simple polygon, then the Euclidean shortest path -(s; xy) from a point s in P to a line segment xy in P is either the shortest path -(s; x) from s to x, the shortest path -(s; y) from s to y, or a polygonal path with l edges such that the first l 0 1 edges are the first l 0 1 edges on either -(s; x) or -(s; y) and the last edge is perpendicular to xy. Proof: The theorem follows from standard geometry results. We sketch the proof here. It is shown in [LP84] that the shortest paths -(s; x) and -(s; y) are polygonal paths whose interior vertices are vertices of P , and if v is the last common point on these two paths, then -(v; x) and -(v; y) are both outward-convex (i.e. the convex hull of each of these subpaths lies outside the region bounded by -(v; x), -(v; y) and the segment xy). As in [LP84] we call the union -(v; x) [ -(v; y) the funnel associated with xy, and we call v the cusp of the funnel. See Figure 8 for an example of a simple polygon with edges of this funnel shown as dashed line segments. The shortest path -(s; xy) has -(s; v) as its initial subpath. To complete the shortest path -(s; xy) we must find a shortest path -(v; xy). If v has a perpendicular line of sight to xy, then this visibility line will be -(v; xy). If v does not have a perpendicular line of sight to xy, then consider the edge e adjacent to v on the funnel that is the closest to perpendicular. Without loss of generality, assume e is the first edge on -(v; y). The path -(v; xy) will follow -(v; y) until it reaches y or it reaches a vertex that has a perpendicular line of sight to xy. ut x y s Figure 8: A simple polygon with shortest paths from s to x, y and xy shown. Using this theorem we can in O(n) time determine the length of the shortest path in P j from the origin o to xy and the closest point on xy to o. We first use the data-structure in [GHL + 87] to determine in O(log n) time the length d x and the last edge e x on the shortest path -(o; x) and the length d y and the last edge e y on the shortest path -(o; y). For each of these edges we check its angle with respect to xy. Note that both of these angles cannot be 90 ffi or greater, or else it would be impossible to form a funnel with -(o; x) and -(o; y). If the angle between e x (e y ) and xy is at least 90 ffi , then we return d x (d y ) as the shortest distance to xy and x (y) as the closest point on xy. If both the angles formed by e x and e y with xy are less than 90 ffi , then the last edge on the shortest path -(o; xy) will be a perpendicular drawn from one of the vertices on the funnel associated with xy. To find this edge we again use the data-structure in [GHL + 87] to examine the edges of the funnel in order, starting with e x . For each edge we calculate the angle formed by its extension with xy. That is, for each edge (u; w) whose extension intersects xy at point z, we calculate the angle 6 uzy. As we move around the funnel these angles increase. When the angle becomes greater than 90 ffi , we have found the vertex from which to drop a perpendicular to xy. It takes O(n) time to find this vertex, and an additional O(log n) time to calculate the distance to xy and the closest point (this is the time it takes to determine the length of the shortest path from o to this vertex). To compute the reference point q C;j , we compute the shortest path distance in P j from the origin to each edge of C. We then choose the smallest distance as the distance to the cell C. For each cell C we will have up to k reference points fq and their corresponding distances )g. We define d points q not within P j . Partition of H For each cell C we compute a partition of H that represents which hypothetical locations can be distinguished from one another by probing from inside C. If two translates P i and P j have the same visibility skeleton for cell C, or if C is outside of both P i and P j , then p i and p j are in the same subset of the partition of H corresponding to cell C. Since the visibility polygon and the visibility skeleton for a point can be computed in time (see [GA81]) and we can compare two visibility skeletons with m vertices in O(m) time to see if they are identical, we can compute the partition of H for C in O(kn m) time, where m is the maximum number of vertices on any of the k visibility skeletons. Although there may be O(n 6 ) cells in the overlay arrangement A, yielding up to O(kn 6 ) reference points, we show in Section 5.4 that only O(k 2 ) reference points are needed for our localization strategy, so we do not need to compute a partition of H for all O(n 6 ) cells. 5 A Greedy Strategy for Localization In this section we present a localizing decision tree strategy, called Minimum Distance Localization Strategy or Strategy MDL for short, for completing the solution of the hypothesis elimination phase of the robot localization problem. Our strategy, which has a greedy flavor, uses the set Q of reference points described in the previous section for choosing probing locations. Strategy MDL has a competitive ratio of k 0 1, where In devising a localizing decision tree strategy, there are two main criteria to consider when deciding where the robot should make the next probe: (1) the distance to the new probe position, and (2) the information to be gained at the new probe position. It is easy to see that a strategy that only considers the second criterion can do arbitrarily worse than an optimal localizing decision tree strategy. Strategy MDL considers (2) only to the extent that it never directs the robot to make a useless probe. Nevertheless, its performance is the best possible. Although it would seem beneficial to weight each possible probe location with the amount of information that could be gained in the worst case by probing at that location, this change will not improve the worst-case behavior of Strategy MDL, as the lower bound example given in Section 5.3 illustrates. Even a strategy that considers both the distance and the information criteria when choosing the next probe position can do poorly. For example, if the robot employs an incremental strategy that at each step tells it to travel to the closest probe location that yields some information, then a map polygon can be constructed such that in the worst case the robot will travel distance 2 k d. Using Strategy MDL for hypothesis elimination, a strategy for the complete robot localization problem can be obtained as follows. Preprocess the map polygon P using a method similar to that in [GMR92]. This preprocessing yields a data structure that stores for each equivalence class of visibility polygons either the location in P yielding that visibility polygon, if there is only one location, or a localizing decision tree that tells the robot how to travel to determine its true initial location. 5.1 Strategy MDL In this subsection we present the details of Strategy MDL. Using the results of Section 4, it is possible to pre-compute Strategy MDL's entire decision tree. However, for ease of exposition we will only describe how the strategy directs the robot to behave on a root-to-leaf path in the tree. In practice, it may also sometimes be preferable not to pre-compute the entire tree, but rather to compute the robot's next move on an interactive basis, as the robot carries out the strategy. Strategy MDL uses the map polygon P , the set H generated in the hypothesis generation phase, and the set Q of reference points defined in Section 4.3. Also, for each point q C;j 2 Q the strategy uses the distance d j (q C;j ) of q C;j from the origin, a path path j (q C;j ) within P j of length d j (q C;j ), and the partition of H associated with cell C, as defined in Section 4.3. Next we describe how Strategy MDL directs the robot to behave. Initially, the set of hypothetical locations used by Strategy MDL is the given set H . As the robot carries out the strategy, hypothetical locations are eliminated from H . Thus in our description of Strategy MDL, we abuse notation and use H to denote the shrinking set of active hypothetical locations; i.e. those that have not yet been ruled out. Similarly, we use Q to denote the shrinking set of active reference points; i.e. those that non-trivially partition the set of active hypothetical locations. We call a path path j (q) active if p j 2 H and q 2 Q are both active. We let d 3 (q3) denote the minimum of f d j (q) j q 2 Q and are active g and let path 3 (q3) denote an active path of length d 3 (q3). From the initial H and Q, an initial path 3 (q3) can be selected. The strategy directs the robot to travel along this path and to make a probe at its endpoint. The robot then uses the information gained at the probe position to update H and Q. The strategy then directs the robot to retrace its path back to the origin and repeat the process until the size of H shrinks to 1. Note that Strategy MDL is well-suited to handling the problem of accumulation of errors caused by successive motions in the estimates of orientation, distance and velocity made by the robot's sensors. This is because the robot always returns to the origin after making a probe, so it can recalibrate its sensors. 5.2 A performance guarantee for Strategy MDL The following theorems show that Strategy MDL is correct and has a competitive ratio of k 0 1. First we show that Strategy MDL never directs the robot to pass through a wall. Then we show that Strategy MDL eliminates all hypothetical locations except the valid one while directing the robot along a path no longer than k 0 1 times the length of an optimal verification tour. A corollary of Theorem 4 is that the localizing decision tree associated with Strategy MDL has a weighted height that is at most 2(k 0 1) times the weighted height of an optimal localizing decision tree. Theorem 3 Strategy MDL never directs the robot to pass through a wall. Proof: The proof is by contradiction. Suppose that p j is the true initial location of the robot and x j is the point on the boundary of P j where the robot would first hit a wall. Furthermore, suppose that when the robot attempts to pass through the wall at x j , the path it has been directed to follow is path i (q). Let C denote the cell of arrangement A that contains the portion of path i (q) just before x j . Since cell C is contained in P j , it contributes a reference point q C;j to the set Q of reference points. In order to arrive at a contradiction, it suffices to show that q C;j is active at the time Strategy MDL chooses path i (q) for the robot to follow. This is because d j (q C;j by definition of since the portion of path i (q) from the origin to x j is contained within P j , and is an intermediate point on path i (q). Thus d j (q C;j MDL would choose path j (q C;j ) rather than path i (q) if q C;j is active. Point q C;j is active when path i (q) is selected because cell C distinguishes between the two active hypothetical locations p i and p j . This is because the skeleton V 3 associated with C relative to P j has a real edge through the point x j , whereas the skeleton V 3 associated with C relative to does not have a real edge through x j . ut Theorem 4 Strategy MDL localizes the robot by directing it along a path whose length is at most and d is the length of an optimal verification tour for the robot's initial position. Proof: Let p t denote the true initial location of the robot. First we show by contradiction that Strategy MDL eliminates all hypothetical initial locations in H except p t . Suppose Q becomes empty before the size of H shrinks to one, and let p i be an active hypothetical location different from p t at the time Q becomes empty. Translates P i and P t are not identical, so there is some point x t on the boundary of P t that does not belong to the boundary of P i . Let C be the cell of arrangement A contained in P t and containing x t . C distinguishes between p i and p t , so q C;t is still in the active set Q - a contradiction. Next we establish an upper bound on the length of the path determined by Strategy MDL. Because the strategy always directs the robot to a probing site that eliminates one or more elements from H , the robot makes at most k 0 1 trips from its initial location to a sensing point and back. To show that each round trip has length at most d, we consider how a robot traveling along an optimal verification tour L would rule out an arbitrary incorrect hypothetical location p i . Then we consider how Strategy MDL would rule out p i . Consider a robot traveling along tour L that eliminates each invalid hypothetical location at the first point x on L where the visibility skeleton of x relative to the invalid hypothetical location differs from the visibility skeleton of x relative to P t . Let x be the first point on L where the robot can eliminate p i . The point x must lie on the boundary of some cell C in the arrangement A that distinguishes p i from p t . Cell C generates a reference point q C;t 2 Q, and d t (q C;t ) - d t (x). Since p t is the true initial location of the robot, the distance d t (x) is no more than the distance along L of x from the origin, as well as the distance along L from x back to the origin. Thus d t (q C;t ) is no more than half the length of L. At the moment Strategy MDL directs the robot to move from the origin to the probing site where it eliminates p i , both p i and p t are active, so point q C;t is active since it distinguishes between them. At this time Strategy MDL directs the robot to travel along path 3 (q3). By definition, the length d 3 (q3) of this path is the minimum over all d j (q) for active In particular, since point q C;t is still active, d 3 (q3) - d t (q C;t ), which is no more than half the length of L. Therefore, Strategy MDL directs the robot to travel along a loop from the origin to some probing position where the robot eliminates p i and back, and the length of this loop is at most d. ut Using the definition of competitive ratio given in Section 1, Theorem 4 can be stated as "Strategy j". Note that if a verifying path is not required to return to its starting point, the bound for Theorem 4 becomes 2(k 0 1)d. Note also that even if the robot were continuously sensing rather than just taking a probe at the end of each path path 3 (q3), a better bound could not be achieved. This is because the robot always goes to the closest point that yields useful information, so no point on path 3 (q3) before q3 will allow it to eliminate any hypothetical locations. Corollary 5 The weighted height of the localizing decision tree constructed by Strategy MDL is at most times the weighted height of an optimal localizing decision tree for the same problem. Proof: Consider the decision tree of Strategy MDL. Let p h denote the initial location associated with the leaf that defines the weighted height of the tree. The weighted height of the tree is thus the distance Strategy MDL will direct the robot to travel to determine that p h is the correct initial location, and by Theorem 4 this distance is at most k 0 1 times the minimum verification tour length for p h . But the minimum verification tour length for p h is at most twice the weight of a path from the root to p h in an optimal localizing decision tree, which is at most the weighted height of the tree. The result follows from these inequalities. ut If the robot is required to return to its initial position, the bound on the weighted height of the localizing decision tree constructed by Strategy MDL drops to k 0 1. It should be clear from the discussions in Sections 4 and 5 that Strategy MDL can be computed and executed in polynomial time. In this paper, we do not comment further on computation time as there are many ways to implement Strategy MDL. Also, if travel times are large compared to computation times, the importance of our results is that they obtain good path lengths. 5.3 Lower bounds In Corollary 5 we proved that the weighted height of the localizing decision tree built by Strategy MDL is no greater than 2(k 0 1)d, where and d is the weighted height of an optimal localizing decision tree. This bound is also a lower bound for Strategy MDL, as illustrated in Figure 9. Consider a map polygon that is a staircase polygon with stairs, such as the one in Figure 3, where each stairstep except the first and last one is similar to the one shown in Figure 9. Each such stairstep has k protrusions placed in a circle, with the end of each protrusion a distance d from the center of the circle. In each stairstep a different protrusion has its end extended, which uniquely identifies the stairstep. Each stairstep also has a longer protrusion, with k smaller protrusions sticking out of it. One of these smaller protrusions is extended to uniquely identify the stairstep. The first small protrusion is a distance d + ffl from the center of the circle, and the last one is a distance d from the center of the circle. For this map polygon, if the robot is initially placed in the center of the circle on one stairstep, Strategy MDL will direct it to travel up the k protrusions of length d until it finds one that has a longer piece at the end, or until it has examined all but one of these protrusions. In the worst case the robot will travel a distance 2(k 0 1)d. An optimal strategy would direct the robot to travel down the protrusion of length examine all the small protrusions coming out of it until it found one that was longer. In the worst case the robot would travel a distance d and ffi can be made arbitrarily small, in the worst case Strategy MDL travels\Omega# times as far as the optimal strategy. Even if we used a strategy that weighted each potential probe location with . d Figure 9: Part of the map polygon that gives lower bound. Not to scale: d ?? ffl; ffi. the amount of information that could be gained from that location in the worst case, we would still build the same decision tree because any probe location in the stairstep yields at most one piece of information in the worst case. Although there are map polygons for which Strategy MDL builds a localizing decision tree whose weighted height is \Omega# times the weighted height of an optimal localizing decision tree, there are other map polygons for which any localizing decision tree strategy builds a tree with weighted height at least k 0 1 times the length of an optimal verification tour. Consider a map polygon that is a staircase polygon with stairs, such as the one in Figure 3, where each stairstep except the first and last one is similar to the one shown in Figure 10. Each such stairstep has k protrusions placed in a circle, with the end of each protrusion a distance d from the center of the circle, and has one protrusion extended at the end to uniquely identify the stairstep. The vertical piece between adjacent stairsteps is longer than 2(k 0 1)d. As with the map polygon shown in Figure 9, Strategy MDL will direct the robot to explore the k protrusions of length d, and in the worst case the robot will travel a distance 2(k 0 1)d. Consider any other localizing decision tree strategy. If it directs the robot to travel to any stairstep besides the one where it starts, then the localizing decision tree that it builds will have weighted height greater than 2(k 0 1)d. The only way to localize the robot while remaining on the initial stairstep is to direct it to examine the protrusions, and in the worst case the robot must travel a distance before it has localized itself (assuming that it must return to the origin at the end). Since no localizing decision tree strategy can build a tree with weighted height less than k times the length of an optimal verification tour for all map polygons, Strategy MDL is the best possible strategy. d Figure 10: Part of the map polygon that shows Strategy MDL is best possible. 5.4 Creating a reduced set of reference points The set Q of reference points has size upper bounded by k times the number of cells in the arrangement A, which may be very large as shown in Section 4.2. In this subsection, we show that when Strategy MDL is run with only a small subset Q of the original reference points, the (k 0 1)d performance guarantee of Section 5.2 still holds. The size of Q 0 will be no more than k(k 0 1). defined as the union of subsets Q there is one Q i for each p i 2 H and Ignoring implementation issues, we define Q i as follows. Initially Q i is empty, and the subset of Q consisting of reference points q C;i generated for translate P i is processed in order of increasing d i (q C;i ). For each successive point q C;i , the partition of H induced by is compared to that induced by Q i alone. If the subset of H containing location p i is further subdivided by the additional reference point q C;i , then q C;i is added to Q i . Conceptually, the reference point q C;i is added if it distinguishes another hypothetical initial location from p i . This process continues contained in a singleton in the partition of H induced by Q i . Since there are only k 0 1 initial locations to be distinguished from p i , Q i will contain at most k 0 1 points. We denote by Strategy MDLR, which stands for Minimum Distance Localization with Reduced reference point set, the strategy obtained by replacing set Q with Q 0 in Strategy MDL. Theorem 6 Strategy MDLR, which uses a set of at most k(k 0 1) reference points, localizes the robot by directing it along a path whose length is at most (k 01)d, where and d is the length of an optimal verification tour for the robot's initial position. Proof: Both the proof that Strategy MDLR directs the robot along a path that determines its initial location and the proof of the (k 0 1)d bound are essentially the same as the proofs of the corresponding results in Theorems 3 and 4 of Section 5.2. The only additional observation needed is that if a reference point q C;i is used in one of the previous proofs to distinguish between two hypothetical initial locations, and if q C;i does not belong to set Q 0 , then Q 0 contains some reference point q C 0 ;j that distinguishes the same pair of locations and that satisfies d j (q C Hence, set Q 0 always contains an adequate substitute for any reference point of Q required by the proofs of Theorems 3 and 4. ut 6 Conclusions and Future Research We have shown that the problem of localizing a robot in a known environment by traveling a minimum distance is NP-hard, and we have given an approximation strategy that achieves a competitive ratio of k 0 1, where k is the number of possible initial locations of the robot. We have also shown that this bound is the best possible. The work in this paper is one part of a strategy for localizing a robot. The complete strategy will preprocess the map polygon and store the decision trees for ambiguous initial positions so that the robot only needs to follow a predetermined path to localize itself. There are many variations to this problem which can be considered. If the robot must localize itself in an environment with obstacles, then the map of the environment can be represented as a simple polygon with holes. If these obstacles are moving, then the problem becomes more difficult. In this paper we assigned a cost of zero for the robot to take a probe and analyze it. In a more general setting we would look for an optimal decision tree, where the edges of a decision tree associated with the outcome of a probe would be weighted with the cost to analyze that probe. A pragmatic variation of the problem would weight reference locations so that those that produce more reliable percepts would be selected first. --R Locating a Robot with Angle Measurements. Decision Trees for Geometric Models. Randomized Robot Navigation Algorithms. Map learning with indistinguishable locations. Mobile Robot Localization Using Landmarks. Efficient Visibility Queries in Simple Polygons. Visibility in Simple Polygons. Homing using combinations of model views. Navigating in Unfamiliar Geometric Terrain. Probing Polygons Minimally is Hard. Representing and Acquiring Geographic Knowledge. Competitive Searching in a Generalized Street. Map validation and self-location in a graph-like world Performance Guarantees for Motion Planning with Temporal Uncertainty. El Gindy and Computers and Intractability The Robot Localization Problem in Two Dimensions. Constructing Optimal Binary Decision Trees is NP-Complete A qualitative approach to robot exploration and map-learning Walking an Unknown Street with Bounded Detour. The Localization Problem for Mobile Robots. Robot Motion Planning. Euclidean Shortest Paths in the Presence of Rectilinear Barriers. Shortest Paths without a Map. Amortized Efficiency of List Update and Paging Rules. Position estimation for an autonomous mobile robot in an outdoor environment. --TR --CTR Sven Koenig , Apurva Mudgal , Craig Tovey, A near-tight approximation lower bound and algorithm for the kidnapped robot problem, Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm, p.133-142, January 22-26, 2006, Miami, Florida Rudolf Fleischer , Kathleen Romanik , Sven Schuierer , Gerhard Trippen, Optimal robot localization in trees, Information and Computation, v.171 n.2, p.224-247, December 15, 2001

navigation;optimization;localization;visibility;positioning;NP-hard;competitive strategy;robot;sensing

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A Controlled Experiment to Assess the Benefits of Procedure Argument Type Checking.

Type checking is considered an important mechanism for detecting programming errors, especially interface errors. This report describes an experiment to assess the defect-detection capabilities of static, intermodule type checking. The experiment uses ANSI C and Kernighan&Ritchie (K&R) C. The relevant difference is that the ANSI C compiler checks module interfaces (i.e., the parameter lists calls to external functions), whereas K&R C does not. The experiment employs a counterbalanced design in which each of the 40 subjects, most of them CS PhD students, writes two nontrivial programs that interface with a complex library (Motif). Each subject writes one program in ANSI C and one in K&R C. The input to each compiler run is saved and manually analyzed for defects. Results indicate that delivered ANSI C programs contain significantly fewer interface defects than delivered K&R C programs. Furthermore, after subjects have gained some familiarity with the interface they are using, ANSI C programmers remove defects faster and are more productive (measured in both delivery time and functionality implemented).

Introduction The notion of data type is an important concept in programming languages. A data type is an interpretation applied to a datum, which otherwise would just be a sequence of bits. The early FORTRAN compilers already used type information to generate efficient code for expressions. For instance, the code produced for the operator "+" depends on the types of its operands. User-defined data types such as records and classes in later programming languages emphasize another aspect: Data types are a tool for modeling the data space of a problem domain. Thus, types can simplify programming and program understanding. A further benefit is type checking: A compiler or interpreter can determine whether a data item of a certain type is permissible in a given context, such as an expression or statement. If it is not, the compiler has detected a defect in the program. It is the defect-detection capability of type checking that is of interest in this paper. There is some debate over whether dynamic type checking is preferable to static type checking, how strict the type checking should be, and whether explicitly declared types are more helpful than implicit ones. However, it seems that overall the benefits of type checking are virtually undisputed. In fact, modern programming languages have evolved elaborate type systems and checking rules. In some languages, such as C, the type-checking rules were even strengthened in later versions. Furthermore, type theory is an active area of research [3]. However, it seems that the benefits of type checking are largely taken for granted or are based on personal anecdotes. For instance, Wirth states [21] that the type-checking facilities of Oberon had been most helpful in evolving the Oberon system. Many programmers can recall instances when type checking did or could have helped them. However, we could find only a single report on a controlled, repeatable experiment testing the benefits of typing [9]. The cost-benefit ratio of type checking is far from clear, because type checking is not free: It requires effort on behalf of the programmer in providing type infor- mation. Furthermore, there are good arguments why relying on compiler type checking may be counter-productive when doing inspections [12, pp. 263-268]. We conclude that the actual costs and benefits of type checking are largely unknown. This situation seems to be at odds with the importance assigned to the con- cept: Languages with type checking are widely used and the vast majority of practicing programmers are affected by the technique in their day-to-day work. The purpose of this paper is to provide initial, "hard" evidence about the effects of type checking. We describe a repeatable and controlled experiment that confirms some positive effects: First, when applied to interfaces, type checking reduced the number of defects remaining in delivered programs. Second, when programmers use a familiar interface, type checking helped them remove defects more quickly and increased their productivity. Knowledge about the effects of type checking can be useful in at least three ways: First, we still lack a useful scientific model of the programming process. Understanding the types, frequencies, and circumstances of programmer errors is an important ingredient of such a model. Second, a better understanding of defect- detection capabilities of type checking may allow us to improve and fine-tune them. Finally, there are still many environments where type checking is missing or incomplete, and confirmed positive effects of type checking may help close these gaps. In this experiment we analyze the effects of type checking when programming against an interface. Subjects were given programming tasks that involve a complex interface (the Motif library). One group of subjects worked with the type checker, the other without. The dependent variables were as follows: Completion Time: The time taken from receiving the task to delivering the program. Functional Units: The number of complete and correct functional units in a program. Each functional unit interfaces to the library and corresponds to one statement in the "gold program" (the model solution). Interface Use Productivity: measured in Functional Units per hour and by Completion Time. Number of Interface Defects: The number of program defects in applying the library interface. Such a defect is either an argument missing, too many, of wrong type, or at incorrect position; or it is the use of an inappropriate function. Interface Defect Lifetime: The total time a particular interface defect is present in the solution during de- velopment. Note that this time may be the sum of one or more time intervals, since a defect may first be eliminated and later reintroduced. We conjecture that type checking makes type defect removal quicker and more reliable, thus also speeding up overall program development. More concretely, we attempt to find support for, or arguments against, the following three hypotheses. checking increases Interface Use Productivity. ffl Hypothesis 2: Type checking reduces the number of Interface Defects in delivered programs. ffl Hypothesis 3: Type checking reduces Interface Defect Lifetimes. Related work We are aware of only two closely related studies. One is the Snickering Type Checking Experiment 1 with the Mesa language. In this work, compiler-generated error messages involving types were diverted to a secret file. A programmer working with this compiler on two different programs was shown the error messages after he had finished the programs and was asked to estimate how much time he would have saved had he seen the messages right away. Interestingly, the programmer had independently removed all the defects detected by the type checker. He claimed that on one program, which was entirely his own work, type checking would not have helped appreciably. On another program which involved interfacing to a complicated library, he estimated that type checking would have saved half of total development time. It is obvious that this type of study has many flaws. But to our knowledge it was never repeated in a more controlled setting. A different approach was taken by the second experi- ment, performed by Gannon [9]. This experiment compares frequencies of errors in programs written in a statically typed and a "type-less" language. Each subject writes the same program twice, once in each lan- guage, but a different order of languages is used for each half of the experiment group. The experiment finds that the typed group has fewer distinct errors, fewer error re-occurrences, fewer compilation runs, and fewer errors remaining in the program (0.21 vs. 0.64 on average). The problem with the experiment is that it was significantly harder to program with the typeless language. The task to be programmed involved strings and the typed language provided this data type, while the type-less language did not. Gannon reports that most of the difficulties encountered by the subjects were actually due to the bit-twiddling required by lack of typing and that "relatively few errors resulted from uses of data of the wrong type" ([9], p.591). Hence the experiment does not tell us how useful type checking is. There is some research on error and defect classifica- tion, which has some bearing on our experiment. Several publications describe and analyze the typical defects in programs written by novices, e.g. [6, 18]. The results are not necessarily relevant for advanced pro- grammers. Furthermore, type errors do not play an important role in these studies. Defect classification has also been performed in larger scale software development settings, e.g. [1, 10]. Type checking was not an explicit concern in these stud- ies, but in some cases related information can be de- rived. For instance, Basili and Perricone [1] report that 39 percent of all defects in a 90.000 line FORTRAN project were interface defects. We conjecture that some fraction of these could have been found by type checking. The defect-detection capabilities of testing methods [2, 8, 22] have received some attention; the corresponding psychological problems were also investigated [20]. There is also a considerable literature about debugging, e.g. [7, 13, 16, 17], and its psychology, e.g. [17, 19]. However, the defects found by testing or debugging are those that already passed the type checks. So the results from these studies would be applicable here only if they focused on defects detectable by type checking which they do not. Several studies have compared the productivity effects of different programming languages, but they either used programmers with little experience and very small programming tasks, e.g. [6], or somewhat larger tasks and experienced programmers, but lacked proper experimental control, e.g. [11]. In addition, all such studies have the inherent problem that they confound too many factors to draw conclusions regarding type checking, even if some of the languages provide type checking and others do not. It appears that the cost and benefits of interface type checking have not yet been studied systematically. 3 Design of the Experiment The idea behind the experiment is the following: Let experienced programmers solve short, modestly complex programming problems involving a complex li- brary. To control for the type-checking/no-type- checking variable, let every subject solve one problem with K&R C, and another with Ansi C. Save the inputs to all compiler runs for later defect analysis. A number of observations regarding the realism of the setup are in order. A short, modestly complex task means that most difficulties observed will stem from using the library, not from solving the task itself. Thus, most errors will occur when interfacing to the library, where the effects of type checking are thought to be most pronounced. Furthermore, using a complex library is similar to the development of a module within a larger project where many imported interfaces must be handled. To ensure that the results would not be confounded by problems with the language, we used experienced programmers familiar with the programming language. However, the programmers had no experience with the library - another similarity with realistic software development, in which new modules are often written within a relatively foreign context. In essence, we used two independent variables: There were two separate problems to be solved as described below) and two alternative treatments (Ansi C and K&R C, i.e., type checking and no type checking). To balance for learning effects, sequence effects, and inter-subject ability differences, we used a counterbalanced design: Each subject had to solve both problems, each with a different language. The groups were balanced with respect to the order of both problem and language, giving a total of four experimental groups (see Table 1). Subjects were assigned to the groups randomly. The design also allows to study a third independent variable, namely experience with the library: In his or her first task the subject has no previous experience while in the second task some experience from the first task is present. The following subsections describe the tasks, the sub- jects, the experiment setup, and the observed variables and discuss internal and external validity of the experi- ment. Detailed information can be found in a technical report [15]. 3.1 Tasks Problem A (2 \Theta 2 matrix inversion): Open a window with four text fields arranged in a 2 \Theta 2 pattern plus an "Invert" and a "Quit" button. See Figure 1. "Quit" exits the program and closes the window. The fields represent a matrix of real values. The values can be entered and edited. When the "Invert" button is pressed, replace the values by the coefficients of the corresponding inverted matrix, or print an error message if the matrix is not invertible. The formula for Problem B (File Browser): Open a window with a menubar containing a single menu. The menu entry "Select file" opens a file-selector box. The entry "Open Table 1: Tasks and compilers assigned to the four groups of subjects first problem A first problem B then problem B then problem A first Ansi C Group 1 Group 2 then K&R C 8 subjects 11 subjects first K&R C Group 3 Group 4 then Ansi C 8 subjects 7 subjects Figure 1: Problem A (2 \Theta 2 matrix inversion) selected file" pops up a separate, scrollable window and displays the contents of the file previously selected in the file selector box. "Quit" exits the program and closes all its windows. See Figure 2. Figure 2: Problem B (File browser) For solving the tasks, the subjects did not use native Motif, but a special wrapper library. The wrapper provides operations similar to those of Motif, but with improved type checking. For instance, all functions have fixed-length parameter lists, while Motif often provides variable-length parameter lists which are not checked. The wrapper also defines types for resource-name con- stants; in Motif, all resources are handled typelessly. Furthermore, the wrapper provides some simplification through additional convenience functions. For in- stance, there is a single function for creating a Row- ColumnManager and setting its orientation and packing mode; Motif requires several calls. The tasks, although quite small, were not at all trivial. The subjects had to understand several important concepts of Motif programming (such as widget , resource, and callback function). Furthermore, they had to learn to use them from abstract documentation only, without example programs; we used no examples as we felt that these would have made the programming tasks too simple. Typically, the subjects took between one and two hours for their first task and about half that time for their second. 3.2 Subjects A total of 40 unpaid volunteers participated in the study. Of those, 6 were removed from the sample: One deleted his protocol files, one was obviously too inexperienced almost 10 times as long as the others), and 4 worked on only one of the two problems. After this mortality, the A/B groups had 8+8 subjects and the B/A groups had 11+7 subjects. We consider this to be still sufficiently balanced [4]. The 34 subjects had the following education. 2 were postdocs in computer science (CS); 19 were PhD students in CS and had completed an MS degree in CS; another subject was also a CS PhD student but held an MS in physics; 12 subjects were CS graduate students with a BS in CS. The subjects had between 4 and 19 years of programming experience and all but 11 of them had written at least 3000 lines in C (all but one at least 300 lines). Only 8 of the subjects had some programming experience with X-Windows or Motif; only 3 of them had written more than 300 lines in X-Windows or Motif. 3.3 Setup Each subject received two written documents and one instruction sheet and was then left alone at a Sun-4 workstation to solve the two problems. The subjects were told to use roughly one hour per problem, but no time limit was enforced. Subjects could stop working even if the programs were not operational. The instruction sheet was a one-page description of the global steps involved in the experiment: "Read sections 1 to 3 of the instruction document; fill in the questionnaire in section 2; initialize your working environment by typing make TC1; solve problem A by. " and so on. The subjects obtained the following mate- rials, most of them both on paper and in files: 1. a half-page introduction to the purpose of the ex- periment 2. a questionnaire about the background of the sub- ject 3. specifications of the two tasks plus the program skeleton for them 4. a short introduction to Motif programming (one page) and to some useful commands (for example to search manuals online) 5. a manual that listed first the names of all types, constants, and functions that might be required, followed by descriptions of each of them including the signature, semantic description, and several kinds of cross-references. The document also included introductions to the basic concepts of Motif and X-Windows. This manual was hand tailored to contain all information required to solve the tasks and hardly anything else. 6. a questionnaire about the experiment (to be filled in at the end) Subjects could also execute a "gold" program for each task. The gold program solved its task completely and correctly and was to be used as a backup for the verbal specifications. Subjects were told to write programs that duplicated the behavior of the gold programs. The subjects did not have to write the programs from scratch. Instead, they were given a program skeleton that contained all necessary #include commands, variable and function declarations, and some initialization statements. In addition, the skeleton contained pseudocode describing step by step what statements had to be inserted to complete the program. The subjects' task was to find out which functions they had to use and which arguments to supply. Almost all statements were function calls. The following is an example of a pseudostatement in the skeleton. /* Register callback-function 'button pushed' for the 'invert' button with the number 1 as 'client data' */ It can be implemented thus: XtAddCallbackF(invert, XmCactivateCallback, button pushed, (XtPointer)1); There were only few variations possible in the implementation of the pseudocode. The programming environment captured all program versions submitted for compilation along with a time stamp and the messages produced by the compiler and linker. A time stamp for the start and the end of the work phase for each problem was also written to the protocol file. The environment was set up to call the standard C compiler of SunOS 4.1.3 using the command cc -c -g for the K&R tasks and version 2.7.0 of the GNU C compiler using gcc -c -g -ansi -pedantic -Wimplicit -Wreturn-type for the Ansi C tasks. 3.4 Dependent variables For hypotheses 2 and 3 we observed when each individual defect in a program was introduced and removed. We also divided the defects in a few non-overlapping classes. We used the following procedure. After the experiment was finished, each program version in the protocol files was annotated by hand. Each different defect that occurred in the programs was identified and given a unique number. For instance, for the call to XtAddCallbackF shown above, there were 15 different defect numbers, including 4 for wrong argument types, 4 for wrong argument objects with correct type, and another 7 for more specialized defects. Each program version was annotated with the defects introduced, removed, or changed into another defect. Additional annotations counted the number of type de- fects, other semantic defects, and syntactic defects that actually provoked one or more error messages from the compiler or linker. The time stamps were corrected for work pauses that lasted more than 10 minutes in order to capture pure programming time only. Summary statistics were computed, for which each defect was classified into one of the following categories: ffl slight: Defects resulting in slightly wrong functionality of the program, but so minor that a programmer may feel no need to correct them. There- fore, this class will also be ignored in order to avoid artifacts in the results. invis: Defects that are invisible, i.e., they do not compromise functionality, but only because of unspecified properties of the library implementation. Changes in the library implementation may result in a misbehaving program. Example: Supplying the integer constant PACK COLUMN instead of the expected Boolean value True works correctly, because (and as long as) the constant happens to have a non-zero value. This rare class of defects will be ignored: invis defects can hardly be detected and thus are not relevant for our experiment ffl invisD: same as invis, except that the defects will be detected by Ansi C parameter type checking (but not by K&R C). The invis class excludes in- visD. severe: Defects resulting in significant deviations from the prescribed functionality. ffl severeD: same as severe, except that the defects will be detected by Ansi C parameter type checking (but not by K&R C). The severe class excludes severeD. These categories are mutually exclusive. Defects that had to be removed before the program would pass even only the K&R C compiler and linker will be ignored. Unless otherwise noted, the defect statistics discussed below are computed based on the sum of severe, sev- ereD, and invisD. Other metrics observed were the number of compilation cycles (versions) and time to delivery, i.e., the time spent by the subjects before delivering the program (whether complete and correct or not). From these metrics and annotations, additional statistics were computed. For instance the frequency of defect insertion and removal, the number of attempts made before a defect was finally removed, the Interface Defect Lifetime, and the number and type of defects remaining in the final program version. See also the definitions in Section 1. For measuring productivity and unimplemented func- tionality, we define a functionality unit (FU) to be a single statement in the gold program. For example, the call to XtAddCallbackF shown in Section 3.3 is one FU. Using the gold programs as a reference normalizes the cases in which subjects produce more than one statement instead. FUs are thus a better measure of program volume than lines of code. Gold program A contains contains 11. We annotated the programs with the number of gaps , i.e., the number of missing FUs. An FU is counted as missing if a subject made no attempt to implement it. 3.5 Internal and external validity The following problems might threaten the internal validity of the experiment, i.e., the correctness of the results 1. For defects where both the K&R and the Ansi C compiler produce an error message, these messages might differ and this might influence productivity. Our subjective judgment here is that for the purposes of this experiment the error messages of both compilers, although sometimes quite different, are overall comparable in quality. Furthermore, none of our subjects were very experienced with one particular compiler and would understand its messages faster than others. 2. There may be annotation errors. To insure consis- tency, all annotations were made by the same per- son. The annotations were cross-checked first with a simple consistency checker (looking whether errors were introduced before removed, times were plausible, etc.), and then some of them were checked manually. The number of annotation mistakes found in the manual check was negligible (about 4%). 3. The learning effect from first to second task might be different for K&R subjects than for Ansi C subjects. This problem, and related ones, is accounted for by the counter-balanced experiment design. The following problems might limit external validity of the experiment, i.e., the generalizability of our results: 1. The subjects were not professional software en- gineers. However, they were quite experienced programmers and held degrees (many of them ad- vanced) in computer science. 2. The results may be domain dependent. This objection cannot be ruled out. This experiment should therefore be repeated in domains other than graphical user interfaces. 3. The results may or may not apply to situations in which the subjects are very familiar with the interfaces used. This question might also be worth a separate experiment. Despite these problems, we believe that the scenario chosen in the experiment is nevertheless similar to many real situations with respect to type-checking errors Another issue is worth discussing here: The learning effect (performance change from first task to second task) is larger than the treatment effect (performance change from K&R C to Ansi C). This would be a problem if the learning reduced the treatment effect [16, pages 106 and 113]. However, as we will see below, in our case the treatment effect is actually increased by learning, making our experiment results conservative ones. We are explicitly considering programmers who are not highly familiar with the interface used. Therefore learning is a natural and necessary part of our setting, not an artifact of improper subject selection. 4 Results and Discussion Many of the statistics of interest in this study have clearly non-normal distributions and sometimes severe outliers. Therefore, we present medians (to be precise: an interpolated 50% quantile) rather than arithmetic means. Where most of the median values are zero, higher quantiles are given. The results are shown in Tables 2 through 4. There are altogether ten different statistics, each appearing in three main columns. The first column shows the statistics for both tasks, independent of order. The second and third columns reflect the observations for those tasks that were tackled first and second, respectively. These columns can be used to assess the learning ef- fect. Each main column reports the medians (or higher quantiles where indicated) for the tasks programmed with Ansi C and K&R C plus the p-value. The p-value is the result of the Wilcoxon Rank Sum Test (Mann- Whitney U Test) and, very roughly speaking, represents the probability (given the observations) that the medians of the two samples are equal. If p - 0:05, the test result is considered statistically significant and we call the distributions significantly different. Significant results are marked in boldface in the tables. When the result is not significant, nothing can be said; there may or may not be a difference. 4.1 Productivity Table shows three measures that describe the over-all time taken and the productivity exhibited by the subjects. Statistic 1, time to delivery, shows no significant difference between Ansi C and K&R C for the first task or for both tasks taken together. Ignoring the programming language, the time spent for the second task is shorter than for the first (p = 0:0012, not shown in the table), indicating a learning effect. In the second task, Ansi C programs are delivered significantly faster than K&R C programs. A plausible explanation is that when they started, programmers did not have a good understanding of the library and were struggling more with the concepts than with the interface itself. This explanation was confirmed by studying the compiler inputs. Type checking is unlikely to help gain a better understanding. Type checks became useful only after programmers had overcome the initial learning hurdle. Statistic 2, the number of program versions compiled, does not show a significant difference; Ansi C programmers compile about as often as K&R C programmers. describes the productivity measured in functional units per hour (FU/h). In contrast to time to de- livery, this value accounts for functionality not implemented by a few of the subjects. Again we find no significant difference for the first task, but a (weakly) significant difference for the second task. There, Ansi C median productivity is about 20% higher than K&R C productivity, suggesting that Ansi C is helpful for programmers after the initial interface learning phase. This observation supports hypothesis 1. The combined (both languages) productivity rises very significantly from the first task to the second task (p = 0:0001, not shown in the table); this was also reported by the subjects and confirms that there is a strong learning effect induced by the sequence of tasks. The actual distri- Ansi K&R Figure 3: Boxplots of productivity (in FU/hour) over both tasks for Ansi C (left boxplot) and K&R C (right boxplot). The upper and lower whiskers mark the 95% and 5% quantiles, the upper and lower edges of the box mark the 75% and 25% quantiles, and the dot marks the 50% quantile (median). All other boxplots following below have the same structure. Ansi K&R Figure 4: Boxplots of productivity (in FU/hour) for first task. Figure 5: Boxplots of productivity (in FU/hour) for second task. butions of productivity measured in FU/h are shown in Figures 3 to 5. We see that Ansi C makes for a more pronounced increase in productivity from the first task to the second (about 78% for the median) than does K&R C (about 26% for the median). Table 2: Overall productivity statistics. Medians of statistics for Ansi C vs. K&R C versions of programs and p-values for statistical significance of Wilcoxon Rank Sum Tests of the two. Values under 0.05 indicate significant differences of the medians. Column pairs are for 1st+2nd, 1st, and 2nd problem tackled chronologically by each subject, respectively. All entries include data points for both problem A and problem B. both tasks 1st task 2nd task Statistic Ansi K&R Ansi K&R Ansi K&R 1 hours to delivery 1.3 1.35 1.6 1.6 0.9 1.3 #versions 15 3 FU/h 8.6 9.7 7.2 8.5 12.8 10.7 Table 3: Statistics on internals of the programming process. See Table 2 for explanations. both tasks 1st task 2nd task Statistic Ansi K&R Ansi K&R Ansi K&R 4 accumul. interf. dfct. lifetime (median) 0.3 1.2 0.5 2.1 0.2 1.1 5 #right, then wrong again (75% quant.) 1.0 1.0 1.0 1.0 0.0 1.0 4.2 Defect lifetimes Table 3 gives some insight into the programming process Statistic 4 is the time from the introduction of an interface defect to its removal (or the end of the experiment) accumulated over all interface defects introduced by a subject. The distributions of this variable over both tasks are also shown as boxplots in Figure 6. As accum. severeD error lifetime2610 Ansi K&R Figure Boxplots of accumulated interface defect lifetime (in hours) over both tasks. we see, the K&R total defect lifetimes are higher and spread over a much wider range; the difference is signifi- cant. Note that the frequency of defect insertion (num- ber of interface defects inserted per hour, not shown in the table) does not show significant differences between the languages, indicating that Ansi C is of little help in defect prevention (as opposed to defect removal). Taken together, these two facts support hypothesis 3: Ansi C helps to remove interface defects quickly. Statistic 5 indicates the number of defects, interface or other, introduced in previously correct or repaired statements of a program. While there is hardly any difference in the first task, the value is significantly higher for K&R C in the second task. We speculate that this happens because the type error messages of Ansi C allow some of the subjects to avoid the trial- and-error defect removal techniques they would have used in K&R C; the effect occurs only in the second task, after the subjects have gained a basic understanding of Motif concepts. 4.3 Defects in delivered programs Table 4 describes the quality of the products delivered by the subjects. Statistic 6 says that there are not more unimplemented functionality units ("gaps") in the K&R C programs. Statistic 7 confirms that there are more defects in the delivered K&R C programs than in the Ansi C pro- grams; see also the distribution as shown in Figure 7. The difference is much more pronounced in the second task, though. Again the reason is probably that the advantages of Ansi C become fully relevant only after most other initial problems have been mastered. Statistics 8 to 10 confirm that the reason for the difference lies indeed in the type checking capabilities of Ansi C: both the rare invisD defects (statistic 8) and the severeD defects (statistic 10, see also Figure 8) are much less frequent in delivered Ansi C programs than in K&R C programs. These defects can be detected by Ansi C type checking. On the other hand, severe defects (statistic 9, see also Figure 9) are about as frequent in delivered Ansi C programs as in K&R C programs. These defects cannot be detected by type checking. As we see in the boxplots, the distributions for severe Table 4: Statistics on the delivered program. See Table 2 for explanations. Lines 6 and 8 do not list medians but other quantiles instead, as indicated. both tasks 1st task 2nd task Statistic Ansi K&R Ansi K&R Ansi K&R 6 #gaps (75% quantile) 0.25 7 #remaining errs in delivered program 1.0 2.0 1.0 2.0 1.0 2.0 9 - for severe only 1.0 1.0 1.0 0.0 1.0 1.0 all remaining errors Figure 7: Boxplots of total number of remaining defects in delivered programs over both tasks. severeD remaining errors Ansi K&R Figure 8: Boxplots of number of remaining severeD defects in delivered programs over both tasks. defects differ only in the upper tail, whereas the distributions for the severeD defects differ dramatically in favor of Ansi C, resulting in a significant overall advantage for Ansi C. These observations support hypothesis 2. severe remaining errors Ansi K&R Figure 9: Boxplots of number of remaining severe defects in delivered programs over both tasks. Detailed analysis of the defects remaining in the delivered programs indicates a slight, but not statistically significant tendency that besides type defects other classes of frequent defects also were reduced in the Ansi C programs: using the wrong variable as a parameter or an assignment target using a wrong constant value as a parameter (p = 0:35). It is unknown whether this is a systematic side-effect of type checking and how it should be explained if it is. There were no significant differences between the two tasks; all of the above results hardly change if one considers the tasks A and B separately. 4.4 Questionnaire results Finally, the subjective impressions of the subjects as reported in the questionnaires are as follows: 26 of the subjects (79%) noted a learning effect from the first program to the second. 9 subjects (27%) reported that they found the Ansi C type checking very helpful, 11 (33%) found it considerably helpful, 4 (12%) found it almost not helpful, 5 (15%) found it not at all helpful. subjects could not decide and 1 questionnaire was lost. 5 Conclusions and further work The experiment results allow for the following statements regarding our hypotheses: ffl Hypothesis 1, Interface Use Productivity: When programming an interface, type checking increases productivity, provided the programmer has gained a basic understanding of the interface. ffl Hypothesis 2, Interface Defects in delivered program: Type checking reduces the number of Interface Defects in delivered programs. ffl Hypothesis 3, Interface Defect Lifetime: Type checking reduces the time defects stay in the program during development. One must be careful generalizing the results of this study to other situations. For instance, the experiment is unsuitable for determining the proportion of interface defects in an overall mix of defects, because it was designed to prevent errors other than interface errors. Hence it is unclear how large the differences will be if defect classes such as declaration defects, initialization defects, algorithmic defects, or control-flow defects are included. Nevertheless, the experiment suggests that for many realistic programming tasks, type checking of interfaces improves both productivity and program quality. Fur- thermore, some of the resources otherwise expended on inspecting interfaces might be allocated to other tasks. As a corollary, library design should strive to maximize the type-checkability of the interfaces by introducing new types instead of using standard types where appropriate. For instance Motif, on which our experiment library was based, is a negative example in this respect. Further work should repeat similar error and defect analyses in different settings (e.g. tasks with complex data flow or object-oriented languages). In particu- lar, it would be interesting to compare productivity and error rates under compile-time type checking, run-time type checking, and type inference. Other important questions concern the influence of a disciplined programming process such as the Personal Software Process [12]. Finally, an analysis of the errors occurring in practice might help devise more effective defect- detection mechanisms. Acknowledgments We thank Paul Lukowicz for patiently guinea-pigging the experimental setup, Dennis Goldenson for his detailed comments on an early draft, and Larry Votta for pointing out an important reference and providing many suggestions on the report. Last, but not least, we thank our subjects. A Solution for Problem A This is the program (ANSI C version) that represents the canonical solution for Problem A. Most of it, including all of the comments, was given to the subjects from the start; they only had to insert the statements marked here with /*FU 1*/ etc. at those places previously held by pseudocode comments as described in Section 3.3 above. The numbers in the FU comments count the functional units as defined in Section 1. #include !stdio.h? #include !stdlib.h? #include "stdmotif.h" void button-pushed (Widget widget, XtPointer client-data, XtPointer call-data); fields for matrix coefficients: 0,1,2,3 for a,b,c,d */ int main (argc, argv) int argc; char *argv[]; manager, /* manager for square and buttons */ square, /* manager for 4 TextFields */ buttons, /* manager for 2 PushButtons */ quit; /* PushButton */ XtAppContext app; XmString invertlabel, quitlabel; /*- 1. initialize X and Motif -*/ /* (already complete, should not be changed) */ globalInitialize ("A"); &argc, argv, fallbacks, NULL); /*- 2. create and configure widgets -*/ 2, False); /*FU 1*/ 2, True); /*FU 2*/ buttons XmStringCreateLocalized ("Invert matrix")); XmStringCreateLocalized ("Quit")); /*- 3. register callback functions -*/ (invert, XmCactivateCallback, button-pushed, /*- 4. realize widgets and turn control to X event loop -*/ /* (already complete, should not be changed) */ XtRealizeWidget (toplevel); return (0); Functions */ void button-pushed (Widget widget, XtPointer client-data, XtPointer call-data) /* this is the callback function to be called when clicking on the PushButtons occurs */ double mat[4], new[4], /* old and new matrix coefficients */ det; /* determinant */ String s; if ((int)client-data == 99) exit (0); /*FU 12*/ else if ((int)client-data == 1) - int for XtGetStringValue (mw[i], XmCvalue, &s); /*FU 13*/ if (det != for XtSetStringValue (mw[i], XmCvalue, ftoa(new[i],8,2)); /*FU 15*/ else matrixErrorMessage("Matrix cannot be inverted",mat,8,2);/*FU 16*/ Solution for Problem B See the description in Appendix A above. #include !stdio.h? #include !stdlib.h? #include "stdmotif.h" void handle-menu (Widget widget, XtPointer client-data, XtPointer call-data); int main (int argc, char *argv[]) menubar, /* the one-entry menu bar */ menu, /* the pulldown menu */ label; /* the label displayed in the work window */ XtAppContext app; /*- 1. initialize X and Motif -*/ /* (already complete, should not be changed) */ globalInitialize ("B"); &argc, argv, fallbacks, NULL); /*- 2. create and configure widgets -*/ XmStringCreate ("File Browser", "LARGE"), 'F'); /*FU 2*/ XmStringCreate ("by Lutz Prechelt", "SMALL")); /*FU 3*/ XtSetWidgetValue (main-w, XmCworkWindow, label); /*FU 4*/ XmStringCreateLocalized ("Select file"), 'f', XmStringCreateLocalized ("Open selected file"), 'O', XmStringCreateLocalized ("Quit"), 'Q', /*- 3. register callback functions -*/ /* (handle-menu was already registered above, nothing to be done) */ /*- 4. realize widgets and turn control to X event loop -*/ /* (already complete, should not be changed) */ XtRealizeWidget (toplevel); return (0); Functions */ void handle-menu (Widget widget, XtPointer client-data, XtPointer call-data) if ((int)client-data == first menu entry selected */ XtManageChild (fs); /*FU 8*/ else if ((int)client-data == 1) - /* second menu entry selected */ toplevel, 25, 80); /*FU 9*/ XtSetStringValue (scrolltext, XmCvalue, readWholeFile (selectedFile())); /*FU 10*/ else if ((int)client-data == 2) - /* third menu entry selected */ exit (0); /*FU 11*/ --R Software errors and complexity: An empirical investigation. Software Testing Techniques. Typing in object-oriented languages: Achieving expressibility and safety Experimental Methodology. Spohrer, editors. Empirical Studies of Program- mers: Fifth Workshop Novice programmer errors: Language constructs and plan composition. Tales of debugging from the front lines. An experimental comparison of the effectiveness of branch testing and data flow testing. An experimental evaluation of data type conventions. Practical results from measuring software quality. Haskell vs. Ada vs. C A Discipline for Software Engi- neering An analysis of the on-line debugging process Empirical Studies of Program- mers: Second Workshop A controlled experiment measuring the impact of procedure argument type checking on programmer productiv- ity The psychological study of program- ming Empirical Studies of Programmers. Analyzing the high frequency bugs in novice programs. Cognitive bias in software engineering. Positive test bias in software testing by professionals: what's right and what's wrong. Gedanken zur Software- Explosion Certification of software components. --TR --CTR Maurizio Morisio , Daniele Romano , Ioannis Stamelos, Quality, Productivity, and Learning in Framework-Based Development: An Exploratory Case Study, IEEE Transactions on Software Engineering, v.28 n.9, p.876-888, September 2002 Adrian Birka , Michael D. Ernst, A practical type system and language for reference immutability, ACM SIGPLAN Notices, v.39 n.10, October 2004 Matthew S. Tschantz , Michael D. Ernst, Javari: adding reference immutability to Java, ACM SIGPLAN Notices, v.40 n.10, October 2005 Robin Abraham , Martin Erwig, Type inference for spreadsheets, Proceedings of the 8th ACM SIGPLAN symposium on Principles and practice of declarative programming, July 10-12, 2006, Venice, Italy Martin Erwig , Deling Ren, An update calculus for expressing type-safe program updates, Science of Computer Programming, v.67 n.2-3, p.199-222, July, 2007 Andreas Zendler, A Preliminary Software Engineering Theory as Investigated by Published Experiments, Empirical Software Engineering, v.6 n.2, p.161-180, June 2001

controlled experiment;defects;productivity;quality;type checking

279143

Optimal Elections in Faulty Loop Networks and Applications.

AbstractLoop networks (or Hamiltonian circulant graphs) are a popular class of fault-tolerant network topologies which include rings and complete graphs. For this class, the fundamental problem of Leader Election has been extensively studied, assuming either a fault-free system or an upper-bound on the number of link failures. We consider loop networks where an arbitrary number of links have failed and a processor can only detect the status of its incident links. We show that a Leader Election protocol in a faulty loop network requires only O(n log n) messages in the worst-case, where n is the number of processors. Moreover, we show that this is optimal. The proposed algorithm also detects network partitions. We also show that it provides an optimal solution for arbitrary nonfaulty networks with sense of direction.

Introduction 1.1 Loop Networks A common technique to improve reliability of ring networks is to introduce link redun- dancy; that is, to have each node connected to two or more additional nodes in the network. With alternate paths between nodes, the network can sustain several nodes and links failures. Several ring networks, suggested in [3, 8, 27, 34, 40] are based on this prin- ciples. The overall topological structure of these redundant rings is always highly regular; in particular, the set of ring edges (regular) and additional edges (bypass) form a Loop Network (since they have at least one hamiltonian cycle). Figure 1: #2, 4# Loop Network (a) with Faulty Links (b) Loop Networks are particular cases of Circulant Graph. Because of an uncoordinated literature, numerous terms have been used to name this topology depending on the model; Circulant Graph, Chordal Ring, or Distributed Loop Computer Networks are the more common. A detailed survey of these topologies is presented in [5]. For sake of simplicity, we will use the term loop network in the remaining of this paper. A loop network C n #d 1 , d 2 , ., d k # of size n and k-chord structure #d 1 , d 2 , ., d k # is a ring R n of n processors {p 0 , each processor is also directly connected to the processors at distance d i and n - d i by additional incident chords. The link connecting two nodes is labeled by the distance which separates these two nodes on the ring, i.e., following the order of the nodes on the ring: the node p i is connected to the node p i+d j mod n through its link labeled d j (as shown in Figure 1(a)). In particular, if a link, between two processors p and q, is labeled by distance d at processor p, this link is labeled by n - d at the other incident processor q, where n is the number of processors. Note that both rings and complete graphs are circulant graphs, denoted as C n # and respectively. It is worth pointing out that some designs for redundant meshes and redundant hypercubes are also circulant graphs, [7]. The distinction between regular and bypass links is purely a functional one. Typically, the bypass links are used strictly for reconfiguration purposes when faults are detected; in the absence of faults, only regular links are used. Special classes of loop networks have been widely investigated to analyze their fault-tolerant properties [3, 7, 8, 9, 27, 33] and solutions have been proposed for reconfiguration after links and/or node failures [30, 39]. In some applications (e.g., distributed systems), all the links (or chords) of a circulant graph are always used to improve the performance of a computation. 1.2 Election In distributed systems, one of the fundamental control problem is the Leader Election [29]. Informally, election is the problem of moving the system from an initial situation where the nodes are in the same computational state, to a final situation where exactly one node is in a distinguished computational state (called leader) and all others are in the same state (called defeated). The election process may be independently started by any subset of the processors. The election problem occurs, for instance, in token-passing when the token is lost or the owner has failed; in such a case, the remaining processors elect a leader to issue a new token. Several other problems encountered in distributed systems can be solved by election; for example: crash recovery (a new server should be found to continue the service when the previous server has crashed), mutual exclusion (where values for election can be defined as the last time the process entered the critical section), group server (where the choice of a server for an incoming request is made through an election among all the available servers managing a replicated resource), etc. Following failures, the network might be partitioned into several disconnected components (as shown in Figure 1(b)). With respect to the election process, a component will be called active if at least one processor in that component independently starts the election process. A leader election protocol must determine a unique element in each active component; such distinguished elements can then determine any additional information (e.g., size of component, etc.) which is needed for the particular application. The nature of such applications is irrelevant to the election process. It is assumed that every processor p i has a distinct id i chosen from some infinite totally ordered set ID; each processor is only aware of its own identity (in particular, it does not know the identities of its neighbours). The processors all perform the same distributed algorithm. A distributed algorithm (or protocol) is a program that contains three types of executable statements: local computations, message send and message receive statements. We assume that the messages on each arc arrive with no error, in a unbounded but finite delay and in a FIFO order. The complexity measure is the maximum number of messages sent during any possible execution. 1.3 Election in a Faulty Loop Network The Leader Election problem in loop networks has been extensively studied assuming that there are no failures in the systems [4, 18, 23, 24, 32]. The problem becomes rather more di#cult if there are failures in the system. In asynchronous systems, in particular, the election problem is unsolvable (i.e., no deterministic solution protocol exists) if failures are undetectable and can occur at any time; this impossibility result holds even if just one processor may fail (in a fail-stop mode) and follows from the result of [10]. The research has thus focused on studying the problem in more restricted environments . (r1) failures are detectable, # Faults (r3) (r1) (r2) Graph Links Nodes Detectability Occurrence Termination Arbitrary Loop Network Complete Complete Complete (r5) < N/2 per node 0 No intermittent Possible [2, 38] Ring Arbitrary Loop Network unbounded unbounded Yes Prior Possible (this paper) Table 1: Impossibility versus Possibility Results (k and t are constants bounding the number of Fail-Stop Faults). . (r2) failure occurs prior to the execution of the election protocol, . (r3) the number of failures is bounded by some constant, . (r4) failures are fail-stop, . (r5) every processor is directly connected to every processor. All the existing results for Election in faulty loop networks have been developed under assumptions (r2), (r3), (r4) and further assuming that the network is either a complete graph (r5) [1, 16, 28, 31, 37] or a ring [14, 41, 42] (see table 1). So far, without detectability, algorithms breaking free of the bounded number of failures assumption (r3) generate an expensive communication complexity (O(n 2 ) messages of O(n) bits, [19]). In this paper, we consider the Election Problem in asynchronous arbitrary loop networks where an arbitrary number of links has failed and a processor can only detect the status of its incident links. That is, we make assumptions (r2) and (r4), and a relaxed version of assumption (r1). Thus, unlike all previous investigations, we do not restrict to complete graphs; we do not make any a priori restriction on the number of failures; we do however assume that a processor can detect the failure of its incidents links. Note that this assumption, the detectability assumption (r1), is required to cope with an unbounded number of faulty components (see table 1). We prove that, under these assumptions, a Leader Election protocol in a faulty loop network requires only O(n log n) messages in the worst-case, where n is the number of processors. Moreover, we show that this is optimal. In case the failures have partitioned the network, the algorithm will detect it and a distinctive element will be determined in each active component; depending on the application, these distinctive elements can thus take the appropriate actions. Both processors and links may fail. In the following, we will assume that if a processor fails all its incident links fail. Thus, without any loss of generality, we can just consider link failures. We emphasize the fact that both regular and bypass links can fail (as shown in Figure 1(b)). A processor can only detect the failure of its incident links. Knowledge that a link is faulty can be either o# line or on line. In the o# line case, the hardware subsystem provides directly such a knowledge to the processors; thus, this information is a priori respect to the execution of the protocol. In the on line case, this knowledge can only be acquired upon an attempt to transmit on a link; if the link is operational, the message will be transmitted, otherwise an error signal will be issued by the system (see Figure 2). From a computational point of view, the on line case is more di#cult than the o# line one. In particular, to transform it into a priori knowledge case (e.g., by a pre-processing phase where each active processor tests its incident links) would cost an additional m messages where is the number of non-faulty links. Thus, our O(n log n) so- lution, for the case where faults are only detected upon transmission attempt, is all the more important since fault-detection is performed only on these links which are used by the computation. Furthermore, this solution can obviously be applied with the same complexity to the case where there is a priori knowledge on the faulty links. Thus, in following, we will only concentrate on the more di#cult case. The algorithm presented here combines known techniques for election in non-faulty networks ([13, 17, 21]) and original routing paradigms based on structural information [12] in order to avoid the faulty components. The algorithm uses asynchronous promotion steps to merge rooted spanning trees. Election Algorithm We present an Election algorithm in loop network, where an arbitrary number of links have failed and where failure of a link is detectable only if an incident node attempts to transmit on it. The full algorithm is given in the Appendix (see also [26]). Any node can independently and spontaneously start the election process (we will model this by having such a node receive a WAKEUP message). If the network is not partitioned, the algorithm will detect it and will elect a leader. In case the failures have partitioned the network, a distinctive element will be determined in each active component and will detect that a partition has occurred; depending on the application, these distinctive elements can thus take the appropriate actions. We will now describe the algorithm as executed in each active component. 2.1 Description In each active component, the algorithm builds a Rooted Spanning Tree or Kingdom by repeatedly combining smaller spanning trees; the final root of the spanning tree is the distinctive element of that component. In the following, we describe the algorithm as executed in one component. The algorithm proceeds in phases and rounds. Initially, each node is a king, and does not know which of its links have crashed. At the end, all nodes are citizen except one which is still a king. During each intermediate phase of the algorithm, each king tries to expand its kingdom (a rooted directed tree) by attacking another kingdom. The attack is carried out by a particular node: the warrior. Each kingdom is a tree with two distinguished nodes: the king and the warrior. Each king is assigned a level, initialized at zero. Each node p stores the identity king p and the level level p of its king, as well as the label of the outgoing chord to its king and to its war- rior. If a node is attacked, it stores the label of the incoming chord from which the attack came. In the algorithm, each warrior p maintains a local view List p of all the others processors with the indication of which of them belong to the kingdom. An attack message is a request message defined by a request status ReqStatus = (reqking, reqlevel, reqList) which contains such a local view reqList. Informally, the attack is carried out only by a warrior; the warrior will select randomly an outgoing link which leads to another kingdom (one connected to a processor which does not belong to its kingdom). It then attempts to transmit a REQUEST message on that link. If the link is faulty, a failure detection signal will notify the warrior of such a situation and the appropriate action (see below) will be taken; otherwise, the REQUEST message will carry the attack to the other kingdom, as shown in Figure 2. Failure ALGORITHM SYSTEM Attempt Request TRANSMISSION Figure 2: Local Failure Detection The attacks by a kingdom follow a Depth First Search strategy. A state S r for each chord is defined to specify if the chord is unused (initially), branched (is part of the spanning tree) or failed (determined after an attempt of transmission). For each branched chord a substate SubS r is introduced to specify if the chord is closed (is faulty or does not lead to another kingdom), or still opened (the incident node has not been completely explored and thus can lead to nodes which have not been reached yet). Initially, all non-faulty chords are opened. It is used to control the backtracking by closing a subtree whose visit has been completed. If a warrior j cannot reach any node outside the kingdom (this is locally determined by the state of its incident links and the local view List j ), then the state of warrior, together with List j , is backtracked to its parent and the chord between them became closed. This strategy has the main advantage to limit the amount of backtracking after a combination compared to a Breadth First Search strategy. A state transition diagram of a chord is shown in Figure 3(a). Each node saves the label {W out , W in , K out } of the incident chord leading to warrior p , the warrior attacking p, and king p respectively. Define the status p of node p as (level p , king p , List p ). Following a lexicographic total order, we say that status p > status j i#: - either (a) level p > level j - or (b) level king p > king j . Our algorithm obeys two main rules: Promotion Rule. A warrior p can only successfully attack a kingdom with status less than its own. Let the attack by warrior p be successful. In case (a), each node in the kingdom which lost is informed of the identity of the new king king p and updates its level to level p (note that the value of level p is unchanged in the attacking kingdom). In case (b), each node in the attacked kingdom receives the identity king p of the new king and all nodes in both kingdoms increases their level by one (the level of a kingdom never decreases). After a successful attack by a warrior p to a warrior j, the warrior of the new kingdom is warrior j . We say that a processor enters a new round when its level changes, (i.e., when its kingdom has been defeated or when its kingdom successfully attacked a kingdom of an identical level). Asynchronous Rule (controls the number of messages during each phase): three di#erent cases are theoretically possible when an attack from a warrior p reaches a node in another kingdom: 1. status p < status j : the warrior is not strong enough to attack this kingdom and, thus, its attack fails: the message is killed and the attacking kingdom is just waiting to get attacked. 2. status p > status j : the attack from p must be forwarded to warrior j . Any subsequent attack by other kingdoms, if not killed, is delayed until this attack is resolved at j (i.e., until j receives a new status). When forwarding an attack, if node i on the path to warrior j has a greater status (i.e., status i > status p ), the request is killed. This situation occurs when the previously visited nodes have not yet been informed that they have become part of a greater kingdom (i.e., the level has increased). When the attack reaches warrior j, if it still has a lower status, then a surrender message is sent back to warrior p and each node on the path waits for the new status. 3. status proved later, this case (i.e., an attack within the same king- dom) cannot occur during the execution of the algorithm. If warrior p receives a message of surrender, it broadcasts the new status to the absorbed kingdom or to both kingdoms, depending on the promotion rule. The new local view List is obtained by merging the two Lists. The initial local view is a list of bits the list is initialized to 10 # (i.e., all bits are set to 0 except List[0] which is set to 1). Concurrency. The number of concurrent incoming attacks in a kingdom must be limited in order to guarantee a message complexity of O(n) for each round. A substate Substate p for each node p is introduced to specify if the node is WaitingForSurrender (has forwarded an attack message), is WaitingForStatus (has forwarded a surrender message and is waiting for its new level), or is Regular (is ready to receive an attack). The state transition diagram of a processor is shown in Figure 3(b). Some substates are introduced to deal with two specific situations which may occur due to the inherent concurrency of the model. First of all, if a citizen j has forwarded an attack to warrior j a subsequent attack with a greater status will be delayed (wait at j), but not killed (asynchronous rule 2). Secondly, an incoming attack can be received before knowing that the kingdom has already absorbed (or been absorbed by) another kingdom: the level may have increased. In both cases, the citizen knows afterwards (when it receives the new status) if the forwarded attack was successful. At this time, if the status of the forwarded attack is smaller than the new received status, the attack will be killed; thus, the citizen can go Branched opened closed Branched Failed (a) CHORDS Unused Regular Status WaitingFor Surrender WaitingFor (b) NODES Figure 3: State Transition Diagrams back to regular substate. Otherwise, the current attack status is still legal; thus, the inhibition waiting substate must be kept. Progress. The problem occurs if a warrior q receives a surrender message from a warrior when it is already engaged in a wait for status process from a warrior w (q has been attacked by w while attacking p). Consistently with the asynchronous rule, the warrior q has to wait for the new status of warrior w before it can send the new status to the warrior p. The extreme case occurs if - a more complicate scenario involving more nodes can be deduced - w is waiting for p (p has attacked w): a deadlock situation. As proved later in Theorem 2.1, the total lexicographic order on the status forbids the creation of such a waiting cycles. Structural Information. The knowledge of the size of the network, the topology, a globally consistent assignment of labels (or, labelings) to interconnection nodes and communication links is used to reduce the communication cost. Since the loop network is a node-symmetric graph (all its nodes are similar to one another), each node can represent the other nodes by their relative distance along the cycle. This is actually available with the edge labeling and can be used to pass the knowledge of the processors (represented by their distances) that have been already reached: when node p 1 receives a message from node p 2 by the incident chord labeled d 1 , it can unambiguously "decode" the information about other nodes contained in the message. Namely, if the message contains information about the node linked to p 2 by a chord d 2 , then this information refers to the node at distance (d 1 +d 2 ) mod n from p 1 in the ring ordering. This fact will be used to determine whether an unused chord (i.e., on which no messages have been sent) is outgoing or not (that is connected to a di#erent kingdom or not). This function combined with the local view of a processor provides the message with a consistent representation of the kingdom which can be passed from processor to processor. This decoding function corresponds to a circular bit shift by the length of the chord, denoted as transpose (the exact code of the function is given at the end of the algorithm). Termination and Partitioning. The algorithm terminates when the kingdom includes all nodes in its connected non-faulty subgraph. The determination of this event may di#er depending on whether the network is disconnected or not. Consider first the case of a partitioned network. Once all reachable nodes have become part of the kingdom, the king will become warrior (because of the bactracking inherent to the depth first search strategy) and all its incident chords will be closed (there is no outgoing link towards a node which does not belong to the kingdom). At this point, it will detect termination; from its local view, it will also determined the size of its kingdom and that a disconnection has occurred. If the network is not disconnected, the termination detection can occur earlier: as soon as a warrior determines, by its local view, that the kingdom includes all the nodes in the network (the list is full, i.e., set to 1 # ). In both cases, the warrior (which is possibly the king) broadcasts along the tree the termination message. Since this message contains the view of the warrior upon termination, every node in the component can determine whether or not the graph is disconnected as well as which other nodes are in this component. In the case of a disconnection, depending on the application, the king can take the appropriate action. An example of an attack is shown in Figure 5, where the kingdom K has a greater status than the kingdom K # (the corresponding loop network C 16 #3, 8# is shown in Figure 4). The result of the successful attack is shown in Figure 6. Messages Used: . (REQUEST,Status): it is an attack by a warrior, and is forwarded to its adversary. This message is also considered as the first ATTEMPT on the chord, and provides the failure detection if the chord is faulty, . (SURRENDER,Status): it is sent by a defeated warrior to inform the winner of its success, . (NEWSTATUS,Status): it is broadcast by the winner on the appropriate tree (de- pending on promotion rule), . (BRANCH): it is sent by a successful warrior on the chord connecting the two trees, . (BACKTRACK,Status): it is sent by the warrior to its parent when all its chords have been closed, that is when all the nodes reachable through this chord are part of the kingdom or are faulty, . (MOVEWARRIOR,Status): it is sent by the warrior to one of its opened chords after a backtracking, . (TERMINATION): it is broadcast by the sole remaining warrior of the connected component to terminate the execution of the algorithm. Any number of processors can spontaneously start the execution of the algorithm; this is modeled by the reception of a WAKEUP message. The active components are those where at least one processor spontaneously start the algorithm (i.e., it receives a WAKEUP message). KING REQUEST BRANCHED a l c e d f Figure 4: Kingdoms in C 2.2 Correctness The protocol is fully asynchronous, the messages received by each processor and the order in which each processor receives its messages depends on the initial input but is non-determinist. However the algorithm is event-driven with messages processed in first- in-first-out order, the order in which each processor processes its communication relies on tree structures and on the asynchronous and progress rules. The correctness follows after establishing the safety (a warrior never attacks a node of its kingdom), the progress (eventually a tree spans all the nodes of a connected compo- nent), and the appropriate termination (there is exactly one elected node in a connected component of the network). In the following, numbers between parentheses refer to corresponding sections of the algorithm in the Appendix. Lemma 2.1 A request message is initiated by a warrior through an unused opened chord. The request message only traversed citizen nodes and branched chords leading to the warrior of the kingdom traversed. Proof The warrior sends the request (if the attempt is successful) through an unused opened arc (4, 5, 7, and procedure attempt at the end of the description of the algorithm). A citizen (or king) can send a request only upon receipt of a request (1) to forward it to its warrior through links labeled W out , that is, a used chord of a citizen. 2 G C O A J O I K' F A A O C A G L request branched warrior king G Figure 5: Two Kingdoms in C Corollary 2.1 The status of a chord becomes used if a warrior has previously sent a request through it, or if the chord has been detected as faulty. Lemma 2.2 The local view List p at a warrior p represents exactly the list of processors which belong to the kingdom of the warrior p. Proof By induction. Clearly, this is true at the initialization when the local view is set to 10 # . Assuming the local view List p at a warrior w is correct and complete before an attack, the warrior modifies its view either after a successful attack (while receiving a surrender message (8): the warrior becomes a citizen, combines the two views, and pass the warrior privilege of the new combined kingdom to the defeated warrior) or after being defeated (while receiving a newstatus message (7): it receives the view of the winning kingdom, and by combination obtains the complete view of the merged kingdom). In both cases, the new local view contains the exact list of processors of the new kingdom, which proves the induction. 2 Lemma 2.3 (Safety.) A warrior never attacks a node of its kingdom. Proof As shown in Lemma 2.2, an attack can only be done upon receipt of a new status which creates the new list of all the nodes which belong to the kingdom (7). All the chords linked to these nodes are closed, any remaining unused chord, even randomly chosen, leads to a processor of a di#erent kingdom. Therefore, no cycle can be created in the kingdom. 2 Several facts and properties can be observed to clarify the correctness. Fact 2.1 From (1) and the asynchronous rule, a waiting citizen, or king, does not process request messages. A A O C A L G A I F request branched warrior king G Figure of Attack in C Fact 2.2 Eventually each node in a kingdom receives the status of its kingdom. Indeed, at the end of any phase or after being defeated (8), the designated warrior broadcasts the new status along the traversed chords. waiting cycle of requests may be created. Proof Immediate since sending a request does not change the regular state of the warrior (7). Therefore, all the requests which wait on a non regular node do not block the warrior which has initiated them. 2 Theorem 2.1 (Progress.) A deadlock may not be introduced by the waiting which arises when some nodes must wait until some condition holds. Proof The message sending is non-blocking. The only case for which a node is blocked waiting for an event is when a warrior waits for a new status message after sending a surrender (1). Similarly, such a surrender message can be deferred at the successful warrior node if it has surrendered to another warrior attack (8). Repeating this setting, a chain of waiting (on surrender) processors can occur. However, this chain cannot become a circular wait: a surrender message is initialized only on a successful attack, that is when the status of an attacking warrior j is strictly lexicographically larger than the status of a defending warrior p. The total ordering on the status defined by the promotion rule forbids such a waiting cycle of processors: status j < . < status p < . < status j contradicts the definition. 2 Corollary 2.2 Eventually, no node is in a waiting substate. Theorem 2.2 A kingdom is a rooted directed tree. Proof By induction. Initially, each kingdom is a one node tree (0). The kingdom is defined by the subgraph composed by the chords marked K out and their incident nodes, and is rooted by the king. It can also be defined by the subgraph composed by the chords marked W out and their incident nodes: in this case the tree is rooted at the warrior. Following a successful attack, the chord connecting the two trees (the absorbing and the absorbed ones) becomes part of the kingdom upon receipt of a NEWSTATUS message initiated by the winner warrior and broadcast through the absorbed kingdom. The outgoing chord to the king is stored in the K out label. The king has a nil value for K out (0). A node (citizen and/or king (3), warrior (7)) changes its label K out only after receiving a new status message announcing the absorption by another kingdom; in this case K out is set to the incoming arc from which such a message is received. This change of orientation guarantees that the tree is rooted at the new king. Note that a similar observation can be repeated for the tree rooted at the warrior. 2 Lemma 2.5 (Appropriate Termination.) The algorithm terminates with a forest of, at most, one rooted spanning tree for each connected components. Proof By the safety Lemma 2.3, the progress Theorem 2.1, and Theorem 2.2. In each connected component where at least one processor initiated the Election protocol, the algorithm builds a rooted spanning tree. 2 The main Theorem is deduced: Theorem 2.3 The algorithm correctly elects a leader. Proof By Theorem 2.1, Theorem 2.2, and Lemma 2.5, the theorem holds. The Election protocol is independently started by any subset of processors electing a particular node in each active connected component (the king (10)). Each group of processors in a (par- titioned or not) active component forms a consistent view (containing the exact list of reachable processors) with a single elected node: the king. Depending on the application, these distinctive elements can thus take the appropriate actions: e.g., promote themselves leader on a majority basis, wait for the recovery of the faulty components, simulate the non-faulty topology by embedding it into the active connected group, form a restricted (connected) working group,. 2 2.3 Analysis The measure of e#ciency analyzed here is the communication complexity (the number and size of messages sent). Lemma 2.6 The number of rounds is at most log k for each kingdom, if k independent nodes start the algorithm. Proof By the promotion rule, based on a tournament, at most n/2 i nodes enter phase i, in fact k/2 i if k independent nodes start the algorithm. The maximum number of rounds is the maximum value of the level of the winning kingdom, i.e., log k. 2 Corollary 2.3 The number of surrender messages sent by a warrior during a particular execution is at most log k, if k independent nodes start the algorithm. Lemma 2.7 For a given round and a given non-faulty chord l in a kingdom, at most two requests will be transmitted through the chord l. Proof For a given round and a given non-faulty chord l in a kingdom, a request passing through this chord will face several possible outcomes: 1. The request is successful with an identical level: it will cause the round to increase in both kingdom. Any forthcoming requests with this previous level will be discarded at the incident node. 2. The request is successful with a di#erent (i.e., larger) level: the level value is updated only in the absorbed kingdom. By Lemma 2.3, only requests sent by a di#erent kingdom may occur. Another request with the same level will behave as described in the case 1 limiting the number of such occurrences to two. 3. the request is unsuccessful: that is, the message has been killed further on the path to the warrior. This implies that the level has been increased by another attack, but the nodes incident on this chord does not know it yet. By the concurrency rule enforcing delay, only one other request can wait at the incident node and will be discarded when the newstatus arrives. A similar argument can be used for a branched chord between two kingdoms. 2 Corollary 2.4 For a given round and a given non-faulty chord l in a kingdom, at most two surrender (resp. new status) messages will be transmitted through the chord l. More precisely, Theorem 2.4 The total number of messages used by the algorithm does not exceed Proof The number of messages of each kind is the following: sent, at a given round, through at most n - 1 non-faulty chords (see Lemma 2.7). Hence, the total number of such request messages sent during the whole execution is bounded by 2 n log k. sent through a path in a kingdom only before a modification of its level. Hence, the total number of such messages sent during the whole execution is also bounded by 2 n log k. broadcast in the kingdom only to increase its level. Hence, the total number of such messages sent during the whole execution is also bounded by sent on each branched chord of the kingdom, i.e., at most n - 1 messages. sent on a branched chord of the kingdom if the subtree cannot reach further nodes. Hence, the total number of such messages is bounded by the size of the spanning tree, i.e., at most n - 1. sent on each opened-branched chord of the kingdom if the node cannot reach further nodes. Hence, the total number of such messages is also bounded by the size of the spanning tree, i.e., at most n - 1. TERMINATION : at most n - 1 messages.Only seven di#erent types of message exists. The status is composed of: the identity of the king which value is at most m, the level which takes at most log n values, and the List which is a n bits array. Therefore, the size of each message is at most n bits. Theorem 2.5 The algorithm has an optimal worst-case message complexity. Proof Given a loop network C, let F (C) denote the set of the possible combination of links failures in C; clearly the cardinality of F (C) is 2 |E| where E is the set of chords of C. Given f # F (C) denote by M(C, f) the number of messages required to solve the election problem in C when the failures described by f have occurred. Then, the worst case complexity WC(C) to solve the election problem in C after an arbitrary number of link failures is where n is the number of processors, and R n is the ring without bypass; the last equality follows from the lower bound by [6] on rings. 2 2.4 Sensitivity to Absence of Failures The algorithm we have presented uses O(n log n) messages in the worst case, regardless of the amount of faults in the system. Consider now the case where no faults have occurred in the system and an Election is required. If all the nodes had a priori knowledge of this absence of failures, then they could execute an optimal Election protocol for non-faulty networks. In this case, depending on the chord structure, a lower complexity (in some cases, O(n)) can be achieved [4, 18, 23, 24, 32]. However, to achieve this complexity, it is required that the absence of failures is a priori known (more specifically, it is common knowledge [15]) to all processors. Now we show how to achieve the same result without requiring this common-knowledge. First observe that the existing optimal algorithms for election in non-faulty loop networks use only a specific subset of the chords to transmit messages. The basic idea is quite sim- ple. A processor "assumes" that its specific incident arcs are non-faulty. Based on this assumption, it starts the corresponding topology-dependent optimal election algorithm A. If a processor x detects a failure when attempting to transmit a message of protocol A, x will start the execution of the algorithm proposed in section 2. Thus, if there is no failures, algorithm A terminates using MA messages; if there are failures, the overall cost of this strategy is MA +O(n log n) which is O(n log n) since MA # O(n log n). The approach actually leads to a stronger result. To obtain the topology-dependent optimal bound MA for the non-faulty case is su#cient that the chords used by A are fault-free. 3 Extensions and Applications We will consider in this section the election problem in a di#erent setting. In fact, we study arbitrary networks with sense of direction in absence of faults. We show how the previous results presented in this paper can be immediately used to prove the positive impact that the availability of "sense of direction" has on the message complexity of distributed problems in arbitrary fault-free networks. 3.1 Sense of Direction The sense of direction refers to the capability of a processor to distinguish between adjacent communication lines, according to some globally consistent scheme [12, 36]. For example, in a ring network this property is also usually referred to as orientation, which expresses the processor's ability to distinguish between ''left'' and ''right'', where ''left'' means the same to all processors. In oriented tori (i.e., with sense of direction), labelings "up" and "down" are added. The existence of an intuitive labeling based on the dimension provides a sense of direction for hypercube, [11]: each edge between two nodes is labeled on each node by the dimension of the bit of the identity in which they di#er. Similarly, the natural labeling for loop networks discussed in the previous section is a sense of direction. For these networks, the availability of sense of direction has been shown to have some impact on the message complexity of the Election problem. In an arbitrary network, we define a globally consistent labeling on the links by extending in a natural way the existing definitions for particular topologies. Fix a cyclic ordering of the processors. The network has a distance sense of direction if at each processor each incident link is labeled according to the distance in the above cycle to the other node reached by this link. In particular, if the link between processors p and q is labeled by distance d at processor p, this link is labeled by n - d at processor q, where n is the number of processors. An example of sense of direction for an arbitrary network is shown in Figure 7. Note that such a definition intrinsically requires the knowledge of the size n of the network, and it includes as special cases the definition of sense of direction for the topologies referred above: the oriented ring ("left" and "right" correspond to 1 and n - 1, respectively), the oriented complete networks (n set to the number of links plus one), and the oriented loop network or circulant graph. Furthermore, in hypercubes, this sense of direction is derivable in O(N) messages from the traditional one [11]. 3.2 Election in Fault-Free Arbitrary Networks We now consider the impact of sense of direction on the message complexity of the Election problem. A G F (a) (b)177A F G Figure 7: Arbitrary Network (a) with Sense of Direction (b) It is obvious that every graph is a subset of the complete graph; that is, any arbitrary network is an "incomplete" complete graph. Less obvious is the fact that: Every arbitrary network with sense of direction is an "incomplete" loop network. That is, every arbitrary network is a loop network where some edges have been removed. This simple observation have immediate important consequences. It implies that an arbitrary graph with sense of direction is just a faulty loop network (compare Figure 1 and Figure 7): the missing links correspond to the faulty ones. Moreover, in this setting, every processor already know which links are faulty (i.e., missing). As a consequence, the algorithm described in Section 2 is also a solution to the election problem in fault-free arbitrary graphs with sense of direction [25]. By theorem 2.4, it follows that if there is sense of direction, a solution with O(n log n) messages exists for the Election problem. log n) is a lower bound on the message complexity for the election problem in bidirectional ring with sense of direction [6], it follows log n) is also a lower bound on the general case. Thus, the bound is tight. In contrast, in arbitrary networks of n processors where the links have no globally consistent labeling (no sense of log n) messages are required to elect a leader [35], and such a bound is achievable [13]. The importance of the result is that it shows the positive impact of sense of direction on the communication complexity of the Election problem in arbitrary network, confirming the existing results for specific topologies. An interesting consequence of our result follows when comparing it to those obtained assuming that each processor knows all the identities of its neighbours [20, 22]. Namely, it shows that it is possible to obtain the same reduction in message complexity requiring much less information (port labels instead of neighbour's name). Concluding Remarks In this paper, we have presented a #(n log n) solution for the Election problem in loop networks where an arbitrary number of links have failed and a processor can only detect the status of its incident links. If the network is not partitioned, the algorithm will detect it and will elect a leader. In case the failures have partitioned the network, a distinctive element will be determined in each active component and will detect that a partition has occurred; depending on the application, these distinctive elements can thus take the appropriate actions. Moreover, the algorithm is worst-case optimal. All previous results have been established only for complete graphs and have assumed an a priori bound on the number of failures. No e#cient solution has been yet developed for arbitrary circulant graphs when failures are bounded but undetectable. Our result is quite general. In fact, our algorithm can be easily modified to solve the Election problem with the same complexity for fault-free arbitrary networks with sense of direction. --R Election in asynchronous complete networks with intermittent link failures. Analysis of chordal ring. Distributed loop computer networks: a survey. New lower bound techniques for distributed leader finding and other problems on rings of processors. Doubly link ring networks. Designing fault-tolerant systems using automorphisms Impossibility of distributed consensus with one faulty process. Optimal elections in labeled hypercubes. Sense of direction: formal definition and properties. A distributed algorithm for minimum spanning tree. Electing a leader in a ring with link failures. Knowledge and common knowledge in a distributed environ- ment Optimal distributed t-resilient election in complete networks A distributed spanning tree algorithm. Towards optimal distributed election on chordal rings. A distributed election protocol for unreliable networks. A modular technique for the design of e Tight lower and upper bounds for a class of distributed algorithms for a complete network of processors. A fully distributed (minimal) spanning tree algorithm. Election in complete networks with a sense of direction. Optimal distributed algorithms in unlabeled tori and chordal rings. On the impact of sense of direction in arbitrary networks. Optimal fault-tolerant leader election in chordal rings Tolerance of double-loop computer networks to multinode failures Optimal fault-tolerant distributed algorithms for election in complete networks with a global sense of direction Distributed Systems. On reliability analysis of chordal rings. Comments on tolerance of double-loop computer networks to multinode failures Reliable loop topologies for large local computer networks. On the message complexity of distributed problems. Sense of direction Optimal asynchronous agreement and leader election algorithm for complete networks with byzantine faulty links. Leader election in the presence of link failures. A multiple fault-tolerant processor network architecture for pipeline computing Design of a distributed fault-tolerant loop network Faults and Fault-Tolerance in Distributed Systems: the Election problem Election on faulty rings with incomplete size information. --TR --CTR Paola Flocchini , Bernard Mans , Nicola Santoro, Sense of direction in distributed computing, Theoretical Computer Science, v.291 n.1, p.29-53, 4 January

fault tolerance;loop networks;interconnection networks;leader election;sense of direction;distributed algorithms

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Logic Testing of Bridging Faults in CMOS Integrated Circuits.

AbstractWe describe a system for simulating and generating accurate tests for bridging faults in CMOS ICs. After introducing the Primitive Bridge Function, a characteristic function describing the behavior of a bridging fault, we present the Test Guarantee Theorem, which allows for accurate test generation for feedback bridging faults via topological analysis of the feedback-influenced region of the faulty circuit. We present a bridging fault simulation strategy superior to previously published strategies, describe the new test pattern generation system in detail, and report on the system's performance, which is comparable to that of a single stuck-at ATPG system. The paper reports fault coverage as well as defect coverage for the MCNC layouts of the ISCAS-85 benchmark circuits.

Introduction In the search for increased quality of integrated circuits, manufacturers must ensure that shipped parts are actually good. To do this, manufacturers must test for the defects that are likely to occur. Shen, Maly, and Ferguson have performed defect simulation experiments showing that the majority of spot defects in MOS technologies cause shorts and opens [13, 23], and Feltham and Maly have shown that the majority of spot defects in current MOS technologies cause changes in the circuit description that result in shorts [11]. The single stuck-at fault model was adopted because it is powerful and simple, but it was never meant to represent the manner in which circuits behave in the presence of defects. A test set that detects 100% of single stuck-at faults may not detect a high percentage of the manufacturing defects. Ferguson and Shen reported that complete single stuck-at test sets failed to detect up to 10% of the probable shorts in the circuits they examined [13]. The need for tests that detect the electrical behavior exhibited by shorts requires a bridging fault model. The first step to generating bridging fault tests is to decide for which of the approximately potential bridging faults to target (where n is the number of nodes in the circuit). Also necessary is a theoretical foundation for bridging fault simulation and test generation that is simple, general, and easily incorporated into current automatic test pattern generation (ATPG) systems, as well as When this work was performed, Brian Chess was with the Computer Engineering Board, University of California, Santa Cruz CA 95064. His current address is Hewlett-Packard Company, 1501 Page Mill Road MS-6UJ, Palo Alto y Tracy Larrabee's address is Computer Engineering Board of Studies, University of California, Santa Cruz 95064. This work was supported by the Semiconductor Research Corporation under Contract 93-DJ-315 and the National Science Foundation under grant MIP-9011254. implementation techniques that will take the theoretical vision to a complete and accurate system comparable in efficiency to single stuck-at ATPG systems. The rest of this paper will describe a theoretical foundation and a practical system that meets these needs. The next section will define crucial terms and briefly review the differences between the single stuck-at fault model and the bridging fault model, and it will describe the work of previous researchers. Section 2 will present the theoretical foundation that makes the implementation possi- ble. After describing the implementation in detail in Section 3 and reporting on its performance in Section 4, Section 5 will finish by summarizing the new work and describing interesting problems that remain open. 1.1 Definitions and Terms A faulty circuit is an isomorphic copy of an associated fault-free circuit except for the introduction of a change known as a fault. Some input combinations, when applied both to the fault-free circuit and to the faulty circuit, will produce identical outputs: in this case the input combination does not produce a logic error on a circuit output, and it is not a logic test for the introduced fault. If there is no input combination that produces an error, the fault, considered in isolation, can never change the logic function of the circuit: in this case the fault is logically undetectable (it is a redundant fault). In the popular stuck-at fault model, it is assumed that a circuit becomes faulty because a wire has lost its ability to switch values; the wire is stuck high (stuck-at 1) or stuck low (stuck-at 0). If this wire has a permanent value of 0 in the faulty circuit, and an input set causes the corresponding wire in the fault-free circuit to take on the value 0, the input set will create no fault effect. Any wire that has a different value in the faulty circuit and the fault-free circuit carries an error, or has been activated, but this may not cause an error at a circuit output. If the input set produces an error on a circuit output, the activated fault has been propagated to a circuit output. A successful test must activate and propagate a fault. Defects are fabrication anomalies. This paper is concerned with local defects-defects affecting only a small portion of the IC. Local or spot defects are often the result of specks of contaminates on the IC or photolithography during manufacturing. The way a defect affects the circuit's behavior is a fault. It is common for local defects to cause a circuit to behave as if the outputs of two gates, which are not connected in the fault-free circuit, are connected. This model of faulty circuit behavior is the bridging fault model [19]. Changes in behavior can be detected as changes in logical function, excess propagation delay, or excess quiescent power supply current (or any combination). This paper is primarily concerned with faults that cause changes in the logical function of the circuit, but bridging fault detection by monitoring excess quiescent power supply current (I DDQ testing) is an important adjunct to logic testing [2]. I DDQ tests for bridging faults are easy to generate but expensive to apply. The results in Section 4 suggest that it would be appropriate to produce I DDQ test patterns for the bridging faults that are either proved untestable or are aborted. This would provide a small number of I DDQ tests that would significantly increase the percentage of tested defects without the cost of the time on the tester that would be necessary to provide I DDQ tests for all bridging faults. A combinational test for a bridging fault shares the same basic characteristics as a test for a stuck-at fault. To introduce an error, the bridge value must be different from one of the gate outputs in the fault-free circuit; to propagate the error, at least one path of fault effects from the bridge to a circuit output must exist. However, the process of activating and propagating the fault is complicated by the possibility of feedback. If a bridging fault creates a feedback loop, a formerly stable combinational circuit may oscillate or take on sequential characteristics that mask the detection of the fault. It is possible to detect some feedback bridging faults that create sequential behavior with sequences of test vectors [19], but in this case extensive analysis may be required to ensure not only that the feedback element can hold state, but that it is guaranteed to hold state. It can be dangerous to assume that the state element introduced by the fault will achieve a stable digital value. As reported by Abramovici and Menon [1], the vast majority of feedback bridging faults can be detected with single combinational tests. When discussing feedback bridging faults, it is useful to refer to the two bridged wires by their locations in the circuit. Given any path that goes from a circuit input to a circuit output and contains the two bridged wires, the back wire is the wire closest to the circuit inputs on this path, and the front wire is the other bridged wire. 1.2 Previous work When the idea of test generation for bridging faults was new, the assumption that the bridging faults caused wired-AND or wired-OR behavior was good. In the dominant technologies of the time (such as TTL), bridging faults did create wired logic. Abramovici and Menon detailed complete theories and techniques to perform ATPG on bridging faults (including bridging faults that introduced in combinational circuits exhibiting wired-logic behavior [1]. However, wired-logic does not accurately reflect the behavior of bridges in static CMOS circuits [4, 12, 20]. The wired-logic model (wired-AND or wired-OR) is the easiest model to implement for simulation and test pattern generation; with the exception of feedback, the wired-logic model is almost as easy for an ATPG system to deal with as the single stuck-at model. A more exact model would assume that the circuit value at the fault site is described in general by a Boolean function of the inputs to the gates driving the bridged wires. This function could be derived in a number of ways-two notable methods are analog simulation [12, 22] and the voting model [3, 4]. Deriving the Boolean function by simulating the two components with the bridged outputs works well at modeling the upstream components from the fault site, but fails to take into account the possible sensitive behavior of downstream components. An optimistic model assumes that the bridge value is always digitally resolvable (in which case the model might not always be correct). A pessimistic model describes the fault behavior with an incomplete Boolean function, where some of the bridge's behavior falls within a gray region within which the model fails to give an answer [12]. Both of these approaches have been implemented in bridging fault simulators and test pattern generators [12, 20]. A more general model assumes that the analog behavior induced by the fault extends for a certain distance beyond the fault site, after which the circuit behavior is digitally resolvable. The idea that a bridge voltage can be interpreted differently by different downstream gates is known as the Byzantine Generals Problem for bridging faults [5]. The EPROOFS simulator [14] implements this via mixed-mode simulation, where a SPICE-like analog simulation of the region around the fault site is incorporated into a digital simulation of the rest of the circuit. This method provides correct answers when previous models might have failed, in particular for many cases involving feedback bridging faults. EPROOFS results are promising, but EPROOFS is slow compared to stuck-at fault simulation, and the use of a mixed-mode simulator precludes adaptation of the technique for test pattern generation. Although EPROOFs is much more accurate than previous simulators, it still may make errors when accurately predicting the behavior of the faulty circuit requires a timing analysis of the digital logic. There are faster simulators that do EPROOFS-like simulation, although they sacrifice some accuracy for speed [18, 22]. There is currently no test pattern generator that implements such sophisticated models. A feedback bridging fault may create an asynchronous sequential circuit in a formerly combinational network. The state of the circuit may prevent stimulation of the fault, or a stimulated fault may cause oscillation, which may prevent a tester from detecting an error at the circuit outputs. Feedback faults cannot be ignored as they can comprise a sizable percentage of realistic bridging faults. Between 10% and 50% of the realistic bridging faults for the MCNC layouts of the ISCAS-85 circuits are feedback bridging faults. Most approaches to generating tests for feedback bridging faults check for tests invalidated by oscillation or sequential behavior by analyzing the inversion parity between the two bridged wires [1, 20]. Because of reconvergent fanout, the inversion parity may change from one input vector to the next. This means that the inversion parity must be recalculated for every input vector, which is inefficient. Previous successful bridging fault test pattern systems-notably that of Millman and Garvey [20]-generate a test as if there is no feedback and then check to make sure that feedback will not invalidate the test. This can be wasteful: a fault that is undetectable because of feedback could have numerous legitimate tests unless feedback is taken into consideration. It is much more efficient to consider feedback as part of the test generation process. The next section describes the theoretical foundation for the Nemesis ATPG system. Nemesis incorporates arbitrary logical behavior of bridged components via the primitive bridge function and prevents feedback complications during test simulation and generation via the Test Guarantee Theorem. Theoretical Foundations Realistic faults have historically been unpopular candidates for test pattern generation. Modeling the behavior of realistic faults frequently requires the circuit to be treated as an electrical entity rather than a logical one; this is not amenable to standard test generation techniques. This section will describe the theoretical foundation for a practical realistic bridging fault ATPG system. 2.1 The Primitive Bridge Function A bridging fault transforms a portion of the circuit around the bridged wires into a single fault block in the faulty circuit. The extent of the circuit replaced by the fault block is a question of the sophistication of the bridging fault model to be used for test pattern generation. The fault block can range from being a replacement for only the two gates with bridged outputs to being a replacement for the two gates with bridged outputs as well as many downstream gates (and perhaps even gates lying along any possible feedback paths). Figure 1 shows how a bridging fault between the outputs of two NAND gates can create a simple two-component fault block in the faulty circuit, and Figure 2 shows a more inclusive fault block for the same fault that will do a better job of modeling varying logic thresholds of downstream gates. A Faulty Circuit Z Y Y Fault-free Circuit Figure 1: A bridging fault between X and Y creates a simple fault block. F A F A Fault-free Circuit Faulty Circuit Y Figure 2: The same bridging fault between X and Y creates a more general fault block. The function of the fault block depends on its size and on the behavior of the bridged components in the chosen technology. The characteristic function of the fault block is the Primitive Bridge Function or PBF. The PBF can be specified as a truth table or other Boolean representation. Table shows three possible PBFs for the introduced fault block from Figure 1. The column labeled ZWAND shows the fault block output if the technology in question follows the wired-AND model, the column labeled ZWOR shows the fault block output if the technology in question follows the wired-OR model, and the one labeled Z SPICE shows the fault block output derived from circuit analysis of the CMOS standard cell components from the MCNC library. This analysis of the two cells driving the bridge to create the PBF is known as two component simulation. Depending on the accuracy required, the fault block may actually have to replace more than two components; it may need to include downstream gates in order to make sure that the outputs of the fault block are digitally resolved [5]. Two-component simulation can also model arbitrary bridge resistance values by treating discrete bridge resistances as separate faults. For bridging faults that do not introduce any feedback, the output of the PBF is computed with wire values from the fault-free circuit. As presented in the next Section, the PBF for bridging faults that do introduce feedback is computed twice: once with fault-free circuit values, and once Table 1: PBFs from wired-AND, wired-OR, and SPICE simulation of MCNC cells with feedback-influenced values. 2.2 The Test Guarantee Theorem Figure 3 shows a feedback bridging fault with the potential for oscillation when using the SPICE- derived PBF from Table 1. In fact, we know that this circuit, implemented with the MCNC cell library, will not oscillate for any set of inputs because the feedback path is too short. Instead, the bridge will settle to an intermediate voltage favoring the back wire's fault-free value. This result is not predicted by the PBF for the bridge and is dependent on the length of the feedback path. The actual behavior of the bridge in this situation is immaterial: because the PBF does not model the behavior, we cannot reliably use it for detection of the fault. When the circuit has the potential for oscillation. Figure 4 shows a feedback bridging fault with the potential for a test being invalidated because of a previous state. For example, when and the SPICE-derived PBF from Table 1 is used, the outputs of the faulty circuit would be different if the feedback loop had a previous value of 0 than if it had a previous value of 1. In this case, if input X is set to 0, the feedback path is broken, and no previous state could invalidate a test. A Z Figure 3: A feedback bridging fault that might oscillate Z A Figure 4: A feedback bridging fault that may hold state Figure 3 illustrates a situation in which, for certain input values, the back wire will not affect the value on the front wire in the fault-free circuit but will affect the output of the fault block in the faulty circuit. Using the SPICE-derived PBF from Table 1, the potential test should be rejected because it may cause the circuit to oscillate, but if the PBF was for the wired-AND model, the circuit could never oscillate. The method for preventing oscillation is the same as the method for preventing sequential behavior-if an error can be propagated from the back wire without altering the inputs to the PBF such that the PBF changes the value on the bridge, then neither oscillation or sequential behavior will prevent a test regardless of which wire carries the fault. This observation leads to: The Test Guarantee Theorem for feedback bridging faults. If a test creates a situation in which the result of propagating either Boolean value from the back wire causes the PBF to assign the same value to the bridge, the test will not be invalidated because of feedback. Given that the PBF correctly models the behavior of a bridge in the absence of feedback, the PBF can be guaranteed to correctly model the the behavior of a bridge in the presence of feedback only when the feedback does not influence the result of the PBF computation. Since the fault-free circuit is acyclic, the sole source of feedback in the faulty circuit is the back wire of the bridge. If the value on the back wire does not affect the result of the computation of the PBF, then no source of feedback can affect the result of the computation of the PBF, and the PBF correctly models the behavior of the bridge. 2 Like the wired-logic theorems of Abramovici and Menon [1], the Test Guarantee Theorem requires that the feedback loop created by the bridge be broken. But unlike the theorems of Abramovici and Menon, this requirement may not be satisfied simply by stipulating that the back wire not sensitize the front wire in the fault-free circuit; the back wire must not be allowed to sensitize the output of the fault block. If the PBF in use is wired-AND, the new theorems will agree with the Abramovici and Menon theorems; if the PBF in use is more complicated, the theorem provides additional accuracy. Enforcing the additional constraints imposed by the Test Guarantee Theorem involves an analysis of the feedback-influenced region of the circuit. A wire is feedback-influenced if it is on any path between the two bridged wires. If an error is to be propagated from the back wire, the feedback influenced region is a subsection of the faulty region, shown in Figure 5. Analysis of the region consists of applying the PBF to faulty circuit values as well as fault-free circuit values and making sure that the results of the two PBF computations agree. If an error is to be propagated from the front wire, the feedback influenced region is disjoint from the faulty region, as shown in Figure 6. Analysis of the feedback region involves propagating the compliment of the fault-free value of the back wire, and applying the PBF to the resulting values. Oscillation and sequential behavior do not need to be prevented by performing a check after test generation. Instead, independence of previous state and the absence of oscillation can be established as a requirement for test generation. Because the method of preventing oscillation and sequential invalidation are the same, there is no need for an analysis of the inversion parity between the bridged wires. Back Wire Front Wire Faulted Region Feedback Region Circuit Inputs Circuit Outputs Figure 5: Error on the back wire: the feedback region is a subset of the faulty region. Carafe, an Inductive Fault Analysis tool, produces a list of realistic bridging faults-bridging faults that could be caused by a single defect connecting two gate outputs. Carafe considers the layout of the circuit and lists the nodes that are adjacent on the same conducting layer of the circuit or that cross each other on layers separated by a single layer of insulating material[15, 16]. This paper is only concerned with Carafe-extracted faults in the interconnect: shorts involving internal cell lines can also be extracted by Carafe, and they present interesting problems [7], but they are beyond the scope of this paper. Previously, bridging fault ATPG was thought to be unwieldy because of the number of feasible bridging faults and the complexity of the bridging fault model. While the number of possible bridging faults is O(n 2 ) where n is the number of nodes in the circuit, the number of realistic bridging faults is a much more manageable O(n) [2]. Also, if the PBFs are derived from two component simulation, the number of different PBFs needed for fault block analysis is not prohibitive because only one PBF is needed for each type of fault block (and the number of different types will be small for synthesized layouts). Section 4 will compare numbers of stuck-at faults, realistic bridging faults, and two-component PBFs for the MCNC layouts of the ISCAS-85 circuits [6]. Carafe reports the likelihood of occurrence for each fault it extracts. This likelihood indicates how likely the fault is to occur relative to all of the other faults in the list. This means that the ATPG system can report not only what percentage of the realistic bridging faults are tested, but what percentage of the probable bridging defects are tested. The defect coverage should be much more indicative than the fault coverage when it comes to relating test quality to defects per million parts shipped (DPM) [25]. After Carafe determines the realistic bridging faults, SPICE simulation is used to determine the PBF for each fault, and then the Nemesis ATPG system[17] generates tests. Figure 7 shows the organization of the total system. e Feedback Region Circuit Inputs Circuit Outputs Faulted Region Front Wire Figure on the front wire: the feedback region is disjoint from the faulty region. Fault Characterization Cell Descriptions Defect Coverage Test Patterns Fault Types PBFs Layout Defect Statistics Carafe Nemesis Figure 7: System organization 3.1 Simulator Unlike the bridging fault simulator of Abramovici and Menon [1], for which pseudocode is given in Figure 8, the Nemesis method of bridging fault simulation, for which pseudocode is given in Figure 9, does not associate bridging faults with wires; instead, wires are tagged with Boolean values representing whether or not an error can be propagated to a primary output [8]. After attempting to propagate an error from a wire, a field in the wire's data structure is set to reflect the success or failure of the propagation. If a bridging fault further down the fault list introduces an error onto the same wire, it can immediately be determined whether or not the fault can be propagated. Nemesis bridging fault simulation is modeled after the Parallel Pattern Single Fault Propagation (PPSFP) simulator of Waicukauski et al. [24]. Note that, given the PBF for a bridge, the bridge value for each of the parallel patterns is evaluated in the same fashion as that of any other gate (each of which can perform an arbitrary combinational function). In parallel bridging fault simulation, faults can be propagated from both of the wires involved in the bridge at the same time. The fault block does not introduce an error on both of the wires for any input pattern: the error is always on one wire or the other. This means that each bit-slice in the pair of faulty and fault-free wire values may represent an error on one wire or the other, but not both. A wire is placed on the simulation Simulate the fault-free circuit with test vector T foreach wire (W ) involved in a bridging fault if T detects a stuck-at fault on W foreach bridging fault (BF ) associated with W if the PBF for BF places an error on W and meets the TGT Accept test T : BF is detected else test T does not detect BF from wire W Figure 8: Pseudo-code for Abromovici and Menon bridging fault simulation Simulate the fault-free circuit with test vector T foreach bridging fault (BF ) if the PBF for BF places an error on a wire (W ) and meets the TGT if a previous simulation of a fault on W can be used use previous simulation data else simulate fault on W introduced by fault block if the fault introduced by the fault block is detectable accept test T for this fault: BF is detected record results of simulation of fault on W for future use Figure 9: Pseudo-code for Wire Memory bridging fault simulation event queue if its faulty and fault-free values differ in any bit-position-regardless of whether the difference represents a value propagated from the fault on the first wire or the second wire. If the two bridged wires share a significant number of downstream components, the number of individual component simulations can be as little as half the number required by simulators that associate bridging faults with wires. For purposes of comparison, we implemented not only the new method of bridging fault simu- lation, but also the method of Abramovici and Menon, and we used each of them in the Nemesis ATPG system. Comparing the two simulation methods, there are a number of reasons for the greater success of the Wire Memory method. The ability to abort simulations when no errors are moving forward, which can only be done in the Wire Memory method, saves a great deal of time. Also, the data structures needed for the Wire Memory method were easily integrated into a system (such as Nemesis) that treats many different types of faults (such as bridge, I DDQ , stuck-at, and delay) in a similar fashion. Data structure manipulation in the older method is more complex because each fault appears twice (once for each wire that may carry an error). 3.2 Test pattern generator There are two types of feedback faults: given a fault, if every path from the back wire to a primary output goes through the front wire, the fault is a feedback fault with no fanout; If some but foreach feedback bridging fault front wire fault-free if test generation is successful (sequential behavior must be prevented via the TGT) FBF is covered, move to the next fault else front wire fault-free if test generation is successful (sequential behavior must be prevented via the TGT) FBF is covered, move to the next fault else if FBF is a feedback fault with no fanout FBF is undetectable, move to the next fault else back wire fault-free if test generation is successful (oscillation must be prevented via the TGT) FBF is covered, move to the next fault else back wire fault-free if test generation is successful (oscillation must be prevented via the TGT) FBF is covered, move to the next fault else FBF is undetectable Figure 10: Pseudo-code for ATPG for bridging faults that may induce feedback not all of the paths from the back wire to a primary output go through the front wire, the fault is a feedback fault with fanout. It is a consequence of the Test Guarantee Theorem that a feedback fault with no fanout can only be detected with the error placed on the front wire. Pseudo-code for the feedback bridging fault test generator is shown in Figure 10. Each attempt to generate a test for a bridging fault enforces constraints on fault-free values for all wires, on faulty values for wires in between the bridge and a circuit output, and, for feedback bridging faults, on feedback-influenced values [9]. Figures 11 through 14 show a sample bridging fault and demonstrate how Nemesis will show that there is no test that will detect the fault. Notice that the inversion parity between the back and front wire can change depending on circuit input values. This makes it crucial to identify both potential oscillation and sequential behavior for the same fault. The Nemesis ATPG system uses Boolean satisfiability, so constraints are not enforced in a particular order [17]. However, for illustration of each of the four test generation attempts, first initial constraints are shown, then derived activation, justification, and propagation values, and finally-if they are required-constraints having to do with the feedback-influenced values. First, Figure 11 shows the attempt to generate a test such that the fault-free value of the front wire is 0 and the fault block output is 1. The first drawing shows the constraints imposed by the values in the attempted test, and the second drawing shows the direct implications of these values, including the value that the application of the PBF to the fault-free circuit values would place on the bridge (the value shown on the dashed line in the illustration). It is not possible to generate a test because the fault-free circuit values cause the PBF to assign a 0 to the bridge, and the first attempt requires a test with a 1 on the bridge. Similarly, Figure 12 shows the attempt to generate a test such that the fault-free value of the front wire is 1 and the fault block output is 0. Once again the first drawing shows the constraints imposed by the values in the attempted test, and the second drawing shows the direct implications of these values, including the value that the application of Figure 11: Front wire stuck-at 11/01/00 0/000 Figure 12: Front wire stuck-at 0 the PBF to the fault-free circuit values would place on the bridge. It is not possible to generate a test here because the fault effect cannot be propagated through the final NAND gate. The test cannot be generated with an error on the front wire. The test generation process must continue because the fault is a feedback with fanout fault, and the fault effect can be propagated to either circuit output using paths that do not include the front wire. Figure 13 shows the attempt to generate a test such that the fault-free value of the back wire is 0 and the fault block output is 1. Once again the first two drawings show the initial constraints and the direct implications of the constrained values, and the added third drawing shows the results of applying the PBF the second time to the feedback-influenced values, as required by the Test Guarantee Theorem. This second PBF application causes the fault block output to change, which causes the test to be rejected. Figure 14 shows the attempt to generate a test such that the fault-free value of the back wire is 1 and the fault block output is 0. The three drawings are analogous to those in Figure 13, and again a test cannot be found because, just as in Figure 13, the circuit has Figure 13: Back wire stuck-at 1 the potential for oscillation. Because each of the four categories of potential tests for this bridging fault is unworkable, the fault is untestable for the given PBF. The Test Guarantee Theorem fits into an ATPG framework elegantly because it allows the check for feedback or sequential invalidation to occur as a requirement of test generation and not as a postprocessing consistency check. 4 Experimental Results This section presents the results for the UCSC system for testing bridging faults. The two-component PBFs were obtained by SPICE simulation. Bridge voltages were converted to digital values by using the logic threshold of the smallest inverter in the MCNC cell library. We compare the performance of the Nemesis bridging fault ATPG system to that of the Nemesis single stuck-at fault ATPG system, and we compare the performance of our simulator to that of our implementation of the Abramovici and Menon simulator. All times given are CPU times in seconds on a Digital Equipment Corporation Decstation 5000/240. Table 2 shows the number of PBFs, the number of stuck-at faults, the number of total realistic bridging faults, the number of bridging faults with no feedback, and the number of feedback bridging faults for the layouts of the ISCAS-85 benchmark circuits using the MCNC cell library. There are three to nine times as many bridging faults as there are stuck-at faults for the given circuits, so an efficient bridging fault ATPG system might take up to 10 times as long to produce tests for all Figure 14: Back wire stuck-at 0 realistic bridging faults as to produce tests for all single stuck-at faults. The number of PBFs is small compared to the number of faults, and in fact, only 309 different PBFs were used in all ten of the MCNC layouts of the ISCAS-85 benchmarks. The number of feedback bridging faults is a significant percentage of the number of bridging faults: Useful fault coverage could not be achieved without accurate tests for the feedback bridging faults. Table 3 compares the new Wire Memory simulation algorithm with the Abramovici and Menon simulation algorithm for PPSFP random test simulation. The comparison is fair, because any of the optimizations that can possibly be applied to advantage for the Abramovici and Menon method is included in our implementation of their method. The Wire Memory method is almost always faster, and the improvement becomes more striking as the size of the circuits increase. Neither method uses much memory: either implementation can run all of the benchmarks on a machine with megabytes of RAM. Tables 4 and 5 show the number of bridging faults covered, proved untestable, or aborted by the bridging fault and single stuck-at fault ATPG systems, as well as the time in seconds necessary to achieve the reported coverage 1 . For all ten circuits, the bridging fault ATPG system takes an average of 4=3 the time per fault as the single stuck-at system takes, but for most circuits 1 The number of single stuck-at faults for each circuit differs from that reported in the literature because the MCNC versions of the ISCAS-85 circuits are technology-mapped implementations using standard cells. Note that not only will the number of single stuck-at faults change, but the number of untestable faults will also be different. Circuit Stuck-At PBFs Bridging faults Total No Feedback Feedback Table 2: Number of faults and PBFs for each circuit Time in Seconds Circuit Faults covered A & M Wire Mem. C1908 4,684 15.3 13.7 C7552 53,271 435.2 287.2 Table 3: Nemesis random parallel simulation (including four of the five largest circuits), the time per processed bridging fault is less than the time per processed single stuck-at fault. This shows that realistic bridging fault ATPG is an efficient and valuable complement to single stuck-at ATPG. Table 4 also shows the fault coverage and bridging defect coverage for the benchmark circuits. For the ten circuits, Nemesis covers an average of 99.39% of the faults and 99.33% of the defects. Nemesis fails to generate tests for or prove untestable very few of the defects. For example, for the C0432, it generated tests for 98.32% of the realistic defects, it proved 1.62% of the realistic defects combinationally untestable, and it failed to process 0.06% of the defects. But many of these faults can still be tested. For example, 1.68% of realistic defects for the C0432 were untestable or were not processed. The addition of only five I DDQ test patterns will leave only 0.19% of the realistic defects for the C0432 untested (the remaining faults are both logically untestable and untestable via detecting excess I DDQ ). This is fewer than one half of the I DDQ test patterns that would be required to test all of the realistic bridging faults. The fault coverage and the defect coverage generally track each other, but they can differ by significant amounts. Using the C7552 as an example, the difference between 99.46% covered faults Circuit Number of Faults Time % Coverage Covered Untestable Aborted (Secs.) Faults Defects Table 4: Bridging fault test pattern generation coverage Circuit Number of Faults Time Covered Untestable Aborted (Secs.) Table 5: Single stuck-at test pattern generation coverage and 98.94% could make a significant difference in DPM estimation [21]. 5 Summary and conclusions The integrated circuit industry changes at a rapid pace, but one element that does not change is the need for quality. The bridging fault model offers additional rigor to the manufacturing test process by modeling the behavior of faults that are likely to occur. In this paper, we have presented the Primitive Bridge Function-a characteristic function describing the behavior of bridged components; we have provided a theoretical foundation for test pattern generation that correctly handles all bridging faults; we have described and reported on the performance of a test pattern simulator that is faster than previously reported simulators and that accurately simulates all realistic bridging faults; and finally, we have described and reported on the performance of our complete ATPG system, which generates tests that cover at least 98.32% of the realistic bridging defects and an average of 99.33% of the realistic bridging defects in our layouts of the MCNC ISCAS-85 benchmark circuits. The time it takes to generate these tests is comparable to the time necessary to generate single stuck-at test sets for the same circuits. We have shown ATPG for realistic bridging faults to be viable and significant. Future experimentation will involve different and more accurate methods for calculating PBFs- methods that address indeterminate logic values and differing downstream gate input thresholds [5]. We are also investigating shorts on the inside of the cell [7] and bridging fault diagnosis [10]. --R A practical approach to fault simulation and test generation for bridging faults. Testing for bridging faults (shorts) in CMOS circuits. Deriving Accurate Fault Models. Accurate modeling and simulation of bridging faults. Fault model evolution for diagnosis: Accuracy vs precision. A neutral netlist of 10 combinatorial benchmark circuits and a target translator in FORTRAN. Testing CMOS logic gates for realistic shorts. Bridge fault simulation strategies for CMOS integrated circuits. Generating test patterns for bridge faults in CMOS ICs. Diagnosis of realistic bridging faults with single stuck-at information Physically realistic fault models for analog CMOS neural networks. Test pattern generation for realistic bridge faults in CMOS ICs. A CMOS fault extractor for inductive fault analysis. EPROOFS: a CMOS bridging fault simulator. Carafe: An inductive fault analysis tool for CMOS VLSI circuits. Carafe: An inductive fault analysis tool for CMOS VLSI circuits. Test pattern generation using Boolean satisfiability. Biased voting: a method for simulating CMOS bridging faults in the presence of variable gate logic thresholds. IEEE Transactions on Computers An accurate bridging fault test pattern generator. Limitations in predicting defect level based on stuck-at fault coverage Fast and accurate CMOS bridging fault simulation. Inductive fault analysis of MOS integrated circuits. Fault simulation for structured VLSI. Defect level as a function of fault coverage. --TR --CTR Ilia Polian , Piet Engelke , Michel Renovell , Bernd Becker, Modeling Feedback Bridging Faults with Non-Zero Resistance, Journal of Electronic Testing: Theory and Applications, v.21 n.1, p.57-69, January 2005 Baradaran Tahoori, Using satisfiability in application-dependent testing of FPGA interconnects, Proceedings of the 40th conference on Design automation, June 02-06, 2003, Anaheim, CA, USA Application-dependent testing of FPGAs, IEEE Transactions on Very Large Scale Integration (VLSI) Systems, v.14 n.9, p.1024-1033, September 2006 Baradaran Tahoori, Application-Specific Bridging Fault Testing of FPGAs, Journal of Electronic Testing: Theory and Applications, v.20 n.3, p.279-289, June 2004 M. Favalli , C. Metra, Bridging Faults in Pipelined Circuits, Journal of Electronic Testing: Theory and Applications, v.16 n.6, p.617-629, Dec. 2000 Donald Shaw , Dhamin Al-Khalili , Come Rozon, Automatic generation of defect injectable VHDL fault models for ASIC standard cell libraries, Integration, the VLSI Journal, v.39 n.4, p.382-406, July 2006

realistic faults;test pattern generation;fault simulation;fault models;bridging faults

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An Unbiased Detector of Curvilinear Structures.

AbstractThe extraction of curvilinear structures is an important low-level operation in computer vision that has many applications. Most existing operators use a simple model for the line that is to be extracted, i.e., they do not take into account the surroundings of a line. This leads to the undesired consequence that the line will be extracted in the wrong position whenever a line with different lateral contrast is extracted. In contrast, the algorithm proposed in this paper uses an explicit model for lines and their surroundings. By analyzing the scale-space behavior of a model line profile, it is shown how the bias that is induced by asymmetrical lines can be removed. Furthermore, the algorithm not only returns the precise subpixel line position, but also the width of the line for each line point, also with subpixel accuracy.

Introduction Extracting curvilinear structures, often simply called lines, in digital images is an important low-level operation in computer vision that has many applications. In photogrammetric and remote sensing tasks it can be used to extract linear features, including roads, railroads, or rivers, from satellite or low resolution aerial imagery, which can be used for the capture or update of data for geographic information systems [1, 2]. In addition it is useful in medical imaging for the extraction of anatomical features, e.g., blood vessels from an X-ray angiogram [3] or the bones in the skull from a CT or MR image [4]. The published schemes for line detection can be classified into three categories. The first approach detects lines by considering the gray values of the image only [5, 6, 7] and uses purely local criteria, e.g., local gray value differences. Since this will generate many false hypotheses for line points, elaborate and computationally expensive perceptual grouping schemes have to be used to select salient lines in the image [8, 9, 10, 7]. Furthermore, lines cannot be extracted with sub-pixel accuracy. The second approach is to regard lines as objects having parallel edges [11, 12, 13]. In a first step, the local direction of a line is determined for each pixel. Then two edge detection filters are applied in the direction perpendicular to the line, where each filter is tuned to detect either the left or right edge of the line. The responses of each filter are combined in a non-linear way to yield the final response of the operator [11]. The advantage of this approach is that since the edge detection filters are based on the derivatives of Gaussian kernels, the procedure can be iterated over the scale-space parameter oe to detect lines of arbitrary widths. However, because special directional edge detection filters that are not separable have to be constructed, the approach is computationally expensive. The final approach is to regard the image as a function z(x; y) and extract lines from it by using various differential geometric properties of this function. The basic idea behind these algorithms is to locate the positions of ridges and ravines in the image function. These methods can be further divided according to which property they use. The first sub-category defines ridges as the point on a contour line of the image, often also called isohypse or isophote, where the curvature of the contour line has a maximum [4, 14, 15]. One way to do this is to extract the contour lines explicitly, to find the points of maximum curvature on them, and to then link the extracted points into ridges [14]. However, this scheme suffers from two main drawbacks. Firstly, since no contour lines will be found for perfectly flat ridges, such ridges will be labeled as an extended peak. Furthermore, for ridges that have a very low gradient the contour lines will become widely separated, and thus hard to link. Another way to extract the maxima of curvature on the contour lines is to give an explicit formula for that curvature and its direction, and to search for maxima in a curvature image [4, 15]. However, this procedure will also fail for perfectly flat ridges. While Sard's theorem [16] tells us that for generic functions such points will be isolated, they occur quite often in real images and lead to fragmented lines without a semantic reason. Furthermore, the ridge positions found by this operator will often be in wrong positions due to the nature of the differential geometric property used, even for images without noise [4, 17]. In the second sub-category, ridges are found at points where one of the principal curvatures of the image assumes a local maximum [18, 15], which is analogous to the approach taken to define ridges in advanced differential geometry [19]. For lines with a flat profile it has the problem that two separate points of maximum curvature symmetric to the true line position will be found [15]. This is clearly undesirable. In the third sub-category, ridges and ravines are detected by locally approximating the image function by its second or third order Taylor polynomial. The coefficients of this polynomial are usually determined by using the facet model, i.e., by a least squares fit of the polynomial to the image data over a window of a certain size [20, 21, 22, 23, 24, 25]. The direction of the line is determined from the Hessian matrix of the Taylor polynomial. Line points are then found by selecting pixels that have a high second directional derivative perpendicular to the line direction. The advantage of this approach is that lines can be detected with sub-pixel accuracy without having to construct specialized directional filters. However, because the convolution masks that are used to determine the coefficients of the Taylor polynomial are rather poor estimators for the first and second partial derivatives, this approach usually leads to multiple responses to a single line, especially when masks larger than 5 \Theta 5 are used to suppress noise. Therefore, the approach does not scale well and cannot be used to detect lines that are wider than the mask size. For these reasons, a number of line detectors have been proposed that use Gaussian masks to detect the ridge points [26, 27, 15]. These have the advantage that they can be tuned for a certain line width by selecting an appropriate oe. It is also possible to select the appropriate oe for each image point by iterating through scale space [26]. However, since the surroundings of the line are not modeled the extracted line position becomes progressively inaccurate as oe increases. Evidently, very few approaches to line detection consider the task of extracting the line width along with the line position. Most of them do this by an iteration through scale-space while selecting the scale, i.e., the oe, that yields the maximum value to a certain scale-normalized response as the line width [11, 26]. However, this is computationally very expensive, especially if one is only interested in lines in a certain range of widths. Furthermore, these approaches will only yield a relatively rough estimate of the line width since, by necessity, the scale-space is quantized in rather rough intervals. A different approach is given in [28], where, lines and edges are extracted in one simultaneous operation. For each line point two corresponding edge points are matched from the resulting description. This approach has the advantage that lines and their corresponding edges can in principle be extracted with sub-pixel accuracy. However, since a third order facet model is used, the same problems that are mentioned above apply. Furthermore, since the approach does not use an explicit model for a line, the location of the corresponding edge of a line is often not meaningful because the interaction between a line and its corresponding edges is neglected. In this paper, an approach to line detection is presented that uses an explicit model for lines, and various types of line profile models of increasing sophistication are discussed. A scale-space analysis is carried out for each of the models. This analysis is used to derive an algorithm in which lines and their widths can be extracted with sub-pixel accuracy. The algorithm uses a modification of the differential geometric approach described above to detect lines and their corresponding edges. Because Gaussian masks are used to estimate the derivatives of the im- age, the algorithm scales to lines of arbitrary widths while always yielding a single response. Furthermore, since the interaction between lines and their corresponding edges is explicitly modeled, the bias in the extracted line and edge position can be predicted analytically, and can thus be removed. Therefore, line position and width will always correspond to a semantically meaningful location in the image. The outline of the paper is as follows. In Section 2 models for lines in 1D and 2D images are presented and algorithms to extract individual line points are discussed. Section 3 presents an algorithm to link the individual line points into lines and junctions. The extraction of the width of the line is discussed in Section 4. Section 5 describes an algorithm to correct the line position and width to their true values. Finally, Section 6 concludes the paper. 2 Detection of Line Points 2.1 Models for Line Profiles in 1D Many approaches to line detection consider lines in 1D to be bar-shaped, i.e., the ideal line of width 2w and height h is assumed to have a profile given by (1) However, due to sampling effects of the sensor lines often do not have this profile. Figure 1 shows a typical profile of a line in an aerial image, where no flat bar profile is apparent. There- fore, let us first consider lines with a parabolically shaped profile because it will make the derivation of the algorithm clearer and provide us with criteria that should be fulfilled for arbitrary line profiles. The ideal line of width 2w and height h is given by (2) The line detection algorithm will be developed for this type of profile, but the implications of applying it to bar-shaped lines will be considered later on. Image data Approximating profile Figure 1: Profile of a line in an aerial image and approximating parabolic line profile. 2.2 Detection of Lines in 1D In order to detect lines with a profile given by (2) in an image z(x) without noise, it is sufficient to determine the points where z 0 (x) vanishes. However, it is usually convenient to select only salient lines. A useful criterion for salient lines is the magnitude of the second derivative z 00 (x) in the point where z 0 Bright lines on a dark background will have z 00 (x) - 0 while dark lines on a bright background will have z 00 (x) AE 0. Please note that for the ideal line profile Real images will contain a significant amount of noise, and thus the scheme described above is not sufficient. In this case, the first and second derivatives of z(x) should be estimated by convolving the image with the derivatives of the Gaussian smoothing kernel since, under certain, very general, assumtions, it is the only kernel that makes the inherently ill-posed problem of estimating the derivatives of a noisy function well-posed [29, 30]. The Gaussian kernels are given by: oe (x) =p 2-oe oe The responses, i.e., the estimated derivatives, will then be: oe oe oe (x) f p (x) s s Figure 2: Scale-space behaviour of the parabolic line f p when convolved with the derivatives of Gaussian kernels for x 2 [\Gamma3; 3] and oe 2 [0:2; 2]. \Gamma2x(OE oe oe oe oe 4 r 00 oe \Gamma2(OE oe oe oe oe 4 (g 000 oe oe where x Z Equations (6)-(8) give a complete scale-space description of how the parabolic line profile will look like when it is convolved with the derivatives of Gaussian kernels. Figure 2 shows the responses for an ideal line with bright line on a dark background, for x 2 [\Gamma3; 3] and oe 2 [0:2; 2]. As can be seen from this figure, r 0 for all oe. Furthermore, r 00 takes on its maximum negative value at Hence it is possible to determine the precise location of the line for all oe. In addition it can be seen that the ideal line will be flattened out as oe increases as a result of smoothing. This means that if large values for oe are used, the threshold to select salient lines will have to be set to an accordingly smaller value. Let us now consider the more common case of a bar-shaped profile. For this type of profile without noise no simple criterion that depends only on z 0 (x) and z 00 (x) can be given since z 0 (x) and z 00 (x) vanish in the interval [\Gammaw; w]. However, if the bar profile is convolved with the derivatives of the Gaussian kernel, a smooth function is obtained in each case. The responses will be: r- b (x,s,1,1) s r- b (x,s,1,1) s -22 Figure 3: Scale-space behaviour of the bar-shaped line f b when convolved with the derivatives of Gaussian kernels for x 2 [\Gamma3; 3] and oe 2 [0:2; 2]. oe r 00 oe oe oe Figure 3 shows the scale-space behaviour of a bar profile with is convolved with the derivatives of a Gaussian. It can be seen that the bar profile gradually becomes "round" at its corners. The first derivative will vanish only at because of the infinite support of g oe (x). However, the second derivative r 00 b (x; oe; w; h) will not take on its maximum negative value for small oe. In fact, for oe - 0:2w it will be very close to zero. Furthermore, there will be two distinct minima in the interval [\Gammaw; w]. It is, however, desirable for r 00 to exhibit a clearly defined minimum at salient lines are detected by this value. After some lengthy calculations it can be shown that has to hold for this. Furthermore, it can be shown that r 00 b (x; oe; w; h) will have its maximum negative response in scale-space for 3. This means that the same scheme as described above can be used to detect bar-shaped lines as well. However, the restriction on oe must be observed. In addition to this, (11) and (12) can be used to derive how the edges of a line will behave in scale-space. Since this analysis involves equations which cannot be solved analytically, the calculations must be done using a root finding algorithm [31]. Figure 4 shows the location of the line and its corresponding edges for w 2 [0; 4] and oe = 1. Note that the ideal edge positions are given by From (12) it is apparent that the edges of a line can never move closer than oe to the real line, and thus the width of the line will be estimated significantly too large for narrow lines. However, since it is possible to invert the map that describes the edge position, the edges can be localized very precisely once they are extracted from an image. The discussion so far has assumed that lines have the same contrast on both sides, which is rarely the case for real images. For simplicity, only asymetrical bar-shaped lines f a (x) =? ! x Line position Edge position True line width Figure 4: Location of a line with width w 2 [0; 4] and its edges for oe = 1. are considered (a 2 [0; 1]). General lines of height h can be obtained by considering a scaled asymmetrical profile, i.e., hf a (x). However, this changes nothing in the discussion that follows since h cancels out in every calculation. The corresponding responses are given by: r a (x; oe; w; a r 00 a The location where r 0 a the position of the line, is given by This means that the line will be estimated in a wrong position whenever the contrast is significantly different on both sides of the line. The estimated position of the line will be within the actual boundaries of the line as long as a The location of the corresponding edges can again only be computed numerically. Figure 5 gives an example of the line and edge positions for It can be seen that the position of the line and the edges is greatly influenced by line asymmetry. As a gets larger the line and edge positions are pushed to the weak side, i.e., the side that posseses the smaller edge gradient. Note that (18) gives an explicit formula for the bias of the line extractor. Suppose that we knew w and a for each line point. Then it would be possible to remove the bias from the line detection algorithm by shifting the line back into its proper position. Section 5 will describe the solution to this problem. Because the asymmetrical line case is by far the most likely case in any given image it is adopted as the basic model for a line in an image. It is apparent from the analysis above that failure to model the surroundings of a line, i.e., the asymmetry of its edges, can result in large errors of the estimated line position and width. Algorithms that fail to take this into account will fail to return very meaningful results. a x Line Position Edge Position True line width True line position Figure 5: Location of an asymmetrical line and its corresponding edges with width and a 2 [0; 1]. 2.3 Lines in 1D, Discrete Case The analysis so far has been carried out for analytical functions z(x). For discrete signals only two modifications have to be made. The first is the choice of how to implement the convolution in discrete space. Integrated Gaussian kernels were chosen as convolutions masks, mainly because the scale-space analysis of Section 2.2 directly carries over to the discrete case. An additional advantage is that they give automatic normalization of the masks and a direct criterion on how many coefficients are needed for a given approximation error. The integrated Gaussian is obtained if one regards the discrete image z n as a piecewise constant function In this case, the convolution masks will be given by: n;oe For the implementation the approximation error is set to 10 \Gamma4 in each case because for images that contain gray values in the range [0; 255] this precision is sufficient. Of course, other schemes, like discrete analogon of the Gaussian [32] or a recursive computation [33], are suitable for the implementation as well. However, for small oe the scale-space analysis will have to be slightly modified because these filters have different coefficients compared to the integrated Gaussian. The second problem that must be solved is the determination of line location in the discrete case. In principle, one could use a zero crossing detector for this task. However, this would yield the position of the line only with pixel accuracy. In order to overcome this, the second order Taylor polynomial of z n is examined. Let r, r 0 , and r 00 be the locally estimated derivatives at point n of the image that are obtained by convolving the image with g n , and g 00 . Then the Taylor polynomial is given by The position of the line, i.e., the point where The point n is declared a line point if this position falls within the pixel's boundaries, i.e., if and the second derivative r 00 is larger than a user-specified threshold. Please note that in order to extract lines, the response r is unnecessary and therefore does not need to be computed. The discussion of how to extract the edges corresponding to a line point will be deferred to Section 4. 2.4 Detection of Lines in 2D Curvilinear structures in 2D can be modeled as curves s(t) that exhibit a characteristic 1D line profile, i.e., f a , in the direction perpendicular to the line, i.e., perpendicular to s 0 (t). Let this direction be n(t). This means that the first directional derivative in the direction n(t) should vanish and the second directional derivative should be of large absolute value. No assumption can be made about the derivatives in the direction of s 0 (t). For example, let z(x; y) be an image that results from sweeping the profile f a along a circle s(t) of radius r. When this image is convolved with the derivatives of a Gaussian kernel, the second directional derivative perpendicular to s 0 (t) will have a large negative value, as desired. However, the second directional derivative along s 0 (t) will also be non-zero. The only remaining problem is to compute the direction of the line locally for each image point. In order to do this, the partial derivatives r x , r y , r xx , r xy , and r yy of the image will have to be estimated, and this can be done by convolving the image with the following kernels x;oe (x; oe oe (y)g oe (x) (26) xx;oe (x; oe xy;oe (x; oe (x) (28) oe The direction in which the second directional derivative of z(x; y) takes on its maximum absolute value will be used as the direction n(t). This direction can be determined by calculating the eigenvalues and eigenvectors of the Hessian matrix r xy r yy The calculation can be done in a numerically stable and efficient way by using one Jacobi rotation to annihilate the r xy term [31]. Let the eigenvector corresponding to the eigenvalue of maximum absolute value, i.e., the direction perpendicular to the line, be given by (n x 1. As in the 1D case, a quadratic polynomial will be used to determine whether the first directional derivative along (n x ; n y ) vanishes within the current pixel. This point will be given by (a) Input image (b) Line points and their response (c) Line points and their direction Figure points detected in an aerial image (a) of ground resolution 2m. In (b) the line points and directions of (c) are superimposed onto the magnitude of the response. where r xx n 2 y Again, (p x is required in order for a point to be declared a line point. As in the 1D case, the second directional derivative along (n x ; n y ), i.e., the maximum eigenvalue, can be used to select salient lines. 2.5 Example Figures 6(b) and (c) give an example of the results obtainable with the presented approach. Here, bright line points were extracted from the input image given in Fig. 6(a) with This image is part of an aerial image with a ground resolution of 2 m. The sub-pixel location of the line points and the direction (n x ; n y ) perpendicular to the line are symbolized by vectors. The strength of the line, i.e., the absolute value of the second directional derivative along (n x ; n y ) is symbolized by gray values. Line points with high saliency have dark gray values. From figure 6 it might appear, if an 8-neighborhood is used, that the proposed approach returns multiple responses to each line. However, when the sub-pixel location of each line point is taken into account it can be seen that there is always a single response to a given line since all line point locations line up prefectly. Therefore, linking will be considerably easier than in approaches that yield multiple responses, e.g., [27, 21, 22], and no thinning operation is needed [34]. 3 Linking Line Points into Lines After individual line pixels have been extracted, they need to be linked into lines. It is necessary to do this right after the extraction of the line points because the later stages of determining line width and removing the bias will require a data structure that uses the notion of a left and right side of an entire line. Therefore, the normals to the line have to be oriented in the same manner as the line is traversed. As is evident from Fig. 6, the procedure so far cannot do this since line points are regarded in isolation, and thus preference between two valid directions n(t) is not made. 3.1 Linking Algorithm In order to facilitate later mid-level vision processes, e.g., perceptual grouping, the data structure that results from the linking process should contain explicit information about the lines as well as the junctions between them. This data structure should be topologically sound in the sense that junctions are represented by points and not by extended areas as in [21] or [23]. Fur- thermore, since the presented approach yields only single responses to each line, no thinning operation needs to be performed prior to linking. This assures that the maximum information about the line points will be present in the data structure. Since there is no suitable criterion to classify the line points into junctions and normal line points in advance without having to resort to extended junction areas another approach has been adopted. From the algorithm in Section 2 the following data are obtained for each pixel: the orientation of the line (n x sin ff), a measure of strength of the line (the second directional derivative in the direction of ff), and the sub-pixel location of the line (p x Starting from the pixel with maximum second derivative, lines will be constructed by adding the appropriate neighbor to the current line. Since it can be assumed that the line point detection algorithm will yield a fairly accurate estimate for the local direction of the line, only three neighboring pixels that are compatible with this direction are examined. For example, if the current pixel is and the current orientation of the line is in the interval [\Gamma22:5 only the points (c x +1; c y \Gamma1), are examined. The choice regarding the appropriate neighbor to add to the line is based on the distance between the respective sub-pixel line locations and the angle difference of the two points. Let be the distance between the two points and j, such that fi 2 [0; -=2], be the angle difference between those points. The neighbor that is added to the line is the one that minimizes In the current implementation, used. This algorithm will select each line point in the correct order. At junction points, it will select one branch to follow without detecting the junction, which will be detected later on. The algorithm of adding line points is continued until no more line points are found in the current neighborhood or until the best matching candidate is a point that has already been added to another line. If this happens, the point is marked as a junction, and the line that contains the point is split into two lines at the junction point. New lines will be created as long as the starting point has a second directional derivative that lies above a certain, user-selectable upper threshold. Points are added to the current line as long as their second directional derivative is greater than another user-selectable lower threshold. This is similar to a hysteresis threshold operation [35]. The problem of orienting the normals n(t) of the line is solved by the following procedure. Firstly, at the starting point of the line the normal is oriented such that it is turned \Gamma90 ffi to the direction the line is traversed, i.e., it will point to the right of the starting point. Then at each line point there are two possible normals whose angles differ by 180 ffi . The angle that minimizes the difference between the angle of the normal of the previous point and the current point is chosen (a) Linked lines and junctions (b) Lines and oriented normals Figure 7: Linked lines detected using the new approach (a) and oriented normals (b). Lines are drawn in white while junctions are displayed as black crosses and normals as black lines. as the correct orientation. This procedure ensures that the normal always points to the right of the line as it is traversed from start to end. With a slight modification the algorithm is able to deal with multiple responses if it is assumed that no more than three parallel responses are generated. For the facet model, for exam- ple, no such case has been encountered for mask sizes of up to 13 \Theta 13. Under this assumption, the algorithm can proceed as above. Additionally, if there are multiple responses to the line in the direction perpendicular to the line, e.g., the pixels in the example above, they are marked as processed if they have roughly the same orientation as termination criterion for lines has to be modified to stop at processed line points instead of line points that are contained in another line. 3.2 Example Figure 7(a) shows the result of linking the line points in Fig. 6 into lines. The results are overlaid onto the original image. In this case, the upper threshold was set to zero, i.e., all lines, no matter how faint, were selected. It is apparent that the lines obtained with the proposed approach are very smooth and the sub-pixel location of the line is quite precise. Figure 7(b) displays the way the normals to the line were oriented for this example. 3.3 Parameter Selection The selection of thresholds is very important to make an operator generally useable. Ideally, semantically meaningful parameters should be used to select salient objects. For the proposed line detector, these are the line width w and its contrast h. However, as was described above, salient lines are defined by their second directional derivative along n(t). To convert thresholds on w and h into thresholds the operator can use, first a oe should be chosen according to (13). Then, oe, w, and h can be plugged into (12) to yield an upper threshold for the operator. Figure 8 exemplifies this procedure and shows that the presented line detector can be scaled (a) Aerial image (b) Detected lines Figure 8: Lines detected (b) in an aerial image (a) of ground resolution 1m. arbitrarily. In Fig. 8(a) a larger part of the aerial image in Fig. 7 is displyed, but this time with a ground resolution of 1 m, i.e., twice the resolution. If 7 pixel wide lines are to be detected, i.e., 3:5, according to (13), a oe - 2:0207 should be selected. In fact, oe = 2:2 was used for this image. If lines with a contrast of h - 70 are to be selected, (12) shows that these lines will have a second derivative of - \Gamma5:17893. Therefore, the upper threshold for the absolute value of the second derivative was set to 5, while the lower threshold was 0:8. Figure 8(b) displays the lines that were detected with these parameters. As can be seen, all of the roads were detected. 4 Determination of the Line Width The width of a line is an important feature in its own right. Many applications, especially in remote sensing tasks, are interested in obtaining the width of an object, e.g., a road or a river, as precisely as possible. Furthermore, the width can, for instance, be used in perceptual grouping processes to avoid the grouping of lines that have incompatible widths. However, the main reason that width is important in the proposed approach is that it is needed to obtain an estimate of the true line width such that the bias that is introduced by asymmetrical lines can be removed. 4.1 Extraction of Edge Points From the discussion in Section 2.2 it follows that a line is bounded by an edge on each side. Hence, to detect the width of the line, for each line point the closest points in the image, to the left and to the right of the line point, where the absolute value of the gradient takes on its maximum value need to be determined. Of course, these points should be searched for exclusively along a line in the direction n(t) of the current line point. Only a trivial modification Figure 9: Lines and their corresponding edges in an image of the absolute value of the gradient. of the Bresenham line drawing algorithm [36] is necessary to yield all pixels that this line will intersect. The analysis in Section 2.2 shows that it is only reasonable to search for edges in a restricted neighborhood of the line. Ideally, the line to search would have a length of 3oe. In order to ensure that almost all of the edge points are detected, the current implementation uses a slightly larger search line length of 2.5oe. In an image of the absolute value of the gradient of the image, the desired edges will appear as bright lines. Figure 9 exemplifies this for the aerial image of Fig. 8(a). In order to extract the lines from the gradient image x y where the following coefficients of a local Taylor polynomial need to be computed: e e xx xy x e e xy yy y e This has three main disadvantages. First of all, the computational load increases by almost a factor of two since four additional partial derivatives with slightly larger mask sizes have to be Figure 10: Comparison between the locations of edge points extracted using the exact formula (black crosses) and the 3 \Theta 3 facet model (white crosses). computed. Furthermore, the third partial derivatives of the image would need to be used. This is clearly undesirable since they are very susceptible to noise. Finally, the expressions above are undefined whenever e(x; However, since the only interesting characteristic of the Taylor polynomial is the zero crossing of its first derivative in one of the principal directions, the coefficients can be multiplied by e(x; y) to avoid this problem. It might appear that an approach to solve these problems would be to use the algorithm to detect line points described in Section 2 on the gradient image in order to detect the edges of the line with sub-pixel accuracy. However, this would mean that some additional smoothing would be applied to the gradient image. This is undesireable since it would destroy the correlation between the location of the line points and the location of the corresponding edge points. Therefore, the edge points in the gradient image are extracted with a facet model line detector which uses the same principles as described in Section 2, but uses different convolution masks to determine the partial derivatives of the image [21, 20, 34]. The smallest possible mask size (3 \Theta 3) is used since this will result in the most accurate localization of the edge points while yielding as little of the problems mentioned in Section 1 as possible. It has the additional benefit that the computational costs are very low. Experiments on a large number of images have shown that if the coefficients of the Taylor polynomial are computed in this manner, they can, in some cases, be significantly different than the correct values. However, the positions of the edge points, especially those of the edges corresponding to salient lines, will only be affected very slightly. Figure 10 illustrates this on the image of Fig. 6(a). Edge points extracted with the correct formulas are displayed as black crosses, while those extracted with the 3 \Theta 3 facet model are displayed as white crosses. It is apparent that because third derivatives are used in the correct formulas there are many more spurious responses. Furthermore, five edge points along the salient line in the upper middle part of the image are missed because of this. Finally, it can be seen that the edge positions corresponding to salient lines differ only minimally, and therefore the approach presented here seems to be justified. 4.2 Handling of Missing Edge Points One final important issue is what the algorithm should do when it is unable to locate an edge point for a given line point. This might happen, for example, if there is a very weak and wide gradient next to the line, which does not exhibit a well defined maximum. Another case where this typically happens are the junction areas of lines, where the line width usually grows beyond the range of 2:5oe. Since the algorithm does not have any other means of locating the edge points, the only viable solution to this problem is to interpolate or extrapolate the line width from neighboring line points. It is at this point that the notion of a right and a left side of the line, i.e., the orientation of the normals of the line, becomes crucial. The algorithm can be described as follows. First of all, the width of the line is extracted for each line point. After this, if there is a gap in the extracted widths on one side of the line, i.e., if the width of the line is undefined at some line point, but there are some points in front and behind the current line point that have a defined width, the width for the current line point is obtained by linear interpolation. This can be formalized as follows. Let i be the index of the last point and j be the index of the next point with a defined line width, respectively. Let a be the length of the line from i to the current point k and b be the total line length from i to j. Then the width of the current point k is given by a This scheme can easily be extended to the case where either i or j are undefined, i.e., the line width is undefined at either end of the line. The algorithm sets w in this case, which means that if the line width is undefined at the end of a line, it will be extrapolated to the last defined line width. 4.3 Examples Figure 11(b) displays the results of the line width extraction algorithm for the example image of Fig. 8. This image is fairly good-natured in the sense that the lines it contains are rather symmetrical. From Fig. 11(a) it can be seen that the algorithm is able to locate the edges of the wider line with very high precision. The only place where the edges do not correspond to the semantic edges of the road object are in the bottom part of the image, where nearby vegetation causes a strong gradient and causes the algorithm to estimate the line width too large. Please note that the width of the narrower line is extracted slightly too large, which is not surprising when the discussion in Section 2.2 is taken into account. Revisiting Fig. 4 again, it is clear that an effect like this is to be expected. How to remove this effect is the topic of Section 5. A final thing to note is that the algorithm extrapolates the line width in the junction area in the middle of the image, as discussed in Section 4.2. This explains the seemingly unjustified edge points in this area. Figure 12(b) exhibits the results of the proposed approach on another aerial image of the same ground resolution, given in Fig. 12(a). Please note that the line in the upper part of the image contains a very asymmetrical part in the center part of the line due to shadows of nearby objects. Therefore, as is predictable from the discussion in Section 2.2, especially Fig. 5, the line position is shifted towards the edge of the line that posesses the weaker gradient, i.e., the (a) Aerial image (b) Detected lines and their width Figure 11: Lines and their width detected (b) in an aerial image (a). Lines are displayed in white while the corresponding edges are displayed in black. (a) Aerial image (b) Detected lines and their width Figure 12: Lines and their width detected (b) in an aerial image (a). upper edge in this case. Please note also that the line and edge positions are very accurate in the rest of the image. 5 Removing the Bias from Asymmetric Lines 5.1 Detailed Analysis of Asymmetrical Line Profiles Recall from the discussion at the end of Section 2.2 that if the algorithm knew the true values of w and a it could remove the bias in the estimation of the line position and width. Equations (15)- give an explict scale-space description of the asymmetrical line profile f a . The position l of the line can be determined analytically by the zero-crossings of r 0 a (x; oe; w; a) and is given in (18). The total width of the line, as measured from the left to right edge, is given by the zero-crossings of r 00 a (x; oe; w; a). Unfortunately, these positions can only be computed by a root finding algorithm since the equations cannot be solved analytically. Let us call these positions e l and e r . Then the width to the left and right of the line is given by v l The total width of the line is . The quantities l, e l , and e r have the following useful property: Proposition 1 The values of l, e l , and e r form a scale-invariant system. This means that if both oe and w are scaled by the same constant factor c the line and edge locations will be given by cl, ce l , and ce r Proof: Let l 1 be the line location for oe 1 and w 1 for an arbitrary, but fixed a. Let oe and . Then l Hence we have l cl 1 Now let e 1 be one of the two solution of r 00 a or e r , and likewise for , with oe 1;2 and w 1;2 as above. This expression can be transformed to (a \Gamma 1)(e 2 =oe 2). If we plug in oe 2 and w 2 , we see that this expression can only be fulfilled for e ce 1 since only then will the factors c cancel everywhere. 2 Of course, this property will also hold for the derived quantities v l , and v. The meaning of Proposition 1 is that w and oe are not independent of each other. In fact, we only need to consider all w for one particular oe, e.g., oe = 1. Therefore, for the following analysis we only need to discuss values that are normalized with regard to the scale oe, i.e., v=oe, and so on. A useful consequence is that the behaviour of f a can be analyzed for oe = 1. All other values can be obtained by a simple multiplication by the actual scale oe. With all this being established, the predicted total line width v oe can be calculated for all w oe and a 2 [0; 1]. Figure 13 displays the predicted v oe for w oe 2 [0; 3]. It be seen that v oe can grow without bounds for w oe # 0 or a " 1. Furthermore, it can be proved that v oe Therefore, in Fig. 13 the contour lines for v oe 2 [2; 6] are also displayed. Section 4 gave a procedure to extract the quantity v oe from the image. This is half of the information required to get to the true values of w and a. However, an additional quantity is needed to estimate a. Since the true height h of the line profile hf a is unknown this quantity needs to be independent of h. One such quantity is the ratio of the gradient magnitude at e r and e l , i.e., the weak and strong side. This quantity is given by a a It is obvious that the influence of h cancels out. Furthermore, it is easy to convince oneself that r also remains constant under simultaneous scalings of oe and w. The quantity r has the advantage that it is easy to extract from the image. Figure 13 displays the predicted r for w oe [0; 3]. It is Predicted line width v s5.65.24.84.443.63.22.82.40 0.51.5 23 line width Predicted gradient ratio r0.80.60.40.20123 (b) Predicted gradient ratio Figure 13: Predicted behaviour of the asymmetrical line f a for w oe 2 [0; 3] and a 2 [0; 1]. (a) Predicted line width v oe . (b) Predicted gradient ratio r. obvious that r 2 [0; 1]. Therefore, the contour lines for r in this range are displayed in Figure 13 as well. It can be seen that for large w oe , r is very close to 1 \Gamma a. For small w oe it will drop to near-zero for all a. 5.2 Inversion of the Bias Function The discussion above can be summarized as follows: The true values of w oe and a are mapped to the quantities v oe and r, which are observable from the image. More formally, there is a function From the discussion in Section 4 it follows that it is only useful to consider v oe However, for very small oe it is possible that an edge point will be found within a pixel in which the center of the pixel is less than 2:5oe from the line point, but the edge point is farther away than this. Therefore, v oe [0; 6] is a good restriction for v oe . Since the algorithm needs to determine the true values (w oe ; a) from the observed (v oe ; r), the inverse f \Gamma1 of the map f has to be determined. Figure 14 illustrates that f is invertible. It displays the contour lines of v oe 2 [2; 6] and r 2 [0; 1]. The contour lines of v oe are U-shaped with the tightest U corresponding to v 2:1. The contour line corresponding to actually only the point (0; 0). The contour lines for r run across with the lowermost visible contour line corresponding to 0:95. The contour line for lies completely on the w oe -axis. It can be seen that, for any pair of contour lines from v oe and r, there will only be one intersection point. Hence, f is invertible. To calculate f \Gamma1 , a multi-dimensional root finding algorithm has to be used [31]. To obtain maximum precision for w oe and a, this root finding algorithm would have to be called at each line point. This is undesirable for two reasons. Firstly, it is a computationally expensive operation. More importantly, however, due to the nature of the function f , very good starting values are required for the algorithm to converge, especially for small v oe . Therefore, the inverse f \Gamma1 is computed for selected values of v oe and r and the true values are obtained by interpolation. The step size of v oe was chosen as 0:1, while r was sampled at 0:05 intervals. Hence, the intersection points of the contour lines in Fig. 14 are the entries in the table of f \Gamma1 . Figure 15 shows the true values of w oe and a for any given v oe and r. It can be seen that despite the fact that f is very a Figure 14: Contour lines of v oe 2 [2; 6] and r 2 [0; 1]. True w s r0.51.52.5(a) True w oe True a2.53.54.55.5v s0.20.610.20.61 (b) True a Figure 15: True values of the line width w oe (a) and the asymmetry a (b). ill-behaved for small w oe f \Gamma1 is quite well-behaved. This behaviour leads to the conclusion that linear interpolation can be used to obtain good values for w oe and a. One final important detail is how the algorithm should handle line points where v oe ! 2, i.e., f \Gamma1 is undefined. This can happen, for example, because the facet model sometimes gives a multiple response for an edge point, or because there are two lines very close to each other. In this case the edge points cannot move as far outward as the model predicts. If this happens, the line point will have an undefined width. These cases can be handled by the procedure given in Section 4.2 that fills such gaps. 5.3 Examples Figure shows how the bias removal algorithm is able to succesfully adjust the line widths in the aerial image of Fig. 11. Please note from Fig. 16(a) that because the lines in this image are fairly symmetrical, the line positions have been adjusted only minimally. Furthermore, it can be seen that the line widths correspond much better to the true line widths. Figure 16(b) shows a) Lines detected with bias removal (b) Detail of (a) (c) Detail of (a) without bias removal Figure Lines and their width detected (a) in an aerial image of resolution 1m with the bias removed. A four times enlarged detail (b) superimposed onto the original image of resolution m. (c) Comparison to the line extraction without bias removal. a four times enlarged part of the results superimposed onto the image in its original ground resolution of 0.25 m, i.e., four times the resolution in which the line extraction was carried out. For most of the line the edges are well within one pixel of the edge in the larger resolution. Figure 16(c) shows the same detail without the removal of the bias. In this case, the extracted edges are about 2-4 pixels from their true locations. The bottom part of Fig. 16(a) shows that sometimes the bias removal can make the location of one edge worse in favor of improving the location of the other edge. However, the position of the line is affected only slightly. a) Lines detected with bias removal (b) Detail of (a) (c) Detail of (a) without bias removal Figure 17: Lines and their width detected (a) in an aerial image of resolution 1m with the bias removed. A four times enlarged detail (b) superimposed onto the original image of resolution m. (c) Comparison to the line extraction without bias removal. Figure 17 shows the results of removing the bias from the test image of Fig. 12. Please note that in the areas of the image where the line is highly asymmetrical the line and edge locations are much improved. In fact, for a very large part of the road the line position is within one pixel of the road markings in the center of the road in the high resolution image. Again, a four times enlarged detail is shown in Fig. 17(b). If this is compared to the detail in Fig. 17(c) the significant improvement in the line and edge locations becomes apparent. The final example in the domain of aerial images is a much more difficult image since it contains much structure. Figure 18(a) shows an aerial image, again of ground resolution 1 m. This image is very tough to process correctly because it contains a large area where the model of the line does not hold. There is a very narrow line on the left side of the image that has a very strong asymmetry in its lower part in addition to another edge being very close. Furthermore, in its upper part the house roof acts as a nearby line. In such cases, the edge of a line can only move outward much less than predicted by the model. Unfortunately, due to space limitations this property cannot be elaborated here. Figure 18(b) shows the result of the line extraction algorithm with bias removal. Since in the upper part the line edges cannot move as far outward as the model predicts, the width of the line is estimated as almost zero. The same holds for the lower part of the line. The reason that the bias removal corrects the line width to near zero is that small errors in the width extraction lead to a large correction for very narrow lines, i.e., if v oe is close to 2, as can be seen from Fig. 13(a). Please note, however, that the algorithm is still able to move the line position to within the true line in its asymmetrical part. This is displayed in Figures 18(c) and (d). The extraction results are enlarged by a factor of two and superimposed onto the original image of ground resolution 0.25 m. Please note also that despite the fact that the width is estimated incorrectly the line positions are not affected by this, i.e., they correspond very closely to the true line positions in the whole image. The next example is taken from the domain of medical imaging. Figure 19(a) shows a magnetic resonance (MR) image of a human head. The results of extracting bright lines with bias removal are displayed in Fig. 19(b), while a three times enlarged detail from the left center of the image is given in Fig. 19(c). The extracted line positions and widths are very good throughout the image. Whether or not they correspond to "interesting" anatomical features is application dependent. Note, however, that the skull bone and several other features are extracted with high precision. Compare this to Fig. 19(d), where the line extraction was done without bias removal. Note that the line positions are much worse for the gyri of the brain since they are highly asymmetrical lines in this image. The final example is again from the domain of medical imaging, but this time the input is an X-ray image. Figure 20 shows the results of applying the proposed approach to a coronary angiogram. Since the image in Fig. 20(a) has very low contrast, Fig. 20(b) shows the same image with higher contrast. Figure 20(c) displays the results of extracting dark lines from Fig. 20(a), the low contrast image, superimposed onto the high contrast image. A three times enlarged detail is displayed in Fig. 20(d). In particular, it can be seen that the algorithm is very succesful in delineating the vascular stenosis in the central part of the image. Note also that the algorithm was able to extract a large part of the coronary artery tree. The reason that some arteries were not found is that very restrictive thresholds were set for this example. Therefore, it seems that the presented approach could be used in a system like the one described in [3] to extract complete coronary trees. However, since the presented algorithm does not generate many false hypotheses, and since the extracted lines are already connected into lines and junctions, no complicated perceptual grouping would be necessary, and the rule base would only need to eliminate false arteries, and could therefore be much smaller. a) Input image (b) Lines detected with bias removal (c) Detail of (b) (d) Detail of (b) without bias removal Figure 18: Lines and their width detected (b) in an aerial image of resolution 1m (a) with bias removal. A two times enlarged detail (c) superimposed onto the original image of resolution m. (d) Comparison to the line extraction without bias removal. 6 Conclusions This paper has presented an approach to extract lines and their widths with very high precision. A model for the most common type of lines, the asymmetrical bar-shaped line, was developed from simpler types of lines, namely the parabolic and symmetrical bar-shaped line. A scale-space analysis was carried out for each of these model profiles. This analysis shows that there is a strong interaction between a line and its two corresponding edges which cannot be ignored. The true line width influences the line width occuring in an image, while asymmetry influ- a) Input image (b) Lines detected with bias removal (c) Detail of (b) (d) Detail of (b) without bias removal Figure 19: Lines and their width detected (b) in a MR image (a) with the bias removed. A three times enlarged detail (c) superimposed onto the original image. (d) Comparison to the line extraction without bias removal. ences both the line width and its position. From this analysis an algorithm to extract the line position and its width was derived. This algorithm exhibits the bias that is predicted by the model for the asymmetrical line. Therefore, a method to remove this bias was proposed. The resulting algorithm works very well for a range of images containing lines of different widths and asymmetries, as was demonstrated by a number of test images. High resolution versions of the test images were used to check the validity of the obtained results. They show that the proposed approach is able to extract lines with very high precision from low resolution images. The extracted line positions and edges correspond to semantically meaningful entities in the im- (a) Input image (b) Higher contrast version of (a) (c) Lines and their widths detected in (a) (d) Detail of (c) Figure 20: Lines detected in the coronary angiogram (a). Since this image has very low con- trast, the results (c) extracted from (a) are superimposed onto a version of the image with better contrast (b). A three times enlarged detail of (c) is displayed in (d). age, e.g., road center lines and roadsides or blood vessels. Although the test images used were mainly aerial and medical images, the algorithm can be applied in many other domains as well, e.g., optical character recognition [23]. The approach only uses the first and second directional derivatives of an image for the extraction of the line points. No specialized directional filters are needed. The edge point extraction is done by a localized search around the line points already found using five very small masks. This makes the approach computationally very efficient. For example, the time to process the MR image of Fig. 19 of size 256 \Theta 256 is about 1.7 seconds on a HP 735 workstation. The presented approach shows two fundamental limitations. First of all, it can only be used to detect lines with a certain range of widths, i.e., between 0 and 2:5oe. This is a problem if the width of the important lines varies greatly in the image. However, since the bias is removed by the algorithm, one can in principle select oe large enough to cover all desired line widths and the algorithm will still yield valid results. This will work if the narrow lines are relatively salient. Otherwise they will be smoothed away in scale-space. Of course, once oe is selected so large that neighboring lines will start to influence each other the line model will fail and the results will deteriorate. Hence, in reality there is a limited range in which oe can be chosen to yield good results. In most applications this is not a very significant restriction since one is usually only interested in lines in a certain range of widths. Furthermore, the algorithm could be iterated through scale-space to extract lines of very different widths. The second problem is that the definition of salient lines is done via the second directional derivatives. However, one can plug semantically meaningful values, i.e., the width and height of the line, as well as oe, into (12) to obtain the desired thresholds. Again, this is not a severe restriction of the algorithm, but only a matter of convenience. Finally, it should be stressed that the lines extracted are not ridges in the topographic sense, i.e., they do not define the way water runs downhill or accumulates [17, 37]. In fact, they are much more than a ridge in the sense that a ridge can be regarded in isolation, while a line needs to model its surroundings. If a ridge detection algorithm is used to extract lines, the asymmetry of the lines will invariably cause it to return biased results. --R Update of roads in GIS from aerial imagery: Verification and multi-resolution extraction An artificial vision system for X-ray images of human coronary trees Detection of roads and linear structures in low-resolution aerial imagery using a multisource knowledge integration technique Tracking roads in satellite images by playing twenty questions. An active testing model for tracking roads in satellite images. The perception of linear structure: A generic linker. Linear delineation. Shape recognition and twenty questions. Multiscale detection of curvilinear structures in 2-d and 3-d image data From step edge to line edge: Combining geometric and photometric information. In So Kweon and Takeo Kanade. Ridges for image analy- sis Curves and singularities: A geometrical introduction to singularity theory. Thin nets and crest lines: Application to satellite data and medical images. Geometric differentiation for the intelligence of curves and surfaces. Curve finding by ridge detection and grouping. Fast recognition of lines in digital images without user-supplied param- eters Direct gray-scale extraction of features for character recog- nition Detection of curved and straight segments from gray scale topography. Computer and Robot Vision The topographic primal sketch. Edge detection and ridge detection with automatic scale selection. Logical/linear operators for image curves. A common framework for the extraction of lines and edges. ter Haar Romney ter Haar Romney Numerical Recipes in C: The Art of Scientific Computing. Discrete derivative approximations with scale-space properties: A basis for low-level feature extraction Recursively implementing the Gaussian and its derivatives. Extracting curvilinear structures: A differential geometric approach. A computational approach to edge detection. Procedural Elements for Computer Graphics. Tracing crease curves by solving a system of differential equations. Automatic Extraction of Man-Made Objects from Aerial and Space Images --TR --CTR Jian Chen , Yoshinobu Sato , Shinichi Tamura, Orientation Space Filtering for Multiple Orientation Line Segmentation, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.22 n.5, p.417-429, May 2000 Markus Mller , Wolfgang Krger , Gnter Saur, Robust image registration for fusion, Information Fusion, v.8 n.4, p.347-353, October, 2007 Nassir Navab , Yakup Genc , Mirko Appel, Lines in One Orthographic and Two Perspective Views, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.25 n.7, p.912-917, July Jan-Mark Geusebroek , Arnold W. M. Smeulders , Hugo Geerts, A Minimum Cost Approach for Segmenting Networks of Lines, International Journal of Computer Vision, v.43 n.2, p.99-111, July 1, 2001 Thierry Graud , Jean-Baptiste Mouret, Fast road network extraction in satellite images using mathematical morphology and Markov random fields, EURASIP Journal on Applied Signal Processing, v.2004 n.1, p.2503-2514, 1 January 2004 Jong Kwan Lee , Timothy S. Newman , G. Allen Gary, Oriented connectivity-based method for segmenting solar loops, Pattern Recognition, v.39 n.2, p.246-259, February, 2006 Andrew K. C. Wong , Peiyi Niu , Xiang He, Fast acquisition of dense depth data by a new structured light scheme, Computer Vision and Image Understanding, v.98 n.3, p.398-422, June 2005 G. J. Streekstra , R. Van Den Boomgaard , A. W. M. Smeulders, Scale Dependency of Image Derivatives for Feature Measurement in Curvilinear Structures, International Journal of Computer Vision, v.42 n.3, p.177-189, May-June 2001 Antonio M. Lpez , Felipe Lumbreras , Joan Serrat , Juan J. Villanueva, Evaluation of Methods for Ridge and Valley Detection, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.21 n.4, p.327-335, April 1999 E. Cernadas , M. L. Durn , T. Antequera, Recognizing marbling in dry-cured Iberian ham by multiscale analysis, Pattern Recognition Letters, v.23 n.11, p.1311-1321, September 2002 Derek C. Stanford , Adrian E. Raftery, Finding Curvilinear Features in Spatial Point Patterns: Principal Curve Clustering with Noise, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.22 n.6, p.601-609, June 2000

lines;medical images;contour linking;feature extraction;aerial images;low-level processing;scale-space;curvilinear structures

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A Generic Grouping Algorithm and Its Quantitative Analysis.

AbstractThis paper presents a generic method for perceptual grouping quality. The grouping method is fairly general: It may be used the grouping of various types of data features, and to incorporate different grouping cues operating over feature sets of different sizes. The proposed method is divided into two parts: constructing a graph representation of the available perceptual grouping evidence, and then finding the "best" partition of the graph into groups. The first stage includes a cue enhancement procedure, which integrates the information available from multifeature cues into very reliable bifeature cues. Both stages are implemented using known statistical tools such as Wald's SPRT algorithm and the Maximum Likelihood criterion. The accompanying theoretical analysis of this grouping criterion quantifies intuitive expectations and predicts that the expected grouping quality increases with cue reliability. It also shows that investing more computational effort in the grouping algorithm leads to better grouping results. This analysis, which quantifies the grouping power of the Maximum Likelihood criterion, is independent of the grouping domain. To our best knowledge, such an analysis of a grouping process is given here for the first time. Three grouping algorithms, in three different domains, are synthesized as instances of the generic method. They demonstrate the applicability and generality of this grouping method.

Introduction This work proposes a generic algorithm for perceptual grouping. The paper presents the new approach, and focuses on analyzing the relation between the information available to the grouping process and the corresponding grouping quality. The proposed generic algorithm may serve to generate domain specific grouping algorithms in different domains, and we implement and test three of them. However, the analysis is domain independent, and thus applies for the specific cases. Visual processes deal with analyzing images and extracting information from them. One reason that makes these processes hard is that only a few subsets of the data items contain the useful information while all others are not relevant. Grouping processes, which rearrange the given data by eliminating the irrelevant data items and sorting the rest into groups each corresponding to a certain object, are indispensable in computer vision [WT83, Gri90, The Gestalt psychologists already noticed that humans use some basic properties, which may be called grouping cues ([Low85]), to recognize the existence of certain structures in a scene and to extract the image elements associated with such structures, even before it is recognized as a meaningful object [Wer50, WT83, Low85, Gor89]. In the field of computer vision, Witkin and Tenenbaum [WT83], suggested that grouping processes should be part of all processing levels. Indeed , grouping was used in many levels, starting from low-level processes such as smoothness based figure-ground discrimination [SU88, HH93], through motion based [AW94, JC94, Sha94] grouping, which may be considered to be mid-level processes, to high-level vision processes such as object recognition [HMS94, ZMFL95]. The proposed method separates between two components of the grouping method: the grouping cues that are used and the grouping mechanism that combines them into a partition of the data set. Like most grouping methods, the grouping mechanism used here is formally defined as the maximization of some consistency function between the group assignments and the given data. This maximization is usually done by various methods, including dynamic programming [SU90], relaxation labeling [PZ89, TJA92], simulated annealing [HH93], and graph clustering [WL93], and may also be done hierarchically [HMS94, MN89, DR92]. The proposed algorithm also maximizes a function: the likelihood of the data available relative to the grouping decision. The crucial difference which exists however between the proposed algorithm and all previous work is in the analysis we provide, which predict the grouping performance based on the reliability of the data. Our specific grouping process is based on representing the unknown partition into groups as in the form of a special graph, in which the vertices are the observed data elements (edges,pixels, etc.) and the arcs contain the grouping information and are estimated by cues. (Others, e.g. [HH93, SU88, WL93], have used graphs for grouping algorithms, but we use it differently here.) The grouping task is divided into two parts: constructing the graph by applying the geometric knowledge on the data, and then finding the "best" partition of the graph into groups. Both stages are implemented using known statistical tools such as Wald's SPRT algorithm and the Maximum Likelihood criterion. Grouping cues are the building blocks of any grouping process, and shall be treated as the only source of information available for this task. They are used, in the first stage of our algorithm, for constructing the graph. In general, they are domain specific and rely on the assumed properties of the sought-for groups. Their choice is essentially made by taste and intuition although more rigorous statistical properties are sometimes taken into account [Low85, Jac88, Cle91, CRH93]. The well-studied task of grouping edge points lying on a smooth boundary, is a good example for the variety of perceptual grouping cues. The typically used cues are collinearity, co-circularity [Sau92], curvature and length [SU90, PZ89], proximity, and some combinations of those [DR92, HH93, GM92]. In other domains, or under different assumptions, other cues are used (e.g. motion based cues [JC94, AW94], symmetry [HMS94], or 3D symmetry-based invariance [ZMFL95]). Although good cues are essential for successful grouping, finding them is not our aim here. Instead we consider the cues as given, and focus on quantifying their reliability, and its relation to the expected grouping quality. We model the cues as random variables, and quantify their reliability using the properties of the corresponding distribution. Moreover, we suggest a general method, denoted cue enhancement, for improving the reliability of a cue, and show a tradeoff between computational efforts and the achieved reliability of the enhanced cue. Although many grouping methods were already suggested and tested, it seems that no solid theoretical background was established. So far, the performance of the grouping algorithms has been assessed by implementing the algorithm, testing it on a small number of simulated or real examples, and then visually evaluating the results. This methodology shows that some of the grouping methods perform well on the examples tested and indeed succeed in partitioning the image elements into seemingly correct subsets. It does not allow us, however, to predict the performance of these algorithms on other images or to compare the algorithms whose assessment was carried out using varied examples. The proposed quantitative analysis of the expected grouping performance, provides some relations between the quality of the available data, the computational effort invested, and the grouping performance, quantified by several measures. The grouping algorithm we propose is generic and domain specific grouping algorithms may be synthesized as instances of it by inserting the appropriate cue and topology specification (in the form of a graph). The analysis, which applies to the generic abstract algorithm, may be useful for predicting the results in the specific domains. The main new contributions of this paper are: a. A fairly general approach to grouping, which is applicable to several domains, and tested in three of them. Most, if not all, previous algorithms were domain specific. b. A quantification of the expected quality of the grouping results. To our best knowledge, such an analysis was not done before. c. A cue enhancement procedure, capable to significantly improve the reliability of many existing grouping cues. The rest of this paper is organized as follows: It starts with a formulation of the grouping task as a graph clustering problem. The graph-based grouping algorithm is follows, and its theoretical analysis is described in Section 4. Section 5 is concerned with the cue enhancement procedure, inluding a short review of Wald's SPRT algorithm. We tested our approach also experimentally, providing three instances of the generic algorithm in three domains, and some comparisons to the theoretical predictions. Some open questions and further research directions are considered in the discussion. 2 The Grouping Task and its Graph Representation 2.1 The Grouping Task be the set of data elements. This data set may consist of the coordinates and grey levels of all pixels in the image, the boundary points in an image, etc. S is naturally divided into several groups (disjoint subsets) so that all data elements in the same group belong to the same object, lie on the same smooth curve, or associated with each other in some other manner. In the context of the grouping task the data set is given but its partition is unknown and should be inferred from indirect information given in the form of grouping cues. Often, only the elements in the last L groups satisfy this description while the elements in the first group, S 0 , are considered as a non-important background. We should also mention, that according to another grouping concept the hypothesized groups are not necessarily disjoint. We do not consider this different task here but we believe that at least some of the tools developed here are useful for analysing it too. 2.2 Grouping Cues Grouping cues are the building blocks of the grouping process and shall be treated as the only source of information available for this task. The grouping cues are domain-dependent and may be regarded as scalar functions C(A) defined over subsets A ae S of the data feature set. Such cue functions should be discriminative. For example, it should be high if the data features in the subset A belong to the same object and low if they do not. Preferably, they should also be invariant to change of the viewing transformation and robust to noise [Low85]. Most of the grouping cues considered in the literature were functions defined over data subsets including only two data elements. (Some exceptions are a convexity cue [Jac88] and an U-shape cue [MN89].) Later, in section 5, we consider Multi-feature cues, defined over data subsets including three data features or more, and show how to integrate the evidence available from them into very reliable bi-feature cues. At this stage, however, we consider only bi-feature cues which may be either the cues used by common grouping processes or the result of the cue enhancement process described later. Following the main goal of this work, to provide a general, domain independent, frame-work for grouping processes, we would like to predict the grouping performance, not relying on a detailed domain-dependent knowledge about a cue, but using only some measure of its reliability. Such reliability measure may be defined by considering the cue function to be a random variable, the distribution of which depends on the features set being in the same group or not. For binary cues, this dependency is simply quantified by two error prob- abilities: p miss is the probability that the cue C(A) indicates that the data features in A do not belong to the same group while in fact they do. p fa is the probability that the cue function indicates that the features of A belong to the same group, while in fact they do not (false alarm). If both is an ideal cue. (For the more general and not necessarily binary cue, this reliability is quantified by the average log likelihood ratio of the cue.) This characterization can sometimes be calculated using analytical models (e.g. [Low85]), and can always be approximated using Monte-Carlo experimentations ([Jac88]). 2.3 Representing Groups and Cues Using Graphs Our approach to the grouping process is based on representing both the unknown partition into groups and the data available from the cues using graphs. The nodes of all the graphs are the observed data elements, but the arcs may take different meanings. The unknown partition, which is to be determined, is represented by the target graph, composed of several disconnected complete subgraphs (cliques). Every such clique represents a different object (or group) and there is no connection (arcs) between nodes which belong to different cliques. A graph with this characterization is called a clique graph and the class of such graphs is denoted G c . The nodes of this graph are available to the grouping algorithm, but its arcs, which contain the grouping information, are hidden and are not directly observable. Knowing that G t belongs to the class of clique graphs, G c , the grouping algorithm should provide a hypothesis graph, G should be as close as possible to G t . The cue information is described by two graphs. The underlying graph, G specifies, by its arcs, the feature pairs which are evaluated (for being in the same group) by the cue function and are available to the grouping algorithm. The second graph, denoted measured graph Gm , specifies the information provided by these cues. That is an arc belongs to Gm iff it belongs to G u and the result of the cue function indicates that the feature pair belongs to the same group. While the underlying graph is specified by the designer, depending on the domain and the computational effort limitations, the measured graph is a result of the cue evaluation process and a part of the grouping process. 3 The Generic Grouping Algorithm The generic grouping algorithm described in this section consists of two main stages: cue evaluation for (many) feature pairs and maximum likelihood graph partitioning. The two stages are general and do not depend on the particular domain in which the grouping is done, except from the obvious choice of a domain-dependent cue and some associated decisions made before the process. 3.1 Some Decisions To Be Made By The Designer The first thing is to choose a grouping cue which naturally depends on the domain and on the assumed characterization of the sought-for groups. The performance analysis, described latter, provide some quantitative means to choose between alternatives cues, which may differ, for example, by the tradeoff between false alarm and miss errors. In principle, all feature pairs, corresponding to a complete underlying graph, (V; E c (V )), should be evaluated. Hypothesis graph G G u G G G (set of groups) Grouping cue underlying Create graph Data features set Grouping by Graph Clustering Underlying graph Measured graph by Graph clustering max. likelihood Decide for each edge Desired target graph Figure 1: The proposed grouping process: The image is a set of data features (edgels in this every one of which is represented by a node of a graph. The first step is to decide about a cue and about the set of feature-pairs to be evaluated using this cue. This set of feature-pairs is specified by the arcs of the underlying graph G The second step is to use grouping cues to decide, for every feature pair in G u , if both data features belong to the same group. These decisions are represented by the a measured graph every arc corresponds to a positive decision (hence Em ' E u ). The known reliability of these decisions is used in the last step to find a maximum likelihood partitioning of the graph, which is represented by the hypothesized (clique) graph G h . A main issue considered in this paper is the relation between this hypothesis G h and the ground truth target graph, G t , which is unknown. Some cues are meaningful, however, only for near or adjacent data elements and are not adequate for evaluating every feature pair. Therefore, the cue evaluation is restricted only a subset of the feature pairs, specified by the spatial extent of the available cue. For example, in order to detect long and smooth curves using co-circularity and proximity cues we may test only close data feature pairs. On the other hand, when testing global cues like affine motion, all feature pairs may be tested and contribute useful information. Another consideration which affects the choice of the underlying graph is the reliability of the grouping process and the computational effort invested in it. As we shall see, the reliability increases with the density of the graph, but so does the computational effort, so some compromise should be made. In this paper we don't investigate the optimal decisions at this stage but just assume that both the cue and the associated adequate "topology" are either given or chosen intuitively. 3.2 First Stage: Evaluate Grouping Cues In the first stage of the grouping process, all feature pairs corresponding to arcs in G are considered, one arc at a time, and the cue function is used to decide whether the two data features belong to the same group (and the arc corresponding to them is in the unobservable graph G t ). A simple decision may be obtained by any binary cue. A more sophisticated and reliable process is to rely on multiple evidence based on other features, as done in our cue enhancement procedure (section 5). The positive decisions are represented by the measured graph decisions are made, Em is an estimate of the projection of the target graph G t on the underlying graph G u . This measured graph carries the information accumulated in the first stage to the second one. Note that it is also possible to postpone the decisions, and mark every arc of the underlying graph with the likelihood of the corresponding pair to be in the same group. Then, the maximum likelihood partition stage proceed similarly. While this approach may yield better results, due to the larger amount of information carried to the second stage, it requires that the actual non-binary cue distributions are given, which is rarely the case, and is not considered further. 3.3 Second Stage: Maximum Likelihood Partition of The Graph Recall that every decision made in the first stage is modeled as a binary random variable, the statistics of which depends on whether the two data features belong to the same group, or whether they not. Therefore, the likelihood that this decision is indeed correct depends on the true and unknown grouping. Therefore, the decisions made in the first stage (and represented by the measured graph Gm ), specify some likelihood for every partition of the graph into subgraphs. Choosing the partition (or a clique graph) which maximizes this likelihood yields an approximation to the required unknown target graph G t , which is one of the clique graphs. In the context of this paper, the cue decisions assumed to be independent and are subject to two types of errors specified uniformly by two error probabilities: ffl miss The error probability pair (ffl miss ; ffl fa ) is identical to the cue probability pair (p miss the common direct use of bi-feature cues and is equal to the error probability of the cue enhancement process (see section 5), which is usually much better. (Making these probabilities nonuniform, and thus associating every arc of G u with an individual pair of error probabil- ities, may be a more accurate model but requires much more accurate knowledge about the error mechanism.) The likelihood of the measurement graph, Gm , for every candidate hypothesis E) 2 G c , is then given by Y LfejEg (2) where the likelihood of each edge is ffl miss if e 2 EnEm We propose now to use the maximum likelihood principle, and to hypothesize the most likely (but not necessarily unique) graph G2Gc LfGm jGg: (4) The maximum likelihood criteria defined by eq. (4) specifies the grouping result, G h , but is not a constructive algorithm. Moreover, this class of optimization problems is known to have high computational complexity (exponential), in the worst case. We therefore address the theoretical aspect and the practical side separately. From the theoretical point of view, we shall now assume that the hypothesis which maximizes the likelihood may be found, and address our main question: "what is the relation between the result G h , and the unknown target graph G t ?" This question is interesting because it is concerned with predicting the grouping performance. If we can show that these two graphs are close in some sense, then it means that algorithms which use the maximum likelihood principle have predictable expected behavior and that even we can't know G t , then the grouping hypothesis G h they produces is close enough to the true partitioning. This question is considered in the next section. From the practical point of view, one should ask if this optimization problem can be solved in a reasonable time. Some people use simulated annealing, or other annealing methods, to solve similar problems [HH93]. Others use heuristic algorithms [Vos92]. We developed a heuristic algorithm which is based on finding seeds of the groups, which form (almost) a clique in Gm . (Random graphs theory [Pal85] implies that cliques of a certain size are most likely to be found inside an object, and are very unlikely to be found elsewhere in the graph). Seeds are found as the highest entries in the square of the adjacency matrix of Gm . Then, these seeds are iteratively modified by making small changes (such as moving one element from one group to another, merging two groups, etc.), using a greedy policy, until a (local) maximum of the likelihood function is obtained. In our experiments (described in section 6), this algorithm performs nicely. More details can be found in [AL95]. 4 Analysis of The Grouping Quality This section quantifies some aspects of the similarity between the unknown scene grouping (represented by G t ), and the hypothesized grouping suggested by our algorithm (represented by G h ). As we shall see, the dissimilarity depends on the error probabilities of the individual arcs, ffl miss ; ffl fa , and on the connectivity , or the density, of G u . The first result demonstrates that good solutions are not rejected. LfGm jG G2Gc provided that ffl miss Proof: For every clique graph, G(V; E) 2 G: LfGm jGg LfGm jG g =@ Y ffl miss e2E u "(E nE) (arcs of E u , which exist in both (or none) of the two sets, E and E , do not affect that ratio, and therefore are not counted) 2 Borrowing the terminology of parameter estimation, this claim shows that maximum likelihood partition is a consistent estimator. That is, arbitrarily reliable labeling of the underlying graph, associated with very good cues, leads to a correct decision. From now on we assume that ffl miss consistency is not ensured. In the more realistic case, where some hypotheses regarding arcs of the underlying graphs may be wrong, we shall show that grouping performance degrades gracefully with the quality (reliability) of the cues and that this performance may be predicted. In general, grouping performance is good for groups which are densely connected within the underlying graph, and is expected to be worse for loosely connected groups. If, for example, a node (data feature) is connected to its group by only one edge in the underlying graph, it may be separated from this group in the hypothesized partition with probability ffl miss , which may be quite high. We now turn into proving a fundamental claim on which most of the other results rely. It is a necessary condition satisfied by any partition selected according to the maximum likelihood principle. Consider two nodes-disjoint subsets of the graph and denote their cut by J(V g. Let l u denote the cut width relative to the underlying graph. Similarly, let l m denote the cut width relative to the measurement graph necessary condition: Let G be the maximum likelihood hypothesis (satisfying eq. (4)), and let log(ffl miss Then, 1. For any disjoint partition of any group V 2. For any two groups l m Proof: The proof technique is similar to that of claim 1. For proving the first part, consider the likelihood ratio between two hypotheses: One is G h and the other, denoted ~ G h , is constructed from G h by separating V i into two different groups, V 0 Y ffl miss l This likelihood ratio is a non-decreasing function of l m Figure 2: The cut involved in splitting a group into two (proof of claim 2) and is larger than 1, for l m - ffl u . Therefore, if the claim is not satisfied, then ~ G h is more likely than G h which contradicts the assumption that (4) holds. The second part of the claim is proved in a similar manner.Qualitatively, the claim shows that a maximum likelihood grouping must satisfy local conditions between many pairs of feature subsets. It further implies that a grouping error, either in the form of adding an alien data feature to a group or deleting its member, requires more than a single false alarm or a single miss, provided that the "connectivity" of the underlying graph is high enough. An addition error, for example, merging a group with an alien node v , requires that a substantial fraction of the edges in J(V i ; fv g), which none of them is in E t , will be included in Em . That is, it requires many false alarms. The parameter ff, specifying the fraction of cut edges required to merge two subsets reflects the expected error if the false alarm probability is equal to the miss probability, then the false alarm probability is higher, so is ff. This condition is now used to show that choosing a sufficiently dense underlying graph can significantly improve the grouping performance. We shall consider two cases: a complete underlying graph, and a locally connected underlying graph. 4.1 Complete Underlying Graphs A complete underlying graph connects every data feature with all others and provides the maximal information to the graph clustering stage. Therefore, it may lead to excellent grouping accuracy. On the other hand, as mentioned before, it is useful only for global grouping cues, such as being on the same straight line, being consistent with an affine motion model, etc. There are many types of grouping inaccuracies, and the following claims consider some of them. be a true data feature group. Then, the probability that a maximum likelihood process will hypothesize a group V containing k nodes of S i and a particular additional node, i=kmin Proof: Use claim 2 with note that l Merging these subsets requires that at least ffl u of the edges connecting then are included in Em . This event happens with a binomial distribution. 2 true group and a maximum likelihood hypothesized group containing at least k nodes of S i . Then, the probability that V contains k 0 nodes or more which are alien to S i , is at most \Gammaaliens - Proof: Use claim 2 with and V to find the probability that a particular data subset V merges. Then, take a worst case approach, and sum these probabilities over all subsets of a certain size j, and over all sizes higher than k 0 . 2 k10203040 Figure 3: Two predictions of the analysis: Left: A k-connected curve-like group (e.g. smooth curve) is likely to brake into a number of sub-groups. The graph shows an upper bound on the expected number of sub-groups versus the minimal cut size in the group, k (eq. 13). Here the group size (length) is 400 elements, ffl miss = 0:14 and ffl (typical values for images like Figure 8). It shows how increasing connectivity quickly reduces the false division of this type of groups. Right: Upper bound on the probability for adding any k 0 alien data features to a group of size k, using a complete underlying graph (claim 4). The error probability is negligible true group and the maximum likelihood hypothesized group containing the maximal number of data features from S i . Then, the probability that V contains jS data features from S i is at most \Gammadeletions - i=kmin miss Proof: For any particular deleted subset S 0 ae use claim 2 with note that l Such a split of S i requires that l m - ffl u . This event happens with a binomial distribution. To find the probability that some subset of size k 0 is deleted, we sum over all subsets, ignoring the dependency between the events which can only decrease this probability. 2 Claims 3,4,5 simply state that if the original group S i is big enough and the miss and false alarm probabilities are small enough, it is very likely that the maximum likelihood partition will include one group for each object, containing most of S i , and very few aliens. The crude bound, plotted in Figure 3(right) shows, as an example, that even for substantial cue errors the probability for hypothesizing highly mixed subsets is small, provided that the group is large enough (k - 15). An even more practical performance measure, which we calculated using some approximations is the expected number of addition and deletion errors. Efk delete i=kmin miss i=kmin where k is the group size, k dffke. Experimental results for these two grouping error types are given in Figure 9(c) and Figure 9(d). The major difficulty we see with the use of a complete underlying graph is that it does not apply to all the cues. It's especially concerns with cues that are meaningful only locally, such as co-circularity for smooth curve detection. Therefore, another option, the locally- dense underlying graph is also proposed. 4.2 Locally Dense Underlying Graphs An intuitive choice of an underlying graph which is less dense than the complete graph is to connect every data feature only to those data features in its neighborhood, either to the closest k data features, or to all data features in a certain radius. When specifying such a graph, it is important to keep a substantial connectivity between the data features of objects so that accidental deletion will be less likely. This connectivity demand is quantified by requiring the projection of every group on the underlying graph, to be k-connected. That is, if any k \Gamma 1 nodes are eliminated, then this projected subgraph remains connected. A nice property of k-connected graphs is that every cut in them contains at least edges. Therefore, a deletion of a node requires at least ffk miss errors. Alien data features are either densely connected to a group, implying that their incorrect addition to a group is prevented with high confidence, or are not connected enough and are not considered at all as candidates for addition. A significant change from the case of complete graph is that ffk miss errors can cause the deletion of a subgroup containing more than one data feature, a fact that demonstrates the relative weakness of the locally connected underlying graph. Being aware of this weakness, we choose to characterize the grouping performance by another measure: the expected number of "large" subgroups to which the group decomposes. Consider a particular cut of size k in the projection of some object on the underlying graph. The probability that the object is divided in this cut into two parts is exactly divide in i=kmin miss Suppose now that we can estimate the number of "potential cuts", and denote this number by N cut . Then, the expected number of group separation will be simply N cut p divide in k\Gammacut . Fortunately, such an estimate may be done for the interesting case of curve like groups. Let S i be a k-connected curve-like group in which the data features are ordered along some curve. A separation of the curve into significant "large" parts, is associated with cuts which separate a group of consecutive curve points from another group of consecutive curve points. The number of such cuts is N Therefore, if we can guarantee that the number of arcs in every one of these cuts is not less than k, then the expected number of parts into which the curve decomposes is not higher than divide in i=kmin miss This number, plotted in Figure generally decrease with increasing the cut size k, but due to the non-constant and non-monotonic nature of the ratio kmin ff, it is not strictly monotonic. Locally connected underlying graphs are used in the 2nd demonstrated instance of the algorithm, which considers grouping of curve like groups based on proximity and smoothness. 5 Cue Enhancement The performance of the grouping algorithm depends very much on the reliability of the cues available to it. In many situations this reliability is predetermined and the grouping algorithm designer can only prefer the more reliable cues from the available variety. This section, however, shows how the reliability of a grouping cue can be significantly improved by using statistical evidence accumulation techniques. This method is not restricted only to our grouping algorithm, and can be used also in other grouping algorithms. Two of the three domain specific grouping algorithm that we implement as examples (the co-linearity and the smoothness) use this procedure. 5.1 The Cue Enhancement Procedure - Overview The cue enhancement procedure considers one pair of data features at a time, and tries to use the other data features in order to estimate the consistency of this pair. We shall say that a subset of data features A is consistent if it is a subset of some true group. The idea behind the following process of evidence accumulation is that a random data subset A that contains the data pair may be consistent only (but not necessarily) if e itself is consistent. Therefore, a multi-feature cues operating on a feature subset A (e 2 carries statistical information on the consistency of e. Although bi-feature cues are easier to calculate and are more straightforward to use, cues which test larger data subsets have several significant advantages: Several useful cues are simply not defined when only one pair of elements is considered (e.g. convexity). Bi-feature cues usually have corresponding multi-feature cues associated with improved reliability. (Observe, for example, that accidental collinearity is less likely if more points are considered while the miss probability should decrease only slightly in this case. More generally, the reliability of the shape-based multi-feature cue of "consistent with some instance of a particular object" clearly increases with the number of data features [GH91, Lin94].) The algorithm is conceptually simple: for every data pair, in the underlying graph, the algorithm draws several random data subsets, A 1 ; A contain the pair e. Then, the corresponding multi-feature cues, are ex- tracted. The cue values are deterministic functions of the subsets A 1 ; A but may be also considered as instances of a random variable, the statistics of which depend on the data pair e, and in particular, on its consistency. The number of random data subsets and their associated cues, required for a conclusive reliable decision on the consistency of e, is determined adaptively and efficiently by a well-known method for statistical evidence integration: Wald's SPRT test. 5.1.1 Wald's SPRT Algorithm and its Application for Cue Enhancement Consider a random variable, x, the distribution of which depends on an unknown binary parameter, which takes the value of ! 0 or ! 1 . Every instance of the random variable carries statistical information on this parameter and integrating this information corresponding to a sequence of the random variable instances will eventually lead to a reliable inference about it. An efficient and accurate procedure for integration the statistical evidence is the Sequential Probability Ratio Test (SPRT) suggested by Wald [Wal52]. This procedure quantifies the evidence obtained from each trial by the log likelihood ratio function of its are the probability functions of the two different populations and x is the value assigned to the random variable in this trial. The log likelihood ratio is high when the value of the random variable x is likely for one hypothesis (! 1 ) and is not likely for the other (! 0 ). It is negative and low when the situation is reversed. If the probabilities of seeing x under both hypotheses are close, then x carries only little information and h(x) - 0. When several trials are taken, the log likelihood function of the composite event should be considered. If, however, the trials are independent then this composite log likelihood function is equal to the sum of the individual log likelihood functions, oe The sum oe n serves as the statistics by which the decision is made. Wald's procedure specifies two limits, upper and lower. If the cumulative log likelihood function crosses one of these limits, a decision is made. Otherwise, more trials are carried out. More formally, denote the decision made by the procedure by let the allowed probabilities of a decision error be The algorithm is given simply by this iterative rule: else test for another subset The upper and lower limits, a depend only on the allowed probability of error (defined in eq. 1), and do not depend on the distribution of the random variable x. We calculate a; b using a practical approximation, proposed by Wald [Wal52], which is very accurate when ffl miss ; ffl fa are small: The basic SPRT algorithm terminates with probability one and is optimal in the sense that it provides the minimum expected number of tests necessary to obtain the required decision error [Wal52]. This expected number of tests is given by: are the conditional expected amounts of evidence from a single its average case optimality, the worst case number of trials required by the SPRT algorithm is not bounded. To deal with this disadvantage, the modified Truncated SPRT [Wal52], which uses a predefined upper bound n 0 on the number of tests, is used. We set n 0 to be few times larger than Efng. In the context of the Cue enhancement procedure, the cue value is regarded as a random variable. Apart from specifying the desired reliability (ffl miss ; ffl fa ) and using equation 16 to calculate the two thresholds a and b, one must supply the two distributions (for consistent and inconsistent feature pairs), from which the log-likelihood ratio can be determined. These distribution should be evaluated carefully: The distributions of the cues taken over the consistent and inconsistent populations and denoted respectively by P con (C(A)) and P incon (C(A)), are usually quite different. It is important however to observe that even if a feature pair (u; v) is consistent, a random set including it may not be. Therefore, these distributions should be modified as follows: A random set containing a feature pair fu; vg ae S i and additional randomly selected data features, v consistent with probability where Therefore, the modified cue distributions, conditioned relative to the consistency of the first two feature are Unfortunately, these distributions are more similar and difficult to distinguish (see Figure 9(a) for such a pair of distributions considered in our experiments). Restricting ourselves to binary cues, the distribution of which is specified by the probabilities the conditional distributions become are .) The log likelihood ratio of the i th randomly-selected subset, A i , becomes: log( p log( 1\Gammap The SPRT based cue enhancement procedure is summarized in Figure 5.1.1. 2. Randomly choose data features x 3. Calculate 4. Update the evidence accumulator For every feature pair (u; v) in the underlying 1. Set the evidence accumulator, oe, and the trials counter, n, to 5. if oe - a or if n - n 0 and oe ? 0, output: (u; v) is consistent. if oe - b or if n - n 0 and oe ! 0, output: (u; v) is inconsistent. else, repeat (2)-(5) Figure 4: The cue enhancement algorithm The success of the cue enhancement procedure relies on the validity of the statistical model, and in particular, on the following two assumptions assumption a: The statistics of the cue values evaluated over all data subsets containing a consistent (inconsistent) arc is approximately the same. assumption b: The cues extracted from two random subsets including the same feature are independent identically distributed random variables. If the assumptions are satisfied, then 6 The cue enhancement procedure described above can identify the consistency of the feature pair within any specified error tolerance irrespective of the reliability of the basic cue and provided that assumptions a and b hold. This surprising conclusion seems to contradict intuition according to which arbitrarily low identification errors are impossible as the amount of data in the image is finite. Indeed, arbitrarily high performance is not possible as it requires a large number of trials leading to a contradiction of the independence assumption. Therefore, the reliability of the basic cue is important because it leads to a lower number of trials, which is both computationally advantageous and important to the validity of the statistical independence assumption. Indeed, our experiments show that the SPRT significantly improves the cue reliability but that the achievable error rate is not arbitrarily small (see experimental results in the next section). For a constant specified reliability (ffl miss ; ffl fa ), the expected running time of the cue enhancement procedure is constant. The total running time for evaluating all the arcs of the underlying graph, G u , is, therefore, linear in the number of arcs. We emphasize here that this enhancement method is completely general and may use any cue that satisfies some benign assumptions as stated in this section. It relies on the distributions of the cues, which should be calculated before and involve certain technicalities described in the full version [AL94]. 6 Simulation and experimentation This section presents three different grouping applications, implemented in different domains, as instances of the generic grouping algorithm described above. To our best knowledge, it is the first time that a generic grouping algorithm is used in multiple domains. For each implementation, the domain, the data features, and the grouping cue are different, but the same grouping mechanism (and computer program) is used (see Table 1). The aim of these examples is to show that useful grouping algorithms may be obtained as instances of the generic approach and to examine the performance predictions against experimental results. We do not expect that our general algorithm will perform as good as domain specific algorithm which were tailored for that domain. Still, in all tested domains, we got grouping results comparable to those obtained from existing, domain specific methods. This is remarkable, because except from the choice of the cues (and the associated underlying graph determined by their extent), the process did not depend on the domain. Moreover, although some of the analysis may help in selecting between different available cues, we did not focus on choosing the best cues, but more on testing our approach using reasonable cues. There- fore, we expect that even better performance will be possible by optimizing cues and their corresponding underlying graphs. (See more results and examples in [AL94, AL95].) 6.1 Example 1: Grouping points by co-linearity cues Given a set of points in R 2 , the algorithm should partition the data into co-linear groups (and one background set). To remove any doubt, we do not intend to propose our grouping approach as an efficient (or even reasonable) method for detecting co-linear clusters. Several common solutions (e.g., Hough transform, RANSAC) exist for this particular task. We have Table 1: The three instances of the generic grouping algorithm The 1st example The 2nd example The 3rd example data elements points in R 2 edgels patches of Affine optical flow grouping cues co-linearity co-circularity consistency with and proximity Affine motion Cue's extent global local global Enhanced cue subsets of 3 points subsets of 3 underlying graph complete graph locally connected graph a complete graph grouping mechanism maximum likelihood graph clustering (same program) chosen this example because it is a characteristic example of grouping tasks associated with globally valid cues (and complete underlying graphs). Moreover, it provides a convenient way for measuring grouping performance, the quantification and prediction of which is our main interest here. The grouping cue is defined over data subsets containing k ? 2 data features (here and is just the second eigenvalue of the associated covariance matrix. Clearly, if this eigenvalue is small, the data subset is closer to linear (see, e.g. [GM92]). The cue is global, hence the underlying graph is the complete graph. To binarize this cue we simply check if its value is lower than a threshold T . We consider synthetic random images containing randomly drawn points (e.g Figure 5(a)). The points are drawn according to a distribution specified by a collection of arbitrary straight lines which are the "objects" associated with the given data, and some additional, uniformly distributed, aliens. With this data source, it is easy to automatically create many data sets with known noise distributions and grouping ground truth (with the exception of a few alien points, located very close to the "objects"). A typical grouping result is shown and explained in Figure 5. We used the co-linearity example to comprehensivly test the performance of the grouping algorithm against its pre- dictions. The first results show the performance of the cue function and the cue enhancement procedure. To examine the cue function, we estimate the two cue-value distributions, differ by the consistency of the included pair fu; vg ae A. This is done by a monte-carlo process over randomly-selected feature-triplets. These two distributions, (defined in eq. 18), tend to be quite similar, as shown in Figure 9(a). In order to make it a binary cue, we proceed with selecting the threshold, T , for the binary cue decision. Any specified threshold determines different binary-cue errors, (p miss ; p fa ). While one is a non-decreasing function of T , the other is a non-increasing function of T , so some compromise is done. The values of (p miss ; p fa ) effects the efficiency of the SPRT algorithm, which is measured by the average number of subsets, Efng, needed to reach a specified error rate Using eq. 17 with P 0 (C(A)); P 1 (C(A)) of Figure 9(a), one can draw Efng in terms of the selected threshold, as shown in Figure 9(b). The optimal threshold is found as the cue-value of the global minimum. Note that the selection of T does NOT effects the resulted grouping quality, but only the computational time needed for the SPRT to reach the desired error of the enhanced cue, (ffl miss ; ffl fa ). This threshold is also optimal in the sense that it provides the maximum information from each evaluated feature-triplet. The measured average number of subsets needed for the SPRT, Efng is given as labels in Figure 9(e), for 100 different pre-specified (ffl miss ; ffl fa ) values, and remarkably agrees with the predicted average (eq. 17), shown by the curves in this graph. It is also shown that the enhanced cue reliability can exceeds 95% (i.e. ffl miss ! 5% , and ffl fa ! 5%), even with the simple cue we used, which has a very low discrimination power by itself. The next results show the overall grouping quality. Regardless the choice of (ffl miss the 5 lines were always detected as the 5 largest groups in our experiments. The selection of does affects, however, the overall grouping quality. This is measured by counting the addition errors and the deletion errors, as shown in Figure 9(c) and 9(d), respectively. Note that while the deletion error is very low, as expected, the addition error is higher than expected. The reason for this discrepancy is some alien data features which are very close to one of the lines and are erroneously added to it. The tradeoff between grouping quality and the computational time of the cue enhancement procedure is obtained by these three As Efng increases (in Figure (e)), the errors decrease (in Figures 9(c) and 9(d)). 6.2 Example 2: Grouping of edgels by smoothness Starting from an image of edgels, (data feature = edge location gradient direction), the algorithm should group edgels which lie on the same smooth curve. This is a very useful grouping task, considered by many researchers (see, e.g [GM92, ZMFL95, HH93, SU90, CRH93]). A crude co-circularity cue function, operating on edgel triples, is used. It is calculated as the maximal angular difference between the gradient direction and the corresponding normal direction to the circular arc passing through the three points. The underlying graph is locally connected and is constructed by connecting every edgel to its K 2 [10; 50] nearest edgels (K is a constant). We test this procedure both on synthetic and real images, and the results are very good in both cases (see Figure 6 and Figure 7). Synthetic images are created by detecting the edges of piecewise constant images which contain grey level smooth blobs (e.g. Figure 6(a)). In the synthetic example, we found that the perimeter of each of the two big blobs splits into 3-4 groups (see Figure 6(e)). It happens in places where the connectivity in G u is low, the minimal connectivity assumption fails, and the split probability increases. (see Figure 3). 6.3 Example 3: Segmentation from Optical Flow using Affine Mo- tion The third grouping algorithm is based on common motion. The data features are pixel blocks, which should be grouped together if their motion obeys the same rule, that is if the given optical flow over them is consistent with one Affine motion model [JC94, AW94]. Technically, every pixel block is represented by its location and six parameters of the local Affine motion model (calculated using Least Squares). The grouping cue is defined over pairs of blocks, and its value is the sum of the optical flow errors of each block when calculating it using the Affine model of the other block. The cue is global and hence a complete underlying graph is used. No cue enhancement is used here, and the cue is not very reliable: typical error probabilities are ffl miss = 0:35 and ffl 0:2. Still, the results are comparable to those obtained by a domain specific algorithm [AW94]. The final clustering result, shown in Figure 8(f), was obtained after a post-processing stage: the obtained grouping is used to calculate an Affine motion model for every group, which is used to classify all the individual pixels in the image into groups. (The same method used in [AW94].) (a) Original image: A set of points. (b) Associated data features: same as original image. (c) Underlying graph Gu : A complete graph. The pixel gray level indicates the number of arcs passing thru. (d) Measured graph Gm . The pixel gray level indicates the number of arcs passing thru. One of the detected groups. Only very few points, if any, were fall in a wrong group. (f) All the detected groups Figure 5: Example 1: grouping of co-linear points. An example to the images used in the experiments. This image is associated with five lines, contains points in the vicinity of each of them, and 150 uniformally distributed additional data features. The grouping result is near-optimal, which not surprise the predictions. It demonstrates the power of a complete underlying graph. Quantitative results of this experiments are shown in Figure 9. (a) Original image: (b) Associated data features: edgels. (c) Underlying graph G locally connected (40 nearest nbrs). The pixel gray level indicates the number of arcs passing thru. The brighter areas correspond to denser regions in G u . (d) Measured graph Gm . The pixel gray level indicates the number of arcs passing thru. Note that the bright groups in the measured graph are no longer correspond to the local density of Gu , but to smoothness. This byproduct can also serve as a saliency map. One of the 14 detected groups. (f) All the 14 detected groups. Figure Example 2-1: Grouping of smooth curves in a synthetic image. Edge detection and gradient where calculated on image (a). 50% of the edge pixels were randomly removed, and 10% of the background pixels were added, as aliens, with uniformly distributed gradient directions. Total number of edgels is about 5,000, and about 110,000 arcs in G u . The (a) Original image: A brain image. (b) Edge detection of (a). The associated data features are edgels. (c) Underlying graph G locally connected (40 nearest nbrs) The pixel gray level indicates the number of arcs passing thru. (d) Measured graph Gm . The pixel gray level indicates the number of arcs passing thru. (e) The five largest detected groups. (f) All the detected groups, superimposed on the original image. Figure 7: Example 2-2: Grouping of smooth curves in a brain image. The underlying graph, G u , is made of 10,400 edgels and 230,000 arcs. The processing time is about 10 minutes on (a) Original image: Flowers sequence. (b) Associated data features: Optical flow (blocks). (c) Underlying graph Gu : A complete graph. The pixel gray level indicates the number of arcs passing thru. (d) Measured graph Gm . The pixel gray level indicates the number of arcs passing thru. The low number of edges in the clouds area indicates that the optical flow in this area does not match with the Affine motion model. (e) The 3 resulted groups, each in a different gray level. Black regions were either eliminated for high error with their Affine model (e.g. on the tree border), or not grouped to any of the groups. (f) A post-processing stage: the obtained grouping is used to calculate an Affine motion model for every group, which is used to classify all the individual pixels in the image into groups (Black pixels were not classified). This result shows that even the groups (e) are not visually nice, they can still capture the correct motion clustering of the image. Figure 8: Example 3: Image segmentation into regions consistent with the same Affine motion parameters. The underlying graph is a complete graph of about 600 nodes (180,000 arcs), and the runtime is about 5 minutes. Cue value Probability densety (a) The distribution of the co-linearity cue values, for subsets include a consistent feature (C(A)),(solid) and for subsets include an inconsistent feature pair, P 1 (C(A)), (dashed). Although these two are very similar, their populations can be distinguished with less then 5% error, as shown in (e). Cue threshold (b) The expected number of trials needed for the cue enhancement procedure as a function of the selected cue threshold. The optimal cue threshold correspond to the minima of this curve. The experimental results are shown in E_fa20101100111111001100110210222111101111221121422411 (c) E_fa42234345556464546577866788981511111312141413816192011111916272115192820 (d) E_miss E_fa (e) The tradeoff between the enhanced cue reliability and the computational effort invested is clearly demonstrated in this figure, showing the error probabilities, ffl miss ; ffl fa , associated with the enhanced cue and the experimental average number of trials, E(n) (the points' labels). The solid lines show the predicted error probabilities, for labels). Every point represents a complete grouping process and is labeled by the resulting number of: deleted points (deletion error) from all 5 lines (c), added points (addition error) to all 5 lines(d). Figure 9: Quantitative results: Comparison between the analysis prediction to the experimental results of example 1. The grouping results tends to reach a near-perfect grouping by using the cue enhancement procedure. The goal of this work is to provide a theoretical framework and a generic algorithm that would apply to various domains and that would have predictable performance. The proposed generic grouping algorithm relies on established statistical techniques such as sequential testing and maximum likelihood. The maximum likelihood principle is close to some previous grouping approaches like the use of densities for evaluating the evidence of certain cues in [Jac88] or the cumulative pairwise interaction score used for Figure-from-Ground discrimination in [HH93]. This paper is distinctive from previous approaches because it provides, for the first time, an analysis of the use of these principles, which relates the expected grouping quality to the cue reliability, the connectivity used, and in some cases the computational effort invested. We did not limit ourselves to the theoretical study. Three grouping appli- cations, every one of which based on a different cue, are implemented as instances of the generic grouping algorithm, and demonstrate its usefulness. Although we made an argument against judging the merits of vision algorithm only by visually comparing their action on a few examples, we would like to indicate here that our results are similar to those obtained by domain specific methods (e.g. [SU90, HH93] for smoothness based grouping and [AW94] for motion based grouping). Note that Gm may also be used to create a saliency map, where the saliency of every data element is its degree in Gm . This saliency map (e.g Figure 7(d),8(d)), is visually comparable with those proposed by other works (e.g. Shashua and Ullman [SU88], Guy and Medioni [GM92]). Its suitablity for figure-ground discrimination is now studied. Some interesting conclusions arise from our analysis and experimentation with grouping algorithms: It is apparent that higher connectivity, provided either by a complete underlying graph or by a high degree locally-connected graph, can enhance the grouping quality. There- fore, the selection of cues for a grouping algorithm, should not be based only on maximizing their reliability but also on their extent. The cue extent determines the connectivity of the valid underlying graph, or in other words, the amount of information which may be extracted by this cue. Another consideration is the cue enhancement possibility: if the cue satisfies the independent random variable assumption, then more reliable cue may be obtained, with a relatively low computational effort. Our analysis of the computational complexity is not complete. Although the requirements of the cue enhancement stage were clearly stated, even as a function of the quality required, we do not have complexity results for the second stage, of finding the maximum likelihood partition. This task is known to be difficult. (simulated annealing is used to solve similar problems [HH93]). We used some heuristics, based on results from random graph theory and on a greedy search, which turned out to work surprisingly good. In the design of a grouping algorithm, one may either invest the computational effort in enhancing the quality of a relatively small number of cues or use a larger number of unreliable cues and merge them by higher connectivity underlying graph. The framework proposed in this paper makes this choice explicit by providing a cue enhancement procedure, independent from the maximum likelihood graph clustering. Making the optimal choice is an interesting open question which we consider. Another research direction is to use our methodology in the context of a different grouping notion, different than partitioning, by which the hypothesized groups are not necessarily disjoint. Acknowledgements We would like to thank John Wang for providing us the Flowers garden optical flow data. --R The construction and analysis of a generic grouping algorithm. Representing moving images with layers. A bayesian mulitple- hypothesis approach to edge grouping and contour segmentation Computing curvilinear structure by token-based grouping Fast spreading metric based approximate graph partitioning algorithms. An algorithm for finding best matches in logarithmic expected time. Grimson and Daniel P. Perceptual grouping using global saliency- enhancing operators Theories of Visual Perception. Extraction of groups for recognition. The Use of Grouping in Visual Object Recognition. Finding structurally consistent motion correspondences. On the amount of data required for reliable recognition. Perceptual Organization and Visual Recognition. Using perceptual organization to extract 3-d structures Graphical Evolution. Trace interface Applications of Spatial Data Structures. Labeling of curvilinear structure across scales by token grouping. Affine Analysis of Image Sequences. Structural saliency: The detection of globally salient structures using locally connected network. Grouping contours by iterated pairing network. Supervised classification of early perceptual structure in dot patterns. Relational Matching. Sequencial Analysis. Laws of organization in perceptual forms. An optimal graph theoretic approach to data clus- tering: Theory and its application to image segmentation On the role of structure in vision. --TR --CTR Shyjan Mahamud , Lance R. Williams , Karvel K. Thornber , Kanglin Xu, Segmentation of Multiple Salient Closed Contours from Real Images, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.25 n.4, p.433-444, April P. Kammerer , R. Glantz, Segmentation of brush strokes by saliency preserving dual graph contraction, Pattern Recognition Letters, v.24 n.8, p.1043-1050, May Jens Keuchel , Christoph Schnrr , Christian Schellewald , Daniel Cremers, Binary Partitioning, Perceptual Grouping, and Restoration with Semidefinite Programming, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.25 n.11, p.1364-1379, November Alexander Berengolts , Michael Lindenbaum, On the Performance of Connected Components Grouping, International Journal of Computer Vision, v.41 n.3, p.195-216, February/March 2001 Jacob Feldman, Perceptual Grouping by Selection of a Logically Minimal Model, International Journal of Computer Vision, v.55 n.1, p.5-25, October Anthony Hoogs , Roderic Collins , Robert Kaucic , Joseph Mundy, A Common Set of Perceptual Observables for Grouping, Figure-Ground Discrimination, and Texture Classification, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.25 n.4, p.458-474, April Song Wang , Joachim S. Stahl , Adam Bailey , Michael Dropps, Global Detection of Salient Convex Boundaries, International Journal of Computer Vision, v.71 n.3, p.337-359, March 2007 A. Engbers , Arnold W. M. Smeulders, Design Considerations for Generic Grouping in Vision, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.25 n.4, p.445-457, April Song Wang , Toshiro Kubota , Jeffrey Mark Siskind , Jun Wang, Salient Closed Boundary Extraction with Ratio Contour, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.27 n.4, p.546-561, April 2005 Bernd Fischer , Joachim M. Buhmann, Path-Based Clustering for Grouping of Smooth Curves and Texture Segmentation, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.25 n.4, p.513-518, April Sudeep Sarkar , Padmanabhan Soundararajan, Supervised Learning of Large Perceptual Organization: Graph Spectral Partitioning and Learning Automata, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.22 n.5, p.504-525, May 2000 Stella X. Yu , Jianbo Shi, Segmentation Given Partial Grouping Constraints, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.26 n.2, p.173-183, January 2004 Sudeep Sarkar , Daniel Majchrzak , Kishore Korimilli, Perceptual organization based computational model for robust segmentation of moving objects, Computer Vision and Image Understanding, v.86 n.3, p.141-170, June 2002

maximum likelihood;generic grouping algorithm;perceptual grouping;grouping analysis;performance prediction;Wald's SPRT;graph clustering

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Decomposition of Arbitrarily Shaped Binary Morphological Structuring Elements Using Genetic Algorithms.

AbstractA number of different algorithms have been described in the literature for the decomposition of both convex binary morphological structuring elements and a specific subset of nonconvex ones. Nevertheless, up to now no deterministic solutions have been found to the problem of decomposing arbitrarily shaped structuring elements. This work presents a new stochastic approach based on Genetic Algorithms in which no constraints are imposed on the shape of the initial structuring element, nor assumptions are made on the elementary factors, which are selected within a given set.

INTRODUCTION MATHEMATICAL morphology [1], [2], [3] concerns the study of shape using the tools of set theory. Mathematical morphology has been extensively used in low-level image processing and analysis appli- cations, since it allows to filter and/or enhance only some characteristics of objects, depending on their morphological shape. A lot of tutorials [3], [2], [4], [1], [5], [6], [7] can be found in the literature. Within the mathematical morphology framework, a binary image A is defined as a subset of the two-dimensional Euclidean space In [3], monadic transforms acting on a generic image A (complement, reflection, and translation) and dyadic operators between sets (dilation, erosion, opening, and closing) are defined. In the following only the definitions of operators used throughout this paper are recalled, such as translation , for some with (2) and dilation for some where A represents the image to be processed, and B is called Structuring Element (SE), namely, another subset of E 2 whose shape parameterizes each operation. An SE B is said to be convex with respect to a given set of morphological operations (e.g., dilation) with a given set of SEs (factors) {F , it can be expressed as a chain of dilations of the F i elements: with for (4) Otherwise B is said to be nonconvex with respect to the same set of SEs, and, thus, it can only be expressed as a chain of Boolean operations (e.g., unions and/or intersections) between convex elements (called . (5) where ( represents any Boolean operation (such as unions <, intersections >, .) and C i are convex elements that can be expressed as chains of dilations, as shown in (4). As discussed in the following section, the decomposition of a binary SE into a chain of operations involving only elementary factors is a key problem. So far, only deterministic solutions have been analyzed and proposed in the literature [8], [9], [10], each relying on different assumptions (such as convex SEs, specific sets of elementary operators, etc.); on the other hand the optimal decomposition (with respect to a given set of optimality criteria) of nonconvex generic SEs with a deterministic approach is still an open problem. This paper addresses this problem utilizing a stochastic ap- proach, based on Genetic Algorithms: starting from a population of potential solutions (individuals) determined through an exhaustive algorithm, an iterative process modifies the existing individuals and/or creates new ones in accordance to a set of genetic operators applied randomly. The individuals that minimize a given cost function tend to replace the others, and, after a sufficient number of iterations, the algorithm tends to converge toward the optimal solution. In particular, the main purpose of this work is to develop a tool able to give a preliminary answer to the problem of optimal decomposition of nonconvex SEs into concatenations of generic elementary operations. This work is organized as follows: Section 2 motivates the need for SE decomposition and discusses some optimality criteria that can drive the decomposition; Section 3 briefly summarizes the Genetic Approach, its terminology and its notations, and describes the implementation of the decomposition algorithm and the data structures involved; Section 4 presents some results while Section 5 concludes the paper with some remarks and a discussion on future developments. 2.1 Motivation The following two subsections motivate the need for SE decomposition on traditional serial systems, in which the use of a large SE is not efficient, and on SIMD cellular systems that allow the execution of only basic operations based on a neighborhood smaller than the size of the SE; the different characteristics of general-purpose (serial) and SIMD cellular (parallel) systems require different techniques in order to exploit the specific hardware characteristics of each system. Hereinafter, a dilation between a generic image A and a complex SE B is considered; due to the different properties of unions and intersections discussed in [3], namely in the following nonconvex SEs are decomposed using chains of unions of convex SEs (using the equality expressed by relation (6), instead of using chains of intersections or other Boolean operations (where no equality relations hold). 2.1.1 Serial Systems General-purpose serial systems have no upper bound to the size of possible SEs: In fact, using a bitmapped image representation, the value of any image pixel can be accessed within a constant time. . The authors are with the Dipartimento di Ingegneria dell'Informazione, Universit- di Parma, I-43100 Parma, Italy. E-mail: broggi@ce.unipr.it. Manuscript received 15 Apr. 1996; revised 17 Apr. 1997. Recommended for acceptance by R. Chin. For information on obtaining reprints of this article, please send e-mail to: tpami@computer.org, and reference IEEECS Log Number 104926. J:\PRODUCTION\TPAMI\2-INPROD\104926\104926_1.DOC correspondence97.dot SB 19,968 12/23/97 2:31 PM 2 8 On the other hand, the computational complexity of a serial implementation of morphological operations depends on the number of elements which form the operands. As an example, the computation of A ! B requires one vector sum and one logical union for each couple of elements a OE A and b OE B, and, thus, where 9(#) indicates the computational complexity (the number of vector operations) of a given operation, and #) represents the number of elements in a set. Using the well-known visual representation of morphological sets [3], the number of vector sums and logical unions required by dilation according to (8), is given by #(A) The structuring element B can be expressed as the dilation between subsets and using the chain rule property [3], . The number of sums required to perform the first step of the processing is given by while the number of sums required to complete the processing (R Thus, the decomposition shown in (10), while incrementing the total number of dilations from one to two, decreases the number of operations performed from 180 to 145. 2.1.2 SIMD Cellular Systems When a bitmapped data representation is used, mathematical morphology operations involve repeated computations over large data structures, thus the use of parallel systems improves the overall performance. Both parallel architectures with spatial parallelism (cellular systems), based on a high number of Processing Elements (PEs) devoted to the simultaneous processing of different image areas, and parallel architectures with operational parallelism (pipeline systems), where the different PEs work in pipeline of the same image area, share common constraints. The planar surface of the silicon chip limits the hardware interconnections, thus reducing the complexity of the elementary operations (the size of the possible SEs) that can be performed by each single PE. This fact is more evident in cellular systems, where the set of all possible operations performed by each single PE (known as Instruction Set, IS) is generally based on the use of 3 - 3 SEs. Thus, since operations based on large SEs cannot be performed, their decomposition into chains of simpler operations belonging to the IS becomes mandatory. The above dilation shows the main difference between serial and cellular systems. On serial systems the dilation can be performed either directly (with a single after the decomposition of B, as a chain of more than one dilation (as shown by (11)), thus leading to a different computational complexity. On the other hand, that dilation cannot be performed directly on a cellular system since it is based on a SE not belonging to the IS. Thus, while in the first case (serial systems), the decomposition may be recommended for a number of reasons (such as the speed-up of the processing), it becomes mandatory in the second case (cellular systems). Assuming a system capable of performing horizontal and vertical dilations and translations in the eight main directions, B (as defined in (9)) is nonconvex with respect to the IS of the system; it may be expressed as a union of convex sets, for example where are convex with respect to the IS of the system and can thus be expressed as: Thus, according to this decomposition, the initial dilation can be performed with six elementary dilations and one logical union. This is a one-level solution, involving only a single level of unions of dilations, also called sum of products (see Fig. 1a). It is obvious that a multilevel solution may lead to a better re- sult. For example using the chain rule property, can be expressed in a two-levels solution: This solution, depicted graphically in Fig. 1b, requires only five dilations and one logical union. 2.2 Optimality Criteria The decomposition of a SE can be aimed to many different goals, such as: . the minimization of the number of decomposing sets (to reduce the number of dilations); . the minimization of the total number of computations (for speed-up reasons); . the minimization of the total number of elements in the decomposing sets (to reduce the size of the data structures and thus also the memory requirements in serial systems); . the possibility to implement complex morphological operations on cellular systems whose IS is based on simple, elementary operations (to overcome the problem caused by the simple interconnection topology that limits the size of possible IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 20, NO. 2, FEBRUARY 1998 3 J:\PRODUCTION\TPAMI\2- correspond ence97.dot . or even the determination of factors with a given shape (to aid the recognition of 2D objects). The optimality criterion addressed in this work can be changed acting on the parameters of a cost function. Genetic Algorithms (GAs), widely used in various fields [11], are optimization algorithms based on a stochastic search [12], operating by means of genetic operators on a population of potential solutions of the considered problem (individuals). The main data structure is the Genome or Chromosome, that is composed of a set of Genes and of a Fitness value. In the population of possible solutions the set of new individuals generated by means of genetic operators is called Offspring. The genetic search is driven by the fitness function: Each individual is evaluated to give some quantitative measure of its fitness, that is the "goodness" of the solution it represents. At each iteration (generation) the fitness evaluation is performed on all individuals. Then, at the following iteration, a new population is generated, starting from the individuals with the highest fitness, and replac- ing, completely or partially, the previous generation. The genetic operators used to generate new individuals are subdivided into two main categories: unary operators, creating new individuals and replacing the existing ones with a modified version of them (e.g., mutation, introduction of random changes of genes), and binary operators, creating new individuals through the combination of data coming from two individuals (e.g., crossover, exchange of genetic material between two individuals). Each iteration step is called generation. The study of GAs led to the more general Evolution Programs (EPs) [13], or Generalized GAs. In "standard" GAs an individual is represented by a fixed-length binary string, encoding the parameters set, which corresponds to the solution it represents; the genetic operators act on these binary codes. In EPs, individuals are represented by generalized data structures without the fixed-length constraint [14], [15]. In addition, ad-hoc operators are defined to act on these data structures. EPs perfectly match the requirements of the SE decomposition problem, since the varying number of elementary items forming a solution does not allow to know a priori the size of a generic solu- tion, that is the length of the coding of a generic individual. In fact, for an efficient implementation, the data structure representing a decomposition must explicitly encode both the number and the shape of each single elementary operation composing the solution. Moreover, this coding must also allow fast and easy processing and evaluation phases. For these reasons it has been necessary to develop an ad hoc EP with specific genetic rules, exploiting a method similar to the one presented in [16] for the solution of the bin-packing problem. Up to now, the number of iterations is chosen by the user, but different termination criteria are under evaluation (such as the percentage of improvement or the number of different individuals) [11], [13]. 3.1 Data Structure As stated above, the data structure representing an individual has to describe in a flexible and compact way the convex elements of (5), showing its shape and decomposition into factors, but it has also to make the evaluation phase fast and simple. This representation has to be variable in length, since the number z of possible partitions involved in the decomposition of a generic individual , has no maximum bound; on the other hand, recalling (4), the number m and the shape of factors F kj , which form element C k , depend directly on C k . For these reasons an individual is represented by an arbitrarily long chain of genes, each gene representing a partition of the input SE (see Fig. 2). The logical union of all genes produces the individual. The simplest implementation consists in representing each individual with a data structure whose fields contain all the above information. Conversely, a more complex, hierarchical data structure has been developed in order both to use a lower amount of memory for each individual and to ease and speed-up the determination of new better solutions. Although the handling of this data structure is definitely complex, it allows to detect possible overlappings among the individuals of a popula- tion. Each level of the hierarchy encodes only the information strictly necessary to that level. Three are the levels of the hierar- chy, as shown in Fig. 2: . Factor level: The basic components are the elementary morphological operations (i.e., the instruction set elements): an integer indicates which element of the IS is used, while a pointer allows to follow the chain of elements. . Gene level: A gene is composed of one or more factors and it corresponds to a dilation chain of factors; an integer gives the origin of the partition described by the gene, thus speci- Fig. 1. (a) Union of dilations, also known as sum of products. (b) A two-levels solution. Fig. 2. The data structure representing two individuals. 4 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 20, NO. 1, JANUARY 1998 J:\PRODUCTION\TPAMI\2-INPROD\104926\104926_1.DOC correspondence97.dot SB 19,968 12/23/97 2:31 PM 4 8 fying the translation required to fit the gene onto the initial (the origin) and a pointer identifies the next gene. . Individual level: One or more genes form the individual that corresponds to a union of dilation chains, corresponding to a decomposition or, more often, to a part of a decomposi- tion. An integer gives the total number of genes forming the individual, a pointer gives the position of the first gene of the chain, while a double precision number contains the fitness value of the individual. 3.2 Initialization of the Population In order to understand this fundamental step, some definitions are introduced. DEFINITION 1. The IS is a set of M factors: DEFINITION 2. Notation H (x,y) stands for H # {(x, y)}, namely, it represents a translation, as in (2). DEFINITION 3. For a generic image H, . terms, 0 DEFINITION 4. For a generic image H, O H represents its origin. In the following, B is the input SE, B i is a generic subset of B with the same origin, and H represents any generic set, convex with respect to the IS: If O Fi OE F i for every F i belonging to the IS, 1 then O H OE H. The process starts with the identification of every element of set &(B), which is defined as for some (19) since every element of &(B) may represent a possible gene; this search has to be deterministic and exhaustive. Since O F OE , the set of possible pairs (h, k) is given by the set of all elements of B; for a generic image H, if (h, Therefore, the algorithm scans all the pixels of the SE and determines which factors can form a legal chain of dilations starting from that pixel. The gene obtained so far needs an additional shift in order to overlap its origin with the origin of B. Since the length of the best solution is not known a priori and since randomly linking together some genes seldom yields a legal solution, we have decided to use each element of &(B), which comprises all the workable genes, as constituting an individual with a chromosome composed of a single gene only. It is also pos- sible, as an option, to include in the population multiple copies of each individual so formed. In this way the set of individuals forming the initial population is obtained. 3.3 The Fitness Function The fitness function f(,) is used to evaluate each individual , in order to drive the algorithm during the search. A cost function f C (,) has been introduced, which is identical to f(,) if and only if is the general partition forming solution , of length N. This function must have several properties: solutions it is equal to f(,); it is defined also for nonlegal solutions (i.e., solutions not covering perfectly the original SE), thus widening the search space; it is easily implemented as a penalty function. 1. In the current implementation, each factor is limited in size to 3 - 3 and O F F OE . Penalty functions are used in highly constrained problems when the need of evaluating nonlegal solutions is met by penalizing them with respect to the legal ones. The cost function thus includes a penalty term: where a is equal to 1 if and only if B B # , as stated above, otherwise a is expressed by a term proportional to the ratio between the number of elements present in the solution and the total number of elements in B. The term b is related, via user-defined parameters, to the current population size, and f P is the penalty function, that is still related to the percentage of elements of B covered by the decomposition contained in the considered individual. Assuming that the goal of the process is to obtain the decomposition that minimizes the number of operations required to compute the dilation of a generic image with SE B, the fitness function f(,) is mainly constituted by the sum of the cost of every partition in addition, it takes into account also the number of logical union operations required, weighted by an appropriate coefficient, and the additional saving allowed by a multilevel solution. The program, in fact, lets the user choose from among three different optimization levels, thus leading to different final results: Level 0 does not perform any optimization; level 1 performs a first pack- ing, based on the methods explained at the end of Section 2.1.2; level 2 tries to pack again the solutions obtained at level 1. The use of optimization levels 1 and 2 becomes of basic importance when the target architecture has the capability to store temporary re- sults, since it allows to achieve solutions with sensibly lower costs. 3.4 The Genetic Search The structure of the algorithm is depicted in Fig. 3; a single processing cycle is composed of four stages: . Selection Operators: Choose two individuals among population for reproduction purpose; . Binary Genetic Operators: Combine, in various ways, the parents' chromosomes in order to get offspring (i.e., one child or two children); . Comparison Operators: Set up a competition between parents and offspring for inclusion in the next generation; . Unary Genetic Operators: Mutate the chromosomes of the individuals winning at the previous stage. When the list containing the individuals of the current population is exhausted, i.e., every individual has been chosen for reproduc- tion, the population just formed undergoes the same processing: this generation-formation process stops when the maximum number of allowed generations is reached. 2 In the following, a brief review of the operators is presented. 3.4.1 Selection Operators The operator is based on the tournament selection scheme as exposed in [17] with a tournament size of two. The scheme implemented makes use of a slightly modified version of the genic selective crowding technique [17] which forces individuals to compete with those who have at least I I I I pixels in common with them. In this way, a pressure is maintained for similar to compete with similar, incrementing in this way the significance of the tournament. 2. As mentioned, other termination criteria are currently under evaluation. IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 20, NO. 2, FEBRUARY 1998 5 J:\PRODUCTION\TPAMI\2- correspond ence97.dot 3.4.2 Binary Genetic Operators In order to obtain individuals that cover the SE in the initial phase we need an operator which varies appreciably the length of indi- viduals' chromosomes. Conversely, after this phase, when the average length of individuals is roughly close to the optimal one, we are concerned about the quality of the chromosome; for this reason two different operators have been conceived. . The first one is based upon the cut and splice operator [18]. Let l be the number of genes constituting a chromosome: This operator cuts the chromosome randomly in correspondence to one of the l possible points. If one of the l - 1 points connecting two consecutive genes is chosen, the chromosome is broken into two parts; otherwise, with probabilityl , it is left unchanged. The two, three, or four chromosome segments of two different individuals are pushed in a stack; the splice stage either merges the first two top elements of the stack, creating in this way a single child, or promotes each element to a full individual. This shows how the number of individuals in the following generation can be altered. An example is shown in Fig. 4a. . The second operator is the dual of the previous one: it attempts to improve the fitness of the parents by mixing their chromosomes, searching for a slight edge of improvement by trial and error. A gene composing the first parent is "injected" into the chromosome of the other parent, replacing one of its genes, thus not changing the length of the chromosome but altering only its content. The procedure is run twice, swapping the two parents' roles. The number of offspring generated is always two, although in some cases they can coincide with a parent. An example is shown in Fig. 4b. 3.4.3 Comparison Operators At this stage the operator chooses among parents and offspring the individuals to be inserted in the next generation. The scheme followed is based upon the Deterministic Crowding scheme presented in [19]. 3.4.4 Unary Genetic Operators In standard GA the unary operator is the mutation, that simply inverts randomly one or more bits of the string representing the chromosome. On the other hand, our implementation of mutation has the primary goal of reinserting genes previously discarded and otherwise definitively lost; typically this is the case of little partitions, whose contribution to the fitness improvement has been underestimated in the previous phases of the execution. This contribution can be essential later to achieve the covering of the whole B. Genes are drawn from an array of genes (containing all possible genes) and stored in memory so that every gene is chosen cyclically. Two operators have been created: . MUTATION 1: This operator compares each gene forming the chromosome of the individual with the gene g coming from the array. The gene g substitutes the most similar one in the chain, that is the gene that maximizes the intersection between the two genes, as in the following example: . MUTATION 2: This operator forces gene g to be included, along with the suppression of those which overlap with it. It can cause a big fitness worsening but it has the advantage to increase diversity in the chromosomes as a whole, as in the following example: The complete process is shown in Fig. 5 where a simple selection (which does not use a tournament scheme) is added. It is possible to traverse the graph following 2 3 paths and the decisions are made according to the value of the respective probability values These parameters are computed at the beginning of every generation starting from parameters describing the status of the current generation; they can be regarded as adaptive parameters [20]. 4.1 Decomposition of Convex SEs In accordance with the way the initial population is generated, the decomposition of a convex SE by means of this approach leads to the same results discussed in the literature (such as in [8]). This is due to the fact that the optimal solution is a member of the initial population, which consists of all possible (i.e., legal) decompositions of the SE, given an arbitrary set of elementary SEs. 4.2 Decomposition of NonConvex SEs Let us now consider the decomposition of the following nonconvex Fig. 3. The generational cycle. Fig. 4. Examples of the cut and splice. (a) Replacing crossover. (b) Operators. 6 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 20, NO. 1, JANUARY 1998 J:\PRODUCTION\TPAMI\2-INPROD\104926\104926_1.DOC correspondence97.dot SB 19,968 12/23/97 2:31 PM 6 8 whose optimal decomposition using the following 3 IS is definitely nontrivial. The stochastic decomposition led to the result shown in the following: The dilation of a generic image A with B is then reduced to the that, considering the IS shown in (23), takes a total of 50 elementary dilations and eight logical unions. Had the algorithm run with the optimization level set to one, (24) could be expressed as a sequence of 22 elementary dilations and eight logical unions: where I is the identity image. In [8] the original SE needs to be convex and it is decomposed using a given set of factors. Conversely, in [9], a wider class of SEs is considered: The original SE can also be nonconvex but must be simply connected and must belong to a specific class ' of decomposable SEs. In that paper, the decomposition of a generic SE S is defined by: S A A A n where A i is a 3 - 3 or less simply connected factor. This represents an optimal decomposition when n is minimized, regardless of the 3. This IS has been chosen to reflect the set of operations available on the PAPRICA system, a special-purpose architecture dedicated to the execution of morphological processings. shape of the A i elements. To compare this algorithm with ours, let us choose a SE belonging to ' as discussed in [9]: Hereinafter, when presenting a SE decomposition, we will use the following where M indicates the decomposition method used, B is the input indicates the function giving the cost associated with each factor belonging to the IS. The optimal decomposition of SE H proposed by Park and Chin in [9] is where f is the four-connected shift cost function mentioned in [8]. Note that using this technique the morphology of the factors is not known a priori, but only at the end. This is in contrast with our approach that requires to specify a set of factors before running the algorithm. To overcome this we can synthesize every factor in (28) with factors belonging to a generic set and use the same set within our program. Obviously, when a factor is not convex with respect to such IS, Boolean operators must be used (in this case, logical unions). The IS used here is a modified version of the set specified in [8]: The resulting decomposition is: According to the four-connected shift cost function [8], D PC (H, f) has cost 14. In the following, the decomposition obtained with our Fig. 5. A schematic representation of the generational process: at each generation individuals are taken in pairs, and in accordance to the respective probability values p1, p2 and p3, selection, binary operators and unary operators are applied. IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 20, NO. 2, FEBRUARY 1998 7 J:\PRODUCTION\TPAMI\2- correspond ence97.dot (Anelli, Broggi, Destri, ABD) stochastic approach is presented, 4 where the superscript stands for the optimization level: with the following partial results: On the other hand, if the cost is set to one for every factor (cost function g), the total cost of D PC (H, g) is equal to the cost of ABD (H, g) becomes: with the following partial results: leading to a total cost of nine. When operating with level 2 of op- timization, A # H can be easily computed replacing the identity image I with the image A in (34) and (36). Table 1 summarizes the cost of the solutions we have obtained for SE H for different optimization levels, and for cost functions f and g, along with the solution presented in [9]. Even though we had to rearrange the decompositions given in [9] in order to fit our requirements (thus altering its cost), this two examples show that the two approaches, although not directly comparable, give solutions with similar cost. In addition, in our approach the freedom of not knowing a priori the shape of the SEs composing the IS is replaced with the possibility to decompose also nonconvex SEs. 5 CONCLUSION This paper presented a new approach to the decomposition of arbitrarily shaped binary morphological structuring elements into chains of elementary factors, using a stochastic technique. The application of this technique to convex structuring elements leads to the optimal decompositions discussed in the literature; in addi- tion, this paper provides a way of decomposing also nonconvex SEs. 4. In these examples, the cost of the union operation has been set to zero. Extensive experimentations (not documented here due to space limitations) have shown that the amount of memory required by the system grows according to the size of the initial SE and with the number and size of the elementary SEs. Elements up to have been decomposed using ISs composed of eight basic operations on a two processors HP 9000 with 128 megabytes of RAM; the decompositions took about six hours of CPU time and computed 200 generations starting with an average of 2,000 individu- als; the computations required about 80 megabytes of memory. Due to the extremely high computational load required by this iterative approach and to the large memory requirements, the genetic engine is now being ported to the MPI parallel environ- ment: the decomposition is managed by a "master" process, which spawns child processes on the different nodes of a cluster of work- stations: each child process is in charge of a specific portion of the processing, which is executed in parallel with all others. This parallel implementation allows to speed up the processing and to decompose very large SEs. Moreover, a graphical interface is also being designed to ease the definition of both the initial SE and the IS, as well as the introduction of parameters. Being based on the Java programming language, its integration into a Web page is straightforward, thus allowing remote users to define and run their own decompositions on our cluster of workstations. In addition, the first release of the complete tool running on many different systems (SunOS, AIX, Linux, HP-UX, DOS, and will be shortly available as public domain software via anonymous FTP to researchers working in the mathematical morphology field. ACKNOWLEDGMENTS The authors would like to thank Prof. Aurelio Piazzi for his valuable suggestions. This work was partially supported by the Italian National Research Council (CNR) under the frame of the "Progetto Finalizzato Trasporti 2." --R "An Evolutionary Algorithm That Constructs Recurrent Neural Networks," "Speeding-Up Mathematical Morphology Computations With Special-Purpose Array Processors," "A New Representation and Operators for Genetic Algorithms Applied to Grouping Problems," "An Introduction to Simulated Evolutionary Optimiza- tion," Genetic Algorithms in Search "Messy Genetic Algorithms: Motivation, Analysis, and First Results," "Messy Genetic Algorithms Revisited: Studies in Mixed Size and Scale," "Image Analysis Using Mathematical Morphology," Adaption Natural and Artificial Systems. "Crossover Interactions Among Niches," Random Sets and Integral Geometry. Genetic Algorithms "Optimal Decomposition of Convex Structuring Elements for a 4-Connected Parallel Array Processor," "Decomposition of Arbitrarily Shaped Morphological Structuring Elements," Image Analysis and Mathematical Morphology. "Adaptive Probabilities of Crossover and Mutation in Genetic Algorithm," "Methods for Fast Morphological Image Transforms Using Bitmapped Binary Images," "Theory of Matrix Morphology," "Morphological Structuring Element Decomposition," --TR --CTR Ronaldo Fumio Hashimoto , Junior Barrera , Carlos Eduardo Ferreira, A Combinatorial Optimization Technique for the Sequential Decomposition of Erosions and Dilations, Journal of Mathematical Imaging and Vision, v.13 n.1, p.17-33, August 2000 Frank Y. Shih , Yi-Ta Wu, Decomposition of binary morphological structuring elements based on genetic algorithms, Computer Vision and Image Understanding, v.99 n.2, p.291-302, August 2005 Ronaldo Fumio Hashimoto , Junior Barrera, A Note on Park and Chin's Algorithm, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.24 n.1, p.139-144, January 2002

genetic algorithms;arbitrarily shaped structuring element decomposition;mathematical morphology

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A Volumetric/Iconic Frequency Domain Representation for Objects With Application for Pose Invariant Face Recognition.

AbstractA novel method for representing 3D objects that unifies viewer and model centered object representations is presented. A unified 3D frequency-domain representation (called Volumetric Frequency RepresentationVFR) encapsulates both the spatial structure of the object and a continuum of its views in the same data structure. The frequency-domain image of an object viewed from any direction can be directly extracted employing an extension of the Projection Slice Theorem, where each Fourier-transformed view is a planar slice of the volumetric frequency representation. The VFR is employed for pose-invariant recognition of complex objects, such as faces. The recognition and pose estimation is based on an efficient matching algorithm in a four-dimensional Fourier space. Experimental examples of pose estimation and recognition of faces in various poses are also presented.

Introduction A major problem in 3-D object recognition is the method of representation, which actually determines to a large extent, the recognition methodology and approach. The large variety of representation methods presented in the literature do not provide a direct link between the 3-D object representation and its 2-D views. These representation methods can be divided into two major categories: object centered and viewer centered (iconic). Detailed discussions are included in [15] and [12]. An object centered representation describes objects in a coordinate system attached to objects. Examples of object centered methods of representation are spatial occupancy by voxels [15], constructive solid geometry (CSG) [15], superquadrics [21] [2], algebraic surfaces [8], etc. However, object views are not explicitly stored in such representations and therefore such datasets do not facilitate the recognition process since the images cannot be directly indexed into such a dataset and need to be matched to views generated by perspective/orthographic projections. Since the viewpoint of the given image is a priori unknown, the recognition process becomes computationally expensive. The second category i.e. viewer centered (iconic) representations, is more suitable for matching a given image, since the model dataset also is comprised of various views of the objects. Examples of viewer centered methods of representation are aspect graphs [16], quadtrees [12], Fourier descriptors [30], moments [13], etc. However, in a direct viewer centered approach, the huge number of views needed to be stored renders this approach impractical for large object datasets. Moreover, such an approach does not automatically provide a 3-D description of the object. For example, in representations by aspect graphs [16], qualitative 2-D model views are stored in a compressed graph form, but the view retrieval requires additional 3-D information in order to generate the actual images from different viewpoints. For recognition purposes, viewer centered representations do not offer a significant advantage over object centered representations. In summation, viewer centered and object centered representations have complementary merits that could be augmented in a merged representation - as proposed in this paper. A first step in unifying object and viewer centered approaches was provided by our recently developed iconic recognition method by Affine Invariant Spectral Signatures (AISS) [6] [5] [4], which was based on an iconic 2-D representation in the frequency domain. However, the AISS is fundamentally different from other viewer centered representations since each 2-D shape representation encapsulates all the appearances of that shape from any spatial pose. It also implies that the AISS enables to recognize surfaces which are approximately planar, invariant to their pose in space. Although this approach is basically viewer centered, it has the advantage of directly linking 3-D model information with image information, thus merging object and viewer centered approaches. Hence, to generalize the AISS it is necessary to extend it from 2-D or flat shapes to general 3-D shapes. Towards this end, we describe in Section 2, a novel representation of 3-D objects by their 3-D spectral signatures which also captures all the 2-D views of the object and therefore facilitates direct indexing of a given image into such a dataset. As a demonstration of the VFR, it is applied for estimating pose of faces and face recognition in Section 3. Range image data of a human head is used to construct the VFR model of a face. We demonstrate that reconstruction from slices of the VFR results are accurate enough to recognize faces from different spatial poses and scales. In Section 3, we describe the matching technique by means of which a gray scale image of a face is directly indexed into the 3-D VFR model based on fast matching by correlation in a 4 dimensional Fourier space. In our experiments (described in Section 5), we demonstrate how the range data generated from a model is used to estimate the pose of a person's face in various images. We also demonstrate the robustness of our 2-D slice matching process by recognizing faces with different poses from a dataset of 40 subjects, and present statistics of the matching experiments. Frequency Representation (VFR) In this section, we describe a novel formulation that merges the 3-D object centered representation in the frequency domain to a continuum of its views. The views are also expressed in the frequency domain. The following formulation describes the basic idea. Given an object O, which is defined by its spatial occupancy on a discrete 3-D grid as a set of voxels fV (x; z)g, we assume without loss of generality, that the object is of equal density. Thus, V (x; otherwise. The 3-D Discrete Fourier Transform (DFT) of the object is given by V(u; v; z)g. The surface of the object is derived from the gradient vector field @ @x @ @y @ @z where k x , k y and k z are the unit vectors along the x; y and z axes. The 3-D Discrete Fourier Transform (DFT) of the surface gradient is given by the frequency domain vector field: F frV (x; Let the object be illuminated by a distant light source 1 with uniform intensity \Upsilon and direction z . We assume that the object O is a regular set [1] and has a constant gradient magnitude K on the object surface, i.e. j rV K. The surface normal is given by . We also assume that O has a Lambertian surface with constant albedo A. Thus points on its surface have a brightness proportional to @ @x @ @y @ @z i are the positive and negative parts. The function z) is not a physically realizable brightness and is introduced only for completeness of Eq. (3). The separation of the brightness function into positive and negative components is used to consider Additional light sources can be handled using superposition. only positive illuminations. The negative components are disregarded in further processing, as this function is separable only in the spatial domain. As elaborated in Section 2.2, can be eliminated using a local Gabor transform. In another approach, the side of the object away from the illumination can be considered as planar and becomes a plane with a negative constant value which does not alter the resulting image. It is also necessary to consider the viewing direction when generating views from the VFR. The brightness function B i (x; y; z) is decomposed as a 3-D vector field by projecting onto the surface normal at each point of the surface. This enables the correct projection of the surface from a given viewpoint. As noted earlier, the surface normal is given by rV Thus, the new vectorial brightness function B i is given by \UpsilonA The 3-D Fourier transform of this model is a complex 3-D vector field V i (u; v; w)=FfB i (x; z)g. The transform is evaluated as: (i x where denotes convolution. Variation in illumination only emphasizes the amplitude of V i in the (i x but does not change its basic structure. The absolute value of defined as the VFR signature. 2.1 Projection Slice Theorem and 2-D Views The function V i (u; v; w) is easily obtained, given the object O. To generate views of the object, we resort to 3-D extensions of the Projection-Slice Theorem [14] [24] that projects the 3-D vector field V i (u; v; w) onto the central slice plane normal to the viewpoint direction. Fig. 1 illustrates the principle by showing the slice derived from the 3-D DFT of a rectangular block. Orthographically viewing the object from a direction results in an image I c which has a 2-D DFT given by Ic To find I c its necessary to project the vector brightness function along the Figure 1: The Projection-Slice Theorem: A slice of the 3-D Fourier Transform of a rectangular block (on the right) is equivalent to the 2-D Fourier Transform of the projection of the image of that block (on the left). viewing direction c after removing all the occluded parts from that viewpoint. The vectorial decomposition of the brightness function along the surface normals as given by Eq. (4) compensates for the integration effects of projections of slanted surfaces. This explains the necessity of using a vectorial frequency domain representation. Removing the occluded surfaces is not a simple task if the object O is not convex or if the scene includes other objects that may partially occlude O. For now, we shall assume that O is convex and is entirely visible. This assumption is quite valid for local image analysis where a local patch can always be regarded as either entirely occluded or visible. Also, for local z) is not a major problem. The visible part of B i (x; y; z) from direction c, denoted by B ic (x; y; z), is given by \UpsilonA where hwr[ff] is the "half wave rectified" value of ff, i.e. Now V ic (u; v; w) can be obtained from B ic (x; simply by calculating the DFT, The image DFT Ic obtained using the Projection-Slice Theorem [14] [24] by slicing V ic (u; v; w) through the origin with a plane normal to c, i.e. derived by sampling V ic (u; v; w) on this plane. An example of such a slicing operation is illustrated in Fig. 1. Note that V ic actually encapsulates both the objects 3-D representation and the continuum of its view-signatures, which are stored as planar sections of j V ic j. As we see from Eq. (5), variations in illumination only emphasizes the amplitude of V i in (i x direction, but do not change its basic structure. Thus, it is feasible to recognize objects that are illuminated from various directions by local signature matching methods as described in Section 2.3, while employing the same signature. 2.2 Local Signature Analysis in 3-D Local signature analysis is implemented by windowing B i with a 3-D Gaussian centered at location proceeding as in Eq. (4) on the windowed object gradient. Such local frequency analysis is implemented by using the Gabor Transform (GT) instead of the DFT. The transition required from the DFT to the GT is quite straightforward. The object O is windowed with a 3-D Gaussian to give oe x oe y oe z The equivalent local VFR is given by The important outcome from this are: 1) The Projection-Slice Theorem [14], [24] can be still employed for local space-frequency signatures of object parts. 2) In local space-frequency almost always does not contain the problematic i part, which can be eliminated by the windowing function. We note that for most local surfaces, [B i \Delta as the local analysis approximates the hwr[\Delta] function with respect to viewing direction c. Hence, the VFR of B ic is a general representation of a local surface patch of V (x; uc +vc x y z e a u' w' slice plane Figure 2: The frequency domain coordinate system in which the slice plane is defined. are the direction cosines of the slice plane normal, which has an azimuth ff and an elevation ffl. Image swing is equivalent to in-plane rotation ', and viewing distance results is variation in the radial frequency r f of the VFR function. 2.3 Indexing using the VFR signature As explained in Section 2.1, the VFR is a continuum of the 2-D DFT of views of the model. To facilitate indexing into the VFR signature data structure, we consider the VFR signature slice plane uc x are the direction cosines of the slice plane normal. We define a 4-D pose space in the frequency domain which consists of the azimuth ff and elevation ffl, defining the slice plane normal with respect to the original axes, the in-plane rotation ' of the slice plane and the scale ae which changes with the distance to the viewed object. Fig. 2 illustrates the coordinate system used. [c x are related to the azimuth ff and elevation ffl as follows6c x c y c z7 sin ff cos ffl sin ffl7\Gamma-=2 - ff -=2 We note again that slices of the VFR signature are planes which are parallel to the imaging plane. Thus the image plane normal and the slice plane normal coincide. By using 3-D coordinate transformations (see Fig. 2) we can transform the frequency domain VFR model to the 4-D pose space (ff; ffl; '; ae). Let (u; v; w) represent the original VFR coordinate system and (-u; - v; - w) be the coordinate system defined by the slice plane. The slice plane is within the 2-D coordinate system (-u; - v), where - w is the normal to the slice plane and corresponds to the viewing direction. The relation between these two systems is given by6u sin ff sin ffl cos ff sin ff cos ffl VFR signature slices, being 2-D DFT's of model views are further transformed to polar coordinates by considering the in-plane rotation ' (equivalent to the image swing or rotation about the optical axis), and the radial frequency r f . - u sin ' \Gamma-=2 -=2 The radial frequency r f is transformed logarithmically to attain exponential variation of r f given by ae = log a . The full transformation of the coordinate system to the 4-D pose space is given by 24 cos ' sin ff sin ffl \Gamma sin ' cos ff sin ' cos ffl7 Thus, the 4-tuple (ff; ffl; '; ae) defines all the points in the 3-D VFR signature frequency space We observe that the space defined by the 4-tuple (ff; ffl; '; ae) is redundant in the sense that infinite number of 4-tuples (ff; ffl; '; ae) may represent the same (u; v; w) point. However, this representation has the important advantage that every (ff; ffl) pair defines a planar slice in V ic (u; v; w). Moreover, every ' defines an image swing and every ae defines another scale. Thus the (ff; ffl; '; ae) representation significantly simplifies the indexing search for the viewing poses and scales. Now, the indexing can be simply implemented by correlation in the frequency domain to immediately determine all pose parameters by linear shifts in space. The significance of this transformation to the 4-D pose space is in using the following properties. The polar coordinate transformation within the slice allows rotated image views to have 2-D frequency domain signatures which shift along the ' axis. Similarly the exponential sampling of the radial frequency r f results is scale changes causing linear shifts along the ae axis. Thus the new coordinate system given by (ff; ffl; '; ae) results in a 2-D frequency domain signature which is invariant to view point and scale and results only in linear shifts in the 4-D pose space so defined. A particular slice corresponding to a particular viewpoint is easily indexed into the transformed VFR signature by using correlation. 3 Pose Estimation and Recognition of Human Faces Recognition of human faces is a hard problem for machine vision, primarily due to the complexity of the shape of a human face. The change in the observed view caused by variation in facial pose is a continuum which needs large numbers of stored models for every face. Since the representation of such a continuum of 3-D views is well addressed by our VFR, we present here, the application of our VFR model for pose-invariant recognition of human faces. First we discuss some of the existing work in face recognition in Section 3.1 followed by our approach to the problem in Section 3.2. We present our results in face pose estimation (Section 4) and face recognition (Section 3) and compare our results in face recognition to some other recent works using the same database [20]. azimuth a elevation e Figure 3: Reconstructions of a model face from slices of the VFR are shown for various azimuths and elevations. Note that all facial features are accurately reconstructed indicating the robustness of the VFR model. 3.1 Face Recognition: A Literature Survey Recent works in face recognition have used a variety of representations including parameterized models like deformable templates of individual facial features [29] [26] [10], 2-D pictorial or iconic models using multiple views [9] [7], matching in eigenspaces of faces or facial features [22] and using intensity based low level interest operators in pictures. Other recent significant approaches have used convolutional neural networks [18] as well as other neural network approaches like [11] and [28]. Hidden Markov Models [25], modeling faces as deformable intensity surfaces [19], and elastic graph matching [17] have also been developed for face recognition. Parameterized models approaches like that of Yuille et al. [29], use deformable template models which are fit to preprocessed images by minimizing an energy functional, while Terzopoulos and Waters [26] used active contour models of facial features. Craw et al. [10] and others have used global head models from various smaller features. Usually deformable models are constructed from parameterized curves that outline subfeatures such as the iris or a lip. An energy functional is defined that attracts portions of the models to pre-processed versions of the image and model fitting is performed by minimizing the functional. These models are used to track faces or facial features in image sequences. A variation is the deformable intensity surface model proposed by Nastar and Pentland [19]. The intensity is defined as a deformable thin plate with a strain energy which is allowed to deform and match varying poses for face recognition. A 97% recognition rate is reported for a database with 200 test images. Template based models have been used by Brunelli and Poggio [9]. Usually they operate by direct correlation of image segments and and are effective only under invariant conditions of scale orientations and illumination. Brunelli and Poggio computed a set of geometrical features such as nose width and length, mouth position and chin shape. They report 90% recognition rate on a database of 47 people. Similar geometrical considerations like symmetry [23] have also been used. A more recent approach by Beymer [7] uses multiple views and a face feature finder for recognition under varying pose. An affine transformation and image warping is used to remove distortion and bring correspondence between test images and model views. Beymer reports a recognition rate of 98% of a database of 62 people, while using 15 modeling views for each face. Among the more well known approaches has been the eigenfaces approach [22]. The principal components of a database of normalized face images is used for recognition. The results report a 95% recognition rate from a database of 3000 face images of about 200 people. However, it must be noted that the database has several face images of each person with very little variation in face pose. More recent reports on a fully automated approach with extensive preprocessing on the FERET database indicate only 1 mistake on a database of 150 frontal views. Elastic graph matching using the dynamic link architecture [17] was used quite successfully for distortion invariant recognition. Objects are represented as sparse graphs. Graph vertices labeled with multi-resolution spectral descriptions and graph edges associated with geometrical distances form the database. A recognition rate of 97.3% is reported for a database of 300 people. Neural network approaches have also been popular. Principal components generated using an autoassociative network have been used [11] and classified using a multilayered perceptron. The database consists of 20 people with no variation in face pose or illumination. Weng and Huang used a hierarchical neural network [28] on a database of 10 subjects. A more recent approach uses a hybrid approach using self organizing map for dimensionality reduction and a convolutional neural networks for hierarchical extraction of successively larger features for classification [18]. The reported results show a 3.8% error rate on the ORL database using 5 training images per person. In [25], a HMM-based approach is used on the ORL database. Error rates of 13% were reported using a top-down HMM. An extension using a pseudo two-dimensional HMM reduces the error to 5% on the ORL database. 5 training and 5 test images were used for each of 40 people under various pose and illumination conditions. 3.2 VFR model of faces In our VFR model, we present a novel representation using dense 3-D data to represent a continuum of views of the face. As indicated by Eq. (7) in Section 2, the VFR model encapsulates the information in the 3-D Fourier domain. This has the advantage of 3-D translation invariance with respect to location in the image coupled with faster indexing to a view/pose of the face using frequency domain scale and rotation invariant techniques. Hence, complete 3-D pose invariant recognition can be implemented on the VFR. Range data of the head is acquired using a Cyberware range scanner. The data consists of 256 \Theta 512 range information from the central axis of the scanned volume. 360 ffi of azimuth is sampled in 512 columns and heights in the range of 25 to 35 cm is sampled in 256 rows. The data is of the heads of subjects looking straight ahead at 0 ffi azimuth and 0 ffi latitude corresponding to the x-axis. This model is then illuminated with numerous sources of uniform illumination thus approximating diffuse illumination. The resulting intensity data in converted from the cylindrical coordinates of the scanner to Cartesian coordinates and inserted in a 3-D surface representation of the head surface as given by Eq. (3). The facial region of interest to us is primarily the frontal region consisting of the eyes, lips and nose. A region corresponding to this area is extracted by windowing the volumetric surface model with a 3-D ellipsoid centered at the nose with a Gaussian fall-off. The parameters of the 3-D volumetric mask are adjusted to ensure that the eyes, nose and lips are contained within it, with the fall off beyond the facial region. The model thus formed is a complex surface which consists of visible parts of the face from an continuous range of view centered around the x-axis or the (0 direction. The resulting model then corresponds to Eq. (6) in our VFR model. Applying Eq. (7), the VFR of the face is obtained. The VFR model is then resampled into the 4-D pose space using Eq. (13) as described in Section 2.3. Reconstructions of a range of viewpoints from a model head, from the VFR slices are shown in Fig. 3. We see from the reconstructions, that all relevant facial characteristics are retained thus justifying our use of the vectorial VFR model. This model is used in the face pose estimation experiments. 3.3 Indexing images into the VFR signature Images of human faces are masked with an ellipse with Gaussian fall-off to eliminate background textures. The resulting image shows the face with the eyes nose and lips. The magnitudes of Fourier transform of the windowed 2-D face images are calculated. The windowing has the effect of focusing on local frequency components (or foveating) on the face. while retaining the frequency components due to facial features. The Fourier magnitude spectrum make the spectral signature translation invariant in the 2-D imaging plane. The spectrum is then sampled in the log-polar scheme similar to the slices of the VFR signa- ture. As most illumination effects are typically lower frequency, band pass filtering is used to compensate for illumination. The spectral signatures from the gray scale images are localized (windowed) log-polar sampled Fourier magnitude spectra. The continuum of slices of the VFR provide all facial poses, and band-passed Fourier magnitude spectrum provides 2-D translation invariant (in the imaging plane) signatures. Log-polar sampling of the 2-D Fourier spectrum allows for scale invariance (translation normal to the imaging plane) and rotation invariance (within the imaging plane). This is because a scaled image manifests itself in Fourier spectrum inversely proportional to the scale and a rotated image has a rotated spectrum. Thus scaled and rotated images have signatures which are only linearly shifted in the log-polar sampled frequency domain. The pose of a given image is determined by correlating the intensity image signature with the VFR in the 4-D pose space. The matching process is based on indexing through the sampled VFR signature slices and maximizing the correlation coefficient for all the 4 pose parameters. The correlation is performed on the signature gradient which reduces dependence of actual spectral magnitudes and as it considers only the shape of the spectral envelope. The results take the form of scale and rotation estimate along with a matching score from 0 to 1. Similar matching methods have been very sucessfully used to match Affine Invariant Spectral Signatures (AISS) [27] [3] [6] [5] [4]. References [27] and [3] already include detailed noise analysis with white and colored noise which shows robustness to noise levels of up to 0 dB SNR for these matching methods. Table 1: Pose estimation errors for faces with known pose. These are the averaged absolute errors for angles and standard deviation of the ratio of estimated size to true size for scale. Azimuth Error Elevation Error Rotation Error Scale Std. Dev. 4 Face Pose Estimation To verify the accuracy of the pose estimation procedure, the method is first tested on images generated from the 3-D face model. 20 images of the face in Fig. 3 are generated using random viewpoints and scales from uniform distributions. The azimuth and elevation are in the range the rotation angle is in the range [\Gamma45 and the scale in the range [0:5; 1:5]. These are indexed in the VFR signature pose space. The results are summarized in Table 1. An example of the correlation peak for the estimated pose in azimuth and elevation is shown in Fig. 4(b) for the test image in Fig. 4(a). The corresponding reconstructed face from the VFR signature slice is shown in Fig. 4(c). a b c Figure 4: (a) A test image with pose parameters The correlation maximum in the azimuth-elevation dimensions of the pose space. The peak is quite discriminative as seen by relative brightness. (c) The reconstructed image from the slice which maximizes the correlation. Pose parameters In addition, we also show the results of pose estimation of face images of the subject with unknown pose and illumination in Fig. 5. Figure 5: Using the VFR model, the pose of the face in the above images is estimated and the faces are recognized. The estimated poses are given in terms of the 4-tuple azimuth ff, the elevation ffl, the relative swing (rotation) ', and the relative scale r 0 a ae . The results are A:(+15 Table 2: Face recognition using the ORL database. Recognition rates are given for 5, 6, 7 and 8 images as VFR signature slices. Number of Slices 5 6 7 8 Recognition Rate 92.5% 95.6% 96.6% 100% 5 Face Recognition Results In this section, we describe experiments on face recognition based on the VFR model. The ORL database [20] is used. The ORL database consists of 10 images of each of 40 people taken in varying pose and illumination. Thus, there are a total of 400 images in the database. Figure Shown are images of a few faces from the set of test images which are used for the face recognition task using our matching scheme. We select a number of these images varying from 5 to 8 as model images and the remaining images form the test set. The model images are windowed with an ellipse with a Gaussian fall-off. The recognition is robust to the window parameters selected, provided the value of oe for the Gaussian fall-off is relatively large. The images are 112 \Theta 92 pixels. The window parameters chosen were for the longer elliptical axis aligned vertically and 22 pixels for the shorter axis aligned horizontally and pixels. Each window is centered at (60,46). This allows for faster processing rather than manually fitting windows to each face image. Thus, the same elliptical Gaussian window was used on all model and test images even though its axes does not align accurately with the axes of all the faces. The windowed images are transformed to the Fourier domain and then sampled in a log-polar format, now correspond to slices in a 4-D VFR signature pose space. The test images are then indexed into the dataset of slices for each person. The recognition rates using 5, 6, 7 and 8 model images are summarized in Table 2. As can be seen, a recognition rate of 92.5% is achieved when using 5 slices. This increases to 100% when using 8 slices in the model. A few of the test images that are recognized are shown in Fig. 6. Computationally each face indexing takes about 320 seconds when using 5 slices and up to about 512 seconds when using 8 slices. The experiments are performed on a 200 MHz Pentium Pro running Linux. 6 Summary and Conclusions We present a novel representation technique for 3-D objects unifying both the viewer and model centered object representation approaches. The unified 3-D frequency-domain representation (called Volumetric Frequency Representation - VFR) encapsulates both the spatial structure of the object and a continuum of its views in the same data structure. We show that the frequency-domain representation of an object viewed from any direction can be directly extracted employing an extension of the Projection Slice theorem. Each view is a planar slice of the complete 3-D VFR. Indexing into the VFR signature is shown to be efficiently done using a transformation to a 4-D pose space of azimuth, elevation, swing (in-plane image rotation) and scale. The actual matching is done by correlation techniques. The application of the VFR signature is demonstrated for pose-invariant face recognition. Pose estimation and recognition experiments is carried out using a VFR model constructed from range data of a person and using gray level images to index into the model. The pose estimation errors are quite low at about 4:05 ffi in azimuth, 5:63 ffi in elevation, 2:68 ffi in rotation and 0:0856 standard deviation in scale estimation. The standard deviation in scale is taken for the ratio of estimated size to true size. Thus it represents the standard deviation assuming a scale of 1.0. Face recognition experiments are also carried out on a large database of 40 subjects with face images in varying pose and illumination. Varying number of model images between 5 and 8 is used. Experimental results indicate recognition rates of 92.5% using 5 model images and goes up to 100% using 8 model images. This compares well with [25] who reported recognition rates of 87% and 95% using the same database with 5 training images. The eigenfaces approach [22] was able to achieve a 90% recognition rate on this database. It also is comparable to the recognition rates of 96.2% reported in [18] again using 5 training images per person from the same database. These are highest reported recognition rates for the ORL database in the literature. The VFR model holds promise as a robust and reliable representation approach that inherits the merits of both the viewer and object centered approaches. We plan future investigations in using the VFR model for robust methods in generic object recognition. --R "Computer Vision," "Superquadrics and Angle Preserving Transformations," "Pictorial Recognition Using Affine Invariant Spectral Sig- natures," "Affine Invariant Shape Representation and Recognition using Gaussian Kernels and Multi-dimensional Indexing," "Iconic Recognition with Affine-Invariant Spectral Signatures," "Iconic Representation and Recognition using Affine-Invariant Spectral Signatures," "Face Recognition Under Varying Pose," "Describing Surfaces" "Face Recognition: Features versus Templates," "Finding face features," "Non-linear dimensionality reduction," "Object Models and Matching," "Visual Pattern recognition by Moment Invariants," "Image Reconstruction from Projections," "Object Recognition," "The Internal Representation of Solid Shape with respect to Vision," "Distortion Invariant Object Recognition in the Dynamic Link Architecture," "Face recognition: A Convolutional Neural Network Approach," Olivetti and Oracle Research Laboratory "Perceptual Organization and the Representation of Natural Form," "View-based and modular eigenspaces for face recognition," "Robust detection of facial features by generalized symme- try," "Parameterisation of a Stochastic Model for Human Face "Analysis of Facial Images using Physical and Anatomical Models," "SVD and Log-Log Frequency Sampling with Gabor Kernels for Invariant Pictorial Recognition," "Learning Recognition and Segmentation of 3- D Objects from 2-D Images," "Feature Extraction from Faces using Deformable Templates," "Fourier Descriptors for Plane Closed Curves," --TR --CTR Ching-Liang Su, Robotic Intelligence for Industrial Automation: Object Flaw Auto Detection and Pattern Recognition by Object Location Searching, Object Alignment, and Geometry Comparison, Journal of Intelligent and Robotic Systems, v.33 n.4, p.437-451, April 2002 Ching-Liang Su, Face Recognition by Using Feature Orientation and Feature Geometry Matching, Journal of Intelligent and Robotic Systems, v.28 n.1-2, p.159-169, June 2000 Yoshihiro Kato , Teruaki Hirano , Osamu Nakamura, Fast template matching algorithm for contour images based on its chain coded description applied for human face identification, Pattern Recognition, v.40 n.6, p.1646-1659, June, 2007 Seong G. Kong , Jingu Heo , Besma R. Abidi , Joonki Paik , Mongi A. Abidi, Recent advances in visual and infrared face recognition: a review, Computer Vision and Image Understanding, v.97 n.1, p.103-135, January 2005

Volumetric frequency representation VFR;pose invariant face recognition;object representation;4D Fourier space;face pose estimation;projection-slice theorem

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An Efficient Solution to the Cache Thrashing Problem Caused by True Data Sharing.

AbstractWhen parallel programs are executed on multiprocessors with private caches, a set of data may be repeatedly used and modified by different threads. Such data sharing can often result in cache thrashing, which degrades memory performance. This paper presents and evaluates a loop restructuring method to reduce or even eliminate cache thrashing caused by true data sharing in nested parallel loops. This method uses a compiler analysis which applies linear algebra and the theory of numbers to the subscript expressions of array references. Due to this method's simplicity, it can be efficiently implemented in any parallel compiler. Experimental results show quite significant performance improvements over existing static and dynamic scheduling methods.

Introduction Parallel processing systems with memory hierarchies have become quite common today. Commonly, most multiprocessor systems have a local cache in each processor to bridge the speed gap between the processor and the main memory. Some systems use multi-level caches [5, 14]. Very often, a copy-back snoopy cache protocol is employed to maintain cache coherence in these multiprocessor systems. Certain supercomputers also use a local memory which can be viewed as a program-controlled cache. When programs with nested parallel loops are executed on such parallel processing systems, it is important to assign parallel loop iterations to the processors in such a way that unnecessary data movement between different caches is minimized. For convenience, in this paper, we call each iteration of a parallel loop a thread. The following loop nest is an example. DO I=1,100 DO K=1,100 ENDDO ENDDO In this example, loop J is a parallel loop because, with the value of I fixed, statement S has no loop-carried dependences, while J varies. Loop J is executed 100 times in the loop nest, creating 10,000 iterations, or 10,000 threads, in total. Each thread addresses 100 elements of array A. Many array elements are repeatedly accessed by these threads as shown in Table 1 and Figure 1, where T i;j denotes the thread corresponding to loop index values I = i and denotes the set of threads created by the index value I = i. As shown in Figure 2, there exist lists of threads: (T 1;1 ), (T 1;2 , T 2;1 ), that each thread modifies and reuses most of the array elements accessed by the neighboring threads in the same list. If the threads in the same list are assigned to different processors, the data of array A will unnecessarily move back and forth between different caches in the system, causing a cache thrashing problem due to true data sharing [12]. The nested loop construct shown in the above example is quite common in parallel code used for scientific computation. J. Fang and M. Lu studied a large number of programs including the LINPACK benchmarks, the PERFECT Club benchmarks, and programs for mechanical CAE, computational chemistry, image and signal processing, and petroleum applications [11]. They reported that almost all of the most time-consuming loop nests contain at least three loop levels, out of which 60% contain at least one parallel loop. Even after using loop interchange to move parallel loops outwards when it was legal, they still found 94% of the parallel loops enclosed by sequential loops. Such loop nests include the cases in which a parallel loop appears in the outermost loop level in a subroutine, but the subroutine is called by a call-statement which is contained by a sequential loop. Most of these loop nests are not perfectly nested, i.e. there exist statements right before or after an inner loop. Fang and Lu proposed a thread alignment algorithm to solve the cache thrashing problem which may be created by such multi-nested loops. Their algorithm, however, assigns threads to processors either by solving linear equations at run time or by storing the precomputed numerical solutions to the equations in the processor's memory. Since storing all the numerical solutions requires quite a large memory space, and since the exact number of threads often cannot be determined statically due to unknown loop bounds, they favor the on-line computation approach. In this paper, we present a method to reduce the run-time overhead in Fang and Lu's algorithm by using a thorough compiler analysis of array references to derive a closed-form formula that solves the thread assignment equations. The thread assignment then becomes highly efficient at run time. Previously, we presented preliminary algorithms [22, 23] to deal with a simple case in which data-dependent array references use the same linear function in the subscripts. No experimental data were given. In this paper, we extend the work by covering multiple linear functions and by clarifying the underlying theory. We report experimental results using a Silicon Graphics (SGI) multiprocessor. Table 1: The elements of A accessed by each thread. With our method, the compiler analyzes the data dependences between the threads and uses that information to restructure the nested loop, perfectly nested or otherwise, in order to reduce or even eliminate true data sharing, which causes cache thrashing. Our method can be efficiently implemented in any parallel compiler, and our experimental results show quite significant improvement over existing static and dynamic scheduling methods. In what follows, we first address related work. We then introduce basic concepts and assumptions. After that, we present solutions to the cache thrashing problem due to true data sharing, and lastly we show the experimental results conducted on an SGI multiprocessor system. Related Work Extensive research regarding efficient memory hierarchies has been reported in the literature. Abu-Sufah, Kuck and Lawrie use loop blocking to improve paging performance by improving the locality of references [2]. Wolfe proposes iteration space tiling as a way to improve data reuse in a cache or a local memory [35]. Gallivan, Jalby and Gannon define a reference window for a dependence as the variables referenced by both the source and the sink of the dependence [15, 16]. After executing the source of the dependence, they save the associated reference window in the cache until the sink has also T1,3 Figure 1: The elements of A accessed by different threads. been executed, which may increase the number of cache hits. Carr, Callahan and Kennedy [7, 8] discuss options for compiler control of a uniprocessor's memory hierarchy. Wolf and Lam develop an algorithm that estimates all temporal and spatial reuse of a given loop permutation [34]. These optimizations all attempt to maximize the reuse of cached data on a single processor. They also have a secondary effect of improving multiprocessor performance by reducing the bandwidth requirement of each processor, thereby reducing contention in the memory system. In contrast, our work considers a multiprocessor environment where each processor has its own local cache or its own local memory, and where different processors may share data. The work by Peir and Cytron [28], Shang and Fortes [30], and by D'Hollander [9] share the common goal of partitioning an index set into independent execution subsets such that the corresponding loop iterations can execute on different processors without interprocessor communication. Their methods apply to a specific type of loop nest called a uniform recurrence or a uniform dependence algorithm, in which the loops are perfectly nested, the loop bounds are constant, the loop-carried dependences have constant distances, and the array subscripts are of the form is a loop index and c an integer constant. Hudak and Abraham [1, 18] develop a static partitioning approach called adaptive data partitioning (ADP) to reduce interprocessor communication for iterative data-parallel loops. They also assume perfectly nested loops. The loop body is restricted to update a single data point A(i; a two-dimensional global matrix A. The subscript expressions of right-hand side array references are restricted to be the sum of a parallel loop index and a small constant, while the subscript expressions of left-hand array references are restricted to contain the parallel loop indices only. Tomko and Abraham T1,3 T100,3 Figure 2: Lists of threads accessing similar array elements. [32] develop iteration partitioning techniques for data-parallel application programs. They assume that there is only one pair of data access functions and that each loop index variable can appear in only one dimension of each array subscript expression. Agarwal, Kranz, and Natarajan [3] propose a framework for automatically partitioning parallel loops to minimize cache coherence traffic on shared-memory multiprocessors. They restrict their discussion to perfectly nested doall loops, and assume rectangular iteration spaces. Unlike these previous works, our work considers nested loops which are not necessarily perfectly nested. Loop bounds can be any variables, and array subscript expressions are much more general. Many researchers have studied the cache false sharing problem [10, 17, 19, 33] in which cache thrashing occurs when different processors share the same cache line of multiple words, although the processors do not share the same word. Many algorithms have been proposed to reduce false sharing by better memory allocation, better thread scheduling, or by program transformations. Our work considers cache thrashing which is due to the true sharing of data words. Our work is most closely related to the research done by Fang and Lu [11, 12, 26, 13]. In their work, the iteration space is partitioned into a set of equivalence classes, and each processor uses a formula to determine which iterations belong to the same equivalence class at execution time. Each processor then executes the corresponding iterations so as to reduce or eliminate cache thrashing. These iterations are the solution vectors of a linear integer system. In Fang and Lu's work, these vectors may either be computed at run time or may be precomputed and later retrieved at run time when loop bounds are known before execution. Both approaches require additional execution time when a processor fetches the next iteration. Unlike Fang and Lu's approaches, we solve the thrashing problem at compile time to reduce run-time overhead, while we achieve the same effect of reducing cache thrashing. Our new method restructures the loops at compile time and is based on a thorough analysis of the relationship between the array element accesses and the loop indices in the nested loop. We have performed experiments on a commercial multiprocessor, namely a Silicon Graphics Challenge Cluster, thereby obtaining real data regarding cache thrashing and its reduction. In contrast, previous data were mainly from simulations [11, 12, 26]. 3 Basic Concepts and Assumptions Data dependences between statements are defined in [6, 24, 4, 25]. If a statement S 1 uses the result of another statement S 2 , then S 1 is flow-dependent on S 2 . If S 1 can safely store its result only after fetches the old data stored in that location, then S 1 is anti-dependent on S 2 . If S 1 overwrites the result of S 2 , then S 1 is output-dependent on S 2 . A dependence within an iteration of a loop is called a loop-independent dependence. A dependence across the iterations of a loop is called a loop-carried dependence. There can be no cache thrashing due to true data sharing if the outermost loop is parallel, because no data dependences will cross the threads. Therefore, in this paper, we consider only loop nests whose outermost loops are sequential. To simplify our discussion, we make the following assumptions about the program pattern: 1) All functions representing array subscript expressions are linear. 2) The loop construct considered here consists of a sequential loop which embraces one or several single-level parallel loops. If there exist multilevel parallel loops, only one level is parallelized, as on most commercial shared-memory multiprocessor systems. Hence, as shown below, a loop nest in our model has three levels: a parallel loop, its immediately enclosed sequential loop, and its immediately enclosing sequential loop: ENDDO ENDDO Figure 3: The loop nest model. are linear mappings from the iteration space N 1 \Theta N 2 \Theta N 3 to the domain space M 1 \Theta M 2 of A: and they can be expressed as: array k). The loops in the above example are not necessarily perfectly nested. Our restructuring techniques, to be presented later, assume arbitrary loop bounds, although we are showing lower bounds of 1 here for simplicity of notation. Multiple array variables and multiple linear subscript functions may exist in the nested loop. Since we are considering cache thrashing due to true data sharing, i.e. due to data dependences between threads, we can also write the loop nest in Figure 3 as: ENDDO ENDDO where A fl m) is an array name appearing in the loop body, ~ h m) are linear mappings from iteration space N 1 \Theta N 2 \Theta N 3 to domain space M 1 fl \Theta M 2 fl of A fl , A m) are potentially dependent reference pairs, and m is the number of such pairs. Fang and Lu [11] reported that arrays involved in nested loops are usually two-dimensional or three-dimensional with a small-sized third dimension. The latter can be treated as a small number of two-dimensional arrays. Nested loops with the parallel loop at the innermost level are degenerate cases of the loop nest in Figure 3. Therefore, our loop nest model seems quite general. Our method can also be applied to a loop nest which contains several separate parallel loops at the middle level. Each of these parallel loops may be restructured according to its own reference patterns, such that the threads in different instances of the same parallel loop are aligned. We currently do not align the threads created by different parallel inner loops. For programs with more complicated loop nests, pattern-matching techniques can be used to identify a loop subnest that matches the nest shown in Figure 3. Other outer- or inner- loops that are not a part of the subnest can be ignored, as long as their loop indices do not appear in the array subscripts. The compiler analysis is based on a simple multiprocessor model in which the cache memory has the following characteristics: 1) it is local to a processor; 2) it uses a copy-back snoopy coherence strategy; and its line size is one word. The transformed code, however, will execute correctly on machines which have a multiword cache line and multilevel caches. Furthermore, as the experimental results will show, the performance of the transformed code is quite good on such realistic machines. Our analysis can also be extended to incorporate more machine parameters such as the cache line size. Solutions In this section, we analyze the relationship between linear functions in the array subscripts. Based on this analysis, we restructure a given loop nest to reduce or eliminate the cache thrashing due to true data sharing. We consider nested loops which are not necessarily perfectly nested and which may have variable loop bounds. For clarity of presentation, in Section 4.1 we first discuss how to deal with dependent reference pairs such that the same subscript function is used in both references in each pair (Different pairs may use different subscript functions). Later in Section 4.2, we will discuss how to deal with more general cases by using simple affine transforms to fit to this model. 4.1 The Basic Model In this subsection, we assume that for each pair of dependent references, the same subscript function is used in both references. Under this assumption, if we extract the subscript function ~ h fl (I; J; K) from each pair of dependent references, then a model for a nested loop which has m pairs of dependent references can be illustrated by the following code segment. ENDDO ENDDO Without loss of generality, suppose that all m linear subscript functions above are different. We assume that each function ~ h m, is of rank 2 and is in the form of ~ h fl (i; j; where Take the following example. ENDDO ENDDO Figure 4: A nested loop with multiple linear subscript functions. In this example, no data dependences exist within loop J . However, two data dependences exist in the whole loop nest, one between the references to A, and the other between those to B. We have two linear functions to consider, one for each dependence: The iteration subspace N 1 \Theta N 2 is called the reduced iteration space because it omits the K loop. In order to find the iterations in the reduced iteration space which may access common memory locations within the corresponding threads, we define a set of elements of array A fl which are accessed within thread using subscript function ~ h fl . Definition 1: Given iteration (i in the reduced iteration space, the elements A fl (f fl (i are accessed within thread . They are denoted by A i 0 ;j 0 g. Definition 2: If we suppose that T i;j and T i 0 ;j 0 are two threads corresponding to (I; J)=(i; ) in the reduced iteration space of the given loop nest such that A i;j m); we say T has a dependence because of ~ h fl , denoted by T i;j Definition 3: If there exists fl, 1 - fl - m, such that T i;j Since both f fl and g fl are linear in terms of i; j and k, the following lemma is obvious. Lemma 1: In the program pattern described above, if there exist fl, 1 - fl - m, and two iterations in the iteration space N 1 \Theta N 2 \Theta N 3 , (i, j, , such that then for any constant n 0 , we have a series of iterations in the space, (i; that satisfy the following equations: The following lemma and two theorems establish the relationship between the loop indexes corresponding to two inter-dependent threads. We will use this index relationship to stagger the loop iteration space such that inter-dependent threads can be assigned to the same processors. Lemma 2: Let T i;j , T i 0 ;j 0 be two threads, . T i;j exist k, k 0 such that a a Theorem 1: Let b fl;1 a exist k, k 0 b fl;1 a fl;2 \Gamma a fl;1 b fl;2 a fl;2 c fl;1 \Gamma a fl;1 c fl;2 b fl;1 a fl;2 \Gamma a fl;1 b fl;2 The proofs of Lemma 2 and Theorem 1 are obvious from Definition 3. We now consider the case of b fl;1 a fl;2 \Gamma a fl;1 b assuming that the loop bounds, N 2 and N 3 , are large enough to satisfy the c fl;1 These assumptions are almost always true in practice [31]. When they are not true, the parallel loops will be too small to be important. With these assumptions, we have the following theorem. Theorem 2 [21]: Let b fl;1 a fl;2 \Gamma a fl;1 b The fact that the J loop at the middle level is a loop guarantees that T i;j (1) a fl;1 (i 0 (2) a fl;2 (i 0 In order to find the threads which have data dependence relations with thread T i;j , we make the following definition. Definition 4: Given iteration (i; j) in the reduced iteration space, we let S i;j denote the following set of iterations in the space: r r where L fl;1 (L fl;1 6= 0) and L fl;2 (1 - fl - m) are defined as: and with GCD fl equal to G:C:D:(b fl;1 c fl;2 \Gammac fl;1 b fl;2 ; a fl;2 c fl;1 \Gammaa fl;1 c fl;2 ; b fl;1 a fl;2 \Gammaa fl;1 b fl;2 ) or equal to \GammaG:C:D: (b fl;1 c fl;2 - c fl;1 b fl;2 , a fl;2 c fl;1 -a fl;1 c fl;2 , b fl;1 a fl;2 -a fl;1 b fl;2 ) to guarantee L fl;1 ? 0; (2) and L with GCD fl equal to G:C:D:(a fl;1 ; b fl;1 ) or equal to \GammaG:C:D:(a fl;1 ; b fl;1 ) to guarantee L fl;1 ? 0; and L with GCD fl equal to G:C:D:(a fl;2 ; b fl;2 ) or equal to \GammaG:C:D:(a fl;2 ; b fl;2 ) to guarantee L fl;1 ? 0; called the staggering parameter corresponding to linear function ~ h fl . If there exist no data dependences between the given pair of references, we define the staggering parameter (L fl;1 ; L fl;2 ) as (0; 0). The staggering parameters for the example in Figure 4 can be calculated to be: (L 1;1 ; L 1;2 and (L 2;1 ; L 2;2 according to Definition 4(1). The following theorem, derived from Theorems 1 and 2 and Definition 4, states that we can use the staggering parameters to uniquely partition the threads into independent sets. Theorem 3: S i;j as defined above satisfies: (1) if (i; The theorem above indicates that S i;j includes all the iterations whose corresponding threads have a data dependence relation with T i;j . We call S i;j an equivalence class of the reduced iteration space. In order to eliminate true data sharing, threads in the same equivalence class should be assigned to the same processor. We want to restructure the reduced iteration space such that threads in the same equivalence class will appear in the same column. Each staggering parameter (L computed for a dependent reference pair tells us that if we stagger the (i row in the reduced iteration space by columns to the right if L 2 ! 0, or to the left if relative to the i-th, then the threads involved in the dependence pair will be aligned in the same column. Different staggering parameters may require staggering the iteration space in different ways. However, if these staggering parameters are in proportion, then staggering by the unified staggering parameter defined below will satisfy all the requirements simultaneously. Definition 5: Given staggering parameters (L , and then we call (g; L1;2 the unified staggering parameter. Lemma 3 [21]: If the condition L k;1 m) in Definition 5 is true, then (a) the iterations (i; belong to two different equivalent classes; and (b) the iterations belong to two different equivalence classes. Theorem 4 [21]: If the condition L k;1 m) in Definition 5 is true, then the reduced iteration space must be staggered according to the unified staggering parameter (g; L1;2 L1;1 g) in order to reduce or eliminate data sharing among the threads, i.e. the (i g)-th row in the reduced iteration space must be staggered by j L1;2 L1;1 gj columns to the right if L 1;2 ! 0, or to the left if L 1;2 ? 0, relative to the i-th row. If a given loop nest satisfies the condition L k;1 m) in Definition 5, then, according to Theorem 4 above, the reduced iteration space can be transformed into a staggered and reduced iteration space (SRIS) by leaving the first g rows unchanged, staggering each of the remaining rows using the unified staggering parameter. There will be no data dependences between different columns in the SRIS. However, if the staggering parameters are not in proportion, i.e, if there exist (j; k) such that , then we can no longer obtain a unique unified staggering parameter. Moreover, staggering alone is no longer sufficient for eliminating data dependences between the different columns in the restructured iteration space. This is because some threads in the same equivalence class are still in different columns. We perform a procedure called compacting which stacks these columns onto each other. We will discuss staggering first. Definition Given staggering parameters (L (L 1;1 , L 2;1 , ., L m;1 ), suppose there exists (j; k) such that 1 - . According to the theory of numbers [27], there exist integers a 1 , a 2 , ., am that satisfy a a fl L fl;2 . We call (g; g 0 unified staggering parameter. Note that since the m-tuple (a 1 , a 2 , ., am ) is not necessarily unique, the (g; g 0 may not be unique either. With Definition 6, a unified staggering parameter (g; g 0 ) of the example in Figure 4 is found to be After staggering by using any unified staggering parameter (g; g 0 ), the resulting SRIS has four possible shapes, as shown in Figure 5(b \Gamma e). Figure 5(a) shows details of one of these shapes. Figure 6(a) and 6(b) show the reduced iteration space for the example in Figure 4 before and after staggering with (3,-3) as the unified staggering parameter. Next, we compute the compacting parameter d using Algorithm 1 and 2, to be presented shortly. We then partition the SRIS into n chunks, where d , which is the total number of columns in the SRIS devided by the compacting parameter d (Figure 5(d \Gamma e)). These d-wide chunks are stacked onto each other to form a compacted iteration space of width d, as shown in Figure 7. As we will explain later, the threads in different columns after compacting the SRIS with d are independent. Moreover, the product of d and g equals the number of equivalence classes. The SRIS shown in Figure 6(b) for our example is transformed by being compacted with into the form shown in Figure 8. The following algorithm computes the compacting parameter d. Algorithm 1: Input: A set of staggering parameters (L Output: The compacting parameter d. Step 1: For each 2-element subset, fL i;1 ; L j;1 g, of fL 1;1 ; L 2;1 ; :::; L m;1 g, compute of all such d 2 hL i;1 ; L j;1 i. Step 2: For each j-element subset, fL pick any element, say a) g'<0, g>1 d d d) g'<0, g>1 e) g'>0, g>1 Figure 5: SRIS and outlines. Using the Euclidean Algorithm, compute integers b Apply Algorithm 2 below to find nonzero integers r 2 ; :::; r j such that r a) Original reduced iteration space (1,1) . Figure The reduced iteration space before and after rearrangement. Let r Step 3: For j from 3 to m, compute Step 4: As will be established later, d is unique regardless of the choice of L i 1 ;1 in Step 2. To calculate the compacting parameter d, non-zero integers r need to be found in Algorithm 1 from the integer coefficients b computed by the Euclidean Algorithm. Algorithm 2 is therefore invoked to derive a group of non-zero integer coefficients from a group of any integer coefficients of a linear expression. Algorithm 2: Input: Non-zero positive integers such that Output: non-zero integers such that 1: If there are an even number of zero coefficients a (0 - 2k - p) among d) d) d) d) d) d) d) d) d) d) d) a Figure 7: Compacted SRIS. Step 2: If there are an odd number of zero coefficients a i 1 (0 Obviously, the non-zero integers computed by Algorithm 2 satisfy For the example in Figure 4, since there are only two linear functions in the loop nest, only Step 1 and Step 4 of Algorithm 1 are used to calculate the compacting parameter d, that is, Next, we need to establish two important facts. First, after compacting with d, the threads in different columns are independent. Second, the compacting parameter d computed by Algorithm 1 is the Figure 8: The reduced iteration space after compacting. largest number of independent columns possible as the result of compacting the SRIS with a constant value. The first fact is established by Theorem 5, Theorem 6, and Corollary 1. To do so, we introduce the following definition. Definition 7: Given an iteration (i; j) in the reduced iteration space, staggering parameters and the unified staggering parameter (g; g 0 are integers that satisfy a a set of iterations S 0 i;j is constructed as follows: (1) For any integer r, iteration (i ) in the space belongs to S 0 a (2) If there exist integers r not all zero, integer r, and iterations (i 0 the space, such that r r and a The following three lemmas and Theorem 5 show that S 0 i;j is the same as the equivalence class S i;j . From the process of constructing S 0 i;j , we immediately have the following lemma. Lemma 4: Given iterations (i; ) in the reduced iteration space, and a unified staggering parameter (g; g 0 ), if there exists integer r such that Lemma 5 [21]: Given iterations (i 0 ) in the reduced iteration space, if there exist integers r all zero, such that r fl L fl;2 r Lemma 6 [21]: Given iterations (i; ) in the reduced iteration space, if (i 0 Theorem 5 [21]: Given staggering parameters (L for any iteration (i; j) in the reduced iteration space. Next, we establish that S 0 i;j is the result of staggering with (g; g 0 followed by compacting with d. This is stated by Corollary 1 below. Lemma 7 [21]: Given staggering parameters (L that are integers that satisfy (1) r (2) if there exist integers r 0 satisfying 1 . For any integers r 00 satisfying r 00 there exists an integer k - 1 such that r 00 Theorem 6 [21]: If d is the compacting parameter determined by Algorithm 1, and d 0 r fl L fl;2 , are integers, not all zero, which satisfy r then there exists an integer k such that d 0 Corollary 1: The set S 0 i;j in Definition 7 satisfies where k and r are integers, (g; g 0 ) is the unified staggering parameter in Definition 6, and d is the compacting parameter computed by Algorithm 1. From the above result, the threads in different columns after compacting the SRIS with d are inde- pendent. Next, we establish with Theorem 7 that any two columns which are d columns apart, where d is computed by Algorithm 1, should be dependent and that, therefore, d is the largest possible number of independent columns as the result of compacting the SRIS with a constant number. Theorem 7: Given (i; j), we have i;j . Proof: According to how d is computed in Algorithm 1, there exist integers r such that By the definition of S 0 To further simplify the process of the staggering and the compacting of the reduced iteration space, the following theorem can be used to replace multiple staggering parameters, which are in proportion, with a single staggering parameter. Theorem 8 [21]: Given staggering parameters (L m) are the staggering parameters satisfying there exists an integer r satisfying We now estimate the time needed by the compiler to compute the staggering parameters, a unified staggering parameter, and the compacting parameter. Suppose there are m reference pairs. The complexity of determining all the staggering parameters is O(m). A unified staggering parameter of these staggering parameters can be determined in O(m) with the Euclidean Algorithm. Let be the number of groups of staggering parameters such that all parameters in the same group are in proportion. m 0 is very small in practice. According to Theorem 8, we only need to consider one representative from each group. The complexity of Algorithm 1 and 2 for computing the compacting parameter is C 2 Lastly, we show the result of restructuring the original loop nest based on staggering and compacting. Note that if all staggering parameters are in proportion, then compacting is unnecessary for data dependence elimination. However, to improve load balance, we compact the SRIS by a compacting factor d, which equals the number of the available processors. The restructured code is parameterized by the loop bounds and by the number of available processors, which can be obtained by a system call at runtime. There is no need to recompile for a different number of available processors. If the given loop nest is perfectly nested, then the resulting code, after staggering and compacting, is shown in Code Segment 1 listed below. Code Segment 1 (The result after restructuring the perfectly nested loop with multiple linear ENDDO ENDDO ENDDO If the given loop nest is not perfectly nested, then the resulting code has two variants, one for and the other for g 0 =0 . We show the code for g 0 listed below (the code for =0 is similar [21]). In this code segment, LB 1 and LB 2 are the lower bounds of I and J , UB 1 and UB 2 are the upper bounds of these two loops, (g; g 0 and d are the unified staggering parameter and the compacting parameter, respectively, which have been determined above. PSI and PSJ are local variables that each processor uses to determine the first iteration J 0 of loop J to be executed on it. Variables J 0 and OFFSET are also local to each processor. PSI and PSJ for each processor are modified every g iterations of the loop I, according to the staggering and the compacting parameters. The values of (PSI; PSJ) are initialized for the d different processors to (LB respectively. We define a function mod* such that x mod* Code Segment 2 (The restructured code for the case of g 0 6= 0): endif endif ENDDO ENDDO ENDDO 4.2 An Extended Model The theory we developed in the previous subsection can be extended to more general cases in which the subscript functions in the same pair of references are not necessary the same. Suppose the following two linear functions ~ and ~ belong to the same pair of references. In order to determine which iterations in the reduced iteration space are dependent due to this reference pair, we consider an affine transformation such that the linear function ~ h 2 can be expressed as ~ a 2;1 a a 2;2 which we denote by ~ ). In order to use the previous results from Section 4.1, we let ~ be identical to ~ which implies a 1;1 a 2;1 a 1;2 a 2;2 c 1;1 c 2;1 c 2;2 and a 2;1 a 2;2 We can now apply the algorithms in Section 4.1 to ~ 2 and ~ which yield a staggering parameter, say For a given iteration (i 0 ). The iteration (i; must have a dependence with (i 00 before the affine transformation if and only if the iteration (i 0 a dependence with (i 00 after the transformation. We denote the distance between (i; as (L 0 2 ), which can be calculated as: or such that L 0 not be constant, meaning that the iterations cannot be aligned with a constant staggering parameter. In common practice, since loop J is DOALL in our loop nest model, the two linear functions ~ will have the same cofficients for loop index variables I and J , which implies that ff 1. In this paper, we will consider the case of ff We now have or We define (L 0 2 ), which are two constants given staggering parameter in this case. If Equations (1) and (2) have a unique solution for we have a unique staggering parameter (L 0 On the other hand, if there exist multiple solutions for then the following theorem shows that under certain conditions, (L 0 determined by different should be in proportion. Theorem 9: Assume ff 1. If the staggering parameter (L of the subscript function ~ after the affine transformation is a solution for Equations (1) and (2), then (L 0 is equal to (fi or to (L proportion with solution (fi Equations (1) and (2). proportion with prove that in proportion to (L supposing that Every solution to Equations (1) and (2) can be written as 0 solution of the homogeneous system associated with Equations (1) and (2), that is, a a So, if a 1;2 b 1;1 \Gamma b 1;2 a 1;1 6= 0, we have a 2;2 b a 1;2 b 1;1 \Gamma b 1;2 a 1;1 a 2;2 b c 1;2 a a 1;2 b 1;1 \Gamma b 1;2 a 1;1 Suppose then we have c 1;2 a according to Definition 4, we have c 1;2 a For the case of a 1;2 b as in Theorem 2, we have: (1) a 1;1 (2) a 1;2 Therefore, according to Definition 4, we also have (- proportion with are in proportion with 2 ) is in proportion with If the condition in Theorem 9 is met, we choose (L 0 as the staggering parameter for the reference pair ~ Table shows examples of staggering parameters for different subscript functions appearing in the dependent reference pair, where the loop index variables are listed in the order from the outermost loop level to the innermost. If we simultaneously consider two reference pairs: A(I; J) with and B(I; J) with then the thread T i;j will share the same array element A(i; thread T i+3;j+1 and the same array element B(i; j) with thread T i+1;j+3 . Using Theorem 9, the staggering parameters (L 0 these two pairs are (3,1) and (1,3) respectively. A unified staggering parameter and compacting parameter can be calculated as (g; g 0 8. Table 2: Examples of different functions in the same dependent reference pair. Loop nest Dependent reference pair ( ~ 5 Experimental Results The thread alignment techniques described in this paper have been implemented as backend optimizations in KD-PARPRO [20], a knowledge-based parallelizing tool which can perform intra- and inter-procedural data dependence analysis and a large number of parallelizing transformations, including loop interchange, loop distribution, loop skewing, and strip mining for FORTRAN programs. To evaluate the effect of the thread alignment techniques on the performance of multiprocessors with memory hierarchies, we experimented with three programs from the LINPACK benchmarks on a SGI Challenge cluster which can be configured to contain up to twenty MIPS 4400 processors. First, the programs were parallelized and optimized using KD-PARPRO. To reduce or eliminate cache thrashing due to true data sharing, our tool recognized the nested loops which may cause the thrashing. It applied the techniques described in the previous section to analyze and restructure the loop nests. The parallelized programs were then compiled using SGI's f77 compiler with the optimization option -O2. The sequential versions of the programs were compiled on the same machine using the same optimization option for f77. The output binary codes were then executed on various configurations with a different number of processors during dedicated time. Each MIPS 4400 processor has a 16K-byte primary data cache and a 4M-byte secondary cache. The cache block size is 32 bytes for the primary data cache and 128 bytes for the secondary cache. A fast and wide split transaction bus POWERpath-2 is used as its coherent interconnect. Cache coherence is maintained with a snoopy write-invalidate strategy. We compared the results obtained by using our algorithm to align the threads with those obtained by using four different loop scheduling strategies provided by SGI system software, namely, simple, interleave, dynamic, and gss. The simple method divides the iterations by the number of processors and then assigns each chunk of consecutive iterations to one processor. The interleave scheduling method divides the iterations into chunks of the size specified by the CHUNK option, and execution of those chunks is statically interleaved among the processes. With dynamic scheduling, the iterations are also divided into CHUNK-sized chunks. As each process finishes a chunk, however, it enters a critical section to grab the next available chunk. With gss scheduling [29] , the chunk size is varied, depending on the number of iterations remaining. None of these SGI-provided methods consider task alignment. The speedup of a parallel execution on shared memory machines just like the SGI cluster can be affected by many factors, including: program parallelism, data locality, scheduling overhead, and load balance. Usually gss, dynamic and interleave schedulings with a small chunk size are supposed to show better load balance than simple scheduling. On the other hand, they tend to incur more scheduling overhead than simple. Furthermore, simple captures more data locality in most cases than other schedulings do. The programs we selected from LINPACK are SGEFA, SPODI, and SSIFA. SGEFA factors a double precision matrix by Gaussian elimination. The main loop structure in this program consists of three imperfectly nested loops. The innermost loop is inside subroutine SAXPY, which multiplies a vector by a constant and then adds the result to another vector. In order to show the array access pattern inside the loop body, we inlined the SAXPY in the code section given below. However, we kept the subroutine call when we applied our techniques to the program. ENDDO ENDDO The SRIS, after staggering the iterations in the reduced iteration space (K; J), is shown in Figure 9. For this program, we only consider the linear function (I; J). The staggering parameter is (1,0) according to Definition 4(4). The number of processors is used to determine the compacting factor. Figure 9: SRIS for SGEFA. SPODI computes the determinant and inverse of a certain double precision symmetric positive-definite matrix. There are two main loop nests in this program, as shown below. We restructured both loop nests. The same as in SGEFA, the innermost loop is contained in the subroutine SAXPY. /* the first loop nest */ ENDDO ENDDO /* the second loop nest */ ENDDO ENDDO The SRISs, after staggering the iterations in the reduced iteration spaces (K; J) and (J; K) for these two loop nests, respectively, are shown in Figure 10(a)(b). The linear functions we considered are: (I; J) and (I; K), respectively. Their staggering parameters are both (1,0), according to Definition 4(4). (a) Loop nest 1 (b) Loop nest 2 Figure 10: SRISs for SPODI. SSIFA factors a double precision symmetric matrix by elimination. The main loop nest in this program is shown below. We view the backward GOTO loop as the outermost sequential loop, within which the value of kstep may change between 1 and 2 in different iterations, based on the input matrix. Depending on the value of kstep, one of the two parallel loop nests inside the outermost sequential loop will be executed for each iteration of the outermost loop. The index step of the outermost loop equals i.e. \Gamma1 or \Gamma2. The array access patterns for these two kstep values are slightly different. The innermost loop is again inside the SAXPY subroutine. 10: CONTINUE IF (K .EQ. ENDDO ENDDO ENDDO 20: CONTINUE The SRISs, after staggering the iterations in the reduced iteration space (K; JJ) for two different ksteps, are shown in Figure 11. For clarity, we use c kstep to denote the value of kstep in the current K iteration, and we use p kstep for its value in the previous K iteration. All threads will be aligned well if we properly align the threads created in the current K iteration with those in the previous K iteration. We need to consider four possible cases of dependences: one is between two references to another is between two references to A(I; the third is from A(I; K \Gamma JJ) to A(I; and the last is from A(I; K \Gamma For all of these cases, the staggering parameter is (L (a) Figure 11: SRISs for SSIFA. The problem sizes we used in our experiments are n=100 and 1000. The performance of the parallel codes transformed by our techniques, compared with the performance achieved by the scheduling methods provided by SGI, are shown in Figures 12-15. Both SGEFA and SSIFA may require pivoting for non-positive-definite symmetric matrices, but not for positive-definite symmetric matrices. We show data for SGEFA with pivoting, and we show data both with and without pivoting for SSIFA. Pivoting may potentially destroy the task alignment.1352 4 8 Number of processors Interleave (chunk Interleave (chunk Dynamic Gss Simple Our method (a) Number of processors Interleave (chunk Interleave (chunk Dynamic Gss Simple Our method (b) n=1000 Figure 12: Gaussian elimination (SGEFA).12345 Number of processors Interleave (chunk Interleave (chunk Dynamic Gss Simple Our method (a) n=10026101418 Number of processors Interleave (chunk Interleave (chunk Dynamic Gss Simple Our method (b) n=1000 Figure 13: Determinant and inverse of a symmetric positive matrix (SPODI). As the figures show, our method always outperforms all of the SGI's scheduling methods, with the exception of program SGEFA. For this program, our method's performance is almost the same as that of simple, although our method outperforms simple by 14% on 16 processors with should be attributed to the reduction of cache thrashing due to true data sharing, a problem that tends to be more severe when more processors are running. The simple scheduling method tends to get better performance than the dynamic, gss, and interleave methods, because it results in better locality and less cache thrashing in most cases, and it also incurs less scheduling overhead. But when the programs Number of processors Interleave (chunk Interleave (chunk Dynamic Gss Simple Our method (a) n=100246810 Number of processors Interleave (chunk Interleave (chunk Dynamic Gss Simple Our method (b) n=1000 Figure 14: Factorization of a symmetric matrix (SSIFA). do not exhibit a good load balance, like SPODI, SSIFA, and other programs in LINPACK, which deal with symmetric matrices, simple's performance results degrade substantially. Our method outperforms simple quite significantly in most cases, especially for SPODI (Figure 13), as well as for SSIFA without pivoting (Figure 15), where our method beats simple by as much as 105%. We are not able to get improvement over simple for program SGEFA when pivoting is much more likely to destroy the locality we try to keep. For the rest of the programs, we attribute our performance gain over simple both to the reduction of cache thrashing due to true data sharing and to a better load balance, although, for SSIFA with pivoting, we believe our method benefits more from load balancing. We note that the SGI system software cannot pick the right scheduling method automatically to fit the particular program. On the other hand, our method seems more capable of delivering good performance for different loop shapes. As the thrashing problem becomes more serious on parallel systems with more processors and greater communication overhead, our method will likely be even more effective. 6 Conclusions This paper presents a method in which the reduced iteration space is rearranged according to the staggering and the compacting parameters. The nested loop (either perfectly nested or imperfectly nested) is restructured to reduce or even eliminate cache thrashing due to true data sharing. This method can be efficiently implemented in any parallel compiler. Although the analysis per se is based on a simple machine model, the resulting code executes correctly on more complex models. Our experimental results show that the transformed code can perform quite well on a real machine. How to extend the techniques proposed in this paper to incorporate additional machine parameters is interesting future work. --R On the performance enhancement of paging systems through program analysis and transformations. Automatic partitioning of parallel loops and data arrays for distributed shared-memory multiprocessors Automatic loop interchange. Multilevel cache hierarchies: organizations Dependence analysis for supercomputing. Improving register allocation for subscripted variables. Compiling scientific code for complex memory hierarchies. Partitioning and labeling of loops by unimodular transformations. Eliminating False Sharing. A solution of cache ping-pong problem in RISC based parallel processing systems Cache or local memory thrashing and compiler strategy in parallel processing systems. An iteration partition approach for cache or local memory thrashing on parallel processing. Performance optimizations On the problem of optimizing data transfers for complex memory systems. Strategies for cache and local memory management by global program transformation. Effects of program parallelization and stripmining transformation on cache performance in a multiprocessor. Compiler techniques for data partitioning of sequentially iterated parallel loops. Reducing false sharing on shared memory multiprocessors through compile-time data transformations The design and the implementation of a knowledge-based parallelizing tool An efficient solution to the cache thrashing problem (Extended Version). Loop restructuring techniques for the thrashing problem. Loop staggering The structure of computers and computations. Dependence graphs and compiler op- timizations A solution of the cache ping-pong problem in multiprocessor systems An introduction to the theory of numbers. Minimum distance: a method for partitioning recurrences for multiproces- sors Guided self-scheduling: a practical scheduling scheme for parallel supercomputers Time optimal linear schedules for algorithms with uniform dependencies. An empirical study of Fortran programs for parallelizing compilers. Iteration partitioning for resolving stride conflicts on cache-coherent multiprocessors False sharing and spatial locality in multiprocessor caches. A data locality optimizing algorithm. More iteration space tiling. --TR

multiprocessors;parallelizing compilers;parallel threads;loop transformations;cache thrashing;true data sharing

279583

Modulo Scheduling with Reduced Register Pressure.

AbstractSoftware pipelining is a scheduling technique that is used by some product compilers in order to expose more instruction level parallelism out of innermost loops. Modulo scheduling refers to a class of algorithms for software pipelining. Most previous research on modulo scheduling has focused on reducing the number of cycles between the initiation of consecutive iterations (which is termed II) but has not considered the effect of the register pressure of the produced schedules. The register pressure increases as the instruction level parallelism increases. When the register requirements of a schedule are higher than the available number of registers, the loop must be rescheduled perhaps with a higher II. Therefore, the register pressure has an important impact on the performance of a schedule. This paper presents a novel heuristic modulo scheduling strategy that tries to generate schedules with the lowest II, and, from all the possible schedules with such II, it tries to select that with the lowest register requirements. The proposed method has been implemented in an experimental compiler and has been tested for the Perfect Club benchmarks. The results show that the proposed method achieves an optimal II for at least 97.5 percent of the loops and its compilation time is comparable to a conventional top-down approach, whereas the register requirements are lower. In addition, the proposed method is compared with some other existing methods. The results indicate that the proposed method performs better than other heuristic methods and almost as well as linear programming methods, which obtain optimal solutions but are impractical for product compilers because their computing cost grows exponentially with the number of operations in the loop body.

Introduction Increasing the instruction level parallelism is an observed trend in the design of current microprocessors. This requires a combined effort from the hardware and software in order to be effective. Since most of the execution time of common programs is spent in loops, many efforts to improve performance have targeted loop nests. Software pipelining [5] is an instruction scheduling technique that exploits the instruction level parallelism of loops by overlapping the execution of successive iterations of a loop. There are different approaches to generate a software pipelined schedule for a loop [1]. Modulo scheduling is a class of software pipelining algorithms that was proposed at the begining of last decade [23] and has been incorporated into some product compilers (e.g. [21, 7]). Besides, many research papers have recently appeared on this topic [11, 14, 25, 13, 28, 12, 26, 22, 29, 17]. Modulo scheduling framework relies on generating a schedule for an iteration of the loop such that when this same schedule is repeated at regular intervals, no dependence is violated and no resource usage conflict arises. The interval between the succesive iterations is termed Initiation Interval (II ). Having a constant initiation interval implies that no resource may be used more than once at the same time modulo II . Most modulo scheduling approaches consists of two steps. First, they compute an schedule trying to minimize the II but without caring about register allocation and then, variables are allocated to registers. The execution time of a software pipelined loop depends on the II , the maximum number of live values of the schedule (termed MaxLive) and the length of the schedule for one iteration. The II determines the issue rate of loop iterations. Regarding the second factor, if MaxLive is not higher than the number of available registers then the computed schedule is feasible and then it does not influence the execution time. Otherwise, some actions must be taken in order to reduce the register pressure. Some possible solutions outlined in [24] and evaluated in [16] are: ffl Reschedule the loop with an increased II . In general, increasing the II reduces MaxLive but it decreases the issue rate, which has a negative effect on the execution time. ffl Add spill code. This again has a negative effect since it increases the required memory bandwidth and it will result in additional memory penalties (e.g. cache misses). Besides, memory may become the most saturated resource and therefore adding spill code may require to increase the II . Finally, the length of the schedule for one iteration determines the cost of the epilogue that should be executed after the main loop in order to finish the last iterarions which have been initiated in the main loop but have not been completed (see section 2.1). This cost may be negligible when the iteration count of the loop is high. Most previous works have focused on reducing the II and sometimes also the length of the schelude for one iteration but they have not considered the register requirements of the proposed schedule, which may have a severe impact on the performance as outlined above. A current trend in the design of new processors is the increase in the amount of instruction level parallelism that they can exploit. Exploiting more instruction level parallelism results in a significant increase in the register pressure [19, 18], which exacerbates the problem of ignoring its effect on the performance of a given schedule. In order to obtain more effective schedules, a few recently proposed modulo scheduling approaches try to minimize both the II and the register requirements of the produced schedules. Some of these approaches [10, 9] are based on formulating the problem in terms of an optimization problem and solve it using an integer linear programming approach. This may produce optimal schedules but unfortunately, this approach has a computing cost that grows exponentially with the number of basic operations in the loop body. Therefore, they are impractical for big loops, which in most cases are the most time consuming parts of a program and thus, they may be the ones that most benefit from software pipelining. Practical modulo scheduling approaches used by product compilers use some heuristics to guide the scheduling process. The two most relevant heuristic approaches proposed in the literature that try to minimize both the II and the register pressure are: Slack Scheduling [12] and Stage Scheduling [8]. Slack Scheduling is an iterative algorithm with limited backtracking. At each iteration the scheduler chooses an operation based on a previouly computed dynamic priority. This priority is a function of the slack of each operation (i.e., a measure of the scheduling freedom for that operation) and it also depends on how much critical the resources used by that operation are. The selected operation is placed in the partial schedule either as early as possible or as late as possible. The choice between these two alternative is made basically by determining how many of the operation's inputs and outputs are stretchable and choosing the one that minimizes the involved values' lifetimes. If the scheduler cannot place the selected operation due to a lack of conflict-free issue slots, then it is forced to a particular slot and all the conflicting operations are ejected from the partial scheduler. In order to limit this type of backtracking, if operations are ejected too many times, the II is incremented and the scheduling is started all over again. Stage Scheduling is not a whole modulo scheduler by itself but a set of heuristic techniques that reduce the register requirements of any given modulo schedule. This objective is achieved by shifting operations by multiples of II cycles. The resulting schedule has the same II but lower register requirements. This paper presents Hypernode Reduction Modulo Scheduling (HRMS) 1 , a heuristic modulo scheduling approach that tries to generate schedules with the lowest II , and from all the possible schedules with such II , it tries to select that with the lowest register requirements. The main part of HRMS is the ordering strategy. The ordering phase orders the nodes before scheduling them, so that only predecessors or successors of a node can be scheduled before it is scheduled (except for recurrences). During the scheduling step the nodes are scheduled as early/late as possible, if their predecessors/successors have been preliminary version of this work appeared in [17]. previously scheduled. The performance of HRMS is evaluated and compared with that of a conventional approach (a top-down scheduler) that does not care about register pressure. For this evaluation we have used over a thousand loops from the Perfect Club Benchmark Suite [4] that account for 78% of its execution time. The results show that HRMS achieves an optimal II for at least 97.5% of the loops and its compilation time is comparable to the top-down approach whereas the register requirements are lower. In addition, HRMS has been tested for a set of loops taken from [10] and compared against two other heuristic strategies. These two strategies are the previously mentioned Slack Scheduling, and FRLC [27], which is an heuristic strategy that does not take into account the register requirements. In addition, HRMS is compared with SPILP [10], which is a linear programming formulation of the problem. Because of the computing requirements of this latter approach, only small loops are used for this comparison. The results indicate that HRMS obtains better schedules than the other two heuristic approaches and its results are very close to the ones produced by the optimal scheduler. The compilation time of HRMS is similar to the other heuristic methods and much lower than the linear programming approach. The rest of this paper is organized as follows. In Section 2, an example is used to illustrate the motivation for this work, that is, reducing the register pressure in modulo scheduled loops while achieving near optimal II . Section 3 describes the proposed modulo scheduling algorithm that is called HRMS. Section 4 evaluates the performance of the proposed approach, and finally, Section 5 states the main conclusions of this work. 2 Overview of modulo scheduling and motivating ex- ample This section includes an overview of modulo scheduling and the motivation for the work presented in this paper. For a more detailed discussion on modulo scheduling refer to [1]. 2.1 Overview of modulo scheduling In a software pipelined loop the schedule for an iteration is divided into stages so that the execution of consecutive iterations that are in distinct stages is overlapped. The number of stages in one iteration is termed stage count(SC). The number of cycles per stage is II . Figure 1 shows the dependence graph for the running example used along this section. In this graph, nodes represent basic operations of the loop and edges represent values generated and consumed by these operations. For this graph, Figure 2a shows the execution of the six iterations of the software pipelined loop with an II of 2 and a SC of 5. The operations have been scheduled assuming a four-wide issue machine, with general-purpose functional units (fully pipelined with a latency of two cycles). The scheduling of each iteration has been obtained using a top-down strategy that gives priority to operations in A G F Figure 1: A sample dependence graph. the critical path with the additional constraint that no resource can be used more than once at the same cycle modulo II . The figure also shows the corresponding lifetimes of the values generated in each iteration. The execution of a loop can be divided into three phases: a ramp up phase that fills the software pipeline, an steady state phase where the software pipeline achieves maximum overlap of iterations, and a ramp down phase that drains the software pipeline. The code that implements the ramp up phase is termed the prologue. During the steady state phase of the execution, the same pattern of operations is executed in each stage. This is achieved by iterating on a piece of code, termed the kernel, that correspods to one stage of the steady state phase. A third piece of code called the epilogue, is required to drain the software pipeline after the execution of the steady state phase. The initiation interval II between two successive iterations is bounded either by loop-carried dependences in the graph (RecMII ) or by resource constraints of the architecture (ResMII ). This lower bound on the II is termed the Minimum Initiation Interval )). The reader is refered to [7, 22] for an extensive dissertation on how to calculate ResMII and RecMII . Since the graph in Figure 1 has no recurrence circuits, its initiation interval is constrained only by the available resources: number of operations divided by number of resources). Notice that in the scheduling of Figure 2a no dependence is violated and every functional unit is used at most once at all even cycles (cycle modulo and at most once at all odd cycles (cycle modulo The code corresponding to the kernel of the software pipelined loop is obtained by ovelapping the different stages that constitute the schedule of one iteration. This is shown in Figure 2b. The subscripts in the code indicate relative iteration distance in the original loop between operations. For instance, in this example, each iteration of the kernel executes an instance of operation A and an instance of operation B of the previous iteration in the initial loop. Values used in a loop correspond either to loop-invariant variables or to loop-variant variables. Loop-invariants are repeatedly used but never defined during loop execution. Loop-invariants, have a single value for all the iterations of the loop and therefore they iteration 1 iteration 2 iteration 3 iteration 4 iteration 5 Prologue Steady Epilogue II Kernel Code iteration 6 a) G F G F G F G F G F G F Figure 2: a) Software pipelined loop execution, b) Kernel, and c) Register requirements. require one register each regardless of the scheduling and the machine configuration. For loop-variants, a value is generated in each iteration of the loop and, therefore, there is a different value corresponding to each iteration. Because of the nature of software pipelining, lifetimes of values defined in an iteration can overlap with lifetimes of values defined in subsequent iterations. Figure 2a shows the lifetimes for the loop-variants corresponding to every iteration of the loop. By overlapping the lifetimes of the different iterations, a pattern of length II cycles that is indefinetely repeated is obtained. This pattern is shown in Figure 2c. This pattern indicates the number of values that are live at any given cycle. As it is shown in [24], the maximum number of simultaneously live values MaxLive is an accurate approximation of the number of register required by the schedule 2 . In this section, the register requirements of a given schedule will be approximated by MaxLive. However, in the experiments section we will measure the actual register requirements after register allocation. Values with a lifetime greater than II pose an additional difficulty since new values are generated before previous ones are used. One approach to fix this problem is to provide some form of register renaming so that successive definitions of a value use distinct registers. Renaming can be performed at compile time by using modulo variable expansion [15], i.e., 2 For an extensive discussion on the problem of allocating registers for software-pipelined loops refer to [24]. The strategies presented in that paper almost always achieve the MaxLive lower bound. In particular, the wands-only strategy using end-fit with adjacency ordering never required more than MaxLive registers. unrolling the kernel and renaming at compile time the multiple definitions of each variable that exist in the unrolled kernel. A rotating register file can be used to solve this problem without replicating code by renaming different instantiations of a loop-variant at execution time [6]. 2.2 Motivating example In many modulo scheduling approaches, the lifetimes of some values can be unnecessarily large. As an example, Figure 2a shows a top-down scheduling, and Figure 3a a bottom-up scheduling for the example graph of Figure 1 and a machine with four general-purpose functional units with a two-cycle latency. In a top-down strategy, operations can only be scheduled if all their predecessors have already been scheduled. Each node is placed as early as possible in order not to delay any possible successors. Similary, in a bottom-up strategy, an operation is ready for scheduling if all its successors have already been scheduled. In this case, each node is placed as late as possible in order not to delay possible predecessors. In both strategies, when there are several candidates to be scheduled, the algorithm chooses the one that is more critical in the scheduling. In the top-down scheduling, node E is scheduled before node F. Since E has no predecessors it can be placed at any cycle, but in order not to delay any possible successor, it is placed as early as possible. Figure 2a shows the lifetimes of loop variants for the top-down scheduling assuming that a value is alive from the beginning of the producer operation to the beginning of the last consumer. Notice that loop variant VE has an unnecessary large lifetime due to the early placement of E during the scheduling. In the bottom-up approach E is scheduled after F, therefore it is placed as late as possible reducing the lifetime of VE (Figure 3b). Unfortunately C is scheduled before B and, in order to not delay any possible predecessor it is scheduled as late as possible. Notice that the VB has an unnecessary large lifetime due to the late placement of C. In HRMS, an operation will be ready for scheduling even if some of its predecessors and successors have not been scheduled. The only condition (to be guaranteed by the pre-ordering step) is that when an operation is scheduled, the partial schedule contains only predecessors or successors or none of them, but not both of them (in the absence of recurrences). The ordering is done with the aim that all operations have a previously scheduled reference operation (except for the first operation to be scheduled). For instance, consider that nodes of the graph in Figure 1 are scheduled in the order fA, B, C, D, F, Gg. Notice that node F will be scheduled before nodes fE, Gg, a predecessor and a successor respectively, and that the partial scheduling will contain only a predecessor (D) of F. With this scheduling order, both C and E (the two conflicting operations in the top-down and bottom-up strategies) have a reference operation already scheduled, when they are placed in the partial schedule. Figure 4a shows the HRMS scheduling for one iteration. Operation A will be scheduled in cycle 0. Operation B, which depends on A, will be scheduled in cycle 2. Then C and later D, are scheduled in cycle 4. At this point, operation F is scheduled as early as possible, G C9 Cycle c) d) Figure 3: Bottom-Up scheduling: a) Schedule of one iteration, b) Lifetimes of variables, c) Kernel, d) Register requirements. i.e. at cycle 6 (because it depends on D), but there are no available resources at this cycle, so it is delayed to cycle 7. Now the scheduler places operation E as late as possible in the scheduling because there is a successor of E previously placed in the partial scheduling, thus operation E is placed at cycle 5. And finally, since operation G has a predecessor previously scheduled, it is placed as early as possible in the scheduling, i.e. at cycle 9. Figure 4b shows the lifetimes of loop variants. Notice that neither C nor E have been placed too late or too early because the scheduler always takes previously scheduled operations as a reference point. Since F has been scheduled before E, the scheduler has a reference operation to decide a late start for E. Figure 4d shows the number of live values in the kernel (Figure 4c) during the steady state phase of the execution of the loop. There are 6 live values in the first row and 5 in the second. In contrast the top-down schedule has simultaneosly live values and the bottom-up schedule has 9. The following section describes the algorithm that orders the nodes before scheduling, and the scheduling step. 3 Hypernode Reduction Modulo Scheduling The dependences of an innermost loop can be represented by a Dependence Graph is the set of vertices of the graph G, where each vertex an operation of the loop. E is the dependence edge set, where each edge (u; v) 2 E represents a dependence between two operations u, v. Edges may correspond to any of the following types of dependences: register dependences, memory dependences or control dependences. The dependence distance ffi (u;v) is a nonnegative integer associated with each edge There is a dependence of distance ffi (u;v) between two nodes u and v if the execution of operation v depends on the execution of operation u at ffi (u;v) iterations before. The latency - u is a nonzero positive integer associated with each node u 2 V and is defined Cycle c) d) Figure 4: HRMS scheduling: a) Schedule of one iteration, b) Lifetimes of variables, c) Kernel, d) Register requirements. as the number of cycles taken by the corresponding operation to produce a result. HRMS tries to minimize the register requirements of the loop by scheduling any operation u as close as possible to their relatives i.e. the predecessors of u, P red(u), and the successors of u, Succ(u). Scheduling operations in this way shortens operand's lifetime and therefore reduces the register requirements of the loop. To software pipeline a loop, the scheduler must handle cyclic dependences caused by recurrence circuits. The scheduling of the operations in a recurrence circuit must not be stretched beyond\Omega \Theta II , where\Omega is the sum of the distances in the edges that constitute the recurrence circuit. HRMS solves these problems by splitting the scheduling into two steps: A pre-ordering step that orders nodes and, the actual scheduling, that schedules nodes (once at a time) in the order given by the pre-ordering step. The pre-ordering step orders the nodes of the dependence graph with the goal of scheduling the loop with an II as close as possible to MII and using the minimum number of reg- isters. It gives priority to recurrence circuits in order not to stretch any recurrence circuit. It also ensures that, when a node is scheduled, the current partial scheduling contains only predecessors or successors of the node, but never both (unless the node is the last node of a recurrence circuit to be scheduled). The ordering step assumes that the dependence graph, -), is connected component. If G is not a connected component it is decomposed into a set of connected components fG i g, each G i is ordered separately and finally the lists of nodes of all G i are concatenated giving a higher priority to the G i with a more restrictive recurrence circuit(in terms of RecMII ). Next the pre-ordering step is presented. First we will assume that the dependence graph function Pre Ordering(G, L, h) fReturns a list with the nodes of G orderedg fIt takes as input: g fThe dependence graph (G) g fA list of nodes partially ordered (L) g fAn initial node (i.e the hypernode) (h) g List return List Figure 5: Function that pre-orders the nodes in a dependence graph without recurrence circuits has no recurrence circuits (Section 3.1), and in Section 3.2 we introduce modifications in order to deal with recurrence circuits. Finally Section 3.3 presents the scheduling step. 3.1 Pre-ordering of graphs without recurrence circuits To order the nodes of a graph, an initial node, that we call Hypernode, is selected. In an iterative process, all the nodes in the dependence graph are reduced to this Hypernode. The reduction of a set of nodes to the Hypernode consists of: deleting the set of edges among the nodes of the set and the Hypernode, replacing the edges between the rest of the nodes and the reduced set of nodes by edges between the rest of the nodes and the Hypernode, and finally deleting the set of nodes being reduced. The pre-ordering step (Figure 5) requires an initial Hypernode and a partial list of ordered nodes. The current implementation selects the first node of the graph (i.e the node corresponding to the first operation in the program order) but any node of the graph can be taken as the initial Hypernode 3 . This node is inserted in the partial list of ordered 3 Preliminary experiments showed that selecting different initial nodes produced different schedules function Hypernode Reduction(V 0 ,G,h) f Creates the subgraph G f And reduces G 0 to the node h in the graph G g for each do for each do else return G 0 Figure Function Hypernode Reduction nodes, then the pre-ordering algorithm sorts the rest of the nodes. At each step, the predecessors (successors) of the Hypernode are determined. Then the nodes that appear in any path among the predecessors (successors) are obtained (function Search All Paths) 4 . Once the predecessors (successors) and all the paths connecting them have been obtained, all these nodes are reduced (see function Hypernode Reduction in Figure to the Hypernode, and the subgraph which contains them is topologically sorted. The topological sort determines the partial order of predecessors (successors), which is appended to the ordered list of nodes. The predecessors are topologically sorted using the PALA algorithm. The PALA algorithm is like an ALAP (As Late As Possible) algorithm, but the list of ordered nodes is inverted. The successors are topologically sorted using an ASAP (As Soon As Possible) algorithm. As an example, consider the dependence graph in Figure 7a. Next, we illustrate the ordering of the nodes of this graph step by step. 1. Initially, the list of ordered nodes is empty (List = fg). We start by designating a node of the graph as the Hypernode (H in Figure 7). Assume that A is the first node of the graph. The resulting graph is shown in Figure 7b. Then A is appended to the that had approximately the same register requirements (there were minor differences caused by resource constraints). 4 The execution time of Search All Paths is O(kV k list of ordered nodes (List = fAg). 2. In the next step the predecessors of H are selected. Since it has no predecessors, the successors are selected (i.e. the node C). Node C is reduced to H, resulting in the graph of Figure 7c, and C is added to the list of ordered nodes (List = fA; Cg). 3. The process is repeated, selecting nodes G and H. In the case of selecting multiple nodes, there may be paths connecting these nodes. The algorithm looks for the possible paths, and topologically sorts the nodes involved. Since there are no paths connecting G and H, they are added to the list (List = fA; C; G; Hg), and reduced to the Hypernode, resulting the graph of Figure 7d. 4. Now H has D as a predecessor, thus D is reduced, producing the graph in Figure 7e, and appended to the list (List = fA; C; G; H;Dg). 5. Then J, the successor of H, is ordered (List = fA; C; G; H;D;Jg) and reduced, producing the graph in Figure 7f. 6. At this point H has two predecessors B and I, and there is a path between B and I that contains the node E. Therefore B, E, and I are reduced to H producing the graph of Figure 7g. Then, the subgraph that contains B, E, and I is topologically sorted, and the partially ordered list fI; E; Bg is appended to the list of ordered circuits. Then, we order this dependence graph as shown in Subsection 3.1. Before presenting the ordering algorithm for recurrence circuits, let us put forward some considerations about recurrences. Recurrence circuits can be classified as: ffl Single recurrence circuits (Figure 8a). I G F I G F I G F I F I F I F a) b) c) d) e) f) g) h) Figure 7: Example of reordering without recurrences. A A A A a) b) c) d) Figure 8: Types of recurrences Recurrence circuits that share the same set of backward edges (Figure 8b). We call recurrence subgraph to the set of recurrence circuits that share the same set of backward edges. In this way Figures 8a and 8b are recurrence subgraphs. ffl Several recurrence circuits can share some of their nodes (Figures 8c and 8d) but have distinct sets of backward edges. In this case we consider that these recurrence circuits are different recurrence subgraphs. All recurrence circuits are identified during the calculation of RecMII . For instance, the recurrence circuits of the graph of Figure 8b are fA, D, Eg and fA, B, C, Eg. Recurrence circuits are grouped into recurrence subgraphs (in the worst case there may be a recurrence subgraph for each backward edge). For instance, the recurrence circuits of Figure 8b are grouped into the recurrence subgraph fA, B, C, D, Eg. Recurrence subgraphs are ordered based on the highest RecMII value of the recurrence circuits contained in each subgraph, in a decreasing order. The nodes that appear in more than one subgraph are removed from all of them excepting the most restrictive subgraph in terms of RecMII . For instance, the procedure Ordering Recurrences(G, L, List, h) fThis procedure takes the dependence graph (G)g fand the simplified list of recurrence subgraphs (L)g fIt returns a partial list of ordered nodes (List)g fand the resulting hypernode (h)g List := Pre Ordering(G 0 , List, h); while L 6= ; do function Generate Subgraph(V , G) fThis function takes the dependence graph (G) and a subset of nodes V g fand returns the graph that consists of all the nodes in V and the edgesg famong themg Figure 9: Procedure to order the nodes in recurrence circuits list of recurrence subgraphs associated with Figure 8c ffA, C, Dg, fB, C, Egg will be simplified to the list ffA, C, Dg, fB, Egg. The algorithm that orders the nodes of a grah with recurrence circuits (see Figure takes as input a list L of the recurrence subgraphs ordered by decreasing values of their RecMII . Each entry in this list is a list of the nodes traversed by the associated recurrence subgraph. Trivial recurrence circuits, i.e. dependences from an operation to itself, do not affect the preordering step since they do not impose scheduling constraints, as the scheduler previously ensured that II - RecMII . The algorithm starts by generating the corresponding subgraph for the first recurrence circuit, but without one of the backward edges that causes the recurrence (we remove the backward edge with higher ffi (u;v) ). Therefore the resulting subgraph has no recurrences and can be ordered using the algorithm without recurrences presented in Section 3.1. The whole subgraph is reduced to the Hypernode. Then, all the nodes in any path between the Hypernode and the next recurrence subgraph are identified (in order to properly use the algorithm Search All Paths it is required that all the backward edges causing recurrences have been removed from the graph). After that, the graph containing the Hypernode, the next recurrence circuit, and all the nodes that are in paths that connect them are ordered applying the algorithm without recurrence circuits and reduced to the Hypernode. If there is no path between the Hypernode and the next recurrence circuit, any node of the recurrence circuit is reduced to the Hypernode, so that the recurrence circuit is now connected to the Hypernode. A F G IH KA F G I KH G I a) b) c) d) e) Figure 10: Example for Ordering Recurrences procedure This process is repeated until there are no more recurrence subgraphs in the list. At this point all the nodes in recurrence circuits or in paths connecting them have been ordered and reduced to the Hypernode. Therefore the graph that contains the Hypernode and the remaining nodes, is a graph without recurrence circuits, that can be ordered using the algorithm presented in the previous subsection. For instance, consider the dependence graph of Figure 10a. This graph has two recurrence subgraphs fA, C, D, Fg and fG, J, Mg. Next, we will illustrate the reduction of the recurrence subgraphs: 1. The subgraph fA, C, D, Fg is the one with the highest RecMII . Therefore the algorithm starts by ordering it. By isolating this subgraph and removing the backward edge we obtain the graph of Figure 10b. After ordering this graph the list of ordered nodes is (List = fA; C; D;Fg). When the graph of Figure 10b is reduced to the Hypernode H in the original graph (Figure 10a), we obtain the dependence graph of Figure 10c. 2. The next step is to reduce the following recurrence subgraph fG, J, Mg. For this purpose the algorithm searches for all the nodes that are in all possible paths between H and the recurrence subgraphs. Then, the graph that contains these nodes is constructed (see Figure 10d). Since backward edges have been removed, this graph has no recurrence circuits, so it can be ordered using the algorithm presented in the previous section. When the graph has been ordered, the list of nodes is appended to the previous one resulting in the partial list (List = fA; C; D;F; I; G; J; Mg). Then, this subgraph is reduced to the Hypernode in the graph of Figure 10c producing the graph of Figure 10e. 3. At this point, we have a partial ordering of the nodes belonging to recurrences, and the initial graph has been reduced to a graph without recurrence circuits (Figure 10e). This graph without recurrence circuits is ordered as presented in Subsection 3.1. So finally the list of ordered nodes is List = fA; C; D;F; I; G; J; M;H;E;B; L; Kg. 3.3 Scheduling step The scheduling step places the operations in the order given by the ordering step. The scheduling tries to schedule the operations as close as possible to the neighbors that have already been scheduled. When an operation is to be scheduled, it is scheduled in different ways depending on the neighbors of these operations that are in the partial schedule. ffl If an operation u has only predecessors in the partial schedule, then u is scheduled as early as possible. In this case the scheduler computes the Early Start of u as: Early Start Where t v is the cycle where v has been scheduled, - v is the latency of v, ffi (v;u) is the dependence distance from v to u, and PSP (u) is the set of predecessors of u that have been previously scheduled. Then the scheduler scans in the partial schedule for a free slot for the node u starting at cycle Early Start u until the cycle Early Start u 1. Notice that, due to the modulo constraint, it makes no sense to scan more than II cycles. ffl If an operation u has only successors in the partial schedule, then u is scheduled as late as possible. In this case the scheduler computes the Late Start of u as: Late Start Where PSS(u) is the set of successors of u that have been previously scheduled. Then the scheduler scans in the partial schedule for a free slot for the node u starting at cycle Late Start u until the cycle Late Start ffl If an operation u has predecessors and successors, then the scheduler scans the partial schedule starting at cycle Early Start u until the cycle min(Late Start ffl Finally, if an operation u has neither predecessors nor successors, the scheduler computes the Early Start of u as: Early Start scans the partial schedule for a free slot for the node u from cycle Early Start u to cycle Early Start u are found for a node, then the II is increased by 1. The scheduling step is repeated with the increased II , which will result in more opportunities for finding slots. An advantage of HRMS is that the nodes are ordered only once, even if the scheduling step has to do several trials. 4 Evaluation of HRMS In this section we present some results of our experimental study. First, the complexity and performance of HRMS are evaluated for a benchmark suite composed of a large number of Number of registers6080100 %of loops L4 HRMS L4 Top-down L6 Top-down Figure cumulative distribution of register requirements of loop variants. innermost DO loops in the Perfect Club [4]. We have selected those loops that include a single basic block. Loops with conditionals in their body have been previously converted to single basic block loops using IF-conversion [2]. We have not included loops with subroutine calls or with conditional exits. The dependence graphs have been obtained using the experimental ICTINEO compiler [3]. A total of 1258 loops, which account for 78% of the total execution time 5 of the Perfect Club, have been scheduled. For these loops, the performance of HRMS is compared with the performance of a Top-Down scheduler. Second, we compare HRMS with other scheduling methods proposed in the literature using a small set of dependence graphs for which there are previously published results. 4.1 Performance evaluation of HRMS We have used two machine configurations to evaluate the performance of HRMS. Both configurations have 2 load/store units, 2 adders, 2 multipliers and 2 Div/Sqrt units. We assume a unit latency for store instructions, a latency of 2 for loads, a latency of 4 (con- figuration L4) or 6 (configuration L6) for additions and multiplications, a latency of 17 for divisions and a latency of roots. All units are fully pipelined except the Div/Sqrt units which are not pipelined at all. In order to evaluate performance the execution time (in cycles) of a scheduled loop has been estimated as the II of this loop times the number of iterations this loop performs (i.e. the number of times the body of the loop is executed). For this purpose the programs of the Perfect Club have been instrumented to obtain the number of iterations of the selected loops. HRMS achieved loops, which means that it is optimal in terms of II for at least 97.5% of the loops. On average, the scheduler achieved an 5 Executed on an HP 9000/735 workstation. HRMS Top-down HRMS Top-down Memory ideal regs. regs. Figure 12: Memory traffic with infinite registers, 64 registers and registers. HRMS Top-down HRMS Top-down L4 L61030Cycles regs. regs. Figure 13: Cycles required to execute the loops with infinite registers, 64 registers and registers. Considering dynamic execution time, the scheduled loops would execute at 98.4% of the maximum performance. Register allocation has been performed using the wands-only strategy using end-fit with adjacency ordering. For an extensive discussion of the problem of allocating registers for software-pipelined loops refer to [24]. Figure 11 compares the register requirements of loop-variants for the two scheduling techniques (Top-down that does not care about register requirements and HRMS) for the two configurations mentioned above. This figure plots the percentage of loops that can been scheduled with a given number of registers without spill code. On average, HRMS requires 87% of the registers required by the Top-down scheduler. Since machines have a limited number of registers, it is also of interest to evaluate the effect of the register requirements on performance and memory traffic. When a loop requires more than the available number of registers, spill code has to be added and the loop has to be re-scheduled. In [16] different alternatives and heuristics are proposed to speed-up the generation of spill code. Among them, we have used the heuristic that spills the variable that maximizes the quotient between lifetime and the number of additional loads and stores required to spill the variable; this heuristic is the one that produces the best results. Figures 12 and 13 show the memory traffic and the execution time respectively of the loops scheduled with both schedulers when there are infinite, 64 and registers available. Notice that in general HRMS requires less memory traffic than Top-down when the number of registers is limited. The difference in memory traffic requirements between both schedulers increases as the number of available registers decreases. For instance, for configuration L6, HRMS requires 88% of the traffic required by the Top-down scheduler if 64 registers are available. If only 32 registers are available, it requires 82.5% of the traffic required by the Top-down scheduler. In addition, assuming an ideal memory system, the loops scheduled by HRMS execute faster than the ones scheduled by Top-down. This is because HRMS gives priority to recurrence circuits, so in loops with recurrences usually produces better results than Top- down. An additional factor that increases the performance of HRMS over Top-down is that it reduces the register requirements. For instance, for configuration L6, scheduling the loops with HRMS produces a speed-up over Top-down of 1.18 under the ideal assumption that an infinite register file is available. The speed-up is 1.20 if the register file has 64 registers and 1.25 if it has only registers. Notice that for both schedulers, the most agressive configuration (L6) requires more registers than the L4 configuration. This is because the degree of pipelining of the functional units has an important effect on the register pressure [19, 16]. The high register requirements of aggressive configurations produces a significant degradation of performance and memory traffic when a limited number of registers is available [16]. For instance, the loops scheduled with HRMS require 6% more cycles to execute for configuration L6 than for L4, if an infinite number of registers is assumed. If only 32 registers are available, L6 requires 16% more cycles than L4. 4.2 Complexity of HRMS Scheduling our testbench consumed 55 seconds in a Sparc-10/40 workstation. This time compares to the 69 seconds consumed by the Top-Down scheduler. The break-down of the scheduler execution time in the different steps is shown in Figure 14. Notice that in HRMS, computing the recurrence circuits consumed only 7%, the pre-ordering step consumed 66%, and the scheduling step consumed 27%. Even though most of the time is spent in the preordering step, the overall time is extremely short. The extra time lost in pre-ordering the nodes, allows for a very simple (and fast) scheduling step. In the Top-Down scheduler, the pre-ordering step consumed a small percentage of the time but the the scheduling step required a lot of time; when the scheduler fails to find an schedule with a given II , the loop has to be rescheduled again with an increased initiation interval, and Top-Down has to re-schedule the loops much more often than HRMS. Time (seconds) HRMS Top-Down Scheduling Priority function Find recurrences and compute MII Figure 14: Time to schedule all 1258 loops for the HRMS and Top-Down schedulers. 4.3 Comparison with other scheduling methods In this section we compare HRMS with three schedulers: an heuristic method that does not take into account register requirements (FRLC [27]), a life-time sensitive heuristic method (Slack [12]) and a linear programming approach (SPILP [10]). We have scheduled 24 dependence graphs for a machine with 1 FP Adder, 1 FP Mul- tiplier, 1 FP Divider and 1 Load/Store unit. We have assumed a unit latency for add, subtract and store instructions, a latency of 2 for multiply and load, and a latency of 17 for divide. Table 1 compares the initiation interval II , the number of buffers (Buf) and the total execution time of the scheduler on a Sparc-10/40 workstation, for the four scheduling methods. The results for the other three methods have been obtained from [10] and the dependence graphs to perform the comparison supplied by its authors. The number of buffers required by a schedule is defined in [10] as the sum of the buffers required by each value in the loop. A value requires as many buffers as the number of times the producer instruction is issued before the issue of the last consumer. In addition, stores require one buffer. In [20], it was shown that the buffer requirements provide a very tight upper bound on the total register requirements. Table 2 summarizes the main conclusions of the comparison. The entries of the table represent the number of loops for which the schedules obtained by HRMS are better (II !), equal (II =), or worse (II ?) than the schedules obtained by the other methods, in terms Application HRMS SPILP Slack FRLC Program II Buf Secs II Buf Secs II Buf Secs II Buf Secs Liver Loop5 3 5 Linpack Whets. Cycle1 4 4 Table 1: Comparison of HRMS schedules with other scheduling methods. of the initiation interval. When the initiation interval is the same, it also shows the number of loops for which HRMS requires less buffers (Buf !), equal number of buffers (Buf =), or more buffers (Buf ?). Notice that HRMS achieves the same performance as the SPILP method both in terms of II and buffer requirements. When compared to the other methods, HRMS obtains a lower II in about 33% of the loops. For the remaining 66% of the loops the II is the same but in many cases HRMS requires less buffers, specially when compared with FRLC. Finally Table 3 compares the total compilation time in seconds for the four methods. Notice that HRMS is slightly faster than the other two heuristic methods; in addition, these methods perform noticeably worse in finding good schedulings. On the other hand, the linear programming method (SPILP) requires a much higher time to construct a scheduling that turns out to have the same performance than the scheduling produced by HRMS. In fact, most of the time spent by SPILP is due to Livermore Loop 23, but even without taking into account this loop, HRMS is over 40 times faster. Slack 7 1 Table 2: Comparison of HRMS performance versus the other 3 methods. HRMS SPILP Slack FRLC Compilation Time 0.32 290.72 0.93 0.71 Table 3: Comparison of HRMS compilation time to the other 3 methods. Conclusions In this paper we have presented Hypernode Reduction Modulo Scheduling (HRMS), a novel and effective heuristic technique for resource-constrained software pipelining. HRMS attempts to optimize the initiation interval while reducing the register requirements of the schedule. HRMS works in three main steps: computation of MII , pre-ordering of the nodes of the dependence graph using a priority function, and scheduling of the nodes following this order. The ordering function ensures that when a node is scheduled, the partial scheduling contains at least a reference node (a predecessor or a successor), except for the particular case of recurrences. This tends to reduce the lifetime of loop variants and thus reduce register requirements. In addition, the ordering function gives priority to recurrence circuits in order not to penalize the initiation interval. We provided an exhaustive evaluation of HRMS using 1258 loops from the Perfect Club Benchmark Suite. We have seen that HRMS generates schedules that are optimal in terms of II for at least 97.4% of the loops. Although the pre-ordering step consumes a high percentage of the total compilation time, the total scheduling time is smaller than the time required by a convential Top-down scheduler. In addition, HRMS provides a significant performance advantage over a Top-down scheduler when there is a limited number of registers. This better performance comes from a reduction of the execution time and the memory traffic (due to spill code) of the software pipelined execution. We have also compared our proposal with three other methods: the SPILP integer programming formulation, Slack Scheduling and FRLC Scheduling. Our schedules exhibit significant improvement in performance in terms of initiation interval and buffer requirements compared to FRLC, and a significant improvement in the initiation interval when compared to Slack lifetime sensitive heuristic. We obtained similar results to SPILP, which is an integer linear programming approach that obtains optimal solutions but has a prohibitive compilation time for real loops. --R Software pipelining. Conversion of control dependence to data dependence. A uniform representation for high-level and instruction-level transformations The Perfect Club benchmarks: Effective performance evaluation of supercomputers. An approach to scientific array processing: The architectural design of the AP120B/FPS-164 family Overlapped loop support in the Cydra 5. Compiling for the Cydra 5. Stage scheduling: A technique to reduce the register requirements of a modulo schedule. Optimum modulo schedules for minimum register requirements. Minimizing register requirements under resource-constrained software pipelining Highly Concurrent Scalar Processing. Circular scheduling: A new technique to perform software pipelining. Software pipelining: An effective scheduling technique for VLIW machines. A Systolic Array Optimizing Compiler. Reducing the Impact of Register Pressure on Software Pipelined Loops. Hypernode reduction modulo scheduling. Register requirements of pipelined loops and their effect on performance. Register requirements of pipelined processors. A novel framework of register allocation for software pipelin- ing Software pipelining in PA-RISC compilers Iterative modulo scheduling: An algorithm for software pipelining loops. Some scheduling techniques and an easily schedulable horizontal architecture for high performance scientific computing. Register allocation for software pipelined loops. Parallelisation of loops with exits on pipelined architectures. Decomposed software pipelining: A new perspective and a new approach. Enhanced modulo scheduling for loops with conditional branches. Modulo scheduling with multiple initiation intervals. --TR --CTR Spyridon Triantafyllis , Manish Vachharajani , Neil Vachharajani , David I. August, Compiler optimization-space exploration, Proceedings of the international symposium on Code generation and optimization: feedback-directed and runtime optimization, March 23-26, 2003, San Francisco, California David Lpez , Josep Llosa , Mateo Valero , Eduard Ayguad, Widening resources: a cost-effective technique for aggressive ILP architectures, Proceedings of the 31st annual ACM/IEEE international symposium on Microarchitecture, p.237-246, November 1998, Dallas, Texas, United States David Lpez , Josep Llosa , Mateo Valero , Eduard Ayguad, Cost-Conscious Strategies to Increase Performance of Numerical Programs on Aggressive VLIW Architectures, IEEE Transactions on Computers, v.50 n.10, p.1033-1051, October 2001 Josep Llosa , Eduard Ayguad , Antonio Gonzalez , Mateo Valero , Jason Eckhardt, Lifetime-Sensitive Modulo Scheduling in a Production Environment, IEEE Transactions on Computers, v.50 n.3, p.234-249, March 2001

software pipelining;register allocation;register spilling;loop scheduling;instruction scheduling

279588

Algorithms for Variable Length Subnet Address Assignment.

AbstractIn a computer network that consists of M subnetworks, the L-bit address of a machine consists of two parts: A prefix si that contains the address of the subnetwork to which the machine belongs, and a suffix (of length L$-$ |si|) containing the address of that particular machine within its subnetwork. In fixed-length subnetwork addressing, |si| is independent of i, whereas, in variable-length subnetwork addressing, |si| varies from one subnetwork to another. To avoid ambiguity when decoding addresses, there is a requirement that no si be a prefix of another sj. The practical problem is how to find a suitable set of sis in order to maximize the total number of addressable machines, when the ith subnetwork contains ni machines. Not all of the ni machines of a subnetwork i need be addressable in a solution: If $n_i > 2^{L-|s_i|},$ then only $2^{L-|s_i|}$ machines of that subnetwork are addressable (none is addressable if the solution assigns no address si to that subnetwork). The abstract problem implied by this formulation is: Given an integer L, and given M (not necessarily distinct) positive integers $n_1, \cdots, n_M,$ find M binary strings $s_1, \cdots, s_M$ (some of which may be empty) such that 1) no nonempty string si is prefix of another string sj, 2) no si is more than L bits long, and 3) the quantity $\sum \nolimits _{|s_k|\ne0} \min \left\{ n_k, 2^{L-|s_k|} \right\}$ is maximized. We generalize the algorithm to the case where each ni also has a priority pi associated with it and there is an additional constraint involving priorities: Some subnetworks are then more important than others and are treated preferentially when assigning addresses. The algorithms can be used to solve the case when L itself is a variable; that is, when the input no longer specifies L but, rather, gives a target integer for the number of addressable machines, and the goal is to find the smallest L whose corresponding optimal solution results in at least addressable machines.

Introduction This introduction discusses the connection between computer networking and the abstract problems for which algorithms are subsequently given. It also introduces some terminology. In a computer network that consists of M subnetworks, the L-bit address of a machine consists of two parts: A prefix that contains the address of the subnetwork to which the machine belongs, and a suffix containing the address of that particular machine within its subnetwork. In the case where the various subnetworks contain roughly the same number of machines, a fixed partition of the L bits into a t-bit prefix, Me, and an (L \Gamma t)-bit suffix, works well in practice: Each subnetwork can then contain up to 2 L\Gammat addressable machines; if it contains more, then only 2 L\Gammat of them will have an address and the remaining ones will be unsatisfied, in the sense that they will have no address. If, in a fixed length partition scheme, some machines are unsatisfied, then the only way to satisfy them is to increase the value of L. However, a fixed length scheme can be wasteful if the M subnetworks consist of (or will eventually consist of) different numbers of machines, say, machines for the ith subnetwork. In such a case, the fixed scheme can leave many machines unsatisfied (for that particular value of L) even though the variable length partition scheme that we describe next could satisfy all of them without having to increase L. In a variable partition scheme, the length of the prefix containing the subnetwork's address varies from one subnetwork to another. In other words, if we let s i be the prefix that is the address of the ith subnetwork, then we now can have js j. However, to avoid ambiguity (or having to store and transmit js i j), there is a requirement that no s i be a prefix of another s j . Variable length subnetwork addressing is easily shown to satisfy a larger total number of addressable machines than the fixed length scheme: There are examples where fixed length subnetwork addressing cannot satisfy all of the addressing can. Furthermore, we are also interested in the cases where even variable length addressing cannot satisfy all of the N machines: In such cases we want to use the L bits available as effectively as possible, i.e., in order to satisfy as many machines as possible. Of course an optimal solution might then leave unsatisfied all the machines of, say, the ith subnetwork; this translates into s i being the empty string, i.e., js solution therefore consists of determining binary strings that maximize the sum A solution completely satisfies the ith subnetwork if it satisfies all of the machines of that subnetwork, i.e., if js no machine of the ith subnetwork is satisfied, and we then say that the ith network is completely unsatisfied. If the solution satisfies some but not all the machines of the ith subnetwork, then that subnetwork is partially satisfied; this happens when in which case only 2 L\Gammajs i j of the machines of that subnetwork are satisfied. An optimal solution can leave some of the subnetworks completely satisfied, others completely unsatisfied, and others partially satisfied. The prioritized version of the problem models the situation where some subnetworks are more important than others. We use the following priority policy. Priority Policy: "The number of satisfied machines of a subnetwork is the same as if all lower-priority subnetworks did not exist." The next section proves some useful properties for a subset of the optimal solutions. We assume the unprioritized case, and leave the prioritized case until the end of the paper. Before proceeding with the technical details of our approach, we should stress that in the above we have provided only enough background and motivation to make this paper self-contained. The reader interested in more background than we provided can find, in references [11, 8, 9, 10, 6, 4, 12], the specifications for standard subnet addressing, and other related topics. For a more general discussion of hierarchical addressing, its benefits in large networks, and various lookup solution methods (e.g., digital trees), see [7, 5]. Finally, what follows assumes the reader is familiar with basic techniques and terminology from the text algorithms and data structures literature - we refer the reader to, for example, the references [1, 2, 3]. Preliminaries The following definitions and observations will be useful later on. We assume, without loss of generality, that . Since the case when n admits a trivial solution (2 L machines are satisfied, all from subnetwork 1), from now on we assume that logarithms are to the base 2. Lemma 1 Let S be any solution (not necessarily optimal). Then there exists a solution S 0 that satisfies the same number of machines as S, uses the same set of subnetwork addresses as S, and in which the completely unsatisfied subnetworks (if there are any) are those that have the k lowest k. In other words, js Proof: Among all such solutions that satisfy the same number of machines as S, consider one that has the smallest number of offending pairs defined as pairs completely unsatisfied, and j is not completely unsatisfied. We claim that the number of such pairs is zero: Otherwise interchanging the roles of subnetworks i and j in that solution does not decrease the total number of satisfied machines, a contradiction since the resulting solution has at least one fewer offending pair. 2 On the other hand, there does not necessarily exist an S 0 of equal value to S and in which all of the (say, completely satisfied subnetworks are those that have the k highest n i values. If, in the optimal solution we seek, we go through the selected subnetworks by decreasing n i values, then we initially encounter a mixture of completely satisfied and partially satisfied subnetworks, but once we get to a completely unsatisfied one then (by the above lemma) all the remaining ones are completely unsatisfied. Lemma 2 Let S be any solution (not necessarily optimal). There exists a solution S 0 that satisfies as many machines as S, uses the same set of subnetwork addresses as S, and is such that js Proof: Among all such solutions that satisfy the same number of machines as S, consider one which has the smallest number of offending pairs defined as pairs i; j such that js We claim that the number of such pairs is zero: Otherwise interchanging the roles of subnetworks i and j in that solution does not decrease the total number of satisfied machines, a contradiction since the resulting solution has at least one fewer offending pair. 2 Let T be a full binary tree of height L, i.e., T has 2 L leaves and 2 nodes. For any solution S, one can map each nonempty s i to a node of T in the obvious way: The node v i of T corresponding to subnetwork i is obtained by starting at the root of T and going down as dictated by the bits of the string s i (where a 0 means "go to the left child" and a 1 means "go to the right child"). Note that the depth of v i in T (its distance from the root) is js i j, and that no v i is ancestor of another v j in T (because of the requirement that no nonempty s i is a prefix of another s j ). For any node w in T , we use parent(w) to denote the parent of w in T , and we use l(w) to denote the number of leaves of T that are in the subtree of w; hence solution completely satisfies subnetwork i iff which case we can extend our terminology by saying that "node v i is completely satisfied by S" rather than the more accurate "the subnetwork i corresponding to node v i is completely satisfied by S." any solution that satisfies Lemmas 1 and 2. Then there is a solution i at the same depth as v i , and is such that implies that v 0 i has smaller preorder number in T than v 0 j (which is equivalent to saying that s 0 i is lexicographically smaller than s 0 can be obtained from S by a sequence of "interchanges" of various subtrees of T , as follows. initially a copy of T , and repeat the following until 1. Perform an "interchange" in T 0 of the subtree rooted at node v i with the subtree rooted at the leftmost node of T 0 having same depth as i is simply the new position occupied by "interchange". 2. Delete from T 0 the subtree rooted at v 0 Performing in T the interchanges done on T 0 gives a new T where the v 0 's have the desired property.The "interchange" operations used to prove the above lemma will not be actually performed by our algorithm - their only use is for the proof of the lemma. Lemma 4 Let S be any solution (not necessarily optimal) that satisfies the properties of Lemmas 1- 3. There exists a solution S 0 that satisfies as many machines as S, that also satisfies the properties of Lemmas 1-3, and is such that any v i that is not the root of T has l(parent(v i the nonempty s i 's of such an S 0 are a subset of the nonempty s i 's of S. Proof: Among all solutions that satisfy the same number of machines as S, let S one that maximizes the integer i which all of v the lemma's property, i.e., they have We claim that that such an S 0 already satisfies the lemma. Suppose to the contrary that i ! k, i.e., that l(parent(v i+1 cannot be completely satisfied since that would imply that l(v i+1 Hence v i+1 is only partially satisfied, i.e., l(v z be the parent of v i+1 and y be the sibling of v i+1 in T ; y must be to the right of v i+1 since otherwise v i is at y and v i too has l(parent(v i which contradicts the definition of i. Also note that the fact that l(z) - n i+1 implies that n i.e., the number of unsatisfied machines in subnetwork promoting v i+1 by "moving it to its parent", one level up the tree T , thus (i) replacing the old s i+1 by a new (shorter) one obtained by dropping the rightmost bit of the old s i+1 , and (ii) deleting from S 0 all of the s j that now have the new s i+1 as a prefix. Note that, for each s j so removed, its corresponding v j was in the subtree of y, hence the removal of these s j 's results in at most l(y) machines becoming unsatisfied, but that is compensated for by l(y) machines of subnetwork that have become newly satisfied as a result of v i+1 's promotion, implying that the new solution S 00 has value that is no less than that of S 0 . However, a v j so deleted from the subtree of y can cause S 00 to no longer satisfy the property of Lemma 1 because of a surviving v t to the right of z having an n t ! n j . We next describe how to modify S 00 so it does satisfy Lemma 1. In the rest of the proof S 0 refers to the solution we started with, before v i+1 was moved up by one level, and S 00 refers to the solution after v i+1 was moved. Let (v denote the set of the deleted v j 's (who were in y's subtree in the original S 0 but are not in S 00 ). If are in S 00 and are to the right of z, hence we need to "repair" S 00 to restore the property of Lemma 1 (if on the other then no such repair is needed). This is done as follows. Simultaneously for each of the elements of the sequence (v do the following: In the tree T , place the element considered (say, v j ) at the place previously (in the original S 0 ) occupied by v j+l+1 (if then that v j cannot be placed and the new solution leaves completely unsatisfied). The S 00 so modified satisfies the same number of machines as the original one, still satisfies Lemmas 1-3, but has "moved" v i+1 one level up the tree T . This can be repeated until v i+1 is high enough that but that is a contradiction to the definition of integer i. Hence it must be the case that S 0 has Lemma 5 There exists an optimal solution S that satisfies the properties of Lemma 4 and in which every subnetwork i has an s i of length equal to either Proof: Let S be an optimal solution satisfying Lemma 4. First, we claim that there is such an S in which every s i satisfies js Suppose to the contrary that, in S, some s i has length less than moving v i from its current position, say node y in T , to a descendant of y whose depth equals e, would leave subnetwork i completely satisfied without affecting the other subnetworks. Repeating this for all i gives a solution in which every s i has length - course moving a v i down to (say) y's left subtree leaves a ``hole'' in y's right subtree in the sense that the right subtree of y is unulitilized in the new solution. The resulting S might have many such unutilized subtrees of T : It is easy to "move them to the right" so that they all lie to the right of the utilized subtrees of T (the details are easy and are omitted). Hence we can assume that S is such that js (Note that the above does not introduce any violation of the properties of Lemma 4.) To complete the proof we must show that js implies that Taking logarithms on both sides gives: which completes the proof. 2 The observations we made so far are enough to easily solve in O(M log M) time the following (easier) version of the problem: Either completely satisfy all M subnetworks, or report that it is not possible to do so. It clearly suffices to find a v i in T for each subnetwork i (since the v i 's uniquely determine the s i 's). This is done in O(M log M) time by the following greedy algorithm, which operates on only that portion of T that is above the v i 's: 1. Sort the n i 's in decreasing order, say n log M) (the log M factor goes away if the n i 's can be sorted in linear time, e.g., if they are integers smaller than M O(1) ). 2. For each n i , compute the depth d i of v i in T : d 3. Repeat the following for on the leftmost node of T that is at depth d i and has none of v no such node exists then stop and output "No Solution Exists"). Time: O(M) by implementing this step as a construction and (simultane- ously) preorder traversal of the relevant portion of T - call it T we start at the root and stop at the first preorder node of depth d 1 , label it v 1 and consider it a leaf of T 0 , then resume until the preorder traversal reaches another node of depth d 2 , which is labeled v 2 and considered to be another leaf of T 0 , etc. Note that in the end the leaves of T 0 are the v i 's in left to right order. Theorem 1 Algorithm greedy solves the problem of finding an assignment of addresses that completely satisfies all subnetworks when such an assignment exists. Its time complexity is O(M) if the are given in sorted order, O(M log M) if it has to sort the n i 's. Proof: The time complexity was argued in the exposition of the algorithm. Correctness of the algorithm follows immediately from Lemmas 1-5. 2 Theorem 2 An assignment that completely satisfies all subnetworks exists if and only if Proof: Observe that algorithm greedy succeeds in satisfying all subnetworks if and only if the inequality is satisfied. 2 Corollary 1 Whether there is an assignment that completely satisfies all subnetworks can be determined in O(M) time, even if the n i 's are not given in sorted order. Proof: The right-hand side of the inequality in the previous theorem can be computed in O(M) time. 2 Would the greedy algorithm solve the problem of satisfying the largest number of machines when it cannot satisfy all of them? That is, when it cannot assign a v i to a node (in Step 3), instead of saying "No Solution Exists", can it accurately claim that the solution produced so far is optimal? The answer is no, as can be seen from the simple example of example the greedy algorithm satisfies 5 machines whereas it is possible to satisfy 7 machines). However, the following holds. Observation 1 The solution returned by the greedy algorithm satisfies a number of machines that is no less than half the number satisfied by an optimal solution. be the number of subnetworks completely satisfied by greedy. Observe that since if we had would have put v i at a greater depth than its current position. Therefore an optimal solution could, compared to greedy, satisfy no more than an additional machines, which is less than the number satisfied by greedy.However, we need not resort to approximating an optimal solution, since the next section will give an algorithm for finding an optimal solution. 3 Algorithm for the Unprioritized Case We assume throughout this section that the greedy algorithm described earlier has failed to satisfy all the machines. The goal then is to satisfy as many machines as possible. We call level ' the 2 ' nodes of T whose depth (distance from the root) is '. We number the nodes of level ' as follows: ('; 1); ('; is the kth leftmost node of level '. Lemma 5 says that v i is either at a depth of d i or of d limits the number of choices for where to place v i to 2 d i choices at depth d i , and 2 d i +1 choices at depth 1. For every to be the maximum number of machines of subnetworks that can be satisfied by using only the portion of T having preorder numbers - the preorder number of (d i ; j), and subject to the constraint that v i is placed at node (d defined analogously but with (d playing the role that played in the definition of C(i; j). The C(i; j)'s and C 0 (i; j)'s will play an important role in the algorithm: Clearly, if we had these quantities for all then we could easily obtain the number of machines satisfied by an optimal solution, simply by choosing the maximum among them: Another notion used by the algorithm is that of the '-predecessor of a node v of T , where ' is an integer no greater than v's depth: It is the node of T at level ' that is immediately to the left of the ancestor of v at level ' (if no such node exists then v has no '-predecessor). In other words, if w is the ancestor of v at level ' (possibly then the '-predecessor of v is the rightmost node to the left of w at level '. The algorithms will implicitly make use of the fact that the '-predecessor of a given node v can be obtained in constant time: If v is represented as a pair (a; b) where a is v's depth and b is the left-to-right rank of b at that depth (i.e., v is the bth leftmost node at depth a), then the '-predecessor of (a; b) is ('; c) where The following algorithm preliminary will later be modified into a better algorithm. The input to the algorithm is L and the n i 's. The output is a placement of the v i 's in T ; recall that this is equivalent to computing the s i 's because the s i 's can easily obtained from the v i 's (in fact each s i can be obtained from v i in constant time, as will be pointed out later). We assume that a preprocessing step has already computed the d i 's. We use pred('; v) or pred('; a; b) interchangeably, to denote the '-predecessor of a node v = (a; b), with the convention that pred('; a; b) is (\Gamma1; \Gamma1) when it is undefined, i.e., when ' ? a or (a; b) has no '-predecessor. 1. For to M in turn, do the following: (a) For with the convention that are 0. be the node of T that "gives C(i; b) its value" in the above maximization, that is, f(i; b) is pred(d pred(d (b) For with the convention that are 0. be the node of T that "gives C 0 (i; b) its value" in the above maximization, that is, f pred(d pred(d 2. Find the largest, over all i and b, of the C(i; b)'s and C 0 (i; b)'s computed in the previous step: Suppose it is C(k; b) (respectively, C 0 (k; b)). Then C(k; b) (respectively, C 0 (k; b)) is the maximum possible number of machines that are satisfied by an optimal solution v To generate a set of assignments that correspond to that optimal solution (rather than just its value), we use the f and f 0 functions obtained in the previous step: Starting at node (d (respectively, we "trace back" from there, and output the nodes of the optimal solution as we go along (in the order v k ; v The details of this "tracing back" are as follows: (a) k. If the largest of the C(i; b)'s and C 0 (i; b)'s computed in the previous step was Then repeat the following until (b) Output "v equal to either f(i; fi) (in case or to f 0 (i; fi) (in case Note. To output the string s i corresponding to a v i node, rather than the (d describing that v i , we modify the above Step 2(b) as follows: If v (a; b) then s i is the binary string consisting of the rightmost a digits in the binary representation of the integer 2 a is the breadth-first number of the node (a; b), and that an empty string corresponds to the root since 2 0 This implies that s i can be computed from the pair (a; b) in constant time. Correctness of the above algorithm preliminary follows from Lemmas 1 - 5. The time complexity of preliminary is unsatisfactory because it can depend on the size of T as well as M , making the worst case take O(M2 L ) time. However, the following simple modification results in an O(M 2 In Steps 1(a) and (respectively) 1(b), replace "For "For iteration bounds for b remain unchanged, at 2 d i for 1(a) and 2 d i +1 for 1(b)). Before arguing the correctness of this modified algorithm, we observe that its time complexity is O(M 2 ), since we now iterate over only M 2 distinct note: The relevant C(i; b)'s need not be explicitly initialized, they can implicitly be assumed to be zero initially; this works because of the particular order in which Step 1 computes them.) Correctness follows from the claim (to be proved next) that there is an optimal solution that, of the 2 a nodes of any level a, does not use any of the leftmost nodes of that level. Let S be an optimal solution that has the smallest possible number (call it t) of violations of the claim, i.e., the smallest number of nodes (a; b) where b ! 2 a \Gamma M and some v i is at (a; b). We prove that by contradiction: Suppose that t ? 0, and let a be the smallest depth at which the claim is violated. Let (a; b) be a node of level a that violates the claim, is placed at (a; b) by optimal solution S. Since there are more than M nodes to the right of v i at level a, the value of S would surely not decrease if we were to modify S by re-positioning all of v in the subtrees of the rightmost nodes of level a (without changing their depth). Such a modification, however, would decrease t, contradicting the definition of S. Hence t must be zero, and the claim holds. The following summarizes the result of this section. Theorem 3 The unprioritized case can be solved in O(M 2 ) time. 4 Algorithm for the Prioritized Case Let the priorities be is the priority of subnetwork k i . In the rest of this section we assume that L is not large enough to completely satisfy all of the M subnetworks (because in the other case, where L is large enough, the priorities do not play a role and Theorem Use greedy (or, alternatively, Corollary 1) in a binary search for the largest i (call it - i) such that the subnetworks k can be completely satisfied; each "comparison" in the binary search corresponds to a call to greedy (or, alternatively, to Corollary 1) - of course it ignores the priorities of the subnetworks k This takes total time O(M log M) even though we may use greedy a logarithmic number of times, because we sort by decreasing n j 's only once, which makes each subsequent execution of greedy cost O(M) time rather than O(M log M ). Let S be the solution, returned by greedy, in which all of subnetworks are completely satisfied. By the definition of - i, it is impossible to completely satisfy all of subnetworks Our task is to modify S so as to satisfy as many of the machines of subnetworks k - as possible without violating the priority policy (hence keeping subnetworks completely satisfied). This is done as follows: 1. set the depth of each k i , 1, to be dlog n k i e. 2. Use greedy log log n k j times to binary search for the smallest depth (call it d) at which v k j can be placed without resulting in the infeasibility (as tested by greedy) of (i) placing all of subnetworks k at their previously fixed depths and (ii) placing k j at depth d (there are log n k j possible values for d, which implies the log log n k j iterations of the binary search). If no such d exists (i.e., if any placement of k j prevents the required placement of proceed to Step 3. If the binary search finds such a d then fix the depth of v j to be d (it stays d in all future iterations), set 2. 3. The solution is described by the current depths of k These fixed depths are then used by a preorder traversal of (part of) T to position v k 1 in T . That the above algorithm respects the priority policy follows from the way we fix the depth of Subnetworks of lower priority do not interfere with it (because they are considered later in the iteration). The time complexity is easily seen to be O(M 2 log L), since n k The following summarizes the result of this section. Theorem 4 The prioritized case can be solved in O(M 2 log L) time. 5 Further Remarks What if L itself is a variable ? That is, consider the situation where instead of specifying L the input specifies a target integer fl for the number of addressable machines; the goal is then to find the smallest L that is capable of satisfying at least fl machines. The algorithms we gave earlier (and that assume a fixed L) can be used as subroutines in a "forward" binary search for the optimal (i.e., smallest) value of L (call it - L) that satisfies at least fl machines: We can use them log - times in a "forward" binary search for - L. So it looks like there is an extra multiplicative log - if L is itself a variable that we seek to minimize, as opposed to the version of the problem that fixes L ahead of time. However, Theorem 2 implies that there is no such log - in the important case where we seek the smallest L that satisfies all the machines: This version of the problem can be solved just as fast as the one where L is fixed and we seek to check whether it can completely satisfy all M subnetworks. Acknowledgement . The authors are grateful to three anonymous referees for their helpful comments on an earlier version of this paper. --R Combinatorial Algorithms on Words Introduction to Algorithms Algorithms "Class A Subnet Experiment" "Parallel searching techniques for routing table lookup," "Class A Subnet Experiment Results and Recommendations" "Fast routing table lookup using CAMs," "Internet standard subnetting procedure" "Broadcasting Internet datagrams in the presence of subnets" "Internet subnets" "Variable Length Subnet Table For IPv4" "On the Assignment of Subnet Numbers" --TR

prefix codes;addressing;algorithms;computer networks

279589

Analysis of Cache-Related Preemption Delay in Fixed-Priority Preemptive Scheduling.

AbstractWe propose a technique for analyzing cache-related preemption delays of tasks that cause unpredictable variation in task execution time in the context of fixed-priority preemptive scheduling. The proposed technique consists of two steps. The first step performs a per-task analysis to estimate cache-related preemption cost for each execution point in a given task. The second step computes the worst case response time of each task that includes the cache-related preemption delay using a response time equation and a linear programming technique. This step takes as its input the preemption cost information of tasks obtained in the first step. This paper also compares the proposed approach with previous approaches. The results show that the proposed approach gives a prediction of the worst case cache-related preemption delay that is up to 60 percent tighter than those obtained from the previous approaches.

INTRODUCTION In real-time systems, tasks have timing constraints that must be satisfied for correct op- eration. To guarantee such timing constraints, extensive studies have been performed on schedulability analysis [1, 2, 3, 4, 5, 6]. They, in many cases, make a number of assumptions to simplify the analysis. One such simplifying assumption is that the cost of task preemption is zero. In real systems, however, task preemption incurs additional costs to process interrupts [7, 8, 9, 10], to manipulate task queues [7, 8, 10], and to actually perform context switches [8, 10]. Many of such direct costs are addressed in a number of recent studies that focus on practical issues related to task scheduling [7, 8, 9, 10]. However, in addition to the direct costs, task preemption introduces indirect costs due to cache memory, which is used in almost all computing systems today. In computing systems with cache memory, when a task is preempted, a large number of memory blocks 1 belonging to the task are displaced from the cache memory between the time the task is preempted and the time the task resumes execution. When the preempted task resumes its execution, it spends a substantial amount of time to reload the cache with the previously displaced memory blocks. Such cache reloading greatly increases the task execution time, which may invalidate the result of schedulability analysis that overlooks the cache-related preemption costs. To rectify this problem, recent studies addressed the issue of incorporating cache-related preemption costs into schedulability analysis [12, 13]. These studies assume that each cache block used by a preempting task replaces from the cache a memory block that is needed by a preempted task. This pessimistic assumption leads to a loose estimation of cache-related preemption delay since the replaced memory block may not be useful to any preempted task. For example, it is possible that the replaced memory block is one that is no longer 1 A block is the minimum unit of information that can be either present or not present in the cache-main memory hierarchy [11]. We assume without loss of generality that memory references are made in the unit of blocks. needed or one that will be replaced without being re-referenced even when there were no preemptions. In this paper, we propose a schedulability analysis technique that considers the usefulness of cache blocks in computing cache-related preemption delay. The goal is to reduce the prediction inaccuracy resulting from the above pessimistic assumption. The proposed technique consists of two steps. In the first step, we perform a per-task analysis to compute the number of useful cache blocks at each execution point in a given task, where a useful cache block at an execution point is defined as a cache block that contains a memory block that may be re-referenced before being replaced by another memory block. The number of useful cache blocks at an execution point gives an upper bound on cache-related preemption cost that is incurred when the task is preempted at that point. The results of this per-task analysis are given in a table that specifies the (worst case) preemption cost for a given number of preemptions of the task. From this table, the second step derives the worst case response times of tasks using a linear programming technique [14] and the worst case response time equation [2, 6]. This paper is organized as follows: In Section II, we survey the related work. Section III describes our overall approach to schedulability analysis that considers cache-related pre-emption cost. Sections IV and V detail the two steps of the proposed schedulability analysis technique focusing on direct-mapped instruction cache memory. Section VI presents the results of our experiments to assess the effectiveness of the proposed approach. Section VII describes extensions of the proposed technique to set-associative cache memory and also to data cache memory. Finally, we conclude this paper in Section VIII. II. RELATED WORK A. Schedulability Analysis in Fixed-priority Scheduling A large number of schedulability analysis techniques have been proposed within the fixed-priority scheduling framework [2, 3, 4, 6]. Liu and Layland [4] show that the rate monotonic priority assignment where a task with a shorter period is given a higher priority is optimal when task deadlines are equal to their periods. They also give the following sufficient condition for schedulability for a task set consisting of n periodic tasks where C i is a worst case execution time (WCET) estimate of - i and T i is its period 2 . This condition states that if the total utilization of the task set (i.e., ) is lower than the given utilization bound (i.e., n (2 1=n \Gamma 1)), the task set is guaranteed to be schedulable under the rate monotonic priority assignment. Later, Lehoczky et al. develop a necessary and sufficient condition for schedulability based on utilization bounds [3]. Another approach to schedulability analysis is the worst case response time approach [2, 6]. The approach uses the following recurrence equation to compute the worst case response where hp(i) is the set of tasks whose priorities are higher than that of - i . In the equation, the term j2hp(i) d R i eC j is the total interferences from higher priority tasks during R i and C i is - i 's own execution time. The equation can be solved iteratively and the iteration terminates when R i converges at a value. This R i value is compared against - i 's deadline These notations will be used throughout this paper along with D i that denotes the deadline of - i where We assume without loss of generality that - i has a higher priority than - to determine the schedulability of - i . Recently, Katcher et al. [10] and Burns et al. [7, 8] provided a methodology for incorporating the cost of preemption into schedulability analysis. In these approaches, preemption costs arising from interrupt handling, task queue manipulation, and context-switching are taken into account in the schedulability analysis. In this paper, we are also interested in incorporating the cost of preemption into schedulability analysis. However, unlike the above studies, our main focus is on indirect preemption costs due to cache memory, which is increasingly being used in real-time computing systems. B. Caches in Real-time Systems Cache memory is used in almost all computing systems today to bridge the ever increasing speed gap between the processor and main memory. However, due to its unpredictable per- formance, cache memory has not been widely used in real-time computing systems where the guaranteed worst case performance is of great importance. The unpredictable performance comes from two sources: intra-task interference and inter-task interference. interference occurs when a memory block of a task conflicts in the cache with another block of the same task. Recently, there has been considerable progress on the analysis of intra-task interference due to cache memory and interested readers are referred to [15, 16, 17, 18, 19]. Inter-task interference, which is the main focus of this paper, occurs when memory blocks of different tasks conflict with one another in the cache. There are two ways to address the unpredictability resulting from inter-task interference. The first way is to use cache partitioning where cache memory is divided into disjoint partitions and one or more partitions are dedicated to each real-time task [20, 21, 22, 23]. In these techniques, each task is allowed to access only its own partitions and thus we need not consider inter-task in- terference. There are two different approaches to cache partitioning: hardware-based and software-based. In hardware-based approaches, extra address-mapping hardware is placed between the processor and cache memory to limit the cache access by each task to its own partitions [20, 21, 22]. On the other hand, in software-based approaches a specialized compiler and linker are used to map each task's code and data only to its assigned cache partitions [23]. Cache partitioning improves the predictability of the system by removing cache-related inter-task interference, but has a number of drawbacks. One common draw-back of both the hardware and software-based approaches is that they require modification of existing hardware or software. Another common drawback is that they limit the amount of cache memory available to individual tasks. Finally, in the case of the hardware-based approach, the extra address-mapping hardware may stretch the processor cycle time, which affects the execution time of every instruction. The other way to address the unpredictability resulting from inter-task interference is to devise an efficient method for analyzing its timing effects. In [12], Basumallick and Nilsen extend the rate monotonic analysis explained in the previous subsection to take into account the inter-task interference. In this approach, the WCET estimate of a task - i is modified as C 0 is the original WCET estimate of - i computed assuming that the task executes without preemption and fl i is the worst case preemption cost that the task - i might impose on preempted tasks. This modification is based on a pessimistic assumption that each cache block used by a preempting task replaces from the cache a memory block that is needed by a preempted task. In the approach, the total utilization of a given task set computed from the modified WCET estimates is compared against the utilization bound given by Equation (1) to determine the schedulability of the task set. One drawback of this approach is that it suffers from a pessimistic utilization bound given by Equation (1), which approaches 0.693 for a large n [4]. Many task sets that have total utilization higher than this bound can be successfully scheduled by the rate monotonic priority assignment [3]. To rectify this problem, Busquets-Mataix et al. in [13] propose a technique based on the response time approach. This technique makes the same pessimistic assumption that each cache block used by a preempting task replaces from the cache a memory block that is needed by a preempted task. This assumption leads to the following equation for computing the worst case response time of a task: d e \Theta (C j is the cache-related preemption cost that task - j might impose on lower priority tasks. The term fl j is computed by multiplying the number of cache blocks used by task - j and the time needed to refill a cache block. Both the utilization bound based and response time based approaches assume that each cache block used by a preempting task replaces from the cache a memory block that is needed by a preempted task. This pessimistic assumption leads to a loose estimation of cache-related preemption delay since it is possible that the replaced memory block is one that is no longer needed or one that will be replaced without being re-referenced even when the lower priority task is executed without preemption. III. OVERALL APPROACH This section overviews our proposed schedulability analysis technique that aims to minimize the overestimation of cache-related preemption delay due to the pessimistic assumption explained in the previous section. For this purpose, the response time equation is augmented as follows in the proposed d where PC i (R i ) is the total cache-related preemption delay of task - i during R i , i.e., the total cache reloading times of - 1 during R i . 22 t 2231 Fig. 1. Example of PC i (R i ). The meaning of PC i (R i ) can be best explained by an example such as the one given in Figure 1. In the example, there are three tasks, Each arrow in the figure denotes a point where a task is preempted and each shaded rectangle denotes cache reloading after the corresponding task resumes execution. With these settings, PC 3 (R 3 ), which is the total cache-related preemption delay of task - 3 during R 3 , is the total sum of cache reloading times of - 1 , - 2 and - 3 during R 3 , which corresponds to the sum of shaded rectangles in the figure. The augmented response time equation can be solved iteratively as follows. d (R 0 . (4) R k+1 (R k As before, this iterative procedure terminates when R m converged R i value is compared against - i 's deadline to determine the schedulability of - i . To compute PC i (R k i ) at each iteration, we take the following two-step approach. 1. Per-task analysis: We statically analyze each task to determine the cache-related preemption cost at each execution point. This is the cost the task pays when it is preempted at the execution point and is upper-bounded by the number of useful cache blocks at that execution point. Based on this information and information about the worst case visit counts of execution points, we construct the following preemption cost table for each task. # of preemptions 1 cost In the table, f k is the k-th marginal preemption cost that is the cost the task pays in the worst case for its k-th preemption over the preemption. 2. Preemption delay analysis: We use a linear programming technique to compute (R k i ) from the preemption cost tables of tasks and a set of constraints on the number of preemptions of a task by higher priority tasks. The following two sections detail the two steps. IV. PER-TASK ANALYSIS OF USEFUL CACHE BLOCKS In this section, we describe a per-task analysis technique to obtain the preemption cost table of each task. We initially focus on the case of direct-mapped 3 instruction cache memory in this section. In Section VII, we discuss extensions for set-associative cache memory and also for data cache memory. As an example of cache-related preemption cost, consider a direct-mapped cache with four cache blocks. Assume that the cache has m 3 at time t in cache blocks c 0 , Further assume that the following memory block references are made after t. 3 In a direct-mapped cache, each memory block can be placed exactly in one cache block whose index is given by memory block number modulo number of blocks in the cache. In this example, the useful cache blocks at time t are cache blocks c 1 and c 2 since they contain memory blocks m 5 and m 6 , respectively, that are re-referenced before being replaced. On the other hand, cache blocks c 0 and c 3 are not useful at time t since they have m 0 and 3 that are replaced by m 4 and m 7 without being re-referenced. If a preemption occurs at time t, the memory blocks m 5 and m 6 contained in cache blocks c 1 and c 2 may be replaced by memory blocks of intervening tasks and thus need to be reloaded into the cache after resumption. The additional time to reload these useful cache blocks is the cache-related preemption cost at time t. Note that this additional cache reload time is not needed if the task is not preempted. In the following, we explain a technique for estimating the number of useful cache blocks at any point in a program. A. Estimation of the Number of Useful Cache Blocks Our technique for estimating the number of useful cache blocks is based on data flow analysis [24] over the task's program expressed in the control flow graph 4 (CFG). To give 4 In a CFG, each node represents a basic block, while edges represent potential flow of control between basic blocks [25] in { } m x { } m a { } (b) cache state at p Useful Useful cache block j cache block i (a) control flow to and from p { } Fig. 2. Analysis on the usefulness of cache blocks. an intuitive idea about this data flow analysis, consider the CFG given in Figure 2. In the figure, a pair (c; m) denotes a reference to memory block m that is mapped to cache block c. The CFG has two incoming paths to the execution point p, i.e., in 1 and in 2 , and two outgoing paths from p, i.e., out 1 and out 2 . If the control flow came through incoming path in 1 , cache block c i would contain memory block m a at point p since m a is the last reference to cache block c i before reaching p. Similarly, cache block c i would have memory block m b at point p if the control has come through the incoming path in 2 . Thus, either m a or m b may reside in cache block c i at p depending on the incoming path. If either of them is the first reference to cache block c i in an outgoing path from p, the cache block may be reused and thus is defined as being useful at point p. The outgoing path out 2 is such a path and thus cache block c i is defined to be useful at p. For a more formal description, we define reaching memory blocks (RMBs) and live memory blocks (LMBs) for each cache block that are similar to reaching definitions and live variables used in traditional data flow analysis [24]. The set of reaching memory blocks of cache block c at point p, denoted by RMB c , contains all possible states of cache block c at point p where a possible state corresponds to a memory block that may reside in the cache block at the point. For a memory block to reside in cache block c, first, it should be mapped to cache block c. Furthermore, it should be the last reference to the cache block in some execution path reaching p. The set of live memory blocks of cache block c at point p, denoted by p , is defined similarly and is the set of memory blocks that may be the first reference to cache block c after p. With these definitions, a useful cache block at point p can be defined as a cache block whose RMBs and LMBs have at least one common memory block. In Figure 2, RMB c i is fm a ; m b g and LMB c i p is fm a g. Thus, cache block c i is defined to be useful at point p. In the following, we explain how to compute RMBs of cache blocks at various execution points of a given program. We initially focus on RMBs at the beginning and end points of basic blocks 5 . The RMBs at other points can easily be derived from the RMBs at the basic block boundaries as we will see later. To formulate the problem of computing RMBs as a data flow problem, we define a set gen c [B]. This set is either null or contains a single memory block. It is null if basic block does not have any reference to memory blocks mapped to cache block c. On the other hand, if the basic block B has at least one reference to a memory block mapped to c, gen c [B] contains as its unique element the memory block that is the last reference to the cache block c in the basic block. Note that in the latter case the memory block in gen c [B] is the one that will reside in the cache block c at the end of the basic block B. Also note that gen c [B] defined in this manner can be computed locally for each basic block. As an example, consider the CFG given in Figure 3. The CFG shows instruction memory block references made in each basic block. Assuming that the instruction cache is direct mapped and has two blocks, gen c 0 [B 1 ] is fm 2 g since m 2 is the memory block whose reference is the last reference to c 0 in B 1 . The gen c [B] sets for other basic blocks and cache blocks can be computed similarly and are given in Figure 3. With gen c [B] defined in this manner, the RMBs of c just before the beginning of B and just after the end of B, which are denoted by RMB c IN [B] and RMB c OUT [B], respectively, can be computed from the following two equations. P a predecessor of B OUT [B] (5) gen c [B] if gen c [B] is not null IN [B] otherwise The first equation states the memory blocks that reach the beginning of a basic block can be derived from those that reach the ends of the predecessors of B. The second equa- 5 A basic block is a sequence of consecutive instructions in which flow of control enters at the beginning and leaves at the end without halt or possibility of branching except at the end [24]. { }3 c } { { }4 { }c 1 3 { } { }10 } c } Fig. 3. Example of gen c [B]. tion states that RMB c OUT [B] is equal to gen c [B] if gen c [B] is not null and RMB c IN [B] otherwise 6 . These data flow equations can be solved using a well-known iterative approach [24]. It starts with RMB c and iteratively converges to the desired values of RMB c IN 's and RMB c OUT 's. The iterative process can be described procedurally as follows. Algorithm 1: Find RMBs of cache blocks at the beginning and end of each basic block assuming that gen c [B] has been computed for each basic block B and cache block c. /* initialize RMB c IN [B] and RMB c OUT for all B's and c's */ for each basic block B do for each cache block c do begin 6 This equation can be rewritten as where set kill c [B] is the set of reaching memory blocks of cache block c killed in basic block B. The set kill c [B] is obtained as follows: (1) it is null if gen c [B] is null, and (2) it is Mc \Gamma gen c [B] if gen c [B] is not null where Mc is the set of all memory blocks mapped to c in the program. This rewritten form is more commonly used in traditional data flow analysis. IN [B] := ;; OUT [B] := gen c [B]; change := true; while change do begin change := false; for each basic block B do for each cache block c do begin IN [B] := P a predecessor of B OUT [P ]; oldout := RMB c OUT [B]; OUT [B] := gen c [B] else RMB c OUT [B] := RMB c IN [B] if RMB c OUT [B] 6= oldout then change := true As we indicated earlier, the RMBs at other points within a basic block can be computed from the RMBs at the beginning of the basic block. Assume that the basic block has the following sequence of instruction memory block references. The references are processed sequentially starting from (c clear that 1 is in cache block c 1 at the point following the reference. No other conflicting memory blocks can be in c 1 at this point. Therefore, the RMBs of c 1 just after the reference is simply fm 1 g. However, the RMBs of other cache blocks are the same as those just before IN [B]. In general, the RMBs of c i after those of other cache blocks are the same as those before (c The problem of computing LMBs can be formulated similarly to the case of RMBs. The difference is that the LMB problem is a backward data flow problem [24] in that the in sets (i.e., LMB c IN [B]) are computed from the out sets (i.e., LMB c OUT [B]) whereas the RMB problem is a forward data flow problem [24] in that the out sets (i.e., RMB c OUT [B]) are computed from the in sets (i.e., RMB c IN [B]). In the LMB problem, the set gen c [B] is either a set with only one element corresponding to the memory block whose reference is the first reference to cache block c in basic block B, or null if none of the references from B are to memory blocks mapped to c. Using gen c [B] defined in this manner, the following two equations relate the LMB c IN [B] and LMB c OUT [B]. S a successor of B IN [S] IN [B] (6) gen c [B] if gen c [B] is not null OUT [B] otherwise An iterative algorithm similar to the one for computing RMBs can be used to solve this backward data flow problem. The difference is that the algorithm starts with LMB c all B's and c's and uses the above two equations instead of those given in Equation (5). After we compute LMBs at the beginning and end of each basic block, the LMBs at other points can be computed analogously to the case of RMBs. The difference is that the processing of references is backward starting from the end of the basic block rather than forward starting from the beginning. In the LMB problem, the LMBs of c i before a reference and those of other cache blocks are the same as those after After the usefulness of each cache block is determined at each point by computing the intersection of the cache block's RMBs and LMBs at the point, it is trivial to calculate the total number of useful cache blocks at the point; we simply have to count the useful cache blocks at that point. By multiplying this total number of useful cache blocks and the time to refill a cache block, the worst case cache-related preemption cost at the point can be computed. B. Derivation of the Preemption Cost Table This subsection explains how to construct the preemption cost table of a task whose k-th entry is the cost the task pays in the worst case for its k-th preemption over the th preemption. The preemption cost table is constructed from two types of information: (1) the preemption cost at each point, and (2) the worst case visit count of each point, which can be directly derived from the CFG of the given program and the loop bound of each loop in the program. The construction assumes the worst case preemption scenario since we cannot predict, in advance, where preemptions will actually occur. The worst case preemption scenario occurs when the first preemption is at the point with the largest preemption cost (i.e., the point with the largest number of useful cache blocks), and then the second preemption at the point with the next largest preemption cost, and so on. This worst case preemption scenario should be assumed for our analysis to be safe. From the above worst case preemption scenario, the entries of the preemption cost table are filled in as follows. First, we pick a point p 1 that has the largest preemption cost. We then fill in the first entry up to the v p 1 -th entry with that preemption cost where v p 1 is the worst case visit count of p 1 . After that, we pick the point that has the second largest preemption cost and perform the same steps starting from the (v p 1 1)-th entry. This process is repeated until the number of entries in the preemption cost table is exhausted. Assuming that the number of entries in the table is K, the K 0 -th marginal preemption cost, where K 0 ? K, can be conservatively estimated to be the same as the K-th marginal preemption cost since the marginal preemption cost is non-increasing. By applying the per-task analysis explained in this section to all the tasks in the task set, we can obtain the following set of preemption cost tables, one for each task, where f i;j is the j-th marginal preemption cost of - i . # of preemptions 1 cost f 1;1 f 1;2 f 1;3 # of preemptions 1 cost f 2;1 f 2;2 f 2;3 # of preemptions 1 cost f 3;1 f 3;2 f 3;3 # of preemptions 1 cost f n;1 f n;2 f n;3 V. CALCULATION OF THE WORST CASE PREEMPTION DELAYS OF TASKS In this section, we explain how to compute a safe upper bound of PC i (R k used in Equation (4) in Section III from the preemption cost table. We formulate this problem as an integer linear programming problem with a set of constraints. We first define g j;l as the number of invocations of - j that are preempted at least l times during a given response time R k . As an example, consider Figure 4 where task - j is invoked three times during the given R k . The first invocation of task - j , i.e., - j1 , is preempted three times, and both the second and third invocations of - j , i.e., - j2 , - j3 , are preempted once. From the definition of g j;l , g j;1 is 3, g j;2 is 1, and g j;3 is 1. Note that since the highest priority task - 1 cannot be preempted, Fig. 4. Definition of g j;l . If we assume that we know the g j;l values that give the worst case preemption scenario among tasks, we can calculate the worst case cache-related preemption delay of - i during (R k (R k where f j;l is the l-th marginal preemption cost of - j . Note that this total cache-related preemption delay of - i includes all the delay due to the preemptions of - i and those of higher priority tasks during R k In general, however, we cannot determine exactly which g j;l combination will give the worst case preemption delay to task - i . For our analysis to be safe, we should conservatively assume a scenario that is guaranteed to be worse than any actual preemption scenario. Such a conservative scenario can be derived from constraints that any valid g j;l combination should satisfy. We give a number of such constraints on g j;l 's in the following. First, g j;l for a given interval R k i cannot be larger than the number of invocations of - j during that interval. Thus, we have In the formulation, N j is the maximum number of preemptions that a single invocation of experience during R k . An upper bound of such an N j value can be calculated as a=1 d R j a=1 d R k are the worst case response times of higher priority tasks - which should be available when the worst case response time of - i is computed. From this, the index l of g j;l can be bounded by N j in the formulation. Second, the number of preemptions of task - j during the given interval R k i cannot be larger than the total number of invocations of - during that interval since only the arrivals of tasks with priorities higher than that of - j can preempt - j . Thus, we have T a More generally, the total number of preemptions of - during the given interval R k cannot be larger than the total number of invocations of - during that interval. Thus, we have Na T a Note that this constraint subsumes the previous constraint. The maximum value of PC i (R k 's satisfying the above constraints is a safe upper bound on the total cache-related preemption delay of task - i during R k i . This problem can be formulated as an integer linear programming problem as follows: maximize (R k subject to Constraint 1 Constraint 2 Na T a At each iteration of the iterative procedure explained in Section III, this integer linear programming problem is solved to compute the PC i (R k application of this iterative procedure is given in the Appendix. VI. EXPERIMENTAL RESULTS To assess the effectiveness of the proposed approach, we predicted the worst case response times of tasks from sample task sets using the proposed technique and compared them with those predicted using previous approaches. For validation purposes, the predicted worst case response times were also compared with measured response times. Our target machine is an IDT7RS383 board with a 20 MHz R3000 RISC CPU, R3010 FPA (Floating Point Accelerator), and an instruction cache and a data cache of 16 Kbytes each. Both caches are direct mapped and have block sizes of 4 bytes. SRAM (static RAM) is used as the target machine's main memory and the cache refill time is 4 cycles. Although the target machine has a timer chip that provides user-programmable timers, their resolution is too low for our measurement purposes. To accurately measure the execution and response times of tasks, we built a daughter board that implements a timer with a resolution of one machine cycle. For our experiments, we also implemented a simple fixed-priority scheduler based on the tick scheduling explained in [7]. The scheduler manages two queues: run queue and delay queue. The run queue maintains tasks that are ready to run and its tasks are ordered by their priorities. The delay queue maintains tasks that are waiting for their next periods and its tasks are ordered by their release times. The scheduler is invoked by timer interrupts that occur every 160,000 machine cycles. When invoked the scheduler scans the delay queue and all the tasks in the delay queue with release times at or before the invocation time of the scheduler are moved to the run queue. If one of the newly moved tasks has a higher priority than the currently running task, the scheduler performs a context switch between the currently running task and the highest priority task. When a task completes its execution, it is placed into the delay queue and the next highest priority task is dispatched from the run queue. To take into account the overheads associated with the scheduler, we used the analysis technique explained in [7]. In this technique, the scheduler overhead S i during response time R i is given by where is the number of scheduler invocations during R i . is the number of times that the scheduler moves a task from the delay queue to the run queue during R i . ffl C int is the time needed to service a timer interrupt (413 machine cycles in our exper- iments). set Task Period WCET # instruction # useful memory blocks cache blocks (unit: machine cycles) (unit: blocks) I Task set specifications. ffl C ql is the time needed to move the first task from the delay queue to the run queue (142 machine cycles in our experiments). ffl C qs is the time needed to move each additional task from the delay queue to the run queue (132 machine cycles in our experiments). A detailed explanation of this equation is beyond the scope of this paper and interested readers are referred to [7]. We used three sample task sets in our experiments and their specifications are given in Table I. The first column of the table is the task set name and the second column lists the tasks in the task set. Four different tasks were used: FFT, LUD, LMS, and FIR. The task FFT performs the FFT and the inverse FFT operations on an array of 8 floating point numbers using the Cooley-Tukey algorithm [26]. LUD solves 10 simultaneous linear equations by the Doolittle's method of LU decomposition [27] and FIR implements a 35 point Finite Impulse Response (FIR) filter [28] on a generated signal. Finally, LMS is a 21 point adaptive FIR filter where the filter coefficients are updated on each input signal [28]. FIRData Section mapped to non-cacheable area LMS FFT LUD Instruction cache Memory Scheduler Scheduler Fig. 5. Code placement for task set T 3 The table also gives in the third and fourth columns the period and the WCET of each task in the task set, respectively. We used the measured execution times of tasks as their WCETs since tight prediction of tasks' WCETs and accurate estimation of cache-related preemption delay are two orthogonal issues. The measured execution time of a task was obtained by executing the task without preemption. This execution time includes the time for initializing the task and also the time for two context switches: one context switch to the task itself and the other from the task to another task upon completion. The table also gives the total number of instruction memory blocks and the maximum number of useful cache blocks of each task in the fifth and sixth columns, respectively. In the experiments, we intentionally placed code for tasks in such a way that caused conflicts among memory blocks from different tasks although the instruction cache in the target machine is large enough to hold all the code used by the tasks. This is because we expect that such a case is typical of large-scale real-time systems. Figure 5 shows such a code placement for task set T 3 . Furthermore, since we consider the preemption delay related to instruction caching only (cf. Section VII), we disabled data caching by mapping data and 4,449,284 1,365,026 3,113,858 29,600 3,113,778 29,520 3,104,178 19,920 3,073,229 (unit: machine cycles) II Worst case response time predictions and measured response times. stack segments of tasks to non-cacheable area. Table II shows the predicted worst case response time of the lowest priority task in each task set. Four different methods were used to predict the worst case response time of the task: A is the method where the worst case preemption cost is assumed to be the cost to completely refill the cache. C is the method explained in [13]. U is the method where the worst case preemption cost is assumed to be the cost to completely reload the code used by a preempted task. Finally, P is the method proposed in this paper where the worst case preemption cost is assumed to be the cost to reload the maximum number of useful cache blocks. In the table, the worst case response time predictions by the above four methods are denoted by RA , RC , RU , R P , respectively. Also denoted by \Delta M is the predicted worst case cache-related preemption delay in method M. It is the difference between the worst case response time predictions by method M with and without cache-related preemption costs. The results show that the proposed technique gives significantly tighter predictions for cache-related preemption delay than the previous approaches. This results from the fact that, unlike the other approaches, the proposed approach considers only useful cache blocks when computing cache-related preemption costs. In one case (task set T 1 ), the proposed technique gives a prediction that is 60% tighter than the best of the previous approaches (1304 cycles vs. 3392 cycles). However, there is still a non-trivial difference between R P and the measured response time. This difference results from a number of sources. First, contrary to our pessimistic assumption that all the useful cache blocks of a task are replaced from the cache between the time the task is preempted and the time the task resumes execution, not all of them were replaced on a preemption during the actual execution. Second, many of actual preemptions occurred at execution points other than the execution point with the maximum number of useful cache blocks. Finally, the worst case preemption scenario assumed in deriving the upper bound on the cache-related preemption delay by the linear programming technique did not occur during the actual execution. Another point we can note from the results is that the cache-related preemption delay (i.e., \Delta) occupies only a small portion of the worst case response time (less than 1% for most cases). This results from the following two reasons. First, the WCETs of tasks were unrealistically large in our experiments since we disabled data caching. This diminished the relative impact of the cache-related preemption delay on the worst case response time. Second, since the target machine uses SRAM as its main memory, the cache refill time is much smaller than that of most current computing systems, which ranges from 8 cycles to more than 100 cycles when DRAM is used as main memory [11]. If DRAM were used instead, the worst case cache-related preemption delay would have occupied a much greater portion of the worst case response time. Furthermore, since the speed improvement of processors is much faster than that of DRAMs [11], we expect that the worst case cache-related preemption delay will occupy an increasingly large portion of the worst case response time in the future. To assess the impact of the cache-related preemption delay on the worst case response time in a more typical setting, we predicted the worst case response time for task set T 1 as we increase the cache refill time while enabling data caching. Figures 6-(a) and (b) show \Delta and \Delta=(W orst Case Response Time), respectively, for this new setting. The results show that as the cache refill time increases, \Delta increases linearly for all the four methods. This results in a wider gap between the cache-related preemption delay predicted by method P Cache Refill Time200000A U (a) Cache Refill Time0.5/ (Worst Case Response A U (b) Fig. 6. Cache refill time vs. \Delta and \Delta=(W orst Case Response Time) and those by the other methods as the cache refill time increases. As a result, the task set is deemed unschedulable by methods A, C, and U when the cache refill time is more than 40, 190, and 210 cycles, respectively. On the other hand, the task set is schedulable by P even when the cache refill time is more than 300 cycles. For methods C and U , there are sudden jumps in \Delta when the cache refill time is about 120 cycles. These jumps occur when an increase in the worst case response time due to increased cache refill time causes additional invocations of higher priority tasks. The results also show that as the cache refill time increases the cache-related preemption delay takes a proportionally larger percentage in the worst case response time. As a result, even for method P , the cache-related preemption delay takes about 10% of the worst case response time when the cache refill time is 100 cycles. For other methods, the cache preemption delay takes a much higher percentage of the worst case response time. VII. EXTENSIONS A. Set Associative Caches In computing the number of useful cache blocks in Section IV, we considered only the simplest cache organization called the direct-mapped cache organization where each memory block can be placed in only one cache block. In a more general cache organization called the n-way set-associative cache organization, each memory block can be placed in any one of the n blocks in the mapped set whose index is given by memory block number modulo number of sets in the cache. This set-associative cache organization requires a policy called the replacement policy that decides which block to replace to make room for a new block among the blocks in the mapped set. The least recently used (LRU) policy, which replaces the block that has not been referenced for the longest time, is typically used for that purpose. In the following, we explain how to compute the maximum number of useful cache blocks for set-associative caches assuming the LRU replacement policy. According to our definition in Section IV, the set RMB c contains all possible states of cache block c at execution point p. In the case of direct-mapped caches, a possible state corresponds to a memory block that cache block c may have at execution point p. This interpretation of a state needs to be extended for set-associative caches since they are indexed in the unit of cache sets rather than in the unit of cache blocks. A state of a cache set for an n-way set-associative cache is defined by a vector (m 1 ;m 2 1 is the least recently referenced block and m n the most recently referenced block. In the following, we formulate the problem of computing RMBs for set-associative caches in data flow analysis terms. As for direct-mapped caches, we initially focus on RMBs at the beginnings and ends of basic blocks. We define the sets RMB c IN [B] and RMB c OUT [B] as the sets of all possible states of cache set c at the beginning and end of basic block B, respectively. The set gen c [B] contains the state of cache set c generated in basic block B. Its element has up to n distinct memory blocks that are referenced in basic block B and are mapped to cache set c. More specifically, it is either empty (when none of the memory blocks mapped to cache set c are referenced in basic block B) or a singleton set whose only element is a vector (gen c In the vector, the component gen c n [B] is the memory block whose last reference in basic block B is the last reference to the cache set c in B. Similarly, the component gen c is the memory block whose last reference in B is the last reference to c in B excepting the references to the memory block gen c n [B]. In general, gen c the memory block whose last reference in B is the last reference to c in B excepting the references to memory blocks gen c As an example, consider a cache with two sets. Assume that the following memory block references are made to cache set 0 in a basic block B. According to the definition of gen c [B], when each cache set has four blocks (i.e., 4-way set-associative cache), the set gen c 0 [B] is f(m Similarly, when each cache set has eight blocks (i.e., 8-way set-associative cache), the set gen c0 [B] is f(null, null, null, With this definition of gen c [B], the sets RMB c IN [B] and RMB c OUT [B], whose elements are now vectors with n memory blocks, are related as follows. P a predecessor of B OUT [B] (12) f(gen c if gen c if gen c if gen c IN [B] if gen c [B] is empty As in the case of direct-mapped caches, the RMBs of cache set c at points other than the beginning and end of basic block B can be derived from RMB c IN [B] and the memory block references within the basic block. Assume that the basic block has the following sequence of memory block references reference to memory block m i that is mapped to cache set c i . As before, the references are processed sequentially starting from (c processings needed for are as follows. For each element (rmb in the RMB of c i , if updated to (rmb since rmb j is now the most recently referenced memory block in the cache set. On the other is updated to (rmb Note that it is only the RMB of c i that needs to be updated by the reference the reference does not affect the states of any other cache sets. The set LMB c for set-associative caches contains all possible reference sequences to cache set c after p and each reference sequence has sufficient information to determine for each block in cache set c whether it is re-referenced before being replaced. For an n-way set-associative cache with the LRU replacement policy, such information corresponds to n distinct memory blocks referenced after p. For this reason, the set gen c [B] for the LMB problem is defined to be either empty or a singleton set f(gen c whose components are the first n distinct memory blocks referenced in basic block B and mapped to cache set c. More specifically, gen c 1 [B] is the memory block whose first reference in B is the first reference to c in B and gen c 2 [B] is the memory block whose first reference in B is the first reference to c in B excepting the references to memory block gen c 1 [B] and so on. The sets LMB c IN [B] and LMB c OUT [B], which correspond to the sets of all possible reference sequences to cache set c after the beginning and end of basic block B, respectively, are related as follows. S a successor of B IN [S] f(gen c if gen c OUT [B] f(gen c if gen c OUT [B] f(gen c if gen c OUT [B] if gen c [B] is empty After the LMBs at the beginnings and ends of basic blocks are computed, the LMBs at other points within basic blocks can be computed in an analogous manner to the case of RMBs. Once the sets RMB c p and LMB c are computed for each execution point p, the calculation of the maximum number of useful blocks in cache set c at p is straightforward. For each element (rmb; lmb) in RMB c , we compute the number of cache hits that would result if references from the memory blocks in lmb are applied to the cache set state defined by rmb. We then pick the element (rmb that yields the largest number of cache hits, which gives the maximum number of useful cache blocks in cache set c at p. The total number of useful cache blocks at p is computed by summing up the maximum numbers of useful cache blocks of all the cache sets in the cache. From this information, the preemption cost table can be constructed as in the case of direct-mapped cache. B. Data Cache Memory Until now, we have focused on preemption costs resulting from the use of instruction cache memory. In this subsection, we explain an extension of the proposed technique needed for data cache memory. Unlike instruction references, some data references have addresses that are not fixed at compile-time. For example, references from a load/store instruction used to implement an array access have different addresses. These data references complicate a direct application of the proposed technique to data cache memory since the technique requires that the addresses of references from each basic block be fixed. Such references also complicate the WCET analysis of tasks and most WCET analysis techniques take a very conservative approach to them. Fortunately, this conservative approach greatly simplifies the adaptation of the proposed technique for data cache memory. We take the extended timing schema approach [19] as our example in the following discussion. In the WCET analysis based on extended timing schema approach, if a load/store instruction references more than one memory block, it is called a dynamic load/store instruction [29] and two cache miss penalties are assumed for each reference from it; one cache miss penalty is because the reference may miss in the cache and the other because it may replace a useful cache block. In the analysis of preemption costs resulting from the use of data cache memory, if a load/store instruction is not dynamic, references from it can be handled in exactly the same way as in the case of instruction references since their addresses in a CFG are fixed. Also, because the extended timing schema approach assumes that all the references from a dynamic load/store instruction miss in the cache, they cannot contribute useful cache blocks. Furthermore, since the approach conservatively assumes that every one of them replaces a useful cache block in deriving the WCET estimate, we can completely ignore them when computing RMBs and LMBs. VIII. CONCLUSION Cache memory introduces unpredictable variation to task execution time when it is used in real-time systems where preemptions are allowed among tasks. We have proposed a new schedulability analysis technique that takes such execution time variation into account. The proposed technique proceeds in two steps. In the first step, a per-task analysis technique constructs for each task a table called the preemption cost table. This table gives for a given number of preemptions an upper bound on the cache-related delay caused by them. Then, the second step computes the worst case response time of each task using a linear programming technique that takes as its input the preemption cost table obtained in the first step. Our experimental results showed that the proposed technique gives a prediction of the worst case cache-related preemption delay that is up to 60% tighter than that obtained from previous approaches. This improved prediction accuracy results from the fact that the proposed technique considers only useful cache blocks in deriving the worst case cache-related preemption delay. A number of extensions are possible for the analysis technique explained in this paper. For example, the per-task preemption cost information can be made more accurate. In the per-task analysis in Section IV, a cache block is considered as useful if it is useful in at least one path. Many such paths, however, cannot be taken simultaneously as the example in Figure shows. In the example, cache block c i is useful only when the flow of control is from in 1 to out 2 . On the other hand, cache block c j is useful only when the flow of control is from in 2 to out 1 . These two flows of control are not compatible with each other and only one of the two cache blocks can be useful at any one time. Nevertheless, both cache blocks are considered as useful in our data flow analysis. In order to rectify this problem, preemption cost should be computed on a path basis. Our initial attempt based on this idea is described in [30]. Another interesting extension to our proposed analysis technique is to consider the intersection of cache blocks used by a preempted task and those used by the higher priority tasks that are released while the former task is preempted [12, 13]. For this purpose, the proposed technique can be augmented as follows: (1) perform the data flow analysis explained in Section IV for the preempted task and (2) count only the useful cache blocks that are mapped to the intersection of cache blocks used by the preempted task and those used by the higher priority tasks released during the preemption. Although this approach is more accurate than the approach explained in this paper, it requires a large number of analyses, i.e., one analysis for each preemption instance. We are currently working on an approximate technique that is similar to the above approach but trades accuracy for low analysis complexity. Appendix Consider a task set consisting of three tasks preemption cost tables are given by # of preemptions 1 cost # of preemptions 1 cost 6 5 4 4 3 3 2 Note that we do not need the preemption cost table for the highest priority task - 1 since it cannot be preempted. The worst case response time of the lowest priority task - 3 can be computed as follows: (R 0 (R 0 (R 0 The PC 3 (R 0 can be computed by solving the following integer linear programming problem. Maximize (R 0 subject to Constraint 1 e R 0T 3 e Constraint 2 e e In the above problem formulation, we use the fact that since task - 1 is the highest priority task, it cannot be preempted and thus g This also gives N which is the maximum number of preemptions a single invocation of task - 2 can experience, can be computed by dividing the worst case response time of - 2 by the period of task - 1 . The worst case response time of - 2 , which is equal to 49, must have been computed beforehand and thus is available when we compute the worst case response time of task - 3 . This gives 2. N 3 , which is the maximum number of preemptions of task - 3 , can be computed by dividing R 0 3 by the periods of tasks - 1 and - 2 (cf. Equation (9)). After solving this integer linear programming problem, we have (PC 3 (R 0 this gives R 1 3 value is used in the next iteration to compute R 2 3 ) is obtained by solving the following integer linear programming problem. Maximize 3;4 \Theta f 3;4 subject to Constraint 1 e e Constraint 2 e e The solution to this integer linear programming problem gives (PC 3 (R 1 and this, in turn, gives R 2 When we repeat the same procedure with R 2 we have R 3 Thus, the procedure converges and R 2 is a safe upper bound on the worst case response time of task - 3 . Since this worst case response time is smaller than task - 3 's deadline (=400), task - 3 is schedulable even when cache-related preemption delay is considered. Acknowledgments The authors are grateful to Jos'e V. Busquets-Mataix for helpful suggestions and comments on an earlier version of this paper. --R "Some Results of the Earliest Deadline Scheduling Al- gorithm," "Finding Response Times in a Real-Time System," "The Rate Monotonic Scheduling Algorithm: Exact Characterization and Average Case Behavior," "Scheduling Algorithms for Multiprogramming in a Hard Real-Time Environment," "Dynamic Scheduling of Hard Real-Time Tasks and Real-Time Threads," "An Extendible Approach for Analyzing Fixed Priority Hard Real-Time Tasks," "Effective Analysis for Engineering Real-Time Fixed Priority Schedulers," "The Impact of an Ada Run-time System's Performance Characteristics on Scheduling Models," "Accounting for Interrupt Handling Costs in Dynamic Priority Task Systems," "Engineering and Analysis of Fixed Priority Schedulers," Computer Architecture A Quantitative Approach. "Cache Issues in Real-Time Systems," "Adding Instruction Cache Effect to Schedulability Analysis of Preemptive Real-Time Systems," Linear and Nonlinear Programming. "Bounding Worst-Case Instruction Cache Performance," "Integrating the Timing Analysis of Pipelining and Instruction Caching," "Worst Case Timing Analysis of RISC Processors: R3000/R3010 Case Study," "Efficient Microarchitecture Modeling and Path Analysis for Real-Time Software," "An Accurate Worst Case Timing Analysis Technique for RISC Pro- cessors," "SMART (Strategic Memory Allocation for Real-Time) Cache Design," "SMART (Strategic Memory Allocation for Real- Time) Cache Design Using the MIPS R3000," "Allocating SMART Cache Segments for Schedulability," "Software-Based Cache Partitioning for Real-time Applications," High Performance Compilers for Parallel Computing. DFT/FFT and Convolution Algorithm: Theory Elementary Numerical Analysis. C Algorithms for Real-Time DSP "Efficient Worst Case Timing Analysis of Data Caching," "Calculating the Worst Case Preemption Costs of Instruction Cache," --TR --CTR Anupam Datta , Sidharth Choudhury , Anupam Basu, Using Randomized Rounding to Satisfy Timing Constraints of Real-Time Preemptive Tasks, Proceedings of the 2002 conference on Asia South Pacific design automation/VLSI Design, p.705, January 07-11, 2002 Yudong Tan , Vincent J. Mooney, III, WCRT analysis for a uniprocessor with a unified prioritized cache, ACM SIGPLAN Notices, v.40 n.7, July 2005 M. Kandemir , G. Chen , W. Zhang , I. Kolcu, Data Space Oriented Scheduling in Embedded Systems, Proceedings of the conference on Design, Automation and Test in Europe, p.10416, March 03-07, Hiroshi Nakashima , Masahiro Konishi , Takashi Nakada, An accurate and efficient simulation-based analysis for worst case interruption delay, Proceedings of the 2006 international conference on Compilers, architecture and synthesis for embedded systems, October 22-25, 2006, Seoul, Korea Mahmut Kandemir , Guilin Chen, Locality-Aware Process Scheduling for Embedded MPSoCs, Proceedings of the conference on Design, Automation and Test in Europe, p.870-875, March 07-11, 2005 Accounting for cache-related preemption delay in dynamic priority schedulability analysis, Proceedings of the conference on Design, automation and test in Europe, April 16-20, 2007, Nice, France Hemendra Singh Negi , Tulika Mitra , Abhik Roychoudhury, Accurate estimation of cache-related preemption delay, Proceedings of the 1st IEEE/ACM/IFIP international conference on Hardware/software codesign and system synthesis, October 01-03, 2003, Newport Beach, CA, USA Hiroyuki Tomiyama , Nikil D. Dutt, Program path analysis to bound cache-related preemption delay in preemptive real-time systems, Proceedings of the eighth international workshop on Hardware/software codesign, p.67-71, May 2000, San Diego, California, United States I. Kadayif , M. Kandemir , I. Kolcu , G. Chen, Locality-conscious process scheduling in embedded systems, Proceedings of the tenth international symposium on Hardware/software codesign, May 06-08, 2002, Estes Park, Colorado Sheayun Lee , Sang Lyul Min , Chong Sang Kim , Chang-Gun Lee , Minsuk Lee, Cache-Conscious Limited Preemptive Scheduling, Real-Time Systems, v.17 n.2-3, p.257-282, Nov. 1999 Yudong Tan , Vincent J. Mooney III, Timing Analysis for Preemptive Multi-Tasking Real-Time Systems with Caches, Proceedings of the conference on Design, automation and test in Europe, p.21034, February 16-20, 2004 Yudong Tan , Vincent Mooney, Timing analysis for preemptive multitasking real-time systems with caches, ACM Transactions on Embedded Computing Systems (TECS), v.6 n.1, February 2007 Jan Staschulat , Rolf Ernst, Scalable precision cache analysis for preemptive scheduling, ACM SIGPLAN Notices, v.40 n.7, July 2005 Zhang , Chandra Krintz, Adaptive code unloading for resource-constrained JVMs, ACM SIGPLAN Notices, v.39 n.7, July 2004 Johan Strner , Lars Asplund, Measuring the cache interference cost in preemptive real-time systems, ACM SIGPLAN Notices, v.39 n.7, July 2004 Jan Staschulat , Rolf Ernst, Multiple process execution in cache related preemption delay analysis, Proceedings of the 4th ACM international conference on Embedded software, September 27-29, 2004, Pisa, Italy Sungpack Hong , Sungjoo Yoo , Hoonsang Jin , Kyu-Myung Choi , Jeong-Taek Kong , Soo-Kwan Eo, Runtime distribution-aware dynamic voltage scaling, Proceedings of the 2006 IEEE/ACM international conference on Computer-aided design, November 05-09, 2006, San Jose, California Chang-Gun Lee , Kwangpo Lee , Joosun Hahn , Yang-Min Seo , Sang Lyul Min , Rhan Ha , Seongsoo Hong , Chang Yun Park , Minsuk Lee , Chong Sang Kim, Bounding Cache-Related Preemption Delay for Real-Time Systems, IEEE Transactions on Software Engineering, v.27 n.9, p.805-826, September 2001 Nikil Dutt , Alex Nicolau , Hiroyuki Tomiyama , Ashok Halambi, New directions in compiler technology for embedded systems (embedded tutorial), Proceedings of the 2001 conference on Asia South Pacific design automation, p.409-414, January 2001, Yokohama, Japan

cache memory;fixed-priority scheduling;preemption;schedulability analysis;real-time system

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Checkpointing Distributed Shared Memory.

Distributed shared memory (DSM) is a very promising programming model for exploiting the parallelism of distributed memory systems, because it provides a higher level of abstraction than simple message passing. Although the nodes of standard distributed systems exhibit high crash rates only very few DSM environments have some kind of support for fault-tolerance.In this article, we present a checkpointing mechanism for a DSM system that is efficient and portable. It offers some portability because it is built on top of MPI and uses only the services offered by MPI and a POSIX compliant local file system.As far as we know, this is the first real implementation of such a scheme for DSM. Along with the description of the algorithm we present experimental results obtained in a cluster of workstations. We hope that our research shows that efficient, transparent and portable checkpointing is viable for DSM systems.

INTRODUCTION Distributed Shared Memory (DSM) systems provide the shared memory programming model on top of distributed memory systems (i.e. distributed memory multiprocessors or networks of workstations). DSM is appealing because it combines the performance and scalability of distributed memory systems with the ease of programming of shared-memory machines. Distributed Shared Memory has received much attention in the past decade and several DSM systems have been presented in the literature [Eskicioglu95][Raina92][Nitzberg91]. However, most of the existing prominent implementations of DSM systems do not provide any support for fault-tolerance [Carter91] [Keleher94] [Li89] [Johnson95]. This is a limitation that we wanted to overcome in our DSM system. When using parallel machines and/or workstation clusters the user should be aware that the likelihood of a processor failure increases with the number of processors, and the failure of just one process(or) will lead to the crash or hang-up of the whole application. Distributed systems represent a cost-effective solution for running scientific computations, but at the same time they are more vulnerable to the occurrence of failures. A study presented in [Long95] shows that we can expect (in average) a failure every 8 hours in a typical distributed system composed by 40 workstations, where each machine exhibits an MTBF of 13 days. If a program that runs on such a system takes more than 8 hours to execute then it would be very difficult to finish its execution, unless there is some fault tolerance support to assure the continuity of the application. Parallel machines are considerably more stable, but even so they present a relatively low MTBF. For instance, the MTBF of a large parallel machine like the Intel Paragon XP/S 150 (with 1024 processors) from Oak Ridge National Laboratory is in the order of 20 hours [ORNL95]. For the case of long-running scientific applications it is essential to have a checkpointing mechanism that would assure the continuity of the application despite the occurrence of failures. Without such mechanism the application would have to be restarted from scratch and this can be very costly for some applications. In this paper, we will present a checkpointing scheme for DSM systems that despite being transparent to the application it is quite general, portable and efficient. The scheme is quite general because it does not depend of any specific feature of our DSM system. In fact, our system implements several protocols of consistency and models of consistency, but the checkpointing scheme was made independent of the protocols. The scheme is also quite portable, since it was implemented on top of MPI and nothing was changed inside the MPI layer. This is why we call it DSMPI [Silva97]. The scheme only requires a POSIX compliant file system, and makes use of the likckpt tool [Plank95] for taking the local checkpoints of processes. That tool works for standard UNIX machines. Finally, DSMPI is an efficient implementation of checkpointing. Usually, efficiency depends on the write latency to the stable storage but also on the characteristics of the checkpointing protocol. While the first feature mainly depends on the underlying system, the second one is under our control. We have implemented a non-blocking coordinated checkpointing algorithm. It does not freeze the whole application while the checkpoint operation is being done, as blocking algorithms do. The only problem of non-blocking algorithms over the blocking ones is the need to record in stable storage some of the in-transit messages that cross the checkpoint line. However, we have exploited the semantics of DSM messages and we achieved an important optimization: no in-transit message has to be logged in stable storage. Some results were taken using a distributed stable storage and we have observed a maximum overhead of 6% for a extremely short interval between checkpoints of 2 minutes. With a more realistic interval, in order of tens of minutes or even hours, the overhead would fall to insignificant values. The rest of the paper is organized as follows: section 2 describes the general organization of DSMPI and its protocols. Section 3 presents the transparent scheme that is based on a non-blocking coordinated checkpointing algorithm. Section 4 compares our algorithm with other schemes. Section 5 presents some performance results, and finally section 6 concludes the paper. 2. OVERVIEW OF DSMPI This section gives a brief description about DSMPI [Silva97]. 2.1 Main Features DSMPI is a parallel library implemented on top of MPI [MPI94]. It provides the abstraction of a globally accessed shared memory: the user can specify some data-structures or variables to be shared and that shared data can be read and/or written by any process of an MPI application. The most important guidelines that we took into account during the design of DSMPI were: 1. assure full portability of DSMPI programs; 2. provide an easy-to-use and flexible programming interface; 3. support heterogeneous computing platforms; 4. optimize the DSM implementation to allow execution efficiency. 5. provide support for checkpointing. For the sake of portability DSMPI does not use any memory-management facility of the operating system, neither requires the use of any special compiler or linker. All shared data and the read/write operations should be declared explicitly by the application programmer. The sharing unit is a program variable or a data structure. DSMPI can be classified as a structure-based DSM as opposed to page-based DSM systems, like IVY [Li89]. It does not incur in the problem of false sharing because the unit of shared data is completely related to existing objects (or data structures) of the application. It allows the use of heterogeneous computing platforms since the library knows the exact format of each shared data object. Most of the other DSM systems are limited to homogeneous platforms. DSMPI allows the coexistence of both programming models (message passing and shared data) within the same MPI application. This has been considered as a promising solution for parallel programming [Kranz93]. Concerning absolute performance, we can expect applications that use DSM to perform worse than their message passing counterparts. However, this is not always true. It really depends on the memory-access pattern of the application and on the way the DSM system manages the consistency of replicated data. We tried to optimize the accesses to shared data by introducing three different protocols of data replication and three different models of consistency that can be adapted to each particular application in order to exploit its semantics. With such facilities we expect DSM programs to be competitive with MPI programs in terms of performance. Some performance results collected so far corroborate this expectation [Silva97]. 2.2 Internal Structure In DSMPI there are two kinds of processes: application processes and daemon processes. The latter ones are responsible for the management of replicated data and the protocols of consistency. Since the current implementations of MPI are not thread-safe we had to implement the DSMPI daemons as separate processes. This is a limitation of the current version of DSMPI that will be relaxed as soon as there is some thread-safe implementation of MPI. All the communication between daemons and application processes is done by message passing. Each application process has access to a local cache that is located in its own address space where it keeps the copies of replicated data objects. The daemon processes maintain the master copies of the shared objects. DSMPI maintains a two-level memory hierarchy: a local cache and a remote shared memory that is located in and managed by the daemons. The ownership of the data objects is implemented through a static distributed scheme. 2.3 DSM Protocols We provided some flexibility in the accesses to shared data by introducing three different protocols of data replication and three different models of consistency that can be adapted to each particular application in order to exploit its semantics. Shared objects can be classified in two main classes: single-copy or multi-copy. The multicopy class replicates the object among all the processes that perform some read request on it. In order to assure consistency of replicated data the system can use a write-invalidate protocol or a write-update protocol [Stumm90A]. This is a parameter that can be tuned by the application programmer. In order to exploit execution efficiency, we have also implemented three different models of consistency: 1. Sequential Consistency (SC), as proposed in the IVY system [Li89]; 2. Release Consistency (RC), that implements the protocol of the DASH multiprocessor 3. Lazy Release Consistency (LRC), that implements a similar protocol as proposed in the TreadMarks system [Keleher94]. It has been shown that the LRC protocol is able to introduce some significant improvements over the other two models. The flexibility provided by DSMPI in terms of different models and protocols is an important contribution to the overall performance of DSMPI. 2.4 Programming Interface The library provides a C interface and the programmer calls the DSMPI functions in the same way it calls any MPI routine. The complete interface is composed of routines: it includes routines for initialization and clean termination, object creation and declaration, read and write operations, and routines for synchronization like semaphores, locks and barriers. The full interface is described in [Silva97]. 3. TRANSPARENT CHECKPOINTING ALGORITHM When devising the transparent checkpointing algorithm for our DSM system we tried to achieve some objectives, like transparency, portability, low performance overhead, low memory overhead and the ability to tolerate partial and total failures of the system. Satisfying all the previous guidelines is not an easy task and has not been possible in other proposals. Most of the existing checkpointing schemes for DSM are mainly concerned with transparency. Transparent recovery implemented at the operating system level or at the DSM- layer is an attractive idea. However, those schemes involve significant modifications in the system which make them very difficult to port to other systems. For the sake of portability, checkpointing should be made independent of the underlying system as much as possible. 3.1 Motivations Our checkpointing scheme is meant to be quite general because it does not depend on any specific feature of our DSM system. Our system implements several protocols of consistency and models of consistency, but the checkpointing scheme was made independent of the protocols. The scheme itself offers some degree of portability. It was implemented on top of MPI that per se already provides a high-level of portability since it was accepted as the standard for message-passing. Nothing was changed inside the MPI layer. The scheme only requires a POSIX compliant file system, and makes use of the libckpt tool [Plank95] for taking the local checkpoints of processes. That tool works for standard UNIX machines. We are taking transparent checkpoints and this means that it is not possible to assure checkpoint migration between machines with different architectures. However, the checkpoint mechanism itself can be ported to other DSM environments. 3.2 Coordinated Checkpointing Our guideline was to adapt some of the checkpointing techniques used in message passing systems, since checkpointing mechanisms have been widely studied in message-passing environments. Two methods for taking checkpoints are commonly used: coordinated checkpointing and independent checkpointing. In the first method, processes have to coordinate between themselves to ensure that their local checkpoints form a consistent system state. Independent checkpointing requires no coordination between processes but it can result in some rollback propagation. To avoid the domino effect and to reduce the rollback propagation message logging is used together with independent checkpointing. Independent checkpointing and message logging was not a very encouraging option, because DSM systems generate more messages than message-passing programs. Thus, we have chosen a coordinated checkpointing strategy for DSMPI. The reasons were manifold: it minimizes the overhead during failure-free operation since it does not need to log messages; it limits the rollback to the previous checkpoint; it avoids the domino-effect; it uses less space in stable storage; it does not require a complex garbage-collection algorithm to discard obsolete checkpoints; and finally, it is the most suitable solution to support job-swapping. It was shown in [Elnozahi92][Plank94] that coordinated checkpointing is a very effective solution for message-passing systems. Some experimental results have shown that the overhead of synchronizing the local checkpoints is negligible when compared with the overhead of writing the checkpoints to disk. Implementing the checkpointing algorithm underneath the DSM layer and in a transparent way to that layer is a possible alternative to provide fault-tolerance as was suggested in [Carter93]. However, this would be a very simplistic approach since the DSM system exchanges several messages not related to the application. Such approach would result in extra overhead and it would not exploit the characteristics of the DSM system. 3.3 System Model We assume that the DSM system only uses message passing to implement its protocols. There is no notion of global time and there is no guarantee that processor clocks are synchronized in some way. Processors are assumed to be fail-stop: there is no provision to detect or tolerate any kind of malicious failures. When a processor fails it stops sending messages and does not respond to other parts of the system. Processor failures are detected by the underlying communication system (MPI) through the use of time-outs. When one of the processes fails, the MPI layer sends a SIGKILL to all the other processes (application and daemons) that will terminate the complete application. MPI makes extensive use of static groups (in fact, the whole set of processes belong at the beginning to a MPI_WORLD_COMM group). If some process fails, some collective operations that involve the participation of the processes will certainly hang-up the application. Thus, it makes sense that a failure of just one process should result in the rollback of the entire application. Communication failures are also dealt by the communication system (MPI). The underlying message-passing system provides a reliable and FIFO point-to-point communication service. Stable storage is implemented on a shared disk that is assumed to be reliable. If the disk is attached to a central file server all the checkpoints become available to the working processors of the system. If stable storage is implemented on the local disk of every processor (assuming that each host has a disk) then the application can only recover if the failed host is also able to recover. 3.4 Checkpoint Contents A checkpoint in a DSM system should include: the DSM data (e.g. pages/objects and the DSM directories) and the private data of each application. Some schemes do not save the private data of processes and thus are not able to recover the whole state of the computation [Stumm90B]. They leave to the application programmer the responsibility of saving the computation state to assure the continuity of the application. Other schemes assume, for the sake of simplicity, that private data and shared data are allocated in DSM [Kermarrec95]. We depart from that assumption and consider that private data is not allocated in the DSM system. Besides, some of the private data like processor registers, program counter and process stack are certainly not part of the DSM data. In our scheme we have to checkpoint the application processes as well as the DSM daemons since they maintain most of the DSM relevant data. DSM daemons save the shared objects and the associated DSM directories. Saved directories reflect the location of the default and current owners of the shared objects. Some optimizations are made by the system: read-only objects are checkpointed only once, and replicated shared objects are checkpointed only by one daemon (the one that maintains its ownership). Each process (application or daemon) saves its checkpoint into a separate file. A global checkpoint is composed by N different checkpoint files and a status_file that keeps the status of the checkpoint protocol execution. This file is maintained by the checkpointing coordinator and is used during the phase of recovery to determine the last committed checkpoint, as well as to ensure the atomicity in the operation of writing a global checkpoint. Checkpointing Algorithm Since checkpointing schemes for message passing systems are very well stabilized, we decided to adopt one of the most used techniques to implement a non-blocking global checkpointing [Elnozahy92][Silva92]. The main difficulty of implementing non-blocking coordinated checkpoint is to guarantee that the global saved state is consistent. Messages in-transit at the time of a global checkpoint are the main concern for saving a consistent snapshot of a distributed application. For instance, messages that are sent after the checkpoint of the process sender and are received before the checkpoint of the receiver are called orphan messages [Silva92]. This sort of messages violates the consistency of the global checkpoint, and thus should be avoided by the algorithm. Other messages that are sent before the checkpoint of the sender and are received after the checkpoint of the destination process are called missing messages. Usually the algorithm should keep track of their occurrence and replay them during the recovery operation. However, we have considered some features of the DSM system that were important for the implementation of the algorithm, namely: the interaction between processes is not done by explicit messages but through object invocations that are RPC-like interactions; there is both shared and replicated data; the DSM system exchanges some additional messages that are only related to the DSM protocols and do not affect the application directly; and finally, there is a DSM directory that is maintained throughout the system. These features were taken into account and some of them were exploited to introduce some optimizations. The resulting scheme presents a novel but important feature over those non-blocking algorithms oriented to message-passing: it does not need to record any cross- checkpoint message in stable storage. The operation of checkpointing is triggered periodically by a timer mechanism. One of the daemons acts like the coordinator (the Master daemon) and is responsible for initiating a global checkpoint and coordinating the steps of the protocol. Only one process is given the right to initiate a checkpointing session in order to avoid multiple sessions and an uncontrolled high-frequency of checkpoint operations. Since there is always one DSMPI daemon that is elected as the Master (during the startup phase) we can guarantee that if this daemon is the checkpoint coordinator, checkpointing will be adequately spaced in time. Each global checkpoint is identified by a monotonically increasing number - the Checkpoint Number (CN). In the first phase of the protocol, the coordinator daemon increments its own CN and broadcasts a "TAKE_CHKP" message to all the other daemons and application processes. Upon receiving this message each of the other processes takes a tentative checkpoint, increments the local CN and sends a message "TOOK_CHKP" to the coordinator. After taking the tentative checkpoint, every process is allowed to continue with its computation. The application does not need to be frozen during the execution of the checkpointing protocol. This is an important feature to avoid interference with the application and to reduce the checkpointing overhead. In the second phase of the protocol, the daemon broadcasts a "COMMIT" message after receiving all the responses (i.e. "TOOK_CHKP") from the participants. Upon receiving a "COMMIT" message the tentative checkpoints are transformed into permanent checkpoints and the previous checkpoint files are deleted. All these phases are recorded into the status_file by the Master Daemon. Usually, the broadcast message "TAKE_CHKP" is received by all the processes in some order that preserves the causality. However, due to the asynchrony of the system it is possible that some situations may violate the causal consistency. An important aspect is that every shared object is tagged with the local CN value and every message sent by the DSM system is tagged with the CN of the sender. The CN value piggybacked in the DSM messages prevents the occurrence of orphan messages: if a daemon receives a message with higher CN than the local one, then it has to take a tentative checkpoint before consuming that message and changing its internal state. If later on, it receives a "TAKE_CHKP" message tagged with an equal CN, then it discards that message since the corresponding tentative checkpoint was already taken. The CN value is also helpful in identifying missing messages: if an incoming message carries a CN value lower than the current local one, it means it was sent in the previous checkpoint interval and is a potential missing message. The checkpointing algorithm has to distinguish between the messages used by the read/write operations and the other messages used by the DSM protocols. Using the semantics of DSM protocols we can avoid the unnecessary logging of some potential missing messages. Daemon processes run in a cycle and when they receive a "TAKE_CHKP" message they take a local snapshot of their internal state, including the DSM directories. Application processes get "TAKE_CHKP" orders when they execute DSMPI routines: whenever they read from the local cache, perform some remote object invocation, or access some synchronization variable. All the invocations on shared objects or synchronization variables involve a two-way interaction: invocation and response. During the period that a process is waiting for a response it remains blocked and thus does not change its internal state. The only interactions that do not fit in this structure are messages related to the DSM protocols. Usually these messages are originated by the daemon processes and do not require an RPC-like interaction. For the sake of clarity, let us distinguish between three different cases: Case 1: Messages Process-to-Daemon These interactions are started by an application process that wants to perform a read/write operation into a shared object or gain access to a synchronization variable. Each process maintains a directory with the location of the owners of the objects, locks, semaphores and barriers. If it wants to access any of them it sends an invocation message to the respective daemon. Then it blocks while waiting for the response. Let us consider two different scenarios that should be handled by the algorithm: 1.1 - The process (P i ) is running in the checkpoint interval N, but the daemon (D k ) is still running in the previous checkpoint interval (i.e. 1.2 - The process (P i ) has its CN equal to (N-1) and the daemon has already taken its N th tentative checkpoint (CN=N); An example of the first scenario is illustrated in Figure 1. Process P i already took its N th checkpoint and performs a read access to a remote object that is owned by daemon D k that has not yet received the corresponding "TAKE_CHKP" message. The "READ" message carries CN equal to N: the daemon checks that and takes a local checkpoint before consuming the message. Instead of a READ operation, it can be a WRITE, LOCK, UNLOCK, WAIT, SIGNAL or BARRIER invocation. All these operations have an associated acknowledge or reply message. Figure 1: Forcing a checkpoint in the DSM daemon. Figure 2 represents the second scenario, where the daemon has already taken its N th checkpoint but the process started a read transaction in the previous checkpoint interval. When it receives the "READ_REPLY" the process realizes it has to take a local checkpoint and increment its local CN before consuming the message and continue with the computation. Figure 2: Forcing a checkpoint in the application process. However, this operation is not enough since the checkpoint CP(i,N) does not record the "READ" invocation message. During recovery the process has to repeat that read transaction again. To do that, the checkpoint routine has to record the "READ" invocation message in the contents of the checkpoint. In some sense, we can say that there is a logical checkpoint immediately before the sending of the "READ" message as represented in Figure 3. At the time of the checkpoint, that invocation message belongs to the address space of the process. Therefore, it is already included in the checkpoint. The only concern was to re-direct the starting point after recovery to some point in the code immediately before the sending of the message. For that purpose, a label was included in every DSMPI routine. After returning from force checkpoint a restart operation the control flow jumps to that label, and the invocation message is re-sent. Thus, we can assure that the read transaction is repeated from its beginning. Figure 3: The notion of a logical checkpoint. Case 2: Messages Daemon-to-Process These messages are started by the daemon processes and are related to the DSM protocols. Usually they are INVALIDATE or UPDATE messages, depending on the replication protocol. These messages do not follow an RPC-like structure and thus do not block the sending daemon. Application processes consume these sort of messages when they execute the cache refresh procedure. We can also identify two different scenarios: 2.1- A protocol message that can be a potential orphan message; 2.2- A protocol message that can be a potential missing message. The first situation is represented in Figure 4: process P i receives a message that carries a higher CN than the local one, and is forced to take a checkpoint before proceeding. Figure 4: Potential orphan message. The second scenario is illustrated in Figure 5: the INVALIDATE or UPDATE message is sent in the previous checkpoint interval and received by the application process after taking its local checkpoint. This is, theoretically, the example of a missing message and in the normal case it would have to be recorded in stable storage in order to be replayed in case of recovery. However, we do not need to log these missing INVALIDATE/UPDATE messages. There is absolutely no problem if the message is not replayed in case of a rollback. The reason is simple: during the recovery procedure, every application process has to clean its local cache. logical checkpoint After that, if the process accesses some shared object it has to perform a remote operation to the owner daemon from where it gets the most up-to-date version of the object. Figure 5: Example of a missing message. Case 3: Messages Daemon-to-Daemon When using a static distributed ownership scheme daemons only communicate between themselves during the startup phase and during the creation of new objects, that also happens during the startup phase. With a dynamic ownership scheme messages between daemons are sent all the time since the ownership of an object can move from daemon to daemon. The current version of DSMPI follows a static distributed scheme but the next version will provide a dynamic distributed scheme as well. We will consider a dynamic distributed ownership scheme similar to the one presented in [Li89] where each shared data object has an associated owner that is changing as data migrates throughout the system. When a process wants to write into a migratory object the system changes its location to the daemon associated with that process. As object location can change frequently during the program execution, every process and daemon has to maintain a guess of the probable owner for each shared object. When a process needs a copy of the data it sends a request to the probable owner. If this daemon has that data object it returns the data, otherwise it forwards the request to the new owner. This forwarding can go further until the current owner is found in the chain of the probable owners. The current owner will send the reply to the asking process that receives the data and updates its value for the probable owner if the reply was not received from the expected daemon. Sometimes this scheme can be inefficient since the request may be forwarded many times before reaching the current owner. This inefficiency can be reduced if all the daemons involved in forwarding a request are given the identity of the current owner. We will consider this optimization, which involves the sending of ownership update messages from the current owner to the daemons belonging to the forward chain. Let us now see what are the implications of this forward-based scheme to the checkpointing algorithm. The messages exchanged between daemons can be divided into three different classes: 3.1- forward messages that are sent on behalf of read/write transactions; 3.2- owner_update messages to announce the current owner of some object; 3.3- the transfer of some object from a current owner to the next owner. The first kind of messages follows the same rule stated for Case 1 (i.e. transactions started by a process). That rule was explained with the help of Figures 1, 2 and 3. The only difference is that all the daemon processes involved in the forwarding chain have to apply that rule. The case of owner_update messages are treated in a similar way to Case 2: if the owner_update message carries a CN higher than the CN at the destination daemon then that message is a potential orphan message and should force a checkpoint at the destination before proceeding. The rule for case 2.1 (explained in Figure 4) applies to this case. If the owner_update message carries a lower CN than the destination daemon it corresponds to potential missing message (like in Figure 5). In the normal case, it should be logged to be replayed in case of recovery. However, once again we realized that there is no problem if we do not log messages. The system will still be able to ensure a consistent recovery of the application. During the recovery procedure the distributed directory will be reconstructed through the use of broadcast to update the current ownership of the objects. This means that those owner_update messages can be lost during recovery. The object directory will be updated in any case. The transfer of data from one daemon to the new owner is included in the forward protocol: when the reply is sent from the current owner to the original process a copy of the data is also sent (or is first sent) to the daemon associated to that process. This daemon will be the new owner. The rules stated previously are also used in this case. Since object ownership can change frequently during the execution of the checkpoint protocol, it is necessary to take care and avoid a shared object to be checkpointed more than once. We solve this problem in a very simple way: the current owner is responsible for checkpointing the shared object. If it transfers the object to another daemon after taking its checkpoint, that object will not be checkpointed again. The CN value tagged with each object is used to prevent a migratory object to be checkpointed more than once. To summarize, our checkpointing algorithm follows a non-blocking coordinated strategy. It avoids the occurrence of orphan messages and detects the potential missing messages. We do not log any missing message in stable storage but the system will still ensure a consistent recovery of the application. To achieve this optimization we have exploited some of the characteristics of the DSM protocols. Our scheme works for sequential consistency and relaxed consistency models. It is also independent of the replication protocol (write-update or write-invalidate). This fact allows a wide applicability of this algorithm to other different DSM systems. 3.6 Recovery Procedure In our case, the application recovery involves the roll back of all the processes to the previous checkpoint. We do not see this as a drawback, but rather as an imposition of the underlying communication system (MPI). Nevertheless, it suits well our goals: to use checkpointing for job-swapping as well, and to tolerate any number of failures. Thus, the recovery procedure is quite simple: all the processes have to roll back to the previously committed checkpoint. The determination of the last committed checkpointed is obtained from the status_file. After restoring the local checkpoints on each process, they still have to perform some actions before re-starting the execution: (i) the object location directory is constructed and updated through all the processes. In the case of a static distribution, this operation can be bypassed; (ii) for every shared object defined as multi-copy the owner daemon resets its associated copy-set list; (iii) each application process cleans its local private cache and updates the object location directory, if necessary. Only after these steps are processes allowed to resume their computation. Cleaning the private cache of the processes during recovery does not introduce a visible overhead, and allows a simpler operation during the checkpoint operation since some potential missing messages exchanged on behalf of the DSM protocols do not need to be logged. 4. Comparison with other schemes Some other coordinated checkpointing algorithms have been proposed in the literature. The algorithm presented in [Janakiraman94] extends the checkpoint/rollback operations only to the processes that have communicate directly or indirectly with the process initiator. That algorithm uses a 2-phase commit protocol during which all the processes participating in the checkpoint session have to suspend their computations, and all the messages in-transit have to be flushed to their destinations. Their algorithm waits for the completion of all on-going read/write operations before proceeding with the checkpointing protocol. Only after all the pending read/write operations have to be terminated the processors begin sending their checkpoints to stable storage. This may result in a higher checkpoint latency and performance overhead since they use a blocking strategy. In [Cabillic95] is presented an implementation of consistent checkpointing in a DSM system. Their approach relies on the integration of global checkpoints with synchronization barriers of the application. The scheme was implemented on top of the Intel Paragon and several optimizations were included, like incremental, non-blocking and pre-flushing checkpointing techniques. They have shown that copy-on-write checkpointing can be an important optimization to reduce the checkpointing overhead. In the recovery operation of that scheme all the processes are forced to roll back to the last checkpoint, as in our case. The only limitation of this scheme is that it does not work with all applications: if there is no barrier() within an application the system is never able to checkpoint. [Costa96] also presents a similar checkpointing scheme, that relies on the garbage collector mechanism to achieve a global consistent state of the system. It is based on a full-blocking checkpointing approach. In [Kaashoek92] was presented a global consistent checkpointing mechanism for the Orca parallel language. It was very easy to implement because that DSM implementation is based on total-order broadcast communication. All the processes receive all broadcast messages in the same order to assure consistency of updates in replicated objects. The checkpointing messages are also broadcasted and inserted in the total order of messages. This ensures the consistency of the global checkpoint. Unfortunately, MPI does not have that characteristic. [Choy95] presented a definition for consistent global states in sequentially consistent shared memory systems. They have also presented a lazy checkpoint protocol that assures global consistency. However, lazy checkpointing schemes may result in a high checkpoint latency, which is not desirable for job swapping purposes. Other different recovery schemes not based on coordinated checkpointing were also presented in the literature. Some of them [Wu89][Janssens93] were based on communication- induced checkpointing: every process is allowed to take checkpoints independently but, before communicating with another one, they are forced to checkpoint in order to avoid rollback propagation and inconsistencies. Communication-induced checkpointing is sensitive to the frequency of inter-process communication or synchronization in the application. This may introduce a high performance overhead and an uncontrolled checkpoint frequency. Another solution for recovery is based on independent checkpointing and message logging [Richard93]. However, we did not find this option very encouraging because DSM systems generate more messages than message passing programs. Even considering some possible optimizations [Suri95], message logging would incur in a significant additional performance and memory overhead. A considerable set of proposals [Wilkinson93][Neves94][Stumm90B][Brown94] [Kermarrec95] are only able to tolerate single processor failures in the system. While this goal is meaningful for distributed systems, where we can expect that machine failures are uncorrelated, the same is not true for parallel machines where total or multiple failures are as likely as partial failures. We require our checkpointing mechanism to be able to tolerate any number of failures. Although those different approaches could be interesting for other systems, we did not find them the most suitable for our system and we decided to adopt a coordinated checkpointing strategy. 5. Performance Results In this section we present some results about the performance and memory overhead of our transparent checkpointing scheme. The results were collected in a distributed system composed of 4 Sun Sparc4 workstations connected by a 10 Mb/s Ethernet. 5.1 Parallel Applications To conduct the evaluation of our algorithm we used the following six typical parallel applications . TSP: solves the Traveling Salesman Problem using a branch-and-bound algorithm. . NQUEENS: solves the placement problem of N-queens in a N-size chessboard. . SOR: solves Laplace's equation on a regular grid using an iterative method. . GAUSS: solves a system of linear equations using the method of Gauss-elimination. . ASP: solves the All-pairs Shortest Paths problem using Floyd's algorithm. . NBODY: this program simulates the evolution of a system of many bodies under the influence of gravitational forces. 1 For lack of space we refer the interested reader to [Silva97] for more details about the applications. 5.2 Performance Overhead We have made some experiments with the transparent checkpointing algorithm in a dedicated network of Sun Sparc workstations. Every processor has a local disk and access to a central file server through an Ethernet network. To take a local checkpoint of each process we used the libckpt tool in its fully transparent mode [Plank95]. None of the optimizations of that tool were used. Two levels of stable storage were used: the first level used the local disks of the processors, while the second level used a central server that is accessible to all the processors through the NFS protocol. Writing checkpoints to the local disks is expected to be much faster than writing to a remote central disk. However, the first scheme of stable storage is only able to recover from transient processor failures. If a processor fails in a permanent way and is not able to restart, then its checkpoint can not be accessed by any other processor of the network and recovery becomes impossible. The central disk does not have this problem (assuming the disk itself is reliable). Considering that stable storage is implemented on a central file server, Table 1 shows the time to commit and the corresponding overhead per checkpoint for all the applications. Usually, the time it takes to commit a global checkpoint is higher than the overhead produced. This is because the algorithm follows a non-blocking approach and the application processes do not need to wait for the completion of the protocol. If the algorithm were based on a blocking approach, the overhead per checkpoint would be roughly equal to the whole time it takes to commit. So, in the Table we can observe that, in the overall, the non-blocking nature of the algorithm allows some reduction in the checkpoint overhead. Application Size Chkp (Kbytes) Time Commit GAUSS (1024) 8500 130.763 128.832 Table 1: Time to commit and overhead per checkpoint using the central disk. The time to take a checkpoint depends basically on four factors: (i) the size of the checkpoint; (ii) the access time to stable storage; (iii) the synchronization structure of the application; (iv) and the granularity of the tasks. Checkpoint operations are only performed inside DSMPI routines. This means that if an application is very asynchronous and coarse-grain it takes some time more to perform a global checkpoint when compared with a more synchronous application. These factors are important but, in practice, the dominant factor is actually the operation of writing the checkpoint files to stable storage. Reducing the size of the checkpoints is a promising solution to attenuate the performance overhead. Another way is to use a stable storage with faster access. Table 2 shows the overhead per checkpoint considering the two different levels of stable storage. As can be seen, the difference between the figures is considerable: in some cases it is more than one order of magnitude. Using the Ethernet and the NFS central file server is really a bottleneck for the checkpointing operation. Nevertheless, it ensures a global accessible stable storage device where checkpoints can be made available even in the occurrence of a permanent failure of some processor. Application Size Chkp (Kbytes) (sec) (local) (sec) (central) GAUSS (1024) 8500 1.186 128.832 GAUSS (2048) 35495 4.284 1127.654 Table 2: Overhead per checkpoint (local vs central disk). Table 3 shows the difference in the overall performance overhead considering the two levels of stable storage and different intervals between checkpoints. We present the results for the SOR application, that was executed for an average time of 4 hours. Application Interval between chkp Table 3: Total performance overhead (local vs. central disk). The average overhead for checkpointing can be tuned by changing the checkpoint interval. In Table 3 we can see that the maximum overhead observed when using the local disk was 6.4%. The corresponding overhead with the central file server was up to 332%. This shows that if we consider a distributed stable storage scheme the performance can become interesting. Nevertheless, two minutes is a very conservative interval between checkpoints. Long-running applications do not need to be checkpointed so often and 20 minutes is a more acceptable interval. For this case, the performance overhead when using the local disks was 0.6%, which is a very small value. The same interval with the central disk as stable storage presented an overhead of 30.7 %. An interesting strategy would be the integration of both stable storage levels: that is, the application is checkpointed periodically to the central server, and in the meantime it can also be checkpointed to the local disks of the processors. If the application fails due to a transient perturbation and all the processors are able to restart, then they can recover from the checkpoints saved in each local disk (if this one correspond to the last committed checkpoint). If some of the processors is affected by a permanent outage then the application can be restarted from the last checkpoint located in the central disk. A possible solution to make the distributed stable storage scheme resilient to a permanent failure of one processor, is to implement a sort of logical ring where each processor should copy its local checkpoint file to the next processor's disk. This can be done after the global checkpoint being committed and in a concurrent way. This lazy update scheme would not introduce any delay in the commit operation: only some additional traffic in the network, that can be regulated if we use a token-based policy and perform each remote checkpoint file copy in a sequential way. Obviously, if we want to tolerate n permanent processor failures we have to replicate each checkpoint file by n+1 locals disks of the network. We measured the performance overhead when using both levels of stable storage and some of the results are presented in Figure 6. For each checkpoint in the central disk we performed K checkpoints to the local disks. The factor K was changed from 0 up to 10. Figure 6 shows the overhead reduction for the SOR application with 512x512 grid points.515253545 Factor K Figure Two-level stable storage (SOR 512). For instance, if the user wants an overhead lower than 5% then the factor K should be 9, 3, 1 and 0 when using a checkpoint interval of 2, 5, 10 and 20 minutes, respectively. If a permanent failure occurs in one of the processors of the system then in the worst case the application will loose approximately 20 minutes of computation in any of the four previous cases. The advantage still goes for an interval of 2 minutes and K equal to 9, since in the occurrence of a transient failure it will lose less computation. Figure 7 shows the corresponding values for the SOR application with 1024x1024 Factor K Figure 7: Two-level stable storage (SOR 1024). The same analysis can be done, considering a watermark of 10% for the performance overhead: when checkpointing the application with an interval of 2, 5, 10 and 20 minutes the factor K should be 11, 4, 1 and 0, respectively. If the user requires an overhead lower than 5% then K should be 8, 4 and 1, with an interval of 5,10 and 20 minutes, respectively. 6. CONCLUSIONS As far as we know, this is the first implementation of a non-blocking coordinated algorithm in a real DSM system. DSMPI provides different protocols and models of consistency, and our algorithm works with all of them. The checkpointing scheme is general-purpose and can be adapted to other DSM systems that use any protocol of replication or model of consistency. Some results were taken considering a distributed stable storage scheme and we have observed a maximum overhead of 6% for an interval between checkpoints of 2 minutes. With a checkpoint interval of 20 minutes the performance overhead was 0.6%. The same interval with the stable storage implemented in a central NFS-file server presented an overhead of 30.7 %. The algorithm herein presented offers an interesting level of portability and efficiency. Though, we plan to enhance some of the features of DSMPI in the next release that will be implemented on MPI-2. We look forward for a thread-safe version of MPI in order to re-design the DSMPI daemons and implement some of the optimization techniques proposed in [Cabillic95]. We hope that this line of research would give some contribution to a standard and flexible checkpointing tool that can be used in real production codes. Acknowledgments The work herein presented was conducted when the first author was a visitor at EPCC (Edinburgh Parallel Computing Centre). The visit was made possible due to the TRACS programme. The first author was supported by JNICT on behalf of the "Programa Ci-ncia" (BD-2083-92-IA). 7. --R "Dynamic Snooping in a Fault-Tolerant Distributed Shared Memory" "The Performance of Consistent Checkpointing in Distributed Shared Memory Systems" "Implementation and Performance of Munin" "Network Multicomputer Using Recoverable Distributed Shared Memory" "On Distributed Object Checkpointing and Recovery" "Lightweight Logging for Lazy Release Consistency Consistent Distributed Shared Memory" "The Performance of Consistent Checkpointing" "A Comprehensive Bibliography of Distributed Shared Memory" "Coordinated Checkpointing-Rollback Error Recovery for Distributed Shared Memory Multicomputers" "Relaxing Consistency in Recoverable Distributed Shared Memory" "CRL: High-Performance All-Software Distributed Shared Memory" "Transparent Fault-Tolerance in Parallel Orca Programs" "TreadMarks: Distributed Shared Memory on Standard Workstations and Operating Systems" "A Recoverable Distributed Shared Memory Integrating Coherence and Recoverability" "Integrating Message-Passing and Shared- Memory: Early Experience" "The Directory-based Cache Coherence Protocol for the DASH Multiprocessor" "Memory Coherence in Shared Virtual Memory Systems" "A Longitudinal Survey of Internet Host Reliability" "A Message Passing Interface Standard" "A Checkpoint Protocol for an Entry Consistent Shared Memory System" "Distributed Shared Memory: A Survey of Issues and Algorithms" data available in: http://www. "Performance Results of ickp - A Consistent Checkpointer on the iPSC/860" "Libckpt: Transparent Checkpointing Under Unix" "Virtual Shared Memory: A Survey of Techniques and Systems" "Using Logging and Asynchronous Checkpointing to Implement Recoverable Distributed Shared Memory" "Global Checkpoints for Distributed Programs" "Implementation and Performance of DSMPI" "Algorithms Implementing Distributed Shared Memory" "Fault-Tolerant Distributed Shared Memory Algorithms" "Reduced Overhead Logging for Rollback Recovery in Distributed Shared Memory" "Implementing Fault-Tolerance in a 64-bit Distributed Operating System" "Recoverable Distributed Shared Virtual Memory: Memory Coherence and Storage Structures" --TR Memory coherence in shared virtual memory systems Algorithms Implementing Distributed Shared Memory Distributed Shared Memory Implementation and performance of Munin Transparent fault-tolerance in parallel Orca programs Integrating message-passing and shared-memory A checkpoint protocol for an entry consistent shared memory system CRL On distributed object checkpointing and recovery Lightweight logging for lazy release consistent distributed shared memory The directory-based cache coherence protocol for the DASH multiprocessor The performance of consistent checkpointing in distributed shared memory systems A longitudinal survey of Internet host reliability A Recoverable Distributed Shared Memory Integrating Coherence and Recoverability Reduced Overhead Logging for Rollback Recovery in Distributed Shared Memory Virtual Shared Memory: A Survey of Techniques and Systems

portability;fault-tolerance;checkpointing;distributed shared memory

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Using permutations in regenerative simulations to reduce variance.

We propose a new estimator for a large class of performance measures obtained from a regenerative simulation of a system having two distinct sequences of regeneration times. To construct our new estimator, we first generate a sample path of a fixed number of cycles based on one sequence of regeneration times, divide the path into segments based on the second sequence of regeneration times, permute the segments, and calculate the performance on the new path using the first sequence of regeneration times. We average over all possible permutations to construct the new estimator. This strictly reduces variance when the original estimator is not simply an additive functional of the sample path. To use the new estimator in practice, the extra computational effort is not large since all permutations do not actually have to be computed as we derive explicit formulas for our new estimators. We examine the small-sample behavior of our estimators. In particular, we prove that for any fixed number of cycles from the first regenerative sequence, our new estimator has smaller mean squared error than the standard estimator. We show explicitly that our method can be used to derive new estimators for the expected cumulative reward until a certain set of states is hit and the time-average variance parameter of a regenerative simulation.

INTRODUCTION The regenerative method is a simulation-output-analysis technique for estimating certain performance measures of regenerative stochastic systems; see [Crane and Iglehart 1975]. The basis of the approach is to divide the sample path into i.i.d. segments (cycles), where the endpoints of the segments are determined by a sequence of stopping times. Many stochastic systems have been shown to be regenerative [Shedler 1993], and the regenerative method results in asymptotically valid confidence intervals. In this paper we propose a new simulation estimator for a performance measure of a regenerative process having two different sequences of regeneration times, and study its small-sample behavior. The idea of our approach is as follows. First simulate a fixed number of regenerative cycles from the first sequence of regeneration times, and compute one estimate. We construct another estimator by dividing up the original sample path into segments with endpoints given by the second sequence of regeneration times, and creating a new sample path by permuting the segments (except for the initial and final segments). We then compute a second estimate of ff from the new permuted path. We show that this estimate has the same distribution as the original one. Our new estimator is finally constructed as the average of the estimates over all possible permutations. This strictly reduces variance when the estimator is not a purely additive function of the sample path. We show that to compute our new estimators, one does not have to actually calculate all permutations and the average over all of them. Instead, we derive formulas for the new estimators, where the expressions can be easily computed by accumulating some extra quantities during the simulation. The storage requirements of our methods are fixed and do not grow as the simulation run length increases. Hence, there is little extra computational effort or storage needed to construct our new estimators. For a run length of any fixed number of cycles from the first regenerative se- quences, the new estimator has the same expected value as the standard estimator and lower variance; thus, it has lower mean squared error. While it turns out that our method has no effect on the standard regenerative ratio estimator for certain steady-state performance measures, the basic technique can still be beneficially applied to a rich class of other performance measures, and in this paper, we consider three specific examples. First, we derive a new estimator for the second moment of the cumulative reward during a regenerative cycle. We show that the standard regenerative variance estimator fits into this framework. Hence, our estimator will result in a variance estimator having no more variability than the standard one. This is important because one measure of the quality of an output-analysis methodology is the variability of the half-width of the resulting confidence interval [Glynn and Iglehart 1987], which is largely influenced by the variance of the variance estimator. We also construct a new estimator for the cumulative reward until some set of states is first hit, which includes the mean time to failure as a special case. Here, the performance measure can be expressed as a ratio of expectations, and we apply Using Permutations in Regenerative Simulations to Reduce Variance \Delta 3 our technique to the numerator and denominator separately. In some sense our method reuses the collected data to construct a new estimator, and as such, it is related to other statistical techniques. For example, the bootstrap [Efron 1979] takes a given sample and resamples the data with replacement. In contrast, one can think of our approach as resampling the data without replacement (i.e., permuting the data), and then averaging over all possible resamples. Other related methods include U-statistics (Chapter 5 of [Serfling 1980]), V -statistics [Sen 1977], and permutation tests (e.g., [Conover 1980]). The rest of the paper is organized as follows. In Section 2, we discuss our assumptions and the standard estimator of a generic performance measure ff. We present the basic idea of how to construct our new estimator using a simple example in Section 3. Section 4 contains a more formal description of our method. Section 5 describes the new estimator for the second moment of the cumulative reward over a regenerative cycle and shows how these results can be used to derive a new estimator of the variance parameter arising in a regenerative simulation. We also discuss here the special case of continuous-time Markov chains. In Section 6 we derive new estimators for the expected cumulative reward until some set of states is hit. We analyze the storage and computational costs of our new estimator in Section 7. We present in Section 8 the results of some simulation experiments comparing our new estimators with the standard ones. Section 9 discusses directions for future re- search. Most of the proofs are collected in Appendix A. Also, we give pseudo-code for one of our estimators in Appendix B. (Calvin and Nakayama [1997] present the basic ideas of our approach, without proofs, in the setting of discrete-time Markov chains.) 2. GENERAL FRAMEWORK Let X be a continuous-time stochastic process having sample paths that are right continuous with left limits on a state space S ae ! d . Note that we can handle discrete-time :g in this framework by letting X btc for all t - 0, where bac is the greatest integer less than or equal to a. be an increasing sequence of nonnegative finite stopping times. Consider the random pair (X; T ) and the shift We define the pair (X; T ) to be a regenerative process (in the classic sense) if (i). f' T (i) (X; are identically distributed; (ii). for any i - 0, ' T (i) (X; T ) does not depend on the "prehistory" See p. 19 of [Kalashnikov 1994] for more details. This definition allows for so-called delayed regenerative processes (e.g., Section 2.6 of [Kingman 1972]). be two distinct increasing sequences of nonnegative finite stopping times such that are both regenerative processes. For example, if X is an ir- reducible, positive-recurrent, discrete-time or continuous-time Markov chain on a countable state space S, then we can define T 1 and T 2 to be the sequences of hitting M. Calvin and M. K. Nakayama times to the states v 2 S and w 2 S, respectively, where we assume that and w 6= v. Our goal is to estimate some performance measure ff, which we will do by generating a sample path segment ~ of a fixed number m 1 of regenerative 1-cycles of our regenerative process. Here, we use the terminology "1-cycles" to denote cycles determined by the sequence T 1 ; i.e., the ith 1-cycle is the path segment (i)g. We similarly define "2-cycles" relative to the sequence T 2 . Now we define the standard estimator of ff based on the sample path ~ 1-cycles to be where h j hm1 is some function. This general framework includes many performance measures of interest. Example 1. Suppose "/ Z for some function g : S ! !, where p - 1. Then we can define h( ~ X) by h( ~ Y (g; where Z for k - 1. Note that here b X) is an unbiased estimator of ff. We will examine this example with in Section 5. Example 2. Suppose that where for k - 1, now is the variance parameter arising from a regenerative simulation. (More details are given in Section 5.1.) Then we can define h( ~ X) by h( ~ where Using Permutations in Regenerative Simulations to Reduce Variance \Delta 5 Note that b X) is the standard regenerative estimator of oe 2 . We will return to this example in Section 5.1. Example 3. Suppose we are interested in computing set of states F ae S. Thus, j is the expected cumulative reward until hitting F conditional on T 1 and the mean time to failure is a special case. It can be shown that where and with a - see [Goyal et al. 1992]. To estimate j, we generate sample paths ~ each consisting of m 1 1-cycles, and we use ~ X 1 to estimate - and ~ X 2 to estimate fl. We can either let ~ independent of ~ We examine the estimation of the numerator and denominator in (6) separately. First, if we want to estimate ff = -, then we define the function h by h( ~ -( ~ where Z with Fg. On the other hand, if we want to estimate then we define the function h by h( ~ where and 1f \Delta g is the indicator function of the event f \Delta g. Thus, the standard estimator of j is -( ~ We will return to this example in Section 6. 6 \Delta J. M. Calvin and M. K. Nakayama 3. BASIC IDEA Our goal now is to create a new estimator for ff. We begin by giving a heuristic explanation of how it is constructed by considering the simple example illustrated in Figure 1. For simplicity, we depict a continuous sample path on a continuous state space S. The T 1 sequence corresponds to hits to the state v, and the T 2 sequence corresponds to hits to state w. The top graph shows the original sample path generated having regenerative 1-cycles. For this path, there are occurrences of stopping times from sequence T 2 . To make it easier to see the individual 2-cycles, each is depicted using a different line style. Now we can construct new sample paths from the original path by permuting the 2-cycles, resulting in (M possible paths. The second graph shows one such permuted path. Here, the original third 2-cycle is now first, the original first 2-cycle is now second, and the original second 2-cycle is now third. The new 1-cycle times are T 0 and the new 2-cycle times are T 0 3. The third graph contains another permuted path, in which the original second 2-cycle is now first, the original third 2-cycle is now second, and the original first 2-cycle is now third. The 1-cycle times are now T 00 and the new 2-cycle times are T 00 3. Note that for each new path, the number of 1-cycles is the same as in the original path, but the paths of some of the 1-cycles have changed. We show in Theorem 1 that all of the paths have the same distribution. For each possible path, we can compute an estimator of ff based on the m 1 new 1-cycles by applying the performance function h j hm1 to it. Our new estimator is then the average over all estimators constructed. It turns out that we do not actually have to construct all permuted paths to calculate the value of our new estimator. The basic reason for this is that we can break up any sample path into a collection of segments of different types. After any permutation, the path changes, but the collection of segments does not. To calculate our new estimator for a given (original) path, we need to determine the different ways the segments can be put together when 2-cycles are permuted. In particular, since we form an estimator based on the 1-cycles for every permutation, we want to understand how 1-cycles are formed from the segments. Another key factor that will allow us to explicitly compute our new estimator without actually computing all permutations is that for the performance measures we consider, the contribution of each 1-cycle to the overall estimator can be expressed as a function of the segments in the cycle. For instance, in Example 1 with 2, we can express the contribution from each 1-cycle as the square of the sum of contributions of the segments in the cycle. We now examine more closely the four different types of path segments that can arise. We focus on the example in Figure 1. (1) The first type is a 1-cycle that does not contain a hit to w. The segments of this type in the original path of the figure are the first and third 1-cycles; i.e., the segment from T 1 (0) to T 1 (1) and the segment from T 1 (2) to T 1 (3). Segments of this type never change under permutation, although they may occur at different times. For example, the third 1-cycle in the original path appears as the fourth 1-cycle in the second permuted path. This segment is the third 1-cycle in the first permuted path, but it occurs at a different time. The first 1-cycle in our Using Permutations in Regenerative Simulations to Reduce Variance \Delta 7 Original Path Another Permuted One Permuted Path Fig. 1. A sample path and some corresponding permuted paths. M. Calvin and M. K. Nakayama example always appears in the same place in all permutations. (2) Now consider any 2-cycle in which state v is not hit, such as the third 2-cycle in the original path in the figure. After any permutation this 2-cycle will be in the interior of some 1-cycle. For example, the third 2-cycle in the original path is in the interior of the fifth 1-cycle in the original path, and in the interior of the second (resp., third) 1-cycle in the first (resp., second) permuted path. (3) The next type of segment goes from w to v before hitting w again. No matter how the 2-cycles are permuted, this type of segment is always the end of some 1-cycle. For example, consider the path segment from T 2 (0) to T 1 (2) in the original sample path. In this path, the segment is the end of the second 1- cycle. In the first permuted path, this segment is again the end of the second 1-cycle, but this new second 1-cycle is different from that in the original path. On the other hand, this segment in the second permuted path is the end of the third 1-cycle. In general, any segment that goes from w to v before hitting w again will be the end of some 1-cycle in any permuted path. (4) The final type of segment goes from v to w before hitting v again. In any permutation, this segment will be the beginning of a 1-cycle. For example, consider the path segment from T 1 (3) to T 2 (1) in the original sample path. In this path, the segment is the beginning of the fourth 1-cycle. In the second permuted path this segment is the beginning of the fifth 1-cycle. In the first permuted path, the segment is again the beginning of the fourth 1-cycle. In general, any segment that goes from v to w before hitting v again will be the beginning of some 1-cycle in any permuted path. Note that the original sample path in Figure 1 consists of segments of types appearing in the following order: 1, 4, 3, 1, 4, 3, 4, 2, 3. In any permutation, the segments will appear in a different order, but the collection of segments never changes. Recall that for each permuted path, we compute an estimate of the performance measure ff based on the m 1 1-cycles. So now we examine how 1-cycles can be constructed from the permutations. For the original path, we divide up the 1- cycles into those that hit state w and those that do not. The ones that do not hit w are type-1 segments and are unaffected by permutations. Now we examine how permutations affect 1-cycles that hit w. These cycles always start with a type-4 segment, followed by some number (possibly zero) of segments of type 2, and end with a type-3 segment. For example, the fifth 1-cycle in the original path starts with the type-4 segment from T 1 (4) to T 2 (2), is followed by the type-2 segment from T 2 (2) to T 3 (3), and concludes with the type-3 segment from Also, the fourth 1-cycle in the original path begins with a type- 4 segment from T 1 (3) to T 2 (1) and terminates with a type-3 segment from T 2 (1) to T 1 (4). This characterization of 1-cycles holds not only for the original sample path, but also for any permuted path. Moreover, for any 2-cycle that hits v in the original path, the type-3 segment and type-4 segment in it will always be in the same 2-cycle in any permutation, and so these two segments can never be in the same 1-cycle since the type-3 segment will always be the end of one 1-cycle and the segment will always be the beginning of the following 1-cycle. For example, the type-3 segment from T 2 (0) to T 1 (2) and the type-4 segment from T 1 (3) to T 2 (1) Using Permutations in Regenerative Simulations to Reduce Variance \Delta 9 are always in the same 2-cycle in any permutation, and as such, they are always in successive 1-cycles. Also, in our example the first type-4 segment from T 1 (1) to and the last type-3 segment from T 2 (3) to T 1 (5) will never be in the same 1-cycle. Any other pair of type-3 segment and type-4 segment will be in the same 1-cycle in some permutation. Thus, to construct all 1-cycles that hit w that are possible under permutations of 2-cycles, we have to consider all valid pairings of the type-4 and type-3 segments, and allocate the type-2 segments among the pairs. The proofs in Sections A.2 and A.3 basically use this reasoning. 4. FORMAL DEVELOPMENT OF GENERAL METHOD Now we more formally show how to construct our new estimator. We begin with some new notation. Let X be the space of paths S that are right continuous with left limits on [0; i(x)). For define a new element and Thus, the new path x 1 obtained by concatenating x 2 on to the end of x 1 . Given the original sample path ~ X, which consists of m 1 1-cycles, we begin by constructing a new sample path ~ X 0 from ~ X such that ~ equality in distribution. This is done by first taking the original sample path ~ X and determining the number of times M 2 that the stopping times from the sequence occur during the m 1 1-cycles. Note that if M then the path ~ X has no 2-cycles. If M 2, then there is only one 2-cycle. Assume now that M 2 - 3. Then for the given path ~ X, we can now look at the (M 2-cycles in the path. We generate a uniform random permutation of the within the path ~ , and this gives us our new sample path ~ which also has m 1 1-cycles. More specifically, define M ~ X . If M 2 - 3, then we break up the path ~ ~ where ~ is the initial path segment until the first time a stopping time from sequence T 2 occurs, ~ is the final path segment from the last time a stopping time from sequence T 2 occurs until the end of the path, and ~ is the kth 2-cycle of the original path ~ 1)) be a uniform random permutation of 1. Then we define our new M. Calvin and M. K. Nakayama path ~ to be ~ which is the original path ~ X with the 2-cycles permuted. Note that ~ have the same number m 1 of 1-cycles, and we prove in Section A.1 that ~ Now for the new sample path ~ , we can calculate which is just the estimator obtained from the new sample path ~ and is based on m 1 1-cycles (recall that h j hm1 ). The number of possible paths ~ 0 we can construct from ~ X is N( ~ which depends on ~ X and is therefore random. We label these paths ~ each of which has the same distribution as ~ X, and for each one we construct b ff( ~ finally define our new estimator for ff to be e h( ~ Another way of looking at our new estimator is as follows. We first generate the original path ~ X and use it to construct the N( ~ X) new paths ~ X (N) . We then choose one of the new paths at random uniformly from ~ this be ~ X, we can think of b standard estimator of ff since it has the same distribution as b X). Then we construct our new estimator e to be the conditional expectation of b respect to the uniform random choice of ~ given the original path ~ X. That is, if E denotes expectation with respect to choosing ~ X 0 from the uniform distribution on ~ then we write e Assuming that E[-ff( ~ the new estimator has the same mean as the original since because ~ X. Moreover, decomposing the variance by conditioning on ~ us implies that the variance of the new estimator e is no greater than that of the original estimator - X). This calculation, combined with the fact that ~ (which will be proved in Section A.1), establishes the following theorem. Theorem 1. Let T 1 and T 2 be two distinct sequences of stopping times, and construct the estimator e X) defined by (11). Assume that E[bff( ~ Using Permutations in Regenerative Simulations to Reduce Variance \Delta 11 X)], and and so the mean squared error of our new estimator e X) is no greater than that of the original estimator b ff( ~ X). Strict inequality is obtained in (12) unless In Theorem 1 we see that there is no variance reduction when for every possible original sample path ~ X, the value of the function h in (1) is unaffected by permutations of the 2-cycles. For example, this is the case in Example 1 with since h( ~ Z Z Z Z Z and so e X). Similarly, by choosing g(x) j 1, we see that permuting 2-cycles does not alter the estimator for E[-(1)]. Thus, our method has no effect on the standard ratio estimator for steady-state performance measures ff that can be expressed as However, for p ? 1 in Example 1, we have in general that h( ~ so typically e X). Also, we usually have that the standard time-average variance estimator in Example 2 for a regenerative simulation will differ from the new estimator defined by (11). Finally, applying the above idea separately to the numerator and denominator in the ratio expression for the mean cumulative reward until hitting some set of states F as in Example 3 will result in a new estimator. 5. ESTIMATING THE SECOND MOMENT OF CUMULATIVE CYCLE REWARD For our new estimator e X) to be computationally efficient, we need to calculate explicitly the conditional expectation in (11) without having to construct all possible permutations. We first do this for Example 1 with and our standard estimator of ff is where we have dropped the dependence of Y on g to simplify the notation. Our new estimator of ff is then e M. Calvin and M. K. Nakayama is the same as Y (k) except that it is for the sample path ~ than ~ X. Now to explicitly calculate (15) in this particular setting, we will divide up the original path into segments using the approach described in Sections 3 and 4. We need some new notation to do this. For our two sequences of stopping times denote the set of indices of the 1-cycles in which a T 2 stopping time occurs, and define the complementary set specifically, H(1; jg. We analogously define the set H(2; 1) with the roles of some lg, which is the first occurrence of a stopping time from sequence T 2 after the 1)st stopping time from the sequence T 1 . Similarly define e some lg, which is the last occurrence of a stopping time from sequence T 2 before the kth occurrence of the stopping-time sequence T 1 . Then, for k 2 H(1; 2), we let which is the contribution to Y (k) until a stopping time from sequence T 2 occurs, and let Y Z e which is the contribution to Y (k) from the last occurrence of a stopping time from sequence T 2 in the kth 1-cycle until the end of the cycle. Also, for l 2 J(2; 1), let Y 22 Z which is the integral of g(X(t)) over the lth 2-cycle in which there is no occurrence of a stopping time from sequence T 1 . We now define B k ae J(2; 1) to be the set of indices of those 2-cycles that do not contain any occurrences of the stopping times from the sequence T 1 and that are between It then follows that for k 2 H(1; 2), Hence, \Theta Y 12 A Using Permutations in Regenerative Simulations to Reduce Variance \Delta 13 Y 22 (l) Y 22 (l) In the last expression, the first term does not change if we replace the original sample path ~ X with the new sample path ~ the last term does change. In Section A.2, we compute explicitly the conditional expectation of (16), when ~ is replaced with ~ with respect to a random permutation given the original path ~ X. The expression for this involves some more notation. Define and Finally, define fi l to be the lth smallest element of the set H(1; 2) for and define fi is the index in H(1; 2) that occurs just before k if k is not the first index and is the last element in H(1; 2) if k is the first element. The following theorem is proved in Section A.2. (Pseudo-code for our estimator is given in Appendix B.) Theorem 2. Suppose we want to estimate ff defined in (13), and assume that our new estimator is given by e and otherwise by e \Theta Y 12 Y Y 22 Y 22 l6=m Y 22 (l)Y 22 (m)C A : (17) The estimator satisfies E[eff( ~ X) is the standard estimator of ff as defined in (14). 5.1 A New Estimator for the Time-Average Variance We can use Theorem 2 to construct a new estimator for the variance parameter in a regenerative simulation of the process X . We start by first giving a more complete explanation of Example 2 in Section 2. cost function. Define Z tf(X(s)) ds: 14 \Delta J. M. Calvin and M. K. Nakayama Since X is a regenerative process, there exists some constant r such that r t ! r as Theorem 2.2 of [Shedler 1993]). Also, r satisfies the ratio formula Assuming that E[Z(f ; exists a finite positive constant oe such that oe as t !1. The constant oe 2 is called the time-average variance of X and is given in (3). Given the central limit theorem described by (18), construction of confidence intervals for r therefore effectively reduces to developing a consistent estimator for oe 2 . The quality of the resulting confidence interval is largely dependent upon the quality of the associated time-average variance estimator. The standard consistent estimator of oe 2 is b X) defined in (4). Note that b oe 2 ( ~ X) can be expressed as Now we define our new estimator e X) to be the conditional expectation of b oe 2 ( ~ with respect to a random permutation of 2-cycles, given the original sample ~ X . Hence, letting b r 0 , Y be the corresponding values of b r, Y (f \Gamma k), and -(k) for the sample path ~ we get that \Theta P m1 since k=1 -(k) is independent of the permutation of 2-cycles. Also, observe that Z r is independent of the permutation of 2-cycles, so \Theta P m1 i.e., we can replace b r 0 with b r. The following is a direct consequence of Theorem 2. Corollary 3. Suppose we want to estimate oe 2 defined in (3), and assume that our new estimator e oe 2 ( ~ X) is given by (19), where the numerator is as in (17) with the function r. The estimator satisfies X)] and Var[eoe 2 ( ~ X)]. 5.2 Continuous-Time Markov Chains We now consider the special case of an irreducible, positive-recurrent, continuous-time Markov chain on a countable state space S having generator matrix be the embedded discrete-time Markov chain, and be the sequence of random holding times of the continuous-time Markov chain; i.e., Wn is the time between the nth and (n 1)st transitions Using Permutations in Regenerative Simulations to Reduce Variance \Delta 15 of X . Define A which is the time of the nth transition. It is well known that conditional on V , the holding time in state Vn is exponentially distributed with mean 1=-(Vn ) and that W i and W j are (conditionally) independent for i 6= j. Assume that the sequences of stopping times T 1 and T 2 correspond to hitting times to fixed states v 2 S and w 2 S, respectively, with w 6= v, and assume that specifically, define - 1 which is the sequence of hitting times to state v for the discrete-time Markov chain. Similarly, define - and Suppose that we want to estimate ff as defined in (13), and our standard estimator of ff is given in (14). Now note that Z Using discrete-time conversion [Hordijk et al. 1976; Fox and Glynn 1990] gives us \Theta W 2 since are conditionally independent given V . For a function f !, let which is the cumulative reward over the kth cycle for the discrete-time chain V , and define the functions to be g 1 Therefore, we get and To create our new estimator of ff, we then compute the conditional expectation of X) with respect to a random uniform permutation of 2-cycles given the original M. Calvin and M. K. Nakayama path ~ X. Define The last term in (20) is independent of permutations of 2-cycles, and so we get the following expression for e - , which follows from Theorems 1 and 2. Theorem 4. Suppose X is an irreducible, positive-recurrent, continuous-time Markov chain on a countable state space S, and we want to estimate ff defined in (13). Assume that T 1 and T 2 correspond to the hitting times to states v and w, respectively, with w 6= v. Assume that E[Y (g our new estimator is given by e - X) is defined by (21), which can be computed from (17) with the function 1 . The estimator satisfies E[e - is the standard estimator of ff as defined in (14). If we had instead first converted to discrete time and then computed - the discrete-time Markov chain and its conditional expectation with respect to the permutation, we would have obtained e - X) as our estimator for ff. However, since E[ - function g 6= 0, E[e - X) is biased. On the other hand, our estimator e - X) is unbiased. 6. EXPECTED CUMULATIVE REWARD UNTIL HITTING A SET Recall that we can express the expected cumulative reward until a hitting time given in (5) as the ratio in (6), and the standard estimator of j is defined in (7). Also, recall that the numerator - is estimated using the sample path ~ and the denominator fl is estimated from path ~ We will examine both the cases when ~ are independent. In the context of estimating the mean time to failure of highly reliable Markovian systems, Goyal, Shahabuddin, Heidelberger, Nicola, and Glynn [1992] and Shahabuddin [1994] estimate - and fl independently; i.e., ~ are indepen- dent. This is useful because then different sampling techniques can be applied to estimate the two quantities. In particular, fl is the probability of a rare event and so it is estimated using importance sampling. On the other hand, we can efficiently estimate - using naive simulation (i.e., no importance sampling). Below, we do not apply importance sampling to estimate fl, but one can also derive a new estimator of fl when using importance sampling. Our new estimator of j is defined as e j( ~ e -( ~ e Using Permutations in Regenerative Simulations to Reduce Variance \Delta 17 where e -( ~ -( ~ e fl( ~ Now to explicitly calculate the numerator and denominator, we will divide up the original path into segments using the approach described in Sections 3 and 4. We need some new notation to do this. For k 2 H(1; 2), let I 12 I 22 I with Fg. Hence, I 12 (k) (resp., I 21 (k)) is the indicator of whether the set F is hit in the initial 1-2 segment (resp., final 2-1 segment) of the 1-cycle with index k 2 H(1; 2). Also, I 22 (l) is the indicator whether the set F is hit in the 2-cycle with index l 2 J(2; 1). We first consider the denominator fl. To derive the new estimator of fl from permuting the 2-cycles, we first write The first term on the right-hand side is independent of permutations of the 2-cycles. For the second term we note that for k 2 H(1; 2), I 12 (k); max l2Bk I 22 (l); I 21 (k) Thus, e I 12 (k); max I 22 (l); I 21 (ae(k)) where ae(k) is the index of the I 21 variable that follows the I 12 (k) variable after a permutation of the 2-cycles, and B 0 l is the same as B l except that B 0 l is after a permutation. We work out in Section A.3 the conditional expectation appearing above. We now examine the estimation of -. Note that the standard estimator of - satisfies -( ~ The first term is not affected by permuting the 2-cycles, but the second term is. M. Calvin and M. K. Nakayama For Y j!l Y where Z T (2) F (l)-T2 (l) Z T (1) e Hence, to compute the new estimator, we need to compute the conditional expectation of the second term in (24), which we can do by using the representation for given in (25); this is done in Section A.3. To present what the new estimator actually works out to, we need some more notation. Define Also, define I 22 (l); r D 22 (l)I 22 (l): Then we have the following result, whose proof is given in Section A.3. Theorem 5. Suppose we want to estimate j in (6), and assume that 1. Then, our new estimator is given by (22) where Using Permutations in Regenerative Simulations to Reduce Variance \Delta 19 (i). e -( ~ -( ~ e -( ~ r (ii). e fl( ~ The estimators e -( ~ -( ~ -( ~ -( ~ -( ~ are the standard estimators of - and fl, respectively. In Theorem 5 the variables used in part (i) are defined for the sample path ~ and the variables in part (ii) are for the sample path ~ . For example, h 12 in part (i) is the cardinality of the set H(1; 2) for the path ~ in part (ii) it is the same but instead for the path ~ Theorem 5 shows that our new estimator for j has unbiased and lower-variance estimators for both the numerator and denominator, but the effect on the resulting ratio estimator is more difficult to analyze rigorously. Instead, we now heuristically examine the bias and variance of the ratio estimator. To do this, we generically let - - fl, and - -fl be estimators of -, fl, and j, respectively. Then using first- and second-order Taylor series expansions, we have the following approximations for the bias and variance of - j: Cov and see p. 181 of [Mood et al. 1974]. We now use these approximations to analyze the standard and new estimators for j. First, consider the case when ~ are independent. Then b -( ~ are independent, so Cov -( ~ Similarly, e -( ~ are independent, so Cov -( ~ it follows from Theorem 5 and (26) and (27) that J. M. Calvin and M. K. Nakayama and where we use the notation a - b to mean that a is approximately no greater than b. Hence, the mean square error of e j( ~ approximately no greater than that of b j( ~ In the case when ~ -( ~ Also, we have that Cov -( ~ -( ~ by Lemma 2.1.1 of [Bratley et al. 1987]. But since the variances of the new estimators of the numerator and denominator are smaller than those for the original esti- mators, we cannot compare the biases and variances of b j( ~ However, we examine this case empirically in Section 8 and find that there is a variance reduction and smaller mean squared error. 7. STORAGE AND COMPUTATION COSTS We now discuss the implementation issues associated with constructing our new estimator e ff( ~ X) given in (17) for the case when ff is defined in (13). First note that the first term in the second line of (17), excluding the factor 2=(h Y Y Also, the last term in the last line of (17) satisfies l6=m Y 22 (l)Y 22 (m) =@ X Y 22 (k)A\Gamma@ X Y 22 Hence, to construct our estimator e X), we need to calculate the following quantities -the sum of the Y (k) 2 over the 1-cycles k 2 J(1; 2); -the sums of the Y 12 (k), Y 21 (k), Y 12 over the 1-cycles k 2 H(1; 2); -the sum of the Y 12 (k)Y 21 (/(k)) over the 1-cycles k 2 H(1; 2); -the sums of the Y 22 (k) and Y 22 (k) 2 over the 2-cycles k 2 J(2; 1). To compute these quantities in a simulation, we do not have to store the entire sample path but rather only need to keep track of the various cumulative sums as the simulation progresses. Also, the amount of storage required is fixed and does not increase with the simulation run length. Therefore, compared to the standard estimator, the new estimator can be constructed with little additional computational effort and storage. (Pseudo-code for this estimator is given in Appendix B.) Using Permutations in Regenerative Simulations to Reduce Variance \Delta 21 A similar situation holds when estimating j using the estimator defined in Theorem 5. We conclude this section with a rough comparison of the work required for the new estimator with that for the standard regenerative method when estimating ff given in (13). Let W s be the (random) amount of work to generate a particular sample path of m 1 1-cycles in a discrete-event simulation, where W s includes the work for the random-variate generation, determining transitions, and appropriately updating data structures needed in the sample-path generation. This quantity is the same for our new method and the standard method. We now study the work needed for the output analysis required by the standard regenerative method. After every transition in the simulation, we need to update the value of the current cycle-quantity Y (g; k); see (2). Let ' 1 denote this (deter- ministic) amount of work, and if there are N total transitions in the sample path, then the total work for updating the Y (g; during the entire simulation is N' 1 . At the end of every 1-cycle, we have to square the current cycle quantity Y (g; add it to its accumulator; see (14). Let ' 2 denote this (deterministic) amount of work required at the end of each 1-cycle, and since there are m 1 1-cycles along the path, the total work for accumulating the sum of the Y (g; Therefore, the cumulative work (including sample-path generation and output analysis) for the standard regenerative method is Now we determine the amount of work needed for the output analysis of our new permutation method. By examining the pseudo-code in Appendix B, we see that after every transition, a single accumulator is updated. Every time a stopping time from either sequence T 1 or T 2 occurs, we compute either a square or a product of two terms and update at most three accumulators, and the amount of work for this is essentially at most 3' 2 . Since the number of times this needs to be done is the cumulative work for our permutation method is Therefore, the ratio of cumulative work of our permutation method relative to the standard regenerative method is Typically in a regenerative simulation, the amount of time W s required to generate the sample path is much greater than the time needed to perform the output analysis, and so RW will usually be close to 1. Hence, the overhead of using our method in a simulation will most likely be very small, and this is what we observed in our experimental results in the next section. 8. EXPERIMENTAL RESULTS Our example is based on the Ehrenfest urn model. The transition probabilities for this discrete-time Markov chain are given by P s s: 22 \Delta J. M. Calvin and M. K. Nakayama In our experiments we take 8. The stopping-time sequences T 1 and T 2 for our regenerative simulation correspond to hitting times to the states v and w, respectively, and so state v is the return state for the regenerative simulation. We ran several simulations of this system to estimate two different performance measures: oe 2 , which is the time-average variance constant from Section 5.1, and j, which is the mean hitting time to a set F from Section 6. For each performance measure, we ran our experiments with several different choices for v, and for each v, we examined all possible choices for w. Choosing no effect on the resulting estimator, so this corresponds the standard estimator. We ran 1,000 independent replications for each choice of v and w. Tables 1-3 and 5-6 present the results from estimating the two performance measures, giving the sample average and sample variance of our new estimator over the 1,000 replications. The average cycle lengths change with different choices of v; in order to make the results somewhat comparable across the tables, we changed the number of simulated cycles for each case so that the total expected number of simulated transitions remains the same. For Table 1, corresponding to simulated 1,000 cycles, and a greater number for the other tables. For example, the expected cycle length is 3:5 times as long for state 1 as for state 2, so in Table 2, we simulated 3,500 cycles. Since our new estimator reduces the variance but at the cost of extra computational effort, we also compare the efficiencies (inverse of the product of the variance and the time to generate the estimator) of our new estimator and the standard one, as suggested by Hammersley and Handscomb [1964] and Glynn and Whitt [1992]. 8.1 Results from Estimating Variance We first examine the results from estimating the time-average variance oe 2 with cost function performed 3 experiments, corresponding to return states and 4, and these results are given in Tables 1-3, respectively. The transition probabilities are symmetric around state 4 (the mean of the binomial stationary distribution), so our first choice of return state is fairly far from the mean. Notice that the variability of the variance estimator is smaller with w near the mean state 4, and that the variance reduction is greater v. The reason for this is that the excursions from v that go below 1 have little variability; because of the strong restoring force of the Ehrenfest model, such excursions tend to be very brief. On the other hand, excursions that get as far as the mean are likely to be quite long (and thus the contribution to the variance estimator tends to have large variability). In the second table we ran the same experiment with obtained similar results. In Table 3 we examine the same model, but now with our return state v chosen to be the stationary mean, 4. The first thing to notice is that, compared with the other choices of the return state, the variance reduction is relatively small. State 4 is the best return state in the sense of minimizing the variance of the regenerative- variance estimator. Therefore, for this example, it appears that our estimator is a significant improvement over the standard regenerative estimator if the standard regenerative estimator is based on a relatively "bad" return state. However, if one is able to choose a near-optimal return state to begin with, our estimator yields a modest improvement. (Unfortunately, there are no reliable rules for choosing a priori a good return state.) Comparing the three tables, we see that the minimum Using Permutations in Regenerative Simulations to Reduce Variance \Delta 23 Table 1. Estimating variance with w Avg. of eoe 2 Sample Var. Table 2. Estimating variance with 2. w Avg. of eoe 2 Sample Var. variability does not change much across tables (0:17 for Table 3, and 0:18 for the other tables). This example suggests that it may be possible to compensate for a bad choice of v by an appropriate choice of w. Finally, we illustrate the computational burden of our method. Table 4 shows the work required for the results obtained in Table 1. The first column gives, for each choice of w, the relative work (CPU time for generating the sample path and output analysis) required for our new estimator, that is, the CPU time with our new estimator divided by the CPU time for the standard regenerative method. (The row with corresponds to the standard regenerative method, so all entries are the other entries are normalized with respect to these.) The second column gives the relative variance; that is, the sample variance of our new estimator divided by the sample variance for the standard regenerative estimator. The last column gives Table 3. Estimating variance with w Avg. of eoe 2 Sample Var. M. Calvin and M. K. Nakayama Table 4. Comparison of efficiency, w Relative Work Relative Var. Relative Efficiency the relative efficiency; that is, the inverse of the product of the relative work from column 1 and the relative variance from the second column. Notice that in the cases where the variance reduction is small, the increase in work is also small. More work is needed when there is a larger variance reduction, but this is still no more than a few percent increase. Note that in the best case the efficiency was improved by nearly a factor of eight. Because little additional work is needed by our method, within each of the other tables, the run times are approximately the same for the different values of w. Therefore, one can roughly approximate the relative efficiencies for Tables 2 and 3 as the ratio of the sample variances for states v and w. It should also be noted that the Ehrenfest model considered here is very simple compared with typical simulation models. The additional work required to compute our new estimators is independent of the model, and so if the work to generate the sample path is much larger than for the Ehrenfest model, the relative increase in work would be correspondingly small. 8.2 Results from Estimating Hitting Times to a Set We now consider estimating j, which is the mean hitting time to a set of states F starting from a state v for our Ehrenfest model with reward function g(x) j 1. We take which is hit infrequently. Tables 5 and 6 show our results from generating 1,000 independent replications for In each replication, we generated a sample path which we used to compute the new estimators for both the numerator and denominator. Hence, using the terminology of Section 6, we let ~ . For each path, we generated 1,000 cycles for and 2,500 cycles for 4, so that the expected sample path length is the same in each case. We calculated (i.e., not using simulation) the theoretical values to be in addition to examining the variance of our new estimator, we can also study the mean squared error. Note that the theoretical value of j depends on the starting state v. First observe that there is no change in the estimates of j and their variances for certain choices of w. This is due to the fact that permuting w-cycles in these cases has no effect on the estimators of either the numerator or denominator. For the other values of w, the (relative) variance reduction is significantly greater when Table than when (Table 5). In addition, although the absolute bias is the greatest for both choices of v, its magnitude is quite small, Using Permutations in Regenerative Simulations to Reduce Variance \Delta 25 Table 5. Estimating expected hitting time to state 7 with w Avg. of ej Sample Var. MSE Table 6. Estimating expected hitting time to state 7 with 2. w Avg. of ej Sample Var. MSE and when examining the mean squared error, the variance reduction overwhelms the effect of the bias. We now explain why the choice of w that results in the most variance reduction is In the original sample path without permuting the w-cycles, the set F is hit a certain number of times. By permuting the w-cycles, we get a variance reduction if there are some v-cycles that hit w but not F and if within a particular v-cycle that hits F , there is more than one w-cycle that hits F and does not hit v. Permuting the w-cycles then can distribute the hits to F to more of the v-cycles. The amount of variance reduction in estimating fl is largely determined by the difference between the maximum and minimum number of v-cycles that hit F from permuting the w-cycles. Choosing results in no variance reduction because we are working with a birth-death process and so the process always hits F no later than it hits w within a v-cycle, and so permuting w-cycles has no effect. Of the remaining choices for maximizes the number of w-cycles that hit F , hence the largest variance reduction. Therefore, in general, we suggest that the state w should be chosen so that w 62 F and it is as "close" as possible to the set F to maximize the number of w-cycles that hit F . 9. DIRECTIONS FOR FUTURE RESEARCH We are currently investigating how to construct confidence intervals based on our new permuted estimators. This is a difficult problem because of the complexity of the estimators. Another area on which we are currently working is determining how to choose the two sequences of regenerative times T 1 and T 2 when there are more than two possibilities. For example, this arises when simulating a Markov chain, 26 \Delta J. M. Calvin and M. K. Nakayama since successive hits to any fixed state form a regenerative sequence. We explored this to some degree experimentally in Section 8, but further study is needed. ACKNOWLEDGMENTS The authors would like to thank the Editor in Chief and the two anonymous referees for their helpful comments on the paper. A. PROOFS A.1 Proof of Theorem 1 We need only prove that ~ that is, the paths have the same distribution when 2-cycles are permuted. Recall M 2 is the number of times that stopping times in the sequence T 2 occur by time as in (8)-(10). For the path ~ to be the number of times a stopping time from sequence T 1 occurs in ~ to be the number of times that stopping times from the sequence T 1 occur outside of the 2-cycles (we do not need to bother with the case since then we do not change the path). If f is a (measurable) function mapping sample paths to nonnegative real values, then i-m1;n i-m1;n for some (measurable) functions f n;i , and so it suffices to show that ~ is invariant under 2-cycle permutations, where the A i are (measurable) sets. Note that ~ \Theta P (M Given M 2 and L, the initial 1-2 segment (i.e., ~ final 2-1 segment (i.e., ~ are conditionally independent of the 2-cycles, so the last probability can be written \Theta P In examining the effect of permutations of the 2-cycles, we need consider only the Using Permutations in Regenerative Simulations to Reduce Variance \Delta 27 last probability, which we rewrite ~ Since we are interested in the effect of permutations, we only look at the numerator of the last expression: and we are finally left with the task of showing that for any permutation oe, P@ ~ But this follows from the fact that ~ ~ Therefore, ~ , and the theorem is proved. Notice that we only used the conditional exchangeability of the cycles, and not the full independence. A.2 Proof of Theorem 2 Recall that in Section 5 we defined for the original sample path ~ X. Using a permuted path ~ instead of the original path ~ X in (16), we get \Theta Y 0 A 22 (l) +@ X 22 (l)A1 28 \Delta J. M. Calvin and M. K. Nakayama where the Y variables and sets are the same as the Y; H; J; B variables and sets, respectively, with ~ replacing ~ X in (16). Recall that E is the conditional expectation operator corresponding to a random (uniform) permutation of 2-cycles (as was done when constructing the path ~ X 0 from ~ X) given the original sample path ~ X. Also, recall that we define our new estimator to be e which we will now show is equivalent to (17). First note that by our construction of the path ~ X 0 from ~ X, the first term in (28) does not change when replacing ~ X with ~ \Theta Y 0 \Theta Y 12 Now we compute the conditional expectation of the second term of (28). We can assume that h 12 - 3. Let ae(k) be the index of the Y 21 segment that follows the segment after a permutation of the 2-cycles. Note that ae(k) 6= /(k) since Y 12 (k) and Y 21 (/(k)) are always in the same 2-cycle, no matter how the 2-cycles are permuted. Any of the other h indices from H(1; 2) are equally likely, however, so that Y For the second summand in the second term of (28), we have Y 22 (l)7 5 Y 22 (l)7 5 Y 22 (l)7 5 Y 22 (l)7 5 Y Y 22 (l) Using Permutations in Regenerative Simulations to Reduce Variance \Delta 29 Y 22 (l)5 Y Y 22 (l)5 Y 22 (l): (31) For the third summand in the second term of (28), we have Y 22 (l) Y 22 (l) 1 fl2Bk gA3 l6=m Now note that Y 22 (l) 2 Y 22 Also, l6=m Y 22 (l)Y 22 (m) 1 fl2Bk;m2Bk g7 5 l6=m Y 22 (l)Y 22 (m) l6=m Y 22 (l)Y 22 (m) Now use the fact that which follows from Lemma 6. Suppose that p white balls, numbered 1 to p, are placed along with q black balls into p+q boxes arranged in a line, with each box getting exactly one ball. Apply a uniformly chosen random permutation to the balls. Then the probability that ball 1 and ball 2 are not separated by a black ball is 2=(2 To apply this lemma to (32), we let be the number of 2-cycles that include one of the T 1 stopping times, and p be the number of remaining 2-cycles. Proof. Let D be the number of boxes in between the boxes containing ball 1 M. Calvin and M. K. Nakayama and ball 2, and let L i be the box number containing ball i Given that the probability that balls 1 and 2 are not separated by at least black ball is the probability that all q black balls are chosen from the p+q boxes that are not between ball 1 and ball 2, which is Thus the desired probability is Hence, Y 22 (l) l6=m Y 22 (l)Y 22 (m): (33) Using Permutations in Regenerative Simulations to Reduce Variance \Delta 31 Finally, putting together (29), (30), (31), and (33), we get that e X) is as in (17). The unbiasedness and variance reduction follow directly from Theorem 1. A.3 Proof of Theorem 5 We first need the following result. Lemma 7. Suppose that q black balls, r white balls, and p red balls are placed at random in q +p+ r boxes arranged in a line (one ball per box). The probability that there are no white balls in an interval formed by two particular black balls (or the start or end of the boxes) is q=(q Proof. Count boxes from the left until a non-red ball is encountered. The desired probability is the probability that the first non-red ball is a black ball. Since of the non-red balls q are black and r are white, this probability is q=(q Now we prove Theorem 5. First recall our definitions of I 12 (k), I 22 (l), I 21 (k), 22 (l), and D 21 (k) given in Section 6. Also recall (23). The first term in (23) is independent of the permutations, and so we now consider the second term. Note that I 12 (i), I 22 (j), and I 21 (k) are independent for any i; j; k. Then I 12 (k); max I 22 (l); I 21 (ae(k)) I 12 (k); max I 22 (l); I 21 (ae(k)) I 12 (k); max I 22 (l); I 21 (ae(k)) I 22 I 22 where we have used Lemma 7. We now examine the estimation of -. Recall (24) and (25). The first term in (24) is not affected by permuting the 2-cycles, but the second term is, and so we now M. Calvin and M. K. Nakayama examine the second term. First note that since the D 12 (k) are unaffected by permutations of 2-cycles. Also, Y Y Y I 22 Finally, after the permutation of the 2-cycles, define R(k) to be the number of 2- cycles in J(2; 1) that immediately follow the path segment corresponding to D 12 (k), and let be the indices of those 2-cycles in J(2; 1) that immediately follow the path segment corresponding to D 12 (k) in the order they appear. Then Y j!l D 22 (ffi l (k)) Y j!l Using Permutations in Regenerative Simulations to Reduce Variance \Delta 33 Lemma 8. D 22 (ffi l r Proof. Suppose m balls are placed in n - m boxes in a line. Let Z be the number of empty boxes on the left end. Then for and (substitute use the identity To get the mean number of D 22 's that do not hit F , we use the above formula with Finally, putting together Lemma 8 and (34), (35), and (36), we get our new estimator for -. The unbiasedness and variance reduction of our two estimators 34 \Delta J. M. Calvin and M. K. Nakayama follow directly from Theorem 1. B. PSEUDO-CODE B.1 Estimator in Theorem 2 Below is the pseudo-code for the estimator in Theorem 2. (The pseudo-code for estimator in Theorem 5 is similar.) Note that it is specifically for a discrete-event simulation of a continuous-time process. Discrete-event processes can be handled by letting the inter-event time \Delta always be 1. The estimator is denoted by alpha. number of regenerative 1-cycles to simulate; k / 0; // counter for number of regenerative 1-cycles cardinality of H(1; 2) sumy12 / 0; // sum of the Y 12 (k) over H(1; 2) sum of the Y 21 (k) over H(1; 2) sumy22 / 0; // sum of the Y 22 (k) over J(2; 1) sumysq / 0; // sum of the Y sumy12sq / 0; // sum of the Y 12 sumy21sq / 0; // sum of the Y 21 sumy22sq / 0; // sum of the Y 22 sum of the Y 12 (k)Y 21 (/(k)) over H(1; 2) accum / 0; // accumulator for Y over the current segment laststoptime / or 2) of last stopping time to occur generate initial state x; do while (k ! m) generate inter-event time \Delta; accum generate next state x; occurs at current time t) then else lasty21 / accum; endif accum / 0; endif occurs at current time t) then Using Permutations in Regenerative Simulations to Reduce Variance \Delta else endif else sumy22 endif accum / 0; endif alpha / sumysq/m; else endif --R A Guide to Simulation A new variance-reduction technique for regenerative simulations of Markov chains Practical Nonparametric Statistics Simulating stable stochastic systems methods: Another look at the jackknife. A joint central limit theorem for the sample mean and regenerative variance estimator. The asymptotic efficiency of simulation estimators. Monte Carlo Methods. Topics on Regenerative Processes. Regenerative Phenomena. Introduction to the Theory of Statistics Some invariance principles relating to jackknifing and their role in sequential analysis. Approximation Theorems of Mathematical Statistics. Importance sampling for highly reliable Markovian systems. Regenerative Stochastic Simulation. --TR A guide to simulation (2nd ed.) Discrete-time conversion for simulating finite-horizon Markov processes The asymptotic efficiency of simulation estimators A Unified Framework for Simulating Markovian Models of Highly Dependable Systems Importance sampling for the simulation of highly reliable Markovian systems --CTR James M. Calvin , Peter W. Glynn , Marvin K. Nakayama, On the small-sample optimality of multiple-regeneration estimators, Proceedings of the 31st conference on Winter simulation: Simulation---a bridge to the future, p.655-661, December 05-08, 1999, Phoenix, Arizona, United States James M. Calvin , Marvin K. Nakayama, Exploiting multiple regeneration sequences in simulation output analysis, Proceedings of the 30th conference on Winter simulation, p.695-700, December 13-16, 1998, Washington, D.C., United States James M. Calvin , Marvin K. Nakayama, Output analysis: a comparison of output-analysis methods for simulations of processes with multiple regeneration sequences, Proceedings of the 34th conference on Winter simulation: exploring new frontiers, December 08-11, 2002, San Diego, California Michael A. Zazanis, Asymptotic Variance Of Passage Time Estimators In Markov Chains, Probability in the Engineering and Informational Sciences, v.21 n.2, p.217-234, April 2007 James M. Calvin , Marvin K. Nakayama, SIMULATION OF PROCESSES WITH MULTIPLE REGENERATION SEQUENCES, Probability in the Engineering and Informational Sciences, v.14 n.2, p.179-201, April 2000 Wanmo Kang , Perwez Shahabuddin , Ward Whitt, Exploiting regenerative structure to estimate finite time averages via simulation, ACM Transactions on Modeling and Computer Simulation (TOMACS), v.17 n.2, p.8-es, April 2007 James M. Calvin , Marvin K. Nakayama, Improving standardized time series methods by permuting path segments, Proceedings of the 33nd conference on Winter simulation, December 09-12, 2001, Arlington, Virginia James M. Calvin , Peter W. Glynn , Marvin K. Nakayama, The semi-regenerative method of simulation output analysis, ACM Transactions on Modeling and Computer Simulation (TOMACS), v.16 n.3, p.280-315, July 2006

variance reduction;permutations;efficiency improvement;regenerative simulation

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Intelligent Adaptive Information Agents.

Adaptation in open, multi-agent information gathering systems is important for several reasons. These reasons include the inability to accurately predict future problem-solving workloads, future changes in existing information requests, future failures and additions of agents and data supply resources, and other future task environment characteristic changes that require system reorganization. We have developed a multi-agent distributed system infrastructure, RETSINA (REusable Task Structure-based Intelligent Network Agents) that handles adaptation in an open Internet environment. Adaptation occurs both at the individual agent level as well as at the overall agent organization level. The RETSINA system has three types of agents. Interface agents interact with the user receiving user specifications and delivering results. They acquire, model, and utilize user preferences to guide system coordination in support of the users tasks. Task agents help users perform tasks by formulating problem solving plans and carrying out these plans through querying and exchanging information with other software agents. Information agents provide intelligent access to a heterogeneous collection of information sources. In this paper, we concentrate on the adaptive architecture of the information agents. We use as the domain of application WARREN, a multi-agent financial portfolio management system that we have implemented within the RETSINA framework.

Introduction Adaptation is behavior of an agent in response to unexpected (i.e., low probability) events or dynamic environments. Examples of unexpected events include the unscheduled failure of an agent, an agent's computational platform, or underlying information sources. Examples of dynamic environments include the occurrence of events that are expected but it is not known when (e.g., an information agent may reasonably expect to become at some point overloaded with information re- quests), events whose importance fluctuates widely (e.g., price information on a stock is much more important while a trans-action is in progress, and even more so if certain types of news become available), the appearance of new information sources and agents, and finally underlying environmental uncertainty (e.g., not knowing beforehand precisely how long it will take to answer a particular query). We have been involved in designing, building, and analyzing multi-agent systems that exist in these types of dynamic and partially unpredictable environments. These agents handle adaptation at several different levels, from the high-level multi-agent organization down to the monitoring of individual method executions. In the next section we will discuss the individual architecture of these agents. Then, in the section entitled "Agent Adaptation" we will discuss the problems and solutions to agent adaptation at the organizational, planning, scheduling, and execution monitoring levels. In particular, we will discuss how our architecture supports organizational and planning-level adaptation currently and what areas are still under active investigation. We will discuss schedule adaptation only in passing and refer the interested reader to work else- where. Finally, we will present a detailed model and some experiments with one particular behavior, agent self-cloning, for execution-level adaptation. Agent Architecture Most of our work in the information gathering domain to date has been centered on the most basic type of intelligent agent: the information agent, which is tied closely to a single data source. The dominant domain level behaviors of an information agent are: retrieving information from external information sources in response to one shot queries (e.g. "retrieve the current price of IBM stock"); requests for periodic information (e.g. "give me the price of IBM every monitoring external information sources for the occurrence of given information patterns, called monitoring requests, (e.g. "notify me when IBM's price increases by 10% over $80"). Information originates from external sources. Because an information agent does not have control over these external information sources, it must ex- tract, possibly integrate, and store relevant pieces of information in a database local to the agent. The agent's information processing mechanisms then process the information in the local database to service information requests received from other agents or human users. Other simple behaviors that are used by all information agents include advertising their capabilities, managing and rebuilding the local database when necessary, and polling for KQML messages from other agents. An information agent's reusable behaviors are facilitated by its reusable agent architecture, i.e. the domain-independent abstraction of the local database schema, and a set of generic software components for knowledge representation, agent control, and interaction with other agents. The generic software components are common to all agents, from the simple information agents to more complex multi-source information agents, task agents, and interface agents. The design of useful basic agent behaviors for all types of agents rests on a deeper specification of agents themselves, and is embodied in an agent architecture. Our current agent architecture is an instantiation of the DECAF (Distributed, Environment-Centered Agent Framework) architecture (Decker et al. 1995). Control: Planning, Scheduling, and Action Execution The control process for information agents includes steps for planning to achieve local or non-local objectives, scheduling the actions within these plans, and actually carrying out these actions. In addition, the agent has a shutdown and an initialization process. The agent executes the initialization process upon startup; it bootstraps the agent by giving it initial objectives to poll for messages from other agents and to advertise its capabilities. The shutdown process is executed when the agent either chooses to terminate or receives an uncontinueable error signal. The shutdown process assures that messages are sent from the terminating agent asserting goal dissolution to client agents and requesting goal dissolution to server agents (see the section on planning adaptation). The agent planning process (see Figure 1) takes as input the agent's current set of goals G (including any new, unplanned- for goals G n ), and the set of current task structures (plan in- stances) T . It produces a new set of current task structures (Williamson, Decker, & Sycara 1996). ffl Each individual task T represents an instantiated approach to achieving one or more of the agent's goals G-it is a unit of goal-directed behavior. Every task has an (optional) deadline. ffl Each task consists of a partially ordered set of subtasks and/or basic actions A. Currently, tasks and actions are related by how information flows from the outcomes of one task or action to the provisions of anther task or action. Sub-tasks may inherit provisions from their parents and provide outcomes to their parents. Each action also has an optional deadline and an optional period. If an action has both a period and a deadline, the deadline is interpreted as the one for the next periodic execution of the basic action. The most important constraint that the planning/plan retrieval algorithm needs to meet (as part of the agent's overall properties) is to guarantee at least one task for every goal until the goal is accomplished, removed, or believed to be unachievable (Cohen & Levesque 1990). For information agents, a common reason that a goal in unachievable is that its specification is malformed, in which case a task to respond with the appropriate KQML error message is instantiated. An information agent receives in messages from other agents three important types of goals: 1. Answering a one-shot query about the associated database. 2. Setting up a periodic query on the database, that will be run repeatedly, and the results sent to the requester each time (e.g., "tell me the price of IBM every 3. Monitoring a database for a change in a record, or the addition of a new record (e.g., "tell me if the price of IBM drops below $80 within 15 minutes of its occurrence"). The agent scheduling process in general takes as input the agent's current set of task structures T , in particular, the set of all basic actions, and decides which basic action, if any, is to be executed next. This action is then identified as a fixed intention until it is actually carried out (by the execution com- ponent). Constraints on the scheduler include: ffl No action can be intended unless it is enabled. ffl Periodic actions must be executed at least once during their period (as measured from the previous execution instance) (technically, this is a max invocation separation constraint, not a "period"). ffl Actions must begin execution before their deadline. ffl Actions that miss either their period or deadline are considered to have failed; the scheduler must report all failed actions. Sophisticated schedulers will report such failures (or probable failures) before they occur by reasoning about action durations (and possibly commitments from other agents) (Garvey & Lesser 1995). ffl The scheduler attempts to maximize some predefined utility function defined on the set of task structures. For the information agents, we use a very simple notion of utility- every action needs to be executed in order to achieve a task, and every task has an equal utility value. In our initial implementation, we use a simple earliest- deadline-first scheduling heuristic. A list of all actions is constructed (the schedule), and the earliest deadline action that is enabled is chosen. Enabled actions that have missed their deadlines are still executed but the missed deadline is recorded and the start of the next period for the task is adjusted to help it meet the next period deadline. When a periodic task is chosen for execution, it is reinserted into the schedule with a deadline equal to the current time plus the action's period. The execution monitoring process takes the agent's next intended action and prepares, monitors, and completes its exe- cution. The execution monitor prepares an action for execution by setting up a context (including the results of previous actions, etc.) for the action. It monitors the action by optionally providing the associated computation-limited resources- for example, the action may be allowed only a certain amount of time and if the action does not complete before that time is up, the computation is interrupted and the action is marked as having failed. Upon completion of an action, results are recorded, downstream actions are passed provisions if so indi- cated, and runtime statistics are collected. Planner Scheduler Execution Monitor Task Structures Plan Library query task montr task montr task run-query run-query send-results register-trigger register-trigger Schedule I.G. task poll-for-msgs Current Action run-query Control Flow Data Flow (ask-all .) (DB-monitor .) (DB-monitor .) site specific external interface code Mirror of External DB extra attributes Registered triggers Goals/Requests Current Activity Information Figure 1: Overall view of data and control flow in an information agent. Agent Adaptation In this section we briefly consider several types of adaptation supported by this individual agent architecture in our current and previous work. These types include organizational, planning, scheduling, and execution-time adaptation. We are currently actively involved in expanding an agent's adaptation choices at the organizational and planning levels-in this short paper we will only describe how our architecture supports organizational and planning-level adaptation, what we have currently implemented, and what directions we are currently pursuing. We have not, in our current work, done much with schedule adaptation; instead we indicate future potential by pointing to earlier work within this general architecture that addresses precisely schedule adaptation. Fi- nally, we present a fairly comprehensive account of one type of execution-time adaptation ("self-cloning"). Organizational Adaptation It has been clear to organizational theorists since at least the 60's that there is no one good organizational structure for human organizations (Lawrence & Lorsch 1967). Organizations must instead be chosen and adapted to the task environment at hand. Most important are the different types and qualities of uncertainty present in the environment (e.g., uncertainty associated with inputs and output measurements, uncertainty associated with causal relationships in the environment, the time span of definitive feedback after making a decision (Scott 1987)). Recently, researchers have proposed that organizations grow toward, and structure themselves around, sources of information that are important to them because they are sources of news about how the future is (evidently) turning out (Stinchcombe 1990). In multi-agent information systems, one of the most important sources of uncertainty revolves around what information is available from whom (and at what cost). We have developed a standard basic advertising behavior that allows agents to encapsulate a model of their capabilities and send it to a "matchmaker" information agent (Kuokka & Harada 1995). Such a matchmaker agent can then be used by a multi-agent system to form several different organizational struc- tures(Decker, Williamson, & Sycara 1996): Uncoordinated Team: agents use a basic shared behavior for asking questions that first queries the matchmaker as to who might answer the query, and then chooses an agent randomly for the target query. Very low overhead, but potentially unbalanced loads, reliability limited by individual data sources, and problems linking queries across multiple ontologies. Our initial implementation used this organization exclusively. Federations: (e.g., (Wiederhold, Wegner, & Cefi 1992; Genesereth & Katchpel 1994; Finin et al. 1994)) agents give up individual autonomy over choosing who they will do business with to a locally centralized "facilitator" (an extension of the matchmaker concept) that "brokers" re- quests. Centralization of message traffic potentially allows greater load balancing and the provision of automatic translation and mediation services. We have constructed general purpose brokering agents, and are currently conducting an empirical study of matchmaking vs. brokering behavior. Of course, a hybrid organization is both possible and compelling in many situations. Economic Markets: (e.g., (Wellman 1993)) agents use price, reliability, and other utility characteristics with which to choose another agent. The matchmaker can supply to each agent the appropriate updated pricing information as new agents enter and exit the system, or alter their advertise- ments. Agents can dynamically adjust their organization as often as necessary, limited by transaction costs. Potentially such organizations provide efficient load balancing and the ability to provide truly expensive services (expen- sive in terms of the resources required). Both brokers and matchmakers can be used in market-based systems (corre- sponding to centralized and decentralized markets, respec- tively). Bureaucratic Functional Units: Traditional manager/employee groups of a single multi-source information agent (manager) and several simple information agent (employees). By organizing into functional units, i.e., related information sources, such organizations concentrate on providing higher reliability (by using multiple underlying sources), simple information integration (from partially overlapping information), and load balancing. "Manag- ing" can be viewed as brokering with special constraints on worker behavior brought about by the manager-worker authority relationship. This is not an exhaustive list. Our architecture has supported other explorations into understanding the effects of organizational structures (Decker 1996). Planning Adaptation The "planner" portion of our agent architecture consists of a new hierarchical task network based planner using a plan formalism that admits sophisticated control structures such as looping and periodic tasks (Williamson, Decker, & Sycara 1996). It has features derived from earlier classical planning work, as well as task structure representations such as TCA/TCX (Simmons 1994) and T-MS (Decker & Lesser 1995). The focus of planning in our system is on explicating the basic information flow relationships between tasks, and other relationships that affect control-flow decisions. Most control relationships are derivative of these more basic rela- tionships. Final action selection, sequencing, and timing are left up to the agent's local scheduler (see the next subsection). Some types of adaptation expressed by our agents at this level in our current implementation include: Adapting to failures: At any time, any agent in the system might be unavailable or might go off-line (even if you are in the middle of a long term monitoring situation with that agent). Our planner's task reductions handle these situations so that such failures are dealt with smoothly. If alternate agents are available, they will be contacted and the subproblem restarted (note that unless there are some sort of partial solutions, this could still be expensive). If no alternate agent is available, the task will have to wait. In the future, such failures will signal the planner for an opportunity to replan. Multiple reductions: Each task can potentially be reduced in several different ways, depending on the current situation. Thus even simple tasks such as answering a query may be result in very different sequences of actions (looking up an agent at the matchmaker; using a already known agent, using a cached previous answer). Interleaved planning and execution: The reduction of some tasks can be delayed until other, "information gathering" tasks, are completed. Previous work has focussed on coordination mechanisms alone. In particular, the Generalized Partial Global Planning family of coordination mechanisms is a domain-independent approach to multi-agent scheduling- and planning-level coordination that works in conjunction with an agent's existing local scheduler to adapt a plan by adding certain constraints (Decker & Lesser 1995). These include commitments to do a task with a minimum level of quality, or commitments to do a task by a certain deadline. If the resulting plan can be successfully scheduled, these local commitments can be communicated to other agents where they become non-local commitments to those agent's local schedulers. Not all mechanisms are needed in all environments. Nagendra-Prasad has begun work on learning which mechanisms are needed in an environment automatically (Prasad & Lesser 1996). Scheduling Adaptation In our current work, we have been using a fairly simple earliest deadline first scheduler that does little adaptation besides adjusting the deadlines of periodic (technically "max invocation separation constrained") actions that miss or are about to miss their initial deadlines. Also, agents can dynamically change their information request periods which affect only the scheduling of the related actions. Earlier work within this architecture has used a more sophisticated "Design-to-Time" scheduling algorithm, which adapts the local schedule in an attempt to maximize schedule quality while minimizing missed deadlines (Garvey & Lesser 1995; Decker & Lesser 1995). In doing so, the scheduler may choose from both "multiple methods" (different algorithms that represent difference action duration/result quality tradeoffs) and anytime algorithms (summarized by du- ration/quality probability distribution tables (Zilberstein & Russell 1992)). Execution Adaptation Within this architecture, previous execution-time adaptation has focussed on monitoring actions (Garvey & Lesser 1995). Recently, we have begun looking at load- balancing/rebalancing behaviors such as agent cloning. Cloning Cloning is one of an information agent's possible responses to overloaded conditions. When an information agent recognizes via self-reflection that it is becoming over- loaded, it can remove itself from actively pursuing new queries ("unadvertising" its services in KQML) and create a new information agent that is a clone of itself. To do this, it uses a simple model of how it's ability to meet new deadlines is related to the characteristics of it's current queries and other tasks. It compares this model to a hypothetical situation that describes the effect of adding a new agent. In this way, the information agent can make a rational meta-control decision about whether or not it should undertake a cloning behavior. This self-reflection phase is a part of the agent's execution monitoring process. The start and finish time of each action is recorded as well as a running average duration for that action class. A periodic task is created to carry out the calculations required by the model described below. The key to modeling the agent's load behavior is its current task structures. Since one-shot queries are transient, and simple repeated queries are just a subcase of database monitoring queries, we focus on database monitoring queries only. Each monitoring goal is met by a task that consists of three activities; run-query, check-triggers, and send-results. Run- query's duration is mostly that of the external query interface function. Check-triggers, which is executed whenever the local DB is updated and which thus is an activity shared by all database monitoring tasks, takes time proportional to the number of queries. Send-results takes time proportional to the number of returned results. Predicting performance of an information agent with n database monitoring queries would thus involve a quadratic function, but we can make a simplification by observing that the external query interface functions in all of the information agents we have implemented so far using the Internet (e.g., stock tickers, news, airfares) take an order of magnitude more time than any other part of the system (including measured planning and scheduling overhead). If we let E be the average time to process an external query, then with n queries of average period p, we can predict an idle percentage of: (1) We validate this model in the next section. When an information agent gets cloned, the clone could be set up to use the resources of another processor (via an 'agent server', or a migratable Java or Telescript program). However, in the case of information agents that already spend the majority of their processing time in network I/O wait states, an overhead proportion O ! 1 of the En time units each period are available for processing. 1 Thus, as a single agent becomes overloaded as it reaches p=E queries, a new agent can be cloned on the same system to handle another queries. When the second agent runs on a separate processor, 1. This can continue, with the i th agent on the same processor handling queries (note the diminishing returns). We also demonstrate this experimentally in the next section. For two agents, the idle percentage should then follow the model I 1+2 (2) It is important to note how our architecture supports this type of introspection and on-the-fly agent creation. The execution monitoring component of the architecture computes and stores timing information about each agent action, so that the agent learns a good estimate for the value of E. The sched- uler, even the simple earliest-deadline-first scheduler, knows the actions and their periods, and so can compute the idle percentage I%. In the systems we have been building, new queries arrive slowly and periods are fairly long, in comparison to E, so the cloning rule waits until there are queries before cloning. In a faster environment, with new queries arriving at a rate r and with cloning taking duration C, the cloning behavior should be begun when the number of queries reaches Execution Adaptation: Experimental Results We undertook an empirical study to measure the baseline performance of our information agents, and to empirically verify the load models presented in the previous section for both a single information agent without the cloning behavior, and an information agent that can clone onto the same processor. We also wanted to verify our work in the context of a real application (monitoring stock prices). Our first set of experiments were oriented toward the measurement of the baseline performance of an information agent. Figure 2 shows the average idle percentage, and the average percentage of actions that had deadlines and that missed Another way to recoup this time is to run the blocking external query in a separate process, breaking run-query into two parts. We are currently comparing the overhead of these two different uni-processor solutions-in any case we stress that both behaviors are reusable and can be used by any existing information agent without reprogramming. Cloning to another processor still has the desired effect. them, for various task loads. The query period was fixed at seconds, and the external query time fixed at 10 seconds (but nothing else within the agent was fixed). Each experiment was run for 10 minutes and repeated 5 times. As expected, the idle time decreases and the number of missed deadlines increases, especially after the predicted saturation point 6). The graph also shows the average amount of time by which an action its deadline. The next step was to verify our model of single information agent loading behavior (Equation 1). We first used a partially simulated information agent to minimize variation factors external to the information agent architecture. Later, we used a completely real agent with a real external query interface (the Security APL stock ticker agent). On the left of Figure 3 is a graph of the actual and predicted idle times for an information agent that monitors a simulated external information source that takes a constant 10 seconds. 2 The information agent being examined was given tasks by a second experiment-driver agent. Each experiment consisted of a sequence of tasks (n) given to the information agent at the start. Each task had a period of 60 seconds, and each complete experiment was repeated 5 times. Each experiment lasted 10 minutes. The figure clearly shows how the agent reaches saturation after the 6th task as predicted by the model 6). The idle time never quite drops below 10% because the first minute is spent idling between startup activities (e.g., making the initial connection and sending the batch of tasks). After adding in this extra base idle time, our model predicts the actual utilization quite well (R R 2 is a measure of the total variance explained by the model). We also ran this set of experiments using a real external in- terface, that of the Security APL stock ticker. The results are shown graphically on the right in Figure 3. 5 experiments were again run with a period of 60 seconds (much faster than normal tasks. Our utilization model also correctly predicted the performance of this real sys- tem, with R and the differences between the model and the experimental results were not significant by either t-tests or non-parametric signed-rank tests. The odd utilization results that occurred while testing were caused by network delays that significantly changed the average value of E (the duration of the external query). However, since the agent's execution monitor measures this value during problem solving, the agent can still react appropriately (the model still fits fine). Finally, we extended our model to predict the utilization for a system of agents with the cloning behavior, as indicated in the previous section. Figure 4 shows the predicted and actual results over loads of 1 to 10 tasks with periods of 60 seconds, repetitions. Agent 1 clones itself onto the same processor when n ? 5. In this case, model R 2 All the experiments described here were done on a standard timesharing Unix workstation while connected to the network. . MD % - MDAmount Number of Periodic Queries Percentages Average Missed Deadline Amount Figure 2: A graph of the average percentage idle time and average percentage of actions with deadlines that missed them for various loads (left Y axis). Superimposed on the graph, and keyed to the right axis, are the average number of seconds by which a missed deadline is missed.0.10.30.50.70.91 Number of Periodic Queries Percentage Idle Time Predicted Actual Information Percentage Idle Time Number of Periodic Queries Predicted Actual Information Agent with 10s simulated External Interface Figure 3: On the left, graph of predicted and actual utilization for a real information agent with a simulated external query interface. On the right, the same graph for the Security APL stock ticker agent. the differences between the model and the measured values are not significant by t-test or signed-ranks. The same graph shows the predicted curve for one agent (from the left side of Figure as a comparison. 3 Number of Periodic Queries Percentage Idle agent Predicted, with cloning (2 agents) Actual, with cloning0.20.40.60.81 Figure 4: Predicted idle percentages for a single non cloning agent, and an agent with the cloning behavior across various task loads. Plotted points are the measured idle percentages from experimental data including cloning agents. Current & Future Work This paper has discussed adaptation in a system of intelligent agents at four different levels: organizational, planning, scheduling, and execution. Work at the organizational and planning levels is a current, active pursuit; we expect to return to schedule adaptation as time and resources permit. Cur- rently, we are conducting an empirical study into matchmak- ers, brokers, and related hybrid organizations. This paper also discussed a fairly detailed model of, and experimentation with, a simple cloning behavior we have im- plemented. Several extensions to this cloning model are being considered. In particular, there are several more intelligent ways with which to divide up the tasks when cloning occurs in order to use resources more efficiently (and to keep queries balanced after a cloning event occurs). These include: ffl Partitioning existing tasks by time/periodicity, so that the resulting agents have a balanced, schedulable set of tasks. ffl Partitioning tasks by client: all tasks from agent 1 end up at the same clone. 3 Since the potential second agent would, if it existed, be totally idle from 1 the idle curve differs there in the cloning case. ffl Partitioning tasks by class/type/content: all tasks about one subject (e.g., the stock price of IBM) end up at the same clone. ffl For multi-source information agents, partitioning tasks by data source: all tasks requiring the use of source A end up at the same clone. Acknowledgements The authors would like to thank the reviewers for their helpful comments. This work has been supported in part by ARPA contract F33615-93-1-1330, in part by ONR contract N00014-95-1-1092, and in part by NSF contract IRI- 9508191. --R Intention is choice with commitment. Designing a family of coordination algorithms. MACRON: an architecture for multi-agent cooperative information gathering Modeling information agents: Advertisements Task environment centered simulation. In Prietula KQML as an agent communication language. Representing and scheduling satisficing tasks. Software agents. Communications of the ACM On using KQML for matchmaking. Organization and Envi- ronment In AAAI Spring Symposium on Adaptation Organizations: Rational Structured control for autonomous robots. Information and Organizations. University of California Press. A market-oriented programming environment and its application to distributed multicommodity flow problems Toward megaprogramming. Unified information and control flow in hierarchical task networks. Constructing utility- driven real-time systems using anytime algorithms --TR --CTR Claudia V. Goldman , Jeffrey S. Rosenschein, Partitioned multiagent systems in information oriented domains, Proceedings of the third annual conference on Autonomous Agents, p.32-39, April 1999, Seattle, Washington, United States Qiuming Zhu , Stuart L. Aldridge , Tomas N. Resha, Hierarchical Collective Agent Network (HCAN) for efficient fusion and management of multiple networked sensors, Information Fusion, v.8 n.3, p.266-280, July, 2007 Marian (Misty) Nodine , Anne Hee Hiong Ngu , Anthony Cassandra , William G. Bohrer, Scalable Semantic Brokering over Dynamic Heterogeneous Data Sources in InfoSleuth", IEEE Transactions on Knowledge and Data Engineering, v.15 n.5, p.1082-1098, September Terrence Harvey , Keith Decker , Sandra Carberry, Multi-agent decision support via user-modeling, Proceedings of the fourth international joint conference on Autonomous agents and multiagent systems, July 25-29, 2005, The Netherlands K. S. Barber , C. E. Martin, Dynamic reorganization of decision-making groups, Proceedings of the fifth international conference on Autonomous agents, p.513-520, May 2001, Montreal, Quebec, Canada Woodrow Barfield, Issues of law for software agents within virtual environments, Presence: Teleoperators and Virtual Environments, v.14 n.6, p.741-748, December 2005 Rey-Long Liu, Collaborative Multiagent Adaptation for Business Environmental Scanning Through the Internet, Applied Intelligence, v.20 n.2, p.119-133, March-April 2004 Keith Decker , Xiaojing Zheng , Carl Schmidt, A multi-agent system for automated genomic annotation, Proceedings of the fifth international conference on Autonomous agents, p.433-440, May 2001, Montreal, Quebec, Canada Alessandro de Luna Almeida , Samir Aknine , Jean-Pierre Briot , Jacques Malenfant, A Predictive Method for Providing Fault Tolerance in Multi-agent Systems, Proceedings of the IEEE/WIC/ACM international conference on Intelligent Agent Technology, p.226-232, December 18-22, 2006 Keith Decker , Jinjiang Li, Coordinating Mutually Exclusive Resources using GPGP, Autonomous Agents and Multi-Agent Systems, v.3 n.2, p.133-157, June 2000 Fabien Gandon, Agents handling annotation distribution in a corporate semantic Web, Web Intelligence and Agent System, v.1 n.1, p.23-45, January K. S. Barber , C. E. Martin, Flexible problem-solving roles for autonomous agents, Integrated Computer-Aided Engineering, v.8 n.1, p.1-15, January 2001 Fabien Gandon, Agents handling annotation distribution in a corporate semantic web, Web Intelligence and Agent System, v.1 n.1, p.23-45, January Rey-Long Liu , Wan-Jung Lin, Incremental mining of information interest for personalized web scanning, Information Systems, v.30 n.8, p.630-648, December 2005 V. Lesser , K. Decker , T. Wagner , N. Carver , A. Garvey , B. Horling , D. Neiman , R. Podorozhny , M. Nagendra Prasad , A. Raja , R. Vincent , P. Xuan , X. Q. Zhang, Evolution of the GPGP/TMS Domain-Independent Coordination Framework, Autonomous Agents and Multi-Agent Systems, v.9 n.1-2, p.87-143, July-September 2004 David Camacho , Ricardo Aler , Daniel Borrajo , Jos M. Molina, A Multi-Agent architecture for intelligent gathering systems, AI Communications, v.18 n.1, p.15-32, January 2005 David Camacho , Ricardo Aler , Daniel Borrajo , Jos M. Molina, A multi-agent architecture for intelligent gathering systems, AI Communications, v.18 n.1, p.15-32, January 2005 Victor R. Lesser, Reflections on the Nature of Multi-Agent Coordination and Its Implications for an Agent Architecture, Autonomous Agents and Multi-Agent Systems, v.1 n.1, p.89-111, 1998 Larry Kerschberg , Doyle J. Weishar, Conceptual Models and Architectures for Advanced Information Systems, Applied Intelligence, v.13 n.2, p.149-164, September-October 2000

intelligent agents;Distributed AI;agent architectures;Multi-Agent Systems;information gathering

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3-D Scene Data Recovery Using Omnidirectional Multibaseline Stereo.

A traditional approach to extracting geometric information from a large scene is to compute multiple 3-D depth maps from stereo pairs or direct range finders, and then to merge the 3-D data. However, the resulting merged depth maps may be subject to merging errors if the relative poses between depth maps are not known exactly. In addition, the 3-D data may also have to be resampled before merging, which adds additional complexity and potential sources of errors.This paper provides a means of directly extracting 3-D data covering a very wide field of view, thus by-passing the need for numerous depth map merging. In our work, cylindrical images are first composited from sequences of images taken while the camera is rotated 360 about a vertical axis. By taking such image panoramas at different camera locations, we can recover 3-D data of the scene using a set of simple techniques: feature tracking, an 8-point structure from motion algorithm, and multibaseline stereo. We also investigate the effect of median filtering on the recovered 3-D point distributions, and show the results of our approach applied to both synthetic and real scenes.

Introduction A traditional approach to extracting geometric information from a large scene is to compute multiple (possibly numerous) 3-D depth maps from stereo pairs, and then to merge the 3-D data [Ferrie and Levine, 1987; Higuchi et al., 1993; Parvin and Medioni, 1992; Shum et al., 1994]. This is not only computationally intensive, but the resulting merged depth maps may be subject to merging er- rors, especially if the relative poses between depth maps are not known exactly. The 3-D data may also have to be resampled before merging, which adds additional complexity and potential sources of errors. This paper provides a means of directly extracting 3-D data covering a very wide field of view, thus by-passing the need for numerous depth map merging. In our work, cylindrical images are first composited from sequences of images taken while the camera is rotated 360 ffi about a vertical axis. By taking such image panoramas at different camera locations, we can recover 3-D data of the scene using a set of simple techniques: feature tracking, 8-point direct and iterative structure from motion algorithms, and multibaseline stereo. There are several advantages to this approach. First, the cylindrical image mosaics can be built quite accurately, since the camera motion is very restricted. Second, the relative pose of the various camera locations can be determined with much greater accuracy than with regular structure from motion applied to images with narrower fields of view. Third, there is no need to build or purchase a specialized stereo camera whose calibration may be sensitive to drift over time-any conventional video camera on a tripod will suffice. Our approach can be used to construct models of building interiors, both for virtual reality applications (games, home sales, architectural remodeling), and for robotics applications (navigation). In this paper, we describe our approach to generate 3-D data corresponding to a very wide field of view (specifically 360 ffi ), and show results of our approach on both synthetic and real scenes. We first review relevant work in Section 2 before delineating our basic approach in Section 3. The method to extract wide-angle images (i.e., panoramic images) is described in Section 4. Section 5 reviews the 8-point algorithm and shows how it can be applied for cylindrical panoramic images. Section 6 describes two methods of extracting 3-D point data: the first relies on unconstrained tracking and using 8-point data input, while the second constrains the search for feature correspondences to epipolar lines. We briefly outline our approach in modeling the data in Section 7-details of this is given elsewhere [Kang et al., 1995a]. Finally, we show results of our approach in Section 8 and close with a discussion and conclusions. Relevant work There is a significant body of work on range image recovery using stereo (a comprehensive survey is given in [Barnard and Fischler, 1982]). Most work on stereo uses images with limited fields of view. One of the earliest work to use panoramic images is the omnidirectional stereo system of Ishigura [Ishigura et al., 1992], which uses two panoramic views. Each panoramic view is created by one of the two vertical slits of the camera image sweeping around 360 ffi ; the cameras (which are displaced in front of the rotation center) are rotated by very small angles, typically about 0.4 ffi . One of the disadvantages of this method is the slow data accumulation, which takes about 10 mins. The camera angular increments must be approximately 1/f radians, and are assumed to be known a priori. Murray [Murray, 1995] generalizes Ishigura et al.'s approach by using all the vertical slits of the image (except in the paper, he uses a single image raster). This would be equivalent to structure from known motion or motion stereo. The advantage is more efficient data acquisition, done at lower angular resolution. The analysis involved in this work is similar to Bolles et al.'s [Bolles et al., 1987] spatio-temporal epipolar analysis, except that the temporal dimension is replaced by that of angular displacement. Another related work is that of plenoptic modeling [McMillan and Bishop, 1995]. The idea is to composite rotated camera views into panoramas, and based on two cylindrical panoramas, project disparity values between these locations to a given viewing position. However, there is no explicit 3-D reconstruction. Our approach is similar to that of [McMillan and Bishop, 1995] in that we composite rotated camera views to panoramas as well. However, we are going a step further in reconstructing 3-D feature points and modeling the scene based upon the recovered points. We use multiple panoramas for more accurate 3-D reconstruction. 3 Overview of approach 3 Omnidirectional multibaseline stereo Recovered 3-D scene points Modeled scene Figure 1: Generating scene model from multiple 360 ffi panoramic views. 3 Overview of approach Our ultimate goal is to generate a photorealistic model to be used in a variety of scenarios. We are interested in providing a simple means of generating such models. We also wish to minimize the use of CAD packages as a means of 3-D model generation, since such an effort is labor-intensive. In addition, we impose the requirement that the means of generating models from real scene be done using commercially available equipment. In our case, we use a workstation with framegrabber (real-time image digitizer) and a commercially available 8-mm camcorder. Our approach is straightforward: at each camera location in the scene, capture sequences of images while rotating the camera about the vertical axis passing through the camera optical center. Composite each set of images to produce panoramas at each camera location. Use stereo to extract 3-D data of the scene. Finally, model the scene using these 3-D data input and render it with the texture provided by the input 2-D image. This approach is summarized in Figure 1. By using panoramic images, we can extract 3-D data covering a very wide field of view, thus by-passing the need for numerous depth map merging. Multiple depth map merging is not only computationally intensive, but the resulting merged depth maps may be subject to merging errors, especially if the relative poses between depth maps are not known exactly. The 3-D data may also have to be resampled before merging, which adds additional complexity and potential sources of s Figure 2: Compositing multiple rotated camera views into a panorama. The '\Theta' marks indicate the locations of the camera optical and rotation center. errors. Using multiple camera locations in stereo analysis significantly reduces the number of ambiguous matches and also has the effect of reducing errors by averaging [Okutomi and Kanade, 1993; Kang et al., 1995b]. This is especially important for images with very wide fields of view, because depth recovery is unreliable near the epipoles 1 , where the looming effect takes place, resulting in very poor depth cues. 4 Extraction of panoramic images A panoramic image is created by compositing a series of rotated camera image images, as shown in Figure 2. In order to create this panoramic image, we first have to ensure that the camera is rotating about an axis passing through its optical center, i.e., we must eliminate motion parallax when panning the camera around. To achieve this, we manually adjust the position of camera relative to an X-Y precision stage (mounted on the tripod) such that the motion parallax effect disappears when the camera is rotated back and forth about the vertical axis [Stein, 1995]. Prior to image capture of the scene, we calibrate the camera to compute its intrinsic camera parameters (specifically its focal length f , aspect ratio r, and radial distortion coefficient -). The camera is calibrated by taking multiple snapshots of a planar dot pattern grid with known depth separation between successive snapshots. We use an iterative least-squares algorithm (Levenberg- 1 For a pair of images taken at two different locations, the epipoles are the location on the image planes which are the intersection between these image planes and the line joining the two camera optical centers. An excellent description of the stereo vision is given in [Faugeras, 1993]. Image 1 Image 2 Image (N-1) Image N Figure 3: Example undistorted image sequence (of an office). Marquardt) to estimate camera intrinsic and extrinsic parameters (except for -) [Szeliski and Kang, 1994]. - is determined using 1-D search (Brent's parabolic interpolation in 1-D [Press et al., 1992]) with the least-squares algorithm as the black box. The steps involved in extracting a panoramic scene are as follow: ffl At each camera location, capture sequence while panning camera around 360 ffi . ffl Using the intrinsic camera parameters, correct the image sequence for r, the aspect ratio, and -, the radial distortion coefficient. ffl Convert the (r; -corrected 2-D flat image sequence to cylindrical coordinates, with the focal length f as its cross-sectional radius. An example of a sequence of corrected images (of an office) is shown in Figure 3. ffl Composite the images (with only x-directional DOF, which is equivalent to motion in the angular dimension of cylindrical image space) to yield the desired panorama [Szeliski, 1994]. The relative displacement of one frame to the next is coarsely determined by using phase correlation [Kuglin and Hines, 1975]. This technique estimates the 2-D translation between a pair of images by taking 2-D Fourier transforms of both images, computing the phase difference at each frequency, performing an inverse Fourier transform, and searching for a peak in the magnitude image. Subsequently, the image translation is refined using local image registration by directly comparing the overlapped regions between the two images [Szeliski, 1994]. ffl Correct for slight errors in the resulting length (which in theory equals 2-f) by propagating residual displacement error equally across all images and recompositing. The error in length is usually within a percent of the expected length. 6 5 Recovery of epipolar geometry Figure 4: Panorama of office scene after compositing. An example of a panoramic image created from the office scene in Figure 3 is shown in Figure 4. 5 Recovery of epipolar geometry In order to extract 3-D data from a given set of panoramic images, we have to first know the relative positions of the camera corresponding to the panoramic images. For a calibrated camera, this is equivalent to determining the epipolar geometry between a reference panoramic image and every other panoramic image. The epipolar geometry dictates the epipolar constraint, which refers to the locus of possible image projections in one image given an image point in another image. For planar image planes, the epipolar constraint is in the form of straight lines. The interested reader is referred to [Faugeras, 1993] for details. We use the 8-point algorithm [Longuet-Higgins, 1981; Hartley, 1995] to extract what is called the essential matrix, which yields both the relative camera placement and epipolar geometry. This is done pairwise, namely between a reference panoramic image and another panoramic image. There are, however, four possible solutions [Hartley, 1995]. The solution that yields the most positive projections (i.e., projections away from the camera optical centers) is chosen. 5.1 8-point algorithm: Basics We briefly review the 8-point algorithm here: If the camera is calibrated (i.e., its intrinsic parameters are known), then for any two corresponding image points (at two different camera placements) in 3-D, we have 5.1 8-point algorithm: Basics 7 The matrix E is called the essential matrix, and is of the form R, where R and t are the rotation matrix and translation vectors, respectively, and [t] \Theta is the matrix form of the cross product with t. If the camera is not calibrated, we have a more general relation between two corresponding image points (on the image plane) (u; v; 1) T and namely F is called the fundamental matrix and is also of rank 2, A, where A is an arbitrary 3 \Theta 3 matrix. The fundamental matrix is the generalization of the essential matrix E, and is usually employed to establish the epipolar geometry and to recover projective depth [Faugeras, 1992; Shashua, 1994]. In our case, since we know the camera parameters, we can recover E. Let e be the vector comprising is the (i,j)th element of E. Then for all the point matches, we have from (1) from which we get a set of linear equations of the form If the number of input points is small, the output of algorithm is sensitive to noise. On the other hand, it turns out that normalizing the 3-D point location vector on the cylindrical image reduces sensitivity of the 8-point algorithm to noise. This is similar in spirit to Hartley's application of isotropic scaling [Hartley, 1995] prior to using the 8-point algorithm. The 3-D cylindrical points are normalized according to the relation With N panoramic images, we solve for sets of linear equations of the form (4). The kth set corresponds to the panoramic image pair 1 and 1). Notice that the solution of e is defined only up to an unknown scale. In our work, we measure the distance between camera positions; this enable us to recover the scale. However, we can relax this assumption by carrying out the following steps: Recovery of epipolar geometry ffl Fix camera distance of first pair (pair 1), to, say unit distance. Assign camera distances for all the other pairs to be the same as the first. ffl Calculate the essential matrices for all the pairs of panoramic images, assuming unit camera distances. ffl For each pair, compute the 3-D points. ffl To estimate the relative distances between of camera positions for pair j 6= 1 (i.e., not the first pair), find the scale of the 3-D points corresponding to pair j that minimizes the distance error to those corresponding to pair 1. Robust statistics is used to reject outliers; specifically, only the best 50% are used. 5.2 Tracking features for 8-point algorithm The 8-point algorithm assumes that feature point correspondences are available. Feature tracking is a challenge in that purely local tracking fails because the displacement can be large (of the order of about 100 pixels, in the direction of camera motion). The approach that we have adopted comprises spline-based tracking, which attempts to globally minimize the image intensity differences. This yields estimates of optic flow, which in turn is used by a local tracker to refine the amount of feature displacement. The optic flow between a pair of cylindrical panoramic images is first estimated using spline-based image registration between the pair [Szeliski and Coughlan, 1994; Szeliski et al., 1995]. In this image registration approach, the displacement fields u(x; y) and v(x; y) (i.e., displacements in the x- and y- directions as functions of the pixel location) are represented as two-dimensional splines controlled by a smaller number of displacement estimates which lie on a coarser spline control grid. Once the initial optic flow has been found, the best candidates for tracking are then chosen. The choice is based on the minimum eigenvalue of the local Hessian, which is an indication of local image texturedness. Subsequently, using the initial optic flow as an estimate displacement field, we use the Shi-Tomasi tracker [Shi and Tomasi, 1994] with a window of size 25 pixels \Theta 25 pixels to further refine the displacements of the chosen point features. Why did we use the approach of applying the spline-based tracker before using the Shi-Tomasi tracker? This approach is used to take advantage of the complementary characteristics of these two trackers, namely: 1. the spline-based image registration technique is capable of tracking features with larger dis- placements. This is done through coarse-to-fine image registration; in our work, we use 6 levels of resolution. While this technique generally results in good tracks (sub-pixel accu- racy) [Szeliski et al., 1995], poor tracks may result in areas in the vicinity of object occlu- sions/disocclusions. 2. the Shi-Tomasi tracker is a local tracker that fails at large displacements. It performs better for a small number of frames and for relatively small displacements, but deteriorates at large numbers of frames and in the presence of rotation on the image plane [Szeliski et al., 1995]. We are considering a small number of frames at a time, and image warping due to local image plane rotation is not expected. The Shi-Tomasi tracker is also capable of sub-pixel accuracy. The approach that we have undertaken for object tracking can be thought of as a "fine-to-finer" tracking approach. In addition to feature displacements, the measure of reliability of tracks is available (according to match errors and local texturedness, the latter indicated by the minimum eigenvalue of the local Hessian [Shi and Tomasi, 1994; Szeliski et al., 1995]). As we'll see later in Section 8.1, this is used to cull possibly bad tracks and improve 3-D estimates. Once we have extracted point feature tracks, we can then proceed to recover 3-D positions corresponding to these feature tracks. 3-D data recovery is based on the simple notion of stereo. 6 Omnidirectional multibaseline stereo The idea of extracting 3-D data simultaneously from more than the theoretically sufficient number of two camera views is founded on two simple tenets: statistical robustness from redundancy and disambiguation of matches due to overconstraints [Okutomi and Kanade, 1993; Kang et al., 1995b]. The notion of using multiple camera views is even more critical when using panoramic images taken at the same vertical height, which results in the epipoles falling within the images. If only two panoramic images are used, points that are close to the epipoles will not be reliable. It is also important to note that this problem will persist if all the multiple panoramic images are taken at camera positions that are collinear. In the experiments described in Section 8, the camera positions are deliberately arranged such that all the positions are not collinear. In addition, all the images are taken at the same vertical height to maximize view overlap between panoramic images. We use three related approaches to reconstruct 3-D from multiple panoramic images. 3-D data recovery is done either by (1) using just the 8-point algorithm on the tracks and directly recovering the 3-D points, or (2) proceeding with an iterative least-squares method to refine both camera pose and 3-D feature location, or (3) going a step further to impose epipolar constraints in performing a full multiframe stereo reconstruction. The first approach is termed as unconstrained tracking and 3-D data merging while the second approach is iterative structure from motion. The third approach is named constrained depth recovery using epipolar geometry. 6.1 Reconstruction Method 1: Unconstrained feature tracking and 3-D data merging In this approach, we use the tracked feature points across all panoramic images and apply the 8- point algorithm. From the extracted essential matrix and camera relative poses, we can then directly estimate the 3-D positions. The sets of 2-D image data are used to determine (pairwise) the essential matrix. The recovery of the essential matrix turns out to be reasonably stable; this is due to the large (360 ffi ) field of view. A problem with the 8-point algorithm is that optimization occurs in function space and not image space, i.e., it is not minimizing error in distance between 2-D image point and corresponding epipolar line. Deriche et al. [Deriche et al., 1994] use a robust regression method called least-median- of-squares to minimize distance error between expected (from the estimated fundamental matrix) and given 2-D image points. We have found that extracting the essential matrix using the 8-point algorithm is relatively stable as long as (1) the number of points is large (at least in the hundreds), and (2) the points are well distributed over the field of view. In this approach, we use the same set of data to recover Euclidean shape. In theory, the recovered positions are only true up to a scale. Since the distance between camera locations are known and measured, we are able to get the true scale of the recovered shape. Note, however, that this approach is not critical upon knowing the camera distances, as indicated in Section 5.1. Let u ik be the ith point of image k, - v ik be the unit vector from the optical center to the panoramic image point in 3-D space, ik be the corresponding line passing through both the optical center and panoramic image point in space, and t k be the camera translation associated with the kth panoramic image (note that t 0). The equation of line ik is then r . Thus, for each point 6.2 Reconstruction Method 2: Iterative panoramic structure from motion 11 (that is constrained to lie on line i1 ), we minimize the error function where N is the number of panoramic images. By taking the partial derivatives of E i with respect to equating them to zero, and solving, we get from which the reconstructed 3-D point is calculated using the relation p v i1 . Note that a more optimal manner of estimating the 3-D point is to minimize the expression A detailed derivation involving (8) is given in Appendix A. However, due to the practical consideration of texture-mapping the recovered 3-D mesh of the estimated point distribution, the projection of the estimated 3-D point has to coincide with the 2-D image location in the reference image. This can be justified by saying that since the feature tracks originate from the reference image, it is reasonable to assume that there is no uncertainty in feature location in the reference image. An immediate problem with the approach of feature tracking and data merging is its reliance on tracking, which makes it relatively sensitive to tracking errors. It inherits the problems associated with tracking, such as the aperture problem and sensitivity to changing amounts of object distortion at different viewpoints. However, this problem is mitigated if the number of sampled points is large. In addition, the advantage is that there is no need to specify minimum and maximum depths and resolution associated with multibaseline stereo depth search (e.g., see [Okutomi and Kanade, 1993; Kang et al., 1995b]). This is because the points are extracted directly analytically once the correspondence is established. 6.2 Reconstruction Method 2: Iterative panoramic structure from motion The 8-point algorithm recovers the camera motion parameters directly from the panoramic tracks, from which the corresponding 3-D points can be computed. However, the camera motion parameters may not be optimally recovered, even though experiments by Hartley using narrow view images indicate that the motion parameters are close to optimal [Hartley, 1995]. Using the output of Omnidirectional multibaseline stereo the 8-point algorithm and the recovered 3-D data, we can apply an iterative least-squares minimization to refine both camera motion and 3-D positions simultaneously. This is similar to work done by Szeliski and Kang on structure from motion using multiple narrow camera views [Szeliski and Kang, 1994]. As input to our reconstruction method, we use 3-D normalized locations of cylindrical image point. The equation linking a 3-D normalized cylindrical image position u ij in frame j to its 3-D position is the track index, is R (k) where P() is the projection transformation; R (k) and t (k) are the rotation matrix and translation vector, respectively, associated with the relative pose of the jth camera. We represent each rotation by a quaternion with a corresponding rotation matrix (alternative representations for rotations are discussed in [Ayache, 1991]). The projection equation is given simply x y x y In other words, all the 3-D points are projected onto the surface of a 3-D unit sphere. To solve for the structure and motion parameters simultaneously, we use the iterative Levenberg-Marquardt algorithm. The Levenberg-Marquardt method is a standard non-linear least squares technique [Press et al., 1992] that works well in a wide range of situations. It provides a way to vary smoothly between the inverse-Hessian method and the steepest descent method. The merit or objective function that we minimize is where F() is given in (9) and 6.3 Reconstruction Method 3: Constrained depth recovery using epipolar geometry 13 is the vector of structure and motion parameters which determine the image of point i in frame j. The weight c ij in (12) describes our confidence in measurement u ij , and is normally set to the inverse variance oe \Gamma2 ij . We set The Levenberg-Marquardt algorithm first forms the approximate Hessian matrix @a @a and the weighted gradient vector @a is the image plane error of point i in frame j. Given a current estimate of a, it computes an increment ffia towards the local minimum by solving where - is a stabilizing factor which varies over time [Press et al., 1992]. Note that the matrix A is an approximation to the Hessian matrix, as the second-derivative terms are left out. As mentioned in [Press et al., 1992], inclusion of these terms can be destabilizing if the model fits badly or is contaminated by outlier points. To compute the required derivatives for (14) and (15), we compute derivatives with respect to each of the fundamental operations (perspective projection, rotation, translation) and apply the chain rule. The equations for each of the basic derivatives are given in Appendix B. The derivation is exactly the same as in [Szeliski and Kang, 1994], except for the projection equation. 6.3 Reconstruction Method 3: Constrained depth recovery using epipolar ge- ometry As a result of the first reconstruction method's reliance on tracking, it suffers from the aperture problem and hence limited number of reliable points. The approach of using the epipolar geometry to limit the search is designed to reduce the severity of this problem. Given the epipolar geometry, 14 8 Experimental results for each image point in the reference panoramic image, a constrained search is performed along the line of sight through the image point. Subsequently, the position along this line which results in minimum match error at projected image coordinates corresponding to other viewpoints is chosen. Using this approach results in a denser depth map, due to the epipolar constraint. This constrain reduces the aperture problem during search (which theoretically only occurs if the direction of ambiguity is along the epipolar line of interest). The principle is the same as that described in [Kang et al., 1995b]. While this approach mitigates the problem of the aperture problem, it suffers from a much higher computational demand. In addition, the recovered epipolar geometry is still dependent on the output quality of the 8-point algorithm (which in turn depends on the quality of tracking). The user has to also specify minimum and maximum depths as well as resolution of depth search. An alternative to working in cylindrical coordinates is to project sections of cylinder to a tangential rectilinear image plane, rectify it, and use the rectified planes for multibaseline stereo. This mitigates the computational demand as search is restricted to horizontal scanlines in the rectified images. However, there is a major problem with this scheme: reprojecting to rectilinear coordinates and rectifying is problematical due to the increasing distortion away from the new center of projection. This creates a problem with matching using a window of a fixed size. As a result, this scheme of reprojecting to rectilinear coordinates and rectifying is not used. 7 Stereo data segmentation and modeling Once the 3-D stereo data has been extracted, we can then model them with a 3-D mesh and texture-map each face with the associated part of the 2-D image panorama. We have done work to reduce the complexity of the resulting 3-D mesh by planar patch fitting and boundary simplification. The displayed models shown in this paper are rendered using our modeling system. A more detailed description of model extraction from range data is given in [Kang et al., 1995a]. 8 Experimental results In this section, we present the results of applying our approach to recover 3-D data from multiple panoramic images. We have used both synthetic and real images to test our approach. As mentioned 8.1 Synthetic scene 15 Figure 5: Panorama of synthetic room after compositing. earlier, in the experiments described in this section, the camera positions are deliberately arranged so that all of the positions are not collinear. In addition, all the images are taken at the same vertical height to maximize overlap between panoramic images. 8.1 Synthetic scene The synthetic scene is a room comprising objects such as tables, tori, cylinders, and vases. One half of the room is textured with a mandrill image while the other is textured with a regular Brodatz pat- tern. The synthetic objects and images are created using Rayshade, which is a program for creating ray-traced color images [Kolb, 1994]. The synthetic images created are free from any radial distor- tion, since Rayshade is currently unable to model this camera characteristic. The omnidirectional synthetic depth map of the entire room is created by merging the depth maps associated with the multiple views taken around inside the room. The composite panoramic view of the synthetic room from its center is shown in Figure 5. From left to right, we can observe the vases resting on a table, vertical cylinders, a torus resting on a table, and a larger torus. The results of applying both reconstruction methods (i.e., unconstrained search with 8-point and constrained search using epipolar geometry) can be seen in Figure 6. We get many more points using constrained search (about 3 times more), but the quality of the 3-D reconstruction appears more degraded (compare Figure 6(b) with (c)). This is in part due to matching occurring at integral values of pixel positions, limiting its depth resolution. The dimensions of the synthetic room are 10(length) \Theta 8(width) \Theta 6(height), and the specified resolution is 0.01. The quality of the recovered 3-D data appears to be enhanced by applying a 3-D median filter 2 . However, the median 2 The median filter works in the following manner: For each feature point in the cylindrical panoramic image, find other feature points within a certain neighborhood radius (20 in our case). Then sort the 3-D depths associated with the neighborhood feature points, find the median depth, and rescale the depth associated with the current feature point such that the new depth is the median depth. As an illustration, suppose the original 3-D feature location is v i , where is the original depth and - v i is the 3-D unit vector from the camera center in the direction of the image point. If d med results (a) Correct distribution (b) Unconstrained 8-point (c) Iterative (d) Constrained search Median-filtered (f) Median-filtered (g) Median-filtered (h) Top view of 8-point iterative constrained 3-D mesh of (e) Figure Comparison of 3-D points recovered of synthetic room. filter also has the effect of rounding off corners. The mesh in Figure 6(f) and the three views in Figure 7 are generated by our 3-D modeling system described in [Kang et al., 1995a]. As can be seen from these figures, the 3-D recovered points and the subsequent model based on these points basically preserved the shape of the synthetic room. In addition, we performed a series of experiments to examine the effect of both "bad" track removal and median filtering on the quality of recovered depth information of the synthetic room. The feature tracks are sorted in increasing order according to the error in matching 3 . We continually is the median depth within its neighborhood, then the filtered 3-D feature location is given by v 0 d med - v i 3 Note that in general, a "worse" track in this sense need not necessarily translate to a worse 3-D estimate. A high 8.1 Synthetic scene 17 (a) View 1 (b) View 2 (b) View 3 Figure 7: Three views of modeled synthetic room of Figure 6(h). remove tracks that have the worst amount of match error, recovering the 3-D point distribution at each instant. From the graph in Figure 8, we see an interesting result: as more tracks are taken out, retaining the better ones, the quality of 3-D point recovery improves-up to a point. The improvement in the accuracy is not surprising, since the worse tracks, which are more likely to result in worse 3-D esti- mates, are removed. However, as more and more tracks are removed, the gap between the amount of accuracy demanded of the tracks, given an increasingly smaller number of available tracks, and the track accuracy available, grows. This results in generally worse estimates of the epipolar ge- ometry, and hence 3-D data. Concomitant to the reduction of the number of points is the sensitivity of the recovery of both epipolar geometry (in the form of the essential matrix) and 3-D data. This is evidenced by the fluctuation of the curves at the lower end of the graph. Another interesting result that can be observed is that the 3-D point distribution that has been median filtered have lower errors, especially for higher numbers of recovered 3-D points. As indicated by the graph in Figure 8, the accuracy of the point distribution derived from just the 8-point algorithm is almost equivalent that that of using an iterative least-squares (Levenberg- Marquardt) minimization, which is statistically optimal near the true solution. This result is in agreement with Hartley's application of the 8-point algorithm to narrow-angle images [Hartley, 1995]. It is also worth noting that the accuracy of the iterative algorithm is best at smaller numbers of input points, suggesting that it is more stable given a smaller number of input data. Table 1 lists the 3-D errors of both constrained and unconstrained (8-point only) methods for the synthetic scenes. It appears from this result that the constrained method yields better results (after match error may be due to apparent object distortion at different viewpoints. Percent of total points0.300.40 RMS error 8-point (known camera distance) 8-point (unknown camera distance) iterative median-filtered 8-point (known camera distance) median-filtered 8-point (unknown camera distance) median-filtered iterative Figure 8: 3-D RMS error vs. number of points. The original number of points (corresponding to 100%) is 3057. The dimensions of the synthetic room are 10(length) \Theta 8(width) \Theta 6(height). original 0.315039 0.393777 0.302287 median-filtered 0.266600 0.364889 0.288079 Table 1: Comparison of 3-D RMS error between unconstrained and constrained stereo results (n is the number of points). 8.2 Real scenes 19 median filtered) and more points (a result of reducing the aperture problem). In practice, as we shall see in the next section, problems due to misestimation of camera intrinsic parameters (specifically focal length, aspect ratio and radial distortion coefficient) causes 3-D reconstruction from real images to be worse. This is a subject of on-going research. 8.2 Real scenes The setup that we used to record our image sequences consists of a DEC Alpha workstation with a J300 framegrabber, and a camcorder (Sony Handycam CCD-TR81) mounted on an X-Y position stage affixed on a tripod stand. The camcorder settings are made such that its field of view is maximized (at about 43 ffi ). To reiterate, our method of generating the panoramic images are as follows: ffl Calibrate camcorder using an iterative Levenberg-Marquardt least-squares algorithm [Szeliski and Kang, 1994]. ffl Adjust the X-Y position stage while panning the camera left and right to remove the effect of motion parallax; this ensures that the camera is then rotated about its optical center. ffl At each camera location, record onto tape an image sequence while rotating the camera, and then digitize the image sequence using the framegrabber. ffl Using the recovered camera intrinsic parameters (focal length, aspect ratio, radial distortion undistort each image. ffl Project each image, which is in rectilinear image coordinates, into cylindrical coordinates (whose cross-sectional radius is the camera focal length). ffl Composite the frames into a panoramic image. The number of frames used to extract a panoramic image in our experiments is typically about 50. We recorded image sequences of two scenes, namely an office scene and a lab scene. A panoramic image of the office scene is shown in Figure 4. We extracted four panoramic images corresponding to four different locations in the office. (The spacing between these locations is about 6 inches and the locations are roughly at the corners of a square. The size of the office is about 10 feet by 15 feet.) The results of 3-D point recovery of the office scene is shown in Figure 9, with three sample Experimental results views of its model shown in Figure 10. As can be seen from Figure 9, the results due to the constrained search approach looks much worse. This may be directly attributed to the inaccuracy of the extracted intrinsic camera parameters. As a consequence, the composited panoramas may actually be not exactly physically correct. In fact, as the matching (with epipolar constraint) is in progress, it has been observed that the actual correct matches are not exactly along the epipolar lines; there are slight vertical drifts, generally of the order of about one or two pixels. Another example of real scene is shown in Figure 11. A total of eight panoramas at eight different locations (about 3 inches apart, ordered roughly in a zig-zag fashion) in the lab are extracted. The longest dimensions of the L-shaped lab is about 15 feet by 22.5 feet. The 3-D point distribution is shown in Figure 12 while Figure 13 shows three views of the recovered model of the lab. As can be seen, the shape of the lab has been reasonably well recovered; the "noise" points at the bottom of Figure 12(a) corresponds to the positions outside the laboratory, since there are parts of the transparent laboratory window that are not covered. This reveals one of the weaknesses of any correlation-based algorithm (namely all stereo algorithms); they do not work well with image reflections and transparent material. Again, we observe that the points recovered using constrained search is worse. The errors that were observed with the real scene images, especially with constrained search, are due to the following practical problems: ffl The auto-iris feature of the camcorder used cannot be deactivated (even though the focal length was kept constant). As a result, there may be in fact slight variations in focal length as the camera was rotated. ffl The camera may not be rotating exactly about its optical center, since the adjustment of the X-Y position stage is done manually and there may be human error in judging the absence of motion parallax. ffl The camera may not be rotating about a unique axis all the way around (assumed to be ver- tical) due to some play or unevenness of the tripod. ffl There were digitization problems. The images digitized from tape (i.e., while the camcorder is playing the tape) contain scan lines that are occasionally horizontally shifted; this is probably caused by the degraded blanking signal not properly detected by the framegrabber. How- ever, compositing many images averages out most of these artifacts. 8.2 Real scenes 21 (a) Unconstrained 8-point (b) Median-filtered version of (a) (c) Iterative (d) Median-filtered version of (c) Constrained search (f) Median-filtered version of (e) (g) 3-D mesh of (b) Figure 9: Extracted 3-D points and mesh of office scene. Notice that the recovered distributions shown in (c) and (d) appear more rectangular than those shown in (a) and (b). (a) View 1 (b) View 2 (b) View 3 Figure 10: Three views of modeled office scene of Figure 9(g) Figure 11: Panorama of laboratory after compositing. ffl The extracted camera intrinsic parameters may not be very precise. As a result of the problems encountered, the resulting composited panorama may not be physically correct. This especially causes problems with constrained search given the estimated epipolar geometry (through the essential matrix). We actually widened the search a little by allowing search as much as a couple of pixels away from the epipolar line; however, this further significantly increases the computational demand and has the effect of loosening the constraints, making this approach less attractive. 9 Discussion and conclusions We have shown that omnidirectional depth data (whose denseness depends on the amount of local texture) can be extracted using a set of simple techniques: camera calibration, image compositing, feature tracking, the 8-point algorithm, and constrained search using the recovered epipolar geom- etry. The advantage of our work is that we are able to extract depth data within a wide field of view simultaneously, which removes many of the traditional problems associated with recovering camera pose and narrow-baseline stereo. Despite the practical problems caused by using unsophisticated equipment which result in slightly incorrect panoramas, we are still able to extract reasonable 3-D data. Thus far, the best real data results come from using unconstrained tracking and the 8-point al- (a) Unconstrained (b) Median-filtered (c) Iterative 8-point version of (a) (d) Median-filtered (e) Constrained (f) Median-filtered version of (c) search version of (e) (g) 3-D mesh of (b) Figure 12: Extracted 3-D points and mesh of laboratory scene. (a) View 1 (b) View 2 (b) View 3 Figure 13: Three views of modeled laboratory scene of Figure 12(g) gorithm (both direct and iterative structure from motion). Results also indicate that the application of 3-D median filtering improves both the accuracy and appearance of stereo-computed 3-D point distribution. To expedite the panorama image production in critical applications that require close to real-time modeling, special camera equipment may be called for. One such possible specialized equipment is Ahuja's camera system (as reported in [Freedman, 1995]), in which the lens can be rotated relative to the imaging plane. However, we are currently putting our emphasis on the use of commercially available equipment such as a cheap camcorder. Even if all the practical problems associated with imperfect data acquisition were solved, we still have the fundamental problem of stereo-that of the inability to match and extract 3-D data in textureless regions. In scenes that involve mostly textureless components such as bare walls and objects, special pattern projectors may need to be used in conjunction with the camera [Kang et al., 1995b]. Currently, the omnidirectional data, while obtained through a 360 ffi view, has limited vertical view. We plan to extend this work by merging multiple omnidirectional data obtained at both different heights and at different locations. We will also look into the possibility of extracting panoramas of larger height extents by incorporating tilted (i.e., rotated about a horizontal axis) camera views. This would enable scene reconstruction of a building floor involving multiple rooms with good vertical view. We are currently characterizing the effects of misestimated intrinsic camera parameters (focal length, aspect ratio, and the radial distortion factor) on the accuracy of the recovered 3-D data. A Optimal point intersection 25 In summary, our set of methods for reconstructing 3-D scene points within a wide field of view has been shown to be quite robust and accurate. Wide-angle reconstruction of 3-D scenes is conventionally achieved by merging multiple range images; our methods have been demonstrated to be a very attractive alternative in wide-angle 3-D scene model recovery. In addition, these methods do not require specialized camera equipment, thus making commercialization of this technology easier and more direct. We strongly feel that this development is a significant one toward attaining the goal of creating photorealistic 3-D scenes with minimum human intervention. Acknowledgments We would like to thank Andrew Johnson for the use of his 3-D modeling and rendering program and Richard Weiss for helpful discussions. A Optimal point intersection In order to find the point closest to all of the rays whose line equations are of the form we minimize the expression where p is the optimal point of intersection to be determined. Taking the partials of E with respect to - k and p and equating them to zero, we have Solving for - k in (18), noting that - substituting - k in (19) yields from which s where is the perpendicular projection operator for ray - is the point along the viewing ray closest to the origin. Thus, the optimal intersection point for a bundle of rays can be computed as a weighted sum of adjusted camera centers (indicated by t k 's), where the weighting is in the direction perpendicular to the viewing ray. A more "optimal" estimate can be found by minimizing the formula with respect to p and - k 's. Here, by weighting each squared perpendicular distance by - \Gamma2 k , we are downweighting points further away from the camera. The justification for this formula is that the uncertainty in - direction defines a conical region of uncertainty in space centered at the cam- era, i.e., the uncertainty in point location (and hence the inverse weight) grows linearly with - k . However, implementing this minimization requires an interative non-linear solver. Elemental transform derivatives The derivative of the projection function (11) with respect to its 3-D arguments and internal parameters is straightforward: @x \Gammaxy \Gammaxz \Gammayz where 3The derivatives of an elemental rigid transformation are @x @t where z 0 \Gammax \Gammay x 0C C C A and (see [Shabana, 1989]). The derivatives of a screen coordinate with respect to any motion or structure parameter can be computed by applying the chain rule and the above set of equations. --R Artificial Vision for Mobile Robots: Stereo Vision and Multisensory Perception. Computational stereo. Robust recovery of the epipolar geometry for an uncalibrated stereo rig. What can be seen in three dimensions with an uncalibrated stereo rig? Integrating information from multiple views. A camera for near In defence of the 8-point algorithm Building 3-D models from unregistered range images Extraction of Concise and Realistic 3-D Models from Real Data A multibaseline stereo system with active illumination and real-time image acquisition Rayshade user's guide and reference manual. The phase correlation image alignment method. A computer algorithm for reconstructing a scene from two projections. Plenoptic modeling: An image-based rendering system Recovering range using virtual multicamera stereo. A multiple baseline stereo. Numerical Recipes in C: The Art of Scientific Computing. Dynamics of Multibody Systems. Projective structure from uncalibrated images: Structure from motion and recognition. Good features to track. Principal component analysis with missing data and its application to object modeling. Accurate internal camera calibration using rotation Image Mosaicing for Tele-Reality Applications Hierarchical spline-based image reg- istration Recovering 3D shape and motion from image streams using nonlinear least squares. A parallel feature tracker for extended image sequences. --TR --CTR Yin Li , Heung-Yeung Shum , Chi-Keung Tang , Richard Szeliski, Stereo Reconstruction from Multiperspective Panoramas, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.26 n.1, p.45-62, January 2004 R. A. Hicks , D. Pettey , K. Daniilidis , R. Bajcsy, Closed Form Solutions for Reconstruction Via Complex Analysis, Journal of Mathematical Imaging and Vision, v.13 n.1, p.57-70, August 2000 Srikumar Ramalingam , Suresh K. Lodha , Peter Sturm, A generic structure-from-motion framework, Computer Vision and Image Understanding, v.103 n.3, p.218-228, September 2006 Nelson L. Chang , Avideh Zakhor, Constructing a Multivalued Representation for View Synthesis, International Journal of Computer Vision, v.45 n.2, p.157-190, November 2001

3-D median filtering;omnidirectional stereo;panoramic structure from motion;scene modeling;8-point algorithm;multibaseline stereo

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Multiple Experiment Environments for Testing.

Concurrent simulation (CS) has been used successfully as a replacement for serial simulation. Based on storing differences from experiments, CS saves storage, speeds up simulation time and allows excellent internal observation of events. In this paper, we introduce Multiple Domain Concurrent Simulation (MDCS) which like concurrent simulation, maintains efficiency by only simulating differences. MDCS also allows experiments to interact with one another and create new experiments through the use of domains. These experiments can be traced and observed at any point, providing insight into the origin and causes of new experiments. While many experiment scenarios can be created, MDCS uses dynamic spawning and experiment compression rather than explicit enumeration to ensure that the number of experiment scenarios does not become exhaustive. MDCS does not require any pre-analysis or additions to the circuit under test. Providing this capability in digital logic simulators allows more test cases to be run in less time. MDCS gives the exact location and causes of every experiment behavior and can be used to track the signature paths of test patterns for coverage analysis.We will describe the algorithms for MDCS, discuss the rules for propagating experiments and describe the concepts of domains for making dynamic interactions possible. We will report on the effectiveness of MDCS for attacking an exhaustive simulation problem such as Multiple Stuck-at Fault simulations for digital logic. Finally, the applicability of MDCS for more general experimentation of digital logic systems will be discussed.

Introduction Concurrent Simulation(CS)[1, 2] has been proven to be powerful and efficient for simulating single stuck-at faults but inadequate for exhaustive applications like Multiple Stuck-at Fault (MSAF) simulations. It was developed as a speedup mechanism over serial simulation, thus experiments are independent and incapable of interacting with one another. Cumulative behaviors which are combinations of experiments cannot be created without approaching exhaustive testing. This usually requires either modifying the circuit [10] or performing some back-tracing [14]. The methods presented here leverage on the efficiencies of concurrent simulation, but do allow scenarios of experiments to be dynamically spawned should independent experiments interact. Any experiment that does not create a state difference from a parent experiment is not propagated. This effectively "compresses" experiments in the simulation. CS[1] has a primary experiment that exhibits the fault free or good behavior of a circuit. This reference experiment exists during the entire sim- ulation. Faulty experiments that differ or diverge from the reference require additional independent bursts of simulation time. Experiment behaviors can be observed and contrasted since each experiment propagated leaves a signature (its identifier). Unfortunately, CS cannot allow independent experiments to interact at all. It is a cost-effective solution for serial simulation, where each fault inserted creates a single experiment that can only see fault effects it creates and the reference exper- iment. As we will show, to modify it to accommodate such functionality would defeat the overall efficiency and would produce inaccurate observation results. Our simulation environment allows multiple domains of experiments to be defined so that combinations of independent experiments may be efficiently simulated. It is not necessary to generate every possible combination of experiments. Only those experiments that arrive at the same node in the network will be tested as candidates for spawning of new behaviors. This eliminates This work is sponsored by the National Science Foun- dation, MIP-9528194 and the National Aeronautics and Space AdministrationLangley Research Center NASA-JOVE. a huge number of potential scenarios that could arise from exhaustive testing. In addition, every experiment in the simulation can be compressed via the same mechanism of concurrent simulation (i.e., only differences are propagated). Unlike CS where there is only a single reference experiment, we allow any number of independent experiments to serve as parents. Details on performing Multiple Stuck-at Fault are presented here as evidence of the framework's success for digital logic. We were motivated to choose the MSAF application because it demonstrates the large (even exhaus- tive) number of experiments needed to be per- formed. It is also a good test bed for verifying the correctness of "compression" and interaction features since digital logic outputs are limited to a finite number of states (0,1,X,Z). Furthermore, the availability of ISCAS benchmark circuits[3, 4] made it possible for us to compare our performance against other applications. This paper is organized as follows. First, we will provide necessary background information on CS. We will then describe some of the inefficiencies of trying to use CS for creating scenarios of experiment combinations. Next, we introduce the concept of domains, parent experiments as dynamic references for compression, the use of identifiers which are needed to observe experiments, and the spawning of new experiments that display cumulative behaviors. Once these fundamental methods are understood, we will then present the Multiple Domain Simulation algorithm. 2. Background Since we leverage on many of the features of Concurrent Simulation, we provide a brief discussion of it here for background information. Consider the following analogy: an engineer is assigned to build an adder circuit. After developing the adder design, the engineer is told she must now build a calculator that does both addition and subtrac- tion. If she took a "serial" approach, she would start designing the adder circuitry all over again, ignore the work she has already done, and then develop the subtractor circuit independently of the adder design. Likewise, in a concurrent approach, the engineer would try to leverage on the adder Multiple Experiment Environments for Testing 31 1101 I 1 I 2 Fault Sources I 1 I 2 Out1Gate Evaluations R= no faults Fig. 1. Conceptual copies indicate all the scenarios to be evaluated design as well as concentrate design efforts on the new and different functionality required, such as integrating a subtractor into the existing design. In concurrent simulation, the simulation of identical behaviors(reference and faulty) is performed only once. Additional simulation time is dedicated to those faulty experiments that are different. Since the goal of CS is to produce simulation results equivalent to simulating each fault separately, it needs to create simulation scenarios for any inserted fault and the reference case. The main idea is to evaluate these scenarios, and then propagate only those that create state dif- ferences. There are many methods to implement Reference Case Fault C1 Fault C2 Fault C3 Fault C4 Current List Look-Ahead List Experiments C R 3 (1) R 3 (1)C (0) R 3 (1) R 3 (1) R R 3 (1) R 3 (1) R (1) R 3 (1) R 3 (1) R 3 (1)R (1) The gate after evaluation of experiment C 1 R 3 (1)R (1) The current list for evaluating fault scenario on Input 1 of the gate.L 1 I 2 I 1 I 2 Fig. 2. Current and Lookahead Lists for processing experiments in a Multiple List Traversal(MLT) algorithm this scheme but the most efficient single CPU algorithm is the Multiple List Traversal(MLT)[5]. Before describing the MLT implementation let us present a basic concurrent fault simulation algorithm Figure 1 shows a single NAND gate to be simulated. The reference value is denoted by R i for each input and output list, where i is the list number. The conceptual copies in Figure 1, shown as dashed NAND gates, indicate all the scenarios that must be evaluated for the single stuck-at faults inserted on the inputs of the NAND gate. Each fault scenario is derived by replacing the reference value with a faulty state value for the specified input. For example, C 1 is a single stuck-at-1 fault on input I 1 of the NAND gate. From the table of fault sources and their respective evaluations in Figure 1, it can be seen that only one of the four faults inserted will produce a state value on the output different from that of the reference output. Fault scenario C 1 produces a state of 0 on the output while, the reference produces a state value of 1. Only C 1 is propagated. The absence of the other fault experiments indicates that they are behaving identically to the reference experiment. The faults are physically stored on the input or output list on which they are deposited, as shown in Figure 2(A). Experiments are denoted in the form identifier(statevalue). For exam- ple, C 1 (1) is concurrent experiment 1 with a state value of 1. The MLT algorithm is straightforward. Each input along with the output list is traversed and an experiment scenario is evaluated based on the type of experiment encountered(reference or fault). The output list must also be traversed to properly insert any experiments propagated or to update states for a specific experiment. The MLT algorithm dynamically creates scenarios to be evaluated by pointing to a single experiment from each input and output list. These pointers are stored in a current list denoted by L c . The reference experiment is the first scenario to be eval- uated. It is created by pointing at the reference states on each list (see the first row of the table in Figure 2). As the input and output lists are tra- versed, the lowest experiment identifier from all lists is chosen as the "next" experiment to be pro- cessed. A lookahead list 1 maintains this information so that when the MLT is ready to process another experiment, it has already determined which experiment is next. For example, in the table of 4 Lentz, Manolakos, Czeck and Heller Figure 2, after the reference case is evaluated, the look-ahead list indicates that input 1 contains the next lowest faulty experiment identifier of all the three lists. Therefore, the next experiment to be processed will be the experiment for fault C 1 . The experiment C 1 is created by pointing to any states present on any list. If present on a list L i , the fault state of C 1 is used to replace the reference value on L i , while using the reference value for any other list on which the fault does not appear. Figure 2A demonstrates this by the dashed lines. These lines show the experiments selected to create L c for generating the experiment for C 1 . C 1 is selected from list L 1 . Since C 1 does not appear on any other list, the current list will point to the reference experiments on these input and output lists. This will create a scenario that will evaluate 1). Thus the evaluated state of C 1 equals zero. Notice that the MLT also maintains a pointer for the output list to quickly determine whether the faulty experiment is already present or whether a comparison against the output reference value should be per- formed. This same information is also depicted in the table row labeled C 1 in Figure 2. After evaluating this case and comparing the state value of to that of the Reference state R 3 (1), we see that there is a difference and that the experiment should be propagated to the output. Since the current list indicates that the output list does not contain storage for experiment C 1 already, the MLT allocates space and updates the output list. This is shown in Figure 2B. To summarize, the MLT traverses all the lists, all inputs and output lists simultaneously. It does a look-ahead to determine which experiment is coming up next on each list so it can quickly determine if the same fault ID is present on more than 1 input or whether it is already present on the output. The look-ahead information is stored in a circular linked list. The experiment scenario to be evaluated is derived from the look-ahead informa- tion. The lowest identifier (the reference value on each list is used to determine the reference state on the output. The next higher id is then chosen from the lookahead information and used to create a scenario for that specific fault experiment. Figure 3 describes the high level algorithm for the simultaneous list traversal portion of the MLT algorithm and the table of Figure 2 enumerates all the scenarios. Events can be scheduled in the future due to any triggered experiment. (i.e., any experiment that is active for the current simulation time). Let concurrent experiment with identifier k, with an current state value of V 0 . 3. Issues Associated with Exhaustive Scenario Since efficient simulation algorithms [2, 6, 11] only look at single stuck-at fault (SSF) scenarios versus the reference experiment behavior, experiment interaction for combinations of SSFs is not possi- ble. For an MLT implementation of CS, modifications would be necessary to accommodate building scenarios of multiple stuck-at faults to simu- late. Recall, the basic MLT algorithm only substitutes the reference value with a single faulty value. Fault scenarios do not interact, nor are they aware of each others' existence. Despite this, to implement MSAF with the CS MLT would require that, for every fault inserted in the network, an N way combination of every fault with every other fault be inserted. This would represent, each 2 way, 3 way.N way combination of a fault with every other fault. In general, given a network with N inputs and one single stuck-at fault deposited on every input that can combine with single-stuck-at faults on another input, the number of multiple stuck-at fault sources which must be inserted can be determined by equation 1, where N equals the number of faults in the set; i is the multiplicity of faults. (1) This shows that the amount of storage for all these combinations of fault scenarios would greatly reduce the efficiency of CS by increasing list lengths and thus list traversal times. Further- more, when these fault scenarios are inserted, they must be assigned a state value to be used by that experiment. Typically, the state value chosen is that of the single stuck-at fault value on the same line, when two or more faults have not yet in- teracted. Figure 4 demonstrates this. There are Multiple Experiment Environments for Testing 5 Initialize network and time wheel While (an event exists on time wheel for T, where T= current time) For each element E i apply Reference or Concurrent fault state value changes For each input and output list if (an R experiment is triggered) call evaluation code for E i using all reference (non-faulty) state values schedule a new event for time propagation delay, if necessary For each Ck present at update current and look-ahead lists experiment identifiers present on all inputs and output lists of For each Input and output List, L j for If L j does not contain experiment CID then Insert a pointer to R i (statevalue) into the current list L c for list L j else Insert a pointer to C i (statevalue) into L c for list L j Update the lookahead list L o by inserting a pointer to the next identifier present on L j Using the state values from experiments pointed at by L c begin evaluation of experiments For scheduled Ck experiments for Time T call evaluation code for if (Ck if (Ck is not on present on the output list) propagate fault Ck as a fault effect schedule event for Ck at time T propagation delay else Ck converge Ck End For each element E i End While Fig. 3. Multiple List Traversal Algorithm (MLT) three single stuck-at faults inserted, one on each input. Combinations of all two-way and three-way combinations must be stored when faults are inserted. A by-product of creating the multiple stuck-at experiments in this manner is the degradation of observation. All experiments are assigned unique identifiers, so that when propa- gated, it can be traced and observed. The presence of any specific identifier on an input or output list should indicate that the experiment has propagated there. Therefore if a MSAF identifier is seen, one would expect it to indicate that the MSAF has been propagated. For instance, in Figure 4, it appears that a three-way MSAF C 1 C 2 C 3 occurred at the output of gate E 1 . In fact, this identifier was only carried forward because it is a copy of the two-way stuck-at C 1 C2. The distinction of whether an identifier's presence is due to a single-stuck-at experiment or a MSAF can not be determined without further detailed analysis or back-tracing. It should be clear from the figure that it is not possible for a three-way stuck-at to occur at the output of this gate since C 3 is a 6 Lentz, Manolakos, Czeck and Heller I 1 I 2 I 1 I 2 I 1 R R R R R R Fig. 4. Storage problems associated with MSAF simulation using Concurrent Simulation. stuck-at on the primary input of E 2 . These issues are eliminated in a Multiple Domain Simula- tion. MDCS creates a separation of experiments into classes called domains and eliminates both the storage problem and observation impairment. 4. Defining the Experiments Through Domain In developing an algorithm that allowed independent experiments to interact while still performing the single set of fault experiments, it was important that the method avoid modifying the circuit by adding additional hardware[10] and avoid back-tracing for observing events or performing analysis[14]. To achieve this, the concept of Domains is introduced. Domains separate the original independent experiments into classes. Different experiments contained within the same domain are by definition not allowed to interact with one another. They are analogous to the independent single-stuck-at fault sources. These experiments define the original parent experiments. For fault simulation, a single domain may contain a set of single-stuck-at fault experiments, ng. Each experiment C i within A will be evaluated independently and will never know of the presence of any other experiment C j within A. In other words, single domain simulation is the same as traditional concurrent fault simulation. If another domain B is added, then experiments contained in domain B are simulated independently. However, interactions between experiments contained in A are allowed to interact and cause cumulative behaviors with those experiments in domain B. This is an example of a two- domain simulation, where two sets of experiments in domains A and B, are simulated independently and any interactions that may occur between experiments in different domains are also simulated. Interacting experiments that cause new behaviors not displayed by any "parent" are propagated or "spawned". Experiments fitting this description are called "offspring" experiments. By using domains, MDCS achieves efficiency in experiment storage and, as will be discussed later, also helps screen experiments before they are simulated, thus saving processing time. Domains minimize storage since only the original set of single stuck-at faults is inserted and does not require additional storage for defining potential combinations of MSAFs. MSAFs experiments will be created dynamically only if two or more fault experiments meet at a node within the network. As an example, if two sets of SSFs, each containing faults were to be simulated for 2 way stuck-at scenarios, MDCS would define two domains and insert N experiments in each domain for a total of 2 \Theta N parent experiments. Each fault in the original sets of SSFs would be simulated in addition to any two-way stuck-at that may arise. We emphasize only two-way stuck-ats that may arise because the input patterns may Multiple Experiment Environments for Testing 7 never provide the stimulus to make these potential interactions occur. In addition to storing the fault experiments, it is still necessary to store the reference experiment. In general, the number of parent experiments necessary for a simulation using Multiple Domain al- is: where m equals the number of defined domains and n i equals the number of experiments contained in domain i. 4.1. Creating Dynamic Scenarios of Experiments It has already been mentioned that different experiments within domains are not allowed to interact with each other. It could be said that these experiments do not "see" each other. However, it is desired that experiments from different domains should be allowed to interact and see one another. This means experiments need to know about the presence of other experiments from different domains should they ever propagate to the same node in a network. From this requirement, a set of rules were derived to determine which experiments should be checked for cumulative new behaviors when this situation does occur. Experiments that satisfy these rules will be called combinable[9]. Other combinations of experiments that may be present but are not combinable will not be simulated. Since MDCS is a discrete-event simulator algo- rithm, it only processes those experiments that are triggered, (active for the current simulation time). Therefore, one requirement for a combinable scenario of experiments is that it contain at least one trigger. In general, a scenario consisting of two or more experiments is created and simulated if the following are satisfied: ffl There must be at least one trigger present in a scenario for it to be evaluated. ffl Experiments do not share any common domains ffl Experiments in the scenario that do have common domains must be related. Either as a parent and offspring experiment Or the scenario must contain the same experiments from common domains. Figure 5 describes the relationship of domains and experiments in a simulation. The reference is always considered to be a parent experiment to any other experiment and is therefore combinable with all other experiments. The reference is the basic experiment to which all other experiment behavior is compared. In addition, offspring experiments should be compared to their parents for similar behavior. If a parent experiment is present and its state value is identical to the evaluated sce- nario, it will be used to suppress the propagation of the offspring created. If experiments are not related as parent and offspring, then they may be related by an identical experiment in a common do- main. For instance, in Figure 5, A are not related as a parent and offspring. They are however, related by the fact that experiment is common to both experiments. Namely, experiment number 1 from Domain B is contained within both A Although experiments share a common domain B, they are not related and therefore no combina- R Common domain B is the same experiment Common domain B does not contain the same experiments and therefore this is not a valid scenario. Unrelated: { { Offspring (Multiple Parent experiments Fig. 5. Experiments can be related, either as a parent offspring relationship or by identical experiments sharing common domains. Related experiments generate valid combinations. 8 Lentz, Manolakos, Czeck and Heller Insert single stuck-at faults into the network and assign a domain for each multiplicity: For an event, scheduled for current time T Evaluate the R experiment and update the new state value, Routput on the output list for element E. Locate all triggers present on all input lists and store them on a list called a trigger list. Begin generation of valid combinations of experiments. valid scenario must contain a trigger and be combinable.) Take an experiment from one input list and determine which other experiments from different inputs of E may be combined by using the rules of combinability. If the experiments are related as: (parent AND offspring) (related by a common experiment in the same domain) (the experiments do not share any common domains) then If any experiment in the scenario is present on the trigger list then the scenario is valid and can be evaluated. Build the current list L C to simulate the scenario. Evaluate the experiment by calling its evaluation function. Begin Parent Checking to see if the scenario can be compressed to a parent experiment present on the output. If Ck where k is the experiment identifier for Ck and no other parent experiments are present on the output with a state value matching Ck (V 0 ). then diverge Ck (V 0 ) as an offspring experiment. End Parent Checking Continue generation of combinations until no more valid combinations of experiments are found. End generation of valid combinations of experiments. Fig. 6. High Level Algorithm For MDCS tion of these experiments will be simulated. Notice that experiments such as A 2 and B 1 could combine since they do not share any common do- main. A high level algorithm is presented here to summarize the major portions of the algorithm. 4.2. High Level Description of the Multiple Domain Algorithm The high level algorithm described in Figure 6 is similar to the MLT algorithm in the manner that a simulation scenario is built via a Current List, evaluated and compared to a reference experi- ment. In MLT, the simulation case was created by selecting all identical experiments present on the inputs and outputs of a gate. Experiments were located through their unique concurrent identifiers (CID) and state values were retrieved. On inputs and outputs where no matching CID could be found, the Reference experiment state value was used. In MDCS, the simulation cases are created by generating valid combinations of experi- ments, opposed to using only a single experiment and the reference. This is the most complex aspect of building a multiple domain environment. There are many checks for compatibility of experiments that utilizes the domain information stored within the C k experiment. Recall that in Concurrent Fault Simulation, the reference was the only parent experiment that all other experiments were compared against. In Multiple Experiment Environments for Testing 9 MDCS, many parent experiments have been de- fined. The reference, and many single stuck-at experiments. Consequently, a parent experiment will sometimes be referred to as a dynamic reference experiment, since when present, it will be used in lieu of the reference for comparisons against any offspring being processed. As an element evaluation begins, the Current periments from the lists of element E i . In order periment must be present in L c , i.e., there must exist an event for the experiment at the current time. MDCS filters out any experiment not triggered for the current event time T . An additional list of "triggers" is maintained for this specific pur- pose. Using this Trigger information, it is possible to drastically reduce the number of combinations that MDCS must explicitly simulate. In Figure 7, two domains are defined, each containing a reference value and two stuck-at values. Faults in domains A and B are injected at I 1 and I 2 respectively. There is a potential of nine possible combinations of experiments, three experiments on I 1 versus the three experiments on I 2 . As will be shown, only three out of the nine cases must be evaluated using the MDCS algorithm. According to the algorithm, the reference experiment is the first to be processed, but notice it is not triggered. Therefore, the activity is due to a triggered faulty experiment. When the triggered experiment A 2 is encountered, it must be processed independently (as a SSF) and then all the combinable experiments present on the other input must be processed against A 2 . All the sce- R (0) R (0) (1) A 3 (1) R (0) I 1 I 2 Satisfy rules of combinability ? (1) Propagate as an offspring (1) A 3A (1)A (1) R (0) yes: evaluate no: invalid yes: evaluate Experiment Scenario Propagate ? Fig. 7. Applying MDCS to Multiple Stuck-at Fault Sim- ulation narios of A 2 (1) versus experiments from L 2 are depicted in the table of Figure 7. Out of these three cases, two of them must be evaluated because they meet the rules of combinability. The result from these evaluations will be propagated to the output only if the experiment produces a different state value from any parent present on the output list L 3 . Since L 3 only contains one experiment (the reference experiment), the results from all evaluated scenarios are compared to it. Row L 3 in the table of Figure 7 shows the results of the evaluations. Case 1 is performing traditional CS (i.e., simulating the SSF A 2 (stuck \Gamma at \Gamma 1) on input I 1 . This case is compressed since it matches the reference state on the output. Only case 3 produces a state value not equal to the reference on the output, R(0), therefore this experiment is propagated to the output of the element. Note that the diverged experiment A 2 B 1 (1) displays a new behavior that would not have been seen by simulating each parent experiment A 2 (1) or B 1 (1) independently. Therefore, an offspring experiment has been spawned. The previous example was shown to provide insight into the method for generating scenarios. The output contained only a reference experiment and there were no other experiments to compare the newly evaluated scenarios against. Let us now demonstrate the MDCS algorithm when other parent experiments besides the reference are present at the output. Figure 8 shows the presence of an experiment (shown in gray that arrived on the output before the current simulation time. If the gate were to be evaluated, all the same scenarios as shown in the table of Figure 7 would still be generated. The difference however, would be that case 3 would see B 1 (1) on L 3 as a parent and check its state value for comparison. Upon verifying that the interaction on the output, no propagation would occur. This is an example of what is called parent checking, where experiments are compressed to dynamic parents opposed to being propagated. 5. Circuit Example Consider Figure 9 in order to demonstrate the storage savings in the MDCS algorithm. This is Czeck and Heller R (0) R (0) (1) A 3 (1) R (0) I 1 I (1) Fig. 8. Using Parent checking on the output before propagating any offspring. Experiment B 1 arrived previous to the current event. the same circuit as given in Figure 4, only now showing all the parent experiments that MDCS would store. There are three domains, A; B; C in Figure 9. Each domain is defined as containing a single stuck-at-1 fault source for each primary input of an element. In this case, domain A, B and C contain values for input I 1 of gate E 1 , I 2 of I 1 of E 2 respectively. Domain definition is very flexible. There are no restrictions on the assignment of domains to signal lines. For instance, all three domains could have contained faults to be inserted on the same input but this would not have been a very interesting case. Using a reference state value of zero, the network is initialized with four parent experiments at E 1 and E 2 . These are the reference R and the three single stuck-at-1 faults A 1 (1),B 1 (1) and C 1 (1). Table 1. Combinations generated for using MDCS include the two SSF and a MSAF Experiment A 1 B 1 . Case 1 Case 2 Case3 Case 4 I 1 R(0) A 1 (1) I 2 R(0) evaluated (1) All experiments for E 1 are triggered. The resulting combinations are shown in Table 1 along02 =Triggered Experiment R R R R R R (1) C (1)Fig. 9. Simulating a Digital Logic Network with three domains. with the contributing parent experiments from each input that produced them. After the signals have propagated to their respective outputs, only the reference and those experiments that have spawned a new behavior have been propagated. The creation of all MSAFs in MDCS are reported along with detection statistics 6. Results In order to demonstrate the potential of MDCS for practical applications, a proof-of-concept prototype was developed. We used the CREATOR Concurrent Fault Simulator [6] as a reference and compared it to our MDCS version for correctness and for measuring overhead associated with our algorithms. These test cases are based on the ISCAS benchmark circuits [3, 4] widely used for evaluating fault simulation techniques. Experimental results are presented for fault (single stuck-at and two-way stuck-at) simulations that were performed using the MDCS scheme. Although MDCS is portable to many platforms, the results presented here were gathered using a VAX 8800 uni-processor machine. The waveforms (test input patterns) were generated using CONTEST [13]. Table 2 contains the information necessary to describe the benchmark circuits used. The circuit number, ISCAS name, number of gates, primary inputs, primary outputs, number of flip- flops, number of single stuck-at faults inserted, and number of input patterns used are provided. The circuits will be referenced by their number from the first column of Table 2 for all graphs. The first letter of the circuit indicates whether it is sequential (S) or combinational (C). The combinational circuits are taken from the ISCAS85 [3] benchmark set and the sequential circuits were taken from the ISCAS89 [4] set. We simulated all the original set of single stuck-at faults plus the combination of potential two-way stuck-at faults. We say "potential" two-way faults because MDCS is not performing exhaustive experimentation, but rather, investigating the whole space for interactions, given a set of input patterns. The results clearly demonstrate the experiment compression feature of MDCS. Not only Multiple Experiment Environments for Testing 11 Table 2. ISCAS benchmark circuit descriptions, ordered in ascending number of faults inserted. Number Circuit # of Gates PInputs POutputs Flops Faults Patterns 9 S526 214 3 6 21 599 1496 14 C6288 2417 are the original single stuck-ats simulated as independent experiments, but if two or more different experiments arrive on different inputs to a gate, then they are tested for interactions. An interaction is counted any time two experiment scenarios are tested for new behavior. This number can include redundant counts for the same combinations of faults due to feedback paths or the application of a new waveform pattern. In contrast, an offspring experiment is one where an interaction creates a new behavior that must be propagated. Given this information, we wanted to find out the overhead in going from single to a multiplicity of 2 faults (and from one domain to two). The CPU times are plotted in Figure 10. Th times show that adding the potential of experiment interaction is fairly cost effective. We know from the basic CS algorithm that many single stuck-at faults are compressed using the reference. The same phenomenon exhibited by MDCS indicates that the other parent experiments must be playing an important role in curtailing the total number of experiments simulated. Otherwise, the complexity in the algorithm would be overwhelming as more experiments become explicit. Circuit C6288 had the largest overhead due to the fact that our code was optimized for sequential circuits. Newer revisions of the algorithm will address this problem The upper bound of all possible two-way experiment scenarios that could arise in an exhaustive simulation was computed by: This value is shown for each benchmark in the column called "Total # of Possible MSAF" in Table 3. The column titled "Total # of SSF+MSAF experiments possible" is the total number of scenarios that could possibly arise. This includes the original single stuck-at faults as well as all possible double fault experiments. This was computed by: , where N is the number of single stuck-at fault sources. ng l e ng l e+MSAF CPU Circuit Number Fig. 10. CPU time for MDCS single stuck-at simulations and single plus double MSAF. 12 Lentz, Manolakos, Czeck and Heller Table 3. Two Domain Simulation (Single and Double) fault simulation performance. Number Circuit # of SSF Total # Total # CPU time (sec) Storage Name inserted of Possible SSF + MSAF SSF SSF+MSAF in Kbytes MSAF experiments possible Ring 6 15 21 0.10 9 S526 599 179101 179700 414.63 695.93 103 14 C6288 7744 29980896 29988640 783.34 2145.32 728 Table 4. Interactions eliminated due to parent checking. No. Circuit Total Poss. # of Interactions Offspring # of Interactions Name MSAF that Occurred Propagated Eliminated Ring 9 S526 179101 1202 53 1149 14 C6288 29980896 52351 972 51379 Unlike other techniques [10], MDCS is not physically inserting all possible fault scenarios or adding additional circuitry. Rather, MDCS allows sets of single stuck-ats to be inserted and then uses test patterns as the stimulus for MSAFs to manifest themselves as interacting experiments. The original purpose for MDCS was not to be used exclusively as a single stuck-at fault simulator, but rather as a test environment for efficiently creating experiment scenarios and observing experiment behavior. However, it is still interesting to compare the overhead of the algorithm to other simulators. Comparing the MDCS underlying concurrent implementation (single stuck-at faults) to the PROOFS simulator[11], the MDCS Multiple Experiment Environments for Testing 13 CPU times for the benchmarks were much better than one would have expected from a concurrent simulation algorithm. This comparison is based on the 32 bit word data as reported in[11]. Although PROOFS demonstrates a massive speed-up over a version of concurrent simulation, the MDCS version of CS uses the MLT for element processing thus closing the gap between the two algorithms. Most commercial concurrent simulators use a two-list traversal[15] mechanism for implementing their concurrent fault simulation and this certainly would impede the speed as reported in PROOFS [11]. One way to measure the performance of MDCS is to investigate the number of experiment interactions that occurred and whether or not they generated offspring. Table 4 shows the statistics gathered on interactions and their relation to the number of offspring experiments. The occurrence of interactions of two or more experiments have an overhead associated with them. For every interaction, a test must be done to determine whether a parent experiment (the reference or other parent) already exists on the output and exhibits the same be- havior. behaving experiments are com- pressed, while different behaviors must be propagated and therefore become offspring. Offspring experiments are a measure of how many new scenarios were propagated. MDCS results show that despite the fact that the number of interacting scenarios can be large, MDCS can eliminate most of them through parent checking. The column of table 4 titled, "# of Interactions that Occurred", is a count of the number of times two or more experiments were tested for cumulative new behaviors. The column called "Offspring Propagated" indicates how many of the interactions actually generated new offspring experiments. Finally, the difference between "# of Interactions that Occurred" and and "Offspring Propagated", is shown in the column called "# number of Interactions Elimi- nated". This column reflects the number of those interacting scenarios that were converged (com- pressed) due to parent checking. Figure 11 shows that parent checking is very ef- fective. Each column of the graph represents the percentage of interactions that were eliminated for each circuit simulated. This not only proves that MDCS curtails the number of experiments spawned, but also that the experiment compression factor is extremely high. This also has a tremendous impact on storage as shown in the last column of Table 3. 6.1. Extension to Larger Domains Digital logic simulation is an excellent test-bed for MDCS because there is a limited number of logic state values an output can assume. As the number of domains was increased to three and then four, it was found that more interactions could be represented by a parent experiment and therefore compressed. CPU time for larger multiplicity of faults grew slightly and remained relatively constant for multiplicities after four. 6.2. Future Directions Since it was demonstrated that MDCS can efficiently create scenarios that can interact, an area of further research interest involves utilizing the function list[2, 7] for creating multiple instances of a model in a single simulation. In fault simulators such as MDCS and CREATOR[6], the function list stores the various Activity Functions that allow a model to assume different behaviors during the simulation. For instance, an AND gate could be forced to act like an OR gate at various times to model some intermittent fault scenario. Including this functionality in the Multiple Domain algorithm will provide the simulator with the ability to create scenarios from combinations of activity functions dynamically and efficiently. In Figure 12, a two input AND gate is shown. The Reference Experiment on the function list[2] contains the activity function for the fault-free experiment, denoted by R 14 . Faulty behaviors are inserted as single stuck-ats when faults are loaded into the simulator. The MDCS algorithm will create and simulate all the single stuck-at fault experiments by creating current lists for simulating the reference experiment, L for simulating input I 1 stuck-at 1, and L The presence of an activity function on the function list causes the model to be evaluated by substi- 14 Lentz, Manolakos, Czeck and Heller INTERACTIONS ELIMINATED Fig. 11. Demonstrating the efficiency of experiment com- pression. Percentage of experiments that were eliminated from the simulation. tuting the stuck-at value on the appropriate input or output list. For example, activity function F 1 will cause the AND gate to be evaluated with input I inputs use the reference state values. The Multiple Domain algorithm traverses the function list to dynamically create fault scenarios of any multiplicity. In the example of Figure 12, the behaviors of activity functions F 1 and F 2 will be tested for interaction against any fault experiments present on the input lists. If a new behavior is detected that is different from the contributing activity functions, then a new activity function is created and an offspring experiment is spawned. Activity functions in the MDCS show much promise for scenario experimentation. The activity functions need not be limited to fault behaviors and may possess more complex model functions such as those used in scenario control in virtual environments[8].R 11 R 12 I Function List AND(I I ) activity Fig. 12. The Function list containing activity functions which allows a model to assume multiple behaviors. 7. Limitations By utilizing dynamic reference (parent) exper- iments, scenarios are compressed into implicit classes, thus reducing the number of explicit experiments that must be stored and propagated. All these features not currently available in other discrete-event simulators can be implemented in a CPU and storage efficient manner. However, one area of concern is the difficulty in establishing appropriate reference experiments beyond the four state simulator, in an application beyond digital logic simulation, such as software programs. In fault simulation, since there are only 4 possible outcomes (0,1,Z,X), offspring experiments are curtailed because their state behaviors can only assume one of these four values. In considering more complex models, where perhaps the model can be an equation, the number of valid output states could be enormous. The algorithm will have to develop a method of choosing appropriate parent experiments to serve as the most appropriate behavior for comparison. One method under investigation is to use valid ranges for specific variables. These are either provided by the user or derived from the model being simulated. Experiments that produce behavior states within the range are compressed while those outside the boundaries are propagated. 8. Conclusions This paper introduced a framework intended to attack large simulation problems where the number of experiments can approach exhaustive test- ing. By using a Multiple Domain Concurrent Simulation algorithm, a methodology was presented for experimentation without explicit representation of all scenarios. In addition, dynamic interactions are allowed to create new cumulative behaviors (offspring), which can be observed through-out the simulation. Any offspring experiments that were spawned during a simulation were detected using input patterns generated from CON- TEST[13] and GENTEST. An advantage of using MDCS is the flexibility and ease of defining experiments. No modifications need to be made to the network and no scenarios (multiple stuck-at faults) need be inserted. Multiple Experiment Environments for Testing 15 Instead, domains are used to insert sets of single stuck-at faults and only those faults that create new behaviors are ever propagated. The algorithms developed for this framework where verified by performing Multiple Stuck-at Fault Simulation for digital logic circuits. This application was chosen because it demonstrates all the features in a MDCS, namely experiment interac- tion, compression, and spawning of new behaviors. Benchmark testing and evaluation using the ISCAS benchmark circuits [3, 4] were performed for multiple stuck-at fault simulations. Our results indicate that MDCS does not create a combinatorial explosion of experiment combinations that need to be stored and simulated. This was evident through the number of experiments converged to a parent. Another interesting feature that is worth noting is the ability to compare the activity of one pattern against another. For instance, because MDCS generates interactions based on pattern stimulus, the more interactions a pattern created seemed to indicate the robustness in exercising the circuit. Signatures of specific patterns could easily be determined and compared for redundancy in test pattern development. Future directions under investigation include using behavioral models, VHDL or other types of models such as HCSM[8] or even BDDs[16] in the Multiple Domain Environment, such that dynamic interactions will be possible within these models. The current MDCS implementation can use VHDL models, but the experiment interaction feature inherent in our framework has not yet been implemented for VHDL models. Using a function list in our framework also shows much promise in simulation applications where orchestrating scenarios is important. It allows many different behaviors to be captured in a single model entity and through the MDCS list traversal mechanism, we can compress and propagate behaviors only of interest. Finally, this methodology shows promise for evaluating the effectiveness of patterns. Coverage and detection of different pattern sets can be performed and help derive new patterns that will be more robust in detecting complex interacting behaviors. Acknowledgements The Authors would like to thank Dr. Pierluca Montesorro for his continued support on the CREATOR simulator, Dr. Fabio Somenzi and Dr. Vishwani Agrawal for their help with benchmarking and Dr.Chung Len Lee for his valuable suggestions Notes 1. Note: This list is also known as the obligation list. --R "The Concurrent Simulation of Nearly Identical Digital Networks" "MOZART: A Concurrent Multi-Level Simulator" "A Neutral Netlist of 10.Combinational Benchmark Circuits and a Target Translator in FORTRAN" "Combina- tional Profiles of Sequential Benchmarks for Sequential Test Generation" "Switch-Level Concurrent Fault Simulation based on a General Purpose List Traversal Mechanism" "Creator: General and Efficient Multilevel Concurrent Fault Simulation" "Creator: New Advanced Concepts in Concurrent Simulation" "HCSM: A framework for Behavior and Scenario Control in Virtual Environments" "Multiple Domain Concurrent Simulation of Interacting Experiments and Its application to Multiple Stuck-at Fault Simulation" "Multiple-fault Simulation and Coverage of Deterministic Single-Fault Test "PROOFS: A Fast, Memory Efficient Sequential Circuit Fault Sim- 16.Lentz, Manolakos, Czeck and Heller ulator" "The Comparative and Concurrent Simulation of Discrete-Event Experiments" "A Directed Search Method for Test Generation Using a Concurrent Simulator" "Sequential circuit fault simulation by fault information tracing algorithm: fit" "An Efficient Method of Fault Simulation for Digital Circuits Modeled from Boolean Gates and Memories" "Symbolic Boolean Manipulation with Ordered Binary Decision Diagrams" He earned his M. --TR --CTR Karen Panetta Lentz , Jamie Heller , Pier Luca Montessoro, System Verification Using Multilevel Concurrent Simulation, IEEE Micro, v.19 n.1, p.60-67, January 1999 Zainalabedin Navabi , Shahrzad Mirkhani , Meisam Lavasani , Fabrizio Lombardi, Using RT Level Component Descriptions for Single Stuck-at Hierarchical Fault Simulation, Journal of Electronic Testing: Theory and Applications, v.20 n.6, p.575-589, December 2004 Maria Hybinette , Richard M. Fujimoto, Cloning parallel simulations, ACM Transactions on Modeling and Computer Simulation (TOMACS), v.11 n.4, p.378-407, October 2001 Maria Hybinette, Just-in-time cloning, Proceedings of the eighteenth workshop on Parallel and distributed simulation, May 16-19, 2004, Kufstein, Austria

scenario;interactive experimentation;concurrent fault simulation;multiple stuck-at

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Gossiping on Meshes and Tori.

AbstractAlgorithms for performing gossiping on one- and higher-dimensional meshes are presented. As a routing model, the practically important wormhole routing is assumed. We especially focus on the trade-off between the start-up time and the transmission time. For one-dimensional arrays and rings, we give a novel lower bound and an asymptotically optimal gossiping algorithm for all choices of the parameters involved. For two-dimensional meshes and tori, a simple algorithm composed of one-dimensional phases is presented. For an important range of packet and mesh sizes, it gives clear improvements upon previously developed algorithms. The algorithm is analyzed theoretically and the achieved improvements are also convincingly demonstrated by simulations, as well as an implementation on the Paragon. On the Paragon, our algorithm even outperforms the gossiping routine provided in the NX message-passing library. For higher-dimensional meshes, we give algorithms which are based on an interesting generalization of the notion of a diagonal. These algorithms are analyzed theoretically, as well as by simulation.

Introduction Meshes and Tori. One of the most thoroughly investigated interconnection schemes for parallel computers is the n\Thetan mesh, in which n 2 processing units (PUs) are connected by a two-dimensional grid of communication links. A torus is a mesh with wrap-around connections. Their immediate generalizations are d-dimensional n\Theta\Delta \Delta \Delta\Thetan meshes and tori. Although these networks have a large diameter in comparison to the various hypercubic networks, they are nevertheless of great importance due to their simple structure and efficient layout. Numerous parallel machines with mesh and torus topologies have been built, and various algorithmic problems have been analyzed on theoretical models of the mesh. Wormhole Routing. Traditionally, algorithms for the mesh have been developed using a store-and- forward routing model in which a packet is treated as an atomic unit that can be transferred between two adjacent PUs in unit time. However, many modern parallel architectures employ wormhole routing instead. Briefly, in this model a packet consists of a number of atomic data units called flits which are routed through the network in a pipelined fashion. As long as there is no congestion in the network, the time to send a packet consisting of l flits between two arbitrary PUs is well approximated by t s +l \Delta t l , where t s is the start-up time (the time needed to initiate the message transmission) and t l is the flit-transfer time (the time required for actually transferring the data). Usually, t s ?? t l , so that it is important to minimize the number of startups when the packet size is small, whereas it is important to minimize the time when the packet size is large. These two goals may conflict, and then trade-offs must be made. Gossiping. Collective communication operations occur frequently in parallel computing, and their performance often determines the overall running time of an application. One of the fundamental communication problems is gossiping (also called total exchange or all-to-all non-personalized communication). Gossiping is the problem in which every PU wants to send the same packet to every other PU. Said differently, initially each of the N PUs contains an amount of data of size L, and finally all PUs know the complete data of size N \Delta L. This is a very communication intensive operation. On a d-dimensional store-and-forward mesh it can be performed trivially in N=d steps, but for wormhole-routed meshes it is less obvious how to organize the routing so that the total cost is minimal. Gossiping appears as a subroutine in many important problems in parallel computation. We just mention two of them. If M keys need to be sorted on N PUs (M ?? N ), then a good approach is to select a set of m splitters [14, 13, 7] which must be made available in all PUs. This means that we have to perform a gossip in which every PU contributes m=N keys. In this case, the cost of gossiping (provided it is performed efficiently) will not dominate the overall sorting time when the input size is large, because the splitters constitute only a small fraction of the data. A second application of gossiping appears in algorithms for solving ordinary differential equations using parallel block predictor-corrector methods [15]. In each application of the method, block point computations corresponding to the prediction are carried out by different PUs, and these values are needed by all PUs for the correction phase, requiring a gossiping of the data. Previous Work. A substantial amount of research has been performed on finding efficient algorithms for collective communication operations on wormhole-routed systems (see, e.g., [1, 4, 12, 3, 17]). However, most papers either deal with very small packets or with very large packets. Both these extreme cases require algorithms optimizing only one parameter. If the packets are small, then the number of start-ups should be minimized. Peters and Syska [12] considered the broadcasting problem on two-dimensional tori and showed that it can be performed in the optimal 2 \Delta dlog 5 ne steps. Their ideas have been generalized to three-dimensional tori in [3]. The algorithms described in these papers can be adapted for the gossiping problem by first concentrating all data into one PU and then performing a broadcast. However, such an approach leads to a prohibitively large transmission time. Another drawback of both approaches is that it is assumed that the routing paths may be selected by the algorithm. The algorithms presented in this paper can also be used if the network only supports dimension-ordered routing. If the packets are large, a store-and-forward approach yields the best results. As mentioned before, on a d-dimensional n \Theta mesh it can be performed trivially in n d =d packet steps. Gossiping in a store-and-forward hypercube model was studied in [8]. There are many other papers on collective communication operations on wormhole-routed meshes and tori. Although these papers do not deal with the same problem, there are some similarities. For exam- ple, Sundar et al. [16] propose a hybrid algorithm for performing personalized all-to-all communication (complete exchange) on wormhole-routed meshes. Briefly, they employ a logarithmic step algorithm until the packet size becomes large, at which point they switch to a linear step algorithm. Our Results. In this paper we focus on the trade-off between the start-up time and the transmission time. This is useful, because there is a large range of mesh sizes, packet sizes and start-up costs, in which neither of the two contributions is negligible. We would like to emphasize that we are not proposing a hybrid algorithm that simply uses the fastest of the gather/broadcast approach and the store-and-forward approach. In an intermediate range of packet sizes, our algorithm is asymptotically better than the best of the two extreme approaches. A non-trivial lower bound shows that our algorithms are close to optimal for all possible values of the parameters involved. For the efficiency of the two-dimensional algorithm, it is essential that data is concentrated in PUs that lie on diagonals. For higher dimensional meshes we give an interesting generalization of the notion of a diagonal, which may be of independent interest. We remark that Tseng et al. [17] also used diagonals in their complete exchange algorithm. However, the generalization of a diagonal given there for three-dimensional tori is rather straightforward. Hyperspaces are used that when projected give back a diagonal in two-dimensional space. We generalize the diagonal in a different way, that gives better performance, and which allows to formulate a generic algorithm that works for arbitrary dimensions (not only dimension three) without problem. We also compare the value of several strategies by substituting parameters in the formulas for their time consumptions. Furthermore, our theoretical results for two-dimensional meshes are completed with measurements of an implementation on the Intel Paragon. The assumed and the real hardware model do not completely coincide, but still we believe that these measurements support our claims in most important points. described. Thereupon, in Section 3, we present several lower bounds for the gossiping problem. The in Section 5 and Section 6, we extend the algorithm to two- and higher-dimensional meshes and tori. Finally, in Section 7, experimental results gathered on the Intel Paragon are presented. 2 Model of Computation A d-dimensional mesh consists of processing units (PUs) laid out in a d-dimensional grid of side length n. Every PU is connected to each of its (at immediate neighbors by a bidirectional communication link. A torus is a mesh with wrap-around connections. We concentrate on the communication complexity, and assume that a PU can perform an unbounded amount of internal computation in a step. It is also assumed that a PU can simultaneously send and receive data over all its connections. This is sometimes called the full-port or all-port model. With minor modifications, the presented algorithms can also be implemented on one-port architectures. For the communication we assume the much considered wormhole routing model (see [6, 11, 5] for some recent surveys). In this model a packet consists of a number of atomic data units called flits. During routing the header flit governs the route of the packet and the other flits follow it in a pipelined fashion. Initially all flits reside in the source PU and finally all flits should reside in the destination PU. At intermediate stages, all flits of a packet reside in adjacent PUs. The packets should be 'expanded' and 'contracted' only once. That is, two or more flits should reside in the same PU only at the source and destination PU. Wormhole routing is likely to produce deadlock unless special care is taken. The reasons to consider wormhole routing instead of the more traditional store-and-forward routing are of a practical nature. On modern MIMD computers (such as the Intel Paragon and the Cray T3D), the time to initiate a packet transmission is considerably larger than the time needed to traverse a connection. Wormhole routing has been developed in response to this fact. The time for sending a packet consisting of l flits over a distance of d connections is given by We refer to t s as the start-up time, t d as the hop time, and t l as the flit-transfer time. Equation (1) is only correct if there is no link contention (in other words, as long as the paths of the packets do not overlap). If paths of various packets overlap, then the transfer time increases. All our algorithms are overlap-free. 3 Lower Bounds We start with a trivial but general lower bound. Thereupon, we give a more detailed analysis, proving a stronger lower bound for special cases. Lemma 1 In any network with N PUs, degree \Delta and diameter D, the time T con (N; \Delta; D) needed to concentrate all information in a single PU satisfies: Proof: The terms are motivated as follows: N \Delta l flits have to be transferred over at most \Delta connections to the PU in which the data is concentrated; one packet must travel over a distance of at least (D=2) to reach the concentration PU; after t steps a PU can hold at most data items by induction. Of course, a lower bound for T con immediately implies the same lower bound for the gossiping problem. The degree of a d-dimensional n \Theta \Delta \Delta \Delta \Theta n mesh is 2 \Delta d, and the diameter equals d \Delta (n \Gamma 1). Usually, t d is comparable to t l while D ! (N=\Delta) \Delta l. Thus we can omit the term (D=2) \Delta t d from the lower bound without sacrificing too much accuracy. By dividing both remaining terms by l \Delta t l , and by setting the following simplified lower bound for concentrating all data in one PU: con In Section 4, gossiping algorithms are presented that match this lower bound up to constant factors for all well as for all n). For the intermediate range there is a considerable deviation from (2). Therefore, these values of r are considered in more detail. Let T 0 gos (n) denote the number of time units (of duration l \Delta t l each) required for gossiping on a circular array with n PUs. Theorem 1 Let Theorem 1 will be proven by two lemmas. Notice that it establishes a smooth transition from the range of small r values (r - n 1\Gammaffl , ffl ? 0) to the range of large r values n)), for which (2) already gives sharp results. First we show that for proving lower bounds, one can concentrate on the dissemination problem: the problem of broadcasting the information that is concentrated in one PU to all other PUs. The number of time units required for this problem is denoted by T 0 dis . Proof: Starting with all data concentrated in a single PU, the initial situation can be established in time con by reversing a concentration. On the other hand, gossiping can be performed by concentrating and subsequently disseminating. As in our case we will prove a dissemination time that is of larger order than the concentration time (e.g., for con For the dissemination problem with certain r, it is easy to see that having full freedom of choosing the size of the packets can be at most a factor two cheaper than when the data are bundled into fixed messages of size r. That is, we may focus on the problem of disseminating n=r messages, residing in PU 0, while sending a message takes 2 time. At most another constant factor difference is introduced if we assume that dissemination has to be performed on a circular array with only rightward connections. By the above argumentation the proof of Theorem 1 is completed by Lemma 3 Consider a circular array with only rightward connections and with n PUs. Initially, PU 0 contains n=r messages of size r. In one step the messages may be sent rightwards arbitrarily far, but the paths of the messages should be disjoint. If r - (n \Gamma 1)=e, then dissemination takes at least n=r \Delta steps. Proof: We speak of the original n=r messages as colors, and the task is to make all colors available in all PUs. We define a cost function F (t) for the distribution of colors after t steps. Consider a PU i and a color c, and let j be the rightmost PU to the left of i holding color c. The contribution of PU i to F (t) by color c is does not contain color c and 0 otherwise. The initial cost is given by We consider how much the cost function can be reduced after a step is performed. It is essential that the paths must be disjoint. One large 'jump' by a message of some color c gives a strong reduction of the contribution by color c, but the following claim shows that the total reduction is at most n= ln(n=r). (n\Gamma1)=e, then after one step the cost function is reduced by at most n= ln(n=r). Moreover, this occurs if we make a jump over distance r with one message from each color. From this, the result of the lemma follows, because then the number of steps required for dissemination is at least r In order to prove the claim, let d c be the maximum jump made by a message of color c, 0 - c - n=r \Gamma 1. Obviously, we must have c d c - n, since the paths of the messages must be disjoint. The reduction of the contribution to F (t) by color c is at most This can be seen as follows. The initial contribution by color c is at most After a step over distance d c , the contribution of the PUs which are within distance d c remains unchanged. This contribution is Furthermore, the contribution of the other PUs becomes The reduction due to the step made by color c is therefore at most From (3), it follows that the total reduction (due to all steps made by all colors) is bounded by \DeltaF - It needs to be shown that this expression is at most n= ln(n=r). 'Powering,' we obtain dc \Deltaln(n=d c d dc d The Lagrange multiplicator theorem (see, e.g., [10, Section 4.3]) gives that the product of factors with a fixed sum is maximal if all factors are equal. Therefore n=r dc It follows that dc d dc Let a r \Deltak We need to show that a k is maximal if k is fixed at its maximum legal value, which is n. Consider the ratio a k+1 =a k . We have a k+1 a k It needs to be shown that a k+1 =a k ? 1. Hence, we must have r r which holds because r - (n \Gamma 1)=e. It follows that the total reduction in cost is at most 4 Linear and Circular Arrays In this section we analyze gossiping on one-dimensional processor arrays. It is assumed that the time for routing a packet is given by (1), as long as the paths of the packets do not overlap. As in the previous section, the distance term, which is of minor importance anyway, is neglected in the rest of this paper. Furthermore, we write express the time needed for gossiping in units of duration l \Delta t l . We only present algorithms for circular arrays. Due to their more regular structure, these are slightly 'cleaner', but with minor modifications all results carry on for linear arrays. 4.1 Basic Approaches For gossiping on a circular array with n PUs, there are two trivial approaches. Each of them is good in an extreme case. 1. Every PU sends a packet containing its data to the left and right. The packets are sent on for bn=2c steps. 2. Recursively concentrate the data packets into a selected PU. After that, disseminate the information to all other PUs by reversing the process. denote the time taken by Approach 1 and Approach 2, respectively. A simple analysis gives Lemma 4 Proof: Approach 1 consists of bn=2c steps, and in each step every packet consists of l flits. The time taken by Approach 2 is determined as follows. During the concentration phase, the packets get three times as heavy in every step: log 3 The expression for the dissemination phase is similar, but in this case the packets consist of n \Delta l flits in every step: log 3 Adding the two contributions and neglecting the lower-order term n=2 gives the stated result. Approach 1 is good when r is small. Comparing it with the lower bound given in (2) shows that it is exactly optimal when goes to infinity, Approach 2 becomes optimal to within a constant factor. It will outperform Approach 1 for many practical values of r. Still, in principle, the time consumption of Approach 2 is not even linear in n. 4.2 Intermixed Approach For log n, both approaches require \Theta(n \Delta log n) time units. This is a factor of log n more than given by the lower bound. For this intermediate range of r-values, we present an algorithm that establishes a better trade-off between the start-up and the transfer time. The algorithm consists of three phases and works with parameters a and b: Algorithm circgos(n, a, b) 1. Concentrate n=a packets in a evenly interspaced PUs, called bridgeheads. 2. For ba=2c steps, send the packets of size n=a among the bridgeheads in both directions, such that afterwards every bridgehead contains the complete data. 3. In dlog a repeatedly increase the number of bridgeheads by a factor of a. This will be done as follows. Let b - ba=2c denote the number of steps allowed in one round, and let Every bridgehead partitions the data into k packets of size n=k each. Thereupon, the packets are broadcast to the new bridgeheads in a pipelined fashion. The packets to the right are sent in order, whereas the packets to the left are sent in reverse order. In Phase 2, the packets are circulated around. The description is pleasant because of the circular structure. In Phase 3, two oppositely directed packet streams are sustained between the bridgeheads. In order to fully exploit the bidirectional communication links, a bridgehead should not send the same packets to the left and right. Rightwards the packets should be sent in order, whereas leftwards they should be sent in reverse order. Figure 1 shows two examples. The total time consumption of algorithm circgos is given in the following lemma. We do not consider all rounding details. cg,f denote the time taken by Phase f of circgos(n; a; b). Then Proof: Phase 1 corresponds to a concentration step on linear arrays of size n=a instead of n. The time needed for Phase 2 follows by multiplying the number of steps by the time taken by each of them. In order to prove the time consumption of Phase 3, it needs to be shown that after b steps every new bridgehead contains the complete data. Consider a new bridgehead and assume it is at distance d - ba=2c to the closest old bridgehead. This new bridgehead receives the first packet after d steps. After that, it receives one more packet in Step d+1 through Step a \Gamma d \Gamma 1. From that point onwards, it receives two packets in every step. After Step a \Gamma d +x, it contains a+ 2 \Delta packets. By setting that after b steps the new bridgehead contains a oe oe oe oe oe oe oe Figure 1: Behavior of one round of Phase 3 of algorithm circgos. Every arrow is labeled with the time steps at which the corresponding packet reaches the PUs. The top figure illustrates the case a = 9 and 5. In this case, 3 packets are routed from the bridgeheads. The bottom figure illustrates the case In this case, the data is partitioned into 4 packets. At first glance, it is not clear what the result of Lemma 5 means. In particular, it is not immediately clear which a and b should be chosen. In order to obtain an impression, we have written a small program which searches for the optimal values. Table 1 lists some typical results. From these results we conclude that ffl For realistic values of n and r, circgos may be several times faster than the best of Approach 1 and Approach 2. Furthermore, it never performs worse. ffl The range of r values for which algorithm circgos is the fastest increases with n. ffl The best choices for a and b increase with n and decrease with r. In this range of n and r, b is approximately given by 729 4397 4493 4973 7373 Table 1: Comparison between the time taken by Approach 1 (top), Approach 2 (middle) and circgos (bottom). The values of the parameters a and b for which the result for circgos was obtained are given behind its time consumption. The cost unit is l \Delta t l . Notice that when a identical to Approach 1. Furthermore, circgos(n; 3; 1) behaves almost identically to Approach 2, but after log 3 n steps the three bridgeheads contain the complete data. The dissemination phase therefore requires one routing step fewer than in Approach 2. This explains why circgos(n; 3; 1) is always faster than Approach 2, which does not profit from the wrap-around connections. Although the exact choice for the parameters is essential for obtaining the best performance (as shown in Table 1), the asymptotic analysis remains unchanged if we take a. The reason for this is that using different parameters can reduce the amount of transferred data by at most a factor of two. For proving asymptotic results this is fine, but for practical applications this is highly undesirable. On the other hand, we might have used more parameters: the factor a by which the number of bridgeheads is increases in every round of Phase 3 might have been chosen differently, together with its corresponding optimal choice of b. Theorem 2 Let r ! n. The number of time units needed by circgos(n; n=r; n=r) is given by Proof: From Lemma 5 it follows that When the second term never dominates because Replacing the factor of ln r in the third term by ln n gives the theorem. Thus, algorithm circgos gives a continuous transition from gossiping times O(n) (as achieved by Approach to gossiping times O(n \Delta log n) (as achieved by Approach 2 when n). For intermediate r values, circgos may be substantially faster: Corollary 1 Algorithm circgos is asymptotically optimal for all values of r. For all log \Omega\Gamma/14 n) times faster than Approach 1 and Approach 2. Proof: The optimality claim follows by comparing the result of Theorem 2 with the lower bound given in Theorem 1. For r - n, optimality was already established before. For r - log n, Approach 1 and Approach 2 both take \Omega\Gamma n \Delta log n) time units. On the other hand, when r - n 1\Gammaffl , circgos has a time consumption of at most O(n \Delta log 4.3 Generalization In the previous section we presented a gossiping algorithm for linear and circular arrays which is optimal to within a constant factor for all values of r. In this section we show that this immediately implies an asymptotically optimal algorithm for 2- and 3-dimensional meshes and tori. In fact, it immediately gives an asymptotically optimal algorithm for d-dimensional meshes, as long as d is constant. The reason to develop gossiping algorithms for 2- and higher-dimensional meshes (as will be done in subsequent sections) is therefore to obtain algorithms with good practical behavior, paying attention to the constants. The algorithm for gossiping on a d-dimensional mesh consists of d phases. In Phase f , the packets participate in a gossip along axis f . For each of these one-dimensional gossips, the most efficient algorithm is taken. As the size of the packets increases in each phase (in Phase f , the packets consist of l \Delta n f =d flits each), this is not necessarily the same algorithm in all phases. The described algorithm will be denoted by high-dim-gos. Theorem 3 For constant d, high-dim-gos has asymptotically optimal performance. Proof: Denote the time taken by Phase f of high-dim-gos by T hdg,f , and the time taken by the optimal gossiping algorithm by T opt . Clearly, T opt exceeds the time required for making all information available in all PUs, starting with the situation at the beginning of Phase d \Gamma 1. In Phase d \Gamma 1, only a fraction of 1=d of the connections is used. Using all connections would make the algorithm faster by at most a factor of d. Thus, T hdg,d f Thus, once again, we would like to emphasize that achieving asymptotically optimal performance is not the real issue, but constructing algorithms with good practical behavior. Algorithm high-dim-gos does not take advantage of the all port communication capability. To do better, the l flits in each PU are colored with d colors: Flit bc \Delta l=dc to Flit b(c+1) \Delta l=d \Gamma 1c are given Color c, d. After that we perform d independent gossiping operations with parameter r where l In Phase f , the packets with Color c participate in an operation along c) mod d. This algorithm will be denoted by high-dim-gos 0 . It has the same start-up time as high-dim-gos, but the transfer time is reduced by a factor of d. 5 Two-Dimensional Arrays In this section we analyze the gossiping problem on two-dimensional n \Theta n tori. First we investigate what can be obtained by overlapping two one-dimensional gossiping algorithms, one along the rows and one along the columns, as sketched in Section 4.3. After that, a truly two-dimensional algorithm is presented, which for some values of n and r performs significantly better. 5.1 Basic Approaches The simplest idea is to apply high-dim-gos 0 with a choice from the presented one-dimensional gossiping algorithms in each phase. Let Approach i-j denote the algorithm in which first Approach i is applied, and then Approach j. Approach 1-2 can be excluded, since it will never outperform the best of the other approaches. Let T 0 i;j denote the number of time steps taken by Approach i-j. Using the results from Section 4, we find Lemma 6 The time consumption for applying the best version of circgos in both phases cannot be fitted in a simple formula, but it is better then the best of the above algorithms by almost the same factors as those found before. In Table 2 some numerical results are given. Because the packets have size n \Delta l during Phase 2 (which dominates the total time consumption), the transition between the various algorithm now occurs for much larger r than in Table 1. 5.2 Intermixed Approach In this section we present a two-dimensional analogue of algorithm circgos. The algorithm as described below does not use the horizontal and vertical connections simultaneously. Such an algorithm is called uni-axial. By applying the coloring technique of high-dim-gos 0 , the transfer time is halved. The algorithm first creates a situation comparable to the one we find after Phase 2 of circgos. For this, three routing phases are required: Algorithm torgos(n, a, b) 1. Each PU i;j where (j \Gamma i) mod designated as a bridgehead. Note that there are a bridgeheads in every row. In each bridgehead concentrate n=a packets from its row. 2. For ba=2c steps, send packets of size n=a along the rows among the bridgeheads in both directions, such that afterwards, every bridgehead contains the complete data of its row. 3. For ba=2c steps, send packets of size n along the columns among the bridgeheads in both directions, such that afterwards, every bridgehead contains a \Delta n data. Now, each bridgehead in Row i, data from every Row i 0 where (i This is the result of the diagonal way in which the bridgeheads were chosen. An example is given in Figure 2. Thus, all data are available on the diagonals of every n=a \Theta n=a submesh. The algorithm proceeds in log a rounds. In each round, the number of bridgeheads is increased by a factor of a. Figure 2: Left: The bridgeheads in a two-dimensional torus for the case 3. After Phase 3, each bridgehead in Row i, all data from every Row i 0 , where (i bridgeheads from nine consecutive rows therefore know the complete data. Right: The bridgeheads (new ones are drawn smaller) during the first round of Phase 4. Hereafter, each bridgehead in Row i knows all data from every Row Invariant 1 At the beginning of Round t, 1 - t - log a n, each PU holds n \Delta a t data, and all data are available on the diagonals of all n=a t \Theta n=a t submeshes. This implies that the gossiping has been completed when log a n. A more formal description of the last phase is given below: Algorithm torgos(n, a, b) (continued) 4. For repeatedly increase the number of bridgeheads by a factor of a by inserting a \Gamma 1 new bridgeheads between any pair of two consecutive bridgeheads in every row. a. The information from the old bridgeheads in a row is passed to the a \Gamma 1 new bridgeheads in steps with packets of size m=(2 \Delta b \Gamma a b. For ba=2c steps, send packets of size m along the columns among the bridgeheads (old as well as new) in both directions, so that afterwards, every bridgehead contains a \Delta m data. The three phases that operate along the rows are identical to those of circgos. Only Phase 3 and Phase 4.b, which add the information of a row to a other rows, are new. The following analogue of Lemma 5 is straightforward: tg,f denote the number of time units needed for Phase f of torgos(n; a; b). Then Proof: The time taken by Phase 1 is given in the proof of Lemma 5. In Phase 2, 3 and 4.b, there are log a n+1 rounds in total, and each round consists of ba=2c routing steps. The size of the packets increases from n=a until n 2 =a over the rounds. The transfer time is therefore bounded by ba=2c \Delta a). The time taken by Phase 4.a is determined analogously. Just as circgos, torgos constitutes a compromise between simplicity and performance. The routing time will be somewhat smaller if the a-values and the corresponding b-values are chosen in dependency of the growing size of the packets. But even the presented basic version of torgos performs fairly well. In Table 2 we compare the performance of all two-dimensional algorithms. It can be seen that torgos is 442 1482 5382 13182 Table 2: Comparison of the results obtained for gossiping on an n \Theta n torus. For every pair of values (n; r), the number of time steps is given (from top to bottom) for Approach 1-1, Approach 2-1, Approach 2-2, high-dim-gos 0 that circgos, and torgos. For the latter two, the values of (the second) a and b for which the result was obtained are indicated. always the most efficient algorithm, but for small r-values the difference with Approach 2-1 is marginal. The performance of the variation of high-dim-gos 0 that utilizes circgos in both phases is better than that of the simple approaches but nevertheless slightly disappointing, particularly if one considers that this algorithm can choose its a and b values in each phase independently. Generally, we can conclude that if one aims for simplicity, one should utilize Approach 2-1. If a slightly more involved algorithm is acceptable, one should use torgos, which may be more than twice as fast. 6 Higher Dimensional Arrays For the success of torgos it was essential that the packets were concentrated on diagonals at all times, as formulated in Invariant 1. Starting in such a situation, the invariant could be efficiently reestablished by copying horizontally (Phase 4.a), and adding together vertically (Phase 4.b). The main problem in the construction of a gossiping algorithm for d-dimensional meshes is that it is unclear how the concept of a diagonal can be generalizes. Once we have such a 'diagonal', we can perform an analogue of torgos. In the following section we describe the appropriate notion of d-dimensional diagonals. After that, we specify and analyze the gossiping algorithm for d - 3. 6.1 Generalized Diagonals The property of a two-dimensional diagonal that must be generalized is the possibility of 'seeing' a full and non-overlapping hyperplane, when looking along any of the coordinate axes. We will try to explain what this means. Let the unit-cube I d 2 R d be defined as I d = [0; 1i \Theta \Delta \Delta \Delta \Theta [0; 1i. When projecting the diagonal of I 2 orthogonally on the x 0 -axes, we obtain the set [0; 1i \Theta 0; when projecting on the x 1 -axes, we obtain 0 \Theta [0; 1i. These projections are bijections (one-to-one mappings) from the diagonal to the sides of I 2 . For algorithm torgos, this means that the information from diagonals can be replicated efficiently. A diagonal behaves like a magical mirror: data received along one axis can be reflected along the other axis. Not only in one direction, but in both directions. This requirement of problem-free copying between diagonals in adjacent submeshes along all coordinate axes leads to the following subset D of I d is called a d-dimensional diagonal if the orthogonal projections of D onto any of the bounding hyperplanes of I d are bijective. We will proof that the union of the following d sets satisfies the property of a d-dimensional diagonal: Notice that D 0)g. On I 2 , the diagonal consists of f(0; 0)g as well as the points in f0 - 1g. The diagonals of I 2 and I 3 are illustrated in Figure 3. On a torus the d partial hyperplanes are connected in a topologically interesting way. 110D Figure 3: Diagonals of I 2 and I 3 . The bounding lines that belong (do not belong) to the considered sets are drawn solid (dashed). The corner points of D 1 are no elements of it. Projecting I 3 downwards maps bijectively on the ground plane. Lemma a diagonal of I d in the sense of Definition 1. Proof: As the D i are completely symmetric, we can concentrate on the projection \Pi 0 along the x 0 -axis. It is easy to check that for all Clearly, these sets are all disjoint, so \Pi 0 is injective. On the other hand, which implies the surjectivity of \Pi 0 . In order to extend the definitions to d-dimensional n \Theta \Delta \Delta \Delta \Theta n cubes, one has to multiply all bounds on every x i by n. Thus, the diagonal Dn can be defined concisely as On grids, only the points with integral coordinates should be taken. An example is given in Figure 4. So, we successfully defined d-dimensional diagonals. The reader is advised to obtain a full understanding of the case 3. For us it was helpful to construct a model of paper (cardboard would have 1 It is easy to see that all d-subsets together form a closed d \Gamma 1-dimensional subspace of the d-dimensional torus. On two-dimensional tori they constitute a circle and on three-dimensional tori a two-dimensional torus. Generally the diagonal of a d-dimensional torus is a d \Gamma 1-dimensional torus, but a proof of this is beyond the scope of the paper. y y y y y y y y y y y y y y y y Figure 4: The diagonal of a 4 \Theta 4 \Theta 4 grid: 16 points, such that if they were occupied by towers in a three-dimensional chess game, none of them could capture an other. been even better). Such a model makes it easy to convince oneself that the required property, that looking along a coordinate axis indeed gives a full but non-overlapping view of the hyperplanes, is satisfied. Though we are not aware of any result in this direction, we are not sure that we are the first to define this concept. Still we are very pleased with the utmost simplicity of Equation (4) and the elegance of the proof of Lemma 8. 6.2 Details of the Algorithm With the defined diagonals, we can now generalize torgos for gossiping on tori of arbitrary dimensions. The algorithm is almost the same as before. With a few extra routing steps, the algorithm can also be applied for meshes. Again, the presented algorithm is uni-axial. By applying the coloring technique of high-dim-gos 0 , the transfer time can be reduced by a factor of d. By rows we mean one-dimensional subspaces parallel to the x 0 -axis. Algorithm cubgos(n, d, a, b) 1. In each row a PUs are designated as bridgeheads, namely the PUs which lie on the diagonal of their n=a \Theta \Delta \Delta \Delta \Theta n=a submesh. Concentrate in each bridgehead n=a packets from their rows. 2. For ba=2c steps, send packets of size n=a along the rows among the bridgeheads in both directions, such that afterwards, every bridgehead contains the complete data of its row. 3. Perform d \Gamma 1 round each consisting of ba=2c routing steps. In Round i, 1 d, packets of size a are routed along the x i -axis among the bridgeheads in both directions, such that afterwards, every bridgehead contains a i \Delta n data. Now each bridgehead in Row data from every Row all data are available on the diagonal of every n=a \Theta n=a submesh. Hereafter, log a further rounds are performed. In each round, the number of bridgeheads is increased by a factor of a. Invariant 2 At the beginning of Round t, 1 - t - log a n, each PU holds n \Delta a (d\Gamma1)\Deltat data, and all data are available on the diagonal of every n=a t \Theta \Delta \Delta \Delta \Theta n=a t submesh. When log a n, this implies that the gossiping has been completed. A more formal description of the last phase is given below. Algorithm cubgos(n, d, a, b) (continued) 4. For repeatedly increase the number of bridgeheads by a factor of a by inserting a \Gamma 1 new bridgeheads between any pair of consecutive bridgeheads in every row. a. The information from the old bridgeheads in a row is passed to the a \Gamma 1 new bridgeheads in steps with packets of size m=(2 \Delta b \Gamma a b. Perform d \Gamma 1 subphases each consisting of ba=2c routing steps. In Subphase i, 1 of size a are routed along the x i -axis among the bridgeheads (old as well as new) in both directions. Afterwards, every bridgehead contains a The following analogue of Lemma 7 is straightforward: Lemma 9 Let T 0 cg,f denote the number of time units needed for Phase f of cubgos(n; d; a; b). Then (a log a Denote the version of the algorithm that utilizes coloring of the packets in order to fully exploit the all-port communication capability by cubgos 0 . Then we get Theorem 4 Let T 0 cg 0 denote the number of time units taken by cubgos 0 (n; d; a; b). Then log a n \Delta r: Proof: In Lemma 9 the third expression dominates the other two by far. 9 280 ( 9, 90824 Table 3: Comparison of the results obtained for gossiping on a three-dimensional n \Theta n \Theta n torus. For every pair of values (n; r), the number of time steps are given (from top to bottom) for application of high-dim-gos 0 with the best choice from Approach 1 and Approach 2 (indicated), for high-dim-gos 0 that utilizes circgos in every phase, and for cubgos 0 . For the latter two, the values of (the last) a and b are also indicated. Thus, the transfer time is within a factor of a=(a \Gamma 1) from optimality for all d, and the start-up time is within a factor of d\Deltaa=2\Deltalog a n log 2\Deltad+1 n d ' a\Delta(log d+1) 2\Deltalog a from optimality. This appears to be a really strong result. From Table 3 it can be seen that for some n and r, cubgos 0 is substantially faster than high-dim-gos 0 , even though the latter has much more freedom of choosing its parameters. Actually, if one is going to apply high-dim-gos 0 , then one can just as well take the best of Approach 1 and Approach 2 in each of the phases. 7 Experiments To validate the efficiency of the developed algorithms, we implemented them on the Intel Paragon [2]. In this section, the experimental results are presented. System Description. The Paragon system used for the experimentation consists of 140 PUs, each consisting of two 50MHz i860 XP microprocessors. One processor, called the message processor, is dedicated to communication, so that the compute processor is released from message-passing operations. Every PU is connected to a Mesh Routing Chip (MRC), and the MRCs are arranged in a 2-dimensional mesh which is 14 nodes high and 10 nodes wide. The links can transfer data at a rate up to 175 MB/s in both directions simultaneously. The algorithms were implemented using the NX message-passing library. NX is the programming interface supplied by Intel. Other communication layers for the Paragon, such as SUNMOS [9], achieve higher bandwidth and lower latency than NX, but were not available. Some features of the Paragon are particularly important in order to understand the performance of the implemented algorithms, namely ffl The MRCs implement dimension order wormhole routing, i.e., packets are first routed along the rows to their destination columns and from there along the columns to their destinations. We employed this fact to embed a circular array into the mesh topology of the Paragon. ffl When a message enters its destination before the receive is posted, the OSF/1 operating system buffers the message in a system buffer. When the corresponding receive is issued, the message is copied from the system buffer to the application buffer. This buffering is very expensive and can be avoided if the recipient first sends a zero-length synchronization message to the sender indicating that it has posted the receive. All implementations make use of this mechanism. ffl In previous experiments on the Paragon, we determined that the startup cost of a message transmission under NX is about 150 -s. Short messages incur a somewhat lower startup cost than long messages, because they are sent immediately whereas long messages wait until sufficient space is available at the destination processor. The experiments also showed that the uni-directional transfer rate from PU to MRC under NX is about 87 MB/s (11.5 ns per byte), whereas the bi-directional transfer rate is approximately 44 MB/s. Furthermore, the bi-directional transfer rate between two MRCs is 175 MB/s. Because of this, the topology of the Paragon can be viewed as a torus. Modifications to the Algorithms. The implemented algorithms deviate slightly from the algorithms described in the previous sections. This was done for two reasons. First, because every PU of the Paragon is connected to an MRC and not directly to its (up to) 4 neighbors, we cannot assume the full-port model in which a PU can send and receive a message in all 4 wind directions simultaneously. Second, as mentioned above, the uni-directional transfer rate of the Paragon using NX is about 87 MB/s, whereas the bi-directional transfer rate is approximately 44 MB/s. This shows that it is more accurate to assume that a PU cannot send and receive simultaneously, although this is not a feature of the Paragon architecture but a feature of NX. We give two examples of how (the analyses of) the algorithms need to be modified in order to reflect these communication characteristics of the Paragon. First, Approach 1 for gossiping on a circular array of n PUs now consists of steps, and in each step every PU must send a message to one of its neighbors and receive a message from its other neighbor. Since this cannot happen simultaneously, the time consumption of Approach 1 is given by Similarly, under the modified model it takes dlog 2 ne instead of dlog 3 ne steps to concentrate all data into one PU and another dlog 2 ne steps to broadcast the data to every other PU. The time taken by Approach 2 is therefore approximately given by ne A detailed analysis of all algorithms under this model is omitted, because the modifications are rather straightforward. Furthermore, the main purpose of this section is to show that the developed techniques actually work in practice, and not that the performance model is accurate. A detailed performance model should incorporate that short messages incur a lower startup cost than long messages, that the send and receive overheads can differ significantly, etc. This is beyond the scope of this paper. Time per byte Message length (bytes) Approach 1 Approach 2 Circgos Figure 5: Performance of the gossiping algorithms on a circular array with 64 PUs. An additional remark concerns the implementations of algorithm circgos and torgos. In the implementations we used 3 parameters: a 1 , a 2 and k, where a 1 is the number of bridgeheads, a 2 is the factor by which the number of bridgeheads is increased in every round, and n=k is the packet size during Phase 3 (4.a) of circgos (torgos). In the descriptions of circgos and torgos, we have set a 2. For asymptotic analysis this is fine, but on such a moderate size platform rounding errors may be introduced which can affect the execution times significantly. Experimental Results. The circular array algorithms were implemented on the Paragon by embedding a circular array into the mesh. Figure 5 compares the performance achieved by algorithm on a circular array with 64 PUs with the performance achieved by Approach 1 and the performance attained by Approach 2. Every data point measured for the implementation of algorithm circgos is labeled with the values of a 1 , a 2 and k for which the result was obtained. In order to place the data on a common scale, we divided the time taken by each algorithm by the message length m. The total execution time is obtained by multiplying the time per byte by the message length. It can be seen that algorithm circgos is always faster than Approach 2. For messages up to 256 bytes, the best results are obtained with a 1. With this set of parameters, the behavior of circgos is almost the same as the behavior of Approach 2, except that it saves 1 startup and the transmission of a packet of size l \Delta n=2 at the end of the concentration phase. When the message size increases, the fastest results are obtained when the number of bridgeheads a 1 also increases, but a 2 and k remain fixed. With these parameters, algorithm circgos first concentrates data in a few selected nodes as in Approach 2, after that it circulates the packets around as in Approach 1, and finally it broadcasts the data to all non-bridgeheads, again as in Approach 2. Other values for a 2 and k always performed worse than a 1. When the message size increases beyond 16 KB, the best results are obtained when a 64. For this value of a, algorithm circgos and Approach 1 behave identically, as can be seen since the data points coincide. Figure 6 shows the performance of 6 gossiping routines on an 8 \Theta 8 configuration of the Paragon: (1) Approach 1-1, (2) Approach 2-2, (3) Approach 2-1, (4) algorithm torgos with parameters a Approach 1-1 using black/white packets, and (6) the gossiping routine gcolx provided by the NX message-passing library. The fifth algorithm implementation does not partition the packet in every PU into a white and a black packet, but first performs a gossip in every 2 \Theta 2 submesh, after which each colors its packet white, and each PU i;j where even colors its packet black. This was done because of the one-port restriction. Furthermore, the first four algorithm implementations do not employ the technique of interleaving horizontal and vertical messages. This was done because on such a moderate size network one does not save many startups by using a concentrate/broadcast approach instead of a store-and-forward approach. Moreover, if the PUs were divided into black and white PUs, Time per byte Message length (bytes) Approach 1-1 Approach 2-2 Approach 2-1 Approach 1-1 w black/white nodes gcolx (a) Messages smaller than 1KB.1.52.53.54.51K 2K 4K 8K 16K 32K 64K Time per byte Message length (bytes) Approach 1-1 Approach 2-2 Approach 2-1 Approach 1-1 w black/white nodes gcolx (b) Messages larger than 1KB. Figure Performance of the gossiping algorithms on a 2-dimensional mesh with 64 PUs. the differences would almost vanish. Because the differences between the various gossiping algorithms are rather small on this moderate size machine, we divided the experimental data into results for messages smaller than 1KB and messages larger than 1KB. Comparing Approach 1-1, 2-2, 2-1 and torgos(2; 2; 1), we find that torgos is the fastest algorithm for messages up to 3KB. For larger messages, Approach 1-1 yields the best results. For a message of 3KB, the ratio between the startup cost of the message transmission and the transmission time of the message is about 4.2, and for such a small ratio Approach 1-1 turns out to be the fastest gossiping algorithm. Furthermore, as was indicated in Section 6, Approach 2-2 and 2-1 have become they never outperform the fastest of Approach 1-1 and torgos(2; 2; 1). Comparing Approach 1-1 and torgos(2; 2; 1) with the gossiping routine gcolx supplied by NX, one can see that gcolx only yields the best results when the message size is very small. For messages larger than about 200 bytes, the fastest of our algorithm implementations always outperforms the vendor supplied routine. The largest relative difference was measured for messages of 1.5KB. For this message length, gcolx requires 3.96 -s/byte, whereas torgos(2; 2; 1) needs 3.04 -s/byte, which corresponds to a performance improvement by a factor of about 1.3. We believe that this supports our claim that the developed algorithms have practical relevance. Also included in Figure 6 are the results obtained for an implementation of Approach 1-1 in which the nodes are divided into white and black nodes, and in which the white nodes route their packets at all times orthogonally to the black ones. It can be seen that except for very small packets, this implementation always produces the best results. As stated before, this is due to the fact that on this moderate size machine one does not save many startups by using a concentrate/broadcast approach instead of a store- and-forward approach, especially when the nodes are split into white and black nodes. The results for this algorithm are mainly included here to show that the idea of interleaving horizontal and vertical packets can be used advantageously. 8 Conclusion We presented gossiping algorithms for meshes of arbitrary dimensions. We optimized the trade-off between contributions due to start-ups and those due to the bounded capacity of the connections. This enabled us to reduce the time for gossiping in theory as well as practice for an important range of the involved parameters. Furthermore, we presented an interesting generalization of a diagonal, which can be applied to arbitrary dimensions. This seems to have wider applicability. Acknowledgments Computational support was provided by KFA J-ulich, Germany. --R 'On the Efficiency of Global Combine Algorithms for 2-D Meshes with Wormhole Routing,' Journal of Parallel and Distributed Computing 'Intel Paragon XP/S - Architecture, Software Environment, and Performance,' Technical Report KFA-ZAM-IB-9409 Advanced Computer Architecture 'Randomized Multipacket Routing and Sorting on Meshes,' Algorith- mica 'Fast Gossiping for the Hypercube,' SIAM J. 'SUNMOS for the Intel Paragon: A Brief User's Guide 'A Survey of Wormhole Routing Techniques in Direct Networks,' IEEE Computer 'Circuit-Switched Broadcasting in Torus Networks,' IEEE Transactions on Parallel and Distributed Systems 'k-k Routing, k-k Sorting, and Cut-Through Routing on the Mesh,' Journal of Algorithms 'A Logarithmic Time Sort for Linear Size Networks,' Journal of the ACM 'Data Communications in Parallel Block Predictor-Corrector Methods for solving ODEs,' Techn. 'Bandwidth-Optimal Complete Exchange on Wormhole-Routed 2D/3D Torus Networks: A Diagonal-Propagation Approach,' IEEE Transactions on Parallel and Distributed Systems --TR --CTR Michal Soch , Paval Tvrdk, Time-Optimal Gossip of Large Packets in Noncombining 2D Tori and Meshes, IEEE Transactions on Parallel and Distributed Systems, v.10 n.12, p.1252-1261, December 1999 Jop F. Sibeyn, Solving Fundamental Problems on Sparse-Meshes, IEEE Transactions on Parallel and Distributed Systems, v.11 n.12, p.1324-1332, December 2000 Ulrich Meyer , Jop F. Sibeyn, Oblivious gossiping on tori, Journal of Algorithms, v.42 n.1, p.1-19, January 2002 Francis C.M. Lau , Shi-Heng Zhang, Fast Gossiping in Square Meshes/Tori with Bounded-Size Packets, IEEE Transactions on Parallel and Distributed Systems, v.13 n.4, p.349-358, April 2002 Yuanyuan Yang , Jianchao Wang, Near-Optimal All-to-All Broadcast in Multidimensional All-Port Meshes and Tori, IEEE Transactions on Parallel and Distributed Systems, v.13 n.2, p.128-141, February 2002 Yuanyuan Yang , Jianchao Wang, Pipelined All-to-All Broadcast in All-Port Meshes and Tori, IEEE Transactions on Computers, v.50 n.10, p.1020-1032, October 2001

wormhole routing;mesh networks;torus networks;gossip;global communication

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An Efficient Algorithm for Row Minima Computations on Basic Reconfigurable Meshes.

AbstractA matrix A of size mn containing items from a totally ordered universe is termed monotone if, for every i, j, 1 i < jm, the minimum value in row j lies below or to the right of the minimum in row i. Monotone matrices, and variations thereof, are known to have many important applications. In particular, the problem of computing the row minima of a monotone matrix is of import in image processing, pattern recognition, text editing, facility location, optimization, and VLSI. Our first main contribution is to exhibit a number of nontrivial lower bounds for matrix search problems. These lower bound results hold for arbitrary, infinite, two-dimensional reconfigurable meshes as long as the input is pretiled onto a contiguous nn submesh thereof. Specifically, in this context, we show that every algorithm that solves the problem of computing the minimum of an nn matrix must take (log log n) time. The same lower bound is shown to hold for the problem of computing the minimum in each row of an arbitrary nn matrix. As a byproduct, we obtain an (log log n) time lower bound for the problem of selecting the kth smallest item in a monotone matrix, thus extending the best previously known lower bound for selection on the reconfigurable mesh. Finally, we show an $\Omega \left( {\sqrt {\log\log n}} \right)$ time lower bound for the task of computing the row minima of a monotone nn matrix. Our second main contribution is to provide a nearly optimal algorithm for the row-minima problem: With a monotone matrix of size mn with mn pretiled, one item per processor, onto a basic reconfigurable mesh of the same size, our row-minima algorithm runs in O(log n) time if 1 m 2 and in $O\!\left( {{{{\log n} \over {\log m}}}\log\log m} \right)$ time if m > 2. In case $m = n^\epsilon$ for some constant $\epsilon,$$(0 < \epsilon \le 1),$ our algorithm runs in O(log log n) time.

Introduction Recently, in an attempt to reduce its large computational diameter, the mesh-connected architecture has been enhanced with various broadcasting capabilities. Some of these involve endowing the mesh with static buses, that is buses whose configuration is fixed and cannot change; more recently, researches have proposed augmenting the mesh architecture with reconfigurable broadcasting buses: these are high-speed buses whose configuration can be dynamically changed in response to specific processing needs. Examples include the bus automaton [25, 26], the reconfigurable mesh [21], the mesh with bypass capability [12], the content addressable array processor [31], the reconfigurable network [7], the polymorphic processor array [16,20], the reconfigurable bus with shift switching [15], the gated-connection network [27, 28], the PARBS [30], and the polymorphic torus network [13, 17] - see the comprehensive survey paper of Nakano [22]. Among these, the reconfigurable mesh and its variants have turned out to be valuable theoretical models that allowed researchers to fathom the power of reconfiguration and its relationship with the PRAM. From a practical standpoint, however, the reconfigurable mesh and its variants [21,30] omit important properties of physical architectures and, consequently, do not provide a complete and precise characterization of real systems. Moreover, these models are so flexible and powerful that it has turned out to be impossible to derive from them high-level programming models that reflect their flexibility and intrinsic power [16, 20]. Worse yet, it has been recently shown that the reconfigurable mesh and the PARBS do not scale and, as a consequence, do not immediately support virtual parallelism [18, 19]. Motivated by the goal of developing algorithms in a scalable model of computation, we adopt a restricted version of the reconfigurable mesh, that we call the basic reconfigurable mesh, (BRM, for short). Our model is derived from the Polymorphic Processor Array (PPA) proposed in [16,20]: the BRM shares with the PPA all the restrictions on the reconfigurability and the directionality of the bus system. The BRM differs from the PPA in that we do not allow torus connections. As a result, the BRM is potentially weaker than the PPA. It is very important to stress that the programming model developed in [16, 20] for the PPA also applies to the BRM. In particular, all the broadcast primitives developed in [16, 20], with the exception of those using torus connections, can be inherited by the BRM. In fact, all the algorithms developed in this paper could have been, just as easily, written using the extended C language primitives of [16,20]. We opted for specifying our algorithm in a more conventional fashion only to make the presentation easier to follow. Consider a two-dimensional array (i.e. a matrix) A of size m \Theta n with items from a totally ordered universe. Matrix A is termed monotone if for every m, the smallest value in row j lies below or to the right of the smallest value in row i, as illustrated in the example above, where the row minima are highlighted. A matrix A is said to be totally monotone if every submatrix of A is monotone. The concepts of monotone and totally monotone matrices may seem artificial and contrived at first. Rather surprisingly, however, these concepts have found dozens of applications to problems in optimization, VLSI design, facility location problems, string editing, pattern recognition, and computational morphology, among many others. The reader is referred to [1-6] where many of these applications are discussed in detail. One of the recurring problem involving matrix searching is referred to as row-minima computation [6]. In particular, Aggarwal et al. [2] have shown that the task of computing the row-minima of an m \Theta n monotone matrix has a sequential lower bound of \Omega\Gamma n log m). They also showed that this lower bound is tight by exhibiting a sequential algorithm for the row-minima problem running in O(n log m) time. In the case matrix is totally monotone, the sequential complexity is reduced to To the best of our knowledge, no time lower bound for the row-minima problem has been obtained in parallel models of computation, in spite of the importance of this problem. The first main contribution of this paper is to propose a number of non-trivial time lower bounds for matrix search problems. These lower bounds hold for general two-dimensional reconfigurable meshes of infinite size, as long as the input is pretiled onto a contiguous submesh of size n \Theta n. Specifically, in this context we show that every algorithm that solves the problem of computing the smallest item of an n \Theta n matrix must take\Omega\Gamma238 log n) time. The same lower bound is shown to hold for the problem of computing the minima in each row of an arbitrary n \Theta n matrix. As a byproduct, we obtain an \Omega\Gamma/13 log n) time lower bound for the problem of selecting the k-th smallest item in a monotone matrix. Previously, Hao et al. [10] have obtained log n) lower bound for selection in arbitrary matrices on finite reconfigurable meshes. Thus, our lower bound extends the result of [10] in two directions: we show that the same lower bound applies to selection on monotone matrices and on a reconfigurable mesh of an infinite size. Finally, we show an almost tight \Omega\Gamma log log n) time lower bound for the task of computing the row minima of a monotone n \Theta n matrix. Our second main contribution is to provide an efficient algorithm for the row-minima problem: with a monotone matrix of size m \Theta n with m - n pretiled, one item per processor, onto a BRM of the same size, our row-minima algorithm runs in O( log n log m log log m) time. In case for some constant ffl, (0 our algorithm runs in O(log log n) time. The remainder of this work is organized as follows: Section 2 introduces the model of computations adopted in this paper; Section 3 discusses a number of relevant lower-bound results; Section 4 presents basic algorithms that will be key in our subsequent row-minima algorithm; Section 5 gives the details of our row-minima algorithm; finally, Section 6 offers concluding remarks and poses open problems. 2 The Basic Reconfigurable Mesh A Basic Reconfigurable Mesh (BRM, for short) of size m \Theta n consists of mn identical SIMD processors positioned on a rectangular array with m rows and n columns. As usual, it is assumed that every processor knows its own coordinates within the mesh: we let P (i; j) denote the processor placed in row i and column j, with P (1; 1) in the north-west corner of the mesh. Figure 1: A basic reconfigurable mesh of size 4 \Theta 4 Each processor P (i; j) is connected to its four neighbors P (i \Gamma exist, and has four ports N, S, E, and W, as illustrated in Figure 1. Local connections between these ports can be established, subject to the following restrictions: 1. In each time unit at most one of the pairs of ports (N, S) or (E,W) can be set; moreover, 2. All the processors that connect a pair of ports must connect the same 3. broadcasting on the resulting subbuses is unidirectional. For example, if the processors set the (E,W) connection, then on the resulting horizontal buses all broadcasting is done either "eastbound" or else "westbound", but not both. Figure 2: Examples of unidirectional horizontal subbuses We refer the reader to Figure 2(a)-(b) for an illustration of several possible unidirectional sub- buses. The BRM is very much like the recently proposed PPA multiprocessor array, except that the BRM does not have the torus connections present in the PPA. In a series of papers [16, 18-20] Maresca and his co-workers demonstrated that the PPA architecture and the corresponding programming environment is not only feasible and cost-effective to implement, it also enjoys additional features that set it apart from the standard reconfigurable mesh and the PARBS. Specifically, these researchers have argued convincingly that the reconfigurable mesh is too powerful and unrestricted to support virtual parallelism under present-day technology. By contrast, the PPA architecture has been shown to scale and, thus, to support virtual parallelism [16, 18]. The BRM is easily shown to inherit all these attractive characteristics of the PPA, including the support of virtual parallelism and the C-based programming environment, making it eminently practical. As in [16], we assume ideal communications along buses (no delay). Although inexact, a series of recent experiments with the PPA [16] and the GCN [27, 28] seem to indicate that this is a reasonable working hypothesis. 3 Lower Bounds The main goal of this section is to demonstrate non-trivial lower bounds for several matrix search problems. Our lower bound arguments do not use the restrictions of the BRM, holding for more powerful reconfigurable meshes that allow any local connections. In fact, our arguments hold for arbitrary two-dimensional reconfigurable meshes of an infinite size, provided that the input is placed into a contiguous n \Theta n submesh thereof. Formally, this section deals with the following problems: Problem 1. Given an n \Theta n matrix pretiled one item per processor onto an n \Theta n submesh of an reconfigurable mesh, find the minimum item in the matrix. Problem 2. Given an n \Theta n matrix pretiled one item per processor onto an n \Theta n submesh of an reconfigurable mesh, find the minimum item of each row. Problem 3. Given an n \Theta n monotone matrix pretiled one item per processor onto an n \Theta n submesh of an 1 \Theta 1 reconfigurable mesh, find the minimum item of each row. Problem 4. Given an n \Theta n totally monotone matrix pretiled one item per processor onto an n \Theta n submesh of an 1 \Theta 1 reconfigurable mesh, find the minimum item of each row. We will show that Problems 1 and 2 have an\Omega\Gamma/29 log n)-time lower bound, and that Problem 3 has an \Omega\Gamma log log n)-time lower bound. The lower bound for Problem 4 is still open. The proofs are based on a technique detailed in [11, 29] that uses the following graph-theoretic result of Tur'an [8]. (Recall that an independent set in a graph is a set of pairwise non-adjacent vertices.) Lemma 3.1 E) be an arbitrary graph. G has an independent set U such that This lemma is used, in an implicit adversary argument, to bound from below the number of items in the matrix that are possible choices for the minimum. Let V be the set of candidates for the minimum at the beginning of the current iteration and let E stand for the set of pairs of candidates that are compared within the current iteration. The situation benefits from being represented by a graph E) with, V and E representing, respectively, the vertices and the edges of the graph. It is intuitively obvious that an adversary can choose the outcome of the comparisons in such a way that the next set of candidates is no larger than the size of an independent set U in G. In other words, for a set V of candidates and for a set E of pairs that are compared by a minimum finding algorithm, items in the independent set U have the potential of becoming the minimum. Consequently, all items in U are still candidates for the minimum after comparing all pairs in E. To make the presentation easier to follow, we assume that each time unit is partitioned into the following three stages: Phase 1 bus reconfiguration: i.e. the processors set local connections; Phase 2 broadcasting: i.e. the processors send at most a data item to each port, and receive a piece of data from each port; Phase 3 local computation: i.e. every processor selects two elements stored in its local memory, compares them and changes its internal status. We begin by proving the following lemma. Lemma 3.2 Every algorithm that solves Problem 1 log n) time. Proof. We will evaluate the number of pairs that can be compared by an algorithm in Phase 3 of time unit t. Notice that in Phase 2 of a time unit, at most 4n items can be sent to the outside of the submesh. Hence, altogether, at most 4nt items can be sent before the execution of Phase 3 of time unit t. Therefore, the outside of the submesh can compare at mostB @ 4nt1 of items. The inside of the submesh can compare at most n 2 pairs in each Phase 3. Consequently, in Phase 3 of time unit t, at most 16n 2 can be compared by the 1 \Theta 1 reconfigurable mesh. Let c t be the number of candidates that can be the minimum after Phase 3 of time unit t. Then, by virtue of Lemma 3.1 we have, By applying the logarithm, we obtain log c t - 2 log c To complete the algorithm at the end of T time units, c T must be less than or equal to 1. Therefore, must hold. In turn, this implies that T 2\Omega\Gamma400 log n), as claimed. 2 It is worth mentioning that Lemma 3.2 implies a similar lower bound for the task of selection in monotone matrices. To see this, note that given an arbitrary matrix A of size n \Theta n we can construct a monotone matrix A 0 of size n \Theta (n + 1) by simply adjoining to A a column vector of all of whose entries are \Gamma1. It is now clear that the minimum item in A is precisely the (n smallest item in A 0 . Thus, we have the following result. Lemma 3.3 Every algorithm that selects the k-th smallest item in a monotone matrix of size n \Theta n requires \Omega\Gamma108 log n) time. Previously, Hao et al. [10] have obtained log n) lower bound for selection in arbitrary matrices on finite reconfigurable meshes. Thus, Lemma 3.3 extends the result of [10] in two directions: first it shows that \Omega\Gammaat/ log n) remains the lower bound for selection on monotone matrices and second, it shows that the lower bound must hold even for infinite reconfigurable meshes. Lemma 3.4 Every algorithm that solves Problem 2 log n) time. Proof. Suppose to the contrary that Problem 2 requires o(log log n) time However, by using the algorithm of Proposition 4.1 in Section 4, the minimum in the matrix can be computed in O(1) further time. This contradicts Lemma 3.2. 2 Lemma 3.5 Every algorithm that solves Problem 3 requires \Omega\Gamma log log n) time. Proof. Since there is an algorithm that solves Problem 3 in O(log log n) time (see Section 5), we can assume that the upper bound for the Problem 3 is O(log log n). Assume that a row-minima algorithm spent time and has found no row-minima so far, and now it is about to execute Phase 3 of time unit t, where t ! ffl log log n for some small fixed ffl ? 0. Proceeding as in the proof of Lemma 3.2, we see that at most 17n 2 t 2 pairs can be compared in Phase 3 of time unit t. Now a simple counting argument guarantees that at most n 1\Gamma1=4 t rows have been assigned at least 17n comparisons each in time unit i, Hence, at time i, at least n \Gamma in 1\Gamma1=4 t rows have been assigned at most 17n 1+1=4 t Assume that the topmost row was assigned at most 17n 1+1=4 t comparisons in each time unit i, be the number of candidates in the top row at the end of Phase 3 of time unit t. By applying the logarithm, we have log c i - 2 log c Hence, for some small fixed ffl ? 0, c ffl log log n ? 1 for large n. Therefore, at least n \Gamma tn 1\Gamma1=4 t rows including the topmost row cannot find the row-minima in Phase 3 of time unit t. Consequently, at most tn 1\Gamma1=4 t rows can find the row-minima in Phase 3 of time unit t. In turn, this implies that there exist n=(tn 1\Gamma1=4 t =t consecutive rows that cannot find the row-minima in Phase 3 of time t. Therefore, we can find a submatrix of size n 1=4 t =t \Theta n 1=4 t =t such that all of the n 1=4 t row-minima are in it but no row-minima is found. Let d t \Theta d t be the size of sub-matrix such that all d t row-minima are in it but no row-minima is found at time t. Then, d t - d =t. In addition, for large t, d holds. Thus, for large t we have: d t - d . By applying the logarithm twice, we can write log log d t - log log d log log log log Hence, in order to have d T - 1 it must be the case that T 2 \Omega\Gamma log log n), and the proof is 4 Preliminaries Data movement operations are central to many efficient algorithms for parallel machines constructed as interconnection networks of processors. The purpose of this section is to review a number of basic data movement techniques for basic reconfigurable meshes. Consider a sequence of n items a 1 , a . We are interested in computing the prefix maxima defined for every j, (1 - j - n), by setting z g. Recently Olariu et al. [23] showed that the task of computing the prefix maxima of a sequence of n numbers stored in the first row of a reconfigurable mesh of size m \Theta n can be solved in O(log n) time if in O( log n log m algorithm is crucial for understanding our algorithm for computing the row minima of a monotone matrix, we now present an adaptation of the algorithm in [23] for the BRM. To begin, we exhibit an O(1) time algorithm for computing the prefix maxima of n items on a BRM of size n \Theta n. The idea of this first algorithm involves checking, for all j, (1 - j - n), whether a j is the maximum of a 1 ; a . The details are spelled out by the following sequence of steps. The reader is referred to Figure 3(a)-(f) where the algorithm is illustrated on the input sequence 7, 3, 8, 6. Algorithm Step 1. Establish a vertical bus in every column j, (1 every processor P (1; j), (1 broadcasts the item a j southbound along the vertical bus in column j; Step 2. Establish a horizontal bus in every row i, (1 every processor P broadcasts the item a n+1\Gammai westbound along the horizontal bus in row Step 3. At the end of Step 2, every processor P (i; j), (i stores the items a n+1\Gammai and sets a local variable b i;j as follows: Step 4. Every processor P (i; j), (i connects its ports E and W; every processor P (i; j), (i broadcasts a 0 eastbound; every processor that receives a 0 from its W port sets b i;n+1\Gammai to 0; Step 5. Every processor P (i; j), (i connects its ports N and S; every processor P northbound on the bus in column i; every processor copies the value received into b 1;i ; to a i ; every processor P (1; i), connects its ports E and W; every processor P (1; i), (1 - i - to the value received from its port W. The correctness of the algorithm above is easily seen. Thus, we have the following result. Proposition 4.1 The prefix maxima of n items from a totally ordered universe stored one item per processor in the first row of a basic reconfigurable mesh of size n \Theta n can be computed in O(1) time. Next, following [23], we briefly sketch the idea involved in computing the prefix maxima of n items a 1 , a a n on a BRM of size m \Theta n with by partitioning the original mesh into submeshes of size m \Theta m, and apply Prefix-Maxima-1 to each such submesh of size m \Theta m. We further combine groups of m consecutive submeshes of size m \Theta m into a submesh of size combine groups of m consecutive submeshes of size m \Theta m 2 into a submesh of size m\Thetam 3 , and so on. Note that if the prefix maxima of a group of m consecutive submeshes are known, then the prefix maxima of their combination can be computed essentially as in Prefix-Maxima-1. For details, we refer the reader to [23]. To summarize the above discussion we state the following result. Proposition 4.2 The prefix maxima of n items from a totally ordered universe stored in one row of a basic reconfigurable mesh of size m \Theta n with can be computed in O( log n log m ) time. Proposition 4.2 has the following important consequence. Proposition 4.3 Let ffl be an arbitrary constant in the range 1. The prefix maxima of n items from a totally ordered universe stored one item per processor in the first row of a basic reconfigurable mesh of size n ffl \Theta n can be computed in O(1) time. For later reference we now solve a particular instance of the row-minima problem, that we call the selective row minima problem. Consider an arbitrary matrix A of size K \Theta N stored, one item per processor, in K consecutive rows of a BRM of size M \Theta N . For simplicity of exposition we assume that A is stored in the first K rows of the platform, but this is not essential. The goal is to compute the minima in rows 1; A. We proceed as follows. Algorithm Selective-Row-Minima; 7,6 (a) (b) (c) (d) Figure 3: Illustrating algorithm Prefix-Maxima-1 r r r r R R i;2 R i;1 Figure 4: Illustrating algorithm Selective-Row-Minima Step 1. Partition the BMR into N=K submeshes R 1 N=K each of size K \Theta K as illustrated in Figure 4; further partition each submesh R i , (1 - i - N=K), into submeshes k each of size K \Theta K; Step 2. Compute the minimum in the first row of each submesh R i;j in O(1) time using Proposition 4.3; let a i;1 ; a K be the minima in the first row of R i;1 by using appropriately established horizontal buses we arrange for every a i;j , to be moved to the processor in the first row and j K-th column of R i;j ; Step 3. We now perceive the original BRM of size M \Theta N as consisting of K submeshes K each of size M \Theta N ; the goal now becomes to compute for every i, (1 - the minimum of row (i \Gamma 1) of A in T i ; it is easy to see that after having established vertical buses in all columns of the BRM, all the partial minima in row (i \Gamma 1) K+1, of A can be broadcast southbound to the first row of T Step 4. Using the algorithm of Proposition 4.2 compute the minimum in the first row of each T i , in O log N \Gammalog K log M \Gammalog O log N log M time. Thus, we have proved the following result. Lemma 4.4 The task of computing the minima in rows 1; of an arbitrary matrix of size K \Theta N stored one item per processor in K rows of a BRM of size M \Theta N can be performed in O log N log M time. 5 The algorithm The goal of this section is to present the details of an efficient algorithm for computing the row- minima of an m \Theta n monotone matrix A. The matrix is assumed pretiled one item per processor onto a BRM R of the same size, such that for every stores A(i; j). We begin by stating a few technical results that will come in handy later on. To begin, consider a subset of the rows of A and let j(i 1 be such that for all k, (1 - k - p), is the minimum in row r k . Since the matrix A is monotone, we must have be the submatrices of A defined as follows: consists of the intersection of the first rows with the first j(i 1 ) columns of A; ffl for every k, consists of the intersection of rows with the columns j(i ffl A p consists of the intersection of rows with the columns j(i p ) through n. The following result will be used again and again in the remainder of this section. Lemma 5.1 Every matrix A k , (1 - k - p) is monotone. Proof. First, let k be an arbitrary subscript with 2 - k - p. and refer to Figure 5. Let B k consist of the submatrix of A consisting of the intersection of rows columns Similarly, let C k be the submatrix of A consisting of the intersection of rows Figure 5: Illustrating the proof of Lemma 5.1 Since the matrix A is monotone and since A(i is the minimum in row i that none of the minima in rows i occur in the submatrix B k . Similarly, since A(i k ; j(i k )) is the minimum in row i k , no minima in rows i in the submatrix C k . It follows that the minima in rows i must occur in the submatrix A k . Consequently, if A k is not monotone, then we violate the monotonicity of A. A perfectly similar argument shows that A 1 and A p are also monotone, completing the proof of the lemma. 2 The matrices A k , (1 - k - p), defined above pairwise share a column. The following technical result shows that one can always transform these matrices such that they involve distinct columns. For this purpose, consider the matrix A 0 k obtained from A k by replacing for every i, (i by dropping column j(i k ). In other words, A 0 k is obtained from A k by retaining the minimum values in its first and next column and then removing the last column. The last matrix A 0 p is taken to be A p . The following result, whose proof is omitted will be used implicitly in our algorithm. Lemma 5.2 Every matrix A 0 In outline, our algorithm for computing the row-minima of a monotone matrix proceeds as follows. First, we solve an instance of the selective row minima, whose result is used to partition the original matrix into a number of monotone matrices as described in Lemmas 5.1 and 5.2. This process is continued until the row minima in each of the resulting matrices can be solved directly. then the problem has a trivial solution running in \Theta(log n) time, which is also best possible even on the more powerful reconfigurable mesh [23]. We shall, therefore, assume that m - 2. exposition we shall assume that c i#1 R c i#1 Figure Illustrating the partition into submeshes T i and R i Algorithm Row-Minima(A); Step 1. Partition R into each of size m \Theta n such that for every m), m of R as illustrated in Figure 6; Step 2. Using the algorithm of Lemma 4.4 compute the minima of the items in the first row of every submesh T i , m), in O( log n log Step 3. Let c m be the columns of R containing the minima in T tively, computed in Step 2. The monotonicity of A guarantees that c 1 - c 2 - c p m . m), be the submesh of consisting of all the processors P (r; c) such that In other words, R i consists of the intersection of rows m with columns c illustrated in Figure 6; c i#1 Figure 7: Illustrating the submeshes S i Step 4. Partition the mesh R into submeshes S 1 illustrated in Figure 7; for log log m iterations, repeat Steps 1-3 above in each submesh S i . The correctness of algorithm being easy to see, we now turn to the complexity. Steps 1-3 have a combined complexity of O log m . In Step 4, c and so, by Lemma 4.4 each iteration of Step 4 also runs in O log m time. Since there are, essentially, log log m such iterations, the overall complexity of the algorithm is O log m log log m . To summarize our findings we state the following result. Theorem 5.3 The task of computing the row-minima of a monotone matrix of size m \Theta n with pretiled one item per processor in a BRM of the same size can be solved in O(log n) and in O log m log log m 2. Theorem 5.3 has the following consequence. Corollary 5.4 The task of computing the row-minima of a monotone matrix of size m \Theta n with pretiled one item per processor in a BRM of the same size can be solved in O(log log n) time. 6 Conclusions and open problems We have shown that the problem of computing the row-minima of a monotone matrix can be solved efficiently on the basic reconfigurable mesh (BRM) - a weaker variant of the recently proposed Polymorphic Processor Array [16]. Specifically, we have exhibited an algorithm that, with a monotone matrix A of size m \Theta n, stored in a BRM of the same size, as input solves the row-minima problem in O(log n) time in case m 2 O(1), and in O log m log log m time otherwise. In particular, if for some fixed ffl, (0 our algorithm runs in O(log log n) time. Our second main contribution was to propose a number of non-trivial time lower bounds for matrix search problems. These lower bounds hold for general two-dimensional reconfigurable meshes of infinite size, as long as the input is pretiled onto an n \Theta n submesh thereof. Specifically, in this context we show that every algorithm that solves the problem of computing the smallest item of an n \Theta n matrix, or the smallest item in each row of an n \Theta n matrix must take\Omega\Gamma453 log n) time. This result implies an \Omega\Gamma/17 log n) time lower bound for the problem of selecting the k-th smallest item in a monotone matrix, extending the result of [10] in two directions: we show that the same lower bound applies to selection on monotone matrices and on a reconfigurable mesh of an infinite size. Finally, we showed an almost tight \Omega\Gamma log log n) time lower bound for the task of computing the row minima of a monotone n \Theta n matrix. These are the first non-trivial lower bounds of this kind known to the authors. A number of problems remain open. First, as noted, there is a discrepancy between the time lower bound we obtained for the task of computing the row-minima of a monotone matrix and the upper bound provided by our algorithm. Narrowing this gap will be a hard problem that we leave for future research. Second, no non-trivial lower bounds for the problem of computing the row-minima of a totally monotone matrix are known to us. This promises to be an exciting area for future research. Yet another problem of interest would be to solve the row-minima problem for the special case of totally monotone matrices. trivially, our algorithm for monotone matrices also works for totally monotone ones. Unfortunately, to this date we have not been able to find a non-trivial lower bound for this problem. Acknowledgement : The authors wish to thank Mike Atallah for many useful comments and for pointing out a number of relevant references. --R Applications of generalized matrix searching to geometric problems Geometric applications of a matrix-searching algorithm Notes on searching in multidimensional monotone arrays Efficient parallel algorithms for string editing and related problems A faster parallel algorithm for a matrix searching problem An efficient parallel algorithm for the row minima of a totally monotone matrix The power of reconfiguration Graphs and Hypergraphs Pattern Classification and Scene Analysis Selection on the reconfigurable mesh An Introduction to Parallel Algorithms IEEE Transactions on Computers Reconfigurable buses with shift switching - concepts and applications IEEE Transactions on Parallel and Distributed Systems Connection autonomy in SIMD computers: a VLSI implementation Virtual parallelism support in reconfigurable processor arrays Hierarchical node clustering in polymorphic processor arrays Hardware support for fast reconfigurability in processor arrays Parallel computations on reconfigurable meshes A bibliography of published papers on dynamically reconfigurable architectures Fundamental data movement on reconfigurable meshes Fundamental algorithms on reconfigurable meshes On the ultimate limitations of parallel processing Bus automata bit serial associate processor The gated interconnection network for dynamic programming Parallelism in comparison problems Constant time algorithms for the transitive closure problem and its applications IEEE Transactions on Parallel and Distributed Systems The image understanding architecture --TR --CTR Schwing , Larry Wilson, Optimal Algorithms for the Multiple Query Problem on Reconfigurable Meshes, with Applications, IEEE Transactions on Parallel and Distributed Systems, v.12 n.9, p.875-887, September 2001 Tatsuya Hayashi , Koji Nakano , Stephen Olariu, An O((log log n)2) Time Algorithm to Compute the Convex Hull of Sorted Points on Reconfigurable Meshes, IEEE Transactions on Parallel and Distributed Systems, v.9 n.12, p.1167-1179, December 1998 R. Lin , K. Nakano , S. Olariu , M. C. Pinotti , J. L. Schwing , A. Y. Zomaya, Scalable Hardware-Algorithms for Binary Prefix Sums, IEEE Transactions on Parallel and Distributed Systems, v.11 n.8, p.838-850, August 2000 Alan A. Bertossi , Alessandro Mei, Time and work optimal simulation of basic reconfigurable meshes on hypercubes, Journal of Parallel and Distributed Computing, v.64 n.1, p.173-180, January 2004

basic reconfigurable meshes;monotone matrices;cellular system design;row minima;search problems;facility location problems;VLSI design;reconfigurable meshes;totally monotone matrices

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Designing Masking Fault-Tolerance via Nonmasking Fault-Tolerance.

AbstractMasking fault-tolerance guarantees that programs continually satisfy their specification in the presence of faults. By way of contrast, nonmasking fault-tolerance does not guarantee as merely guarantees that when faults stop occurring, program executions converge to states from where programs continually (re)satisfy their specification. We present in this paper a component based method for the design of masking fault-tolerant programs. In this method, components are added to a fault-intolerant program in a stepwise manner, first, to transform the fault-intolerant program into a nonmasking fault-tolerant one and, then, to enhance the fault-tolerance from nonmasking to masking. We illustrate the method by designing programs for agreement in the presence of Byzantine faults, data transfer in the presence of message loss, triple modular redundancy in the presence of input corruption, and mutual exclusion in the presence of process fail-stops. These examples also serve to demonstrate that the method accommodates a variety of fault-classes. It provides alternative designs for programs usually designed with extant design methods, and it offers the potential for improved masking fault-tolerant programs.

Introduction In this paper, we present a new method for the design of "masking" fault-tolerant systems [1-4]. We focus our attention on masking fault-tolerance because it is often a desirable -if not an ideal- property for system design: masking the effects of faults ensures that a system always satisfies its problem specification and, hence, users of the system always observe expected behavior. By the same token, when the users of the system are other systems, the design of these other systems becomes simpler. To motivate the design method, we note that designers of masking fault-tolerant systems often face the potentially conflicting constraints of maximizing reliability while minimizing overhead. As a result, designers avoid methods that yield complex designs, since the complexity itself may result in reduced reliability. Moreover, they avoid methods that yield inefficient implementations, since system users are generally unwilling to pay a significant cost in price or performance for the sake of masking fault-tolerance. Therefore, a key goal for our method is to yield well-structured -and hence more reliable- systems, while still offering the potential for efficient implementation. Other goals of the method include the ability to deal with a variety of fault-classes and the ability to provide designs -albeit alternative ones- for masking tolerant systems which are typically designed by using classical methods such as replication, exception handling, and recovery blocks. With these goals in mind, our method is based on the use of components that add tolerance properties to a fault-intolerant system. It divides the complexity of designing fault-tolerant system into that of designing relatively simpler components and that of adding the components to the fault- intolerant system. And, by focusing attention on the efficient implementation of the components themselves, it offers the potential for efficient implementation of the resulting system. We call the components added in the first stage correctors and those added in the second stage detectors. Efficient implementation of correctors and detectors is important, as noted above, for offering the potential for efficient masking fault-tolerant implementations. To manage the complexity of adding components to a system, the method proceeds in a stepwise fashion. Informally speaking, instead of adding the components which will ensure that the problem specification is satisfied in the presence of faults all at once, the method adds components in two stages. In the first stage, the method merely adds components for nonmasking fault-tolerance. By nonmasking fault-tolerance we intuitively mean that, when faults stop occurring, the system execution eventually reaches a "good" state from where the system continually "satisfies" its problem specification. In the second stage, the method adds components that additionally ensure that the problem specification is not "violated" until the program reaches these good states. It follows that the fault-tolerance of the system is enhanced from nonmasking to masking. As in any component based design, to prove the correctness of the resulting composite system, we need to ensure that the components do not interfere with each other, i.e., they continue to accomplish their task even if they are executed concurrently with the other components. To this end, in the first stage, we ensure that fault-intolerant system and the correctors added to it do not interfere with each other. And, in the second stage, we ensure that the resulting nonmasking fault-tolerant system and the detectors added to it do not interfere with each other. We demonstrate that our method accommodates a variety of fault-classes, by using it to design programs that are masking fault-tolerant to Byzantine faults, input corruption, message loss, and fail-stop failures. More specifically, we design: (1) a Byzantine agreement program whose processes are subject to Byzantine faults; (2) an alternating-bit data transfer program whose channel messages may be lost; (3) a triple modular redundancy (TMR) program whose inputs may be corrupted; and (4) a new token-based mutual exclusion program whose processes may fail-stop in a detectable manner. The TMR and Byzantine agreement examples also serve to provide alternative designs for programs usually associated with the method of replication. The alternating-bit protocol example serves to provide an alternative design for a program usually associated with the method of exception handling or that of rollback recovery. The mutual exclusion case study serves to demonstrate that, by focusing on the addition of efficient components, the method enables the design of improved programs. We proceed as follows. First, in Section 2, we recall a formal definition of programs, faults, and what it means for programs to be masking or nonmasking fault-tolerant. Then, in Section 3, we present our two-stage method for design of masking fault-tolerance. Next, in Section 4, we illustrate the method by designing standard masking fault-tolerant programs for Byzantine agreement, data transfer, and TMR. In Section 5, we present our case study in the design of masking fault-tolerant token-based mutual exclusion. Finally, we compare our method with extant methods for designing masking fault-tolerant programs and make concluding remarks in Section 6. Programs, Faults, and masking and Nonmasking Tolerances In this section, we recall formal definitions of masking and nonmasking fault-tolerance of programs [5] in order to characterize a relationship between these two tolerance types, and to motivate our design method which is presented in Section 3. Programs. A program p is defined recursively to consist of a (possibly empty) program q, a set of "superposition variables", and a set of "superposition actions". The superposition variables of p are disjoint from the remaining variables of p, namely the variables of q. Each superposition action of p has one of two forms: hnamei :: hguardi \Gamma! hstatementi , or hnamei :: haction of qi k hstatementi A guard is a boolean expression over the variables of p. Thus, evaluating a guard may involve accessing the variables of q. Note that there is a guard in each action of p: in particular, the guard of the actions of the second (i.e., k) form is the same as that of the corresponding action of q. A statement is an atomic, terminating update of zero or more superposition variables of p. Thus, the superposition actions of the first form do not update the variables of q, whereas those of the second may since they are based on an action of q. Note that, since statements of p do not update the variables of q, the only actions of p that update the variables of q are the actions of q. Thus, programs are designed by superposition of variables and actions on underlying programs [6]. Superposition actions may access, but not update, the underlying variables, whereas the underlying actions may not access or update the superposition variables. Operationally speaking, the superposition actions of the first form execute independently (asynchronously) of other actions and those of the second form execute in parallel (synchronously) with the underlying action they are based upon. State. A state of a program p is defined by a value for each variable of p, chosen from the predefined domain of the variable. A "state predicate" of p is a boolean expression over the variables of p. An action of p is enabled in a state iff its guard is true at that state. We use the term "S state" to denote a state that satisfies the state predicate S. Closure. An action "preserves" a state predicate S iff in any state where S holds and the action is enabled, executing all of the statements in the action instantaneously in parallel yields a state where S holds. S is "closed" in a set of actions iff each action in that set preserves S. It follows from this definition that if S is closed in (the actions of) p then executing any sequence of actions of p starting from a state where S holds yields a state where S holds. Computation. A computation of p is a fair, maximal sequence of steps; in every step, an action of p that is enabled in the current state is chosen and all of its statements are instantaneously executed in parallel. (Recall that actions of the second form consist of multiple statements composed in parallel.) Fairness of the sequence means that each action in p that is continuously enabled along the states in the sequence is eventually chosen for execution. Maximality of the sequence means that if the sequence is finite then the guard of each action in p is false in the final state. Problem Specification. The problem specification that p satisfies consists of a "safety" specification and a "liveness" specification[7]. A safety specification identifies a set of "bad" finite computation prefixes that should not appear in any program computation. Dually, a liveness specification identifies a set of "good" computation suffixes such that every computation has a suffix that is in this set. We assume that the problem specification is suffix closed, i.e., if a computation satisfies the problem specification, so do its suffixes. (Remark: Our definition of liveness is stronger than Alpern and Schneider's definition [7]: the two definitions become identical if the liveness specification is fusion closed; i.e., if computations hff; x; fli and hfi; x; ffii satisfy the liveness specification then computations hff; x; ffii and hfi; x; fli also satisfy the liveness specification, where ff; fi are finite computation prefixes, fl; ffi are computation suffixes, and x is a program state.) Invariant. An invariant of p is a state predicate S such that S 6=false, S is closed in p, and every computation of p starting from a state in S satisfies the problem specification of p. Informally, an invariant of p includes the states reached in fault-free executions of p. Note that may have multiple invariants. Techniques for the design of invariants have been articulated by Dijkstra [8], using the notion of auxiliary variables, and by Gries [9], using the heuristics of state predicate ballooning and shrinking. Techniques for the mechanical calculation of invariants have been discussed by Alpern and Schneider [10]. Convergence. A state predicate Q "converges to" R in p iff Q and R are closed in p and, starting from any state where Q holds, every computation of p has a state where R holds. Note that the converges-to relation is transitive. Lemma 2.1. If Q converges to R in p and every computation of p starting from states where R holds satisfies a liveness specification, then every computation of p starting from states where Q holds satisfies that liveness specification. Proof. Consider a computation c of p starting from a Q state. Since Q converges to R in p, c has a suffix c 1 starting from an R state. Since every computation of p starting from an R state satisfies the liveness specification, c 1 has a suffix c 2 that is identified by the liveness specification. is also a suffix of c, it follows that c also satisfies the liveness specification. Thus, every computation of p starting from a Q state satisfies that liveness specification. Faults. The faults that a program is subject to are systematically represented by actions whose execution perturbs the program state. We emphasize that such representation is possible notwithstanding the type of the faults (be they stuck-at, crash, fail-stop, omission, timing, performance, or Byzantine), their nature (be they permanent, transient, or intermittent), their observability (be they detectable or not), or their repairability (be they correctable or not). In some cases, such representation of faults introduces auxiliary variables. For example, to represent a fail-stop fault as a state perturbation, we introduce an auxiliary variable up. Each action is restricted to execute only when up is true. The fail-stop fault is represented by the action that changes up from true to false, thereby disabling all the actions in a detectable manner. Moreover, the repair of a fail-stopped program can be represented by the fault action that changes up from false to true, and initializes the state of j. (This initialization may retain the state before the process fail-stopped provided that information is on a non-volatile storage, or it may initialize it to some predetermined value. We ignore these details as they depend on the problem at hand.) In other words, fail-stop and repair faults are respectively represented by the following fault actions: Fail-stop :: up \Gamma! up := false Repair :: :up \Gamma! up := true; f initialize the state of the process g To represent a Byzantine fault as a state perturbation, we introduce an auxiliary variable b. The specified actions of the program are restricted to execute only when b is false, i.e., when the program is non-Byzantine. If b is true, i.e., if the program is Byzantine, the program is allowed to execute actions that can change its state arbitrarily. Thus, the Byzantine fault is represented by the action that changes b from false to true, thereby enabling the program to enter a mode where it executes actions that change its state arbitrarily. In other words, Byzantine fault is represented by the following Byzantine :: :b \Gamma! b := true Fault-span. A fault-span of program p for a fault-class F is a predicate T such that T is closed in p and F . Informally, the fault-span includes the set of states that p reaches when executed in the presence of actions in F . Note that p may have multiple fault-spans for F . If program p with invariant S is subject to a fault-class F , the resulting states of p may no longer satisfy S. However, these states satisfy fault-span of p, say T . Moreover, every state in S also satisfies T . Fault-Tolerance: masking and Nonmasking. We are now ready to give a formal definition of fault-tolerance [5]. Instantiations of this definition yield definitions of masking and nonmasking fault-tolerance. Let p be a program, F be a set of fault actions, and S be an invariant of p. We say that "p is F -tolerant for S" iff there exists a state predicate T of p such that the following three conditions hold: Closure: T is closed in p and F Convergence: T converges to S in p This definition may be understood as follows. At any state where the invariant, S, holds, executing an action in p yields a state where S continues to hold, but executing an action in F may yield a state where S does not hold. Nonetheless, the following three facts are true about this last the fault-span, holds, (ii) subsequent execution of actions in p and F yields states where T holds, and (iii) when actions in F stop executing, subsequent execution of actions in p alone eventually yields a state where S holds, from which point the program resumes its intended execution. When the definition is instantiated so that the fault-span T is identical to the invariant S, we get that p is masking F -tolerant for S. And when the definition is instantiated so that T differs from S, we get that p is nonmasking F -fault-tolerant for S. In the rest of this paper, the predicate S p denotes an invariant of program p. Moreover, the predicate T p denotes a fault-span predicate for a program p that is F -tolerant for S p . Finally, when the fault-class F is clear from the context, we omit mentioning F ; thus, "masking tolerant" abbreviates "masking F -tolerant". 3 A Method for Designing masking Tolerance From the definitions in the previous section, we observe that masking and nonmasking fault-tolerance are related as follows. Theorem 3.1. For any program p, If there exists S p and T p such that p is nonmasking F -tolerant for S p and every computation of p starting from a state where T p holds satisfies the safety specification of p Then there exists S p such that p is masking F -tolerant for S p . Proof. Let S np and T np be state predicates satisfying the antecedent. Then every computation of p starting from a state where S np holds satisfies its problem specification, and starting from a state where T np holds satisfies its safety specification. From Lemma 2.1, it follows that every computation of p starting from a T np state satisfies its problem specification. Thus, choosing S p =T np satisfies the consequent. The Method. Theorem 3.1 suggests that an intolerant program can be made masking tolerant in two stages: In the first stage, the intolerant program is transformed into one that is nonmasking tolerant for, say, the invariant S np and the fault-span T np . In the second stage, the tolerance of resulting program is enhanced from nonmasking to masking, as follows. The nonmasking tolerant program is transformed so that every computation upon starting from a state where T np holds, in addition to eventually reaching a state where S np holds, also satisfies the safety specification of the problem at hand. We address the details of both stages, next. Stage 1. For a fault-intolerant program, say p, the problem specification is satisfied by computations of p that start at a state where its invariant holds but not necessarily by those that start at a state where its fault-span holds. Hence, to add nonmasking tolerance to p, a program component is added to p that restores it from fault-span states to invariant states. We call the program component added to p for nonmasking tolerance a corrector. Well-known examples of correctors include reset procedures, rollback-recovery, forward recovery, error correction codes, constraint (re)satisfaction, voters, exception handlers, and alternate procedures in recovery blocks. The design of correctors has been studied extensively in the literature. We only note that correctors can be designed in a stepwise and hierarchical fashion; in other words, a large corrector can be designed by parallel and/or sequential composition of small correctors. One simple parallel composition strategy is to superpose small correctors on others. An example of a sequential composition strategy, due to Arora, Gouda, and Varghese [11], is to order the small correctors in a linear manner (or, more generally, a well-founded manner) such that each corrector does not interfere with the recovery task of the correctors lower than it in the chosen ordering. For a detailed discussion of corrector compositions, we refer the reader to [12]. Stage 2. For a nonmasking program, say np, even though the problem specification is satisfied after computations of np converge to invariant states, the safety specification need not be satisfied in all computations of np that start at fault-span states. Therefore, in the second stage, we restrict the actions of np so that the safety specification is preserved during the convergence of computations of np to invariant states. By Theorem 3.1, it follows that the resulting program is masking tolerant. To see that restriction of actions of np is sufficient for preserving safety during convergence, recall that the safety specification essentially rules out certain finite prefixes of computation of np. Now consider any prefix of a computation of np that is not ruled out by the safety specification: Execution of an action following this prefix increases the length of the computation prefix by one. As long as the elongated prefix is not one of the prefixes ruled out by the safety specification, safety is not violated. In other words, it suffices that whenever an action is executed, the resulting prefix be one that is not ruled out by the safety specification. It follows that there exists, for each action of np, a set of computation prefixes for which execution of that action preserves the safety specification. Assuming the existence of auxiliary state (which in the worst case would record the history of the computation steps), for each action of np, there exists a state predicate that is true in exactly those states where the execution of that action preserves safety. We call this state predicate the safe predicate of that action. It follows that if an action is executed in a state where its safe predicate is satisfied, safety is preserved. The restriction of the actions of np so as to enhance the tolerance of np to masking can now be stated precisely. Each action of np is restricted to execute only when its safe predicate holds. Moreover, for each action of np, the detection of its safe predicate may require the addition of a program component to np. We call a program component added to np for detecting that the safe predicate of an action holds a detector. Well-known examples of detectors include snapshot procedures, acceptance tests, error detection codes, comparators, consistency checkers, watchdog programs, snooper programs, and exception conditions. Analogous to the compositional design of large correctors, large detectors can be designed in a stepwise and hierarchical fashion, by parallel and/or sequential composition of small detectors. Thus, in sum, the second stage adds at most one detector per action of np and restricts each action of np to execute only when the detector of that action witnesses that its safe predicate holds. Before concluding our discussion of this stage, we make three observations about its application: 1. The safe predicate of several program actions is trivially true; 2. The safe predicate of most other actions requires only simple detector components, which introduce only little history state to check the safe predicate, and 3. If the problem specification is fusion closed and suffix closed, then no history state is required to check the safe predicate. Observation (1) follows from the fact that the actions of masking tolerant programs can be conceptually characterized as either "critical" or "noncritical", with respect to the safety specification. Critical actions are those actions whose execution in the presence of faults can violate the safety specification; hence, only they require non-trivial safe predicates. In other words, the safe predicate of all non-critical actions is merely true. For example, in terminating programs, e.g. feed-forward circuits or database transactions, only the actions that produce an output or commit a result are critical. In reactive programs, e.g. operating systems or plant controllers, only the actions that control progress while maintaining safety are critical. In the rich class of "total" programs for distributed systems [13], e.g. distributed consensus, infima finding, garbage collection, global function computation, reset, routing, snapshot, and termination detection, only the "decider" actions that declare the outcome of the computation are critical. Observation (2) follows from the fact that conventional specification languages typically yield safety specifications that are tested on the current state only or on the current computation step only; i.e., the set of finite prefixes that their safety specifications rule out can be deduced from the last or their last two states of computation prefixes. Thus, most safety specifications in practice do not require maintenance of unbounded "history" variables for detection of the safe predicates of each action. Observation (3) follows from the fact that if the problem specification is fusion closed and suffix closed then the required history information already exists in the current state. A proof of this observation is presented in [12]. Verification obligations. The addition of corrector and detector components as described above may add variables and actions to an intolerant program and, hence, the invariant and the fault-span of the resulting program may be different from those of the original program. The addition of corrector and detector components thus creates some verification obligations for the designer. Specifically, when a corrector is added to an intolerant program, the designer has to ensure that the corrector actions and the intolerant program actions do not interfere with each other. That is, even if the corrector and the fault-intolerant program execute concurrently, both accomplish their tasks: The corrector restores the intolerant program to a state from where the problem specification of the intolerant program is (re)satisfied. And starting from such a state, the intolerant program satisfies its problem specification. Similar obligations are created when detectors are added to a nonmasking program. Even if the detectors and the nonmasking program are executed concurrently, the designer has to ensure that the detector components and the components of the nonmasking program all accomplish their respective tasks. Another set of verification obligations is due to the fact that the corrector and detector components are themselves subject to the faults that the intolerant program is subject to. Hence, the designer is obliged to show that these components accomplish their task in spite of faults. More precisely, the corrector tolerates the faults by ensuring that when fault actions stop executing it eventually restores the program state as desired. In other words, the corrector is itself nonmasking tolerant to the faults. And each detector tolerates the faults by never falsely witnessing its detection predicate, even in the presence of the faults. In other words, each detector is itself masking tolerant to the faults. As can be expected, our two-stage design method can itself be used to design masking tolerance in the detectors, if their original design did not yield masking tolerant detectors. Adding detectors components by superposition. One way of simplifying the verification obligations is to add components to a program by superposing them on the program: if a program p is designed by a superposition on the program q, then it is trivially true that p does not interfere with q (although the converse need not be true, i.e., q may interfere with p). In particular, superposition is well-suited for the addition of detector components to a nonmasking tolerant program, np, in Stage 2, since detectors need only to read (but not update) the state of np. (It is for this reason that we have stated the definition of programs in Section 2 in terms of superposition.) Thus, the detectors do not interfere with the tasks of the corrector components in np. When superposition is used, the verification of the converse obligation, i.e. that np does not interfere with the detectors, may be handled as follows. Ensure that the corrector in np terminates after it restores np to an invariant state and that as long as it has not terminated it prevents the detectors from witnessing their safe predicate. Aborting the detectors during the execution of the corrector guarantees that the detectors never witness their safe predicate incorrectly, and the eventual termination of the corrector guarantees that eventually detectors are not prevented from witnessing their safe predicate. More specifically, the simplified verification obligations resulting from superposition are explained from Theorems 3.2 and 3.3. Let program p be designed by superposition on q such that T p ) T q Theorem 3.2. If q is nonmasking F -tolerant for S q , then T p converges to S q in p. Theorem 3.3. If q is nonmasking F -tolerant for S q , then converges to S p in p) converges to S p in p) Proof: Since q is nonmasking fault-tolerant, T q converges to S q in q. Since p is designed by a superposition on q, it follows that (T p - T q converges to T p - S q ). Since the converges-to relation is transitive and (T p - S q converges to S p - S q ), it follows that (T p - T q converges to S p - S q ), i.e., converges to S p in p. Theorems 3.2 and 3.3 imply that if p is designed by superposition on a nonmasking tolerant program q, then to reason about p, it suffices to assume that q always satisfies its invariant S q , even in the presence of faults. For a discussion of alternative strategies for verifying interference freedom, we refer the reader to [12]. In this section, we demonstrate that our method is well suited for the design of classical examples of masking tolerance, which span a variety of fault-classes. Specifically, our examples of masking tolerance achieve Byzantine agreement in the presence of Byzantine failure, data transfer in the presence of message loss in network channels, and triple modular redundancy (TMR) in the presence of input corruption. Notation. For convenience in presenting these designs, we will partition the actions of a program into "processes". 4.1 Example agreement Recall the Byzantine agreement problem: A unique process, the general, g, asserts a binary value d:g. Every process j in the system is required to eventually finalize its decision such that the following two conditions hold: (1) if g is non-Byzantine, the final decision reached by every non- Byzantine process is identical to d:g; and (2) even if g is Byzantine, the final decisions reached by all non-Byzantine processes are identical. Faults corrupt processes permanently and undetectably such that the corrupted processes are Byzantine. It is well known that masking tolerant Byzantine agreement is possible iff there are at least 3f+1 processes, where f is the number of Byzantine processes [14]. For ease of exposition, we will restrict our attention to the case where the total number of processes (including g) is 4 and, hence, f is 1. A generalization for multiple Byzantine faults is presented elsewhere [15]. As prescribed by our method, we will design the masking tolerant solution to the Byzantine agreement problem in two stages. Starting with an intolerant program for Byzantine agreement, we will first transform that program to add nonmasking tolerance, and subsequently enhance the tolerance to masking. Intolerant Byzantine agreement. The following simple program suffices for agreement but not for tolerance to faults: Process g is assumed to have a priori finalized its decision d:g. Each process j other than g receives the value d:g from process g and then finalizes its decision to that value. To this end, the program maintains two variables for each process j: a boolean f:j that is true iff j has finalized its decision, and d:j whose value denotes the decision of j. The program has two actions for each process j. The first action, IB1, copies d:g into the decision variable d:j: to denote that j has not yet copied d:g, we add a special value ? to the domain of d:j; thus, j copies d:g only if d:j is ?. The second action, IB2, finalizes the decision d:j: if j has copied its decision by truthifying f:j. Formally, the actions of the intolerant program, IB, are as follows: Invariant. In program IB, g has a priori finalized its decision. Moreover, when a process finalizes its decision, d:j is different from ?, and the final decision of each non-Byzantine process is identical to d:g. Hence, the invariant of program IB is S IB , where Fault Actions. The faults in this example make one process Byzantine, provided that no process is Byzantine. As discussed in Section 2, these faults would be represented by the following fault action at each j : Nonmasking tolerant Byzantine agreement. Program IB is intolerant because if g becomes Byzantine before all processes have finalized their decisions, g may keep changing d:g arbitrarily and, hence, the final decisions reached by the non-Byzantine processes may differ. We now add nonmasking tolerance to IB so that eventually the decisions reached by all non-Byzantine processes are identical. Since IB eventually reaches a state where the decisions of all processes differ from ? (i.e., are 0 or 1), it follows that eventually the decisions of at least two of the three processes other than g will be identical. Hence, if all of these processes ensure that their decision is the same as that of the majority, the resulting program will be nonmasking tolerant. Our nonmasking tolerant program consists of four actions for each process j: the first two are identical to the actions of IB. The third action, NB3, is executed by j only if it is Byzantine: this action nondeterministically changes d:j to either 0 or 1 and f:j to either true or false. The fourth action, NB4, changes the decision of j to the majority of the three processes. Formally, the actions of the nonmasking program, NB, are as follows: \Gamma! d:j; f:j := 0j1; truejfalse majdefined - d:j 6=maj \Gamma! d:j := maj where, majdefined Remark. A formula (operation may be read as the value of obtained by performing the (commutative and associative) operation on the X:j values for all j (in this case j is a process) that satisfy R:j. As a special case, when operation is a conjunction, we when operation is a disjunction, we may be read as if R:j is true then so is X:j, and may be read as there exists a process where both R:j and X:j are true. Moreover, if R:j is true, i.e., X:j is computed for all processes, we omit R:j. This notation is generalized from [16]. Invariant and fault-span. As in program IB, in program NB, when any non-Byzantine process finalizes its decision, d:j 6= ?. Also, if g remains non-Byzantine, all other non-Byzantine processes reach the same decision value as process g. Hence, the fault-span of NB, TNB is: Observe that if g is non-Byzantine, starting from any state in TNB , the nonmasking program works correctly. Also, if g is Byzantine, the nonmasking program works correctly if it starts in a state where all processes have correctly finalized their decisions. Hence, the invariant of program NB is Enhancing the tolerance to masking. Program NB is not yet masking tolerant as a non- Byzantine process j may first finalize its decision incorrectly, and only later correct its decision to that of the majority of the other processes. Hence, to enhance the tolerance of NB to masking, it suffices that j finalize its decision only when d:j is the same as the majority. The masking program thus consists of four actions at each process j: the three actions are identical to actions NB1;NB3 and NB4, and the fourth action, MB2 is restricted so that j finalizes its decision only when d:j is the same as the majority. Formally, the actions of the masking program, MB, are as follows: majdefined - d:j =maj \Gamma! f:j := true Invariant. The fault-span of the nonmasking program, TNB , is implied by the invariant, SMB , of the masking program. Also, in SMB , j finalizes its decision only when d:j is the same as that of the majority. Thus, SMB is: Theorem 4.1 The Byzantine agreement program MB is masking fault-tolerant for invariant SMB . 4.2 Example 2 : Data transfer Recall the data transfer problem: An infinite input array at a sender process is to be copied, one array item at a time, into an infinite output array at a receiver process. The sender and receiver communicate via a bidirectional channel that can hold at most one message in each direction at a time. It is required that each input array item be copied into the output array exactly once and in the same order as sent. Moreover, eventually the number of items copied by the receiver should grow unboundedly. Data transfer is subject to the faults that lose channel messages. As before, we will design the masking tolerance for data transfer in two stages. The resulting program is the well known alternating-bit protocol. Intolerant program. Iteratively, a simple loop is followed: sender s sends a copy of one array item to receiver r. Upon receiving this item, r sends an acknowledgment to s, which enables the next array item to be sent by s and so on. To this end, the program maintains binary variables rs in s and rr in rs is 1 if s has received an acknowledgment for the last item it sent, and rr is 1 if the r has received an item but has not yet sent an acknowledgment. The 0 or 1 items in transit from s to r are denoted by the sequence cs, and the 0 or 1 acknowledgments in transit from r to s are denoted by the sequence cr. Finally, the index in the input array corresponding to the item that s will send next is denoted by ns, and the index in the output array corresponding to the item that r last received is denoted by nr. The intolerant program contains four actions, the first two in s and the last two in r. By ID1, s sends an item to r, and by ID2, s receives an acknowledgment from r. By ID3, r receives an item from s, and by ID4, r sends an acknowledgment to s. Formally, the actions of the intolerant program, ID, are as follows (where c1 ffi c2 denotes concatenation of sequences c1 and c2): ID2 :: cr 6=hi \Gamma! rs; cr; ns := ns Remark. For brevity, we have ignored the actual data transfered between the sender and the receiver: we only use the array index of that data. Invariant. When r receives an item, ns holds, and this equation continues to hold until s receives an acknowledgment. When s receives an acknowledgment, ns is exactly one larger than nr and this equation continues to hold until r receives the next item. Also, if cs is nonempty, cs contains only one item, hnsi. Finally, in any state, exactly one of the four actions is enabled. Hence, the invariant of program ID is, S ID , where rs Fault Actions. The faults in this example lose either an item sent from s to r or an acknowledgment sent from r to s. The corresponding fault actions are as follows: cs 6=hi \Gamma! cs := tail(cs) cr 6=hi \Gamma! cr := tail(cr) Nonmasking tolerant program. Program ID is intolerant as it deadlocks when a fault loses an item or an acknowledgment. Hence, we add nonmasking tolerance to this fault by adding an action by which s detects that an item or acknowledgment has been lost and recovers ID by retransmitting the item. Thus, the nonmasking program consists of five actions; four actions are identical to the actions of program ID, and the fifth action retransmits the last item that was sent. This action is executed when both channels, cs and cr, are empty, and rs and rr are both zero. In practice, this action can be implemented by waiting for a some predetermined timeout so that the sender can be sure that either the item or the acknowledgment is lost, but we present only the abstract version of the action. Formally, the actions of the nonmasking program, ND, are as follows: \Gamma! cs := cs ffi hnsi Fault-span and invariant. If an item or an acknowledgment is lost, the program reaches a state where cs and cr are empty and rs and rr are both equal to zero. Also, even in the presence of faults, if cs is nonempty, it contains exactly the item whose index in the input array is hnsi. Thus, the fault-span of the nonmasking program is rs and the invariant is the same as the invariant of ID, i.e., Enhancing the tolerance to masking. Program ND is not yet masking tolerant, since r may receive duplicate items if an acknowledgment from r to s is lost. Hence, to enhance the tolerance to masking, we need to restrict the action ID3 so that r copies an item into the output array iff it is not a duplicate. Upon receiving an item, if r checks that nr is exactly one less than the index number received with the item, r will receive every item exactly once. Thus, we can enhance its tolerance to masking by adding such a check to program ND. However, this check forces the size of the message sent from the s to r to grow unboundedly. However, we can exploit the fact that in ND, ns and nr differ by at most 1, in order to simulate this check by sending only a single bit with the item as follows. Process s adds one bit, bs, to every item it sends such that the bit values added to two consecutive items are different and the bit values added to an item and its duplicates are the same. Thus, to detect that a message is duplicate, r maintains a bit, br, that denotes the sequence number of the last message it received. It follows that an item received by r is a duplicate iff br is the same as the sequence number in that message. The masking program consists of five actions. These actions are as follows: \Gamma! rs; cs := 0; cs ffi hns; bsi MD2 :: cr 6=hi \Gamma! rs; cr; ns; bs := ns \Gamma! cs := cs ffi hns; bsi \Gamma! if ((head(cs)) 2 6=br) then cs; rr := tail(cs); 1 \Gamma! Remark. Observe that in the masking program, the array index ns and nr need not be sent on the channel as it suffices to send the bits bs and br. With this modification, the resulting program is the alternating bit protocol. Invariant. In any state reached in the presence of program and fault actions, if cs is nonempty, cs has exactly one item, hns; bsi. Also, when r receives an item, nr =ns holds, and this equation continues to hold until s receives an acknowledgment. Moreover, bs is the same as ns mod 2, br is the same as nr mod 2, and exactly one of the five actions is enabled. Finally, nr is the same as ns or nr is one less than ns. Thus, the invariant of the masking program is SMD , where rs bs=(ns mod ns Theorem 4.2. The alternating-bit program MD is masking tolerant for invariant SMD . 4.3 Example 3 : TMR Recall the TMR problem: Three processes share an output, out. A binary value is input to in:j, for each process j. It is required that the output be set to this binary value. Faults corrupt the input value of any one of the three processes. Intolerant TMR. In the absence of faults, it suffices that out be set to in:j, for any process j. Hence, the action of program IR in each process j is as follows (where out =? denotes that the output has not yet been set): Fault actions. In this example, the faults corrupt the input value in:j of at most one process. They are represented by the following fault actions, one for each j (where k also ranges over the Nonmasking TMR. Program IR is intolerant since out may be set incorrectly from a corrupted in:j. Therefore, to add nonmasking tolerance to IR, we add a corrector that eventually corrects out. Since at most one in:j is corrupted, the correct output can differ from at most one in:j. Hence, if out differs from the in:j of two processes, the corrector resets out to the in:j value of those two. Thus, the nonmasking program, NR, consists of two actions in each process j: action NR1 is the same as IR1 and action NR2 is the corrector. Formally, these two actions are as follows (where \Phi denotes modulo 3 addition): \Gamma! out := in:j \Gamma! out := in:j Enhancing the tolerance to masking. Program NR is not yet masking tolerant since out may be set incorrectly before being corrected. Therefore, to enhance the tolerance to masking, we restrict the action NR1 so that the output is always set to an uncorrupted in:j. A safe predicate for this restriction of action NR1 is Restricting action NR1 with this safe predicate yields a stronger version of action NR2, thus the resulting masking tolerant program MR consists of only one action for each j: \Gamma! out := in:j Invariant. In program MR, if out is equal to in:j for some j, then there exists another process whose input value is the same as in:j. Hence, the invariant of program MR is, SMR , where Theorem 4.3 The triple modular redundancy program MR is masking tolerant for invariant SMR . In this section, we design a new and improved masking tolerant solution for the mutual exclusion problem using our two-stage method. Recall the mutual exclusion problem: Multiple processes may each access their critical sections provided that at any time at most one process is accessing its critical section. Moreover, no process should wait forever to access its critical section, assuming that each process leaves its critical section in finite time. We assume that the processes have unique integer ids. At any instant, each process is either "up" or "down". Only up processes can execute program actions. Actions executed by an up process j may involve communication only with the up processes connected to j via channels. Channels are bidirectional. A fault fail-stops one of the processes, i.e., renders an up process down. Fail-stops may occur in any (finite) number, in any order, at any time, and at any process as long as the set of up processes remains connected. One class of solutions for mutual exclusions is based on tokens. In token-based solutions, a unique token is circulated between processes, and a process enters its critical section only if (but not necessarily if) it has the token. To ensure that no process waits forever for the token, a fair strategy is chosen by which if any process requests access to its critical section then it eventually receives the token. An elegant token-based program is independently due to Raymond [17] and Snepscheut [18]; this program uses a fixed tree to circulate the token. The case study is organized as follows. In Section 5.1, we recall (an abstract version of) the intolerant mutual exclusion program of Raymond and Snepscheut. In Section 5.2, we transform this fault-intolerant program into a nonmasking tolerant one by adding correctors. Finally, in Section 5.3, we enhance the tolerance to masking by adding detectors. The resulting solution is compared with other masking tolerant token-based mutual exclusion solutions in the next section. 5.1 The Fault-Intolerant Program The processes are organized in a tree. Each process j maintains a variable P:j, to denote the parent of j in this tree; a variable h:j, to denote the holder process of j which is a neighbor of j in the direction of the process with the token; and a variable Request:j, to denote the set of requests that were received from the neighbors of j in the tree and that are pending at j. The program consists of three actions for each process, the first for making or propagating to the holder process a request for getting the token; the second for transmitting the token to satisfy a pending request from a neighbor; and the third for accessing the critical section when holding the token. The actions are as follows: (j needs to request critical section - Request:j 6= OE) \Gamma! \Gamma! h:k; h:j := j; j; Request:k \Gamma! access critical section These actions maintain the holder relation so that it forms a directed tree rooted at the process that has the token. The holder relation, moreover, conforms to the parent tree; i.e., if k is the holder of are adjacent in the tree. Thus, the invariant of the fault-intolerant program is S IM , where (j where P N 5.2 A Nonmasking Tolerant Version In the presence of faults, the parent tree used by IM may become partitioned. As a result, the holder relation may also become inconsistent. Moreover, the token circulated by IM may be lost, e.g., when the process that has the token (i.e., whose holder equals itself) fail-stops. Hence, to add nonmasking tolerance to fail-stops, we need to add a corrector that restores the parent tree and the holder tree. We build this corrector by superposing two correctors: NT which corrects the parent tree and NH which corrects the holder tree. In particular, we ensure that in the presence of fail-stops eventually the parent tree is constructed, the holder relation is identical to the parent relation and, hence, the root process has the token. 5.2.1 Designing a Corrector NT for the parent Tree For a corrector that reconstructs the parent tree, we reuse Arora's program [19] for tree main- tenance. This program allows faults to yield program states where there are multiple trees and unrooted trees. Continued execution of the program ensures convergence to a fix-point state where there is exactly one rooted spanning tree. To deal with multiple trees, the program has actions that merge trees. The merge actions use an integer variable root:j, denoting the id of the process that j believes to be its tree root, as follows. A process j merges into the tree of a neighboring process k when root:k ?root:j. Upon merging, j sets root:j to be equal to root:k and P:j to be k. Also, j aligns its holder relation along the parent relation by setting h:j to k. Observe that, by merging thus, no cycles are formed and the root value of each process remains at most the root value of its parent. When no merge actions are enabled, it follows that all rooted processes have the same root value. To deal with unrooted trees, the program has actions that inform all processes in unrooted trees that they have no root process. These actions use a variable col:j, denoting the color of j, as follows. When a process detects that its parent has failed or the color of its parent is red, the process sets its color to red. When a leaf process obtains the color red, it separates from its tree and resets its color to green, thus forming a tree consisting only of itself. When a leaf separates from its tree, it aligns its holder relation along the parent relation by setting its holder to itself. Formally, the actions of the corrector NT for process j are as follows (Adj:j denotes the set of up neighbors of process j): \Gamma! col:j := red \Gamma! P:j; root:j; h:j := k; root:k; k Fault Actions. Formally, the fail-stop action for process j is as follows: failstop:: up:j \Gamma! up:j := false Fault-span and Invariant. In the presence of faults, the actions of NT preserve the acyclicity of the graph of the parent relation as well as the fact that the root value of each process is at most the root value of its parent. They also preserve the fact that if a process is colored red then its parent is also colored red. Thus, the fault-span of corrector NT is the predicate TNT , where the graph of the parent relation is a forest - After faults stop occurring, eventually the program ensures that if a process is colored red then all its children are colored red, i.e., all processes in any unrooted tree are colored red. Furthermore, the program reaches a state where all processes are colored green, i.e., no process is in an unrooted tree. Finally, the graph of parent relation forms a rooted spanning tree. In particular, the root values of all processes are identical. Remark. Henceforth, for brevity, we use the term ch:j to denote the children of j; the term j is a root to denote that the parent of j is j, col:j is green, and j is up; and the term nbrs(X) to denote the set of processes adjacent to processes in the set of processes X (including X). Formally, j is a root j (P:j =j - col:j =green - up:j), and 5.2.2 Designing a Corrector NH for the holder Tree After the parent tree is reconstructed, the holder relation may still be inconsistent, in two ways. (1) The holder of j need not be adjacent to j in the parent tree, or (2) the holder of j may be adjacent to j in the tree but the holder relation forms a cycle. Hence, the corrector NH that restores the holder relation consists of two actions: Action NH1 corrects the holder of j when (1) holds, by setting h:j to P:j. Action NH2 corrects the holder of j when (2) holds: if the parent of k is j, holder of j is k and the holder of k is j, j breaks this cycle by setting h:j to P:j. The net effect of executing these actions is that eventually the holder relation is identical to the parent relation and, hence, the root process has the token. \Gamma! h:j := P:j \Gamma! h:j := P:j Fault-Span and Invariant. The corrector NH ensures that the holder of j is adjacent to j in the parent tree and for every edge (j; P:j) in the parent tree, either h:j is the same as P:j, or h:(P:j) is the same as j, but not both. Thus, NH corrects the program to a state where 5.2.3 Adding the Corrector : Verifying Interference Freedom As described earlier, the corrector we add to IM is built by superposing two correctors NT and NH. NH updates only the holder relation and NT does not read the holder relation. Therefore, NH does not interfere with NT . Also, after NT reconstructs the tree and satisfies SNT , none of its actions are enabled. Therefore, NT does not interfere with NH. IM updates variables that are not read by NT . Therefore, IM does not interfere with NT . Also, NH reconstructs the holder relation by satisfying the predicates SNH1 :j and SNH2 :j for each process j. Both SNH1 :j and SNH2 :j are respectively preserved by IM . Therefore, IM does not interfere with NH. Finally, after the tree and the holder relation is reconstructed and (S NT - SNH ) is satisfied, actions of NT and NH are disabled. Therefore, NT and NH do not interfere with IM . It follows that the corrector consisting of both NT and NH ensures that a state satisfying (S NT - SNH ) is reached, even when executed concurrently with IM . Since (S NT - SNH may add the corrector to IM to obtain the nonmasking tolerant program NM , whose actions at process j are as follows: Fault-Span and Invariant. The invariant of program NM is the conjunction of SNT , and SNH . Thus, the invariant of NM is The fault-span of program NM is equal the TNT , i.e., Theorem 5.1. The mutual exclusion program NM is nonmasking fault-tolerant for SNM . 5.3 Enhancing the Tolerance to masking Actions NM5;NM7 and NM8 can affect the safety of program execution only when the process executing them sets the holder to itself, thereby generating a new token. The safe predicate that should hold before generation of a token is therefore the condition "no process has a token". Towards detection of this safe predicate, we exploit the fact that NM is nonmasking tolerant: if the token is lost, NM eventually converges to a state where the graph of the parent relation is a rooted tree and the holder of each processes is its parent. Hence, it suffices to check whether the program is at such a state. To perform this check, we let j initiate a diffusing computation whenever j executes action NM5;NM7 or NM8. Only when j completes the diffusing computation successfully, does it safely generate a token. Actions NM2 and NM3, which respectively let process k transmit a token to process j and its critical section, can affect the safety of program execution only if they involve a spurious token generated in the presence of fail-stops. The safe predicate that should hold before these actions execute would certify that the token is not spurious. Towards detection of this safe predicate, we exploit the fact that fail-stops are detectable faults and, hence, we can let the fail-stop of a process force its neighboring processes to participate in a diffusing computation. Recalling from above that a new token is safely generated only after a diffusing computation completes, we can define the safe predicate for NM2 to be "k is not participating in a diffusing computation" and for action NM3 to be "j is not participating in a diffusing computation". Observe that the safe predicate detection to be performed for the first set of actions (NM5;NM7; and NM8) is global, in that it involves the state of all processes, whereas the safe predicate detection to be performed for the second set of actions (NM2 and NM3) is local. We will design a separate detector for each set of actions, such that superposition of these detectors on NM yields a masking fault-tolerant program. 5.3.1 Designing the Global Detector, GD As discussed above, the global detector, GD, uses a diffusing computation to check if some process has a token. Only a root process can initiate a diffusing computation. Upon initiation, the root propagates the diffusing computation to all of its children. Each child likewise propagates the computation to its children, and so on. It is convenient to think of these propagations as a propagation wave. When a leaf process receives the propagation wave, it completes and responds to its parent. Upon receiving responses from all its children, the parent of the leaf likewise completes and responds to its parent, and so on. It is convenient to think of these completions as a completion wave. In the completion wave, a process responds to its parent with a result denoting whether the subtree rooted at that process has a token. Thus, when the root receives a completion wave, it can decide whether some process has a token by inspecting the result. The diffusing computation is complicated by the following situations: multiple (root) processes may initiate a diffusing computation concurrently, processes may fail-stop while the diffusing computation is in progress, and a process may receive a token after responding to its parent in a diffusing computation that it does not have the token. To deal with concurrent initiators, we let only the diffusing computation of the highest id process to complete successfully; those of the others are aborted, by forcing them to complete with the result false. Specifically, if a process propagating a diffusing computation observes another diffusing computation initiated by a higher id process, it starts propagating the latter and aborts the former diffusing computation by setting the result of its former parent (the process from it received the former diffusing computation) to false. This ensures that the former parent completes the diffusing computation of the lower id process with the result false. To deal with the fail-stop of a process, we abort any diffusing diffusing computations that the neighboring processes may be propagating: Specifically, if j is waiting for a reply from k to complete in a diffusing computation and k fail-stops then j cannot decide if some descendent of k has a token. Hence, upon detecting the fail-stop of k, j aborts its diffusing computation by setting its result to false. Finally, to deal with the potential race condition where a diffusing computation "misses" a token because the token is sent to some process that has already completed in the diffusing computation with the result true, we ensure that even if this occurs the diffusing computation completes at the initiator only with the result false. Towards this end, we modify the global detector as follows: A process completes in a diffusing computation with the result true only if all its neighbors have propagated that diffusing computation. And, the variable result is maintained to be false if the process ever had a token since the last diffusing computation was propagated. To see why this modification works, consider the first process, say j, that receives a token after it has completed in a diffusing computation with the result true. Let l denote the process that sent the token to j. It follows that l has at least propagated the diffusing computation and its result is false. Moreover, since j is the first process to receive a token after completing the diffusing computation with the result true, l can only complete that diffusing computation with the result false. Since the result of l is propagated towards the initiator of the diffusing computation in the completion wave, the initiator is guaranteed to complete the diffusing computation with the result false. In sum, the diffusing computation deals with each of these complications via an abort mechanism that, by setting the result of the appropriate processes to false, fails the appropriate diffusing computations. When the initiator of a diffusing computation completes with the result false, it starts yet another diffusing computation. Towards this end, the diffusing computation provides an initiation mechanism that lets a root process initiate a new diffusing computation. To distinguish between the different computations initiated by some process, we let each process maintain a sequence number that is incremented in every diffusing computation. Furthermore, when a process propagates a new diffusing computation, it resets its result to true provided that it does not have the token. From the above discussion, process j needs to maintain a phase, phase:j, a sequence number, sn:j, and a result, res:j. The phase of j is either prop or comp and denotes whether j is propagating a diffusing computation or it has completed its diffusing computation. The sequence number of distinguishes between successive diffusing computations initiated by a root process. Finally, the result of j denotes whether j completed its diffusing computation correctly or it aborted its diffusing computation. Actions for the global detector. The global detector consists of four actions, viz INIT , PROP , COMP , and ABORT . INIT lets process j initiate a diffusing computation by incrementing its sequence number. We specify here only the statement of INIT ; the conditions under which j executes INIT are specified later. propagate a diffusing computation when j and P:j are in the same tree and sn:j is different from sn:(P:j). If the holder relation of j is aligned along the parent relation and P:j is in the propagate phase, j propagates that diffusing computation and sets its result to true. Otherwise, completes that diffusing computation with the result false. COMP lets j complete a diffusing computation if all children have completed the diffusing computation and all neighbors have propagated or completed that diffusing computation. The result computed by j is set to true iff the result returned by all its children is true, all neighbors of j have propagated that diffusing computation, and the result of j is true. If the root completes a diffusing computation with the result true, the safe predicate has been detected and the root process can proceed to safely generate a new token and, consequently, change its result to false. ABORT lets j complete a diffusing computation prematurely with the result false. When j aborts a diffusing computation, j also sets the result of its parent to false to ensure that the parent of j completes its diffusing computation with the result false. We specify here only the statement of the conditions under which j executes ABORT are specified later. Formally, the actions of detector GD for process j are as follows: INIT (j) :: if (P:j =j) then phase:j; sn:j := prop; newseq(); res:j := true \Gamma! sn:j :=sn:(P:j); phase:j; res:j := prop; true else res:j := false COMP (j) :: phase:j =prop - \Gamma! res:j if (P:j else if (P:j =j - res:j) then res:j :=false if (P:j 2Adj:j) then res:(P:j) := false Remark. In the ABORT action, j synchronously updates the state of its parent in addition to its own. This action can be refined, since the parent of j completes its diffusing computation only after j completes its diffusing computation, so that j only updates its own state and P:j reads the state of j later. Fault Actions. When a process fail-stops, all of its neighbors abort any diffusing computation that they are propagating. Moreover, if the initiator aborts its diffusing computation it initiates a new one. Hence, the fault action is as denotes that the statement X:l is executed at all processes that satisfy R:l): \Gamma! up:j := false; Invariant. We relegate the invariant SGD of the global detector to Appendix A1. 5.3.2 Designing the Local Detector LD The safe predicate for action NM2 is "k is not participating in a diffusing computation"; that comp. The safe predicate for action NM3 is "j is not participating in a diffusing computation"; that is, phase:j =comp. Therefore, these actions are modified as follows: \Gamma! h:k; h:j := j; j; Request:k \Gamma! access critical section 5.3.3 Adding the Detectors : Verifying Interference Freedom Actions NM5;NM7; and NM8 are restricted to execute INIT , to initiate a diffusing computation whose successful completion, i.e. execution of COMP with result true will generate a new token. And, as described above, actions NM2 and NM3 are restricted with the local detectors, to obtain LD1 and LD2, respectively. We still need to verify that the composition is free from interference. Note that the global detector, GD, is a superposition on NM and, hence, GD does not interfere with NM . To ensure that NM does not interfere with GD, we restrict all actions of NM , other than NM5;NM7; and NM8, to execute ABORT . (The alert reader will note that this last restriction is overkill: some actions of NM need not be thus restricted, but we leave that optimization as an exercise for the reader.) As long as the correctors of NM are executing, GD is safely aborted. Once the correctors of NM terminate, GD makes progress. Hence, NM does not interfere with GD. Also, execution of GD eventually reaches a state where the phase of all processes is comp. Thus, LD does not interfere with NM , and since LD detects the safe predicate atomically, it is not interfered by NM and GD. Formally, the actions of the resulting masking tolerant program MM are as follows: Fault Actions. The fault action is identical to the fault action described in Section 5.3.1. Invariant. The invariant of program MM is the conjunction of TNM and SGD . Thus, the invariant of MM is Theorem 6.3. The mutual exclusion program MM is masking tolerant for SMM . Remark. A leader election program can be easily extracted from our mutual exclusion case study. To this end, we drop the variables h and Request from program MM . Thus, the resulting program consists of the corrector NT (actions MM4\Gamma6) and the detector GD (actions MM9 and MM10). In this program, a process is a leader iff it is a root and its phase is comp. This program is derived by adding detector GD to the nonmasking tolerant program NT . The detailed design of such a leader election program is presented in [20]. 6 Discussion and Concluding Remarks In this paper, we presented a compositional method for designing masking fault-tolerant programs. First, by corrector composition, a nonmasking fault-tolerant program was designed to ensure that, once faults stopped occurring, the program eventually reached a state from where the problem specification was satisfied. Then, by detector composition, the program was augmented to ensure that, even in the presence of the faults, the program always satisfied its safety specification. We demonstrated the method by designing classical examples of masking fault-tolerant programs. Notably, the examples covered a variety of fault-classes including Byzantine faults, message faults, input faults and processor fail-stops and repairs. Also, they illustrated the generality of the method, in terms of its ability to provide alternative designs for programs usually associated with other well-known design methods for masking fault-tolerance: Specifically, the TMR and Byzantine examples are usually associated with the method of replication or, more generally, the state-machine-approach for designing client-server programs [21]. The alternating-bit protocol example is usually associated with the method of exception handling or that of rollback-recovery -with the "timeout" action, MD5, being the exception-handler or the recovery-procedure. We found that judicious use of this method offers the potential for the design of improved masking tolerant solutions, measured in terms of the scope of fault-classes that are masked and/or the performance of the resulting programs. This is because, in contrast to some of the well-known design methods, the method is not committed to the overhead of replication; instead, it encourages the design of minimal components for achieving the required tolerance. And, in contrast to the sometimes ad hoc treatment of exception-handling and recovery procedures, it focuses attention on the systematic resolution of the interference between the underlying program and the added tolerance components. One example of an improved masking tolerant solution designed using the method is our token-based mutual exclusion program. In terms of performance, in the absence of faults, our program performs exactly as its fault-intolerant version (due to Raymond [17] and Snepscheut [18]) and thus incurs no extra overhead in this case. By way of contrast, the acyclic-graph-based programs of Dhamdhere and Kulkarni [22] and Chang, Singhal, and Liu [23] incur time overhead for providing fault-tolerance, even in the absence of faults. Also, in the tree based program of Agrawal and Abbadi [24], the amount of work performed for each critical section may increase when processes fail (especially when the failed processes are close to the tree root); in our program, failure of a process causes an overhead only during the convergence phase, but not after the program converges. Moreover, in terms of tolerance, our program is more tolerant than that of [24] (which in the worst case is intolerant to more than log n process fail-stops). We note in passing that our mutual exclusion program can be systematically extended to tolerate process repairs as well as channel failures and repairs. Also, it can be systematically transformed so that processes cannot access the state of their neighbors atomically but only via asynchronous message passing. For other examples of improved solutions designed using the method, the interested reader is referred to our designs for leader election [20], termination detection [20], and distributed reset [25]. We also note that although superposition was used for detector composition in our example designs, superposition is only one of the possible strategies for detector composition. The advantage of superposing the detectors on the underlying nonmasking tolerant program is the immediate guarantee that the detectors did not interfere with the closure and convergence properties of the underlying program. One useful extension of the method would be to design programs that are nonmasking tolerant to one fault-class and masking tolerant to another or, more generally, that possess multiple tolerance properties (see [12, 25, 26]). The design of such multitolerant programs is motivated by the insight that the fault-span of a program need not be unique [5]. Hence, multiple fault-spans may be associated with a program, for instance, if the program is subject to multiple fault-classes. It follows that the program can be nonmasking tolerant to one of these fault-classes and masking tolerant to another. More generally, we find that multitolerance has several practical applications [12]. Another useful extension would be to augment the method to allow "tolerance refinement", i.e., to allow refinement of a tolerant program from an abstract level to a concrete level while preserving its tolerance property. Tolerance refinement is orthogonal to the "tolerance addition" considered in the paper, which adds the desired masking tolerance directly at any desired (but fixed) level of implementation. With this extension we could, for instance, refine our mutual exclusion program so that neighboring processes communicate only via asynchronous message passing within the scope of the method itself. Finally, alternative design methods based on detector and corrector compositions would be worth studying. An alternative stepwise method would be to first perform detector composition and then perform corrector composition, which we view as designing masking tolerance via fail-safe tolerance [12]. Another alternative (but not stepwise) method would be to compose detectors and correctors simultaneously. It would be especially interesting to compare these methods with respect to design-complexity versus performance-complexity tradeoffs. Acknowledgments . We are grateful to Ted Herman for helpful comments on a preliminary version of this paper and thank the anonymous referees for their detailed, constructive suggestions. --R A compositional framework for fault tolerance by specification transformation. System structure for software fault tolerance. Dependable computing and fault tolerance: Concepts and terminology. Closure and convergence: A foundation of fault-tolerant comput- ing Parallel Program Design: A Foundation. Defining liveness. A Discipline of Programming. The Science of Programming. Proving boolean combinations of deterministic properties. Constraint satisfaction as a basis for designing nonmasking fault-tolerance Component based design of multitolerance. Structure of Distributed Algorithms. The Byzantine generals problem. Compositional design of multitolerant repetitive byzantine agree- ment Predicate calculus and program semantics. A tree based algorithm for mutual exclusion. Fair mutual exclusion on a graph of processes. Efficient reconfiguration of trees: A case study in the methodical design of nonmask- ing fault-tolerance Designing masking fault-tolerance via nonmasking fault-tolerance Implementing fault-tolerant services using the state machine approach: A tutorial A token based k resilient mutual exclusion algorithm for distributed systems. A fault tolerant algorithm for distributed mutual exclusion. An efficient fault-tolerant solution for distributed mutual exclusion Multitolerance in distributed reset. Multitolerant barrier synchronization. --TR --CTR Anil Hanumantharaya , Purnendu Sinha , Anjali Agarwal, A component-based design and compositional verification of a fault-tolerant multimedia communication protocol, Real-Time Imaging, v.9 n.6, p.401-422, December Ted Herman, Superstabilizing mutual exclusion, Distributed Computing, v.13 n.1, p.1-17, January 2000 I-Ling Yen , Farokh B. Bastani , David J. Taylor, Design of Multi-Invariant Data Structures for Robust Shared Accesses in Multiprocessor Systems, IEEE Transactions on Software Engineering, v.27 n.3, p.193-207, March 2001 Meng Yu , Peng Liu , Wanyu Zang, Specifying and using intrusion masking models to process distributed operations, Journal of Computer Security, v.13 n.4, p.623-658, July 2005 Sushil Jajodia , Paul Ammann , Catherine D. McCollum, Surviving Information Warfare Attacks, Computer, v.32 n.4, p.57-63, April 1999 Felix C. Grtner, Fundamentals of fault-tolerant distributed computing in asynchronous environments, ACM Computing Surveys (CSUR), v.31 n.1, p.1-26, March 1999

masking and nonmasking fault-tolerance;correctors;stepwise design formal methods;distributed systems;component based design;detectors

284742

Local Convergence of the Symmetric Rank-One Iteration.

We consider conditions under which the SR1 iteration is locally convergent. We apply the result to a pointwise structured SR1 method that has been used in optimal control.

Introduction . The symmetric rank-one (SR1) update [1] is a quasi-Newton method that preserves symmetry of an approximate Hessian (optimization problems) or Jacobian (nonlinear equations). The analysis in this paper is from the nonlinear equations point of view. Our purpose is to prove a local convergence result using the concept of uniform linear independence from [5], extend that result to structured updates where part of the Jacobian can be computed exactly, and then apply those results to the pointwise SR1 update considered in [14] in the context of optimal control. We we begin with a nonlinear equation in R N . We make the standard assumptions in nonlinear equations. Assumption 1.1. F has a root x . There is ffi ? 0 such that the Jacobian F 0 (x) exists and is Lipschitz continuous in the set with Lipschitz constant fl. F 0 (x ) is nonsingular. Later in this paper we will assume that F 0 (x) is symmetric near x and use the SR1 update to maintain symmetric approximations to F 0 (x ). From current approximations iteration computes a new point x+ by computing a search direction, and updating x c , Version of May 17, 1995. y North Carolina State University, Department of Mathematics and Center for Research in Scientific Computation, Box 8205, Raleigh, N. C. 27695-8205 (Tim Kelley@ncsu.edu). The research of this author was supported by National Science Foundation grant #DMS-9321938 and North Atlantic Treaty Organization grant #CRG 920067. Computing was partially supported by an allocation of time from the North Carolina Supercomputing Center. z Universit?t Trier, FB IV - Mathematik and Graduiertenkolleg Mathematische Optimierung, 54296 Trier, Germany (sachs@uni-trier.de). The research of this author was supported by North Atlantic Treaty Organization grant #CRG 920067. is updated to form B+ by setting and, if (y updating B c to obtain the update for B is skipped, so Observations have been reported [15], [4], [5], [19], [10], [18], [11], [20], [14], that indicate that SR1 can outperform BFGS in the context of optimization, where either the approximate Hessians can be expected to be positive definite or a trust-region framework is used [3], [4], [5]. One modification of SR1 that allows for a convergence analysis has been proposed in [17]. The SR1 update was considered in [2] where it was shown that the local superliner convergence theory in that paper did not apply, because the updates could become undefined. Preservation of symmetry is an attractive feature, as is the possibility of storing one vector per iterate in a matrix-free limited memory implementation. However, implementations that store a single vector for each iteration are known for Broyden's method [8], [13], and the BFGS method [21], which have much better understood convergence properties. The advantage of the SR1 method over others is in a reduction in the number of iterations. Such a reduction has been observed by many authors as we indicated above. As is standard, we update the approximate Jacobian only if for some oe 2 (0; 1) fixed. In many recent papers [5], [15], [3], on SR1, (1.5), with an arbitrary choice of oe, is one of the assumptions used to prove convergence of B n to The numerical results presented in [5], [15], and [3] use very small values of oe. In this paper, as was done in [14], we use a larger value of oe than in other treatments of the SR1 iteration as a way to improve stability. The estimates in x 2 can be used as a rough guide in selection of an appropriate value of oe. The numerical results in x 5 illustrate the benefits of varying oe. In this paper we show that if the initial approximations to the solution and Jacobian are sufficiently good and if the sequence of steps that satisfy (1.5) also satisfy a uniform linear independence condition [5], then the iteration will be locally linearly convergent and the sequence fB k g will remain near to F 0 (x ). A stronger uniform linear independence condition implies k-step superlinear convergence for some integer k. Thus, our goal is different from that in [5], [15], and [3], where convergence of the iteration to the solution was an assumption and conditions were given that guaranteed convergence of the approximate Jacobians to F 0 (x ). While the uniform linear independence condition may seem strong, it is a reasonable condition for very small problems. Such problems arise as part of the pointwise SR1 method proposed in [14] for certain optimal control problems and the results in this paper give insight that we use to improve the performance of the pointwise SR1 method. In x 2 we state and prove our basic convergence results. We consider structured updates in x 3 and the application to pointwise updates in x 4. Finally we report numerical results in x 5. 2. Basic Lemmas and Convergence Results. As we deal with local convergence only in this paper, we take full steps use the fact that which is a direct consequence of the definition and the equation for the quasi-Newton step B c s Hence the SR1 update can be We use the notation for errors in Jacobian and solution approximations. It may happen that many iterations take place between updates of B. We must introduce notation to keep track of those iterations that result in a change in B. We say that s n is a major step, x n+1 is a major iteration, and B n+1 a major update 1). In this case B n+1 6= B n . A step is a minor step if it is not a major step (and hence no update of B takes place local convergence theory must show that the new approximation B n+1 to the Jacobian at the solution is nonsingular. We do this by proving an analog of the "bounded deterioration" results used in the study of other quasi-Newton methods. However, an inherent instability must be kept in check and this is where the uniform linear independence becomes important. 2.1. Stability. We base our main result on several lemmas. The first simply summarizes some well known results in nonlinear equations [6], [13], [16], and has nothing to do with an assumption of symmetry of F 0 (x ) or any particular quasi-Newton method. Lemma 2.1. Let Assumption 1.1 hold and let ae 2 (0; 1) be given. Then there are ffl 0 and ffi 0 such that if x c and B c satisfy then B c is nonsingular and Moreover Lemma 2.2. Let the hypotheses of Lemma 2.1 hold. Then there is C 1 such that if c is a major step, then ks c k and ks c Proof. By (2.2) and hence, using (2.6), ks c k - ks c k This is (2.7) with C We apply (2.9) again to obtain and hence (2.8) follows from (2.6) and the the fact that C 1 ? fl. At this point we need to consider a sequence of major steps. Several minor steps in which (1.5) fails could lie between any two major steps, but they have no effect on the estimates of the approximate Jacobians. This is also the first place where symmetry of E (and hence of F 0 (x )) plays an important role. We remark that (2.11) and (2.12) differ from estimates of kE k k, used in other recent papers [5], [15] in that the assumption of good approximations to x and F 0 (x ) is used in a crucial way. The next lemma uses the general approach of [5] and the observation from [5] and [15] that only the major steps need be considered in the estimates. Lemma 2.3. Let Assumption 1.1 hold, let ae 2 (0; 1), and let ffl 0 and ffi 0 be such that the conclusions of Lemma 2.1 hold. Let - k - 0, x 0 , and B 0 be such that F 0 symmetric and Then at least - k major steps can be taken with B k+1 nonsingular for all Moreover, there is are the sequence of the first k - k major steps, iterations, and updates, then for any 1 - k - k, Also, for any 0 - ks Proof. We set We note that implies that ks We prove the lemma by induction on k. For and (2.14). We obtain (2.12) from (2.8) and (2.13). If (2.11) holds for all k ! K - k, then from (2.7) and (2.14), We use the induction hypothesis and C to obtain which proves the first inequality in (2.11). The second inequality from (2.10). apply Lemma 2.1 to conclude that at least - k major steps can be taken and k. Hence, for ks ks l k: Assuming that (2.12) holds for we note that (2.12) is a consequence of Lemma 2.2 if oeks K Every E k is symmetric because E 0 is. Hence, if we write then Combining (2.16) and (2.6) yields Hence, by (2.15), ks ks As in the proof of (2.11) we have and We apply (2.18), (2.19), and the induction hypothesis to (2.17) to get ks verifying (2.12). The estimates in Lemma 2.3 are analogs to the bounded deterioration results common in the quasi-Newton method literature. However, in this case the deviation of B k from F 0 (x ) is exponentially increasing and, at least according to the bounds in Lemma 2.3, can eventually become so large that convergence may be lost. The SR1 update has, however, a self-correcting property in the linear case [9] that has been exploited in much of the recent work on the method [15], [3], [5]. This self-correction property overcomes the instability indicated in the estimates (2.11) and (2.12). 2.2. Uniform Linear Independence. Our uniform linear independence assumption differs slightly from that in [5]. We only consider major steps and are not concerned at this point with the total number (major steps required to form the sequence of linearly independent major steps. Assumption 2.1. There are c min ? 0 and - k - n such that the hypotheses of Lemma 2.3 hold. Moreover from each of the sets of columns of normalized major steps a subsequence fv p can be extracted such that the matrix S p with columns fv p has minimum singular value at least c min . 2.3. Convergence Results. Using the assumptions above we can prove q-linear convergence directly using Lemma 2.3 and the methods for exploitation of uniform linear independence from [5]. Theorem 2.4. Let Assumptions 2.1 and the hypotheses of Lemma 2.3 hold. Let ae 2 (0; 1) be given. Then if sufficiently small the SR1 iterates converge q-linearly to x with q-factor at most ae. Proof. The proof is based on the simple observation that Lemma 2.3 states that the iteration may proceed through - major steps that then Assumption 2.1 implies that the iteration may continue. 2.3 implies that k and that by (2.14) ks ks k k Note that min Now enough so that then are replaced by E- k and e- k . Hence, we may continue the iteration by Lemmas 2.1 and 2.3, obtaining q-linear convergence with q-factor at most ae. 3. Structured Updates. In order to apply the convergence results to optimal control problems in the context of pointwise updates, as a next step, we extend the statements from the previous sections to the case of structured updates. We use the notational conventions of [7] in the formulation. Suppose that the Jacobian F 0 of F can be split into a Lipschitz continuous computable part C(x) and a part A(x) which will be approximated by a SR1 update: We define the SR1 update by where the step is computed by solving If we choose the secant condition (B+ holds and we obtain from (3.2) and (3.3) Hence the SR1 update can be written with a perturbation ~ F+ of F (x+ ) ~ c ~ We use the notation for errors in the Jacobian. Next we apply Lemma 2.1 to obtain a similar estimate Lemma 3.1. Let Assumption 1.1 hold and let ae 2 (0; 1) be given. Then there are ffl 0 and ~ ffi 0 such that if x c and B c for a structured SR1 update satisfy then defined and satisfies Moreover ks c Proof. Note that with a Lipschitz constant fl C for C. Then holds and since Lemma 2.1 holds for arbitrary approximations of the Jacobian, x+ exists and (3.8) holds. Observing that ks c k 2 proves (3.9) and completes the proof. The next lemma is an extension of Lemma 2.2. Before we can state the lemma we define when an update is skipped. We update B only if for some oe 2 (0; 1) fixed. The definition of minor and major steps is the same as that in x 2 with (3.10) playing the role of (1.5). Lemma 3.2. Let the hypotheses of Lemma 3.1 hold. Then there is C 1 such that if c is a major step, then ks c k and ks c Proof. For the structured updates we have ~ If we note that ks c k and (3.10) has been changed accordingly then the proof is the same as for Lemma 2.2 with and F (x) replaced by ~ F . In the next lemma symmetry is assumed which causes some additional changes for the proof. Lemma 3.3. Let Assumption 1.1 and the hypotheses of Lemma 3.1 hold. Let - be such that Then at least - k major steps can be taken with B k+1 nonsingular for all Moreover, there is are the sequence of the first k - k major steps, iterations, and updates, then for any 1 - k - k, Also, for any 0 - ks Proof. The first part of the induction proof for (3.15) is identical to the one for Lemma 2.3 and therefore omitted. Assuming that (3.16) holds for we note that (3.16) is a consequence of Lemma 3.2 if we have from (3.13) oeks K If we write ~ then by (3.9) and ~ Combining (3.18) and (3.9) yields Note that by assumption hence by definition (3.6) Hence, by (3.17), ks where, as in x 2 As in the proof of (2.11) we have and We apply (3.20), (3.21), and the induction hypothesis to (3.19) to get ks verifying (3.16). Using the definition of uniform linear independence from x 2.2 we may state the structured analog of Theorem 2.4. The proof is essentially identical to that of Theorem 2.4. Theorem 3.4. Let Assumptions 2.1 and the hypotheses of Lemma 3.3 hold. Let ae 2 (0; 1) be given. Then if sufficiently small the SR1 iterates converge q-linearly to x with q-factor at most ae. 4. Pointwise Structured Updates for Optimal Control. In order to apply the convergence results to optimal control problems in the context of pointwise updates, as a next step, we extend the statements from the previous sections to the case of pointwise structured updates. Our nonlinear equations represent the necessary conditions for the optimal control problem minimize over We set solves the adjoint equation We seek to solve the nonlinear system @ H u (x; u; t)C A =B A for which satisfy the boundary conditions We use the following assumption Assumption 4.1. f; L and their first and second partial derivatives with respect to x and u are continuous on IR N \Theta IR M \Theta [0; T ]. Under the above assumption, F is Fr'echet -differentiable and the Fr'echet -derivative is given by @ A =B @ @ A with dt and all other components as multiplication operators. into two parts F 0 containing all information from first derivatives @ A and A(z) consisting of second order derivatives @ All entries depend on time t so that A(z) is typically approximated by a family of quasi Newton updates depending also on time t @ A with We use a pointwise analog of (3.10) We update B at each t 2 [0; T ] by a structured SR1 update In order to justify the use of pointwise updates, we state the next lemma. Lemma 4.1. If B 0 is given of the form (4.3), then all B k defined by (4.6) are also multiplication operators with Proof. The proof is via induction and we show the step from B c to B+ . We write Note that by (4.4) Since the differentiation operator D appears linearly in F and in C it cancels in H This implies with the differentiability assumption on the data that pointwise holds for some OE 2 L 1 [0; T ]. Since B c is also in L 1 we have The decision in the update formula (4.6) shows that the components of B+ are measurable functions. They are also essentially bounded because either or using the choice of oe in (4.6) and (4.7) This completes the proof. The next Lemma gives pointwise estimates on the error in the Jacobian in the context of pointwise updates which will be used later for uniform estimates. Lemma 4.2. Assume that for some t 2 [0; T is a multiplication operator and let is a major step, then E+ (t) is also a multiplation operator and the following estimates hold: ks c (t)k and ks c (t)k)ks c (t)k: Proof. Observe that we can rewrite (4.4) pointwise By assumption, the last term is a multiplication operator and therefore does not contain the differentiation operator D. In first and the second term in parentheses the D cancels, so that we can estimate pointwise under the given smoothness assumptions on the data: ks c Hence we obtain from (3.13) using (4.12) that pointwise ks c (t)k ks c (t)k To estimate kE+ s c k recall that from the secant condition and therefore (4.9) holds. Furthermore, from (4.11) and ks c k 2 The next Lemma describes a linear rate estimate in a uniform norm. Lemma 4.3. For ae 2 (0; 1) there are ffl 0 and ffi 0 such that if s c is a major step for all t and B c for the pointwise structured SR1 update satisfies defined and satisfies Moreover for some fl; C ks c k1 ks c k 2(4.15) and Proof. The assumptions imply that is small so that Lemma 2.1 can be invoked to yield (4.13). From this we deduce ks c k1 . (4.8) gives ks c k1 which is (4.14). In the same way we use (4.9) and (4.10) to obtain (4.15) and (4.16), resp. 5. Numerical Results. We present some numerical results which illustrate the observations from the previous sections. Let us consider the following class of examples: First we set Furthermore, we set x . The initial data are given by @ We did update if is true. In [14] we used The discretization parameter comes from the discretization of the two-point boundary value problem by the trapezoid finite difference scheme used with a Richardson extrapolation to achieve 4th order accurate results, see e.g. [12]. This indicates that the termination criterion should be We approximate the norm in the discrete case also with accuracy of order 4. The numbers in column [No Upd %] give the percentage of the 121 3 \Theta 3-matrices which have not been updated at iteration k. The difference in the matrices is computed as follows: where kB(t)k F denotes the Frobenius-Norm in R 3\Theta3 . In Table 5.1 we report on the results for the SR-1 update where the choice of oe is based on the truncation error of the discretization scheme (h 4 - 0:6 \Theta 10 \Gamma5 ). Tables 5.2 and 5.3 show the effects of more conservative updating strategies. The analysis in the preceding sections indicates that a larger value of oe will keep smaller and might thereby allow for a more monotone iteration. While an increase by a factor of 100 did reduce the size of kE k k, it did not lead to any improvement in overall performance (see Table 5.2). Increasing oe by a factor of 1000 (see Table 5.3) led to a performance improvement of about 20%. In Table 5.2 we used a less stringent requirement for not updating the Hessians. In this table oe was increased by a factor of 10 2 . Note the reduced error in the Hessian updates in the early phase of the algorithm. Table 9 0.72865D-04 0.002 1.7 332.568 Table Table --R Analysis of a symmetric rank-one trust region method Testing a class of methods for solving minimization problems with simple bounds on the variables Numerical Methods for Nonlinear Equations and Unconstrained Optimization Convergence theorems for least change secant update methods Fast secant methods for the iterative solution of large nonsymmetric linear systems John Wiley and Sons An Algorithm for Optimizing Functions with Multiple Minima Numerical Solution of Two Point Boundary Value Problems Iterative Methods for Linear and Nonlinear Equations A pointwise quasi-Newton method for unconstrained optimal control problems A theoretical and experimental study of the symmetric rank one update Iterative Solution of Nonlinear Equations in Several Variables A new approach to the symmetric rank-one updating algorithm Yield optimization using a GaAs process simulator coupled to a physical device model On large scale nonlinear least squares calculations Compact storage of Broyden-class quasi-Newton matrices --TR --CTR P. Spellucci, A Modified Rank One Update Which Converges Q-Superlinearly, Computational Optimization and Applications, v.19 n.3, p.273-296, September 2001

optimal control;SR1 update;pointwise quasi-Newton method

284954

Approximation Algorithms for the Feedback Vertex Set Problem with Applications to Constraint Satisfaction and Bayesian Inference.

A feedback vertex set of an undirected graph is a subset of vertices that intersects with the vertex set of each cycle in the graph. Given an undirected graph G with n vertices and weights on its vertices, polynomial-time algorithms are provided for approximating the problem of finding a feedback vertex set of G with smallest weight. When the weights of all vertices in G are equal, the performance ratio attained by these algorithms is 4-(2/n). This improves a previous algorithm which achieved an approximation factor of $O(\sqrt{\log n})$ for this case. For general vertex weights, the performance ratio becomes $\min\{2\Delta^2, 4 \log_2 n\}$ where $\Delta$ denotes the maximum degree in G. For the special case of planar graphs this ratio is reduced to 10. An interesting special case of weighted graphs where a performance ratio of 4-(2/n) is achieved is the one where a prescribed subset of the vertices, so-called blackout vertices, is not allowed to participate in any feedback vertex set.It is shown how these algorithms can improve the search performance for constraint satisfaction problems. An application in the area of Bayesian inference of graphs with blackout vertices is also presented.

Introduction E) be an undirected graph and let be a weight function on the vertices of G. A cycle in G is a path whose two terminal vertices coincide. A feedback vertex set of G is a subset of vertices F ' V (G) such that each cycle in G passes through at least one vertex in F . In other words, a feedback vertex set F is a set of vertices of G such that by removing F from G, along with all the edges incident with F , a forest is obtained. A minimum feedback vertex set of a weighted graph (G; w) is a feedback vertex set of G of minimum weight. The weight of a minimum feedback vertex set will be denoted by -(G; w). The weighted feedback vertex set (WFVS) problem is defined as finding a minimum feedback vertex set of a given weighted graph (G; w). The special case where w is the constant function 1 is called the unweighted feedback vertex set (UFVS) problem. Given a graph G and an integer k, the problem of deciding whether -(G; 1) - k is known to be NP-Complete [GJ79, pp. 191-192]. Hence, it is natural to look for efficient approximation algorithms for the feedback vertex set problem, particularly in view of the recent applications of such algorithms in artificial intelligence, as we show in the sequel. Suppose A is an algorithm that finds a feedback vertex set FA for any given undirected weighted graph (G; w). We denote the sum of weights of the vertices in FA by w(FA ). The performance ratio of A for (G; w) is defined by RA (G; -(G; performance ratio r A (n; w) of A for w is the supremum of RA (G; w) over all graphs G with vertices and for the same weight function w. When w is the constant function 1, we call r A (n; 1) the unweighted performance ratio of A. Finally, the performance ratio r A (n) of A is the supremum of r A (n; w) over all weight functions w defined over graphs with n vertices. An approximation algorithm for the UFVS problem that achieves an unweighted performance ratio of 2 log 2 n is essentially contained in a lemma due to Erd-os and P'osa [EP62]. This result was improved by Monien and Schulz [MS81], where they achieved a performance ratio of O( log n). In Section 2, we provide an approximation algorithm for the UFVS problem that achieves an unweighted performance ratio of at most 4 \Gamma (2=n). Our algorithm draws upon a theorem by Simonovits [Si67] and our analysis uses a result by Voss [Vo68]. Actually, we consider a generalization of the UFVS problem, where a prescribed subset of the vertices, called blackout vertices, is not allowed to participate in any feedback vertex set. This problem is a subcase of the WFVS problem wherein each allowed vertex has unit weight and each blackout vertex has infinite weight. Our interest in graphs with blackout vertices is motivated by the loop cutset problem and its application to the updating problem in Bayesian inference which is explored in Section 4. In Section 3, we present two algorithms for the WFVS problem. We first devise a primal-dual algorithm which is based on formulating the WFVS problem as an instance of the set cover problem. The algorithm has a performance ratio of 10 for weighted planar graphs and 4 log 2 n for general weighted graphs. This ratio is achieved by extending the Erd-os- P'osa Lemma to weighted graphs. The second algorithm presented in Section 3 achieves a performance ratio of weighted graphs, where \Delta(G) is the maximum degree of G. This result is interesting for low degree graphs. A notable application of approximation algorithms for the UFVS problem in artificial intelligence due to Dechter and Pearl is as follows [DP87, De90]. We are given a set of its values from a finite domain D i . Also, for every are given a constraint subset R i;j ' D i \Theta D j which defines the allowable pairs of values that can be taken by the pair of variables Our task is to find an assignment for all variables such that all the constraints R i;j are satisfied. With each instance of the problem we can associate an undirected graph G whose vertex set is the set of variables, and for each constraint R i;j which is strictly contained in D i \Theta D j (i.e., R i;j there is an edge in G connecting x i and x j . The resulting graph G is called a constraint network and it is said to represent a constraint satisfaction problem. A common method for solving a constraint satisfaction problem is by backtracking, that is, by repeatedly assigning values to the variables in a predetermined order and then backtracking whenever reaching a dead end. This approach can be improved as follows. First, find a feedback vertex set of the constraint network. Then, arrange the variables so that variables in the feedback vertex set precede all other variables, and apply the backtracking procedure. Once the values of the variables in the feedback vertex set are determined by the backtracking procedure, the algorithm switches to a polynomial-time procedure solve-tree that solves the constraint satisfaction problem in the remaining forest. If solve-tree succeeds, a solution is found; otherwise, another backtracking phase occurs. The complexity of the above modified backtracking algorithm grows exponentially with the size of the feedback vertex set: If a feedback vertex set contains k variables, each having a domain of size 2, then the procedure solve-tree might be invoked up to 2 k times. A procedure solve-tree that runs in polynomial-time was developed by Dechter and Pearl, who also proved the optimality of their tree algorithm [DP88]. Consequently, our approximation algorithm for finding a small feedback vertex set reduces the complexity of solving constraint satisfaction problems through the modified backtracking algorithm. Furthermore, if the domain size of the variables varies, then solve-tree is called a number of times which is bounded from above by the product of the domain-sizes of the variables whose corresponding vertices participate in the feedback vertex set. If we take the logarithm of the domain size as the weight of a vertex, then solving the WFVS problem with these weights optimizes the complexity of the modified backtracking algorithm in the case where the domain size is allowed to vary. 2 The Unweighted Feedback Vertex Set Problem The best approximation algorithm prior to this work for the UFVS problem attained a performance ratio of O( log n) [MS81]. We now use some results of [Si67] and [Vo68] in order to obtain an approximation algorithm for the UFVS problem which attains a performance In fact, we actually consider a slight generalization of the UFVS problem where we mark each vertex of a graph as either an allowed vertex or a blackout vertex. In such graphs, feedback vertex sets cannot contain any blackout vertices. We denote the set of allowed vertices in G by A(G) and the set of blackout vertices by B(G). Note that when problem reduces to the UFVS problem. A feedback vertex set can be found in a graph G with blackout vertices if and only if every cycle in G contains at least one allowed vertex. A graph G with this property will be called a valid graph. The motivation for dealing with this modified problem is clarified in Section 4 where we use the algorithm developed herein to reduce the computational complexity of Bayesian inference. Throughout this section, G denotes a valid graph with a nonempty set of vertices V (G) which is partitioned into a nonempty set A(G) of allowed vertices, a possibly empty set B(G) of blackout vertices, and a set of edges E(G) possibly with parallel edges and self- loops. We use - a (G) as a short-hand notation for -(G; w) where w assigns unit weight to each allowed vertex and an infinite weight to each blackout vertex. A neighbor of v is a vertex which is connected to v by an edge in E(G). The degree \Delta G (v) of v in G is the number of edges that are incident with v in G. A self-loop at a vertex v contributes 2 to the degree of v. The degree of G, denoted \Delta(G), is the largest among all degrees of vertices in G. A vertex in G of degree 1 is called an endpoint . A vertex of degree 2 is called a linkpoint and a vertex of any higher degree is called a branchpoint. A graph G is called rich if every vertex in V (G) is a branchpoint. The notation \Delta a (G) will stand for the largest among all degrees of vertices in A(G) (a degree of a vertex in A(G) takes into account all incident edges, including those that lead to neighbors in B(G)). In a rich valid graph we have \Delta a (G) - 3. Two cycles in a valid graph G are independent if their vertex sets share only blackout vertices. Note that the size of any feedback vertex set of G is bounded from below by the largest number of pairwise independent cycles that can be found in G. A cycle \Gamma in G is called simple if it visits every vertex in V (G) at most once. Clearly, a set F is a feedback vertex set of G if and only if it intersects with every simple cycle in G. A graph is called a singleton if it contains only one vertex. A singleton is called self-looped if it contains at least one self loop; for a singleton we have -(G; it is self-looped and -(G; otherwise. A graph G is connected if for every two vertices there is a connecting path in G. Every graph G can be uniquely decomposed into isolated connected components Similarly, every feedback vertex set F of G can be partitioned into feedback vertex sets such that F i is a feedback vertex set of G i . Hence, - a A 2-3-subgraph of a valid graph G is a subgraph H of G such that the degree in H of every vertex in A(G) is either 2 or 3. The degree of a vertex belonging to B(G) in H is not restricted. A 2-3-subgraph exists in any valid graph which is not a forest. A maximal 2-3-subgraph of G is a 2-3-subgraph of G which is not a subgraph of any other 2-3-subgraph of G. A maximal 2-3-subgraph can be easily found by applying depth-first-search (DFS) on G. A linkpoint v in a 2-3-subgraph H is called a critical linkpoint if v is an allowed vertex, and there is a cycle \Gamma in G such that fvg. We refer to such a cycle \Gamma in G as a witness cycle of v. Note that we can assume a witness cycle to be simple and, so, verifying whether a linkpoint v in H is a critical linkpoint is easy: Remove the set of vertices G, with all incident edges, and apply a breadth-first-search (BFS) to check whether there is a cycle through v in the remaining graph. A cycle in a valid graph G is branchpoint-free if it does not pass through any allowed branchpoints; that is, a branchpoint-free cycle passes only through allowed linkpoints and blackout vertices of G. The rest of this section is devoted to showing that the following algorithm correctly outputs a vertex feedback set and achieves an unweighted performance ratio less than 4. Algorithm SubG-2-3 (Input: valid graph G; Output: feedback vertex set F of G); if G is a forest then F / ;; else begin: Using DFS, find a maximal 2-3-subgraph H of G; Using BFS, find the set X of critical linkpoints in H; Let Y be the set of allowed branchpoints in H; Find a set W that covers all branchpoint-free cycles of H which are not covered by X; end. Note that if are isolated cycles in H and so W consists of one vertex of each such cycle. We elaborate on how the set W is computed when B(G) 6= ;. Let H 0 be a graph obtained from H by removing the set X along with its incident edges. Let H b be the subgraph of induced by the allowed linkpoints and blackout vertices of H 0 . For every isolated cycle in H b , we arbitrarily choose an allowed linkpoint from that cycle to W . Next, we replace each maximal (with respect to containment) chain of allowed linkpoints in H b by an edge, resulting in a graph H b . We assign unit cost to all edges corresponding to a chain of allowed linkpoints, and a zero cost to all other edges, and compute a minimum-cost spanning forest T of H b . We now add to W one linkpoint from each chain of allowed linkpoints in H b that corresponds to an edge in H It is now straightforward to verify that the complexity of SubG-2-3 is linear in jE(G)j. The following two lemmas, which generalize some claims used in the proof of Theorem 1 in [Si67], are used to prove that SubG-2-3 outputs a feedback vertex set of a valid graph G. H be a maximal 2-3-subgraph of a valid graph G and let \Gamma be a simple cycle in G. Then, one of the following holds: (a) \Gamma is a witness cycle of some critical linkpoint of H, or - allowed branchpoint of H, or - (c) \Gamma is a cycle in H that consists only of blackout vertices or allowed linkpoints of H. Proof. Let \Gamma be a simple cycle in G and assume to the contrary that neither of (a)-(c) holds. This implies in particular that \Gamma cannot be entirely contained in H. We distinguish between two cases: (1) \Gamma does not intersect with H; and (2) \Gamma intersects with H only in blackout vertices and allowed linkpoints of H. Case 1: In this case we could join \Gamma and H to obtain a 2-3-subgraph H of G that contains H as a proper subgraph. This however contradicts the maximality of H. Case 2: If \Gamma intersects with H only in blackout vertices, then as in case 1, we can join \Gamma and H and contradict the maximality of H. Suppose now that \Gamma intersects with H in some allowed linkpoints of H. Note that in such a case \Gamma must intersect with H in at least two distinct allowed linkpoints of H, or else \Gamma would be a witness cycle of the only intersecting (critical) linkpoint. Since \Gamma is not contained in H by assumption, we can find two allowed linkpoints v 1 and v 2 in V (\Gamma) " V (H) that are connected by a path P along \Gamma such that and P is not entirely contained in H. Joining P and H, we obtain a 2-3-subgraph of G that contains H as a proper subgraph, thus contradicting the maximality of H. H be a maximal 2-3-subgraph of G and let \Gamma 1 and \Gamma 2 be witness cycles in G of two distinct critical linkpoints in H. are independent cycles, namely, Proof. Let v 1 and v 2 be the critical linkpoints associated with respectively, and assume to the contrary that V contains an allowed vertex Then, there is a path P in G that runs along parts of the cycles \Gamma 1 and \Gamma 2 , starting from passing through u, and ending at v 2 . are witness cycles, we have g. And, since v 1 and v 2 are distinct critical linkpoints, the vertex u cannot possibly coincide with either of them. Therefore, the path P is not entirely contained in H. Joining P and H we obtain a 2-3-subgraph of G that contains H as a proper subgraph, thus reaching a contradiction. Theorem 3 For every valid graph G, the set F computed by SubG-2-3 is a feedback vertex set of G. Proof. Let \Gamma be a simple cycle in G. We follow the three cases of Lemma 1 to show that (a) \Gamma is a witness cycle of some critical linkpoint of H. By construction, all critical linkpoints of H are in F . allowed branchpoint of H. By construction, all allowed branchpoints of H are in F . (c) \Gamma is a cycle in H that consists only of blackout vertices or allowed linkpoints of contains a critical linkpoint, then SubG-2-3 selects that linkpoint into the feedback vertex set F . Otherwise, the cycle \Gamma must be entirely contained in the graph H b that was used to create W . We now show that W covers all cycles in H b . Assume the contrary and let \Gamma be a cycle in H b that is not covered by W . Recall that in the construction of H b , each chain of allowed linkpoints in \Gamma was replaced by an edge with a unit cost. Let be the resulting cycle in H b . Since W does not cover \Gamma, all unit-cost edges in \Gamma were necessarily chosen to the minimum-cost spanning forest T . On the other hand, since T does not contain any cycles, there must be at least one zero-cost edge of \Gamma which is not contained in T . Hence, by deleting one of the unit-cost edges of \Gamma from T and inserting instead a particular zero-cost edge of \Gamma into T , we can obtain a new spanning forest T 0 for b . However, the cost of T 0 is smaller than that of T , which contradicts our assumption that T is a minimum-cost spanning forest. A reduction graph G 0 of an undirected graph G is a graph obtained from G by a sequence of the following transformations: ffl Delete an endpoint and its incident edge. ffl Connect two neighbors of a linkpoint v (other than a self-looped singleton) by a new edge and remove v from the graph with its two incident edges. A reduction graph of a valid graph G is not necessarily valid, since the reduction process may generate a cycle consisting of blackout vertices only. We will be interested in reduction sequences in which each transformation yields a valid graph. Lemma 4 Let G 0 be a reduction graph of G. If G 0 is valid, then - a Proof. Let G; be a sequence of reduction graphs where each H i is obtained by a removal of one linkpoint and possibly some endpoints from H i\Gamma1 . Since G 0 is valid, each H i is a valid graph as well. Let v i be the linkpoint that is removed from H i to obtain H i+1 . First we show that - a (G 0 ) - a (G). Suppose F is a feedback vertex set of H i+1 for some be a cycle in H i that passes through v i . A reduction of \Gamma obtained by replacing the linkpoint v i on \Gamma by an edge connecting the neighbors of v i yields a cycle in H i+1 . The vertex set of - \Gamma intersects the set F . Hence, F is also a feedback vertex set of H i which implies that - a (H Now we show that - a (G 0 ) - a (G). Suppose F is a minimal feedback vertex set of H i . If F does not contain v i , then F is also a feedback vertex set of H i+1 . Otherwise, write We claim that the set F 0 cannot fail to cover more than one cycle in H i+1 . If it failed, then there would be two distinct cycles \Gamma 1 and \Gamma 2 in H i that contain v i , in which case the cycle in H i induced by (V would not be covered by F , thus contradicting the fact that F is a feedback vertex set of H i . It follows by this and the minimality of F that the set F 0 fails to cover exactly one cycle in H i+1 . This cycle contains at least one allowed vertex u because H i+1 is a valid graph. Therefore, the set F 0 [ fug is a feedback vertex set of H i+1 . Hence, - a (H A reduction graph G of a graph G is minimal if and only if G is a valid graph and any proper reduction graph G 0 of G is not valid. Lemma 5 If G is a minimal reduction graph of G, then G does not contain blackout linkpoints, and every feedback vertex set of G contains all allowed linkpoints of G . Proof. Recall that G is a valid graph. If G contains a blackout linkpoint, then its removal creates a valid reduction graph which contradicts the minimality of G . Now assume F is a feedback vertex set of G and v is an allowed linkpoint which is not in F . If the removal of v yields a graph that is not valid, then v must have been included in F . If the removal of v yields a valid graph, then G is not minimal. The next lemma is needed in order to establish the performance ratio of SubG-2-3. It is a variant of Lemma 4 in [Vo68]. Lemma 6 Let G be a valid graph with no blackout linkpoints and such that no vertex has degree less than 2. Then, for every feedback vertex set F of G which contains all linkpoints of G, Proof. Suppose (G). In this case we have jV (G)j - 3jV and, therefore, the lemma holds trivially. So we assume from now on that jF j ! jV (G)j. denote the set of edges in E(G) whose terminal vertices are all vertices in F . denote the set of edges in E(G) whose terminal vertices are all vertices in X. Also, let E F;X denote the set of those edges in G that connect vertices in F with vertices in X. Clearly, E F , EX , and E F;X form a partition on E(G). Now, the graph obtained by deleting F from G is a nonempty forest on X and, therefore, jE However, each vertex in X is a branchpoint in G because all linkpoints are assumed to be in F and there are no vertices of degree less than 2. Therefore, i.e., On the other hand, Combining the last two inequalities we obtain The main claim of this section now follows. Theorem 7 The unweighted performance ratio of SubG-2-3 is at most 4 \Gamma (2=jV (G)j). Proof. Let F be the feedback vertex set computed by SubG-2-3 for a valid graph G which is not a forest. We show that jF j - 4 - a 2. The theorem follows immediately from this inequality. Let H, X, Y , and W be as in SubG-2-3. Suppose - a cycles in G pass through some allowed vertex v in G and, so, no vertex other than v can be a critical linkpoint in H. Now, if v is a linkpoint in H, then H is a cycle. Otherwise, one can readily verify that H must contain exactly two branchpoints. In either case we have jF j - 2. We assume from now on that - a (G) - 2. For every v i 2 X, let \Gamma i be some witness cycle of v i in G. By Lemma 2, the cycles \Gamma i are pairwise independent. Therefore, the minimum number of vertices needed to cover such cycles is jXj. Let f\Gamma j g be the set of branchpoint-free cycles in H that do not contain any critical linkpoints of H. Note that each cycle \Gamma j is independent with any witness cycle \Gamma i . We now claim that any smallest set W 0 of vertices of V (H) that intersects with the vertex set of each \Gamma j must be of size jW j. To see this, note that W 0 contains only allowed linkpoints of H. If we remove from H b all the edges that correspond to linkpoints belonging to W 0 , then we clearly end up with a forest. By construction, the minimum number of edges (or allowed linkpoints), needed to be removed from H b so as to make it into a forest, is jW j. Recalling that every cycle \Gamma j is independent with any witness cycle \Gamma i , the set W 0 cannot possibly intersect with any of the cycles \Gamma i . Hence, in order to cover the cycles f\Gamma in G, we will need at least jXj vertices. Therefore, On the other hand, we recall that jF We distinguish between the following two cases. Case 1: Here we have, Case 2: be a feedback vertex set of G of size - a (G) and let W 0 be a smallest subset of F that intersects with the vertex set of each \Gamma . Clearly, W 0 consists of allowed linkpoints of H, and, as we showed earlier in this proof, jW H 1 be the subgraph of H obtained by removing all critical linkpoints of H and all linkpoints in W 0 . With each deleted linkpoint, we also remove recursively all resulting endpoints from H while obtaining H 1 . Thus, a deletion of a linkpoint from H can decrease the number of branchpoints by 2 at most. Hence, the number of branchpoints left in H 1 is at least Furthermore, the graph H 1 does not contain any endpoints. 1 be a minimal reduction graph of H 1 and let H 2 be a valid graph obtained by removing all singleton components from H 1 . Since H 1 does not contain any endpoints, the number of branchpoints of H 1 is preserved in H 1 and in H 2 . Therefore, the graph H 2 contains at least jY branchpoints. On the other hand, since H 1 is a minimal reduction and due to Lemma 5, there are no blackout linkpoints in H 1 and every feedback vertex set of H 1 contains all allowed linkpoints of H 1 . Furthermore, the graphs H do not contain any endpoints. It follows that we can apply Lemma 6 to H 2 and any feedback vertex set of H 2 , thus obtaining where the equality is due to Lemma 4. Therefore, Recall that W 0 was chosen as a subset of a smallest feedback vertex set F of G. Let X 0 be a smallest subset of F that covers the witness cycles f\Gamma i g and let Z 0 be a smallest subset of F that covers the cycles of H 1 . Since H 1 does not contain any of the critical linkpoints of H, each witness cycle \Gamma i is independent with any cycle in H 1 and, so, we have It also follows from our previous discussion that In addition, by construction of H 1 we have It thus follows that Combining with (1), we obtain the desired result. Weighted Feedback Vertex Set In this section, we consider the approximation of the WFVS problem described in Section 1. That is, given an undirected graph G and a weight function w on its vertices, find a feedback vertex set of (G; w) with minimum weight. As in the previous section, we assume that G may contain parallel edges and self-loops. A weighted reduction graph G 0 of an undirected graph G is a graph obtained from G by a sequence of the following transformations: ffl Delete an endpoint and its incident edge. ffl Let u and v be two adjacent vertices such that w(u) - w(v) and v is a linkpoint. Connect u to the other neighbor of v, and remove v from the graph with its two incident edges. The following lemma can be easily verified. (See, e.g., the proof of Lemma 4). weighted reduction graph of (G; w). Then, -(G A weighted reduction graph G of a graph G is minimal if and only if any weighted reduction graph G 0 of G is equal to G . A graph is called branchy if it has no endpoints and, in addition, its set of linkpoints induces an independent set, i.e., each linkpoint is either an isolated self-looped singleton or connected to two branchpoints. Clearly, any minimal weighted reduction graph must be branchy. We note that the complexity of transforming a graph into a branchy graph is linear in jE(G)j. We are now ready to present our algorithms for finding an approximation for a minimum-weight feedback vertex set of a given weighted graph. In Section 3.1 we give an algorithm that achieves a performance ratio of 4 log 2 jV (G)j. In Section 3.2 we present an algorithm that achieves a performance ratio of 3.1 The primal-dual algorithm The basis of the first approximation algorithm is the next lemma which generalizes a lemma due to Erd-os and P'osa [EP62, Lemma 3]. That lemma was obtained by Erd-os and P'osa while estimating the smallest number of edges in a graph which contains a given number of pairwise independent cycles. Later on, in [EP64], they provided bounds on the value of -(G; 1) in terms of the largest number of pairwise independent cycles in G. Tighter bounds on -(G; 1) were obtained by Simonovits [Si67] and Voss [Vo68]. Lemma 9 The shortest cycle in any branchy graph G with at least two vertices is of length Proof. Let t be the smallest even integer such that 2 \Delta 2 t=2 ? jV (G)j. Apply BFS on G of depth t starting at some vertex v. We now claim that the search hits some vertex twice and so there exists a cycle of length - 2t in G. Indeed, if it were not so, then the induced BFS tree would contain at least 2 \Delta 2 t=2 distinct vertices of G, which is a contradiction. In each iteration of the proposed algorithm, we first find a minimal weighted reduction graph, and then find a cycle \Gamma with the smallest number of vertices in the minimal weighted reduction graph. The algorithm sets ffi to be the minimum among the weights of the vertices in V (\Gamma). This value of ffi is subtracted, in turn, from the weight of each vertex in V (\Gamma). Vertices whose weight becomes zero are added to the feedback vertex set and deleted from the graph. Each such iteration is repeated until the graph is exhausted. Algorithm MiniWCycle (Input: (G; w); Output: feedback vertex set F of (G; w)); While H is not a forest do begin: Find a minimal weighted reduction graph Find a cycle \Gamma 0 in H 0 with the smallest number of vertices; Remove X (with all incident edges) from H end. Finding a shortest cycle can be done by running BFS from each vertex until a cycle is found and then selecting the smallest. A more efficient approach for finding the shortest cycle is described in [IR78]. It is not hard to see that MiniWCycle computes a feedback vertex set of G. We now analyze the algorithm employing techniques similar to those used in [Ho82], [Ho83], and [KhVY94]. We note that the algorithm can also be analyzed using the Local Ratio Theorem of Bar-Yehuda and Even [BaEv85]. Theorem 10 The performance ratio of algorithm MiniWCycle is at most 4 log 2 jV (G)j. Proof. We assume that jV (G)j ? 1. Given a feedback vertex set F of (G; w), let be the indicator vector of F , namely, x We denote by C the set of cycles in G. The problem of finding a minimum-weight feedback vertex set of (G; w) can be formulated in terms of x by an integer programming problem as follows: minimize ranging over all nonnegative integer vectors (2) Let C v denote the set of cycles passing through vertex v in G and consider the following integer programming packing problem: maximize ranging over all nonnegative integer vectors \Gamma2C such that Clearly, the linear relaxation of (3) is the dual of the linear relaxation of (2), with being the dual variables. weighted reduction graph computed at some iteration of algorithm MiniWCycle. Then, for each cycle as follows: If all vertices in V (\Gamma 0 ) belong to G, then Otherwise, we "unfold" the transformation steps performed in obtaining H 0 from H in backward order, i.e., from H 0 back to H: In each such step we add to \Gamma 0 chains of linkpoints (connecting vertices in that were deleted. When this process finishes, the cycle \Gamma 0 of H 0 transforms into a cycle \Gamma of G. We now show that MiniWCycle can be interpreted as a primal-dual algorithm. We first show that it computes a dual feasible solution for (3) with a certain maximality prop- erty. The initial dual feasible solution is the one in which all the dual variables y \Gamma are zero. i be a cycle chosen at iteration i of MiniWCycle and let \Gamma i be the associated cycle in G. We may view the computation of iteration i of MiniWCycle as setting the value of the dual variable y \Gamma i to the weight ffi of a lightest vertex in V (\Gamma 0 ). The updated weight wH 0 (v) of every precisely the slack of the dual constraint that corresponds to v. It is clear that by the choice of ffi, the values of the dual variables y \Gamma at the end of iteration i of MiniWCycle satisfy the dual constraints (4) corresponding to vertices i ). It thus follows that the dual constraints hold for all vertices Let v be a vertex that was removed from H to obtain H 0 in iteration i of MiniWCycle. It remains to show that the dual constraint (4) corresponding to such a vertex holds in each iteration j of the algorithm for every j - i. We show this by backward induction on j. By the previous discussion it follows that the constraints corresponding to vertices that exist in the last iteration all hold. Suppose now that the dual constraints corresponding to vertices in V (H 0 ) in iteration j are not violated. We show that the dual constraints corresponding to vertices in V in that iteration are also not violated. Let c be a chain of linkpoints in H in iteration j, and let v 1 and v 2 be the two branchpoints adjacent to c. Let u be a vertex of minimum weight among v 1 , v 2 , and the vertices in c. We note that the weighted reduction procedure deletes all vertices in c except possibly for one representative, depending on whether u is in c or is one of its adjacent branchpoints. We now observe that the set of cycles that pass through a linkpoint in c is the same for all linkpoints in c, and is contained in the set of cycles that pass through v 1 , and is also contained in the set of cycles that pass through v 2 . This implies that if the dual constraint corresponding to u is not violated, then the dual constraints corresponding to any vertex in c is also not violated. The algorithm essentially constructs a primal solution x from the dual solution y: It selects into the feedback vertex set all vertices for which: (i) the corresponding dual constraints are tight; and (ii) in the iteration the constraint first became tight, the corresponding vertex belonged to the graph. As stated earlier, this construction yields a feasible solution. Let x denote the optimal primal and dual fractional solutions, respectively. It follows from the duality Theorem that w(v) w(v) \Gamma2C y \Gamma2C Hence, to prove the theorem, it suffices to bound the ratio between the LHS and the RHS of (5). First note that y \Gamma 6= 0 only for cycles \Gamma in G that are associated with cycles \Gamma 0 that were chosen at some iteration of MiniWCycle. By the above construction of x, it is clear that the dual variable y \Gamma of each such cycle \Gamma contributes its value to at most V vertices. Hence, \Gamma2C Now, in each iteration, the graph H 0 is a branchy graph. Therefore, by Lemma 9, we have that jV (\Gamma 0 )j - 4 log 2 jV (G)j. Hence the theorem is proved. Proposition 11 For planar graphs, the weighted performance ratio of MiniWCycle is at most 10. Proof. We first notice that the weighted reduction process preserves planarity and, there- fore, at each iteration of algorithm MiniWCycle we remain with a planar graph. We claim that every rich planar graph G must contain a face of length at most 5. Assume the contrary. By summing up the lengths of all the faces, we get that 2jEj - 6jZj, where Z denotes the set of faces of G. By Euler's formula, Hence, However, since the degree of each vertex is at least 3, we get that which is a contradiction. Furthermore, this implies that a branchy planar graph must contain a cycle of length at most 10. 3.2 Low-degree graphs The algorithm presented in this section is based on the following variant of Lemma 6. Lemma 12 Let G be a branchy graph. Then, for every feedback vertex set F of G, Proof. Let F be a feedback vertex set of G. We can assume without loss of generality that F contains only branchpoints, since this assumption can only decrease jF j. Let G 0 be the minimal (unweighted) reduction graph of G, i.e., G 0 contains only branchpoints or isolated self-looped singletons. Clearly, F is also a feedback vertex set of G 0 . Thus, G 0 and F satisfy the conditions of Lemma 6 (\Delta a = \Delta), yielding that, Since G 0 is a branchy graph, the number of linkpoints in G can be at most \Delta(G 0 ) Hence, We now present a weighted greedy algorithm for finding a feedback vertex set in a graph G. Algorithm WGreedy (Input: (G; w); Output: feedback vertex set F of (G; w)); while H is not a forest do begin: Find a minimal weighted reduction graph (H 0 of (H; wH ); F remove U i from H 0 i with its incident edges; end. For a subset S ' V , let w(S) denote the sum of weights of the vertices in S. We now prove the following theorem. Theorem 13 Let G be a branchy graph. Denote by F the feedback vertex set computed by algorithm WGreedy, and by F a minimum-weight feedback vertex set in G. Then, Proof. Assume that the number of iterations the while loop is executed in algorithm WGreedy is p. We define the following weight functions w (G). The weight function w i is defined, for 1 - i - p, as follows: For all For a subset S, let w i (S) denote the sum of weights of the vertices in S, where the weight function is w i . Clearly, Suppose that at one of the weighted reduction steps of algorithm WGreedy, a chain c of equal weight linkpoints was reduced to a single vertex, say, v, which either belongs to c or is one of the two branchpoints adjacent to c. Suppose further that v was added to F . If F also contains a vertex from the chain c, then without loss of generality, we can assume that this vertex can be replaced by v. Let Obviously, 1 . We claim that if p. Assume this is not the case. Then, with respect to the order in which vertices entered F in algorithm WGreedy, let u be the first vertex such that u 2 F , was removed from the graph in a weighted reduction step. This means that u was at the time of its removal a linkpoint that had an adjacent vertex u 0 with smaller weight. But then, by exchanging u for u 0 in F , we obtain a feedback vertex set which has smaller weight, contradicting the optimality of F . Hence, for a vertex Therefore, Notice that in the graph H 0 i , the weight function w i assigns the same weight to all vertices. Hence, by Lemma 12, we have that w i the theorem follows. It follows from Lemma 8 that the performance ratio of algorithm WGreedy for (G; w) is at most 2\Delta 2 (G) for any graph G. 4 The Loop Cutset Problem and its Application In section 4.1 we consider a variant of the WFVS problem for directed graphs and in section 4.2 we describe its application to Bayesian inference. 4.1 The loop cutset problem The underlying graph of a directed graph D is the undirected graph formed by ignoring the directions of the edges in D. A loop in D is a subgraph of D whose underlying graph is a cycle. A vertex v is a sink with respect to a loop \Gamma if the two edges adjacent to v in are directed into v. Every loop must contain at least one vertex that is not a sink with respect to that loop. Each vertex that is not a sink with respect to a loop \Gamma is called an allowed vertex with respect to \Gamma. A loop cutset of a directed graph D is a set of vertices that contains at least one allowed vertex with respect to each loop in D. Our problem is to find a minimum-weight loop cutset of a given directed graph D and a weight function We denote by -(D; w) the sum of weights of the vertices in such a loop cutset. Greedy approaches to the loop cutset problem have been suggested by [SuC90] and [St90]. Both methods can be shown to have a performance ratio as bad as \Omega\Gamma n=4) in certain planar graphs [St90]. An application of our approximation algorithms to the loop cutset problem in the area of Bayesian inference is described later in this section. The approach we take is to reduce the weighted loop cutset problem to the weighted feedback vertex set problem solved in the previous section. Given a weighted directed graph (D; w), we define the splitting weighted undirected graph (D s ; w s ) as follows. Split each vertex v in D into two vertices v in and v out in D s such that all incoming edges to v become undirected incident edges with v in , and all outgoing edges from v become undirected incident edges with v out . In addition, we connect v in and v out by an undirected edge. Set w s (v in and w s (v out w(v). For a set of vertices X in D s , we define /(X) as the set obtained by replacing each vertex v in or v out in X by the respective vertex v in D from which these vertices originated. Our algorithm can now be easily stated. Algorithm LoopCutset (Input: (D; w); Output: loop cutset F of (D; w)); Construct (D s Apply MiniWCycle on (D s ; w s ) to obtain a feedback vertex set X; F / /(X). Note that each loop in D is associated with a unique cycle in D s , and vice-versa, in a straightforward manner. Let I (\Gamma) denote the loop image of a cycle \Gamma in D s , and I -1 (K) denote the cycle image of a loop K in D. It is clear that the mapping I is The next lemma shows that algorithm LoopCutset outputs a loop cutset of (D; w). Lemma 14 Let (D; w) be a directed weighted graph and (D s ; w s ) be its splitting graph. Then: (i) If F is a feedback vertex set of (D s ; w s ) having finite weight, then /(F ) is a loop cutset of (D; w), and w s U is a loop cutset of D, then the set U s obtained from U by replacing each vertex v 2 U by vertex v out s is a feedback vertex set of D s , and Proof. We prove (i). The proof of (ii) is similar. Let \Gamma be a loop in D. To prove the lemma we show that an allowed vertex with respect to \Gamma belongs to /(F ). Let I -1 (\Gamma) be the unique cycle image of \Gamma in D s . Since F is a cycle cover of D s having finite weight, there must be a vertex v out 2 F in I -1 (\Gamma). Now, it is clear that vertex v 2 \Gamma from which v out originated is an allowed vertex with respect to \Gamma as needed. To complete the proof, by the finiteness of must have w s for each vertex in F . It follows from Lemma 14 that -(D; In addition, due to Theorem 10 applied to the graph D s , and since the number of vertices in D s is twice the number of vertices in D, we get the following bound on the performance ratio of algorithm LoopCutset. Theorem 15 The performance ratio of LoopCutset is at most 4 log 2 (2jV (D)j). We now show that in the unweighted loop cutset problem, we can achieve a performance ratio better than 4. In this case, for each vertex v 2 D, the weight of v in 2 D s is one unit, and the weight of v out 2 D s is 1. This falls within the framework considered in Section 2, since vertices with infinite weight in D s can be treated as blackout vertices. We can therefore apply SubG-2-3 in the LoopCutset algorithm instead of applying MiniWCycle and obtain the following improved performance ratio. Theorem using SubG-2-3, the unweighted performance ratio of LoopCutset is at most 4 \Gamma (2=jV (D)j). Proof. We have, where the equality is due to Lemma 14, and the inequality is due to Theorem 7. Since (D)j, the claim is proved. 4.2 An application We conclude this section with an application of approximation algorithms for the loop cutset problem. Let P distribution where each u i draws values from a finite set called the domain of u i . A directed graph D with no directed cycles is called a Bayesian network of P if there is a 1-1 mapping between fu and vertices in D, such that associated with vertex i and P can be written as follows: Y are the source vertices of the incoming edges to vertex i in D. It is worth noting that Bayesian networks are useful knowledge representation schemes for many artificial intelligence tasks. Bayesian networks allow a wide spectrum of independence assumptions to be considered by a model builder so that a practical balance can be established between computational needs and adequacy of conclusions. For a complete exploration of this subject see [Pe88]. Suppose now that some variables fv among fu are assigned specific values respectively. The updating problem is to compute the probability In principle, such computations are straightforward because each Bayesian network defines the joint probability distribution conditional probabilities can be computed by dividing the appropriate sums. However, such computations are inefficient both in time and space unless they use conditional independence assumptions defined by Eq. (6). We shall see next how our approximation algorithms for the loop cutset problem reduce the computations needed for solving the updating problem. A trail in a Bayesian network is a subgraph whose underlying graph is a simple path. A vertex b is called a sink with respect to a trail t if there exist two consecutive edges a ! b and b / c on t. A trail t is active by a set of vertices Z if (1) every sink with respect to t either is in Z or has a descendant in Z and (2) every other vertex along t is outside Z. Otherwise, the trail is said to be blocked by Z. Verma and Pearl [VePe88] have proved that if D is a Bayesian network of P and all trails between a vertex in fr and a vertex in fs are blocked by g, then the corresponding sets of variables fu r 1 are independent conditioned on fu t 1 g. Furthermore, Geiger and Pearl [GP90] proved a converse to this theorem. Both results are presented and extended in [GVP90]. Using the close relationship between blocked trails and conditional independence, Kim and Pearl [KiP83] developed an algorithm update-tree that solves the updating problem on Bayesian networks in which every two vertices are connected with at most one trail. update-tree views each vertex as a processor that repeatedly sends messages to each of its neighboring vertices. When equilibrium is reached, each vertex i contains the conditional probability distribution P computations reach equilibrium regardless of the order of execution in time proportional to the length of the longest trail in the network. Pearl [Pe86] solved the updating problem on any Bayesian network as follows. First, a set of vertices S is selected, such that any two vertices in the network are connected by at most one active trail in S [ Z, where Z is any subset of vertices. Then, update-tree is applied once for each combination of value assignments to the variables corresponding to S, and, finally, the results are combined. This algorithm is called the method of conditioning and its complexity grows exponentially with the size of S. Note that according to the definition of active trails, the set S in Pearl's algorithm is a loop cutset of the Bayesian network. In this paper we have developed approximation algorithms for finding S. When the domain size of the variables varies, then update-tree is called a number of times which is bounded from above by the product of the domain sizes of the variables whose corresponding vertices participate in the loop cutset. If we take the logarithm of the domain size as the weight of a vertex, then solving the weighted loop cutset problem with these weights optimizes Pearl's updating algorithm in the case where the domain sizes are allowed to vary. It is useful to relate the feedback vertex set problem with the vertex cover problem in order to establish lower bounds on the performance ratios attainable for the feedback vertex set problem. A vertex cover of an undirected graph is a subset of the vertex set that intersects with each edge in the graph. The vertex cover problem is to find a minimum weight vertex cover of a given graph. There is a simple polynomial reduction from the vertex cover problem to the feedback vertex set problem: Given a graph G, we extend G to a graph H by adding a vertex v e for each edge e 2 E(G), and connecting v e with the vertices in G with which e is incident in G. It is easy to verify that there always exists a minimum feedback vertex set in H whose vertices are all in V (G) and this feedback vertex set is also a minimum vertex cover of G. In essence, this reduction replaces each edge in G with a cycle in H, thus transforming any vertex cover of G to a feedback vertex set of H. Due to this reduction, it follows that the performance ratio obtainable for the feedback vertex set problem cannot be better than the one obtainable for the vertex cover problem. The latter problem has attracted a lot of attention over the years but has so far resisted any approximation algorithm that achieves in general graphs a constant performance ratio less than 2. We note that the above reduction retains planarity. However, for planar graphs, Baker [Bak94] provided a Polynomial Approximation Scheme (PAS) for the vertex cover problem. For the UFVS problem, there are examples showing that 4 is the tightest constant performance ratio of algorithm SubG-2-3. Another consequence of the above reduction is a lower bound on the unweighted performance ratio of the following greedy algorithm, GreedyCyc, for the feedback vertex set problem. In each iteration, GreedyCyc removes a vertex of maximal degree from the graph, adds it to the feedback vertex set, and removes all endpoints in the graph. A similar greedy algorithm for the vertex cover problem is presented in [Jo74] and in [Lo75]. The latter algorithm was shown to have an unweighted performance ratio no better than \Omega\Gammahan jV (G)j) [Jo74]. Due to the reduction to the cycle cover problem, the same lower bound holds also for GreedyCyc, as demonstrated by the graphs of [Jo74]. A tight upper bound on the worst-case performance ratio of GreedyCyc is unknown. Finally, one should notice that the following heuristics may improve the performance ratios of our algorithms. For example, in each iteration MiniWCycle chooses to place in the cover all zero-weight vertices found on the smallest cycle. This choice might be rather poor especially if many weights are equal. It may be useful in this case to perturb the weights of the vertices before running the algorithm. Similarly, in algorithm SubG- 2-3, there is no point in taking blindly all branchpoints of H. An appropriate heuristic here may be to pick the branchpoints one by one in decreasing order of residual degrees. Furthermore, the subgraph H itself should be constructed such that it contains as many high degree vertices as possible. Remark In a preliminary version of this paper, presented in [BaGNR94], we conjectured that a constant performance ratio is attainable by a polynomial time algorithm for the WFVS problem. This has been recently verified in [BeG94, BaBF94] where a performance ratio of 2 has been obtained. Acknowledgment We would like to thank David Johnson for bringing [EP62] to our attention, and Samir Khuller for helpful discussions. --R Constant ratio approximations of the weighted feedback vertex set problem for undirected graphs Approximation algorithms for NP-complete problems on planar graphs A local-ratio theorem for approximating the weighted vertex cover problem Approximation Algorithms for the Feedback Vertex Set Problem with Applications to Constraint Satisfaction and Bayesian Inference Approximation algorithms for the loop cutset prob- lem The cycle cutset method for improving search performance in AI Enhancement schemes for constraint processing: backjumping On the maximal number of disjoint circuits of a graph On the independent circuits contained in a graph On the logic of causal models independence in Bayesian networks Approximation algorithms for set covering and vertex covering problems Efficient bounds for the stable set Finding a minimum circuit in a graph Approximation algorithms for combinatorial problems A primal-dual parallel approximation technique applied to weighted set and vertex cover A computational model for combined causal and diagnostic reasoning in inference systems On the ratio of optimal integral and fractional covers Four approximation algorithms for the feedback vertex set problem Probabilistic reasoning in intelligent systems: Networks of plausible infer- ence A new proof and generalizations of a theorem by Erd-os and P'osa on graphs without k On heuristics for finding loop cutsets in multiply connected belief networks Cooper G. Semantics and expressiveness Some properties of graphs containing k independent circuits --TR --CTR Paola Festa , Panos M. Pardalos , Mauricio G. C. Resende, Algorithm 815: FORTRAN subroutines for computing approximate solutions of feedback set problems using GRASP, ACM Transactions on Mathematical Software (TOMS), v.27 n.4, p.456-464, December 2001 Rudolf Berghammer , Alexander Fronk, Exact computation of minimum feedback vertex sets with relational algebra, Fundamenta Informaticae, v.70 n.4, p.301-316, April 2006 Rudolf Berghammer , Alexander Fronk, Exact Computation of Minimum Feedback Vertex Sets with Relational Algebra, Fundamenta Informaticae, v.70 n.4, p.301-316, December 2006 Ioannis Caragiannis , Christos Kaklamanis , Panagiotis Kanellopoulos, New bounds on the size of the minimum feedback vertex set in meshes and butterflies, Information Processing Letters, v.83 n.5, p.275-280, 15 September 2002 Maw-Shang Chang , Chin-Hua Lin , Chuan-Min Lee, New upper bounds on feedback vertex numbers in butterflies, Information Processing Letters, v.90 n.6, p.279-285, Camil Demetrescu , Irene Finocchi, Combinatorial algorithms for feedback problems in directed graphs, Information Processing Letters, v.86 n.3, p.129-136, 16 May Rastislav Krlovi , Peter Ruika, Minimum feedback vertex sets in shuffle-based interconnection networks, Information Processing Letters, v.86 n.4, p.191-196, 31 May Jiong Guo , Jens Gramm , Falk Hffner , Rolf Niedermeier , Sebastian Wernicke, Compression-based fixed-parameter algorithms for feedback vertex set and edge bipartization, Journal of Computer and System Sciences, v.72 n.8, p.1386-1396, December, 2006 Venkatesh Raman , Saket Saurabh , C. R. Subramanian, Faster fixed parameter tractable algorithms for finding feedback vertex sets, ACM Transactions on Algorithms (TALG), v.2 n.3, p.403-415, July 2006 Reuven Bar-Yehuda , Keren Bendel , Ari Freund , Dror Rawitz, Local ratio: A unified framework for approximation algorithms. In Memoriam: Shimon Even 1935-2004, ACM Computing Surveys (CSUR), v.36 n.4, p.422-463, December 2004

bayesian networks;combinatorial optimization;approximation algorithms;constraint satisfaction;vertex feedback set

284962

Surface Approximation and Geometric Partitions.

Motivated by applications in computer graphics, visualization, and scientific computation, we study the computational complexity of the following problem: given a set S of n points sampled from a bivariate function f(x,y) and an input parameter $\eps > 0$, compute a piecewise-linear function $\Sigma(x,y)$ of minimum complexity (that is, an xy-monotone polyhedral surface, with a minimum number of vertices, edges, or faces) such that $| \Sigma(x_p, y_p) \; - \; z_p | \:\:\leq\:\: \eps$ for all $(x_p, y_p, z_p) \in S$. We give hardness evidence for this problem, by showing that a closely related problem is NP-hard. The main result of our paper is a polynomial-time approximation algorithm that computes a piecewise-linear surface of size O(Ko log Ko), where Ko is the complexity of an optimal surface satisfying the constraints of the problem.The technique developed in our paper is more general and applies to several other problems that deal with partitioning of points (or other objects) subject to certain geometric constraints. For instance, we get the same approximation bound for the following problem arising in machine learning: given n "red" and m "blue" points in the plane, find a minimum number of pairwise disjoint triangles such that each blue point is covered by some triangle and no red point lies in any of the triangles.

Introduction In scientific computation, visualization, and computer graphics, the modeling and construction of surfaces is an important area. A small sample of some recent papers [2, 3, 5, 7, 10, 13, 20, 21] on this topic gives an indication of the scope and importance of this problem. The first author has been supported by National Science Foundation Grant CCR-93-01259 and an NYI award. Rather than delve into any specific problem studied in these papers, we focus on a general, abstract problem that seems to underlie them all. In many scientific and computer graphics applications, computation takes place over a surface in three dimensions. The surface is generally modeled by piecewise linear (or, sometimes piecewise cubic) patches, whose vertices lie either on or in the close vicinity of the actual surface. In order to ensure that all local features of the surface are captured, algorithms for an automatic generation of these models generally sample at a dense set of regularly spaced points. Demands for real-time speed and reasonable performance, however, require the models to have as small a combinatorial complexity as possible. A common technique employed to reduce the complexity of the model is to somehow "thin" the surface by deleting vertices with relatively "flat" neighborhoods. Only ad hoc and heuristic methods are known for this key step. Most of the thinning methods follow a set of local rules (such as deleting edges or vertices whose incident faces are almost coplanar), which are applied in an arbitrary order until they are no longer applicable. Not surprisingly, these methods come with no performance guarantee, and generally no quantitative statement can be made about the surface approximation computed by them. In this paper, we address the complexity issues of the surface approximation problem for surfaces that are xy-monotone. These surfaces represent the graphs of bivariate functions f(x; y), and they arise quite naturally in many scientific applications. One possible approach for handling arbitrary surfaces is to break them into monotone pieces, and apply our algorithm individually on each piece. Let us formally define the main problem studied in our paper. Let f be a function of two variables x and y, and let S be a set of n points sampled from f . A piecewise linear function \Sigma is called an "-approximation of f , for " ? 0, if for every point Given S and ", the surface approximation problem is to compute a piecewise linear function \Sigma that "-approximates f with a minimum number of breakpoints. The breakpoints of \Sigma can be arbitrary points of R 3 , and they are not necessarily points of S. In many applications, f is generally a function hypothesized to fit the observed data-the function \Sigma is a computationally efficient substitute for f . The parameter " is used to achieve a complexity-quality tradeoff - smaller the ", higher the fidelity of the approximation. (The graph of a piecewise linear function of two variables is also called a polyhedral terrain in computational geometry literature; the breakpoints of the function are the vertices of the terrain.) The state of theoretical knowledge on the surface approximation problem appears to be rather slim. The provable performance bounds are known only for convex surfaces. For this special case, an O(n 3 presented by Mitchell and Suri [16] for computing an "-approximation of a convex polytope of n vertices in R 3 . Their algorithm produces an approximation of size O(K o log n), where K o is the size of an optimal "-approximation. Extending their work, Clarkson [6] has proposed an O(K randomized algorithm for computing an approximation of size O(K o log K can be an arbitrarily small positive number. In this paper, we study the approximation problem for surfaces that correspond to graphs of bivariate functions. We show that it is NP-Hard to decide if a surface can be "-approximated using k vertices (or facets). The main result of our paper, however, is a polynomial-time algorithm for computing an "-approximating surface of a guaranteed qual- ity. If an optimal "-approximating surface of f has K our algorithm produces a surface with O(K vertices. Observe that we are dealing with two approximation measures here: ", which measures the absolute z difference between f and the "simplified" surface \Sigma, and the ratio between the sizes of the optimal surface and the output of our algorithm. For the lack of better terminology, we use the term "approximation" for both measures. Notice though that " is an input (user-specified) parameter, while log K o is the approximation guarantee provided by the analysis of our algorithm. The key to our approximation method is an algorithm for partitioning a set of points in the plane by pairwise disjoint triangles. This is an instance of the geometric set cover problem, with an additional disjointness constraint on the covering objects (triangles). Observe that the disjointness condition on covering objects precludes the well-known greedy method for set covering [11, 14]; in fact, we can show that a greedy solution has size in the worst-case. Let us now reformulate our surface approximation problem as a constrained geometric partitioning problem. p denote the orthogonal projection of a point p 2 R 3 onto the xy-plane z = 0. In general, for any set A ae R 3 , we use - A to denote the orthogonal projection of A onto the xy-plane. Then, in order to get an "-approximation of f , it suffices to find a set of triangles in 3-space such that (i) the projections of these triangles on plane z = 0 are pairwise disjoint and they cover the projected set of points - S, and (ii) the vertical distance between a triangle and any point of S directly above/below it is at most ". Our polynomial-time algorithm produces a family of O(K We stitch together these triangles to produce the desired surface \Sigma. The "stitching" process introduces at most a constant factor more triangles. The geometric partition framework also includes several extensions and generalizations of the basic surface approximation problem. For instance, we can formulate a stronger version of the problem by replacing each sample point by a horizontal triangle (or, any polygon). Specifically, we are given a family of horizontal triangles (or polygons) in 3- space, whose projections on the xy-plane are pairwise disjoint. We want a piecewise linear, "-approximating surface whose maximum vertical distance from any point on the triangles is ". Our approximation algorithm works equally well for this variant of the problem-this variant addresses the case when some local features of the surface are known in detail; unfortunately, our method works only for horizontal features. Finally, let us mention the planar bichromatic partition problem that is of independent interest in the machine learning literature: Given a set R of 'red' points and another set B of 'blue' points in the plane, find a minimum number of pairwise disjoint triangles such that each blue point lies in a triangle and no red point lies in any of the triangles. Our algorithm gives a solution with O(K The running time of our algorithms, though polynomial, is quite high, and at the moment has only theoretical value. These being some of the first results in this area, however, we expect that the theoretical time complexity of these problems would improve with further work. Perhaps, some of the ideas in our paper may also shed light on the theoretical performance of some of the "practical" algorithms that are used in the trade. A Proof of NP-Hardness We show that both the planar bichromatic partition problem and the surface approximation problem are NP-Hard , by a reduction from the planar 3-SAT. We do not know whether they are in NP since the coordinates of the triangles in the solution may be quite large. We recall that the 3-SAT problem consists of n variables x clauses each with three literals C x k . The problem is to decide whether the boolean formula has a truth assignment. An instance of 3-SAT is called planar if its variable-clause graph is planar. In other words, F is an instance of the planar 3-SAT if the graph G(F E) is planar (see [12]), where V and E are defined as follows: appears in C Theorem 2.1 The planar bichromatic partition problem is NP-Hard. Proof: Our construction is similar to the one used by Fowler et al. [9], who prove the intractability of certain planar geometric covering problems (without the disjointness con- dition); see also [4, 8] for similar constructions. We first describe our construction for the bichromatic partition problem. To simplify the proof, our construction allows three or more points to lie on a line-the construction can be modified easily to remove these degeneracies. Let F be a boolean formula, and let E) be a straight-line planar embedding of the graph G(F ). We construct an instance of the red-blue point partition problem whose solution determines whether F is satisfiable. a 11 (iv) Figure 1: (i) An instance of planar 3-SAT: instance of bichromatic partition, (iii) Details of P 1 and C 1 , only some of the red points lying near P 1 and C 1 are shown, (iv) Two possible coverings of blue points on P 2 . We start by placing a 'blue' point at each clause node C m. Let x i be a variable node, and let e i1 il be the edges incident to it. In the plane embedding of G, the edges e ij form a "star" (see Figure 1 (i)). We replace this star by its Minkowski sum with a disk of radius ffi , for a sufficiently small ffi ? 0. Before performing the Minkowski sum, however, we shrink all the edges of the star at x i by 2ffi, so that the "star-shaped polygons" meeting at a clause node do not overlap (see Figure 1 (ii)). Let P i denote the star-shaped polygon corresponding to x i . In the polygon P i , corresponding to each edge e ij , there is a tube, consisting of two copies of e ij , each translated by distance ffi, plus a circular arc s ij near the clause node C j . We place an even number of (say 2k i ) 'blue' points on the boundary of P i , as follows. We put two points a ij and b ij on the circular arc s ij near its tip. If C j contains the literal six points on the straight-line portion of P i 's boundary, three each on translated copies of the edge e ij . On each copy, we move the middle point slightly inwards so as to replace the original edge of P i by a path of length two. On the other hand, if C j contains the literal - four points on straight-line portion of P i 's boundary, two each on translated copies of the edge e ij . Thus, the number of blue points added for edge e ij is either six of eight. (2k i is the total number of points put along P i .) Let B denote the set of all blue points placed in this way, and let Finally, we scatter a large (but polynomially bounded) number of 'red' points so that (i) any segment connecting two blue points that are not adjacent along the boundary of some contains a red point, and (ii) any triangle with three blue points as its vertices contains at least one red point unless the triangle is defined by a Figure (iii).) Let R be a set of red points satisfying the above two properties. We claim that the set of blue points B can be covered by k disjoint triangles, none of which contains any red point, if and only if the formula F has a truth assignment. Our proof is similar to the one in Fowler et al. [9]; we only sketch the main ideas. The red points act as enforcers, ensuring that only those blue points that are adjacent on the boundary of a P i can be covered by a single triangle. Thus, the minimum number of triangles needed to cover all the points on P i is k i . Further, there are precisely two ways to cover these points using k i triangles- in one covering, a ij and b ij are covered by a single triangle for those clauses only in which x and, in the other covering, a ij and b ij are covered by a single triangle for those clauses only in which - Figure 4 (iv). We regard the first covering as setting x and the second covering as setting x Suppose 1. For any clause C j that contains x i , the points a ij and b ij are covered by a single triangle, and we can cover the clause point corresponding to C j by the same triangle. The same holds if x and the clause C j contains - In other words, the clause point of C j can be covered for free if C j is satisfied. Thus, the set of blue points B can be covered by k triangles if and only if the clause point for each clause C j is covered for free, that is, the formula F has a truth assignment. This completes our proof of NP-Hardness of the planar bichromatic partition problem. 2 Remark: The preceding construction is degenerate in that most of the red points lie on segments connecting two blue points. There are several ways to remove these collinearities; we briefly describe one of them. For each polygon P i , replace every blue point b on P i by two blue points b placed very close to b. (We do not make copies of 'clause points' C j , m.) For every pair of blue points b l that we did not want to cover by a single triangle in the original construction, we place a red point in the convex hull of b 0 l . If there are 4k i blue points on the boundary of P i , they can be covered by k i triangles, and there are exactly two ways to cover these blue points by k i triangles, as earlier. Following a similar, but more involved, argument, we can prove that the set of all blue points can be covered by triangles if and only if F is satisfiable. Theorem 2.2 The 3-dimensional surface approximation problem is NP-Hard. Proof: Our construction is similar in spirit to the one for the bichromatic partition problem, albeit slightly more complex in detail. We use points of three colors: red, white and black. The 'white' points lie on the plane z = 0, the `black' points lie on the plane z = 2A, and the 'red' points lie between A is a sufficiently large constant. To maintain a connection with the previous construction, the black and white points play the role of blue points, while the red points play the role of enforcers as before, restricting the choice of "legal" triangles that can cover the black or white points. We will describe the construction in the xy-plane, which represents the orthogonal projection of the actual construction. The actual construction is obtained simply by vertically translating each point to its correct plane, depending on its color. We start out again by putting a 'black' point at each clause node C j . Then, for each variable x i , we construct the "star-shaped" polygon P i ; this part is identical to the previous construction. We replace each of the two straight-line edges of P i by "concave chains," bent inward, and also make a small "dent" at the tip of the circular arc s ij , as shown in Figure 2. We place 12 points on each arm of P i , alternating in color black and white, as follows. At the tip of the circular arc s ij , we put a white point c ij at the outer endpoint of the dent and a black point d ij at the inner endpoint of the dent (Figure 2 (ii)). The rest of the construction is shown in Figure 2 (i) - we put two more points a ij on the circular arc and 4 points ff l on each of the two concave chains. The two points surrounding namely, a ij and b ij , are such that any segment connecting them to any point on the two concave chains lies inside P i . Next, corresponding to each edge e ij of the graph G(F ), we put a 'white' point c 0 ij on the segment joining c ij and the clause point C j , very close to c ij such that (See Figure 2 (iii).) This condition says that, in the final construction when the black and white points have been translated to their correct z-plane, the vertical distance between ij and the segment C no more than "-recall that " is the input measure of approximation. This completes the placement of white and black points. The only remaining part of the construction is the placement of 'red' points, which we now describe. a 22 a d 21 c 0c 0d 22 c 22 e c 0(iii) c 0b21 c 21 ff 1a 21 Figure 2: Placing points on the polygon P 2 corresponding to Figure 1: (i) Modified P 2 and points on P 2 , (ii) points and triangles in the neighborhood of c points and triangles near C 1 . We add a set of triangles, each containing a large (but polynomially bounded) number of red points-the role of these triangles is to restrict the choice of legal triangles that can cover black/white points. The set of triangles associated with P i is labeled T i . The construction of T i is detailed in Figure 2 (i). Specifically, for an edge e ij , if x then we put a small triangle that intersects the segment b ij c 0 ij but not b ij c ij . On the other hand, if - then we put a small triangle that intersects a ij c 0 ij but not a ij c ij . Next, we put a small number of triangles inside P i , near its concave chains, so that at most three consecutive points along P i may be covered by one triangle without intersecting any triangle of T i . We ensure that one of these triangles intersects the triangle 4ff 1 so that fff 1 cannot be covered by a single triangle. We also place three triangles near each clause C j , each containing a large number of red points; see Figure 2 (iii). Finally, we translate black and white points in the z-direction, as described earlier. Let f- be the set of all 'red' triangles. We move all points in - i vertically to the plane z There are two ways to cover the points of P i with 2k i legal (non-intersecting) triangles-one in which are covered by a single triangle, and the one in which b ij are covered by a triangle. These coverings are associated with the true and false settings of the variable x i . Let P denote the set of all points constructed, let t denote the total number of 'red' triangles, and let We claim that there exist a polyhedral terrain with 3(k+m+t) vertices that "-approximates P if and only if F has a truth assignment, provided that " is sufficiently small-recall that m is the number of clauses in F . The claim follows from the observations that it is always better to cover all red points lying in a horizontal triangle - i by - i itself, and that a clause C j requires one triangle to cover its points if and only if one of the literals in C j is set true; otherwise it requires two triangles. (For instance, if C j contains the literal x i and x i is set true, then the triangle a ij can be enlarged slightly to cover c 0 ij . The remaining three points for the clause C j can be covered by one additional triangle.) The rest of the argument is the same as for the bichromatic partition problem. Finally, we can perturb the points slightly so that no four of them are coplanar.In the remainder of the paper, we develop our approximation algorithms. 3 A Canonical Trapezoidal Partition We introduce an abstract geometric partitioning problem in the plane, which captures the essence of both the surface approximation problem as well as the bichromatic points partition problem. The abstract problem deals with trapezoidal partitions under a boolean constraint function C satisfying the "subset restriction" property. More precisely, let C be a boolean function from compact, connected subsets of the plane to f0; 1g satisfying the following property: For technical reasons, we choose to work with "trapezoids" instead of triangles, where the top and bottom edges of each trapezoid are parallel to the X-axis. The trapezoids and triangles are equivalent for the purpose of approximation-each triangle can be decomposed into two trapezoids, and each trapezoid can be decomposed into two triangles. Given a set of n points P in the plane, a family of trapezoids is called a valid trapezoidal partition (a trapezoidal partition for brevity) of P with respect to a boolean constraint function C if the following conditions hold: (ii) \Delta covers all the points: P ae (iii) The trapezoids in \Delta have pairwise disjoint interiors. We can cast our bichromatic partition problem in this abstract framework by setting set of 'blue' points) and, for a trapezoid - ae R 2 , defining only if - is empty of red points, that is, - In the surface approximation problem, we set (the orthogonal projection of S on the plane z = 0) and a trapezoid - ae R 2 has only if - can be vertically lifted to a planar trapezoid - in R 3 so that the vertical distance between - - and any point of S directly above/below it is at most ". The space of optimal solutions for our abstract problem is potentially infinite-the vertices of the triangles in our problem can be anywhere in the plane. For our approximation results, however, we show that a restricted choice of trapezoids suffices. Given a set of n points P in the plane, let L(P ) denote the set consisting of the following lines: the horizontal lines passing through a point of P , and the lines passing through two points of P . Thus, jL(P We will call the lines of L(P ) the canonical lines determined by P . We say that a trapezoid \Delta ae R 2 is canonical if all of its edges belong to lines in L(P ). A trapezoidal partition \Delta is canonical if all of its trapezoids are canonical. The following lemma shows that by limiting ourselves to canonical trapezoidal partitions only, we sacrifice at most a constant (multiplicative) factor in our approximation. Figure 3: A canonical trapezoidal partition Lemma 3.1 Any trapezoidal partition of P with k trapezoids can be transformed into a canonical trapezoidal partition of P with at most 4k trapezoids. Proof: We give a construction for transforming each trapezoid \Delta 2 \Delta into four trapezoids 4, with pairwise disjoint interiors, so that \Delta i together cover all the points in P " \Delta. By (3.1), the new set of trapezoids is a valid trapezoidal partition of P . Our construction works as follows. Consider the convex hull of the points P "\Delta. If the convex hull itself is a trapezoid, we return that trapezoid. Otherwise, let '; denote the left, right, top and bottom edges of \Delta, as shown in Figure 4 (i). We perform the following four steps, which constitute our transformation algorithm. r (iv) r Figure 4: Illustration of the canonicalization. (i) We shrink the trapezoid \Delta by translating each of its four bounding edges towards the interior, until it meets a point of P \Delta . Let \Delta 0 ' \Delta denote the smaller trapezoid thus obtained respectively, denote a point of P \Delta lying on the left, right, top, and bottom edge of \Delta 0 ; we break ties arbitrarily if more than one point lies on an edge. (ii) We partition \Delta 0 into two trapezoids, \Delta L and \Delta R , by drawing the line segment p u as shown in (Figure 4 (ii). (iii) We next partition \Delta L into two trapezoids \Delta LU and \Delta LB , by drawing the maximal horizontal segment through p ' . Let p 0 ' denote right endpoint of this segment. Similarly, we partition \Delta R into \Delta RU and \Delta RB , lying respectively above and below the horizontal line segment p r p 0 r . (iv) We rotate the line supporting the left boundary of \Delta LU around the point p ' in clock-wise direction until it meets a point of the set denote the intersection of this line and the top edge of \Delta LU . We set Figure 4 (iv)). (If a triangle, which we regard as a degenerate trapezoid; e.g. \Delta 4 in Figure 4 (iii).) The top and bottom edges of \Delta 1 contain p u and p ' , respectively, the left edge contains p ' and q ' , and the right edge is determined by the pair of points p u and p b . Thus, the trapezoid \Delta 1 is in canonical position. The three remaining trapezoids are constructed similarly. In the above construction, if any of the four trapezoids \Delta i does not cover any point of P \Delta , then we can discard it. Thus, an arbitrary trapezoid of the partition \Delta can be transformed into at most four canonical trapezoids. This completes the proof of the lemma.4 Greedy Algorithms At this point, we can obtain a weak approximation result using the canonical trapezoidal partition. Roughly speaking, we can use the greedy set covering heuristic [6, 14], ignoring the disjointness constraint, and then refine the heuristic output to produce disjoint trapezoids. Unfortunately, the last step can increase the complexity of the solution quite significantly. Theorem 4.1 Given a set P of n points in the plane and a boolean constraint function C satisfying (3.1) that can be evaluated in polynomial time, we can compute an O(K log K size trapezoidal partition of P respecting C in polynomial time, where K o is the size of an optimal trapezoidal partition. Proof: Consider the set \Xi of all valid, canonical trapezoids in the plane-the set \Xi has O(n 6 ) trapezoids. We form the geometric set-system X can be computed by testing each \Delta 2 \Xi whether it is valid. We compute a set cover of X using the greedy algorithm [11, 14] in polynomial time. The greedy algorithm returns a set R consisting of O(K necessarily disjoint. In order to produce a disjoint cover, we first compute the arrangement A(R) of the plane induced by R. Then, we decompose each face of A(R) into trapezoids by drawing a horizontal segment through each vertex until the segment hits an edge of the arrangement. The resulting partition is a trapezoidal partition of P . The number of trapezoids in the partition is O((K - since the arrangement A(R) has this size. The total running time of the algorithm is polynomial. 2 Remark: (i) The canonical form of trapezoids is used only to construct a finite family of trapezoids to search for an approximate solution. A direct application of the definition in the previous subsection gives a family of O(n 6 ) canonical trapezoids. By using a slightly different canonical form, we can reduce the size of canonical triangles to O(n 4 ). In another paper [1], we present an near-quadratic time algorithm for finding an approximation of size (ii) One can show that the number of trapezoids produced by the above algorithm is \Omega\Gamma K in the worst case. A Recursively Separable Partition 13 5 A Recursively Separable Partition We now develop our main approximation algorithm. The algorithm is based on dynamic programming, and it depends on two key ideas-a recursively separable partition and a compliant partition. These partitions are further specializations of the canonical trapezoidal partition introduced in the previous section, and they are central to our algorithm's performance. A trapezoidal partition \Delta is called recursively separable if the following holds: ffl \Delta consists of a single trapezoid, or ffl there exists a line ' not intersecting the interior of any trapezoid in \Delta such that (i) are both nonempty, where ' are the two half-planes defined by ', and (ii) each of \Delta recursively separable. The following key lemma gives an upper bound on the penalty incurred by our approximation algorithm if only recursively separable trapezoidal partitions are used. Lemma 5.1 Let P be a finite set of points in the plane and let \Delta be a trapezoidal partition of P with k trapezoids. There exists a recursively separable partition \Delta ? of P with O(k log trapezoids. In addition, each separating line is either a horizontal line passing through a vertex of \Delta or a line supporting an edge of a trapezoid in \Delta. Proof: We present a recursive algorithm for computing \Delta ? . Our algorithm is similar to the binary space partition algorithm proposed by Paterson and Yao [17]. We assume that the boundaries of the trapezoids in \Delta are also pairwise disjoint-this assumption is only to simplify our proof. At each recursive step of the algorithm, the subproblem under consideration lies in a trapezoid T . (This containing trapezoid may degenerate to a triangle, or it may even be unbounded.) The top and bottom edges of T (if they exist) pass through the vertices of \Delta, while the left and right edges (if they exist) are portions of edges of \Delta. Initially T is a set to an appropriately large trapezoid containing the family \Delta. Let \Delta T denote the trapezoidal partition of P " T obtained by intersecting \Delta with T , and let V T be the set of vertices of lying in the interior of T . An edge of \Delta T cannot intersect the left or right edge of T , because they are portions of the edge of T . Therefore, each edge of \Delta T either lies in the interior of T , or intersects only the top and bottom edges of T . If stop. Otherwise, we proceed as follows. If there is a trapezoid 4 2 \Delta T that completely crosses T (that is, its vertices lie on the top and bottom edges of T ), then we do the following. If 4 is the leftmost trapezoid of \Delta T , then we partition T into two trapezoids by drawing a line through the right edge of \Delta, so that T 1 contains 4 and T 2 contains the remaining trapezoids of \Delta T . If 4 is not the leftmost trapezoid of \Delta T , then we partition T into by drawing a line through the left edge of \Delta. If every trapezoid \Delta in \Delta T has at least one vertex in the interior of T , and so the previous condition is not met, then we choose a point with a median y-coordinate. We partition T into trapezoids T by drawing a horizontal line ' v passing through v. Each trapezoid partitioned into two trapezoids by adding the segment . At the end of this dividing step, let \Delta 1 and \Delta 2 be the set of trapezoids of that lie in T 1 and T 2 , respectively. We recursively refine \Delta 1 and \Delta 2 into separable partitions respectively, and return \Delta ? 2 . This completes the description of the algorithm. We now prove that \Delta satisfies the properties claimed in the lemma. It is clear that \Delta is recursively separable and that each separating line of \Delta either supports an edge of \Delta or it is horizontal. To bound the size of \Delta , we charge each trapezoid of \Delta to its bottom-left vertex. Each such vertex is either a bottom-left vertex of a trapezoid of \Delta, or it is an intersection point of a left edge of a trapezoid of \Delta with the extension of a horizontal edge of another trapezoid of \Delta. There are only k vertices of the first type, so it suffices to bound the number of vertices of the second type. Since the algorithm extends a horizontal edge of a trapezoid of \Delta T only if every trapezoid of \Delta T has at least one vertex in the interior of T , and we always extend a horizontal edge with a median y-coordinate, it is easily seen that the number of vertices of the second type is O(k log k). This completes the proof. 2 Remark: Given a family \Delta of k disjoint orthogonal rectangles partitioning P , we can find a set of O(k) recursively separable rectangles that forms a rectangular partition of P -this uses the orthogonal binary space partition algorithm of Paterson and Yao [18]. 6 An Approximation Algorithm Lemma 5.1 applies to any trapezoidal partition of P . In particular, if we start with a canonical trapezoidal partition \Delta, then the output partition \Delta ? is both canonical and recursively separable, and each separating line in \Delta ? belongs to the family of canonical lines L(P ). For the lack of a better term, we call a trapezoidal partition of P that satisfies these conditions a compliant partition. Lemmas 3.1 and 5.1 together imply the following useful theorem. Theorem 6.1 Let P be a set of n points in the plane and let C be a boolean constraint function satisfying the condition (3.1). If there is a trapezoidal partition of P respecting C with k trapezoids, then there is a compliant partition of P also respecting C with O(k log trapezoids. In the remainder of this section, we give a polynomial-time algorithm, using dynamic programming, for constructing an optimal compliant partition. By Theorem 6.1, this partition has O(K trapezoids. Recall that the set consists of all canonical lines determined by P . Consider a subset of points R ' P , and a canonical trapezoid \Delta containing R. Let oe(R; \Delta) denote the size of an optimal compliant partition of R in \Delta; the size of a partition is the number of trapezoids in the partition. Theorem 6.1 gives the following recursive definition of oe: min where the minimum is over all those lines ' 2 L that are either horizontal and intersect \Delta, or intersect both the top and bottom edges of \Delta; ' denote the positive and negative half-planes induced by . The goal of our dynamic programming algorithm is to compute oe(P; T ), for some canonical trapezoid T enclosing all the points P . We now describe how the dynamic programming systematically computes the required partial answers. Every canonical trapezoid \Delta in the plane can be described (uniquely) by a 6-tuple consisting of integers between 1 and n. The first two numbers fix two points p i and p j through which the lines containing the top and bottom edges of \Delta pass; the second pair fixes the points p k 1 through which the line containing the left edge of passes; and the third pair fixes the points p l 1 through which the line containing the right edge of \Delta passes. (In case of ties, we may choose the points closest to the corners of \Delta.) We use the notation \Delta(i; for the trapezoid associated with the 6-tuple If the 6-tuple does not produce a trapezoid, then \Delta(i; undefined. We use the abbreviated notation to denote the size of an optimal compliant partition for the points contained in \Delta(i; The quantity oe(i; undefined if the trapezoid \Delta(i; If the points in P are sorted in increasing order of their y-coordinates, then \Delta(i; is defined only for i - j. Our dynamic programming algorithm computes the oe values as follows. If C (\Delta(i; Otherwise, min where the last minimum varies over all pairs of points such that the line passing through them intersects both the top and the bottom edge of \Delta(i; If \Xi denotes the set of all canonical trapezoids, then )-each 6-tuple is associated with at most one unique trapezoid. If Q(n) denotes the time to decide whether for an arbitrary trapezoid \Delta, then we can initially compute all trapezoids for which C these trapezoids, we initially set For all the remaining trapezoids in \Xi, we use the recursive formula presented above to compute their oe. Computing oe for a trapezoid requires computing the minimum of O(n 2 ) quantities. Thus the total running time of the algorithm is O(n 8 ). The following theorem states the main result of our paper. Theorem 6.2 Given a set P of n points in the plane and a boolean constraint C satisfying condition (3.1), we can compute a geometric partition of P with respect to C using the number of trapezoids in an optimal partition. Our algorithm runs in worst-case time O(n 8 +n 6 Q(n)), where Q(n) is the time to decide whether any subset R ' P . Remark: By computing oe's in a more clever order and exploiting certain geometric properties of a geometric partition, the time complexity of the above algorithm can be improved by one order of magnitude. This minor improvement, however, doesn't seem worth the effort needed to explain it. Theorem 6.2 immediately implies polynomial-time approximation algorithms for the surface approximation and the planar bichromatic partition problem. In the case of the surface approximation problem, deciding C (\Delta) for a trapezoid \Delta requires checking whether there is a plane - in R 3 such that the vertical distance between - and the points covered by \Delta is at most ". This problem can be solved in linear time using the fixed-dimensional linear programming algorithm of Megiddo [15]. more practical algorithm, running in time O(n log n), is the following. Let A ' P denote the set of points covered by the trapezoid \Delta. For a point respectively, denote the point p translated vertically up and down by ". Let A only if sets A + and A \Gamma can be separated by a plane. The two sets are separable if their convex hulls are disjoint. This can be tested in O(n log n) time-for instance, see the book by Preparata and Shamos [19].) Theorem 6.3 Given a set S of n points in R 3 and a parameter " ? 0, in polynomial time we can compute an "-approximate polyhedral terrain \Sigma with O(K K o is the number of vertices in an optimal terrain. Our algorithm runs in O(n 8 ) worse-case time. In the planar bichromatic partition problem, deciding whether C checking whether \Delta contains any point from the red set R. This can clearly be done in O(n) time. Actually, with a preprocessing requiring O(n 2 log O(1) n) time, this test can be made in O(log n) time for any trapezoid \Delta. Our main purpose, however, is to show the polynomial time for the approximation algorithm. Theorem 6.4 Given a set R of 'red' points and another set B of `blue' points in the plane, we can find in polynomial time a set of O(K disjoint triangles that cover B but do not contain any red point; K o is the number of triangles in an optimal solution. Remark: In view of the remark following Theorem 6.1, given a set R of 'red' points and another set B of 'blue' points in the plane, we can find in polynomial time a set of O(K disjoint orthogonal rectangles that cover B but do not contain any red point. In this case, the time complexity improves by a few orders of magnitude, because there are only rectangle and each rectangle is subdivided into two rectangles by drawing a horizontal or vertical line passing through one of the points. Omitting all the details, the running time in this case is O(n 5 ). Extensions We can extend our algorithm to a slightly stronger form of surface approximation. In the basic problem, the implicit function (surface) is represented by a set of sample points S. What if the sample consists of two-dimensional compact, connected pieces? In this section, we show an extension of our algorithm that deals with the case when the sample consists of a set T of n horizontal triangles with pairwise disjoint xy-projection. (Since any polygon can be decomposed into triangles, this case also handles polygons.) Our goal is to compute a polyhedral terrain \Sigma, so that the vertical distance between any point in T is at most ". We produce a terrain \Sigma with O(K is the number of vertices in an optimal such surface. be the input set of n horizontal triangles in R 3 with the property that their vertical projections on the plane are pairwise disjoint. We will consistently use the following notational convention: for an object s 2 R 3 , - s denotes its orthogonal projection on the plane z = 0, and for a subset g ' - denotes the portion of s in R 3 such that g. Abusing the notation slightly, we say that a set \Xi of trapezoids (or triangles) in R 3 "-approximates T within a region Q ' R 2 if the vertical distance between T and \Xi in Q is at most " and the vertical projections of trapezoids of \Xi are disjoint on z = 0. Let S denote the set of vertices of the triangles in T , and let - S be their orthogonal projection on z = 0. We set S, as the set of points in our abstract problem. The constraint function is defined as follows. Given a trapezoid \Delta 2 R 3 , we have C and only if the vertical distance between \Delta and any point in directly above/below \Delta is at most ". (Thus, while the point set P includes only the vertices of T , the constraint set takes into consideration the whole triangles.) The constraint C satisfies (3.1), and it can be computed in polynomial time. It is also clear that the size of an optimal trapezoidal partition of P with respect to C is a lower bound on the size of a similar partition for T , the set of triangles. We first apply Theorem 6.3 to obtain a family \Delta of O(k log trapezoids that "-approximates P with respect to C; clearly k - K . The next step of our algorithm is to extend \Delta to a polyhedral terrain that "-approximates the triangles of T . Care must to be exercised in this step if one wants to add only O(k log new trapezoids. In the second step, we work with the projection of T and \Delta in the plane (i) (ii) (iii) Figure 5: (i) - T and - \Delta, (ii) R 1 and G; (iii) R 2 and Q i 's. \Deltag and - g. Let R be the set of connected components of the closure of \Delta. That is, R is the portion of T lying in the common exterior of - \Delta, as shown in Figure 5 (i). R is a collection of simple polygons, each of which is contained in a triangle of - T . Since the corners of the triangles of - are covered by - \Delta, the vertices of all polygons in R lie on the boundary of - \Delta, and each edge of R is contained in an edge of - \Delta or of - T . Let R 1 ' R be the subset of polygons that touch at least three edges of trapezoids in - \Delta, and let R For each polygon P i 2 R 1 , we compute a set of triangles that "-approximate T within . For a vertex lying on the boundary of a trapezoid - \Delta, let - v denote the point on \Delta whose xy-projection is v. Let - T i be the triangle containing P i . We triangulate P i , and, for each triangle 4abc in the triangulation, we pick 4-a - b-c. Since T i is parallel to the xy-plane, it can be proved that the maximum vertical distance between 4-a - b-c and T i is ". We repeat this step for all polygons in R 1 . The number of triangles generated in this step is proportional to the number of vertices in the polygons of R 1 , which we bound in the following lemma. Lemma 7.1 The total number of vertices in the polygons of R 1 is O(k log k). Proof: Each vertex of a polygon in R 1 is either (i) a vertex of a trapezoid in - \Delta, or (ii) an intersection point of an edge of - \Delta with an edge of a triangle in T . There are only O(k log vertices of type (i), so it remains to bound the number of vertices of type (ii). We construct an undirected graph be the set of edges in - \Delta. 1 To avoid confusion, we will call the edges of \Gamma segments and those of E arcs. For each segment fl i , we place a point i close to fl i , inside the trapezoid bounded by . The set of resulting points forms the node set V . If there is an edge pq of a polygon in R 1 such that we add the arc (i; j) to E; see Figure 5 (ii). It is easily seen that G is a planar graph, and that log k). Fix a pair of segments that (1; be the set of edges in R 1 , sorted either left to right or top to bottom, as the case may be, that are incident to fl 1 and fl 2 . Let jE 12 Assume that for every 1 - lies on fl 1 and q i lies on fl 2 . The number of vertices of type (ii) is obviously 2 We call two edges equivalent if the interior of the convex hull of p i q i and p j q j does not intersect any trapezoid of - \Delta. This equivalence relation partitions E 12 into equivalent classes, each consisting of a contiguous subsequence of E 12 . Let - ij denote the number of equivalence classes in E ij . 1: There are at most two edges in each equivalence class of E 12 . Proof: Assume for the sake of a contradiction that three edges belong to an equivalence class. Further, assume that the triangle - T bounded by p i q i lies below p i q i (see Figure 6 (i)). Since the quadrilateral Q defined by p i q i and q does not contain any trapezoid of - \Delta, p i+1 q i+1 is also an edge of - T . But then Q is a connected component of R and it touches only two edges of - \Delta, thereby implying that is not an edge of a polygon of R 1 , a contradiction. 2 Thus, 1 The segments of \Gamma may overlap, because the trapezoids of - can touch each other. If a segment fl i of \Gamma is an edge of two trapezoids, then no edge of R1 can be incident to fl i . Figure Edges in an equivalent class of E 12 , and (ii) edges in different equivalent classes Next, we bound the quantity 12 be two consecutive equivalent classes of E 12 , let p i q i be the bottom edge of E j 12 , and let p i+1 q i+1 be the top edge of E j+1 12 . The quadrilateral Q contains at least one trapezoid - of - \Delta. We call the triangle edges of Q, and p i the trapezoidal edges of Q. The triangle edges of Q are adjacent in E 12 . Let be the set of resulting quadrilaterals. Since suffices to bound the number of quadrilaterals in Q. Consider the planar subdivision induced by Q and call it A(Q). For each bounded face Q(f) be the smallest quadrilateral of Q that contains f . Since the boundaries of quadrilaterals do not cross, Q(f) is well defined. 2: Every face f of A(Q) can be uniquely associated with a trapezoid \Delta such that Proof: The claim is obviously true for the unbounded face, so assume that f is a bounded does not contain any other quadrilateral of Q, then so by definition f contains a trapezoid of - \Delta. If Q does contain another quadrilateral of Q, let Q j be the largest trapezoid that lies inside Q i -that is, @Q j is a part of @f . If none of the trapezoidal edges of Q j lies in the interior of Q i , then the trapezoidal edges of both lie on the same segments of \Gamma, say, . Consequently, the triangle edges of both belong to E 12 , which is impossible because then the triangle edges of Q i are not adjacent in E 12 . Hence, one of the trapezoidal edges of Q j lies in the interior of Q i . Let \Delta be the trapezoid bounded by this edge. Since the triangle edges of Q j lie outside - the interior of - \Delta does not intersect any edge of R 1 , - lies in f . This completes the proof of Claim 2. 2 By Claim 2, the number of faces in A(Q) is at most j - log k). This completes the proof of the lemma. 2 Next, we partition the polygons of R 2 into equivalence classes in the same way as we Closing Remarks 21 partitioned the edges of E 12 in the proof of Lemma 7.1. That is, we call two polygons endpoints lie on the same pair of edges in - \Delta, and (ii) the interior of the convex hull of does not intersect any trapezoid of - \Delta. Using the same argument as in the proof of the above lemma, the following lemma can be established. Lemma 7.2 The edges of R 2 can be partitioned into O(k log equivalence classes. For each equivalence class be the convex hull of E i -observe that Q i is a convex quadrilateral, as illustrated in Figure 5 (iii). Each quadrilateral Q i can be "- approximated using at most three triangles in R 3 in the same way as we approximated each polygon P i of R 1 . By Lemma 7.2, the total number of triangles created in this step is also O(k log k). Putting together these pieces, we obtain the following lemma. Lemma 7.3 The family of trapezoids \Delta can be supplemented with O(k log additional trapezoids in R 3 so that all the triangles of T are "-approximated. The orthogonal projection of all the trapezoids on the plane disjoint. The area not covered by the projection of trapezoids found in the preceding lemma, of course, can be approximated without any regards to the triangles of T . The final surface has O(K trapezoids and it "-approximates the family of triangles T . We finish with a statement of our main theorem in this section. Theorem 7.4 Given a set of n horizontal triangles in R 3 , with pairwise disjoint projection on the plane z = 0, and a parameter " ? 0, we can compute in polynomial time a "- approximate polyhedral terrain of size O(K o log K is the size of an optimal "-approximate terrain. 8 Closing Remarks We presented an approximation technique for certain geometric covering problems with a disjointness constraint. Our algorithm achieves a logarithmic performance guarantee on the size of the cover, thus matching the bound achieved by the "greedy set cover" heuristic for arbitrary sets and no disjointness constraint. Applications of our result include polynomial time algorithms to approximate a monotone, polyhedral surface in 3-space, and to approximate the disjoint cover by triangles of red-blue points. We also proved that these problems are NP-Hard . The surface approximation problem is an important problem in visualization and computer graphics. The state of theoretical knowledge on this problem appears to be rather slim. Except for the convex surfaces, no approximation algorithms with good performance guarantees are known [6, 16]. For the approximation of convex polytopes, it turns out that one does not need disjoint covering, and therefore the greedy set cover heuristic works. We conclude by mentioning some open problems. An obvious open problem is to reduce the running time of our algorithm for it to be of any practical value. Finding efficient heuristics with good performance guarantees seems hard for most of the geometric partitioning problems, and requires further work. A second problem of great practical interest is to "-approximate general polyhedra-this problem arises in many real applications of computer modeling. To the best of our knowledge, the latter problem remains open even for the special case where one wants to find a minimum-vertex polyhedral surface that lies between two monotone surfaces. The extension of our algorithm presented in Section 7 does not work because we do not know how to handle the last fill-in stage. --R Fast greedy algorithms for geometric covering and other problems An algorithm for piecewise linear approximation of an implicitly defined two-dimensional surfaces An algorithm for piecewise linear approximation of an implicitly defined manifold Decision trees for geometric models Polygonization of implicit surfaces Algorithms for polytope covering and approximation Simplification of objects rendered by polygonal approximations Several hardness results on problems of point separation and line stabbing Optimal packing and covering in the plane are NP-complete Piecewise linear approximations of digitized space curves with applications Scientific Visualization of Physical Phenomena pp. Approximation algorithms for combinatorial problems Computing 11 A high resolution 3D surface construction algo- rithm On the ratio of optimal integral and fractional cover Separation and approximation of polyhedral surfaces Efficient binary space partitions for hidden-surface removal and solid modeling Optimal binary space partitions for orthogonal objects Computational Geometry: An Introduction Decimation of triangle meshes An adaptive subdivision method for surface fitting from sampled data On some link distance problems in a simple polygon --TR --CTR Gabriel Peyr , Stphane Mallat, Surface compression with geometric bandelets, ACM Transactions on Graphics (TOG), v.24 n.3, July 2005 David Cohen-Steiner , Pierre Alliez , Mathieu Desbrun, Variational shape approximation, ACM Transactions on Graphics (TOG), v.23 n.3, August 2004 Pankaj K. Agarwal , Boris Aronov , Vladlen Koltun, Efficient algorithms for bichromatic separability, ACM Transactions on Algorithms (TALG), v.2 n.2, p.209-227, April 2006 Pankaj K. Agarwal , Boris Aronov , Vladlen Koltun, Efficient algorithms for bichromatic separability, Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms, January 11-14, 2004, New Orleans, Louisiana Michiel Smid , Rahul Ray , Ulrich Wendt , Katharina Lange, Computing large planar regions in terrains, with an application to fracture surfaces, Discrete Applied Mathematics, v.139 n.1-3, p.253-264, Mark de Berg , Micha Streppel, Approximate range searching using binary space partitions, Computational Geometry: Theory and Applications, v.33 n.3, p.139-151, February 2006 Bernard Chazelle , C. Seshadhri, Online geometric reconstruction, Proceedings of the twenty-second annual symposium on Computational geometry, June 05-07, 2006, Sedona, Arizona, USA Joseph S. B. Mitchell , Joseph O'Rourke, Computational geometry, ACM SIGACT News, v.32 n.3, p.63-72, 09/01/2001

simplification;visualization;approximation algorithms;levels of detail;dynamic programming;machine learning;terrains

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Computational Complexity and Knowledge Complexity.

We study the computational complexity of languages which have interactive proofs of logarithmic knowledge complexity. We show that all such languages can be recognized in ${\cal BPP}^{\cal NP}$. Prior to this work, for languages with greater-than-zero knowledge complexity only trivial computational complexity bounds were known. In the course of our proof, we relate statistical knowledge complexity to perfect knowledge complexity; specifically, we show that, for the honest verifier, these hierarchies coincide up to a logarithmic additive term.

Introduction The notion of knowledge-complexity was introduced in the seminal paper of Goldwasser Micali and Rackoff [GMR-85, GMR-89]. Knowledge-complexity (KC) is intended to measure the computational advantage gained by interaction. Satisfactory formulations of knowledge- complexity, for the case that it is not zero, have recently appeared in [GP-91]. A natural suggestion, made by Goldwasser, Micali and Rackoff, is to classify languages according to the knowledge-complexity of their interactive-proofs [GMR-89]. We feel that it is worthwhile to give this suggestion a fair try. The lowest level of the knowledge-complexity hierarchy is the class of languages having interactive proofs of knowledge-complexity zero, better known as zero-knowledge. Actually, there are three hierarchies extending the three standard definitions of zero-knowledge; namely An extended abstract of this paper appeared in the 26th ACM Symposium on Theory of Computing (STOC 94), held in Montreal, Quebec, Canada, May 23-25, 1994. y Department of Applied Mathematics and Computer Science, Weizmann Institute of Science, Rehovot, Israel. E-mail: oded@wisdom.weizmann.ac.il. Supported by grant no. 92-00226 from the United States Israel Binational Science Foundation, Jerusalem, Israel. z Computer Science Division, University of California at Berkeley, and International Computer Science Institute, Berkeley, CA 94720. E-mail: rafail@cs.Berkeley.EDU. Supported by an NSF Postdoctoral Fellowship and ICSI. x Computer Science Department, Technion - Israel Institute of Technology, Haifa 32000, Israel. E-mail: erez@cs.technion.ac.il. perfect, statistical and computational. Let us denote the corresponding hierarchies by PKC(\Delta), SKC(\Delta), and CKC(\Delta). Assuming the existence of one-way functions, the third hierarchy collapses, namely differently, the zero level of computational knowledge-complexity extends to the maximum possible. Anyhow, in the rest of this paper we will be only interested in the other two hierarchies. Previous works have provided information only concerning the zero level of these hierar- chies. Fortnow has pioneered the attempts to investigate the computational complexity of (perfect/statistical) zero-knowledge [F-89], and was followed by Aiello and Hastad [AH-87]. Their results can be summarized by the following theorem that bounds the computational complexity of languages having zero-knowledge proofs. Theorem [F-89, AH-87]: co-AM Hence, languages having statistical zero-knowledge must lie in the second level of the polynomial-time hierarchy. Needless to say that function k and in particular for k j 0. On the other hand, if we allow polynomial amount of knowledge to be revealed, then every language in IP can be proven. Theorem [LFKN-90, Sh-90]: As indicated in [GP-91], the first equality is a property of an adequate definition (of knowledge complexity) rather than a result. In this paper we study the class of languages that have interactive-proofs with logarithmic knowledge-complexity. In particular, we bound the computational complexity of such languages, showing that they can be recognized by probabilistic polynomial-time machines with access to an NP oracle. Main Theorem: We recall that BPP NP is contained in the third level of the polynomial-time hierarchy (PH). It is believed that PH is a proper subset of PSPACE. Thus, assuming PH ae PSPACE, our result yields the first proof that there exist languages in PSPACE which cannot be proven by an interactive-proof that yields O(log n) bits of knowledge. In other words, there exist languages which do have interactive proofs but only interactive proofs with super-logarithmic knowledge-complexity. Prior to our work, there was no solid indication 1 that would contradict the possibility that all languages in PSPACE have interactive-proofs which yield only one bit of knowledge. Alas, if one had been willing to assume that all languages in PSPACE have interactive proofs of logarithmically many rounds, an assumption that we consider unreasonable, then the result in [BP-92] would have yielded a proof that PSPACE is not contained in SKC(1), provided (again) that PH ae PSPACE . The only attempt to bound the computational complexity of languages having interactive proofs of low knowledge-complexity was done by Bellare and Petrank. Yet, their work refers only to languages having interactive proofs that are both of few rounds and of low knowledge complexity [BP-92]. Specifically, they showed that if a language L has a r(n)-round interactive-proof of knowledge-complexity O( log n then the language can be recognized in BPP NP . Our proof of the Main Theorem consists of two parts. In the first part, we show that the procedure described by Bellare and Petrank [BP-92] suffices for recognizing languages having interactive proofs of logarithmic perfect knowledge complexity. To this end, we use a more careful analysis than the one used in [BP-92]. In the second part of our proof we transform interactive proofs of statistical knowledge complexity k(n) into interactive proofs of perfect knowledge complexity k(n)+log n. This transformation refers only to knowledge-complexity with respect to the honest verifier, but this suffices since the first part of our proof applies to the knowledge-complexity with respect to the honest verifier. Yet, the transformation is interesting for its own sake, and a few words are in place. The question of whether statistical zero-knowledge equals perfect zero-knowledge is one of the better known open problems in this area. The question has been open also for the case of zero-knowledge with respect to the honest verifier. We show that for every poly-time computable function k : N7!N (and in particular for k j This result may be considered an indication that these two hierarchies may collide. Techniques Used As stated above, the first part of our proof consists of presenting a more careful analysis of an existing procedure, namely the procedure suggested by Bellare and Petrank in [BP-92]. Their procedure, in turn, is a culmination of two sequences of works discussed bellow. The first sequence originates in Fortnow's definition of a simulator-based prover [F-89]. Fortnow [F-89], and consequently Aiello and Hastad [AH-87], used the simulator-based prover in order to infer, by way of contradiction, bounds on the sizes of specific sets. A more explicit usage of the simulator-based prover was introduced by Bellare, Micali and Ostrovsky specifically, they have suggested to use a PSPACE-implementation of the simulator-based prover, instead of using the original prover (of unbounded complexity) witnessing the existence of a zero-knowledge interactive proof system. (Thus, they obtained a bound on the complexity of provers required for zero-knowledge proof systems.) Ostrovsky [Ost-91] suggested to use an implementation of the interaction between the verifier and the simulation-based prover as a procedure for deciding the language. Furthermore, assuming that one-way functions do not exist, he used "universal extrapolation" procedures of [ILu-90, ILe-90] to approximate the behavior of the simulator-based prover. (Thus, assuming that one-way function do not exists, he presented an efficient procedure that decides languages in SKC(0) and inferred that one-way functions are essential to the non-triviality of statistical zero-knowledge). Bellare and Petrank distilled the decision procedure from the context of one-way functions, showing that the simulator-based prover can be implemented using a perfect universal extrapolator (also known as a "uniform generation" procedure) [BP-92]. The error in the implementation is directly related to the deviation of the uniform generation procedure. The second sequence of works deals with the two related problems of approximating the size of sets and uniformly generating elements in them. These problems were related by Jerrum et. al. [JVV-86]. Procedures for approximating the size of sets were invented by Sipser [Si-83] and Stockmeyer [St-83], and further improved in [GS-89, AH-87], all using the "hashing paradigm". The same hashing technique, is the basis of the "universal extrapo- lation" procedures of [ILu-90, ILe-90]. However, the output of these procedures deviates from the objective (i.e., uniform distribution on the target set) by a non-negligible amount (i.e., 1=poly(T ) when running for time T ). On the other hand, Jerrum et. al. have also pointed out that (perfect) uniform generation can be done by a BPP \Sigma P Bellare and Petrank combined the hashing-based approximation methods with the ideas of [JVV-86] to obtain a BPP NP -procedure for uniform generation with exponentially vanishing error probability [BP-92]. Actually, if the procedure is allowed to halt with no output with constant (or exponentially vanishing) probability, then its output distribution is exactly uniform on the target set. Motivation for studying KC In addition to the self-evident fundamental appeal of knowledge complexity, we wish to point out some practical motivation for considering knowledge-complexity greater than zero. In particular, cryptographic protocols that release a small (i.e., logarithmic) amount of knowledge may be of practical value, especially if they are only applied once or if one can obtain sub-additive bounds on the knowledge complexity of their repeated executions. Note that typically, a (single application of a) sub-protocol leaking logarithmically many bits (of knowl- edge) does not compromise the security of the entire protocol. The reason being that these (logarithmically many) bits can be guessed with non-negligible probability, which in turn means that any attack due to the "leaked bits" can be simulated with non-negligible probability without them. But why use low knowledge-complexity protocols when one can use zero-knowledge ones (see, [GMW-86, GMW-87])? The reason is that the non-zero-knowledge protocols may be more efficient and/or may require weaker computational assumptions (see, for example, [OVY-91]). Remarks A remark concerning two definitions. Throughout the paper, SKC(k(\Delta)) and PKC(k(\Delta)) denote the classes of knowledge-complexity with respect to the honest verifier. Note that the Main Theorem is only strengthen by this, whereas the transformation (mentioned above) is indeed weaker. Furthermore, by an interactive proof we mean one in which the error probability is negligible (i.e., smaller than any polynomial fraction). A few words of justification appear in Section 2. A remark concerning Fortnow's paper [F-89]. In course of this research, we found out that the proof that SKC(0) ' co-AM as it appears in [F-89] is not correct. In particular, there is a flaw in the AM-protocol presented in [F-89] for the complement language (see Appendix A). However, the paper of Aiello and Hastad provides all the necessary machinery for proving Fortnow's result as well [AH-87, H-94]. Needless to say that the basic approach presented by Fortnow (i.e., looking at the "simulator-based prover") is valid and has inspired all subsequent works (e.g., [AH-87, BMO-90, Ost-91, BP-92, OW-93]) as well as the current one. Preliminaries Let us state some of the definitions and conventions we use in the paper. Throughout this paper we use n to denote the length of the input x. A function f called negligible if for every polynomial p and all sufficiently large n's p(n) . 2.1 Interactive proofs Let us recall the concept of interactive proofs, presented by [GMR-89]. For formal definitions and motivating discussions the reader is referred to [GMR-89]. A protocol between a (computationally unbounded) prover P and a (probabilistic polynomial-time) verifier V constitutes an interactive proof for a language L if there exists a negligible function ffl such that 1. Completeness: If x 2 L then 2. Soundness: If x 62 L then for any prover P Remark: Usually, the definition of interactive proofs is robust in the sense that setting the error probability to be bounded away from 1does not change their expressive power, since the error probability can be reduced by repetitions. However, this standard procedure is not applicable when knowledge-complexity is measured, since (even sequential) repetitions may increase the knowledge-complexity. The question is, thus, what is the right definition. The definition used above is quite standard and natural; it is certainly less arbitrary then setting the error to be some favorite constant (e.g., 1) or function (e.g., 2 \Gamman ). Yet, our techniques yield non-trivial results also in case one defines interactive proofs with non-negligible error probability (e.g., constant error probability). For example, languages having interactive proofs with error probability 1=4 and perfect knowledge complexity 1 are also in BPP NP . For more details see Appendix B. Also note that we have allowed two-sided error probability; this strengthens our main result but weakens the statistical to perfect transformation 2 . Suppose you had a transformation for the one-sided case. Then, given a two-sided interactive proof of some statistical knowledge complexity you could have transformed it to a one-sided error proof of the same knowledge complexity (cf., [GMS-87]). Applying the transformation for the one-sided case would have yielded an even better result. 2.2 Knowledge Complexity Throughout the rest of the paper, we refer to knowledge-complexity with respect to the honest verifier; namely, the ability to simulate the (honest) verifier's view of its interaction with the prover. (In the stronger definition, one considers the ability to simulate the point of view of any efficient verifier while interacting with the prover.) We let denote the random variable that represents V 's view of the interaction with P on common input x. The view contains the verifier's random tape as well as the sequence of messages exchanged between the parties. We begin by briefly recalling the definitions of perfect and statistical zero-knowledge. A perfect zero-knowledge (resp., statistical zero-knowledge) over a language L if there is a probabilistic polynomial time simulator M such that for every x 2 L the random variable M(x) is distributed identically to the statistical difference between M(x) and (P; V )(x) is a negligible function in jxj). Next, we present the definitions of perfect (resp., statistical) knowledge-complexity which we use in the sequel. These definitions extend the definition of perfect (resp., statistical) zero- knowledge, in the sense that knowledge-complexity zero is exactly zero-knowledge. Actually, there are two alternative formulations of knowledge-complexity, called the oracle version and the fraction version. These formulations coincide at the zero level and differ by at most an additive constant otherwise [GP-91]. For further intuition and motivation see [GP-91]. It will be convenient to use both definitions in this paper 3 . By the oracle formulation, the knowledge-complexity of a protocol is the number of oracle (bit) queries that are needed to simulate the protocol efficiently. Definition 2.1 (knowledge complexity - oracle version): Let k: N ! N. We say that an interactive proof language L has perfect (resp., statistical) knowledge complexity k(n) in the oracle sense if there exists a probabilistic polynomial time oracle machine M and an oracle A such that: 1. On input x 2 L, machine M queries the oracle A for at most k(jxj) bits. 2. For each x 2 L, machine M A produces an output with probability at least 1, and given that M A halts with an output, M A (x) is identically distributed (resp., statistically close) to In the fraction formulation, the simulator is not given any explicit help. Instead, one measures the density of the largest subspace of simulator's executions (i.e., coins) which is identical (resp., close) to the Definition 2.2 (knowledge complexity - fraction version): Let ae: N ! (0; 1]. We say that an interactive proof language L has perfect (resp., statistical) knowledge-complexity log 2 (1=ae(n)) in the fraction sense if there exists a probabilistic polynomial-time machine M with the following "good subspace" property. For any x 2 L there is a subset of M's possible random tapes S x , such that: 3 The analysis of the [BP-92] procedure is easier when using the fraction version, whereas the transformation from statistical to perfect is easier when using the oracle version. 1. The set S x contains at least a ae(jxj) fraction of the set of all possible coin tosses of M(x). 2. Conditioned on the event that M(x)'s coins fall in S x , the random variable M(x) is identically distributed (resp., statistically close) to )(x). Namely, for the perfect case this means that for every - c where M(x;!) denotes the output of the simulator M on input x and coin tosses sequence !. As mentioned above, these two measures are almost equal. Theorem [GP-91]: The fraction measure and the oracle measure are equal up to an additive constant. Since none of our results is sensitive to a difference of an additive constant in the measure, we ignore this difference in the subsequent definition as well as in the statement of our results. Definition 2.3 (knowledge complexity classes): languages having interactive proofs of perfect knowledge complexity k(\Delta). languages having interactive proofs of statistical knowledge complexity k(\Delta). 2.3 The simulation based prover An important ingredient in our proof is the notion of a simulation based prover, introduced by Fortnow [F-89]. Consider a simulator M that outputs conversations of an interaction between a prover P and a verifier V . We define a new prover P , called the simulation based prover, which selects its messages according to the conditional probabilities induced by the simulation. Namely, on a partial history h of a conversation, P outputs a message ff with probability denotes the distribution induced by M on t-long prefixes of conversations. (Here, the length of a prefix means the number of messages in it.) It is important to note that the behavior of P is not necessarily close to the behavior of the original prover P . Specifically, if the knowledge complexity is greater than 0 and we consider the simulator guaranteed by the fraction definition, then P and P might have quite a different behavior. Our main objective will be to show that even in this case P still behaves in a manner from which we can benefit. 3 The Perfect Case In this section we prove that the Main Theorem holds for the special case of perfect knowledge complexity. Combining this result with the transformation (Theorem 2) of the subsequent section, we get the Main Theorem. Theorem 1 PKC(O(log n)) ' BPP NP Our proof follows the procedure suggested in [BP-92], which in turn follows the approach of [F-89, BMO-90, Ost-91] while introducing a new uniform generation procedure which builds on ideas of [Si-83, St-83, GS-89, JVV-86] (see introduction). Suppose that is an interactive proof of perfect knowledge complexity O(log n) for the languages L, and let M be the simulator guaranteed by the fraction formulation (i.e., Definition 2.2). We consider the conversations of the original verifier V with the simulation-based-prover P (see definition in Section 2.3). We are going to show that the probability that the interaction (P accepting is negligible if x 62 L and greater than a polynomial fraction if x 2 L. Our proof differs from [BP-92] in the analysis of the case x 2 L (and thus we get a stronger result although we use the same procedure). This separation between the cases x 62 L and x 2 L can be amplified by sequential repetitions of the protocol (P remains to observe that we can sample the (P in probabilistic polynomial-time having access to an NP oracle. This observation originates from [BP-92] and is justified as follows. Clearly, V 's part of the interaction can be produced in polynomial-time. Also, by the uniform generation procedure of [BP-92] we can implement by a probabilistic polynomial time machine that has access to an NP oracle. Actually, the implementation may fail with negligible probability, but this does not matter. Thus, it remains only to prove the following lemma. Lemma 3.1 1. If x 2 L then the probability that (P outputs an accepting conversation is at least2 2. If x 62 L then the probability that (P outputs an accepting conversation is negligible. Remark: In [BP-92], a weaker lemma is proven. Specifically, they show that the probability that (P output an accepting conversation (on x 2 L) is related to 2 \Gammak \Deltat , where t is the number of rounds in the protocol. Note that in our proof t could be an arbitrary polynomial number of rounds. proof: The second part of the lemma follows from the soundness property as before. We thus concentrate on the first part. We fix an arbitrary x 2 L for the rest of the proof and allow ourselves not to mention it in the sequel discussion and notation. Let q be the number of coin tosses made by M . We denote q the set of all possible coin tosses, and by S the "good subspace" of M (i.e., S has density 2 \Gammak in\Omega and for ! chosen uniformly in S the simulator outputs exactly the distribution of the interaction Consider the conversations that are output by the simulator on ! 2 S. The probability to get such a conversation when the simulator is run on ! uniformly selected in \Omega\Gamma is at least 2 \Gammak . We claim that the probability to get these conversations in the interaction (P is also at least 2 \Gammak . This is not obvious, since the distribution produced by (P not be identical to the distribution produced by M on a uniformly selected ! 2 \Omega\Gamma Nor is it necessarily identical to the distribution produced by M on a uniformly selected ! 2 S. However, the prover's moves in (P are distributed as in the case that the simulator selects ! uniformly in \Omega\Gamma whereas the verifier's moves (in (P are distributed as in the case that the simulator selects ! uniformly in S. Thus, it should not be too surprising that the above claim can be proven. However, we need more than the above claim: It is not enough that the (P conversations have an origin in S, they must be accepting as well. (Note that this is not obvious since M simulates an interactive proof that may have two-sided error.) Again, the density of the accepting conversations in the "good subspace" of M is high (i.e., need to show that this is the case also for the (P Actually, we will show that the probability than an (P conversation is accepting and "has an origin" in S is at least Let us begin the formal argument with some notations. For each possible history of the interaction, h, we define subsets of the random tapes of the simulator (i.e., subsets of \Omega\Gamma as h is the set of ! 2\Omega which cause the simulator to output a conversation with prefix h. S h is the subset of !'s in\Omega h which are also in S. A h is the set of !'s in S h which are also accepting. Thus, letting M t (!) denote the t-message long prefix output by the simulator M on coins !, we get A h Let C be a random variable representing the (P be an indicator so that the conversation - c is accepting and Our aim is to prove that . Note that -c -c The above expression is exactly the expectation value of jAc j . Thus, we need to show that: where the expectation is over the possible conversations - c as produced by the interaction Once Equation (1) is proven, we are done. Denote the empty history by -. To prove Equation (1) it suffices to prove that since using jA - j The proof of Equation (2) is by induction on the number of rounds. Namely, for each round i, we show that the expected value of jA h j over all possible histories h of i rounds (i.e., length i) is greater or equal to the expected value of this expression over all histories h 0 of rounds. In order to show the induction step we consider two cases: 1. the current step is by the prover (i.e., P ); and 2. the current step is by the verifier (i.e., V ). In both cases we show, for any history h, where the expectation is over the possible current moves m, given history h, as produced by the interaction (P Technical Claim The following technical claim is used for deriving the inequalities in both cases. positive reals. Then, Proof: The Cauchy-Schwartz Inequality asserts: a i! Setting a i can do this since y i is positive) and b i a i , and rearranging the terms, we get the desired inequality. 2 Prover Step - denoted ff Given history h, the prover P sends ff as its next message with probability . Thus, ff ff The inequality is justified by using the Technical Claim and noting that and Verifier Step - denoted fi By the perfectness of the simulation, when restricted to the good subspace S, we know that given history h, the verifier V sends fi as its next message with probability jS hffifi j . Thus, The inequality is justified by using the Technical Claim and noting that and j\Omega h j. Having proven Equation (3) for both cases, Equation (2) follows and so does the lemma. 2 4 The Transformation In this section we show how to transform statistical knowledge complexity into perfect knowledge complexity, incurring only a logarithmic additive term. This transformation combined with Theorem 1 yields the Main Theorem. Theorem 2 For every (poly-time computable) k : N 7! N, We stress again that these knowledge complexity classes refer to the honest verifier and that we don't know whether such a result holds for the analogous knowledge complexity classes referring to arbitrary (poly-time) verifiers. proof: Here we use the oracle formulation of knowledge complexity (see Definition 2.1). We start with an overview of the proof. Suppose we are given a simulator M which produces output that is statistically close to the real prover-verifier interaction. We change both the interactive proof and its simulation so that they produce exactly the same distribution space. We will take advantage of the fact that the prover in the interactive proof and the oracle that "assists" the simulator are both infinitely powerful. Thus, the modification to the prover's program and the augmentation to the oracle need not be efficiently computable. We stress that the modification to the simulator itself will be efficiently computable. Also, we maintain the original verifier (of the interactive proof), and thus the resulting interactive proof is still sound. Furthermore, the resulting interaction will be statistically close to the original one (on any x 2 L) and therefore the completeness property of the original interactive proof is maintained (although the error probability here may increase by a negligible amount). Preliminaries be the guaranteed interactive proof. Without loss of gener- ality, we may assume that all messages are of length 1. This message-length convention is merely a matter of encoding. Recall that Definition 2.1 only guarantees that the simulator produces output with probability - 1. Yet, employing Proposition 3.8 in [GP-91], we get that there exists an oracle machine M , that after asking k(n) log log n queries, always produces an output so that the output is statistically close to the interaction of (P; V ). Let A denote the associated or- acle, and let be the simulation-based prover and verifier 4 induced by M 0 (i.e., In the rest of the presentation, we fix a generic input x 2 L and omit it from the notation. notations: Let [A; B] i be a random variable representing the i-message (i-bit) long prefix of the interaction between A and B (the common input x is implicit in the notation). We denote by A(h) the random variable representing the message sent by A after interaction-history h. Thus, if the i th message is sent by A, we can write [A; B] Y we denote the fact that the random variables X and Y are statistically close. Using these notations we may write for every h 2 f0; 1g i and oe 2 f0; 1g: and similarly, 4.1 The distribution induced by (P statistically close to the distributions induced by both M proof: By definition, the distributions produced by M are statistically close. Thus, we have s We prove that [P statistically close to [P by induction on the length of the interaction. Assuming that [P s we wish to prove it for i + 1. We distinguish two cases. In case the i st move is by the prover, we get s (use the induction hypothesis for s =). In case the i st move is by the verifier, we get s s s 4 A simulator-based verifier is defined analogously to the simulator-based prover. It is a fictitious entity which does not necessarily coincide with V . where the first s is justified by the induction hypothesis and the two others by Eq. (4). We stress that since the induction hypothesis is used only once in the induction step, the statistical distance is linear in the number of induction steps (rather than exponential). 2 Motivating discussion: Note that the statistical difference between the interaction (P the simulation M due solely to the difference between the proper verifier (i.e., and the verifier induced by the simulator (i.e., V 0 ). This difference is due to V 0 putting too much probability weight on certain moves and thus also too little weight on their sibling messages (recall that a message in the interaction contains one bit). In what follows we deal with two cases. The first case is when this difference between the behavior of V 0 (induced by M 0 ) and the behavior of the verifier V is "more than tiny". This case receives most of our attention. We are going to use the oracle in order to move weight from a verifier message fi that gets too much weight (after a history h) to its sibling message fi \Phi 1 that gets too little weight (after the history h) in the simulation. Specifically, when the new simulator M 00 invokes M 0 and comes up with a conversation that has h ffi fi as a prefix, the simulator M 00 (with the help of the oracle) will output (a different) conversation with the prefix h ffi (fi \Phi 1) instead of outputting the original conversation. The simulator M 00 will do this with probability that exactly compensates for the difference between V 0 and V . This leaves one problem. How does the new simulator M 00 come up with a conversation that has a prefix h ffi (fi \Phi 1)? The cost of letting the oracle supply the rest of the conversation (after the known prefix hffi(fi \Phi1)) is too high. We adopt a "brutal" solution in which we truncate all conversations that have as a prefix. The truncation takes place both in the interaction (P stops the conversation after fi \Phi 1 (with a special STOP message) and in the simulation where the oracle recognizes cases in which the simulator M 00 should output a truncated conversation. These changes make M 00 and V behave exactly the same on messages for which the difference between V 0 and V is more than tiny. Naturally, V immediately rejects when P 00 stops the interaction abruptly, so we have to make sure that this change does not foil the ability of P 00 to convince V on an input x 2 L. It turns out that these truncations happen with negligible probability since such truncation is needed only when the difference between V and V 0 is more than tiny. Thus, P 00 convinces V on x 2 L almost with the same probability as P 0 does. The second possible case is that the difference between the behavior of V and V 0 is tiny. In this case, looking at a full conversation - c, we get that the tiny differences sum up to a small difference between the probability of - c in the distributions of M 0 and in the distribution of We correct these differences by lowering the probabilities of all conversations in the new simulator. The probability of each conversation is lowered so that its relative weight (relatively to all other conversations) is equal to its relative weight in the interaction (P Technically, this is done by M 00 not producing an output in certain cases that M 0 did produce an output. Technical remark: The oracle can be used to allow the simulator to toss bias coins when the simulator does not "know" the bias. Suppose that the simulator needs to toss a coin so that it comes-up head with probability N and both N and m are integers. The simulator supplies the oracle with a uniformly chosen r 2 f0; 1g m and the oracle answers head if r is among the first N strings in f0; 1g m and tail otherwise. A similar procedure is applicable for implementing a lottery with more than two a-priori known values. Using this procedure, we can get extremely good approximations of probability spaces at a cost related to an a-priori known upper bound on the size of the support (i.e., the oracle answer is logarithmic in the size of the support). O(t) where t is the number of rounds in the interaction ffl Let h be a partial history of the interaction and fi be a possible next move by the verifier. We say that fi is weak with respect to h if ffl A conversation - with respect to it is i-good. (Note that a conversation can be i-weak only if the i th move is a verifier move.) ffl A conversation - it is i-weak but j-good for every A conversation - i-co-critical if the conversation obtained from - c, by complementing (only) the i th bit, is i-critical. (Note that a conversation can be i-critical only for a single i, yet it may be i-co-critical for many i's.) ffl A conversation is weak if it is i-weak for some i, otherwise it is good. conversations with negligible probability. proof: Recall that [P and that the same holds also for prefixes of the conver- sations. Namely, for any 1 - i - t, [P s us define a prefix h 2 f0; 1g i of a conversation to be bad if either or ffl' The claim follows by combining two facts. Fact 4.3 The probability that (P outputs a conversation with a bad prefix is negligible. to be the set of bad prefixes of length i. By the statistical closeness of we get that for some negligible fraction fl. On the other hand, \Delta can be bounded from bellow by which by definition of B i is at least Thus, and the fact follows. 2 Fact 4.4 If a conversation - contains a bad prefix. proof: Suppose that fi is a bad prefix then we are done. Otherwise it holds that Using the fact that fi is weak with respect to h, we get which implies that h ffi fi is a bad prefix of - c. 2 Combining Facts 4.3 and 4.4, Claim 4.2 follows. 2 conversation. Then, the probability that - c is output by M 0 is at least (1 \Gamma ffl) dt=2e \Delta Prob([P is i-good for every proof: To see that this is the case, we write the probabilities step by step conditioned on the history so far. We note that the prover's steps happen with equal probabilities in both sides of the inequality, and therefore can be reduced. Since the relevant verifier's steps are not weak, we get the mentioned inequality. The actual proof proceeds by induction on k \Gamma l. Clearly, the claim holds. We note that if k \Gamma l = 1 the claim also holds since step k in the conversation is either a prover step or a k-good verifier step. To show the induction step we use the induction hypothesis for 2. Namely, include one prover message and one verifier message. Assume, without loss of generality, that the prover step is k \Gamma 1. Since P 0 is the simulator based prover, we get Since step k of the verifier is good, we also have: Combining Equations 5, 6, and 7, the induction step follows and we are done. 2 Dealing with weak conversations We start by modifying the prover P 0 , resulting in a modified prover, denoted P 00 , that stops once it gets a verifier message which is weak with respect to the current history; otherwise, Namely, Definition (modified prover - P 00 STOP if fi is weak with respect to We assume that the verifier V stops and rejects immediately upon receiving an illegal message from the prover (and in particular upon receiving this STOP message). Next, we modify the simulator so that it outputs either good conversations or truncated conversations which are originally i-critical. Jumping ahead, we stress that such truncated i-critical conversations will be generated from both i-critical and i-co-critical conversations. The modified simulator, denoted M 00 , proceeds as follows 5 . First, it invokes M 0 and obtains a conversation - queries the augmented oracle on - c. The oracle answers probabilistically and its answers are of the form (i; oe), where i 2 f1; :::; tg and oe 2 f0; 1g. The probability distribution will be specified below, at this point we only wish to remark that the oracle only returns pairs (i; oe) for which one of the following three conditions holds 1. - c is good, is good and is not i-co-critical for any i's then the oracle always answers this way); 2. - c is i-critical and 3. - c is i-co-critical and oe = 1. Finally, the new simulator (M 00 ) halts outputting which in case not a prefix of - c. Note that i may be smaller than t, in which case M 00 outputs a truncated conversation which is always i-critical; otherwise, M 00 outputs a non-truncated conversation. Note that this oracle message contains at most 1 log t bits where t is the length of the interaction between P 0 and V . It remains to specify the oracle's answer distribution. Let us start by considering two special cases. In the first case, the conversation generated by M 0 is i-critical, for some i, but is not j-co-critical for any j ! i. In this case the oracle always answers (i; 0) and consequently the simulator always outputs the i-bit long prefix. However, this prefix is still being output with too low probability. This will be corrected by the second case hereby described. In this ("second") case, the conversation - c generated by M 0 is good and i-co-critical for a single i. This means that the i-bit long prefix is given too much probability weight whereas the prefix obtained by complimenting the i th bit gets too little weight. To correct this, the oracle outputs (i; 1) with probability q and (t; q will be specified. What happens is that the M 00 will output the "i-complimented prefix" with higher probability than with which it has appeared in M 0 . The value of q is determined as follows. Denote Then, setting q so that allows the simulator to output the prefix with the right probability. 5 We stress that P 00 is not necessarily the simulator-based prover of M 00 . In the general case, the conversation generated by M 0 may be i-co-critical for many i's as well as j-critical for some (single) j. In case it is j-critical, it can be i-co-critical only for us consider the sequence of indices, (i 1 ; :::; i l ), for which the generated conversation is critical or co-critical (i.e., the conversation is i k -co-critical for all k ! l and is either i l -critical or i l -co-critical). We consider two cases. In both cases the q k 's are set as in the above example; namely, q \Phi 1) and \Phi 1). 1. The generated conversation, - -co-critical for every k ! l and is i l - critical. In this case, the distribution of the oracle answers is as follows. For every l, the pair (i k ; 1) is returned with probability ( the pair appears with probability We stress that no other pair appears in this distribution. 6 2. The generated conversation, - -co-critical for every k - l. In this case, the distribution of the oracle answers is as follows. For every k - l, the pair (i is returned with probability ( the pair (t; 0) appears with probability appears in this distribution. 1. [P 2. Each conversation of (P )-conversation or a truncated (i.e., critical) one, is output by M 00 with probability that is at least a 3fraction of the probability that it appears in [P proof: The weak conversations are negligible in the output distribution of (P 4.2). The only difference between [P originates from a different behavior of P 00 on weak conversations, specifically P 00 truncates them while P 0 does not. Yet, the distribution on the good conversations remains unchanged. Therefore the distribution of [P statistically close to the distribution of [P and we are done with Part (1). For Part (2) let us start with an intuitive discussion which may help reading through the formal proof that follows. First, we recall that the behavior of the simulation M 0 in prover steps is identical to the behavior of the interaction (P steps. This follows simply from the fact that P 0 is the simulation based prover of M 0 . We will show that this property still holds for the new interaction (P and the new simulation M 00 . We will do this by noting two different cases. In one case, the prover step is conducted by P 00 exactly as it is done by P 0 and then M 00 behaves exactly as M 0 . The second possible case is that the prover step contains the special message STOP. We shall note that this occurs with exactly the same probability in the distribution (P in the distribution of M 00 . Next, we consider the verifier steps. In the construction of M 00 and P 00 we considered the behavior of M 0 and V on verifier steps and made changes when these differences were not "tiny". We called a message fi weak with respect to a history h, if the simulator assigns the message fi (after outputting h) a probability which is smaller by a factor of more than from the probability that the verifier V outputs the message fi on history h. We did not 6 Indeed the reader can easily verify that these probabilities sum up to 1. make changes in messages whose difference in weight (between the simulation M 0 and the interaction were smaller than that. In the proof, we consider two cases. First, the message fi is weak with respect to the history h. Clearly, the sibling message fi \Phi 1 is getting too much weight in the simulation M 0 . So in the definition of M 00 we made adjustments to move weight from the prefix h ffi (fi \Phi 1) to the prefix h ffi fi. We will show that this transfer of weight exactly cancels the difference between the behavior of V and the behavior of M 0 . Namely, the weak messages (and their siblings) are assigned exactly the same probability both in M 00 and by V . Thus, we show that when a weak step is involved, the behavior of and the behavior of M 00 are exactly equivalent. It remains to deal with messages for which the difference between the conditional behavior of V and M 0 is "tiny" and was not considered so far. In this case, M 00 behaves like M 0 . However, since the difference is so tiny, we get that even if we accumulate the differences throughout the conversation, they sum up to at most the multiplicative factor 3=4 stated in the claim. Let us begin the formal proof by writing again the probability that (P 00 c as the product of the conditional probabilities of the t steps. Namely, Y We do the same for the probability that M 00 outputs a conversation c. We will show by induction that each step of any conversation is produced by M 00 with at least times the probability of the same step in the (P )-interaction. Once we have shown this, we are done. Clearly this claim holds for the null prefix. To prove the induction step, we consider the two possibilities for the party making the st step. st step is by the prover: Consider the conditional behavior of M 00 given the history so far. We will show that this behavior is identical to the behavior of P 00 on the same partial history. A delicate point to note here is that we may talk about the behavior of M 00 on a prefix only if this prefix appears with positive probability in the output distribution [M 00 However, by the induction hypothesis any prefix that is output by [P appears with positive probability in [M 00 We partition the analysis into two cases. 1. First, we consider the case in which the last message of the verifier is weak with respect to the history that precedes it. Namely, and fi is weak with respect to h 0 . In this case, both in the interaction (P in the simulation M 00 , the next message of the prover is set to STOP with probability 1. Namely, 2. The other possible case is that the last message of the verifier is not weak with respect to its preceding history. In this case, the simulator M 00 behaves like M 0 and the prover (Note that the changes in critical and co-critical steps apply only to verifier steps.) Thus, To summarize, the conditional behavior of M 00 in the prover steps and the conditional behavior of P 00 are exactly equal. st step is by the verifier: Again, we consider the conditional behavior of M 00 given the history so far. Let us recall the second modification applied to M 0 when deriving M 00 . This modification changes the conditional probability of the verifier steps in the distribution of M 0 in order to add weight to steps having low probability in the simulation. We note that this modification is made only in critical or co-critical steps of the verifier. Consider a history h i which might appear in the interaction (P possible response fi of V to h i . Again, by the induction hypothesis, h i has a positive probability to be output by the simulation M 00 and therefore we may consider the conditional behavior of M 00 on this history h i . There are three cases to be considered, corresponding to whether either fi or fi \Phi 1 or none is weak with respect to h i . We start with the simplest case in which neither fi nor fi \Phi 1 is weak (w.r.t. h i ). In this case, the behavior of M 00 is identical to the behavior of M 0 since the oracle never sends the message (i in this case. However, by the fact that fi is not weak, we get that and we are done with this simple case. We now turn to the case in which fi is weak (w.r.t. h i ). In this case, given that M 00 has produced the prefix h i , it produces h i ffifi whenever M 0 produces the prefix h i ffifi. Furthermore, with conditional probability q (as defined above), M 00 produces the prefix h i ffi fi also in case produces the prefix h i ffi (fi \Phi 1). As above, we define is the simulation (M 0 ) based verifier, we may also write Also, recall that q was defined as p\Gammap 0 using these notations: Using Equation (8), we get Finally, we turn to the case in which fi \Phi 1 is weak (w.r.t. h i ). Again, this means that fi is co-critical in - c. Given that M 00 has produced the prefix h i , it produces h i ffi fi only when M 0 produces the prefix h i ffi fi, and furthermore, M 00 does so only with probability q is again as defined above). We denote p and p 0 , with respect to the critical message fi \Phi 1. Namely, Thus, recalling that This completes the proof of Claim 4.6. 2 Lowering the probability of some simulator outputs After handling the differences between M 0 and (P which are not tiny, we make the last modification, in which we deal with tiny differences. We do that by lowering the probability that the simulator outputs a conversation, in case it outputs this conversation more frequently than it appears in (P 00 ; V ). The modified simulator, denoted M 000 , runs M 00 to obtain a conversation - c. (Note that M 00 always produces output.) Using the further-augmented oracle, M 000 outputs - c with probability c Note that p - c - 1 holds due to Part 2 of Claim 4.6. 1. M 000 produces output with probability 3; 2. The output distribution of M 000 (i.e., in case it has output) is identical to the distribution proof: The probability that M 000 produces an output is exactly: As for part (2), we note that the probability that a conversation - c is output by M 000 is exactly4 the simulator halts with an output with probability exactly 3, we get that given that M 000 halts with an output, it outputs - c with probability exactly and we are done. 2 An important point not explicitly addressed so far is whether all the modifications applied to the simulator preserve its ability to be implemented by a probabilistic polynomial-time with bounded access to an oracle. Clearly, this is the case with respect to M 00 (at the expense of additional regarding the last modification there is a subtle points which needs to be addressed. Specifically, we need to verify that the definition of M 000 is implementable; namely, that M 000 can (with help of an augmented oracle) "sieve" conversations with exactly the desired probability. Note that the method presented above (in the "technical remark") may yield exponentially small deviation from the desired probability. This will get very close to a perfect simulation, but yet will not achieve it. To this end, we modify the "sieving process" suggested in the technical remark to deal with the specific case we have here. But first we modify P 00 so that it makes its random choices (in case it has any) by flipping a polynomial number of unbiased coins. 7 This rounding does change a bit the behavior of P 00 , but the deviation can be made so small that the above assertions (specifically Claim 4.6) still hold. Consider the specific sieving probability we need here. c=d , where a d observation is that c is the number of coin tosses which lead M 00 to output - c (i.e., using the notation of the previous section, j). Observing that b is the size of probability space for [P using the above modification to P 00 , we rewrite p - c as 3ad c are some non-negative integers. We now note, that the oracle can allow the simulator to sieve conversations with probability e c in the following way. M 000 sends to the oracle the random tape ! that it has tossed for M 00 , and the oracle sieves only e out of the possible c random tapes which lead M 00 to output - c. The general case of p - c2 f is deal by writing c To implement this sieve, M 000 supplies the oracle with a uniformly chosen f-bit long string (in addition to !). The oracle sieves out q random-tapes (of M 00 ) as before, and uses the extra bits in order to decide on the sieve in case ! equals a specific (different) random-tape. Combining Claims 4.1, 4.6 (part 1), and 4.7, we conclude that (P 00 is an interactive proof system of perfect knowledge complexity O(log n) for L. This completes the proof of Theorem 2. 7 The implementation of P 00 was not discussed explicitly. It is possible that P 00 uses an infinite number of coin tosses to select its next message (either 0 or 1). However, an infinite number of coin tosses is not really needed since rounding the probabilities so that a polynomial number of coins suffices, causes only exponentially small rounding errors. Concluding Remarks We consider our main result as a very first step towards a classification of languages according to the knowledge complexity of their interactive proof systems. Indeed there is much to be known. Below we first mention two questions which do not seem too ambitious. The first is to try to provide evidence that NP-complete languages cannot be proven within low (say logarithmic or even constant) knowledge complexity. A possible avenue for proving this conjecture is to show that languages having logarithmic knowledge complexity are in co-AM, rather than in BPP NP (recall that NP is unlikely to be in co-AM - see also [BHZ-87]). The second suggestion is to try to provide indications that there are languages in PSPACE which do not have interactive proofs of linear (rather than logarithmic) knowledge complexity. The reader can easily envision more moderate and more ambitious challenges in this direction. Another interesting question is whether all levels greater then zero of the knowledge- complexity hierarchy contain strictly more languages than previous levels, or if some partial collapse occurs. For example, it is open whether constant or even logarithmic knowledge complexity classes do not collapse to the zero level. Regarding our transformation of statistical knowledge complexity into perfect knowledge complexity (i.e., Theorem 2), a few interesting questions arise. Firstly, can the cost of the transformation be reduced to bellow O(log n) bits of knowledge? A result for the special case of statistical zero-knowledge will be almost as interesting. Secondly, can one present an analogous transformation that preserves one-sided error probability of the interactive proof? (Note that our transformation introduces a negligible error probability into the completeness condition.) Finally, can one present an analogous transformation that applies to knowledge complexity with respect to arbitrary verifiers? (Our transformation applies only to knowledge complexity with respect to the honest verifier.) 6 Acknowledgement We thank Leonard Shulman for providing us with a simpler proof of Claim 3.2. --R The (True) Complexity of Statistical Zero-Knowledge Making Zero-Knowledge Provers Efficient The Complexity of Perfect Zero-Knowledge Interactive Proof Systems: Provers that never Fail and Random Selection. "Proofs that Yield Nothing But their Validity and a Methodology of Cryptographic Protocol Design" "How to Play any Mental Game or a Completeness Theorems for Protocols of Honest Majority" Quantifying Knowledge Complexity. The Knowledge Complexity of Interactive Proofs. The Knowledge Complexity of Interactive Proofs. Public Coins in Interactive Proof Systems Better Ways to Generate Hard NP Instances than Picking Uniformly at Random Direct Minimum-Knowledge computations Random Generation of Combinatorial Structures from a Uniform Distribution. Algebraic Methods for Interactive Proof Systems. Fair Games Against an All-Powerful Adversary A Complexity Theoretic Approach to Randomness. The Complexity of Approximate Counting. --TR --CTR Amos Beimel , Paz Carmi , Kobbi Nissim , Enav Weinreb, Private approximation of search problems, Proceedings of the thirty-eighth annual ACM symposium on Theory of computing, May 21-23, 2006, Seattle, WA, USA Oded Goldreich , Salil Vadhan , Avi Wigderson, On interactive proofs with a laconic prover, Computational Complexity, v.11 n.1/2, p.1-53, January Amit Sahai , Salil Vadhan, A complete problem for statistical zero knowledge, Journal of the ACM (JACM), v.50 n.2, p.196-249, March

knowledge complexity;interactive proofs;randomness;cryptography;complexity classes;zero knowledge

284976

Guaranteeing Fair Service to Persistent Dependent Tasks.

We introduce a new scheduling problem that is motivated by applications in the area of access and flow control in high-speed and wireless networks. An instance of the problem consists of a set of persistent tasks that have to be scheduled repeatedly. Each task has a demand to be scheduled "as often as possible." There is no explicit limit on the number of tasks that can be scheduled concurrently. However, such limits are imposed implicitly because some tasks may be in conflict and cannot be scheduled simultaneously. These conflicts are presented in the form of a conflict graph. We define parameters which quantify the fairness and regularity of a given schedule. We then proceed to show lower bounds on these parameters and present fair and efficient scheduling algorithms for the case where the conflict graph is an interval graph. Some of the results presented here extend to the case of perfect graphs and circular-arc graphs as well.

Introduction In this paper we consider a new form of a scheduling problem which is characterized by two features: Persistence of the tasks: A task does not simply go away once it is scheduled. Instead, each task must be scheduled innitely many times. The goal is to schedule every task as frequently as possible. Dependence among the tasks: Some tasks con ict with each other and hence cannot be scheduled concur- rently. These con icts are given by a con ict graph. This graph imposes constraints on the sets of tasks that may be scheduled concurrently. Note that these constraints are not based simply on the cardinality of the sets, but rather on the identity of the tasks within the sets. Extended summary y IBM { Research Division, T. J. Watson Research Center, Yorktown Heights, NY 10598. Email: famotz,sbar,madhug@watson.ibm.com. z Dept. of Computer Science, Columbia University, New York, NY 10027. Email: mayer@cs.columbia.edu. Part of this work was done while the author was at the IBM T. J. Watson Research Center. Partially supported by an IBM Graduate Fellowship, NSF grant CCR-93-16209, and CISE Institutional Infrastructure Grant CDA-90-24735 We consider both the problems of allocation, i.e., how often should a task be scheduled and regularity, i.e., how evenly spaced are lengths of the intervals between successive scheduling of a specic task. We present a more formal description of this problem next and discuss our primary motivation immediately afterwards. While all our denitions are presented for general con ict graphs, our applications, bounds, and algorithms are for special subclasses { namely, perfect graphs, interval graphs and circular arc-graphs 1 . Problem statement An instance of the scheduling problem consists of a con ict graph G with n vertices. The vertices of G are the tasks to be scheduled and the edges of G dene pairs of tasks that cannot be scheduled concurrently. The output of the scheduling algorithm is an innite sequence of subsets of the vertices, I 1 ; I lists the tasks that are scheduled at time t. Notice that for all t, I t must be an independent set of G. In the form above, it is hard to analyze the running time of the scheduling algorithm. We consider instead a nite version of the above problem and use it to analyze the running time. Input: A con ict graph G and a time t. Output: An independent set I t denoting the set of tasks scheduled at time unit t. The objective of the scheduling algorithm is to achieve a fair allocation and a regular schedule. We next give some motivation and describe the context of our work. As we will see, none of the existing measures can appropriately capture the \goodness" of a schedule in our framework. Hence we proceed to introduce measures which allow for easier presentation of our results. 1 A graph is perfect if for all its induced subgraphs the size of the maximum clique is equal to the chromatic number (cf. [11]). A graph is an interval graph (circular-arc graph) if its vertices correspond to intervals on a line (circle), and two vertices are adjacent if the corresponding intervals intersect (cf. [20]). A. Bar-Noy, A. Mayer, B. Schieber, and M. Sudan 1.1 Motivation Session scheduling in high-speed local-area net- works. MetaRing ([7]) is a recent high-speed local-area ring-network that allows \spatial reuse", i.e., concurrent access and transmission of user sessions, using only minimal intermediate buering of packets. The basic operations in MetaRing can be approximated by the following: if some node has to send data to some other node a session is established between the source and the destination. Sessions typically last for a while and can be active only if they have exclusive use of all the links in their routes. Hence, sessions whose routes share at least one link are in con ict. These con icts need to be regulated by breaking the data sent in a session into units of quotas that are transmitted according to some schedule. This schedule has to be e-cient and fair. E-cient means that the total number of quotas transmitted (throughput) is maximized whereas fair means that the throughput of each session is maximized, and that the time between successive activation of a session is minimized, so that large buers at the source nodes can be avoided. It has been recognized ([5]) that the access and control in such a network should depend on locality in the con ict graph. However, no rm theoretical basis for an algorithmic framework has been proposed up to now. To express this problem as our scheduling problem we create a circular-arc graph whose vertices are the sessions, and in which vertices are adjacent if the corresponding paths associated with the sessions intersect in a link. Time sharing in wireless networks. Most indoor designs of wireless networks are based on a cellular architecture with a very small cell size. (See, e.g., [13].) The cellular architecture comprises two levels { a stationary level and a mobile level. The stationary level consists of xed base stations that are interconnected through a backbone network. The mobile level consists of mobile units that communicate with the base stations via wireless links. The geographic area within which mobile units can communicate with a particular base station is referred to as a cell. Neighboring cells overlap with each other, thus ensuring continuity of communications. The mobile units communicate among themselves, as well as with the xed information networks, through the base stations and the back-bone network. The continuity of communications is a crucial issue in such networks. A mobile user who crosses boundaries of cells should be able to continue its communication via the new base-station. To ensure this, base-stations periodically need to transmit their identity using the wireless communication. In some implementations the wireless links use infra-red waves. Therefore, two base-station whose cells overlap are in con ict and cannot transmit their identity simulta- neously. These con icts have to be regulated by a time-sharing scheme. This time sharing has to be ecient and fair. E-cient means that the scheme should accommodate the maximal number of base stations whereas fair means that the time between two consecutive transmissions of the same base-station should be less then the time it takes a user to cross its corresponding cell. Once again this problem can be posed as our graph-scheduling problem where the vertices of the graph are the base-stations and an edge indicates that the base stations have overlapping cells. 1.2 Relationship to past work Scheduling problems that only consider either persistence of the tasks or dependence among the tasks (but not both) have been dealt with before. The task of scheduling persistent tasks has been studied in the work of Baruah et al. [2]. They consider the problem of scheduling a set of n tasks with given (arbitrary) frequencies on m machines. (Hence, yields an instance of our problem where the con ict graph is a clique.) To measure \regularity" of a schedule for their problem they introduce the notion of P -fairness. A schedule for this problem is P -fair (proportionate-fair) if at each time t for each task i the absolute value of the dierence in the number of times i has been scheduled and f i t is strictly less than 1, where f i is the frequency of task i. They provide an algorithm for computing a P -fair solution to their problem. Their problem fails to capture our situation due to two reasons. First, we would like to constrain the sets of tasks that can be scheduled concurrently according to the topology of the con ict graph and not according to their cardinality. Moreover, in their problem every \feasible" frequency requirement can be scheduled in a P -fair manner. For our scheduling problem we show that such a P -fair schedule cannot always be achieved. To deal with feasible frequencies that cannot be scheduled in a P -fair manner, we dene weaker versions of \regularity". The dependency property captures most of the work done based on the well-known \Dining Philoso- phers" paradigm, see for example [9], [18], [6], [1], [8], and [4]. In this setting, Lynch [18] was the rst to explicitly consider the response time for each task. The goal of successive works was to make the response time of a node to depend only on its local neighborhood in the con ict graph. (See, e.g., [4].) While response time in terms of a node's degree is adequate for \one-shot" tasks, it does not capture our requirement that a task Guaranteeing Fair Service to Persistent Dependent Tasks 3 should be scheduled in a regular and fair fashion over a period of time. 1.3 Notations and denitions A schedule S is an innite sequence of independent sets I 1 ; I We use the notation S(i; t) to represent the schedule: S(i; lim inf t!1 ff (t) g. We refer to f i as the frequency of the i-th task in schedule S. Definition 1.1. A vector of frequencies ^ feasible if there exists a schedule S such that the frequency of the i-th task under schedule S is at least f i . Definition 1.2. A schedule S realizes a vector of f if the frequency of the i-th task under schedule S is at least f i . A schedule S c-approximates a vector of frequencies ^ f if the frequency of the i-th task under schedule S is at least f i =c. A measure of fairness Fairness is determined via a partial order that we dene on the set of frequency vectors. Definition 1.3. Given two frequency vectors ^ f is less fair there exists an index j and a threshold f such that f j < f g j and for all i such that g i f , Definition 1.4. A vector of frequencies ^ f is max-min fair if no feasible vector ^ g. Less formally, in a max-min fair frequency vector one cannot increase the frequency of some task at the expense of more frequently scheduled tasks. This means that our goal is to let task i have more of the resource as long as we have to take the resource away only from tasks which are better o, i.e., they have more of the resource than task i. Measures of regularity Here, we provide two measures by which one can evaluate a schedule for its reg- ularity. We call these measures the response time and the drift. Given a schedule S, the response time for task i, denoted r i , is the largest interval of time for which the i-th task waits between successive schedulings. More precisely, For any time t, the number of expected occurrences of task i can be expressed as f i t. But note that if r i is larger than 1=f i , it is possible that, for some period of time, a schedule allows a task to \drift away" from its expected number of occurrences. In order to capture this, we introduce a second measure for the regularity of a schedule. We denote by d i the drift of a task i. It indicates how much a schedule allows task i to drift away from its expected number of scheduled units (based on its frequency): Note that if a schedule S achieves drift d i < 1 for all then it is P-fair as dened in [2]. Finally, a schedule achieves its strongest form of regularity if each task i is scheduled every 1=f i time-units (except for its rst appearance). Hence we say that a schedule is rigid if for each task i there exists a starting point s i such that the task is scheduled on exactly the time units 1.4 Results In Section 2 we motivate our denition of max-min fairness and show several of its properties. First, we provide an equivalent alternate denition of feasibility which shows that deciding feasibility of a frequency vector is computable. We prove that every graph has a unique max-min fair frequency vector. Then, we show that the task of even weakly-approximating the max-min fair frequencies on general graphs is NP-hard. As we mentioned above many practical applications of this problem arise from simpler networks, such as buses and rings (i.e., interval con ict graphs and circular-arc con ict graphs). For the case of perfect- graphs (and hence for interval graphs), we describe an e-cient algorithm for computing max-min fair frequencies. We prove that the period T of a schedule realizing such frequencies satises and that there exist interval graphs such that n) . The rest of our results deal with the problem of nding the most \regular" schedule (under the above mentioned measures) that realizes any feasible frequency vector. Section 3 shows the existence of interval graphs for which there is no P -fair schedule that realizes their max-min fair frequencies. In Section 4 we introduce an algorithm for computing a schedule that realizes any given feasible frequencies on interval graphs. The schedule computed by the algorithm achieves response-time of d4=f i e and drift of O( log slight modication of this algorithm yields a schedule that 2-approximates the given frequencies. The advantage of this schedule is that 4 A. Bar-Noy, A. Mayer, B. Schieber, and M. Sudan it achieves a bound of 1 on the drift and hence a bound of d2=f i e on the response time. In Section 5 we present an algorithm for computing a schedule that 12-approximates any given feasible frequencies on interval graphs and has the advantage of being rigid. All algorithms run in polynomial time. In Section 6 we show how to transform any algorithm for computing a schedule that c-approximates any given feasible frequencies on interval graphs into an algorithm for computing a schedule that 2c-approximates any given feasible frequencies on circular-arc graphs. (The response-time and drift of the resulting schedule are doubled as well.) Finally, in Section 7 we list a number of open problems and sketch what additional properties are required to obtain solutions for actual net- works. Due to space constraints some of the proofs are either omitted or sketched in this extended summary. Allocation Our denition for max-min fair allocation is based on the denition used by Jae [14] and Bertsekas and Gallager [3], but diers in one key ingredient { namely our notion of feasibility. We study some elementary properties of our denition in this section. In particular, we show that the denition guarantees a unique max-min fair frequency vector for every con ict graph. We also show the hardness of computing the frequency vector for general graphs. However, for the special case of perfect graphs our notion turns out to be the same as of [3]. The denition of [14] and [3] is considered the traditional way to measure throughput fairness and is also based on the partial order as used in our denition. The primary dierence between our denition and theirs is in the denition of feasibility. Bertsekas and Gallager [3] use a denition, which we call clique feasible, that is dened as follows: A vector of frequencies (f feasible for a con ict graph G, if for all cliques C in the graph G. The notion of max-min fairness of Bertsekas and Gallager [3] is now exactly our notion, with feasibility replaced by clique feasibility. The denition of [3] is useful for capturing the notion of fractional allocation of a resource such as bandwidth in a communication networks. However, in our application we need to capture a notion of integral allocation of resources and hence their denition does not su-ce for our purposes. It is easy to see that every frequency vector that is feasible in our sense is clique feasible. However, the converse is not true. Consider the case where the con ict graph is the ve-cycle. For this graph the vector (1=2; 1=2; 1=2; 1=2; 1=2) is clique feasible, but no schedule can achieve this frequency. 2.1 An alternate denition of feasibility Given a con ict graph G, let I denote the family of all independent sets in G. For I 2 I, let (I) denote the characteristic vector of I . Proposition 2.1. A vector of frequencies ^ f is feasible if and only if there exist weights f I g I2I , such that I2I I f . The main impact of this assertion is that it shows that the space of all feasible frequencies is well behaved (i.e., it is a closed, connected, compact space). Immediately it shows that determining whether a frequency vector is feasible is a computable task (a fact that may not have been easy to see from the earlier denition). We now use this denition to see the following connection: Proposition 2.2. Given a con ict graph G, the notions of feasibility and clique feasibility are equivalent if and only if G is perfect. Proof (sketch): The proof is follows directly from well-known polyhedral properties of perfect graphs. (See [12], [16].) In the notation of Knuth [16] the space of all feasible vectors is the polytope STAB(G) and the space of all clique-feasible vectors is the polytope QSTAB(G). The result follows from the theorem on page 38 in [16] which says that a graph G is perfect if and only if 2.2 Uniqueness and computability of max-min fair frequencies In the full paper we prove the following theorem. Theorem 2.3. There exists a unique max-min fair frequency vector. Now, we turn to the issue of the computability of the max-min fair frequencies. While we do not know the exact complexity of computing max-min fair frequencies 2 it does seem to be a very hard task in gen- eral. In particular, we consider the problem of computing the smallest frequency assigned to any vertex by a max-min allocation and show the following: Theorem 2.4. There exists an > 0, such that given a con ict graph on n vertices approximating the In particular, we do not know if deciding whether a frequency vector is feasible is in NP [ coNP Guaranteeing Fair Service to Persistent Dependent Tasks 5 smallest frequency assigned to any vertex in a max-min fair allocation, to within a factor of n , is NP-hard. Proof (sketch): We relate the computation of max-min fair frequencies in a general graph to the computation of the fractional chromatic number of a graph. The fractional chromatic number problem (cf. [17]) is dened as follows: To each independent set I in the graph, assign a weight w I , so as to minimize the quantity I w I , subject to the constraint that for every vertex v in the graph, the quantity I3v w I is at least 1. The quantity I w I is called the fractional chromatic number of the graph. Observe that if the w I 's are forced to be integral, then the fractional chromatic number is the chromatic number of the graph. The following claim shows a relationship between the fractional chromatic number and the assignment of feasible frequencies. 2.5. Let (f be a feasible assignment of frequencies to the vertices in a graph G. Then is an upper bound on the fractional chromatic number of the graph. Conversely, if k is the fractional chromatic number of a graph, then a schedule that sets the frequency of every vertex to be 1=k is feasible. The above claim, combined with the hardness of computing the fractional chromatic number [17], suf- ces to show the NP-hardness of deciding whether a given assignment of frequencies is feasible for a given graph. To show that the claim also implies the hardness of approximating the smallest frequency in the max-min fair frequency vector we inspect the Lund- Yannakakis construction a bit more closely. Their construction yields a graph in which every vertex participates in a clique of size k such that deciding if the (fractional) chromatic number is k or kn is NP-hard. In the former case, the max-min fair frequency assignment is 1=k to every vertex. In the latter case at least some vertex will have frequency smaller that 1=(kn ). Thus this implies that approximating the smallest frequency in the max-min fair frequencies to within a factor of n is NP-hard. 2 2.3 Max-min fair frequencies on perfect graphs We now turn to perfect graphs. We show how to compute in polynomial time max-min fair frequencies for this class of graphs and give bounds on the period of a schedule realizing such frequencies. As our main focus of the subsequent sections will be interval graphs, we will give our algorithms and bounds rst in terms Figure 1: An interval graph for which n) . of this subclass and then show how to generalize the results to perfect graphs. We start by describing an algorithm for computing max-min fair frequencies on interval graphs. As we know that clique-feasibility equals feasibility (by Proposition 2.2), we can use an adaptation of [3]: Algorithm 1: Let C be the collection of maximal cliques in the interval graph. (Notice that C has at most n elements and can be computed in polynomial time.) For each clique C 2 C the algorithm maintains a residual capacity which is initially 1. To each vertex the algorithm associates a label assigned/unassigned. All vertices are initially unassigned. Dividing the residual capacity of a clique by the number of unassigned vertices in this clique yields the relative residual capacity. Iteratively, we consider the clique with the smallest current relative residual capacity and assign to each of the clique's unassigned vertices this capacity as its frequency. For each such vertex in the clique we mark it assigned and subtract its frequency from the residual capacity of every clique that contains it. We repeat the process till every vertex has been assigned some frequency. It is not hard to see that Algorithm 1 correctly computes max-min fair frequencies in polynomial- time. We now use its behavior to prove a tight bound on the period of a schedule for an interval graph. The following theorem establishes this bound. (See also Figure 1.) Theorem 2.6. Let f be the frequencies in a max-min fair schedule for an interval graph G, are relatively prime. Then, the period for the schedule Furthermore, there exist interval graphs for which T =n) . 6 A. Bar-Noy, A. Mayer, B. Schieber, and M. Sudan It is clear that Algorithm 1 works for all graphs where clique feasibility determines feasibility, i.e., perfect graphs. However, the algorithm does not remain computationally e-cient. Still, Theorem 2.6 can be directly extended to the class of perfect graphs. We now use this fact to describe a polynomial-time algorithm for assigning max-min fair frequencies to perfect graphs. Algorithm 2: This algorithm maintains the labelling procedure assigned/unassigned of Algorithm 1. At each phase, the algorithm starts with a set of assigned frequencies and tries to nd the largest f such that all unassigned vertices can be assigned the frequency f . To compute f in polynomial time, the algorithm uses the fact that deciding if a given set of frequencies is feasible is reducible to the task of computing the size of the largest weighted clique in a graph with weights on vertices. The latter task is well known to be computable in polynomial-time for perfect graphs. Using this decision procedure the algorithm performs a binary search to nd the largest achievable f . (The binary search does not have to be too rened due to Theorem 2.6). Having found the largest f , the algorithm nds a set of vertices which are saturated under f as follows: Let be some small number, for instance is su-cient. Now it raises, one at a time, the frequency of each unassigned vertex to f +, while maintaining the other unassigned frequencies at f . If the so obtained set of frequencies is not feasible, then it marks the vertex as assigned and its frequency is assigned to be f . The algorithm now repeats the phase until all vertices have been assigned some frequency. 3 Non-existence of P-fair allocations Here we show that a P -fair scheduling realizing max-min fair frequencies need not exist for every interval graph. Theorem 3.1. There exist interval graphs G for which there is no P-fair schedule that realizes their max-min frequency assignment. In order to prove this theorem we construct such a graph G as follows. We choose a parameter k and for every permutation of the elements dene an interval graph G . We show a necessary condition that must satisfy if G has a P-fair schedule. Lastly we show that there exists a permutation of 12 elements which does not satisfy this condition. Given a permutation on k elements, G consists of 3k intervals. For the Figure 2: The graph G for that the max-min frequency assignment to G is the following: All the tasks B(i) have frequency 1=k; all the tasks A(i) have frequency all the tasks C(i) have frequency i=k. (See Figure 2.) We now observe the properties of a P-fair schedule for the tasks in G . (i) The time period is k. (ii) The schedule is entirely specied by the schedule for the tasks B(i). (iii) This schedule is a permutation of k elements, where (i) is the time unit for which B(i) is scheduled. To see what kind of permutations constitute P-fair schedules of G we dene the notion of when a permutation is fair for another permutation. Definition 3.1. A permutation 1 is fair for a permutation 2 if for all the conditions cond ij dened as follows: 3.2. If a permutation is a P-fair schedule for G then is fair for the identity permutation and for permutation . be a permutation on 12 elements. In the full paper we show that no permutation is fair to both and the identity. Realizing frequencies exactly In this section we rst show how to construct a schedule that realizes any feasible set of frequencies (and hence in particular max-min frequencies) exactly on an interval graph. We prove its correctness and demonstrate a bound of d4=f i e on the response time for each interval i. We then proceed to introduce a potential function that is used to yield a bound of O(n 1+ ) on the drift for every interval. We also prove that if the feasible frequencies are of the form then the drift of the schedule can be bounded by 1 and thus the waiting time can be bounded by We use this property to give an algorithm for Guaranteeing Fair Service to Persistent Dependent Tasks 7 computing a schedule that 2-approximates any feasible set of frequencies with high regularity. Input to the Algorithm: A unit of time t and a con ict graph G which is an interval graph. Equivalently, a set of intervals on the unit interval [0; 1] of the x-coordinate, where I Every interval I i has a frequency f with the following constraint: I For simplicity, we assume from now on that these constraints on the frequencies are met with equality and that t g. Output of the Algorithm: An independent set I t which is the set of tasks scheduled for time t such that the scheduled S, given by fI t g T realizes frequencies f i . The algorithm is recursive. Let s i denote the number of times a task i has to appear in T time units, i.e., s . The algorithm has log T levels of recursion. (Recall that log T is O(n) for max-min fair frequencies.) In the rst level we decide on the occurrences of the tasks in each half of the period. That is, for each task we decide how many of its occurrences appear in the rst half of the period and how many in the second half. This yields a problem of a recursive nature in the two halves. In order to nd the schedule at time t, it su-ces to solve the problem recursively in the half which contains t. (Note that in case T is odd one of the halves is longer than the other.) Clearly, if a task has an even number of occurrences in T it would appear the same number of times in each half in order to minimize the drift. The problem is with tasks that have an odd number of occurrences s i . Clearly, each half should have at least bs i c of the occurrences. The additional occurrence has to be assigned to a half in such a way that both resulting sub-problems would still be feasible. This is the main di-culty of the assignment and is solved in the procedure Sweep. Procedure Sweep: In this procedure we compute the assignment of the additional occurrence for all tasks that have an odd number of occurrences. The input to this procedure is a set of intervals I (with odd s i 's) with the restriction that each clique in the resulting interval graph is of even size. (Later, we show how to overcome this restriction.) The output is a partition of these intervals into two sets such that each clique is equally divided among the sets. This is done by a sweep along the x-coordinate of the intervals. During the sweep every interval will be assigned a variable which at the end is set to 0 or 1 (i.e., rst half of the period or second half of the period). Suppose that we sweep point x. We say that an interval I i is active while we sweep point x if x 2 I i . The assignment rules are as follows. For each interval I i that starts at x: If the current number of active intervals is even: A new variable is assigned to I i (I i is unpaired). If the current number of active intervals is odd: I i is paired to the currently unpaired interval I j and it is assigned the negation of I j 's variable. Thus no matter what value is later assigned to this variable, I i and I j will end up in opposite halves. For each interval I i that ends at x: If the current number of active intervals is even: Nothing is done. If the current number of active intervals is odd: If I i is paired with I I j is now being paired with the currently unpaired interval I k . Also, I j 's variable is matched with the negation of I k 's variable. This will ensure that I j and I k are put in opposite halves, or equivalently, I i and I k are put in the same halves. If I i is unpaired: Assign arbitrarily 0 or 1 to I i 's variable. These operations ensure that whenever the number of active intervals is even, then exactly half of the intervals will be assigned 0 and half will be assigned this will be proven later. Recall that we assumed that the size of each clique is even. Let us show how to overcome this restriction. For this we need the following simple lemma. For by C x the set of all the input intervals (with odd and even s i 's) that contain x; C x will be referred to as a clique. Lemma 4.1. The period T is even if and only if is oddgj is even for every clique C. This lemma implies that if T is even then the size of each clique in the input to procedure Sweep is indeed even. If T is odd, then a dummy interval I n+1 which extends over all other intervals and which has exactly one occurrence is added to the set I before calling Sweep. Again, by Lemma 4.1, we are sure that in this modied set I the size of all cliques is even. This would increase the period by one. The additional time unit will be allotted only to the dummy interval and thus can be ignored. We note that to produce the schedule at time t we just have to follow the recursive calls that include t in their period. Since there are no more than log T such calls, the time it takes to produce this schedule is polynomial in n for max-min fair frequencies. 8 A. Bar-Noy, A. Mayer, B. Schieber, and M. Sudan Lemma 4.2. The algorithm produces a correct schedule for every feasible set of frequencies. Lemma 4.3. If the set of frequencies is of the form then the drift can be bounded by 1 and hence the response time can be bounded by d2=f i e. Proof: Since our algorithm always divides even s i into equal halves, the following invariant is maintained: At any recursive level, whenever s i > 1, then s i is even. Also note that thus we can express each f i as 2 i k . Now, following the algorithm, it can be easily shown that there is at least one occurrence of task i in each time interval of size Hence c and P -fairness Lemma 4.4. The response time for every interval I i is bounded by d4=f i e. Proof: The proof is based on the Lemma 4.3. This lemma clearly implies the case in which the frequencies are powers of two. Moreover, in case the frequencies are not powers of two, we can virtually partition each task into two tasks with frequencies p i and r i respectively, so that f a power of two, and r i < p i . Then, the schedule of the task with frequency p i has drift 1. This implies that its response time is d2=p i e d4=f i e. 2 We remark that it can be shown that the bound of the above lemma is tight for our algorithm. We summarize the results in this section in the following theorem: Theorem 4.5. Given an arbitrary interval graph as con ict graph, the algorithm exactly realizes any feasible frequency-vector and guarantees that r i 4.1 Bounding the drift Since the algorithm has O(log T ) levels of recursion and each level may increase the drift by one, clearly the maximum drift is bounded by O(log T ). In this section we prove that we can decrease the maximum drift to be O( log xed , where n is the number of tasks. By Theorem 2.6 this implies that in the worst case the drift for a max-min fair frequencies is bounded by O(n 1+ ). Our method to get a better drift is based on the following observation: At each recursive step of the algorithm two sets of tasks are produced such that each set has to be placed in a dierent half of the time-interval currently considered. However, we are free to choose which set goes to which half. We are using this degree of freedom to decrease the drift. To make the presentation clearer we assume that T is a power of two and that the time units are Consider a sub-interval of size T=2 j starting after time and ending at t for 1. In the rst j recursion levels we already xed the number of occurrences of each task up to t ' . Given this number, the drift d ' at time t ' is xed. Similarly, the drift d r at time t r is also xed. At the next recursion level we split the occurrences assigned to the interval thus xing the drift dm at time t Optimally, we would like the drifts after the next recursion level at each time unit to be the weighted average of the drifts d ' and d r . In other words, let would like the drift at time t to be d r In particular, we would like the drift at t m to be (d ' This drift can be achieved for t m only if the occurrences in the interval can be split equally. However, in case we have an odd number of occurrences to split, the drift at t m is (d ' depending on our decision in which half interval to put the extra occurrence. Note that the weighted average of the drifts of all other points changes accordingly. That is, if the new dm is (d ' then the weighted average in t 2 d r +(1 )d ' +2x, where and the weighted average in t 2 [(t r d r +(1 )d ' +(2 2)x, where 1=2. Consider now the two sets of tasks S 1 and S 2 that we have to assign to the two sub-intervals (of the same size) at level k of the recursion. For each of the possible two assignments, we compute a \potential" based on the resulting drifts at time t m . For a given possibility let D[tm ; i; k] denote the resulting drift of the i-th task at t m after k recursion levels. Dene the potential of t m after k levels as xed even constant . We choose the possibility with the lowest potential. Theorem 4.6. Using the policy described above the maximum drift is bounded by O( log T n ), for any xed . Realizing frequencies rigidly In this section we show how to construct a schedule that 12-approximates any feasible frequency-vector in a rigid fashion on an interval graph. We reduce our Rigid Schedule problem to the Dynamic Storage Allocation problem. The Dynamic Storage Allocation Guaranteeing Fair Service to Persistent Dependent Tasks 9 problem is dened as follows. We are given objects to be stored in a computer memory. Each object has two parameters: (i) its size in terms of number of cells needed to store it, (ii) the time interval in which it should be stored. Each object must be stored in adjacent cells. The problem is to nd the minimal size memory that can accommodate at any given time all of the objects that are needed to be stored at that time. The Dynamic Storage Allocation problem is a special case of the multi-coloring problem on intervals graphs which we now dene. A multi-coloring of a weighted graph G with the weight function such that for all v 2 V the size of F (v) is w(v), and such that if (v; u) 2 E then F (v) \ F ;. The multi-coloring problem is to nd a multi-coloring with minimal number of colors. This problem is known to be an NP-Hard problem [10]. Two interesting special cases of the Multi-Coloring problem are when the colors of a vertex either must be adjacent or must be \spread well" among all colors. We call the rst case the AMC problem and the second case the CMC problem. More formally, in a solution to AMC if F for all 1 i < k. Whereas in a solution to CMC which uses T colors, if F divides T , and (ii) x and It is not hard to verify that for interval graphs the AMC problem is equivalent to the Dynamic Storage Allocation problem described above. Simply associate each object with a vertex in the graph and give it a weight equal to the number of cells it requires. Put an edge between two vertices if their time intervals inter- sect. The colors assigned to a vertex are interpreted as the cells in which the object is stored. On the other hand, the CMC problem corresponds to the Rigid Schedule problem as follows. First, we replace the frequency f(v) by a weight w(v). Let Now, assume that the output for the CMC problem uses T colors and let the colors of v be fx 1 < < x k g We interpret this as follows: v is scheduled in times x It is not di-cult to verify that this is indeed a solution to the Rigid Scheduling problem. Although Dynamic Storage Allocation problem is a special case of the multi-coloring problem it is still known to be an NP-Hard problem [10] and for similar reasons the Rigid Scheduling problem is also NP-Hard. Therefore, we are looking for an approximation algorithm. In what follows we present an approximation algorithm that produces a rigid scheduling that 12-approximates the given frequencies. For this we consider instances of the AMC and CMC problems in which the input weights are powers of two. Definition 5.1. A solution for an instance of AMC is both aligned and contiguous if for all In [15], Kierstead presents an algorithm for AMC that has an approximation factor 3. A careful inspection of this algorithm shows that it produces solutions that are both aligned and contiguous for all instances in which the weights are power of two. We show how to translate a solution for such an instance of the AMC problem that is both aligned and contiguous into a solution for an instance of the CMC problem with the same input weights. For be the k-bit number whose binary representation is the inverse of the binary representation of x. Lemma 5.1. For 1)g. Consider an instance of the CMC problem in which all the input weights are powers of two. Apply the solution of Kierstead [15] to solve the AMC instance with the same input. This solution is both aligned and contiguous, and uses at most 3T 0 colors where T 0 is the number of colors needed by an optimal coloring. Let be the smallest power of 2 that is greater than T 0 . It follows that T 6T 0 . Applying the transformation of Lemma 5.1 on the output of the solution to AMC yields a solution to CMC with at most T colors. This in turn, yields an approximation factor of at most 12 for the Rigid Scheduling problem, since w(v)=T f(v)=2. Theorem 5.2. The above algorithm computes a rigid schedule that 12-approximates any feasible frequency-vector on an interval graph. 6 Circular-Arc graphs In this section we show how to transform any algorithm A for computing a schedule that c-approximates any given feasible frequency-vector on interval graphs into an algorithm A 0 for computing a schedule that 2c-approximates any given feasible frequencies on circular-arc graphs. A. Bar-Noy, A. Mayer, B. Schieber, and M. Sudan f be a feasible frequency-vector on a circular-arc graph G. 1: Find the maximum clique C in G. is an interval graph. 2 be the frequency-vectors resulting from restricting f to the vertices of G 0 and C, respectively. Note that ^ are feasible on G 0 and C, respectively Step 2: Using A, nd schedules S 1 and S 2 that c- g 2 on G 0 and C, respectively. Step 3: Interleave S 1 and S 2 . Clearly, the resulting schedule 2c-approximates ^ f on the circular-arc graph G. 7 Future research Many open problems remain. The exact complexity of computing a max-min fair frequency assignment in general graphs is not known and there is no characterization of when such an assignment is easy to compute. All the scheduling algorithms in the paper use the inherent linearity of interval or circular-arc graphs. It would be interesting to nd scheduling algorithms for the wider class of perfect graphs. The algorithm for interval graphs that realizes frequencies exactly exhibits a considerable gap in its drift. It is not clear from which direction this gap can be closed. Our algorithms assume a central scheduler that makes all the decisions. Both from theoretical and practical point of view it is important to design scheduling algorithms working in more realistic environments such as high-speed local-area networks and wireless networks (as mentioned in Section 1.1). The distinguishing requirements in such an environment include a distributed implementation via a local signaling scheme, a con ict graph which may change with time, and restrictions on space per node and size of a signal. The performance measures and general setting, however, remain the same. A rst step towards such algorithms has been recently carried out by Mayer, Ofek and Yung in [19]. Acknowledgment . We would like to thank Don Coppersmith and Moti Yung for many useful discussions --R A Dining Philosophers Algorithm with Polynomial Response Time. A Notion of Fairness in Resource Allocation. Data Networks. Distributed Resource Allocation Algorithms. A Local Fairness Algorithm for Gigabit LANs/MANs with Spatial Reuse. The Drinking Philosophers Problem. Hierarchical Ordering of Sequential Processes. Computers and Intractability Algorithmic Graph Theory and Perfect Graphs. Cellular Packet Communica- tions Bottleneck Flow Control. A Polynomial Time Approximation Algorithm for Dynamic Storage Allocation. The Sandwich Theorem On the Hardness of Approximating Minimization Problems. Fast Allocation of Nearby Resources in a Distributed System. Distributed Scheduling Algorithm for Fairness with Minimum Delay. Matrix characterizations of circular-arc graphs --TR --CTR Sanjoy K. Baruah , Shun-Shii Lin, Pfair Scheduling of Generalized Pinwheel Task Systems, IEEE Transactions on Computers, v.47 n.7, p.812-816, July 1998 Francesco Lo Presti, Joint congestion control: routing and media access control optimization via dual decomposition for ad hoc wireless networks, Proceedings of the 8th ACM international symposium on Modeling, analysis and simulation of wireless and mobile systems, October 10-13, 2005, Montral, Quebec, Canada Sandy Irani , Vitus Leung, Scheduling with conflicts on bipartite and interval graphs, Journal of Scheduling, v.6 n.3, p.287-307, May/June Ami Litman , Shiri Moran-Schein, On distributed smooth scheduling, Proceedings of the seventeenth annual ACM symposium on Parallelism in algorithms and architectures, July 18-20, 2005, Las Vegas, Nevada, USA Tracy Kimbrel , Baruch Schieber , Maxim Sviridenko, Minimizing migrations in fair multiprocessor scheduling of persistent tasks, Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms, January 11-14, 2004, New Orleans, Louisiana Tracy Kimbrel , Baruch Schieber , Maxim Sviridenko, Minimizing migrations in fair multiprocessor scheduling of persistent tasks, Journal of Scheduling, v.9 n.4, p.365-379, August 2006 Violeta Gambiroza , Edward W. Knightly, Congestion control in CSMA-based networks with inconsistent channel state, Proceedings of the 2nd annual international workshop on Wireless internet, p.8-es, August 02-05, 2006, Boston, Massachusetts

dining philosophers problem;fairness;interval graphs;scheduling

284994

An Algorithm for Finding the Largest Approximately Common Substructures of Two Trees.

AbstractOrdered, labeled trees are trees in which each node has a label and the left-to-right order of its children (if it has any) is fixed. Such trees have many applications in vision, pattern recognition, molecular biology and natural language processing. We consider a substructure of an ordered labeled tree T to be a connected subgraph of T. Given two ordered labeled trees and an integer d, the largest approximately common substructure problem is to find a substructure U1 of T1 and a substructure U2 of T2 such that U1 is within edit distance d of U2 and where there does not exist any other substructure V1 of T1 and V2 of T2 such that V1 and V2 satisfy the distance constraint and the sum of the sizes of V1 and V2 is greater than the sum of the sizes of U1 and U2. We present a dynamic programming algorithm to solve this problem, which runs as fast as the fastest known algorithm for computing the edit distance of two trees when the distance allowed in the common substructures is a constant independent of the input trees. To demonstrate the utility of our algorithm, we discuss its application to discovering motifs in multiple RNA secondary structures (which are ordered labeled trees).

Introduction Ordered, labeled trees are trees in which each node has a label and the left-to-right order of its children (if it has any) is fixed. 1 Such trees have many applications in vision, pattern recognition, molecular biology and natural language processing, including the representation of images [12], patterns [2, 10] and secondary structures of RNA [14]. They are frequently used in other disciplines as well. A large amount of work has been performed for comparing two trees based on various distance measures recently generalized one of the most commonly used distance measures, namely the edit distance, for both rooted and unrooted unordered trees. These works laid out a foundation that is useful for comparing graphs [15, 24]. In this paper we extend the previous work by considering the largest approximately common sub-structure problem for ordered labeled trees. Various biologists [5, 14] represent RNA secondary structures as trees. Finding common patterns (also known as motifs) in these secondary structures helps both in predicting RNA folding [5] and in functional studies of RNA processing mechanisms [14]. Previous methods for detecting motifs in the RNA molecules (trees) are based on one of the following two approaches: (1) transforming the trees to sequences and then using sequence algorithms [13]; (2) representing the molecules using a highly simplified tree structure and then searching for common nodes in the trees [5]. Neither of the two approaches satisfactorily takes the full tree structure into account. By contrast, utilizing the proposed algorithm for pairs of trees enables one to locate tree-structured motifs occurring in multiple RNA secondary structures. Our experimental results concerning RNA classification show the significance of these motifs [23]. Preliminaries 2.1 Edit Distance and Mappings We use the edit distance [17] to measure the dissimilarity of two trees. There are three types of edit operations, i.e., relabeling, delete, and insert a node. Relabeling node n means changing the label on n. Deleting a node n means making the children of n become the children of the parent of n and removing n. Insert is the inverse of delete. Inserting node n as the child of node n 0 makes n the parent of a consecutive subsequence of the current children of n 0 . Fig. 1 illustrates the edit operations. For the purpose of this work, we assume that all edit operations have a unit cost. The edit distance, or simply the distance, from tree T 1 to tree T 2 , denoted is the cost of a minimum cost sequence of edit operations transforming to T 2 [17]. The notion of edit distance is best illustrated through the concept of mappings. A mapping is a graphical specification of which edit operations apply to each node in the two trees. For example, the mapping in Fig. 2 shows a way to transform T 1 to T 2 . The transformation includes deleting the two nodes labeled a and m in T 1 and inserting them into T 2 1 Throughout the paper, we shall refer to ordered labeled trees simply as trees when no ambiguity occurs. a c r a b r r r a f c r r f e f f Fig. 1. (i) Relabeling: To change one node label (a) to another (b). (ii) Delete: To delete a node; all children of the deleted node (labeled b) become children of the parent (labeled r). (iii) Insert: To insert a node; a consecutive sequence of siblings among the children of the node labeled r (here, f and the left g) become the children of the newly inserted node labeled b. r a m b c d e pT r a m b c d e pT Fig. 2. A mapping from tree T1 to tree T2 . We use a postorder numbering of nodes in the trees. Let t[i] represent the node of T whose position in the left-to-right postorder traversal of T is i. When there is no confusion, we also use t[i] to represent the label of node t[i]. Formally, a mapping from T 1 to T 2 is a triple (M; simply M if the context is clear), where M is any set of ordered pairs of integers (i; (ii) For any pair of (i (one-to-one is to the left of t 1 ] is to the left of t 2 preservation is an ancestor of t 1 is an ancestor of t 2 [j 2 ] (ancestor order preservation condition). The cost of M is the cost of deleting nodes of T 1 not touched by a mapping line plus the cost of inserting nodes of T 2 not touched by a mapping line plus the cost of relabeling nodes in those pairs related by mapping lines with different labels. It can be proved [17] that ffi(T 1 equals the cost of a minimum cost mapping from tree T 1 to tree T 2 . 2.2 Cut Operations We define a substructure U of tree T to be a connected subgraph of T . That is, U is rooted at a node n in T and is generated by cutting off some subtrees in the subtree rooted at n. Formally, let T [i] represent the subtree rooted at t[i]. The operation of cutting at node t[i] means removing T [i]. A set S of nodes of T [k] is said to be a set of consistent subtree cuts in T [k] if (i) t[i] 2 S implies that t[i] is a node in T [k], and (ii) t[i]; t[j] 2 S implies that neither is an ancestor of the other in T [k]. Intuitively, S is the set of all roots of the removed subtrees in T [k]. We use Cut(T ; S) to represent the tree T with subtree removals at all nodes in S. Let Subtrees(T ) be the set of all possible sets of consistent subtree cuts in T . Given two trees T 1 and T 2 and an integer d, the size of the largest approximately common root-containing substructures within distance [i] and T 2 [j], denoted fi(T 1 [j]; (or simply fi(i; j; when the context is clear), is maxfjCut(T 1 )jg subject to ffi(Cut(T 1 [j]). Finding the largest approximately common substructure (LACS), within distance d, of T 1 [i] and T 2 [j] amounts to calculating max 1-u-i;1-v-j ffi(T 1 [v]; d)g and locating the Cut(T 1 [v]) that achieve the maximum size. The size of LACS, within distance d, of T 1 and T 2 is d)g. We shall focus on computing the maximum size. By memorizing the size information during the computation and by a simple backtracking technique, one can find both the maximum size and one of the corresponding substructure pairs yielding the size in the same time and space complexity. 3 Our Algorithm 3.1 Notation We use desc(i) to represent the set of postorder numbers of the descendants of the node t[i] and l(i) denotes the postorder number of the leftmost leaf of the subtree T [i]. When T [i] is a leaf, is an ordered forest of tree T induced by the nodes numbered i to j inclusive (see Fig. 3). If i ? j, then ;. The definition of mappings for ordered forests is the same as for trees. Let F 1 and F 2 be two forests. The distance from F 1 to F 2 , denoted \Delta(F 1 the cost of a minimum cost mapping from to F 2 [25]. set S of nodes of F is said to be a set of consistent subtree cuts in F if (i) t[p] 2 S implies that i - p - j, and (ii) t[p]; t[q] 2 S implies that neither is an ancestor of the other in F . We use Cut(F; S) to represent the sub-forest F with subtree removals at all nodes in S. Let Subtrees(F ) be the set of all possible sets of consistent subtree cuts in F . Define the size of the largest approximately common root-containing substructures, within distance k, of F 1 and F 2 , denoted \Psi(F 1 ; k), to be subject to \Delta(C ut(F 1 [l(i)::s] and F 2 [l(j)::t], we also represent there is no confusion. [10] [3] [5] [6] [8] [6] [3] Fig. 3. An induced ordered forest. 3.2 Basic Properties Lemma 3.1. Suppose s 2 desc(i) and t 2 desc(j). Then Proof. Immediate from definitions. Lemma 3.2. Suppose s 2 desc(i) and t 2 desc(j). Then for all k, 1 - k - d, ae \Psi(T 1 ae \Psi(;; T 2 Proof. (i) follows from the definition. For (ii), suppose [l(i)::s]) is a smallest set of consistent subtree cuts that maximizes jCut(T 1 )j where \Delta(Cut(T 1 of the following two cases must hold: (1) t 1 . If (1) is true, then \Psi(T 1 [l(i)::s], ;, 1. (iii) is proved similarly as for (ii). Lemma 3.3. Suppose s 2 desc(i) and t 2 desc(j). If (l(s) 6= l(i) or l(t) 6= l(j)), then Proof. Suppose [l(i)::s]) and S 2 2 Subtrees(T 2 [l(j)::t]) are two smallest sets of consistent subtree cuts that maximize jCut(T 1 )j where \Delta(C ut(T 1 at least one of the following cases must hold: Case 1. t 1 (i.e., the subtree T 1 [s] is removed). So, \Psi(l(i)::s; l(j)::t; Case 2. t 2 (i.e., the subtree T 2 [t] is removed). So, \Psi(l(i)::s; l(j)::t; Case 3. t 1 and t 2 [t] (i.e., neither T 1 [t] is removed) (Fig. 4). Let M be the mapping (with cost 0) from Cut(T 1 ) to Cut(T 2 In M , T 1 [s] must be mapped to [t] because otherwise we cannot have distance zero between Cut(T 1 Therefore \Psi(l(i)::s; l(j)::t; Since these three cases exhaust all possible mappings yielding \Psi(l(i)::s; l(j)::t; 0), we take the maximum of the corresponding sizes, which gives the formula asserted by the lemma. Fig. 4. Illustration of the case in which one of T1 [l(i)::s] and T2 [l(j)::t] is a forest and neither nor T2 [t] is removed. Lemma 3.4. Suppose s 2 desc(i) and t 2 desc(j). Suppose both T 1 [l(j)::t] are trees (i.e., Proof. Since [t]. First, consider the case where t 1 are two smallest sets of consistent subtree cuts that maximize jCut(T 1 )j where ffi(Cut(T 1 [t], M be the mapping (with cost 0) from Cut(T 1 Fig. 5). Clearly, in M , t 1 [s] must be mapped to t 2 [t]. Furthermore, the largest common root-containing substructure of must be the largest common root-containing substructure of T 1 [s] and T 2 [t]. This means that \Psi(l(i)::s; l(j)::t; 2, where the 2 is obtained by including the two nodes t 1 [t]. Next consider the case where t 1 [s] 6= t 2 [t] (i.e., the roots of the two trees T 1 [s] and T 2 [t] differ). In order to get distance zero between the two trees after applying cut operations to them, we have to remove both trees entirely. Thus, \Psi(l(i)::s; l(j)::t; Fig. 5. Illustration of the case in which both T1 [l(i)::s] and T2 [l(j)::t] are trees and t1 Lemma 3.5. Suppose s 2 desc(i) and t 2 desc(j). If (l(s) 6= l(i) or l(t) 6= l(j)), then for all k, 1 - k - d, Proof. Suppose are two smallest sets of consistent subtree cuts that maximize jCut(T 1 )j where \Delta(C ut(T 1 k. Then at least one of the following cases must hold: Case 1. t 1 . So, \Psi(l(i)::s; l(j)::t; Case 2. Case 3. t 1 and t 2 [t] . Let M be a minimum cost mapping from Cut(T 1 ) to There are three subcases to examine: [s] is not touched by a line in M . Then, \Psi(l(i)::s; l(j)::t; (b) t 2 [t] is not touched by a line in M . Then, \Psi(l(i)::s; l(j)::t; are both touched by lines in M . Then (s; . So, there exists an h such that \Psi(l(i)::s; l(j)::t; h). The value of h ranges from 0 to k. Therefore we take the maximum of the corresponding sizes, i.e., \Psi(l(i)::s; l(j)::t; h)g. Lemma 3.6. Suppose s 2 desc(i) and t 2 desc(j). Suppose both T 1 [l(i)::s] and T 2 [l(j)::t] are trees (i.e., where ae Proof. Since [l(j)::t] are trees, T 1 [t]. We first show that removing either T 1 [s] or T 2 [t] would not yield the maximum size. There are three cases to be considered: Case 1. Both T 1 are removed. Then, \Psi(l(i)::s; l(j)::t; cutting at just the children of t 1 [t] would cause \Psi(l(i)::s; l(j)::t; 2. Therefore removing both T 1 and T 2 cannot yield the maximum size. Case 2. Only T 1 [s] is removed. Then, \Psi(l(i)::s; l(j)::t; [t]; k). Assume without loss of generality that jT 2 k. The above equation implies that we have to remove some subtrees from T 2 so that there are no more than k nodes left in T 2 [t]. Thus, \Psi(l(i)::s; l(j)::t; On the other hand, if we just cut at the children of t 1 [s] and leave t 1 [s] in the tree, we would map t 1 [s] to t 2 [t]. This would lead to \Psi(l(i)::s; l(j)::t; 1. Thus, removing T 1 alone cannot yield the maximum size. Case 3. Only T 2 [t] is removed. The proof is similar to that in Case 2. The above arguments lead to the conclusion that in order to obtain the maximum size, neither T 1 nor [t] can be removed. Now suppose [s]) and S 2 2 Subtrees(T 2 [t]) are two smallest sets of consistent subtree cuts that maximize jCut(T 1 )j where ffi(Cut(T 1 be a minimum cost mapping from Cut(T 1 ) to Cut(T 2 ). Then at least one of the following cases must hold: Case 1. t 1 [s] is not touched by a line in M . So, \Psi(l(i)::s; l(j)::t; Case 2. t 2 [t] is not touched by a line in M . So, \Psi(l(i)::s; l(j)::t; Case 3. Both t 1 are touched by lines in M . By the ancestor order preservation and sibling order preservation conditions on mappings (cf. Section 2.1), (s; t) must be in M . Thus, if t 1 [t], we have \Psi(l(i)::s; l(j)::t; mapping t 1 [s] to t 2 costs 1, we have \Psi(l(i)::s; l(j)::t; 2. 3.3 The Algorithm From Lemma 3.4 and Lemma 3.6, we observe that when s is on the path from l(i) to i and t is on the path from l(j) to j, we need not compute fi(s; t; k), 0 - k - d, separately, since they can be obtained during the computation of fi(i; j; k). Thus, we will only consider nodes that are either the roots of the trees or having a left sibling. Let keynodes(T ) contain all such nodes of a tree T , i.e., ? k such that )g. For each Fig. 6 computes fi(s; t; Procedure Find-Largest-2 in Fig. 6 computes fi(s; t; d. The main algorithm is summarized in Fig. 6. Now, to calculate the size of the largest approximately common substructures (LACSs), within distance d, of T 1 [i] and T 2 [j], we build, in a bottom-up fashion, another array fl(i; using fi(i; j; d) as follows. Let are the postorder numbers of the children of t 1 [i] or [i] is a leaf. Let are the postorder numbers of the children of t 2 [j] or Rg. The size of LACSs, within distance d, of T 1 [i] and T 2 [j] is fl(i; j; d). The size of LACSs, within distance d, of Consider the complexity of the algorithm. We use an array to hold \Psi, fi and fl, respectively. These arrays require O(d \Theta jT 1 j \Theta jT 2 space. Regarding the time complexity, given fi(i; calculating fl(i; j; d) requires O(jT 1 j \Theta jT 2 time. For a fixed i and j, Procedure Find-Largest- the total time is bounded by O( [i]j \Theta jT 2 [i]j \Theta From [25, Theorem 2], the last term above is bounded by O(d 2 \Theta jT 1 j \Theta jT 2 j \Theta min(H 1 is the height of T i and L i is the number of leaves in T i . When d is a constant, this is the same as the complexity of the best current algorithm for tree matching based on the edit distance [11, 25], even though the problem at hand appears to be harder than tree matching. Note that to calculate max 1-i-jT1 j;1-j-jT2 j ffi(i; j; 0)g, one could use a faster algorithm that runs in time O(jT 1 j \Theta jT 2 j). However, the reason for considering the keynodes and the formulas as specified in Lemmas 3.3 and 3.4 is to prepare the optimal sizes from forests to forests and store these size values in the array to be used in calculating fi(s; t; could incorporate the faster algorithm into the Find-Largest algorithm, the overall time complexity would not be changed, because the calculation of fi(s; t; dominates the cost. 4 Implementation and Discussion We have applied our algorithm to find motifs in multiple RNA secondary structures. In this experiment, we examined three phylogenetically related families of mRNA sequences chosen from GenBank [1] pertaining to the poliovirus, human rhinovirus and coxsackievirus. Each family contained two sequences, as shown in Table 1. Algorithm Find-Largest Input: Trees T 1 and an integer d. Output: fi(i; j; := 1 to jkeynodes(T1)j do := 1 to jkeynodes(T2)j do begin run Procedure Find-Largest-1 on input (i; j; 0); run Procedure Find-Largest-2 on input (i; j; d); Procedure Find-Largest-1 Input: Output: fi(s; t; for s := l(i) to i do for t := l(j) to j do for s := l(i) to i do for t := l(j) to j do if (l(s) 6= l(i) or l(t) 6= l(j)) then compute \Psi(l(i)::s; l(j)::t; 0) as in Lemma 3.3; else begin /* compute \Psi(l(i)::s; l(j)::t; 0) as in Lemma 3.4; fi(s; t; Procedure Find-Largest-2 Input: Output: fi(s; t; for k := 1 to d do for k := 1 to d do for s := l(i) to i do compute \Psi(T 1 [l(i)::s]; ;; k) as in Lemma 3.2 (ii); for k := 1 to d do for t := l(j) to j do compute \Psi(;; T2 [l(j)::t]; k) as in Lemma 3.2 (iii); for k := 1 to d do for s := l(i) to i do for t := l(j) to j do if (l(s) 6= l(i) or l(t) 6= l(j)) then compute \Psi(l(i)::s; l(j)::t; k) as in Lemma 3.5; else begin /* compute \Psi(l(i)::s; l(j)::t; k) as in Lemma 3.6; fi(s; t; Fig. 6. Algorithm for computing fi(i; j; k). Family Sequence # of trees File # poliovirus polio3 sabin strain 3,026 file 1 pol3mut 3,000 file 2 human rhinovirus rhino 2 3,000 file 3 coxsackievirus cox5 3,000 file 5 Table 1. Data used in the experiment. Under physiological conditions, i.e., at or above the room temperature, these RNA molecules do not take on only a single structure. They may change their conformation between structures with similar free energies or be trapped in local minima. Thus, one has to consider not only the optimal structure but all structures within a certain range of free energies. On the other hand, a loose rule of thumb is that the "real" structure of an RNA molecule appears in the top 5% - 10% of suboptimal structures of the sequence based on the ranking of their energies with the minimum energy one (i.e. the optimal one) being at the top. Therefore, we folded the 5' non-coding region of the selected mRNA sequences and collected (roughly) the top 3,000 suboptimal structures for each sequence. We then transformed these suboptimal structures into trees using the algorithms described in [13, 14]. Fig. 7 illustrates an RNA secondary structure and its tree representation. The structure is decomposed into five terms: stem, hairpin, bulge, internal loop and multi-branch loop [14]. In the tree, H represents hairpin nodes, I represents internal loops, B represents bulge loops, M represents multi-branch loops, R represents helical stem regions (shown as connecting arcs) and N is a special node used to make sure the tree is connected. The tree is considered to be an ordered one where the ordering is imposed based upon the 5' to 3' nature of the molecule. The resulting trees for each mRNA sequence selected from GenBank were stored in a separate file, where the trees had between 70 and 180 nodes (cf. Table 1). Each tree is represented by a fully parenthesized notation where the root of every subtree precedes all the nodes contained in the subtree. Thus, for example, the tree depicted in Fig. 7(ii) is represented as (N(R(I(R(M(R(B(R(M(R(H))(R(H))))))(R(H))))))). For each pair of trees T 1 in a file, we ran the algorithm Find-Largest on T 1 , finding the size of the largest approximately common substructures, within distance 1, for each subtree pair T 1 [i] and T 2 [j], locating one of the corresponding substructure pairs yielding the size. These substructures constituted candidate motifs. Then we calculated the occurrence number 2 of each candidate motif M by adding variable length don't cares (VLDCs) to M as the new root and leaves to form a VLDC pattern V and then comparing V with each tree T in the file using the pattern matching technique developed in [26]. (conventionally denoted by " ") can be matched, at no cost, with a path or portion of a path in T . The technique calculates the minimum distance between V and T after implicitly computing an optimal substitution for the VLDCs in V , allowing zero or more cuttings at nodes from T (see Fig. 8).) This way we can locate the motifs approximately occurring in all (or the majority 2 The occurrence number of a motif M with respect to distance k refers to the number of trees of the file in which M approximately occurs (i.e. these trees approximately contain M) within distance k. of) the trees in the file. 3U A U A A A U G C A U U A A U A U G U A U A A A U U A G G A A G A G G G G U U G U U G A C C G U A G A U A U GU U A A U U I A A A G C A A G U U C A U U U C G C C A U U A A GFig. 7. Illustration of a typical RNA secondary structure and its tree representation. (i) Normal polygonal representation of the structure. (ii) Tree representation of the structure. Table 2 summarizes the results where the motifs occur within distance 0 in at least 350 trees in the corresponding file. The table shows the number of motifs discovered for each sequence, the number of distinct motifs found in common between both sequences of each family, and the minimum and maximum sizes of these common motifs. Table 3 shows some big motifs found in common in all the three families and the number of each sequence's secondary structures that contain the motifs. These motifs serve as a starting point to conduct further study of common motif analysis [3, 22]. 3 One can speed up this method by encoding the candidate motifs into a suffix tree and then using the statistical sampling and optimization techniques described in [23] to find the motifs. a r a b d Fig. 8. Matching a VLDC pattern V and a tree T (both the pattern and tree are hypothetical ones solely used for illustration purposes). The root in V would be matched with nodes and the two leaves in V would be matched with nodes in T , respectively. Nodes would be cut. The distance of V and T would be 1 (representing the cost of changing c in V to d in T ). Family Sequence # of motifs found # of common motifs min size max size poliovirus polio3 sabin strain 836 347 3 101 pol3mut 793 rhinovirus rhino 2 287 coxsackievirus cox5 306 136 3 20 cvb305pr 391 Table 2. Statistics concerning motifs discovered from the secondary structures of the mRNA sequences used in the experiment. Motifs found polio3 pol3mut rhino 2 rhino 14 cox 5 cvb305pr (R(B(R(B(R(B(R))))))) 2,272 1,822 3,000 2,252 2,997 2,979 Table 3. Motifs found in common in the secondary structures of the poliovirus, human rhinovirus and coxsack- ievirus sequences. The motifs are represented in a fully parenthesized notation where the root of every subtree precedes all the nodes contained in the subtree. For each motif, the table also shows the number of each sequence's suboptimal structures that contain the motif. The proposed algorithm and the discovered motifs have also been applied to RNA classification successfully [23]. Our experimental results showed that one can get more intersections of motifs from sequences of the same family. This indicates that closeness in motif corresponds to closeness in family. Another application of our algorithm is to apply it to a tree T and itself and calculate fi(i; j; This allows one to find repeatedly occurring substructures (or repeats for short) in T . Finding repeats in secondary structures across different RNA sequences may help understand the structures of RNA. Readers interested in obtaining these programs may send a written request to any one of the authors. Our work is based on the edit distance originated in [17]. This metric is more permissive than other worthy metrics (e.g. [18, 19, 20]) and therefore helps to locate subtle motifs existing in RNA secondary structures. The algorithm presented here assumes a unit cost for all edit operations. In practice, a more refined non-unit cost function can reflect more subtle differences in the RNA secondary structures [14]. It would then be interesting to score the measures in detecting common substructures or repeats in trees. Another interesting problem is to find a largest consensus motif T 3 in two input trees T 1 and T 2 where T 3 is a largest tree such that each of T 1 and T 2 has a substructure that is within a given distance to T 3 . A comparison of the different types of common substructures (see also [6, 7, 8]), probably based on different metrics (e.g. [18, 19, 20]), as well as their applications remains to be explored. Acknowledgments We wish to thank the anonymous reviewers for their constructive suggestions and pointers to some relevant papers. We also thank Wojcieok Kasprzak (National Cancer Institute), Nat Goodman (Whitehead Institute of MIT) and Chia-Yo Chang for their useful comments and implementation efforts. This work was supported by the National Science Foundation under Grants IRI-9224601, IRI-9224602, IRI-9531548, IRI-9531554, and by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0046373. --R Nucleic Acids Research Waveform correlation by tree matching. Secondary structure computer prediction of the polio virus 5 Alignment of trees - An alternative to tree edit RNA secondary struc- tures: Comparison and determination of frequently recurring substructures by consensus A largest common similar substructure problem for trees embedded in a plane. Largest common similar substructures of rooted and unordered trees. The largest common similar substructure problem. A tree system approach for fingerprint pattern recognition. A unified view on tree metrics. Distance transform for images represented by quadtrees. An algorithm for comparing multiple RNA secondary structures. Comparing multiple RNA secondary structures using tree comparisons. Structural descriptions and inexact matching. Exact and approximate algorithms for unordered tree matching. The tree-to-tree correction problem The metric between rooted and ordered trees based on strongly structure preserving mapping and its computing method. A metric between unrooted and unordered trees and its bottom-up computing method "A metric on trees and its computing method." The tree-to-tree editing problem The cardiovirulent phenotype of coxsackievirus B3 is determined at a single site in the genomic 5 Automated discovery of active motifs in multiple RNA secondary structures. An algorithm for graph optimal monomorphism. Simple fast algorithms for the editing distance between trees and related problems. Approximate tree matching in the presence of variable length don't cares. On the editing distance between undirected acyclic graphs. --TR --CTR Roger Keays , Andry Rakotonirainy, Context-oriented programming, Proceedings of the 3rd ACM international workshop on Data engineering for wireless and mobile access, September 19-19, 2003, San Diego, CA, USA M. Vilares , F. J. Ribadas , J. Graa, Approximately common patterns in shared-forests, Proceedings of the tenth international conference on Information and knowledge management, October 05-10, 2001, Atlanta, Georgia, USA Rolf Backofen , Sven Siebert, Fast detection of common sequence structure patterns in RNAs, Journal of Discrete Algorithms, v.5 n.2, p.212-228, June, 2007 S. Bhowmick , Wee Keong Ng , Sanjay Madria, Constraint-driven join processing in a web warehouse, Data & Knowledge Engineering, v.45 n.1, p.33-78, April D. C. Reis , P. B. Golgher , A. S. Silva , A. F. Laender, Automatic web news extraction using tree edit distance, Proceedings of the 13th international conference on World Wide Web, May 17-20, 2004, New York, NY, USA Ada Ouangraoua , Pascal Ferraro , Laurent Tichit , Serge Dulucq, Local similarity between quotiented ordered trees, Journal of Discrete Algorithms, v.5 n.1, p.23-35, March, 2007 Thomas Kmpke, Distance Patterns in Structural Similarity, The Journal of Machine Learning Research, 7, p.2065-2086, 12/1/2006 N. Bourbakis , P. Yuan , S. Makrogiannis, Object recognition using wavelets, L-G graphs and synthesis of regions, Pattern Recognition, v.40 n.7, p.2077-2096, July, 2007 S. Bhowmick , Sanjay Kumar Madria , Wee Keong Ng, Detecting and Representing Relevant Web Deltas in WHOWEDA, IEEE Transactions on Knowledge and Data Engineering, v.15 n.2, p.423-441, February Dmitriy Bespalov , Ali Shokoufandeh , William C. Regli , Wei Sun, Scale-space representation of 3D models and topological matching, Proceedings of the eighth ACM symposium on Solid modeling and applications, June 16-20, 2003, Seattle, Washington, USA Jiang , Andreas Munger , Horst Bunke, On Median Graphs: Properties, Algorithms, and Applications, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.23 n.10, p.1144-1151, October 2001 Yanhong Zhai , Bing Liu, Web data extraction based on partial tree alignment, Proceedings of the 14th international conference on World Wide Web, May 10-14, 2005, Chiba, Japan Jun Huan , Wei Wang , Deepak Bandyopadhyay , Jack Snoeyink , Jan Prins , Alexander Tropsha, Mining protein family specific residue packing patterns from protein structure graphs, Proceedings of the eighth annual international conference on Resaerch in computational molecular biology, p.308-315, March 27-31, 2004, San Diego, California, USA

trees;pattern recognition;dynamic programming;pattern matching;computational biology

285006

On the Power of Finite Automata with both Nondeterministic and Probabilistic States.

We study finite automata with both nondeterministic and random states (npfa's). We restrict our attention to those npfa's that accept their languages with a small probability of error and run in polynomial expected time. Equivalently, we study Arthur--Merlin games where Arthur is limited to polynomial time and constant space.Dwork and Stockmeyer [SIAM J. Comput., 19 (1990), pp. 1011--1023] asked whether these npfa's accept only the regular languages (this was known if the automaton has only randomness or only nondeterminism). We show that the answer is yes in the case of npfa's with a 1-way input head. We also show that if L is a nonregular language, then either L or $\bar{L}$ is not accepted by any npfa with a 2-way input head.Toward this end, we define a new measure of the complexity of a language L, called its 1-tiling complexity. For each $n$, this is the number of tiles needed to cover the 1's in the "characteristic matrix" of L, namely, the binary matrix with a row and column for each string of length $\le n$, where entry [x,y]=1 if and only if the string $xy \in L$. We show that a language has constant 1-tiling complexity if and only if it is regular, from which the result on 1-way input follows. Our main result regarding the general 2-way input tape follows by contrasting two bounds: an upper bound of polylog(n) on the 1-tiling complexity of every language computed by our model and a lower bound stating that the 1-tiling complexity of a nonregular language or its complement exceeds a function in $2^{\Omega (\sqrt{\log n})}$ infinitely often.The last lower bound follows by proving that the characteristic matrix of every nonregular language has rank n for infinitely many n. This is our main technical result, and its proof extends techniques of Frobenius and Iohvidov developed for Hankel matrices [Sitzungsber. der Knigl. Preuss. Akad. der Wiss., 1894, pp. 407--431], [Hankel and Toeplitz Matrices and Forms: Algebraic Theory, Birkhauser, Boston, 1982].

Introduction The classical subset construction of Rabin and Scott [25] shows that finite state automata with just nondeterministic states (nfa's) accept exactly the regular languages. Results of Rabin [24], Dwork and Stockmeyer [7] and Kaneps and Freivalds [17] show that the same is true of probabilistic finite state automata which run in polynomial expected time. Here and throughout the paper, we restrict attention to automata which accept languages with error probability which is some constant ffl less than 1=2. However, there has been little previous work on finite state automata which have both probabilistic and nondeterministic states. Such automata are equivalent to the Arthur-Merlin games of Babai and Moran [3], restricted to constant space, with an unbounded number of rounds of communication between Arthur and Merlin. In this paper, we refer to them as npfa's. In the computation of an npfa, each transition from a probabilistic state is chosen randomly according to the transition probabilities from that state, whereas from a nondeterministic state, it is chosen so as to maximize the probability that an accepting state is eventually reached. We let 1NPFA and 2NPFA-polytime denote the classes of languages accepted by npfa's which have a 1-way or a 2-way input head, respectively, and which run in polynomial expected time. Dwork and Stockmeyer [8] asked whether 2NPFA-polytime is exactly the set of regular languages, which we denote by Regular. In this paper, we prove the following two results on npfa's. Theorem 1.1 Theorem 1.2 If L is nonregular, then either L or - L is not in 2NPFA-polytime. Thus, we resolve the question of Dwork and Stockmeyer for npfa's with 1-way head, and in the case of the 2-way head model, we reduce the question to that of deciding whether 2NPFA-polytime is closed under complement. Theorem 1.1 also holds even if the automaton has universal, as well as nondeterministic and probabilistic states. Moreover, Theorem 1.2 holds even for Arthur-Merlin games that use o(log log n) space. In proving the two results, we introduce a new measure of the complexity of a language L called its 1-tiling complexity. Tiling complexity arguments have been used previously to prove lower bounds for communication complexity (see e.g. Yao [29]). With each language L ' \Sigma , we associate an infinite binary matrix ML , whose rows and columns are labeled by the strings of \Sigma . Entry ML [x; y] is 1 if the string xy 2 L and is 0 otherwise. Denote by ML (n) the finite submatrix of ML , indexed by strings of length - n. Then, the 1-tiling complexity of L (and of the matrix ML (n)) is the minimum size of a set of 1-tiles of ML (n) such that every 1-valued entry of ML (n) is in at least one 1-tile of the set. Here, a 1-tile is simply a submatrix (whose rows and columns are not necessarily contiguous) in which all entries have value 1. In Section 3, we prove the following theorems relating language acceptance of npfa's to tiling complexity. The proofs of these theorems build on previous work of Dwork and Stockmeyer [8] and Rabin [24]. Theorem [3.1] A language L is in 1NPFA only if the 1-tiling complexity of L is O(1). Theorem [3.3] A language L is in 2NPFA-polytime only if the 1-tiling complexity of L is bounded by a polynomial in log n. What distinguishes our work on tiling is that we are interested in the problem of tiling the matrices ML (n), which have distinctive structural properties. If L is a unary language, then ML (n) is a matrix in which all entries along each diagonal from the top right to the bottom left are equal. Such a matrix is known as a Hankel matrix. An elegant theory on properties of such Hankel matrices has been developed [15], from which we obtain strong bounds on the rank of ML (n) if L is unary. In the case that L is not a unary language, the pattern of 0's and 1's in ML (n) is not as simple as in the unary case, although the matrix still has much structure. Our main technical contribution, presented in Section 4, is to prove new lower bounds on the rank of ML (n) when L is not unary. Our proof uses techniques of Frobenius and Iohvidov developed for Hankel matrices. Theorem [4.4] If L is nonregular, then the rank of ML (n) is at least n+ 1 infinitely often. By applying results from communication complexity relating the rank of a matrix to its tiling complexity, we can obtain a lower bound on the 1-tiling complexity of non-regular languages. Theorem [4.5] If L is nonregular, then the 1-tiling complexity of either L or - L exceeds a function in log n) infinitely often. However, there are nonregular languages, even over a unary alphabet, with 1-tiling complexity O(log n) (see Section 4). Thus the above lower bound on the 1-tiling complexity of L or - L does not always hold for L itself. A simpler theorem holds for regular languages. Theorem [4.1] The 1-tiling complexity of L is O(1) if and only if L is regular. By combining these theorems on the 1-tiling complexity of regular and non-regular languages with the theorems relating 1-tiling complexity to acceptance by npfa's, our two main results (Theorems 1.1 and 1.2) follow as immediate corollaries. The rest of the paper is organized as follows. In Section 2, we define our model of the npfa, and the tiling complexity of a language. We conclude that section with a discussion of related work on probabilistic finite automata and Arthur-Merlin games. In Section 3, we present Theorems 3.1 and 3.3, which relate membership of a language L in the classes 1NPFA and 2NPFA-polytime to the 1-tiling complexity of L. A similar theorem is presented for the class 2NPFA, in which the underlying automata are not restricted to run in polynomial expected time. In Section 4, we present our bounds on the tiling complexity of both regular and nonregular languages. Theorems 1.1 and 1.2 are immediate corollaries of the main results of Sections 3 and 4. Extensions of these results to alternating automata and to Turing machines with small space are presented in Section 5. Conclusions and open problems are discussed in Section 6. Preliminaries We first define our npfa model in Section 2.1. This model includes as special cases the standard models of nondeterministic and probabilistic finite state automata. In Section 2.2 we define our notion of the tiling complexity of a language. Finally, in Section 2.3, we discuss previous work on this and related models. 2.1 Computational Models and Language Classes A two-way nondeterministic probabilistic finite automaton (2npfa) consists of a set of states Q, an input alphabet \Sigma, and a transition function ffi, with the following properties. The states Q are partitioned into three subsets: the nondeterministic states N , the probabilistic (or random) states R, and the halting states H . H consists of two states: the accepting state q a and the rejecting state q r . There is a distinguished state q 0 , called the initial state. There are two special symbols 2 \Sigma, which are used to mark the left and right ends of the input string, respectively. The transition function ffi has the form For each fixed q in R, the set of random states, and oe 2 (\Sigma [ f6 c; $g), the sum of ffi(q; oe; q over all q 0 and d equals 1. The meaning of ffi in this case is that if the automaton is in state q reading symbol oe, then with probability ffi(q; oe; q d) the automaton enters state q 0 and moves its input head one symbol in direction d (left if stationary if 0). For each fixed q in N , the set of nondeterministic states, and oe all q 0 and d. The meaning of ffi in this case is that if the automaton is in state q reading symbol oe, then the automaton nondeterministically chooses some q 0 and d such that ffi(q; oe; q enters state q 0 and moves its input head one symbol in direction d. Once the automaton enters state q a (resp. q r ), the input head moves repeatedly to the right until the right endmarker is read, at which point the automaton halts. In other words, for q 2 fq a ; q r g, ffi(q; oe; q; for all oe On a given input, the automaton is started in the initial configuration, that is, in the initial state with the head at the left end of the input. If the automaton halts in state q a on the input, we say that it accepts the input, and if it halts in state q r , we say that it rejects the input. Fix some input string strategy (or just strategy) on w is a function such that ffi(q; oe; q oe. The meaning of Sw is that if the automaton is in state q 2 N reading w j , then if Sw (q; the automaton enters state q 0 and moves its input head one symbol in direction d. The strategy indicates which nondeterministic choice should be made in each configuration. A language L ' \Sigma is accepted with bounded error probability if for some constant ffl ! 1=2, 1. for all w 2 L, there exists a strategy Sw on which the automaton accepts with probability 2. for all 2 L, on every strategy Sw , the automaton accepts with probability - ffl. Language acceptance could be defined with respect to a more general type of strategy, in which the nondeterministic choice made from the same configuration at different times may be different. It is known (see [4, Theorem 2.6]) that if L is accepted by an npfa with respect to this more general definition, then it is also accepted with respect to the definition above. Hence, our results also hold for such generalized strategies. A one-way nondeterministic probabilistic finite automaton (1npfa) is a 2npfa which can never move its input head to the left; that is, ffi(q; oe; q Also, a probabilistic finite automaton (pfa) and a nondeterministic finite automaton (nfa) are special cases of an npfa in which there are no nondeterministic and no probabilistic states, respectively. We denote by 1NPFA and 2NPFA the classes of languages accepted with bounded error probability by 1npfa's and 2npfa's, respectively. If, on all inputs w and all nondeterministic strategies, the 2npfa halts in polynomial expected time, we say that L is in the class 2NPFA- polytime. The classes 1PFA, 2PFA and 2PFA-polytime are defined similarly, with pfa replacing npfa. Finally, Regular denotes the class of regular languages. Our model of the 2npfa is equivalent to an Arthur-Merlin game in which Arthur is a 2pfa, and our classes 2NPFA and 2NPFA-polytime are identical to the classes AM(2pfa) and AM(ptime- 2pfa), respectively, of Dwork and Stockmeyer [8]. 2.2 The Tiling Complexity of a Language We adapt the notion of the tiling complexity of a function, used in communication complexity theory, to obtain a new measure of the complexity of a language. Given a finite, two-dimensional matrix M , a tile is a submatrix of M in which all entries have the same value. A tile is specified by a pair (R; C) where R is a nonempty set of rows and C is a nonempty set of columns. The entries in the tile are said to be covered by the tile. A tile is a b-tile if all entries of the submatrix are b. A set of b-tiles is a b-tiling of M if every b-valued entry of M is covered by at least one tile in the set. If M is a binary matrix, the union of a 0-tiling and a 1-tiling of M is called a tiling of M . Let T (M) be the minimum size of a tiling of M . Let T 1 (M) be the minimum size of a 1-tiling of M , and let T 0 (M) be the minimum size of a 0-tiling of M . Then, Note that in these definitions it is permitted for tiles of the same type to overlap. We can now define the tiling complexity of a language. Associated with a language L over alphabet \Sigma is an infinite binary matrix ML . The rows and columns of ML are indexed (say, in lexicographic order), by the strings in \Sigma . Entry ML [x; only if xy 2 L. Let L n be the strings of L of length - n. Let ML (n) be the finite submatrix of ML whose rows and columns are indexed by the strings of length - n. The 1-tiling complexity of a language L is defined to be the function T 1 Similarly, the 0-tiling complexity of L is and the tiling complexity of L is A tiling of a matrix M is disjoint if every entry [x; y] of M is covered by exactly one tile. The disjoint tiling complexity of a matrix M , ~ is the minimum size of a disjoint tiling of M . Also, the disjoint tiling complexity of a language, ~ (n), is ~ T(ML (n)). Tilings are often used in proving lower bounds in communication complexity. Let f : 1g. The function f is represented by a matrix M f whose rows are indexed by elements of X and whose columns are indexed by elements of Y , such that M f [x; Let T f denote T(M f ). Suppose that two cooperating parties, P 1 and P 2 , get inputs x 2 X and respectively, and want to compute f(x; y). They can do so by exchanging information according to some protocol (precise definitions of legal protocols can be found in [13]). If the protocol is deterministic, then the worst case number of bits that need to be exchanged (that is, the deterministic communication complexity) is bounded below by log ~ If the protocol is non-deterministic, then the lower bound is log T f [1]. Finally, if the object of the non-deterministic protocol is only to verify that f(x; that is indeed the case), then the lower bound on the number of bits exchanged is log T 1 f . 2.3 Related Work Our work on npfa's builds on a rich literature on probabilistic finite state automata. Rabin [24] was the first to consider probabilistic automata with bounded error probability. He showed that However, with a 2-way input head, pfa's can recognize nonregular languages. This was shown by Freivalds [10], who constructed a 2pfa for the language f0 n 1 Greenberg and Weiss [12] showed that exponential expected time is required by any 2pfa accepting this language. Dwork and Stockmeyer [7] and independently Kaneps and Freivalds [17] showed that in fact any 2pfa which recognizes a nonregular language must run in exponential expected time. It follows that 2PFA-polytime = Regular. Roughly, Rabin's proof shows that any language L accepted by a 1pfa has only finitely many equivalence classes. Here, two strings x; x 0 are equivalent if and only if for all y, xy 2 L. The Myhill-Nerode theorem [14] states that a language has a finite number of equivalence classes if and only if it is regular. This, combined with Rabin's result, implies that decades later, this idea was extended to 2pfa's. A strengthened version of the Myhill-Nerode theorem is needed for this extension. Given a language L, we say that two strings x; x 0 are pairwise n-inequivalent if for some y, xy 2 L , x 0 y 62 L, and furthermore, (the nonregularity of L) be size of the largest set of pairwise n- inequivalent strings. Kaneps and Freivalds [16] showed that NL (n) - b(n + 3)=2c for infinitely many n. (It is interesting to note that to prove their bound, Kaneps and Freivalds first showed that NL (n) equals the number of states of the minimal deterministic 1-way finite automaton that accepts all words of length - n that are in L and rejects all words of length - n that are not in L. Following Karp [19], we denote the latter measure by OE L (n). Karp [19] previously proved that OE L (n) infinitely many n. Combining this with the fact that NL (n) and OE L (n) are equal, it follows immediately that NL (n) ? n=2+1 for infinitely many n. This is stronger (by 1) for even n than Kaneps and Freivalds' lower bound. We also note that Dwork and Stockmeyer [7] obtained a weaker bound on NL (n) without using OE L (n).) Using tools from Markov chain theory, Dwork and Stockmeyer [7] and Kaneps and Freivalds [17] showed that if a language is accepted by a 2pfa in polynomial expected time, then the language has "low" nonregularity. In fact, NL (n) is bounded by some polynomial in log n. This, combined with the result of Kaneps and Freivalds, implies that 2PFA-polytime = Regular. Models of computation with both nondeterministic and probabilistic states have been studied intensively since the work of Papadimitriou [23] on games against nature. Babai and Moran [3] defined Arthur-Merlin games to be Turing machines with both nondeterministic and probabilistic states, which accept their languages with bounded error probability. Their work on polynomial time bounded Arthur-Merlin games laid the framework for the remarkable progress on interactive proof systems and their applications (see for example [2] and the references therein). Space bounded Arthur-Merlin games were first considered by Condon and Ladner [6]. Condon [4] showed that AM(log-space), that is, the class of languages accepted by Arthur- Merlin games with logarithmic space, is equal to the class P. However, it is not known whether the class AM(log-space, polytime) - the subclass of AM(log-space) where the verifier is also restricted to run in polynomial time - is equal to P, or whether it is closed under complement. Fortnow and Lund [9] showed that NC is contained in AM(log-space,poly-time). Dwork and Stockmeyer [8] were the first to consider npfa's, which are Arthur-Merlin games restricted to constant space. They described conditions under which a language is not in the classes 2NPFA or 2NPFA-polytime. The statements of our Theorems 3.2 and 3.3 generalize and simplify the statements of their theorems, and our proofs build on theirs. In communication complexity theory terms, their proofs roughly show that languages accepted by npfa's have low "fooling set complexity". This measure is defined in a manner similar to the tiling complexity of a language, based on the following definition. Define a 1-fooling set of a binary matrix A to be a set of entries The size of a 1-fooling set of a binary matrix is always at most the 1-tiling complexity of the matrix, because no two distinct entries in the 1-fooling set, [x can be in the same tile. However, the 1-tiling complexity may be significantly larger than the 1-fooling set in fact, for a random n \Theta n binary matrix, the expected size of the largest 1-fooling set is O(log n) whereas the expected number of tiles needed to tile the 1-entries is \Omega\Gamma n= log n) 3 NPFA's and Tiling Three results are presented in this section. For each of the classes 1NPFA, 2NPFA and 2NPFA- polytime, we describe upper bounds on the tiling complexity of the languages in these classes. The proof for 1NPFA's is a natural generalization of Rabin's proof that The other two proofs build on previous results of Dwork and Stockmeyer [8] on 2npfa's. 3.1 1NPFA and Tiling Theorem 3.1 A language L is in 1NPFA only if the 1-tiling complexity of L is O(1). Proof: Suppose L is accepted by some 1npfa M with error probability ffl ! 1=2. Let the states of M be cg. Consider the matrix ML . For each 1-entry [x; y] of ML , fix a nondeterministic strategy that causes the string xy to be accepted with probability at least 1 \Gamma ffl. With respect to this strategy, define two vectors of dimension c. Let p xy be the state probability vector at the step when the input head moves off the right end of x. That is, the i'th entry of the vector is the probability of being in state i at that moment, assuming that the automaton is started at the left end of the input 6 cxy$ in the initial state. Let r xy be the column vector whose i'th entry is the probability of accepting the string xy, assuming that the automaton is in state i at the moment that the head moves off the right end of x. Then the probability of accepting the string xy is the inner product ffl)=c. Partition the space [0; 1] c into cells of size - \Theta - \Theta \Delta \Delta \Delta \Theta - (the final entry in the cross product should actually be less than - if 1 is not a multiple of -). Associate each 1-entry [x; y] with the cell containing the vector p xy ; we say that [x; y] belongs to this cell. With each cell C, associate the rectangle RC defined as fxj there exists y such that [x; y] belongs to Cg \Theta fyj there exists x such that [x; y] belongs to Cg: This is the minimal submatrix that covers all of the entries associated with cell C. We claim that RC is a valid 1-tile - that is, RC covers only 1-entries. To see this, suppose . If [x; y] belongs to C, then it must be a 1-entry. Otherwise, there exist x 0 and y 0 such that [x; y 0 belong to C; that is, xy are in the same cell. We claim that xy is accepted with probability at least 1=2 on some strategy, namely the strategy that while reading x, uses the strategy for xy 0 , and while reading y, uses the strategy for x 0 y. To see this, note that c c -c our choice of -: Hence, the probability that xy is accepted on the strategy described above is Because xy is accepted with probability greater than ffl on this strategy, it cannot be that xy 62 L. Hence, for all [x; y] 2 RC , xy must be in L. Therefore RC is a 1-tile in ML . Every 1-entry [x; y] is associated with some cell C, and is covered by the 1-tile RC that is associated with C. Thus, every 1-entry of ML is covered by some RC . Hence L can be 1-tiled using one tile per cell, which is a total of d1=-e 3.2 2NPFA and Tiling We next show that if L 2 2NPFA, then T 1 L (n) is bounded by a polynomial. Theorem 3.2 A language L is in 2NPFA only if the 1-tiling complexity of L is bounded by a polynomial in n. Proof: Suppose L is accepted by some 2npfa M with error probability c be the number of states of M . As in Theorem 3.1, for each 1-entry [x; y] of ML (n), fix a nondeterministic strategy that causes M to accept the string xy with probability at least 1 \Gamma ffl. We construct a stationary Markov chain H xy that models the computation of M on xy using this strategy. This Markov chain has states. 2c of the states are labeled (q; l), where q is a state of M and l 2 f0; 1g. The other states are labeled Initial, Accept, Reject, and Loop. The state (q; 0) of H xy corresponds to M being in state q while reading the rightmost symbol of 6 cx. The state (q; 1) of H xy corresponds to M being in state q while reading the leftmost symbol of y$. The state Initial corresponds to the initial configuration of M . The states Accept, Reject, and Loop are sink states of H xy . A single step of the Markov chain H xy corresponds to running M on input xy (using the fixed nondeterministic strategy) from the appropriate configuration for one or more steps, until M enters a configuration corresponding to one of the chain states (q; l). If M halts in the accepting (resp., rejecting) state before entering one of these configurations, H xy enters the Accept (resp., Reject) state. If M does not halt and never again reads the rightmost symbol of 6 cx or the leftmost symbol of y$, then H xy enters the Loop state. The transition probabilities are defined accordingly. Consider the transition matrix of H xy . Collect the rows corresponding to the chain states Initial and (q; 0) (for all q) and call this submatrix P xy . Collect the rows corresponding to the chain states (q; 1) and call this submatrix R xy . Then the transition matrix looks like this: R xy Initial Accept Reject Loop where I 3 denotes the identity matrix of size 3. (We shall engage in a slight abuse of notation by using H xy to refer both to the transition matrix and to the Markov chain itself.) Note that the entries of P xy depend only on x and the nondeterministic strategy used; these transition probabilities do not depend on y. This assertion appears to be contradicted by the fact that our choice of nondeterministic strategy may depend on y; however, the idea here is that if we replace y with y 0 while maintaining the same nondeterministic strategy we used for xy, then will be identical to P xy , because the transitions involved simulate computation of M on the left part of its input only. Similarly, R xy depends only on y and the strategy, and not on x. We now show that if jxj - n and if p is a nonzero element of P xy , then a second Markov chain K(6 cx) with states of the form (q; l), where q is a state of M and 1. The chain state (q; l) with l - j6 cxj corresponds to M being in state q scanning the l'th symbol of 6 cx. Transition probabilities from these states are obtained from the transition probabilities of M in the obvious way. Chain states of the form (q; cxj + 1) are sink states of K(6cx) and correspond to the head of M falling off the right end of 6 cx with M in state q. Now consider a transition probability p in P xy . Suppose that, in the Markov chain H xy , p is the transition probability from (q; 0) to (q 0 ; 1). Then p 2 f0; 1=2; 1g, since if H xy makes this transition, it must be simulating a single computation step of M . Suppose p is the transition probability from (q; 0) to (q 0 ; 0). If p ? 0, then there must be some path of nonzero probability in K(6 cx) from state (q; cxj) to (q 0 ; cxj) that visits no state (q 00 ; cxj), and since K(6 cx) has at most cn states that can be on this path, there must be such a path of length at most cn + 1. Since 1/2 is the smallest nonzero transition probability of M , it follows that p - 2 \Gammacn\Gamma1 . The cases where p is a transition probability from the Initial state are similar. Similarly, if jyj - n and if r is a nonzero element of R xy , then r - 2 \Gammacn\Gamma1 . Next we present a lemma which bounds the effect of small changes in the transition probabilities of a Markov chain. This lemma is a slight restatement of a lemma of Greenberg and Weiss [12]. This version is due to Dwork and Stockmeyer [8]. If k is a sink state of a Markov chain R, let a(k; R) denote the probability that R is (eventually) trapped in state k when started in state 1. Let fi - 1. Say that two numbers r and r 0 are fi-close if either (i) chains i;j=1 and R are fi-close if r ij and r 0 are fi-close for all pairs Lemma 3.1 Let R and R 0 be two s-state Markov chains which are fi-close, and let k be a sink state of both R and R 0 . Then a(k; R) and a(k; R 0 ) are fi 2s -close. The proof of this lemma is based on the Markov chain tree theorem of Leighton and Rivest [20], and can be found in [8]. Our approach is to partition the 1-entries of ML (n) into equivalence classes, as in the proof of Theorem 3.1, but this time we will make entries [x; y] and [x equivalent only if the corresponding Markov chains H xy and H x 0 y 0 are fi-close, where fi will be chosen small enough that we can use Lemma 3.1 to show that xy 0 and x 0 y are accepted with high probability by combining the strategies for xy and x 0 y 0 . If [x; y] is a 1-entry such that jxj - n and jyj - n, then for any nonzero p of P xy (or r of R xy By partitioning each coordinate interval into subintervals of length -, we divide the space into at most d(cn cells, each of size at most - \Theta - \Theta \Delta \Delta \Delta -. Partition the 1-entries in ML (n) into equivalence classes by making xy and x 0 y 0 equivalent have the property that for each state transition, if p and p 0 are the respective transition probabilities, either log p and log p 0 are in the same (size -) subinterval of Note that the number of equivalence classes is at most (d(cn We claim that if - is chosen small enough, these equivalence classes induce a 1-tiling of ML (n) of size at most the number of equivalence classes. As in Theorem 3.1, we associate with each equivalence class C the rectangle RC defined by fxjthere exists y such that [x; y] 2 Cg \Theta fyj there exists x such that [x; y] 2 Cg. We claim that for each [x; y] in RC , xy 2 L. That is, all entries in the rectangle are 1, so the rectangle forms a 1-tile. Let [x; y] be in RC . There must be some y 0 such that [x; y 0 and some x 0 such that [x 0 ; y] 2 C. Consider the associated Markov chains H xy 0 and H x 0 y , and in particular, consider the transition submatrices P xy 0 and R x 0 y . The first is associated with a particular nondeterministic strategy on x, namely one which assumes the input is xy 0 and tries to cause xy 0 to be accepted with high probability. The second is associated with a particular nondeterministic strategy on y, namely one which assumes the input is x 0 y and tries to cause x 0 y to be accepted with high probability. The two matrices P xy 0 and R x 0 y taken together correspond to a hybrid strategy on xy: while reading x, use the strategy for xy 0 , and while reading y, use the strategy for x 0 y. We will argue that this hybrid strategy causes xy to be accepted with probability - 1=2. We construct a hybrid Markov chain H xy using P xy 0 and R x 0 y . This chain models the computation of M on xy using the hybrid strategy. Since the 1-entries [x; y 0 ] and [x 0 ; y] are in the same equivalence class C, it follows that if p and p 0 are corresponding transition probabilities in the Markov chains H xy 0 and H x 0 y , then either Therefore, H xy 0 and H x 0 y are 2 -close, and it immediately follows that H xy is 2 -close to H xy 0 (and to H x 0 y ). Let a xy 0 be the probability that M accepts input xy 0 on the strategy for xy 0 , and let a xy be the probability that M accepts input xy using the hybrid strategy. Then a xy 0 (resp., a xy ) is exactly the probability that the Markov chain eventually trapped in the Accept state, when started in the Initial state. Now xy 0 2 L implies a xy are 2 -close, Lemma 3.1 implies that a xy a xy 0 which implies a xy - Since ffl and d are constants, and since ffl ! 1=2, we can choose - to be a constant so small that a xy - 1=2. Therefore xy must be in L. Since each 1-entry [x; y] is in some equivalence class, the matrix ML (n) can be 1-tiled using at most (d(cn tiles. Therefore, Since c; d, and - are constants independent of n, this shows that T 1 L (n) is bounded by a polynomial in n. 2 3.3 2NPFA-polytime and Tiling We now show that if L 2 2NPFA-polytime, then T 1 L (n) is bounded by a polylog function. Theorem 3.3 A language L is in 2NPFA-polytime only if the 1-tiling complexity of L is bounded by a polynomial in log n. Proof: Suppose L is accepted by some 2npfa M with error probability ffl ! 1=2 in expected time at most t(n). Let c be the number of states of M . For each 1-entry [x; y] of ML (n), fix a nondeterministic strategy that causes M to accept the string xy with probability at least 1 \Gamma ffl. We construct the Markov chain H xy just as in Theorem 3.2. Say that a probability p is small if p ! t(n) \Gamma2 ; otherwise, p is large. Note that if p is a large transition probability, then dividing the 1- entries of ML (n) into equivalence classes, make xy and x 0 y 0 equivalent if H xy and H x 0 y 0 have the property that for each state transition, if p and p 0 are the respective transition probabilities, either p and p 0 are both small, or log p and log p 0 are in the same (size -) subinterval of This time the number of equivalence classes is at most (d2 log Model the computation of M on inputs x 0 y, xy 0 , and xy by Markov chains H x 0 y , H xy 0 , and H xy , respectively, as before. If p and p 0 are corresponding transition probabilities in any two of these Markov chains, then either p and p 0 are 2 -close or p and p 0 are both small. Let E x 0 y be the event that, when H x 0 y is started in state Initial, it is trapped in state Accept or Reject before any transition labeled with a small probability is taken; define E xy 0 and E xy similarly. Since M halts in expected time at most t(n) on the inputs x 0 y, xy 0 , and xy, the probabilities of these events go to 1 as n increases. Therefore, by changing all small probabilities to zero, we do not significantly change the probabilities that H x 0 y , H xy 0 , and H xy enter the Accept state, provided that n is sufficiently large. A formal justification of this argument can be found in Dwork and Stockmeyer [8]. After these changes, we can argue that a xy - and choose - so that a xy - 1=2, as before. It then follows that (1) for all sufficiently large n, establishing the result. 2 4 Bounds on the Tiling Complexity of Languages In this section, we obtain several bounds on the tiling complexity of regular and nonregular languages. In Section 4.1, we prove several elementary results. First, all regular languages have constant tiling complexity. Second, the 1-tiling complexity of all nonregular languages is at least log infinitely often. We also present an example of a (unary) non-regular language which has 1-tiling complexity O(log n). In Section 4.2, we use a rank argument to show that for all nonregular languages L, either L or its complement has "high" 1-tiling complexity infinitely often. 4.1 Simple Bounds on the Tiling Complexity of Languages The following lemma is useful in proving some of the theorems in this section. Its proof is implicit in work of Melhorn and Schmidt [21]; we include it for completeness. Lemma 4.1 Any binary matrix A that can be 1-tiled with m tiles has at most 2 m distinct rows. Proof: Let A be a binary matrix that can be 1-tiled by m tiles fT For each row r of A, let g. Suppose r 1 and r 2 are rows such that I(r 1 We show that in this case, rows r 1 and r 2 are identical. To see this, consider any column c of A. Suppose that entry [r 1 ; c] has value 1, and is covered by some tile T therefore r 2 2 R j and [r 2 ; c] is covered by tile T j . Hence entry [r must have value 1, since T j is a 1-tile. Hence, if [r 1 ; c] has value 1, so does [r 2 ; c]. Similarly, if [r 2 ; c] has value 1, then so does entry [r 1 ; c]. Therefore r 1 and r 2 are identical rows. Since there are only 2 m possible values for I(r), A can have at most 2 m distinct rows. 2 Theorem 4.1 The 1-tiling complexity of L is O(1) if and only if L is regular. Proof: By the Myhill-Nerode theorem [14, Theorem 3.6], L is regular if and only if ML has a finite number of distinct rows. Suppose L is regular. Then by the above fact there exists a constant k such that ML has at most k distinct rows. Consider any (possibly infinite) set R of identical rows in ML . Let C b be the set of columns which have bit b in the rows of R, for 1. Then the subset specified by (R; C b ) is a b-tile and covers all the b-valued entries in the rows of R. It follows that the 1-valued entries of R can be covered by a single tile, and hence there is a 1-tiling of ML (n) of size k. (Similarly, there is a 0-tiling of ML (n) of size k.) Suppose L is not regular. Since L is not regular, ML has an infinite number of distinct rows. It follows immediately from Lemma 4.1 that M cannot be tiled with any constant number of tiles. 2 The above theorem uses the simple fact that the 1-tiling complexity L (n) of a language L is a lower bound on the number of distinct rows of ML (n). In fact, the number of distinct rows of ML (n), for a language L, is closely related to a measure that has been previously studied by many researchers. Dwork and Stockmeyer called this measure non-regularity, and denoted the non-regularity of L by NL (n) [7]. NL (n) is the maximum size of a set of n-dissimilar strings of L. Two strings, w and w 0 , are considered n-dissimilar if jwj - n and jw 0 j - n, and there exists a string v such that jwvj - n, It is easy to show that the number of distinct rows of ML (n) is between NL (n) and NL (2n). Previously, Kaneps and Freivalds [16] showed that NL (n) is equal to the number of states of the minimal 1-way deterministic finite state automaton which accepts a language L 0 for which L 0 is the set of strings of L of length - n. Shallit [28] introduced a similar measure: the nondeterministic nonregularity of L, denoted by NNL (n), is the minimal number of states of a 1-way nondeterministic finite automaton which accepts a language L 0 for which L 0 In fact, it is not hard to show that To see this, suppose that M is an automaton with NNL (2n) states, which accepts a language We construct a 1-tiling of ML (n) with one tile T q per state q of M , where entry [x; y] is covered by T q if and only if there is an accepting path of M on xy which enters state q as the head falls off the rightmost symbol of x. It is straightforward to verify the set of tiles defined in this way is indeed a valid 1-tiling of ML (n). A similar argument was used by Schmidt [27] to prove lower bounds on the number of states in an unambiguous nfa. We next turn to simple lower bounds on the 1-tiling complexity of nonregular languages. From Theorem 4.1, it is clear that if L is nonregular, then T 1 L (n) is unbounded. We now use a known lower bound on the nonregularity of nonregular languages to prove a lower bound for (n). Theorem 4.2 If L is not regular, then T 1 infinitely many n. Proof: Kaneps and Freivalds [16] proved that if L is not regular, then NL (n) - b(n+3)=2c for infinitely many n. By the definition of NL (n), the matrix ML (n) must have at least NL (n) distinct rows. Therefore, by Lemma 4.1, T 1 (n). The lemma follows immediately.We next present an example of a unary nonregular language, with 1-tiling complexity O(log n). Thus, the lower bound of Theorem 4.2 is optimal to within a constant factor. Theorem 4.3 Let L be the complement of the language fa 1-tiling complexity O(logn). Proof: We show that the 1-valued entries of ML (n) can be covered with O(log n) 1-tiles. Let lg n denote blog 2 binary numbers, of length at most lg n. Number the bits of these numbers from right to left, starting with 1, so that for example . For any binary number q, lg q is the maximum index i such that if q is equal to 2 1. The next fact follows easily. only if for all j such that j - maxflg x; lg yg, Roughly, we construct a 1-tiling of ML (n), corresponding to the following nondeterministic communication protocol. The party P 1 guesses an index j and sends j and x j to P 2 . Also P 1 sends indicating whether or not j - lg x. If j - lg x, then P 2 checks that y checks that j - lg y and that y or equivalently, that y In either case, can conclude that y of ML (n) is 1. The number of bits sent from 2. We now describe the 1-tiling corresponding to this protocol. It is the union of two sets of tiles. The first set has one tile T j;b for each j; b such that lg n - The second set of tiles has one tile S j;0 , for all j such that dlog ne - j - 1. To see that all the 1's in the matrix are covered by one of these tiles, note that if entry [a x ; a y ] of the matrix is 1, then by the Fact, there exists an index j such that j - maxflg x; lg yg and either x y, and j is such that covered by tile T j;0 . 2 The nondeterministic communication protocol in the above proof is a slight variation of a simple (and previously known) protocol for the complement of the set distinctness problem. In the set distinctness problem, the two parties each hold a subset of must determine whether the subsets are distinct. In our application, the problem is to determine, whether the subset of whose corresponding values in x are 0, is distinct from the subset of whose corresponding values in y are 1. 4.2 Lower Bounds on the Tiling Complexity of Nonregular Languages In this section we prove that if a language L is nonregular, then the 1-tiling complexity of either L or - L is "high" infinitely often. To prove this, we first prove lower bounds on the rank of ML when L is nonregular. We then apply theorems from communication complexity relating rank to tiling complexity. The proofs of the lower bounds on the rank of ML are heavily dependent on distinctive structural properties of ML . Consider first the case where L is a unary language over the alphabet fag. In this case, for all It follows that for every n, ML (n) is such that its auxiliary diagonal (the diagonal from the top right to the bottom left) consists of equal elements, as do all diagonals parallel to that diagonal. An example is shown in Figure 1. Such matrices are classically known as Hankel matrices, and have been extensively studied [15]. In fact, a direct application of known results on the rank of Hankel matrices shows that if L is nonregular, then infinitely often. This was first proved by Iohvidov (see [15, Theorem 11.3]), based on previous work of Frobenius [11]. ffl a 1 a 2 a 3 a 4 a 5 a 6 a a a a a Figure 1: The Hankel matrix ML (6) for If L is a non-unary language, then ML does not have the simple diagonal structure of a Hankel matrix. Nevertheless, ML still has structural properties that we are able to exploit. In fact, the term Hankel matrix has been extended from its classical meaning to refer to matrices ML of non-unary languages (see [26]). In what follows, we generalize the results on the rank of classical Hankel matrices, and prove that for any nonregular language L, over an arbitrary alphabet, rank(ML (n)) 4.2.1 Notation and basic facts Let L be a language over an arbitrary alphabet, and let Consider a row of M indexed by a string w. This row corresponds to strings that have the prefix w. For any string s, row ws corresponds to strings with the prefix ws. Thus the entries in row ws can be determined by looking at those entries in row w whose columns are indexed by strings beginning with s (see Figure 2). In what follows, we consider this relationship between the rows of M more formally. Let M(n;m) denote the set of vectors (finite rows) of M which are indexed by strings x of length - n and whose columns are indexed by strings of length - m. Let - M(n; m) denote the subset of vectors of M(n;m) which are indexed by strings x of length exactly n. If v 0 is row x of M(n;m+ i), where i ? 0 and v is row x of M(n;m), then v 0 is called an extension of v. Suppose s be a string over \Sigma of length - m (possibly the empty string, ffl). Define split (s) (v) to be the subvector formed from v by selecting exactly those columns whose labels have s as a prefix. Also, relabel the columns of split (s) (v) by removing the prefix s. Note that split (ffl) v. Note also that if \Sigma is unary, say foeg, then split (oe) (v) is v with the first column removed. Let jvj denote the dimension (number of entries) of vector v. If \Sigma is binary and oe 2 \Sigma, then Figure 2: The matrix M(3) for is a palindromeg. The bold entries in row 110 are determined by the bold entries in row 11. The bold entries in row 110 comprise split (0) (11) for M(2; 3). More generally, if c Also, the vector v consists of the first entry (indexed by the empty string, ffl), plus an "interleaving" of the entries of split (oe) (v), for each oe 2 \Sigma. More precisely, we have the following Fact 4.1 Let We generalize the definition of the split function to sets of vectors. If V is a set of vectors in M(n;m), and jsj - m, let split g. Then we have the following. Fact 4.2 [ jsj=i split (a) - In what follows, the vectors we consider are assumed to be elements of vector spaces over an arbitrary field F (e.g. our proofs will hold if F is taken to be the field of rationals F). All references to rank, span, and linear independence apply to vector spaces over F. Lemma 4.2 Suppose that b and that where the ff i are in the field F. Suppose that for 1 k is an extension in M(n;m+ 1) of b k and that v 0 is an extension of v to the same length as the b 0 k . Suppose also that for some it is the case that for all s of length i, split . is a string of length - m. Consider a string j 0 of length m+ 1. Let j Also, By the hypothesis of the lemma, split Putting the last three equalities together, v 0 [j Let rank(M(n; m)) be the rank of the set of vectors M(n;m) and let span(M(n; m)) be the vector space generated by the vectors in M(n;m). The next lemma follows immediately from the definitions. Lemma 4.3 If v 0 2 span(M(n; m)); m? 0 and 4.2.2 A Lower Bound on the Rank of M(n) when L is Nonregular A trivial lower bound on the rank of M(n) is given by the following fact. Fact 4.3 L is nonregular if and only if there is an infinite sequence of integers p r satisfying This is easily shown using the Myhill-Nerode theorem. Clearly, such a sequence exists if and only if the rank of M(n) (as n increases) is unbounded. Moreover, the rank of M(n) is unbounded if and only if the number of distinct rows in M(n) is unbounded. The Myhill-Nerode theorem states that the number of equivalence classes of L (equivalently, the number of distinct rows of M) is finite if and only if L is regular. It follows that L is nonregular if and only if the rank of M(n) is unbounded. This conclusion has already been noted (see Sections II.3 and II.5 of the book by Salomaa and Soittola [26], which describes results from the literature on rational power series and regular languages). The above lower bound is very weak. In what follows, we significantly improve it by using the special structure of M(n). Namely, we show that there is an infinite sequence of values of n such that rank(M(n)) - n + 1. We define the first value of n in our sequence to be the length of the shortest word in L (clearly this case). To construct the remainder of the sequence, we show (in Lemma 4.5) that because L is nonregular, for any value of n, there is some m - n such that rank(M(n prove (in Lemma 4.6 and the proof of Theorem 4.4) that if n is such that rank(M(n)) - n+ 1, and we choose the smallest m - n such that rank(M(n in fact rank(M(m 2. We begin with the following useful lemma. Lemma 4.4 Let n - 0; m - 1. Suppose that M(n Proof: By induction on i. The result is true by hypothesis of the lemma in the case and that the lemma is true for It follows from the induction hypothesis that if v 2 M(n must also be the case that if v 2 M(n+ then It remains to consider the vectors in - 1). By Fact 4.2 (a), each such vector v is of the form split (oe) (v 0 ), where for some oe; 1. By the inductive hypothesis, Then, by Fact 4.2 (b), all of the vectors in split (oe) (M(n; are in M(n+1;m \Gamma i+1). Hence, Finally, by the hypothesis of the lemma, span(M(n Corollary 4.1 For any n - 0, if rank(M(n+1; r. Proof: If n - p then M(p) is a submatrix of M(n; 2p) so the result follows trivially. Otherwise, choose i so that n a submatrix of M(n and hence by Lemma 4.4, the rows of M(p) are contained in span(M(n; p)). Thus again The following lemma shows the existence of an m - n such that rank(M(n Lemma 4.5 Let L be a nonregular language. Then for any n, there exists an m - n such that Proof: Let r be the number of strings of length - n. Clearly, rank(M(n; m)) - r for all m, since there are r rows in M(n;m). Let r as in Fact 4.3, that is, Hence, by Corollary 4.1, it must be the case that rank(M(n 2p is one possible value of m that satisfies the lemma. 2 It remains to show that if n is such that rank(M(n)) - n+ 1, and m is the smallest number such that m - n and rank(M(n+1; m+1)) ? rank(M(n; m+1)), then rank(M(m+1)) - m+2. This is clearly true if for all in this case rank(M(n; m+ 1)) - m+ 2. The difficult case is when there exist values of i such that To help deal with this case, we prove the following lemma. Lemma 4.6 Suppose that the following properties hold: 2. m is the smallest number ? n such that M(n 3. i is a number in the range Then, there is some vector in M(n+ i which is not in span(M(n; is the extension of some Then, we claim that for some s; split Fact 4.2 (b), this is sufficient to prove the lemma. Suppose to the contrary that for all s of length i, split be a basis of M(n;m). Let fb 0 p g be an extension of this basis in 1). By Properties 1 and 2 of the lemma, v is in span(M(n; m)). Let applying Fact 4.1, we see that for all s; split We want to show that for all s of length i, split It follows from this and from Lemma 4.2 that contradicting the fact that v 0 62 span(M(n; m+ 1)). Consider the vectors split k ). These are in M(n+ Fact 4.2 (b). If this is clearly in span(M(n; m+ 1))). If and by Property 2 of this lemma, these vectors are in span(M(n; l be a basis for span(M(n; and for 1 - k - l, let c 0 k be an extension in M(n;m . Clearly the set fc 0 l g is also linearly independent, and since rank(M(n; set is a basis for span(M(n; split Then, also split Also, since v 2 M(n m), from Fact 4.2 (b) it must be that the vectors split (s) (v) are in M(n Hence, again by Property 2 of this lemma, and by Lemma 4.4, these vectors are in span(M(n; l is a basis for span(M(n; follows that there exists a unique sequence of coefficients - l such that split Also, by combining Equation 2 with Equation 4, we see that split 1;l 2;l p;l c l ]: Thus p;k for all k 2 We claim split l 2;l l l ]: We now justify the claim. By our initial assumption, split Thus for some unique coefficients - 0 l , split l c 0 Each c 0 k is an extension of c k , and there is a unique linear combination of c l that is equal to split (s) (v). It follows that each - 0 This proves the claim. Combining the claim with Equation 3 yields split as desired. 2 We now prove the lower bound. Theorem 4.4 If L is nonregular, then Proof: The base case is n such that the shortest word in the language is of length n. Suppose that rank(M(n)) - fixed n. Let m be the smallest number - n such that rank(M(n there is such an m. We claim that rank(M(m 2. the claim is clearly true. Suppose m ? n. be a basis for M(n; k), n - k - m+ 1, where the extensions of all vectors in B k are in B k+1 . Let B 0 denote the subset of B k which are extensions of vectors in B k\Gamma1 . We construct a set of m linearly independent vectors in M(m + 1) as follows. For k from n to m+ 1, we define a linearly independent set C k of vectors in M(m+ 1; k), of size at least k + 1. Then, Cm+1 is the desired set. Let C . This is by definition a linearly independent set, and it has size - n because (by our initial assumption) rank(M(n)) - n + 1. Suppose that n - that C k is already constructed and is linearly independent. Construct C k+1 as follows. k be the set of extensions in M(m+ of the vectors in C k . Add C 0 k to C k+1 . to C k+1 . (Thus, C k+1 is expanded to contain those vectors in B k+1 which are not in B 0 .) (iii) Finally, suppose nothing is added to C k+1 in step (ii); that is, rank(M(n; is such that then this is equivalent to: rank(M(n; Thus, we can apply Lemma 4.6 to obtain a vector v which is not in span(M(n; but is not in k ).) Add v 0 to C k+1 . We claim that the vectors in C k+1 are linearly independent. Clearly the set C 0 k is linearly independent. Consider each vector u 0 added to C k+1 , which is not in C 0 k . By the construction, u 0 is not in span(B 0 be the extension of vector u in M(m+ 1; k). We claim that the vector u must be linearly dependent on the set B k . This is true if u 0 is added in step (ii), since in this case u is in M(n; is a basis for M(n; k). It is also true in the case that u the vector added in step (iii), since then by Lemma 4.4, Hence, Moreover, u can be expressed as a unique linear combination of the vectors of C k , with non-zero coefficients only on those vectors in B k . If u 0 were in span(C 0 k ), then since it is an extension of u, it would also be expressible as a unique linear combination of the vectors of C 0 k , with non-zero coefficients only on those vectors in B 0 k . But that contradicts the fact that u 0 62 span(B 0 4.2.3 The Tiling Complexity Lower Bound Theorem 4.5 If L is nonregular, then the 1-tiling complexity of either L or - L is at leastp log infinitely often. Proof: Melhorn and Schmidt, and independently Orlin, showed that for any binary matrix [21, 22]. Their result holds for A over any field. Halstenberg and Reischuk, refining a proof of Aho et. al., showed that dlog ~ By Theorem 4.4, if L is nonregular, then the rank of M(n) is at least n It follows that for infinitely many n, log 5 Variations on the Model In this section, we discuss extensions of our main results to other related models. We first show that Theorem 1.1 also holds for the following "alternating probabilistic" finite state automaton model. In this model, which we call a 2apfa, the nondeterministic states N are partitioned into two subsets, NE and NU of existential and universal states, respectively. Accordingly, for a fixed input, there are two types of strategy, defined as follows for a fixed input string An existential (universal) strategy on w is a function such that ffi(q; oe; q A language L ' \Sigma is accepted with bounded error probability if for some constant ffl ! 1=2, 1. for all w 2 L, there exists an existential strategy Ew on which the automaton accepts with probability strategies Uw , and 2. for all 2 L, on every existential strategy Ew , the automaton accepts with probability - ffl on some universal strategy Uw . The complexity classes 1APFA, 1APFA-polytime, and so on, are defined in the natural way, following our conventions for the npfa model. Theorem 5.1 Proof: As in Theorems 1.1 and 3.1, we show that if L is a language accepted by a 1APFA, then the tiling complexity of L is bounded. We first extend the notation of Theorem 3.1. If E is an existential strategy on xy and U is a universal strategy on xy, let p xy (E; U) be the state probability (row) vector at the step when the input head moves off the right end of x, on the strategies E; U . Let r xy (E; U) be the column vector whose i'th entry is the probability of accepting the string xy, assuming that the automaton is in state i at the moment that the head moves off the right end of x, on the strategies E; U . For each 1-entry [x; y] of ML , fix an existential strategy E xy , that causes xy to be accepted with probability at least 1 \Gamma ffl, for all universal strategies. Partition the space [0; 1] c into cells of size - \Theta - \Theta -, as before. Let C be a nonempty subset of the cells. We say that entry [x; y] of ML belongs to C if xy 2 L, and C is the smallest set of cells which contain all the vectors p xy strategies U . With each nonempty subset C of the cells, associate a rectangle R C defined as follows. fx j there exists y such that [x; y] belongs to Cg \Theta fy j there exists x such that [x; y] belongs to Cg: R C is a valid 1-tile. To see this, suppose that [x; y] 2 R C . If [x; y] belongs to C, then it must be a 1-entry. Otherwise, there exist x 0 and y 0 such that [x; y 0 belong to C. Consider the strategy E that while reading x, uses the strategy E xy 0 , and while reading y, uses the strategy E x 0 y . We claim that xy is accepted with probability at least 1=2 on existential strategy E and any universal strategy U on xy. The probability that xy is accepted on strategies E; U is belong to the same set of cells C, are in the same cell, for some universal strategy U 0 . Moreover, This is because this quantity is the probability that x 0 y is accepted on existential strategy and a universal strategy which is a hybrid of U and U 0 ; also by definition of E x 0 y , the probability that x 0 y is accepted with respect to E x 0 y and any universal strategy is -c our choice of -: Hence, the probability that xy is accepted on the strategies E; U is Since U is arbitrary, it follows that there is an existential strategy E such that on all strategies U , the probability that xy is accepted on the strategies E; U is greater than ffl, and so it cannot be that xy 62 L. Hence, for all [x; y] 2 R C , xy must be in L. Therefore R C is a 1-tile in ML . The proof is completed as in Theorem 3.1. 2 In the same way, Theorem 3.3 can also be extended to obtain the following. Theorem 5.2 A language L is in 2APFA-polytime only if the 1-tiling complexity of L is bounded by 2 polylog(n) . Thus, for example, the language Pal, consisting of all strings over f0; 1g which read the same forwards as backwards, is not in the class 2APFA-polytime. To see this, consider the submatrix of ML (n), consisting of all rows and columns labeled by strings of length exactly n. This matrix contains a fooling set of size 2 n ; hence a 1-tiling of ML (n) requires at least 2 n tiles. We next extend Theorem 1.2 to automata with o(log log n) space. We refer to these as Arthur-Merlin games, since this is the usual notation for such automata which are not restricted to a finite number of states [7]. The definition of an Arthur-Merlin game is similar to that of an npfa, except that the machine has a fixed number of read/write worktapes. The Arthur-Merlin game runs within space s(n) if on any input w with jwj - n, at most s(n) tape cells are used on any worktape. Thus, the number of different configurations of the Arthur-Merlin game is Theorem 5.3 Let M and - M be Arthur-Merlin games which recognize a nonregular language L and its complement - L, respectively, within space o(log log n). Suppose that the expected running time of both M and - M is bounded by t(n). Then, for all b ! 1=2, log log t(n) - (log n) b . In particular, t(n) is not bounded by any polynomial in n. Proof: The proof of Theorem 1.2 can be extended to space bounded Arthur-Merlin games, to yield the following generalization of Equation 1. Let c(n) be an upper bound on the number of different configurations of M on inputs of length n, and let sufficiently large n, the number of 1-tiles needed to cover ML (n) is at most Since M uses o(log log n) space, for any constant c ? 0, d(n) - (log n) c , for sufficiently large n. Now, suppose to the contrary that for some b ! 1=2, log log t(n) ! (log n) b for sufficiently large n. Then, log n): Hence, the number of tiles needed to cover the 1-valued entries of ML (n) is 2 o( log n) . The same argument for - M shows that also for for sufficiently large n, the number of tiles needed to cover the 1-valued entries of M- L (n) is 2 o( log n) . Hence, by Theorem 4.5 L must be regular, contradiction. 2 Finally, we consider a restriction of the 2npfa model, which, given polynomial time, can only recognize regular languages. A restricted 2npfa is a 2npfa for which there is some ffl ! 1=2 such that on all inputs w and strategies Sw , the probability that the automaton accepts is either Theorem 5.4 Any language accepted by a restricted 2npfa with bounded error probability in polynomial time is regular. Proof: Let L be accepted by a 2npfa M with bounded error probability in polynomial expected time. Let \Sigma be the alphabet, ffi the transition function, the set of states and N ae Q the set of nondeterministic states of M . Without loss of generality, let g. We first define a representation of strategies as strings over a finite alphabet. Let \Sigma loss of generality, assume that \Sigma"\Sigma string S corresponds to a strategy on 6 cw$, where is of the and to be the set of strings of the form oe each oe i is in the alphabet \Sigma, each S i is in the alphabet \Sigma 0 , and furthermore, corresponds to a strategy of M on input causes w to be accepted. Then, L 0 is accepted by a 2pfa with bounded error probability in polynomial time. Thus, L 0 is regular [7]. Moreover, note that a string of the form is in L if and only if for some choice of S 0 is in L 0 . Let M 0 be a one-way deterministic finite state automaton for L 0 , and assume without loss of generality that the set of states in which M 0 can be when the head is at an even position, is disjoint from the set of states in which M 0 can be when the head is at an odd position. Then, from M 0 we can construct a one-way nondeterministic finite state automaton for L, by replacing the even position states by nondeterministic states. Hence, L is regular. 2 6 Conclusions We have introduced a new measure of the complexity of a language, namely its tiling complexity, and have proved a gap between the tiling complexity of regular and nonregular languages. We have applied these results to prove limits on the power of finite state automata with both probabilistic and nondeterministic states. An intriguing question left open by this work is whether the class 2NPFA-polytime is closed under complement. If it is, we can conclude that 2NPFA-polytime = Regular. Recall that the class 2NPFA does contain nonregular languages, since it contains the class 2PFA, and Freivalds [10] showed that f0 n 1 is in this class. However, Kaneps [18] showed that the class 2PFA does not contain any nonregular unary language. Another open question is whether the class 2NPFA contains any nonregular unary language. It is also open whether there is a nonregular language in 2APFA-polytime. There are several other interesting open problems. Can one obtain a better lower bound on the tiling complexity of nonregular languages than that given by Theorem 4.5, perhaps by an argument that is not based on rank? We know of no nonregular language with tiling complexity less n) infinitely often, so the current gap is wide. --R On notions of information transfer in VLSI circuits Proof verification and hardness of approximation problems Computational Models of Games On the Power of finite automata with both nondeterministic and probabilistic states Probabilistic game automata Finite state verifiers I: the power of interaction Interactive proof systems and alternating time-space complexity Probabilistic two-way machines A lower bound for probabilistic algorithms for finite state machines On different modes of communication Introduction to Automata Theory Hankel and Toeplitz Matrices and Forms: Algebraic Theory Minimal nontrivial space complexity of probabilistic one-way Turing machines Running Time to Recognize Nonregular Languages by 2- Way Probabilistic Automata Regularity of one-letter languages acceptable by 2-way finite probabilistic au- tomata Some bounds on the storage requirements of sequential machines and Turing machines The Markov chain tree theorem Las Vegas is better than determinism in VLSI and distributed computing Contentment in Graph Theory: Covering Graphs with Cliques. Games against nature Probabilistic automata Finite automata and their decision problems Succinctness of description of context free Automaticity: properties of a measure of descriptional complexity Some complexity questions related to distributed computing Lower bounds by probabilistic arguments --TR --CTR Lutz Schrder , Paulo Mateus, Universal aspects of probabilistic automata, Mathematical Structures in Computer Science, v.12 n.4, p.481-512, August 2002 Bala Ravikumar, On some variations of two-way probabilistic finite automata models, Theoretical Computer Science, v.376 n.1-2, p.127-136, May, 2007

nondeterministic probabilistic finite automata;interactive proof systems;matrix tiling;hankel matrices;arthur-merlin games

285057

Formal verification of complex coherence protocols using symbolic state models.

Directory-based coherence protocols in shared-memory multiprocessors are so complex that verification techniques based on automated procedures are required to establish their correctness. State enumeration approaches are well-suited to the verification of cache protocols but they face the problem of state space explosion, leading to unacceptable verification time and memory consumption even for small system configurations. One way to manage this complexity and make the verification feasible is to map the system model to verify onto a symbolic state model (SSM). Since the number of symbolic states is considerably less than the number of system states, an exhaustive state search becomes possible, even for large-scale sytems and complex protocols.In this paper, we develop the concepts and notations to verifiy some properties of a directory-based protocol designed for non-FIFO interconnection networks. We compare the verification of the protocol with SSM and with the Stanford Mur 4 , a verification tool enumerating system states. We show that SSM is much more efficient in terms of verification time and memory consumption and therefore holds that promise of verifying much more complex protocols. A unique feature of SSM is that it verifies protocols for any system size and therefore provides reliable verification results in one run of the tool.

Introduction Caching data close to the processor dynamically is an important technique for reducing the latency of memory accesses in a shared-memory multiprocessor system. Because multiple copies of the same memory block may exist, a cache coherence protocol often maintains coherence among all data copies [29]. In large-scale systems directory-based protocols [4, 5, 19] remain the solution of choice: They do not rely on efficient broadcast mechanisms, and moreover they can be optimized and adapt to various sharing patterns. The current trend is towards more complex protocols, usually implemented in software on a protocol processor [18]. Because of this flexibility, proposals even exist to let users define their own protocol on a per-application basis [28]. One major problem is to prove that a protocol is correct. When several coherence transactions bearing on the same block are initiated at the same time by different processors, messages may enter a race condition, from which the protocol behavior is often hard to predict [2] because the protocol designer can not visualize all possible temporal interleavings of coherence messages. Automated procedures for verifying a protocol are therefore highly desirable. There are several approaches to verify properties of cache protocols. A recent paper surveys these approaches [24]. One important class of verification techniques derives from state enumeration methods (reachability or perturbation analysis), which explore all possible system states [7, 15]. Generally, the method starts with a system model in which finite state machines specify the behavior of components in the protocol. A global state is the composition of the states of all components. A state expansion process starts in a given initial state and exercises all possible transitions leading to a number of new states. The same process is applied repeatedly to every new state until no new state is generated. At the end, a global state transition diagram or a reachability graph showing the transition relations among global states is reported. The major drawback of state enumeration approaches is that the size of the system state space increases quickly with the number and complexity of the components in the protocol, often creating a state space explosion problem [15]. Verifying a system with increasing numbers of caches becomes rapidly impractical in terms of computation time and memory requirement. As protocols become more complex, it is not clear whether verifying a small-scale system model can provide a reliable error coverage for all system sizes [25]. Recently, we have introduced a new approach called SSM (Symbolic State Model) to address the state space explosion problem [23] and we have applied it to simple snoopy protocols on a single bus [2]. SSM is a general framework for the verification of systems composed of homogeneous, communicating finite state machines and thus is applicable to the verification of cache protocols in homogeneous shared-memory multiprocessors. SSM takes advantage of equivalences among global states. More precisely, with respect to the properties to verify such as data consistency, SSM exploits an abstract correspondence relation among global states so that a met- 3astate can represent a very large set of states [23,]. Based on the observation that the behavior of all caches are characterized by the same finite state machine, caches in the same state are combined into an class; a global state is then composed of classes. Moreover, the number of caches in a state class is abstractly represented by a set of repetition constructors indicating 0, 1, or multiple instances of caches in that class. An abstracted global state represents a family of global states and can be efficiently expanded because expanding an abstracted state is equivalent to expanding a very large set of states. SSM verifies properties of a protocol for any system size and therefore the verification is more reliable than verification relying on state enumeration for small system sizes. We have developed a tool to apply this new approach. To illustrate its application in a concrete case, we verify in this paper three important coherence properties of a protocol designed for non-FIFO interconnection networks. A non-FIFO network is a network in which messages between two nodes can be received in a different order than they are sent. Therefore, the number of possible races among coherence messages is much larger than in a system with a FIFO net- work. To demonstrate the efficiency of our tool, we compare it with Murj [8]. We show that SSM is much more efficient in terms of verification time and memory consumption and therefore holds the promise of verifying much more complex protocols. The paper is structured as follows. Section 2 provides an outline of the protocol for non-FIFO networks. Verification model, correctness issues and mechanisms for detecting various types of errors are discussed in section 3. We then develop the verification method in sections 4 and 5. Results of our study are in section 6. Section 7 contains the conclusion. Directory-Based Protocol for Non-FIFO Networks The protocol is inspired from Censier and Feautrier's write-invalidate protocol [4]. Every memory block is associated with a directory entry containing a vector of presence bits, each of which indicates whether a cache has a copy of the block. The presence bit is set when the copy is first loaded in cache and is reset when the copy is invalidated or replaced. When multiple copies exist in different caches, they must be identical to the memory copy. An extra dirty bit per block in the directory entry indicates whether or not a dirty cached copy exists. In this case, there cannot be more than one cached copy and we say that the copy is Exclusive. The cache with the exclusive copy is also often called the Owner of the line. To enforce ownership of the block, invalidations must be sent to caches with their presence bits set. Finally the replacement of a Shared copy is silent in the sense that the presence bit is not reset at the memory. This protocol is applicable in general to CC-NUMAs (Cache-Coherent Non-Uniform Memory Access machines). In a CC-NUMA the shared memory is distributed equally among processor nodes and is cached in the private cache of each processor. In this case, a directory is attached to each memory partition and covers the memory blocks allocated to it. Thus each block has a Home memory where its directory entry resides. The implementation of this (conceptually) simple protocol requires careful synchronizations between caches and directory, involving many cache states, memory states and messages between caches and memory. A complete specification of the design can be found in [26]. The coherence messages exchanged between memory and caches are given in table 1. Messages are basically of two types: control and data. Control messages include requests and acknowledgments. These messages and their role are self-explanatory, except for SAck, a synchronization message whose role will become clear later. In the following, we only describe the salient features of the protocol used in the verifica- tion. To simplify the following description we refer to the "state of the block in the cache" as the "state of the cache". The same convention applies to the state of the block in the directory and in other state machines throughout the paper. 2.1 Cache States Caches can be in three stable states: Invalid (I), Shared (S; clean copy potentially shared with other caches), and Owner (O; modified and only cached copy-also called Exclu- sive). However, since cache state transitions are not instantaneous, three transient states are added to keep track of requests issued by the cache but not yet completed. 1. Read-Miss-Pending (RMP) state: the block frame is empty pending the reception of the block after a read miss. 2. Write-Miss-Pending (WMP) state: the block frame is empty pending the reception of the block with ownership after a write miss. 1. Coherence Messages Type Message Action Memory To Inv Request to invalidate the local copy. InvO Request to invalidate the local copy and write it back to memory. UpdM Request to update the main memory copy and change the local copy to the Shared state. O-ship Ownership grant. Data Block copy supplied by the memory controller. NAck Negative acknowledgment indicating that a request was rejected because of a locked directory entry. Cache To ReqSC Request a Shared copy. ReqO Request Ownership. ReqOC Request Ownership and block copy. DxM Block copy supplied by an owner in response to an UpdM message from memory. DOxMR Block copy supplied by an owner after replacement. DOxMU Block copy supplied by an owner in response to an InvO message from memory. IAck Acknowledgment indicating invalidation complete. SAck Synchronization message. 3. Write-Hit-Pending (WHP) state: the block frame contains a shared copy pending the reception of ownership rights to complete a write access. These states are sufficient in a system with a FIFO network. With a non-FIFO network, possible races exist between coherence requests sent at the same time by two different caches to the same block. Such requests are serialized by the home node but the responses they generate may enter a race and reach caches out of order. Consider the following case. Assume that two processing nodes p 1 and p 2 issue both a request for an exclusive copy of a block at the same time. The request from p 1 reaches the home node first and is granted the copy. Then the request from p 2 is processed by the home and invalidations are sent to p 1 . In a non-FIFO network, it is possible that the invalidation will reach p 1 before the exclusive copy. A similar scenario can occur if p 1 requests a shared copy at the same time as p 2 wants an exclusive copy or if p 1 requests an exclusive copy while p 2 wants a shared copy. To deal with these three race conditions the protocol uses three additional transient cache states which synchronize the interactions between caches and memory. These states are: Transient Owner-to-Invalid (TxOI), Transient Shared-to-Invalid (TxSI), and Transient Owner-to-Shared (TxOS). To resolve the race between two processing nodes requesting an exclusive copy, a cache in state WMP moves to state TxOI when it receives an invalidation so that, when it receives the data block, it executes its pending write, writes the block back to memory and invalidates its copy to end up in state I. States TxSI and TxOS solve the other two races in a similar fashion. 2.2 Memory States The stable memory states are indicated by the presence and the dirty bits in the directory and are Shared, Exclusive or Uncached. When a memory block is in a stable state it is free or unlocked, meaning that the memory controller may accept new requests for the block. However memory state transitions are not instantaneous. Between the time the directory controller starts processing an incoming request and the time it considers the request completed, the directory entry is in a transient state and is locked to maintain a semi-critical section on each memory block [30]. Requests reaching a locked (busy) directory entry are nacked. The three corresponding transient memory states are: XData, XOwn, XOwnC, which indicate that the transaction in progress is for a shared copy, for ownership rights or for an exclusive copy (respectively), and are typical in systems with FIFO networks. The protocol is based on intervention forwarding: When a processing node requests a copy and the block is exclusive in a remote cache, the home node first requests the copy of the dirty block, updates the memory, and forwards it to the requester. When an owner victimizes its modified copy for replacement, the memory state remains Exclusive until the write-back message reaches the memory controller. Between the transmission and the reception of the write-back message the memory controller may receive a request for a shared copy issued by another cache and forward it to the owner. When the memory controller receives the block copy sent at the time of replacement, it will "believe" that the block copy was sent in response to its forwarded request; meanwhile, the forwarded request is still pending. This problem was also identified in [2]. The solution suggested in [2] counts on the presumed owner to ignore the forwarded request. How- ever, in a large-scale system with unpredictable network delays, intractable problems can be caused by the forwarded request if it is further outpaced by other messages [25]. 1. Directory state transitions to synchronize owner and memory. To solve the problem of ambiguous write-back messages, we use different message IDs for a cache write back caused by a replacement (DOxMR) or by an invalidation (DOxMU) (see table 1). Moreover we add two transient states to the directory: Synch1 and Synch2. The memory controller unlocks the directory entry only when it has received both the replaced data block and the synchronization message. For example, when the memory controller receives a request for a shared copy (ReqSC), the request is forwarded to the owner and the memory state is changed to XData. If the presumed owner has written back the block, it replies with a synchronization message when it receives the request forwarded by the memory. If the memory controller receives a block message of type DOxMR or a synchronization message (SAck) from the owner, the directory enters the transient state Synch2 and waits for the synchronization message (SAck) or for the write-back message (DOxMR) from the owner respectively. State Synch1 takes care of the similar case when the original request message was ReqOC, as shown in figure 1. 3 Protocol Model and Correctness Properties 3.1 System Model The first step in any verification is to construct a system model with manageable verification complexity. The model should leave out the details which are peculiar to an implementation, while retaining the features essential to the properties to verify. In the early stage of protocol design, this approach also facilitates the rapid design and modifications of the model. In this section we describe the system model used in the verification of the protocol. First, a single block is modeled, which is sufficient to check properties related to cache coherence [20]. Replacements can take place at any time and are modeled as processor accesses. Second, we abstract the directory-based CC-NUMA architecture by the system model of figure 2.a, which is appropriate since we model a single block. The model consists of a directory XData ReqOC DOxMU SAck,or SAck,or SAck,or SAck,or ReqSC DOxMR DOxMR DOxMR DOxMR and of multiple processor-cache pairs. Each processor is associated with one message sending channel (CH!) and one message receiving channel (CH?) to model the message flow between caches and main memory. The message channels do not preserve the execution order of memory accesses (in order to model non-FIFO interconnections). Messages are never lost but they may be received in a different order than they were issued. When the cache protocol does not treat differently messages from the local processor and from remote processors, the model of figure 2.a is equivalent to the model of figure 2.b, in which the home memory is modeled as an independent active entity. We will use the system model of figure 2.b throughout this paper. 2. Verification model for directory-based CC-NUMA architectures. Third, values of data copies are tracked by the same abstraction as we first proposed in [22]. A cache may have a data block in one of three status: nodata (the cache has no valid copy), fresh (the cache has an up-to-date copy), and obsolete (the cache has an out-of-date copy); the CH? CH! Home Memory & Directory (full-map CH? CH! CH? CH! base machine CH? CH! Home Memory CH? CH! CH? CH! home node (a) Abstract Model for a CC-NUMA Multiprocessors (single block) (b) Refined Model. protocol machine memory copy is either fresh or obsolete. During the course of verification, the expansion process keeps track of the status of all block copies in conformance with the protocol semantics. This third abstraction is necessary, as we discovered in the verification of the S3.mp protocol [21, 25] (which is different from the protocol used in this paper). Consider the protocol transactions illustrated in figure 3. Initially, cache A has a dirty copy of the block, replaces it, and performs a write-back to the home node. Cache A keeps a valid copy of the block until it receives an acknowledgment from the memory in order to guarantee that the memory receives the block safely. Meanwhile, cache B sends a request for an exclusive copy to home. Subsequently, cache A processes the data-forward-request from home, considers it as an acknowledgment for the prior write-back request and sends the block to B. B then executes its write and replaces the block due to some other miss. As shown in figure 3, a race condition exists between the two write-back requests. If the write-back from B wins the race, the stale write-back from A overwrites the values updated by B. Note that in this example all state transitions are permissible. To overcome this problem, the verification model need to maintain a global variable to remember which write-back carries the latest value. 3. A Stale Write-back Error. 3.2 Formal Protocol Model Given the architectural model of figure 2, we now formally define the constituent finite state machines interacting in the protocol. A convenient language to specify such machines is CSP [14]. Message transmission is represented by the postfix '!', and message reception by the postfix '?'. Definition 1. (Receiving Channel) The receiving channel machine recording the messages received from the memory and in transit to the cache has structure RChM= (Q r , S r , Xm !, d1 r , Xc ?, set of memory-to-cache messages (table 1), (messages from memory), Home A 1. write-back 2. exclusive-request 3. data-forward-request 4. exclusive-forward 5. write-back (messages to cache), are the messages issued by the memory controller and consumed by the cache, respectively. Definition 2. (Sending Channel) The sending channel machine recording the messages issued by the cache and in transit to the memory controller has structure SChM= (Q s , S s , Xc !, d1 s , Xm ?, d2 s ), where set of cache-to-memory messages (table 1), ,(messages from cache), (messages to memory), are the messages issued by the cache and consumed by the memory, respectively. The state of a channel machine is made of all the messages in transit. At each state expansion step, a receiving (or sending) channel may record the command sent by the memory (or its cache), or may propagate a command to its cache (or the memory). The behavior of each cache controller is given in definition 3. Definition 3. (Cache Machine) The state machine characterizing the cache behavior has structure CM= (Q c , S r , S s , Xc ?, d1 c , Xc !, d2 c ), where coherence messages as defined in definitions 1 and 2, Xc ? and Xc ! are the messages consumed and produced by the cache, respectively. Upon receiving a message, a cache controller may or may not respond by generating response messages according to d1 c . Additionally, we embed the processor machine in the cache machine. The processor may issue accesses to its local cache, which may cause cache state changes and issuance of coherence messages as specified by d2 c . The finite state machines for main memory and for the protocol are formalized as follows. Definition 4. (Memory-Directory) The main memory machine keeping the directory has structure messages as defined in definitions 1 and 2, Xm ? and Xm ! are the caches-to-memory and memory-to-caches commands respectively. When the memory machine consumes a message, response messages may or may not be sent to caches. Q BM denotes the set of possible states of a base machine as defined below. Definition 5. (Base Machine) The base machine is the composition of the cache machine and of its two corresponding channel machines, that is, BM Definition 6. (Protocol Machine) The protocol machine is defined as the composition of all base machines and of the memory machine, that is, PM= with n caches. The state tables used in the verification for d1 c , d2 c , and d m can be found in [26]. The memory controller consumes messages from caches and responds according to the block state and the message type. Finally, the state of the protocol machine is also referred to as the global state in this paper. 3.3 Correctness Properties of the Protocol In this paper we verify three properties: data consistency, incomplete protocol specification and livelock, with the following definitions. 3.3.1 Data Consistency The basic condition for cache coherence was given in [4]: All loads must always return the value which was updated by the latest store with the same address. We formulate this condition within the framework of the reachability expansion as follows. Definition 7. (Data Consistency) With respect to a particular memory location, the protocol preserves data consistency if and only if the following condition is always true during the reachability analysis: the family of global states originated from G', including G' itself, consistently return on a load the value written by a STORE access t which writes the most recent value to the memory location and brings a global state G to G' or the value written by STORE transitions after t. That is, states reached by expanding G' are not allowed to access the old value defined before t. In the architectural model of figure 2 memory accesses are made of several consecutive events and thus are not atomic. We do not constrain in any way the sequences of access generated by processors. Moreover the hardware does not distinguish between synchronization instructions and regular load/store instructions. So, in this paper, latency tolerance mechanisms in the processors and in the caches are not modeled and we assume that the mechanisms are correct and enforce proper sequencing and ordering of memory accesses in cooperation with the software. Based on the model of data values in section 3.1, data inconsistency is detected when a processor is allowed to read data with obsolete values. Definition 8. (Detection of Data Inconsistency) All data copies are tagged with values in the set {nodata, fresh, obsolete} and data transfers are emulated in the expansion. Data inconsistency is detected when a processor is allowed to read data with obsolete values. 3.3.2 Incomplete Protocol Specification Because of unforeseen interleavings of events, the protocol specification is often incom- plete, especially in the early phases of its development. This flaw manifests itself as an unspecified message reception, i.e., some entity in the protocol receives a message which is unexpected given its current state [31]. State machine models are very effective at detecting unspecified message reception. The procedure is simple and is directly tied to the structure of the reachability graph: an unspecified message reception is detected when the system is in a state and a message is received for which no transition out of the state is specified in the protocol description. Besides detecting the error, the state enumeration shows the path leading to the erroneous state. 3.3.3 Deadlock and Livelock A protocol is deadlocked when it enters a global state without possible exit. In a livelock situation the processes interacting in the protocol could theoretically make progress but are trapped in a loop of states (e.g, a processor keeps on re-trying a request which is always rejected by another processor). Deadlocks are easy to detect in a state enumeration since they are states without exits to other states, but it is very difficult to detect livelocks. At the level of abstraction adopted in this paper, protocol components communicate via messages. Thus we can only detect deadlocks and livelocks derived from the services (functional- ity) provided by the cache coherence protocol. An example of a protocol-intrinsic livelock is a blocked processor waiting for a message (e.g., an invalidation acknowledgment) which is never sent by another processor in the protocol specification. Deadlock and livelock conditions due to a particular implementation of the protocol such as finite message buffers or the fairness of serving memory requests cannot be detected at this level of abstraction. Definition 9. (Livelock) In the context of coherence protocols, a livelock is a condition in which a given block is locked by some processor(s) so that some processor is permanently prevented from accessing the block [20]. In our state expansion process, we check the following conditions in order to detect livelocks and correct the protocol: Conditions for Livelock-freeness: (a) The protocol can visit every state in the global state transition diagram infinitely many times, that is, the global state transition diagram is strongly-connected. Given a global state, every other state in the global state transition diagram is reachable [2]. (b) If a processor issues a memory access to a block, this memory access must eventually be satisfied (e.g., a value is always returned on a load to resume processor execution). Specifi- cally, given an initial global state in which a cache is in an "invalid" state, there must exist reachable global states in which the cache state becomes "shared" or "dirty" after a read miss or a access [20]. Conditions (a) and (b) are sufficient to avoid livelocks as defined in definition 9 because they assure that every processor can read and modify a block an arbitrary number of times. Condition (a) is stronger than necessary because it assumes that the cache protocol operates in steady state. A cache protocol machine might start from an initial state and never return to it later. In this case, the global state graph would comprise two sub-graphs: One sub-graph consisting of the initialization state would have exits to a second sub-graph corresponding to the steady state operation of the protocol, but not vice versa. This special case can be identified by careful analysis of the state graph after a livelock is reported. 4 The Verification Method In the models of figure 2 the order of the states of base machines in a global state representation is irrelevant to protocol correctness. Because of this symmetry, the size of state space can be reduced by a factor n!, given a system with n processors. The symbolic state model (SSM) exploits more powerful abstraction relations than the symmetric relation in order to further reduce the size of the state space. To be reliable, the new abstraction must be equivalent to the system model with respect to the properties to verify. 4.1 Equivalence Between State Transition Systems In general, we formalize the system to verify as a finite state transition system: Definition 10. (Finite State Transition System) With respect to a cache block, the behavior of a cache system with m local cache automata is modeled by a finite state transition system M:(s 0 , A, s 0 is the initial state , A is the set of state symbols, S is the global state space (a subset of ), A A - . A A - . S is the set of operations causing state transitions, and d represents the state transition function, . Consider a state transition system M: (s 0 , A, S, S, d). With respect to the properties P to verify, we want to find a more abstract state transition system corresponds to M, S r is smaller than S, and error states of M are mapped into error states of M r . Definition 11. (Correspondence) Given two state transition systems M: (s 0 , A, S, S, d) and M r : corresponds to M if there exists a correspondence relation j such that: 1. s 0r corresponds to s 0 , i.e., s 0r js 0 , 2. For each , at least one state corresponds to s, i.e., s r js. 3. If M in state s makes a transition to state t on an enabled operation t, and state s r of M r corresponds to state s, then there exists a state t r such that M r moves from s r to t r by t and t r corresponds to t. Figure 4 illustrates this correspondence relation. 4. Correspondence Relation. Definition 12. (Equivalence) Two state transition systems M and M r such that M r corresponds to are equivalent with respect to a property P to verify iff the following conditions are verified at any step during the expansion of M r . Let s r be the current (correct) state of M r . and let t r be the next state of M r after a transition t. 1. If P is verified in t r then P holds in all states t of M such that t r jt. 2. If P does not hold in t for some t such that t r jt then P does not hold in t r 3. If P does not hold in t r then there must exist states s and t such that s r js and t r jt and P does not hold in t. The first condition of definition 12 establishes that if the expansion of M r completes without violating property P, then the expansion of M would also complete without violating P. The second and third conditions of definition 12 ensure that an error state is discovered in the expansion of M r iff an error state exists in M and the error state of M r corresponds to the error state of M. In the following we first specify an abstract machine M r corresponding to the protocol machine M of definition 6. We will then prove that M r is equivalent to M with respect to the cor- rectness properties of section 3.3 4.2 Abstract SSM Models with Atomic Memory Accesses The SSM method was first introduced in [22] under the assumption of atomic memory accesses. We developed an abstraction relation among global states based on the observation that, in order to model cache protocols, the state must keep track of whether there exists 0, 1, or multiple copies in the Exclusive state which has the latest copy of the data. On the other hand, the number of read-only shared data copies do not affect protocol behavior, provided there is at least one cached copy. Symbolic states can be represented by using repetition constructors. Definition 13. (Repetition Constructors-Atomic Memory Accesses) 1. The Null (0) specifies zero instance. 2. The Singleton (1) specifies one and only one instance. This constructor can be omitted in the state representation. 3. The Plus (+) specifies one or multiple instances. 4. The Star (*) specifies zero, one or multiple instances. With these repetition constructors we can represent for example the set of global states such that "one or multiple caches are in the Invalid state, and zero, one or multiple caches are in the Shared state" as metastate such as corresponds to a set of explicit states in M. 5. Ordering Relations among Repetition Constructors. Repetition constructors are ordered by the sets of states they represent. Thus, 1 < (figure 5). These ordering relations extend to the metastates (called composite states in [22]) such that, for example, contained by because the set of global states represented by the first composite state is a subset of those represented by the second composite state. Because of this containment relation among composite states, only the composite states which are not contained by any other composite state are kept during the verification. At the end of the state expansion, the state space of M is collectively represented by a relatively small number of essential composite states in M r [23]. 4.3 SSM Models with Non-atomic Memory Accesses To model protocols with non-atomic accesses, we need to define the elements forming the basis for the repetition abstraction and to add a new repetition constructor called the Universe Constructor. In the model of figure 2, base machines naturally form the units of abstraction repetition. Henceforth, a set of base machines in the same state will be represented by , in which C is the cache state, p is the value of the presence bit in the directory, r is the number of base machines in the set (specified by one of the repetition constructors above), R is the state of the receiving channel, and S is the state of the sending channel. R and S are specified by the messages in transit in the channels. Since the channels model non-FIFO networks, the order of messages in each channel is irrelevant. Often, when there is no confusion, part of the notation may be omitted. For example, we will use the notation , where q combines the cache state with the states of its two message channels. Although the singleton, the plus, and the star are useful to represent an unspecified number of instances of a given construct (such as base machines in a given global state), they are not precise enough to model intermediate states in complex protocol transactions triggered by event counting. Consider an abstract state (S * , I + ). When a write miss occurs, all caches in shared state S must be invalidated and the ultimate state is (O, I + ) in which the processor in state O has the exclusive dirty copy. At the behavioral level [23], this state transition is done in one step because memory accesses are assumed atomic. However, when accesses are no longer atomic, invalidations are sent to caches in state S and the number of shared copies is counted down one-by-one upon receiving invalidation acknowledgments. As a result, we need to distinguish the two states contained by metastate (S * , I these two states correspond to the cases where either some or no caches are in state S. To deal with this problem, we first define the inval- idation-set: Definition 14. (Invalidation-Set) The invalidation-set (Inv-Set) contains the set of caches with their presence bits set and which must be invalidated before the memory grants an exclusive copy. When a request for an exclusive copy (such as request-for-ownership ReqO or request-for- owner-copy ReqOC in our protocol) is pending at the memory, copies must be invalidated and the state expansion process needs to keep track of whether the invalidation-set is empty. Since all caches in the same state are specified by repetition constructors, the exact number of caches in a particular state is unknown and using the * constructor alone to represent any number of copies may prevent the expansion of some possible states. Consider the following composite state with the invalidation-set between brack- ets: and where Q denotes all other base machines with their presence bits reset. 6. Expansion Steps with Null and Non-null Instances Covered by the * Constructor. When the memory receives the request for an exclusive copy (ReqOC) from the cache in state C, it cannot determine whether the invalidation-set is empty because the definition of * includes the cases of null and non-null instances. One way to solve this shortcoming in the notation is to explore both cases in the expansion process. When the global state is expanded, two states, corresponding to an empty and to a non-empty invalidation-set are generated. The expansion steps are shown in figure 6. expansion step is q 1 * ), which means that some machines in state q 1 change state to q 2 and others remain in q 1 . with C1 C2 C f ReqOC f ReqOC Data f s3: QC1 f IAck f IAck s4: QC2- Data f f IAck if caches in C1 do not acknowledge invalidation requests (C1 changes to C1') the memory receives IAcks from caches in C2' caches in C2 respond to Inv (C2 changes to C2') the memory receives IAcks from caches in C2' 1. In s0, suppose that memory receives a request for an exclusive copy from the cache in state C. Two states corresponding to an empty and to a non-empty invalidation-set are generated. In s2, invalidations are sent to caches in the invalidation-set, whereas in s1, the requester obtains an exclusive copy (the new owner) and the invalidation-set is empty. 2. In the expansion of s2, caches in state C2 receive invalidations, respond with an invalidation acknowledgment, and change state to C2'. 3. When the memory receives invalidation acknowledgments from caches in state C2' in s3, two states with an empty and a non-empty invalidation-sets are again generated. 4. In global state s5, assume that caches in state C1 do not acknowledge invalidations because of an incorrect design. In s6, when the acknowledgment messages from caches in state C2' are received by the memory, the expansion may consider the invalidation-set as empty again and make a transition to s4. However, the case where the invalidation-set is not empty is also covered by * and must also be expanded. Either the process never stops or some errors go undetected In order to solve this problem, the expansion process needs to remember which expansion path it followed. In figure 6, transitions (s0 - s1) and (s0 - s2) correspond to an empty and a non-empty invalidation-set respectively. However, the invalidation-set in s2 should in fact cover only three cases: (1) , (2), and (3) . Unfortu- nately, splitting states such as s2 results in a combinatorial explosion of the state space. A more efficient solution is to work on state s2 and keep track of whether the invalidation-set is empty. To this end, we introduce a new constructor called the universe constructor or u construc- tor. When a transition is applied to a non-empty invalidation-set of the form the null case is not generated. Rather the components inside the invalidation-set are expanded one by one without considering the null case. To keep track of the fact that we have expanded a component at least once without considering the null case, we use the u constructor. The component expansion is of the type . The u constructor is similar to * except that the transition to the null case can now be exercised in the expansion of the inval- idation-set. An invalidation-set may be considered empty if and only if it has the form . Let's examine the expansion steps using the u constructor to see how the procedure works (figure 7). 1. In global state s1, the expansion process explores the path in which some caches respond to invalidations. In global state s2, the constructor * is replaced by u for the class of caches * . x n . q 2 . q remaining in C2, so that the next time it is expanded, the expansion process will consider the null case. 7. Resolution Provided by the u Constructor. 2. In global state s3, the expansion process chooses to expand the class of caches in state C1, considering only the case of the non-empty set. If caches in state C1 do not acknowledge invalida- tions, the process moves into state s4. According to the condition for emptiness of the invalidation-set, the pending request for an exclusive copy in state s4 is never resumed (because of the caches in class C1'). This situation can be easily detected as a livelock situation in which the protocol is trapped in a loop. s2: QC1 f IAck s3: QC1 f IAck f IAck f IAck C2 f IAck f IAck C2 f IAck Data f if caches in C1 don't acknowledge Inv if caches in C1 acknowledge Inv (C1 changes to C1') caches in C2 respond to Inv (C2 changes to C2') the memory receives IAcks from caches in C2' the memory receives IAcks from caches in C1' the memory receives IAcks from caches in C1' or C2' pending request will f ReqOC never be resumed 3. On the other hand, if all caches in the invalidation-set acknowledge memory, the expansion process takes another path through states s4', s5 and s6. The only occurrence of event counting in write-invalidate protocols is the collection of acknowledgments for invalidations. In write-update protocols updates must be acknowledged in the same way: the equivalent of the invalidation-set could be called the update-set. 4.4 Symbolic State Model Combining the basic framework of section 4.2 and the refinement of section 4.3, in a system with an unspecified number of caches, we group base machines in the same state into state classes and specify their number in each class by one of the following repetition constructors. Definition 15. (Enhanced Set of Repetition Constructors) 1. The Null (0) specifies zero instance. 2. The Singleton (1) specifies one and only one instance. This constructor can be omitted in the state representation. 3. The Plus (+) specifies one or multiple instances. 4. The Star (*) specifies zero, one or multiple instances. However, the case of "zero instance" is not explored for transactions depending on event counting in the expansion. 5. The Universe (u) specifies zero, one or multiple instances. The case of "zero instance" is explored for transactions depending on event counting in the expansion. Definition 16. (Composite State) A composite state represents the state of the protocol machine for a system with an arbitrary number of base machines. It is constructed over state classes of the form , where n=|Q BM | is the number of states of a base machine, q i -Q BM , r i -[0, 1, +, *, u] and q MM -Q m is the memory machine state. Repetition constructors are again ordered by the possible states they specify; namely, 1 < 8). This order leads to the definition of state containment. 8. Ordering Relations Among the Repetition Constructors (Enhanced Set). . q n Definition 17. (Containment) We say that composite state S 2 contains composite state S 1 , or S 1 and q The consequence of containment is that, if then the family of states represented by S 2 is a superset of the family of states represented by S 1. Therefore, S 1 can be discarded during the verification process provided we We will prove that the expansion process based on the expansion rules of section 4.4.1 is a monotonic operator on the set of composite states S, that is, is a memory event applied to S 1 and S 2 . 4.4.1 Rules for the Expansion Process The set of operators applicable to composite states during the state generation process is defined as follows, where '/' stands for ``or'' and `-' shows a state transition. 1. Aggregation: (q 0 , q r 2. Coincident Transition: q 1 r , where r -[1, +, *, u] and - t is an observed transition. 3. One-step Transition: (a) (Q, q 1 (b) (Q, q 1 where all machines not in state q 1 are denoted by Q in the tuple, - t is a transition applied to the base machine in state q 1 such that q 1 - t q 2 , and t causes all other base machines in q 1 to move to state q 3 . After the transition, some machines in Q may be affected as shown by the change from Q to Q'. 4. N-steps Transitions: This rule specifies the repetitive application of the same transition N times, where N is an arbitrary positive integer. (a) (Q, q 1 , q MM (b) (Q, q 1 (c) (Q, q 1 5. Progress Transitions: Provided q 1 - t q 2 , we have * , q MM ), and $ such that q r 1 * , q MM '). where the states between bracket form the invalidation-set and base machines not in the invalida- tion-set are denoted by Q in the tuples. Aggregation rules group base machines in the same state. One example of a coincident transition is when the memory controller sends an invalidation signal to every cache with a valid copy. A one-step transition occurs for example when the memory receives a request for an exclusive copy from a base machine in class q 1 ; this base machine changes its state to q 2 (the request message is removed from the sending channel) because the memory normally processes only one request for an exclusive copy at a time; in this case all other machines in q 1 and in Q may stay in the same state or may change state because new invalidation messages are sent to their receiving channel. Rules (b) and (c) of an N-steps transition correspond to two chains of transitions: and The same transition q 1 - t q 2 can be applied an unlimited number of times as long as there are base machines in state q 1 . The transition - t has no effect on other machines (denoted by Q in the tuple). Typical examples are: (1) processors replacing their copy in state shared, (2) processors receiving the same type of messages, and (3) processors issuing the same memory access independently Two additional rules with similar interpretation as N-steps transitions are required for the progress of the expansion process. These progress transitions deal with protocol transactions involving event counting as explained in section 4.3 and correspond to two chains of transitions: * , q MM ), and * , q MM '). In our protocol, they model the processing of a request for an exclusive copy at the memory. Transition - t is the reception of an invalidation acknowledgment (such as IAck in table 1). Inv-Set is the set of caches with their presence bits set at the memory and which must be invalidated before the memory grants an exclusive copy. Rule (a) applies during the invalidation process whereas rule (b) applies after the successful invalidation of all copies. During the state expansion process, all cache transactions possible in the current state are explored. A state expansion step has two phases. First, a new composite state is produced by applying one of the above transition rules to the current state. Second, the aggregation rule is applied to lump base machines in the same state (see for example figure 11). 4.5 Monotonicity of the State Expansion In general, a system to verify is composed of finite state machines so that one machine can communicate with all other machines directly and a composite state of any SSM is of the form are the possible states of each machine and r i s are repetition constructors. A partial order exists among repetition constructors such as the one in figure 8. State expansion rules include aggregation rules, one-step transition rules and compound transition rules corresponding to multiple applications of the same one-step transition rule. Aggregation rules are the rules used to represent symbolic states as compactly as possible, based on the partial order on the repetition constructors of the abstract state representation. Containment of composite states is based on the partial order among constructors. In this context, we can prove that the expansion rules in SSM are monotonic operators: namely, Intuitively, if an SSM state S 1 is contained by S 2 and if an expansion step is done correctly, then the next states of all the states included in S 1 must also be contained in S 2 . This expansion and containment of the abstract states in SSM are independent of the properties to verify. Properties such as data consistency (see definition 7) are formulated by users, and then are checked on the reduced state space. Lemma 1. The aggregation process is monotonic, that is, if and , then we have , where q is a possible state of each state machine and all r i are repetition constructors. Proof: The proof follows from the ordering relation among the repetition constructors and from checking all possible combinations of r 11 , r 12 , r 21 and r 22 subject to the constraints of this lemma and to the aggregation rule. q Lemma 2. The immediate successor S 1 originated from state is contained by state S 2 originated from state if the same expansion rule taken on the same memory event t is . q n q r 21 . q n . q n applied to S 1 and S 2 . Proof: We only need to consider the effect of applying t to machines in state q i in S 1 and S 2 . To simplify the notation, all classes q j (j - i) are lumped in Q. Provided q i - t q k , the following two states are generated when a one-step transition rule is applied to S 1 and S 2 . Q' means that the transition may cause state changes of other machines. Since includes the case of a single base machine, must contain the case of zero base machine. It is clear that S 1 - containment relation is also true when compound rules involving multiple one-step transitions such as the N-steps rule and the Progress rule are applied to S 1 and S 2 . q Lemma 3. The claim S 1 - S 2 holds if Proof: Because the aggregation process is monotonic by lemma 1, lemma 3 simply extends the results of lemma 2. q Theorem 1. (Monotonicity) If S 1 - S 2 , then for every S 1 reachable from S 1 there exists S 2 reachable from S 2 such that S 1 - S 2. Proof: This is an immediate result of lemma 3. q The algorithm for the state expansion process is shown in figure 9. Two lists keep track of non-expanded and visited states. At each step, a new state is produced and states which are contained by any other states are pruned. The final output is a set of essential states. Definition 18. (Essential State) Composite state S is essential if and only if there does not exist a composite state S such that S - S. Readers should be aware of the fact that the generation of all essential states is successful only when the verified system is correct. If the system is incorrect, expanding error states which lead to unpredictable states is practically meaningless. We assume that the state expansion process terminates whenever an error is detected. As illustrated in figure 10, the state space reported at the end of a error-free expansion process is partitioned into several families of states (which may be r r overlapping) represented by essential composite states [23]. 9. Algorithm for Generating Essential States. Theorem 2. The essential composite states generated by the algorithm of figure 9 are complete. They symbolically represent all states which can be produced by a basic state enumeration method with no state abstraction. Proof: Consider states s, t such that s t in the state enumeration method, and composite SSM states s r , t r and s r t r in the symbolic form such that s r covers s. The resulting next state t r also covers t, because during the generation of composite states from s to t the same transition functions are applied and the same information is accumulated as in the expansion of s into t. q 4.5.1 Uniqueness of the Set of Essential States The set of essential states is unique provided the state graph connecting the essential states is strongly connected; namely, there exists at least one path from every essential state to all other essential states. Algorithm: Essential States Generation. W: list of working composite states. H: list of visited composite states.(output:essential states) while (W is not empty) do begin get current state A from W. for all state class v - A for all applicable operations t on v for any state P - W and Q - H then discard A'. else begin remove P from W if P - A'. remove Q from H if Q - A'. add A' to W. if discard A and terminate all FOR loops starting a new run. insert A to H if A is fully expanded and is not contained. end. Theorem 3. If a successful run of the verification starting with a legal initial state generates a set of essential states ES such that the state transition graph formed by essential states in ES is strongly connected, then the set ES is unique in the sense that the state expansion process always produces the same set of essential states ES if it starts with any legal and reachable state S. Proof: The set of essential states defines a fixpoint where the state expansion process terminates. From theorem 2, the states in ES represents all possible configurations that the system can reach. Therefore, S must be contained by at least one S e in ES. Because the symbolic state expansion is monotonic, all states derived from S are contained by states derived from S e . When the state transition graph of ES is strongly connected, there must exist at least one path from S e to all other essential states. It is impossible to reach an essential state S e - ES from S. q 10. Representation of the State Space by Essential States. Theorem 3 does not hold when the state graph is not strongly connected. Consider the simple case in which the state graph consists of two subgraphs, G1 and G2. G1 and G2 are individually strongly connected and paths exist from G1 to G2, but not vice versa. If the state expansion process starts from a state which is contained by states in G2 but not by states in G1, then only subgraph G2 is produced. In order to generate the entire state graph, the state expansion must start with a state in G1. However, a livelock error that G2 has no transition to G1 may be reported in the above case according to the conditions in section 3.3.3. To overcome the problem, we can isolate the subgraphs and analyze them. Protocol designers cannot determine whether the state graph is strongly connected in advance. It is, however, normally safe to start the state expansion process with an initial state in which all caches are invalid because this is usually the state when the system is turned on. 4.6 Accumulation of State Information The accumulation and compaction of state information in composite states is a major strength of the SSM method over other approaches. Consider the simple state transition caused by a read misses under the assumption of atomic memory accesses: essential state Initially, no processor has a copy of the block. On each read miss, the caches receive shared data copies and all other caches remain in the Invalid state. In order to reach the state (Shared, Shared, Invalid), which is covered by (Shared traditional state enumeration method would need to model at least three caches. In general, it is difficult to predict the number of caches needed in a model to reach all the possible states of a protocol. The SSM method eliminates this uncertainty since it verifies a protocol model independently of the number of processors 5 Correspondence between State Enumeration and SSM Models We have shown that the SSM expansion is monotonic. We still need to prove that the abstract SSM state transition system M r : (s 0r , A, S r , S, d r ) is equivalent to the explicit state transition system M:(s 0 , A, S, S, d) with respect to the properties of section 3.3. The correspondence relation j in SSM is as follows. Definition 19. (Correspondence Relation) State corresponds to state , i.e. s r js, if s is one of the states abstractly represented by s r , where a i is the state of the local automaton i and . The number of local automata of s in state must be a case covered by the repetition constructor r j , namely, We can always find an abstract initial state s 0r which corresponds to the initial state s 0 in the explicit model. For instance, it is normal to start the verification with an initial state in which no cached copy exist. In this case, all caches are invalid and (Invalid Invalid,.,Invalid). Theorem 4. Consider the state transition system M:(s 0 , A, S, S, d) of an explicit model with (an arbitrary number of) m local automata and the abstract state transition model d r ) in the SSM. Consider two states and , where a i is the state of the local automaton i and . Given s r js and , we can find such that and t r jt.Then, t is a state which violates a properties of section 3.3, iff t r is an error state in M r . Proof: (1) In regard to data consistency and completeness of specification, the proof is a direct conse- r2 . q n rn s: a 1 a 2 . a m a i:1.m q j:1.n , A a i:1.m q j a i:1.m q j s: a 1 a 2 . a m r2 . q n rn a i:1.m q j:1.n , A quence of theorems 1 and 2. Because s is one of the states represented by s r (i.e. s r js), the monotonic operation of SSM guarantees that t is a state characterized by t r . Furthermore, data consistency and completeness of specification are properties checked on the current states independent of other states (for instance, data inconsistency is found when a processor is allowed to read stale data; definition 8). Thus, t r must be an error state if t is an error state, and vice versa. (2) To show absence of simple deadlocks and livelocks as defined in definition 9, we need to show that processors are never trapped and are able to complete their reads and writes eventually (sec- tion 3.3). Consider that the explicit model M is trapped in a subset of states: (s1 -> s2 -> s3 ->. -> sn -> s1). In the abstract SSM model M r , we must have a corresponding set of states (s1 r -> s2 r -> s3 r -> sn r -> s1 r ) such that si r jsi for all i because of theorem 1 and theorem 2. Suppose that the circular loop (s1 r -> s2 r -> s3 r -> sn r -> s1 r ) is broken because some enabled transition from si r to t r . A corresponding exit from si to t must exist and ti r jt because both M and M r have the same constituent finite state machines. q 6 Protocol Error Detection Since unexpected message reception errors are easy to detect, we only describe the model and the procedure for detecting inconsistencies. We also present a subtle livelock error found during the course of this verification. Finally, we compare the performance of the SSM method with Murj in terms of time complexity and memory usage. In all verification results reported here, the expansion process starts with an initial state with no cached copy, empty message channels and state. In the SSM method, the initial state is ( , free). 6.1 Data Inconsistency The detection mechanism for data inconsistency is based on the model described in section 3.3.1. A status variable is added to caches and channel messages carrying data with possible values of nodata (n), fresh (f), and obsolete (o). The status of the memory copy can be fresh (f) or obsolete (o). Movement of data copies are modeled by assigning the status of one variable to another variable. In Figure 11 the state of each class has been augmented in between parentheses by the status associated with every data value. The figure illustrates the state transitions triggered by a read miss request (ReqSC) and a transition ending with an owner copy in a cache. In accordance with definition 7, the owner has the fresh copy, whereas all other copies, including the memory copy become obsolete. Data inconsistency is detected whenever a processor can read an obsolete data. 6.2 Livelock The expansion steps leading to a livelock in the original protocol are now described. Ini- tially, consider a system state with an owner and no request in progress (directory entry is free). The state has the form (., free). 11. Data Transfer and Detection of Data Inconsistency in SSM. Consider the following scenario. 1. The owner replaces its copy and writes the block back to memory. The state is (., indicating that a write-back message is in the output channel. O I n Data f f ReqSC WMP n A fresh copy is cached in S state A fresh copy is in propagation Caches do not have a copy Directory is free to accept new requests and the memory copy is fresh I n Data f Data f f ReqSC WMP n State transition: memory responds to ReqSC requests I n Data f f ReqSC WMP n Load data from the memory Aggregation after the state transition I n )Data f f ReqSC WMP n Many intermediate states are not shown A write-miss request loads data from the memory after successfully invalidating other cached copies I n f ReqSC WMP n State transition: receiving the data and executing the pending write New owner with fresh data memory copy becomes obsolete 2. Next, the same cache experiences a write miss and sends a request for an exclusive copy. The new state is (., free) and a race exists between the write-back and the ownership messages in the case of a non-FIFO network. 12. Livelock Detection in SSM 3. The memory receives the ownership message before the write-back message; in this case the memory state is changed to XOwnC and an invalidation (InvO) is sent to the cache because the memory still records that the cache is an owner. The resulting state is (., XOwnC). 4. The cache receives the InvO message and changes its state to TxOI. The system state becomes (., XOwnC). WMP f DOxMR ReqOC I NAck f RMP f ReqSC WMP NAck f WMP f ReqOC WMP f ReqO WMPInvO DOxMR XOwnC I NAck f RMP f ReqSC WMP NAck f WMP f ReqOC WMP f ReqO WMPInvO f Synch1 I NAck f RMP f ReqSC WMP NAck f WMP f ReqOC WMP f ReqO TxOIf DOxMR XOwnC I NAck f RMP f ReqSC WMP NAck f WMP f ReqOC WMP f ReqO TxOIf f Synch1 I NAck f RMP f ReqSC WMP NAck f WMP f ReqOC WMP f ReqO WMPf DOxMR ReqOC initial state: loop forever in this sink state memory receives the ReqOC read miss miss miss read miss read miss read miss miss miss cache receives the NAck memory receives and aborts the ReqSC cache receives the NAck cache receives the NAck cache receives the NAck memory receives and aborts the ReqSC memory receives and aborts the ReqSC memory receives and aborts the ReqSC memory receives and aborts the request memory receives and aborts the request memory receives and aborts the request memory receives and aborts the request cache receives the NAck cache receives the NAck cache receives the NAck cache receives the NAck memory receives the DOxMR cache receives the InvO memory receives the DOxMR WMP InvO DOxMR TxOI 5. Finally, when the memory receives the write-back message, it enters the synchronization state Synch1 and expect a synchronization message SAck from the cache (figure 1). The system state is (., Synch1). However, the synchronization message will never be sent by the cache, which locks the directory entry forever. In the SSM method, this error was successfully detected by reporting that a cycle exists between four global states without exit to a state outside the loop, as shown in figure 12 (the global state transition diagram is not strongly connected). This error was not detected by the present Murj system because checking the connectivity of the global state diagram is overwhelmingly complex when the size of the global state diagram is large. The livelock condition originates from the fact that memory does not check the presence bits when it receives an ownership request. The livelock can be removed by the following correction to the protocol. When the memory receives a ReqOC message, it checks whether the processor identifier of the message corresponds to the current owner. If it does, the memory state is changed to the synchronization state Synch1 directly (following the state diagram in figure 1). Later, when the write-back message arrives, the memory updates its copy of the block, supplies the cache with the copy of the block and unlocks the directory entry. 6.3 Comparison with the Murj System The Murj system, developed by Dill et al. [8], is based on state enumeration. There are two versions of Murj: the non-symmetric Murj system (Murj-ns) and the symmetric Murj system (Murj-s). In Murj-ns, two system states are equivalent if and only if they are identical whereas Murj-s exploits the symmetry of the system by using a characteristic state to represent states which are permutations of each other [16]. For example, two system states composed of three local cache states, (shared, shared, invalid) and (invalid, shared, shared), are deemed equivalent because the order of cache states in the global state representation is irrelevant to the correctness of the protocol. The time complexity and memory usage of a verification are closely related to the size of the system state space. Generally, an exhaustive search algorithm performs three fundamental operations. 1. Generate a new state, if there is any left; otherwise terminate and report the final set of global states. 2. Compare the new state against the set of previously visited states. 3. Keep the new state for future expansion if the new state was not visited before. The most time-consuming step is comparing the new state to previously visited states. The time complexity grows in proportion to the size of the search space (the set of states generated and analyzed during the procedure), while the memory usage increases with the size of the global TxOI state space (the set of states saved and reported at the end). Since the search space is a direct expansion of the global state space, reducing the size of the global state space is particularly important. Murj incorporates state encoding to reduce memory usage and hash tables to speed up the search and comparison operations. These optimizations are not implemented in SSM. Table shows performance comparisons between Murj-ns, Murj-s, and SSM running on Model MBytes of memory for the verification of the protocol. We make the following observations. First, for small-scale systems with less than five processors, the time complexity and the memory usage of Murj-ns and Murj-s are tolerable. Second, the sizes of both the global state space and the search space of Murj-s are significantly less than those of Murj-ns, but there is little difference in the times taken by both methods. In the case of four processor systems, we observe that Murj-s takes longer than Murj-ns. (The extra overhead due to the state permutation mappings in Murj-s may explain this.) As more processors are added in the model, the verification time and the memory usage increase drastically in both cases. As compared to Murj, SSM is very efficient. The verification based on SSM runs in 0.9 seconds with Mbytes of memory and the state space (123 global states) is comparatively very small. The fact that the performance of classical enumeration techniques is acceptable for small system sizes raises the question of whether more elaborate approaches such as SSM are really needed. Since the final set of essential states reported in the SSM covers all possible states the system can reach, essential states with the maximum number of base machines in different states represent the most complex states of the system. In the verification using SSM, the most complex essential states consisted of 25 base machines in different states. This means that a system model of at least 25 processors is required to obtain a 100% error coverage in a state enumeration method. In the case of Murj-ns, we observe (roughly) a times increase in the size of the search space each time one more process is added to the model. If this trend continues up to 25 proces- sors, the search space could reach a size of 10 37 states for a model with 25 processors. The time 2. Comparison between SSM, Murj-ns and Murj-s Method Number of processors Size of global state space Size of search space Verification time Memory usage (Mbytes) Murj-ns 5 excessive memory usage (over 200Mbytes) Murj-s SSM any n >1 123 4,205 0.9 0.02 and the memory space needed by a verification of such complexity would be prohibitive on any existing machine. In the protocol verified in this paper, the number of messages floating in any one of the message channels at any time is bounded in spite of the fact that the number of processors in the model is arbitrary. However, the SSM method does not preclude the possibility that a protocol may allow processors to send multiple or even an arbitrary number of messages of the same type [25]. As a result, the model for message channels may need to be adapted by using some finite variables to represent infinite system behavior [13]. In such cases, repetition constructors might be useful to keep track abstractly of the number of messages of the same type. The SSM method can detect protocol-intrinsic livelocks (section 3.3.3). Because the number of global states reported is relatively small (123 states in this case), the time complexity of checking the connectivity of the global state transition diagram is more manageable than in Murj. 7 Conclusion Cache coherence protocols designed for systems assuming non-FIFO networks are required in systems with adaptive routing and fault-tolerant interconnection networks. In this paper, we have verified a directory-based cache coherence protocol for non-FIFO networks. The verification of the protocol was done by the Murj system and the SSM method. Generally speak- ing, from the study, we have found that the Murj system is effective in verifying small-scale systems with manageable complexity. However, we have shown that, for the protocol verified in this paper, a system model with at least 25 processors is required in order to reach 100% error cover- age. With this many processors, the complexity of the state space search would be prohibitive for the Murj system, whereas the performance of SSM shows that it could deal with much more complex protocols than the one used in this paper. Overall, the SSM method offers three advantages over classic state enumeration methods with no state abstraction. First, it overcomes the state explosion problem. Second, since the entire global state space is symbolically represented by a small number of essential states, the time complexity of checking the connectivity of the global state transition diagram (needed for livelock detection) is manageable. Third, it verifies the protocol for any system size. Recently, Ip and Dill have integrated a variation of the SSM method in Murj [17]. Their tool expands explicit states and then infers abstract states based on generated explicit states, whereas our tool works directly on the abstract states. Therefore, the new Murj tool may require multiple runs (adding one more processor to the model in each consecutive run) to reach the complete verification results obtained with our method. Their experience confirms that classical state enumeration approaches will be sufficient to verify protocols for systems with small numbers of processors, whereas methods based on symbolic state representations such as SSM will be critical in the future for the design of complex protocols in large-scale multiprocessors. In the architectural model of figure 2 memory accesses are made of several consecutive events and thus are not atomic. We do not constrain in any way the sequences of access generated by processors. Moreover the hardware does not distinguish between synchronization instructions and regular load/store instructions. So, in this paper, latency tolerance mechanisms in the processors and in the caches are not modeled and we assume that the mechanisms are correct and enforce proper sequencing and ordering of memory accesses [9]. However, the methodology of SSM does not preclude the verification of consistency in the presence of latency tolerance hard- ware. In order to include latency tolerance hardware. synchronization accesses must be modeled and the sequence of accesses generated by the processors are constrained by the memory consistency model [11]. This approach was applied in [26] and in [27] for the delayed consistency protocol specified in [10]. Whereas state enumeration approaches are appropriate for verifying coherence properties, they do not seem to be applicable to the verification of memory access orders. The reason is that no one has found a way so far to formulate the verification property for memory order over the state enumeration graph. Thus, the verification of memory access orders must still rely on testing procedures [6] or on manual proofs [1, 12]. Acknowledgments This research was supported by the National Science Foundation under Grant No. CCR- 9222734. We also want to acknowledge the contributions of David L. Dill and C. Norris Ip who provided invaluable information on the Murj system. --R "A Lazy Cache Algorithm" "The Cache Coherence Problem in Shared-Memory Multiprocessors" "Cache Coherence Protocols: Evaluation Using a Multiprocessor Simulation Model" "A new solution to coherence problems in multicache sys- tems" "Directory-Based Cache Coherence in Large Scale Multiprocessors" Reasoning About Parallel Architectures "Protocol Representation with Finite-State Models" "Protocol Verification as a Hardware Design Aid" "Memory Access Buffering in Multiprocessors" "Delayed Consistency and Its Effects on the Miss Rate of Parallel Programs" "Mem- ory Consistency and Event Ordering in Shared-Memory Multiprocessors" "Proving Sequential Consistency of High Performance Shared Memories" "Verification of a Distributed Cache Memory by Using Abstractions" "Communicating Sequential Processes" "Algorithms for Automated Protocol Verification" "Better Verification Through Symmetry" "Verifying Systems with Replicated Components in Murj" "The Stanford Flash Multiprocessor Design," "The Directory-Based Cache Coherence Protocol for the DASH Multiprocessor" "Formal Verification of the Gigamax Cache Consistency Protocol" "The S3.mp Scalable Shared Memory Multiprocessor" "The Verification of Cache Coherence Protocols," "A New Approach for the Verification of Cache Coherence Proto- cols" "A Survey of Techniques for Verifying Cache Coherence Proto- cols" "Verifying Distributed Directory-based Cache Coherence Protocols: S3.mp, a Case Study" "Symbolic State Model: A New Approach for the Verification of Cache Coherence Protocols," "Formal Verification of Delayed Consistency Protocols" "Tempest and Typhoon: User-Level Shared Memory," "A Survey of Cache Coherence Schemes for Multiprocessors" "Data Coherence Problem in a Multicache System" "Towards Analyzing and Synthesizing Protocols" --TR Cache coherence protocols: evaluation using a multiprocessor simulation model Memory access buffering in multiprocessors The cache coherence problem in shared-memory multiprocessors A lazy cache algorithm A Survey of Cache Coherence Schemes for Multiprocessors Directory-Based Cache Coherence in Large-Scale Multiprocessors Proving sequential consistency of high-performance shared memories (extended abstract) Delayed consistency and its effects on the miss rate of parallel programs Reasoning about parallel architectures The verification of cache coherence protocols The Stanford FLASH multiprocessor Tempest and typhoon Symbolic state model Verification techniques for cache coherence protocols Communicating sequential processes A New Approach for the Verification of Cache Coherence Protocols Protocol Verification as a Hardware Design Aid Formal Verification of Delayed Consistency Protocols Verifying Distributed Directory-Based Cahce Coherence Protocols Verification of a Distributed Cache Memory by Using Abstractions Verifying Systems with Replicated Components in Murphi Better Verification Through Symmetry

state enumeration methods;formal methods;cache coherence protocols;shared-memory multiprocessors;state abstraction

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Property testing and its connection to learning and approximation.

In this paper, we consider the question of determining whether a function f has property P or is &egr;-far from any function with property P. A property testing algorithm is given a sample of the value of f on instances drawn according to some distribution. In some cases, it is also allowed to query f on instances of its choice. We study this question for different properties and establish some connections to problems in learning theory and approximation.In particular, we focus our attention on testing graph properties. Given access to a graph G in the form of being able to query whether an edge exists or not between a pair of vertices, we devise algorithms to test whether the underlying graph has properties such as being bipartite, k-Colorable, or having a p-Clique (clique of density p with respect to the vertex set). Our graph property testing algorithms are probabilistic and make assertions that are correct with high probability, while making a number of queries that is independent of the size of the graph. Moreover, the property testing algorithms can be used to efficiently (i.e., in time linear in the number of vertices) construct partitions of the graph that correspond to the property being tested, if it holds for the input graph.

Introduction We are interested in the following general question of Property Let P be a fixed property of functions, and f be an unknown function. Our goal is to determine (possibly if f has property P or if it is far from any function which has property P, where distance between functions is measured with respect to some distribution D on the domain of f . Towards this end, we are given examples of the form (x; f(x)), where x is distributed according to D. We may also be allowed to query f on instances of our choice. The problem of testing properties emerges naturally in the context of program checking and probabilistically checkable proofs as applied to multi-linear functions or low-degree polynomials [14, 7, 6, 19, 21, 36, 5, 4, 10, 11, 8, 9]. Property testing per se was considered in [36, 35]. Our definition of property testing is inspired by the PAC learning model [37]. It allows the consideration of arbitrary distributions rather than uniform ones, and of testers which utilize randomly chosen instances only (rather than being able to query instances of their own choice). Full version available from http://theory.lcs.mit.edu/~oded/ y Dept. of Computer Science and Applied Math., Weizmann Institute of Science, ISRAEL. E-mail: oded@wisdom.weizmann.ac.il. On sabbatical leave at LCS, MIT. z Laboratory for Computer Science, MIT, 545 Technology Sq., Cambridge, MA 02139. E-mail: shafi@theory.lcs.mit.edu. x Laboratory for Computer Science, MIT, 545 Technology Sq., Cambridge, MA 02139. E-mail: danar@theory.lcs.mit.edu. Supported by an NSF postdoctoral fellowship. We believe that property testing is a natural notion whose relevance to applications goes beyond program checking, and whose scope goes beyond the realm of testing algebraic prop- erties. Firstly, in some cases one may be merely interested in whether a given function, modeling an environment, (resp. a given program) possesses a certain property rather than be interested in learning the function (resp. checking that the program computes a specific function correctly). In such cases, learning the function (resp., checking the program) as means of ensuring that it satisfies the property may be an over-kill. Secondly, learning algorithms work under the postulation that the function (representing the environment) belongs to a particular class. It may be more efficient to test this postulation first before trying to learn the function (and possibly failing when the postulation is wrong). Similarly, in the context of program checking, one may choose to test that the program satisfies certain properties before checking that it computes a specified function. This paradigm has been followed both in the theory of program checking [14, 36] and in practice where often programmers first test their programs by verifying that the programs satisfy properties that are known to be satisfied by the function they compute. Thirdly, we show how to apply property testing to the domain of graphs by considering several classical graph properties. This, in turn, offers a new perspective on approximation problems as discussed below. THE RELEVANT PARAMETERS. Let F be the class of functions which satisfy property P. Then, testing property P corresponds to testing membership in the class F . The two parameters relevant to property testing are the permitted distance, ffl, and the desired confidence, ffi . We require the tester to accept each function in F and reject every function which is further than ffl away from any function in F . We allow the tester to be probabilistic and make incorrect positive and negative assertions with probability at most ffi . The complexity measures we focus on are the sample complexity (the number of examples of the function's values that the tester requires), the query complexity (the number of function queries made - if at all), and the running time of the tester. 1.1. Property Testing and Learning Theory As noted above, our formulation of testing mimics the standard frameworks of learning theory. In both cases one is given access to an unknown target function (either in the form of random instances accompanied by the function values or in the form of oracle access to the function). A semantic difference is that, for sake of uniformity, even in case the functions are Boolean, we refer to them as functions rather than con- cepts. However, there are two important differences between property testing and learning. Firstly, the goal of a learning algorithm is to find a good approximation to the target function testing algorithm should only determine whether the target function is in F or is far away from it. This makes the task of the testing seem easier than that of learning. On the other hand, a learning algorithm should perform well only when the target function belongs to F whereas a testing algorithm must perform well also on functions far away from F . Furthermore, (non-proper) learning algorithms may output an approximation ~ f of the target f 2 F so that ~ f 62 F . We show that the relation between learning and testing is non-trivial. On one hand, proper (representation dependent) learning implies testing. On the other hand, there are function classes for which testing is harder than (non-proper) learning, . Nonetheless, there are also function classes for which testing is much easier than learning. Further details are given in Subsection 2.2. In addition, the graph properties discussed below provide a case where testing (with queries) is much easier than learning (also with queries). 1.2. Testing Graph Properties In the main technical part of this paper, we focus our attention on testing graph properties. We view graphs as Boolean functions on pairs of vertices, the value of the function representing the existence of an edge. We mainly consider testing algorithms which use queries and work under the uniform dis- tribution. That is, a testing algorithm for graph property P makes queries of the form "is there an edge between vertices u and v" in an unknown graph G. It then decide whether G has property P or is "ffl-away" from any graph with property P, and is allowed to err with probability 1=3. Distance between two N -vertex graphs is defined as the fraction of vertex-pairs which are adjacent in one graph but not in the other. We present algorithms of poly(1=ffl) query-complexity and running-time 1 at most exp( ~ testing the following graph properties: k-Colorability for any fixed k - 2. (Here the query- complexity is poly(k=ffl), and for the running-time is ~ ae-Clique for any ae ? 0. That is, does the N -vertex graph has a clique of size aeN . ae-CUT for any ae ? 0. That is, does the N -vertex graph has a cut of size at least aeN 2 . A generalization to k-way cuts works within query-complexity poly((log k)=ffl). ae-Bisection for any ae ? 0. That is, does the N -vertex graph have a bisection of size at most aeN 2 . 1 Here and throughout the paper, we consider a RAM model in which trivial manipulation of vertices (e.g., reading/writing a vertex name and ordering vertices) can be done in constant time. Furthermore: ffl For all the above properties, in case the graph has the desired property, the testing algorithm outputs some auxiliary information which allows to construct, in poly(1=ffl) \Delta N - time, a partition which approximately obeys the property. For example, for ae-CUT, we can construct a partition with at least (ae \Gamma ffl)N 2 crossing edges. ffl Except for Bipartite (2-Colorability) testing, running-time of poly(1=ffl) is unlikely, as it will imply NP ' BPP . ffl None of these properties can be tested without queries when using o( examples. ffl The k-Colorability tester has one-sided error: it always accepts k-colorable graphs. Furthermore, when rejecting a graph, this tester always supplies a poly(1=ffl)-size sub-graph which is not k-colorable. All other algorithms have two-sided error, and this is unavoidable within o(N ) query- complexity. ffl Our algorithms for k-Colorability, ae-Clique and ae-Cut can be easily extended to provide testers with respect to product distributions: that is, distributions the form \Pi(u; is a distribution on the vertices. In contrast, it is not possible to test any of the graph properties discussed above in a distribution-free manner. GENERAL GRAPH PARTITION. All of the above properties are special cases of the General Graph k-Partition property, parameterized by a set of lower and upper bounds. The parameterized property holds if there exists a partition of the vertices into k disjoint subsets so that the number of vertices in each subset as well as the number of edges between each pair of subsets is within the specified lower and upper bounds. We present a testing algorithm for the above general property. The algorithm uses ~ queries, runs in time exponential in its query-complexity, and makes two-sided error. Approximating partitions, if they exist, can be efficiently constructed in this general case as well. We note that the specialized algorithms perform better than the general algorithm with the appropriate parameters. OTHER GRAPH PROPERTIES. Going beyond the general graph partition problem, we remark that there are graph properties which are very easy to test (e.g., Connectivity, Hamiltonicity, and Planarity). On the other hand, there are graph properties in NP which are extremely hard to test; namely, any testing algorithm must inspect at of the vertex pairs. In view of the above, we believe that providing a characterization of graph properties according to the complexity of testing them may not be easy. OUR TECHNIQUES. Our algorithms share some underlying ideas. The first is the uniform selection of a small sample and the search for a suitable partition of this sample. In case of k-Colorability certain k-colorings of the subgraph induced by this sample will do, and these are found byk-coloring a slightly augmented graph. In the other algorithms we exhaustively try all possible partitions. This is reminiscent of the exhaustive sampling of Arora et. al. [3], except that the partitions considered by us are always directly related to the combinatorial structure of the problem. We show how each possible partition of the sample induces a partition of the entire graph so that the following holds. If the tested graph has the property in question then, with high probability over the choice of the sample, there exists a partition of the sample which induces a partition of the entire graph so that the latter partition approximately satisfies the requirements established by the property in ques- tion. For example, in case the graph has a ae-cut, there exists a 2-way-partition of the sample inducing a partition of the entire graph with (ae \Gamma ffl)N 2 crossing edges. On the other hand, if the graph should be rejected by the test, then by definition no partition of the entire graph (and in particular none of the induced partitions) approximately obeys the requirements. The next idea is to use an additional sample to approximate the quality of each such induced partition of the graph, and discover if at least one of these partitions approximately obeys the requirements of the property in question. An important point is that since the first sample is small (i.e., of size poly(1=ffl)), the total number of partitions it induces is only exp poly(1=ffl). Thus, the additional sample must approximate only these many partitions (rather than all possible partitions of the entire graph) and it suffices that this sample be of size poly(1=ffl), The difference between the various algorithms is in the way in which partitions of the sample induce partitions of the entire graph. The simplest case is in testing Bipartiteness. For a partition of the sample, all vertices in the graph which have a neighbor in S 1 are placed on one side, and the rest of the vertices are placed on the other side. In the other algorithms the induced partition is less straightforward. For example, in case of ae-Clique, a partition (S 1 of the sample S with induces a candidate clique roughly as follows. Consider the set T of graph vertices each neighboring all of Then the candidate clique consists of the aeN vertices with the highest degree in the subgraph induced by T. In the, ae- Cut, ae-Bisection and General Partition testing algorithms, we use auxiliary guesses which are implemented by exhaustive search. 1.3. Testing Graph Properties and Approximation The relation of testing graph properties to approximation is best illustrated in the case of Max-CUT. A tester for the class ae-Cut, working in time T (ffl; N ), yields an algorithm for approximating the maximum cut in an N -vertex graph, up to additive error fflN 2 , in time 1 ffl \DeltaT (ffl; N ). Thus, for any constant ffl ? 0, we can approximate the size of the max-cut to within fflN 2 in constant time. This yields a constant time approximation scheme (i.e., to within any constant relative error) for dense graphs, improving on Arora et. al. [3] and de la Vega [17] who solved this problem in polynomial-time (O(N 1=ffl 2 )-time and exp( ~ In both works the problem is solved by actually constructing approximate max-cuts. Finding an approximate max-cut does not seem to follow from the mere existence of a tester for ae-Cut; yet, as mentioned above, our tester can be used to find such a cut in time linear in N (i.e., ~ One can turn the question around and ask whether approximation algorithms for dense instances can be transformed into corresponding testers as defined above. In several cases this is possible. For example, using some ideas from our work, the Max-CUT algorithm of [17] can be transformed into a tester of complexity comparable to ours. We do not know whether the same is true with respect to the algorithms in [3]. Results on testing graph properties can be derived also from work by Alon et. al. [1]. That paper proves a constructive version of the Regularity Lemma of Szemer-edi, and obtains from it a polynomial-time algorithm that given an N -vertex graph, finds a subgraph of size f(ffl; k) which is not k-colorable, or omits at most fflN 2 edges and k-colors the rest. Noga Alon has observed that the analysis can be modified to yield that almost all subgraphs of size f(ffl; k) are not k-colorable, which in turn implies a tester for k-Colorability. In comparison with our k-Colorability Tester, which takes a sample of O(k 2 log k=ffl 3 ) vertices, the k-colorability tester derived (from [1]) takes a much bigger sample of size equaling a tower of (k=ffl) 20 exponents (i.e., log f(ffl; A DIFFERENT NOTION OF APPROXIMATION FOR MAX-CLIQUE. Our notion of ae-Clique Testing differs from the traditional notion of Max-Clique Approximation. When we talk of testing "ae-Cliqueness", the task is to distinguish the case in which an N -vertex graph has a clique of size aeN from the case in which it is ffl-far from the class of N -vertex graphs having a clique of size aeN . On the other hand, traditionally, when one talks of approximating the size of Max-Clique, the task is to distinguish the case in which the max-clique has size at least aeN from, say, the case in which the max-clique has size at most aeN=2. Whereas the latter problem is NP-Hard, for ae - 1=64 (see [9, Sec. 3.9]), we've shown that the former problem can be solved in exp(O(1=ffl 2 ))-time, for any ae; ffl ? 0. Furthermore, Arora et. al. [3] showed that the "dense-subgraph" problem, a generalization of ae-cliqueness, has a polynomial-time approximation scheme (PTAS) for dense instances. TESTING k-COLORABILITY VS. APPROXIMATING k- COLORABILITY. Petrank has shown that it is NP-Hard to distinguish 3-colorable graphs from graphs in which every 3-partition of the vertex set violates at least a constant fraction of the edges [30]. In contrast, our k-Colorability Tester implies that solving the same promise problem is easy for dense graphs, where by dense graphs we mean N -vertex graphs edges. This is the case since, for every ffl ? 0, our tester can distinguish, in exp(k 2 =ffl 3 )-time, between k- colorable N -vertex graphs and N -vertex graphs which remain non-k-colorable even if one omits at most fflN 2 of their edges. 2 We note that deciding k-colorability even for N -vertex graphs of minimum degree at least k\Gamma3 k\Gamma2 \Delta N is NP-complete (cf., Edwards [18]). On the other hand, Edwards also gave a polynomial-time algorithm for k-coloring k-colorable N - vertex graphs of minimum degree at least ffN , for any constant k\Gamma2 . 1.4. Other Related Work PROPERTY TESTING IN THE CONTEXT OF PCP: Property testing plays a central role in the construction of PCP systems. Specif- ically, the property tested is being a codeword with respect to a specific code. This paradigm explicitly introduced in [6] has shifted from testing codes defined by low-degree polynomials [6, 19, 5, 4] to testing Hadamard codes [4, 10, 11, 8], and recently to testing the "long code" [9]. PROPERTY TESTING IN THE CONTEXTOF PROGRAM CHECKING: There is an immediate analogy between program self-testing [14] and property-testing with queries. The difference is that in self-testing, a function f (represented by a program) is tested for being close to a fully specified function g, whereas in property-testing the test is whether f is close to any function in a function class G. Interestingly, many self-testers [14, 36] work by first testing that the program satisfies some properties which the function it is supposed to compute satisfies (and only then checking that the program satisfies certain constraints specific to the function). Rubinfeld and Sudan [36] defined property testing, under the uniform distribution and using queries, and related it to their notion of Robust Char- acterization. Rubinfeld [35] focuses on property testing as applied to properties which take the form of functional equations of various types. PROPERTY TESTING IN THE CONTEXT OF LEARNING THEORY: Departing from work in Statistics regarding the classification of distributions (e.g., [24, 16, 41]), Ben-David [12] and Kulkarni and Zeitouni [28] considered the problem of classifying an unknown function into one of two classes of functions, given labeled examples. Ben-David studied this classification problem in the limit (of the number of examples), and Kulkarni and Zeitouni studied it in a PAC inspired model. For any fixed ffl, the problem of testing the class F with distance parameter ffl can be casted as such a classification problem (with F and the set of functions ffl-away from F being the two classes). A different variant of the problem was considered by Yamanishi [39]. TESTING GRAPH PROPERTIES. Our notion of testing a graph property P is a relaxation of the notion of deciding the graph As noted by Noga Alon, similar results, alas with much worse dependence on ffl, can be obtained by using the results of Alon et. al. [1]. property P which has received much attention in the last two decades [29]. In the classical problem there are no margins of error, and one is required to accept all graphs having property P and reject all graphs which lack it. In 1975 Rivest and Vuillemin [33] resolved the Aanderaa-Rosenberg Conjecture [34], showing that any deterministic procedure for deciding any non-trivial monotone N -vertex graph property must ex- entries in the adjacency matrix representing the graph. The query complexity of randomized decision procedures was conjectured by Yao to be \Omega\Gamma N 2 ). Progress towards this goal wasmade by Yao [40], King [27] and Hajnal [23] culminating in an \Omega\Gamma N 4=3 ) lower bound. Our results, that some non-trivial monotone graph properties can be tested by examining a constant number of random locations in the matrix, stand in striking contrast to all of the above. APPROXIMATION IN DENSE GRAPHS. As stated previously, Arora et. al. [3] and de la Vega [17] presented PTAS for dense instances of Max-CUT. The approach of Arora et. al. uses Linear Programming and Randomized Rounding, and applies to other problems which can be casted as a "smooth" Integer Programs. 3 The methods of de la Vega [17] are purely combinatorial and apply also to similar graph partition prob- lems. Following the approach of Alon et. al. [1], but using a modification of the regularity Lemma (and thus obtaining much improved running times), Frieze and Kannan [20] devise PTAS for several graph partition problems such as Max-Cut and Bisection. We note that compared to all the above re- sults, our respective graph partitioning algorithms have better running-times. Like de la Vega, our methods use elementary combinatorial arguments related to the problem at hand. Still our methods suffice for dealing with the General Graph Partition Problem. Important Note: In this extended abstract, we present only two of our results on testing graph properties: the k-Colorability and the ae-Clique testers. The definition and theorem regarding the General Graph Partition property appears in Subsection 3.3. All other results as well as proofs and further details can be found in our report [22]. 2. General Definitions and Observations 2.1. Definitions fFng be a parameterized class of functions, where the functions 4 in Fn are defined over f0; 1g n and let be a corresponding class of distributions (i.e., Dn is a distribution on f0; 1g n ). We say that a function f defined on f0; 1g n is ffl-close to Fn with respect to Dn if there exists a function g 2 Fn such that 3 In [2], the approach of [3] is extended to other problems, such as Graph Isomorphism, using a new rounding procedure for the Assignment Problem. 4 The range of these functions may vary and for many of the results and discussions it suffices to consider Boolean function. Otherwise, f is ffl-far from Fn (with respect to Dn ). We shall consider several variants of testing algorithms, where the most basic one is defined as follows. Definition 2.1 (property testing): Let A be an algorithm which receives as input a size parameter n, a distance parameter confidence Fixing an arbitrary function f and distribution Dn over f0; 1g n , the algorithm is also given access to a sequence of f-labeled examples, where each x i is independently drawn from the distribution Dn . We say that A is a property testing algorithm (or simply a testing algorithm) for the class of functions F if for every n, ffl and ffi and for every function f and distribution Dn over f0; 1g n the following holds with probability at least 1 \Gamma ffi (over the examples drawn from Dn and the possible coins tosses of A), A accepts f (i.e., outputs 1); ffl if f is ffl-far from Fn (with respect to Dn ) then with probability at least rejects f (i.e., outputs 0). The sample complexity of A is a function of n; ffl and ffi bounding the number of labeled examples examined by A on input Though it was not stated explicitly in the definition, we shall also be interested in bounding the running time of a property testing algorithm (as a function of the parameters n; ffi; ffl, and in some case of a complexity measure of the class F ). We consider the following variants of the above definition: (1) Dn may be a specific distribution which is known to the algorithm. In particular, we shall be interested in testing with respect to the uniform distribution; (2) Dn may be restricted to a known class of distributions (e.g., product distributions); (3) The algorithm may be given access to an oracle for the function f , which when queried on x 2 f0; 1g n , returns f(x). In this case we refer to the number of queries made by A (which is a function of n, ffl, and ffi ), as the query complexity of A. 2.2. Property Testing and PAC Learning A Probably Approximately Correct (PAC) learning algorithm [37] works in the same framework as that described in Definition 2.1 except for the following (crucial) differences: (1) It is given a promise that the unknown function f (referred to as the target function) belongs to F ; (2) It is required to output (with probability at least h which is ffl-close to f , where closeness is as defined in Equation (1) (and ffl is usually referred to as the approximation parameter). Note that the differences pointed out above effect the tasks in opposite directions. Namely, the absence of a promise makes testing potentially harder than learning, whereas deciding whether a function belongs to a class rather than finding the function may make testing easier. In the learning literature, a distinction is made between proper (or representation dependent) learning and non-proper learning [31]. In the former model, the hypothesis output by the learning algorithm is required to belong to the same function class as the target function f , i.e. h 2 F , while in the latter model, no such restriction is made. We stress that a proper learning algorithm (for F ) may either halt without output or output a function in F , but it may not output functions not in F . 5 There are numerous variants of PAC learning (including learning with respect to specific distributions, and learning with access to an oracle for the target function f ). Unless stated otherwise, whenever we refer in this section to PAC learning we mean the distribution-free no-query model described above. The same is true for references to property testing. In addition, apart from one example, we shall restrict our attention to classes of Boolean functions. TESTING IS NOT HARDER THAN PROPER LEARNING. Proposition 2.1 If a function class F has a proper learning algorithm A, then F has a property testing algorithm A 0 such that mA 0 (n; ffl; Furthermore, the same relation holds between the running times of the two algorithm. The proof of this proposition, as well as of all other propositions in this section, can be found in our report [22]. The above proposition implies that if for every n, Fn has polynomial (in n) VC-dimension [38, 15], then F has a tester whose sample complexity is poly(n=ffl) \Delta log(1=ffi). The reason is that classes with polynomial VC-dimension can be properly learned from a sample of the above size [15]. However, the running time of such a proper learning algorithm, and hence of the resulting testing algorithm might be exponential in n. Corollary 2.2 Every class which is learnable with a poly(n=ffl) sample is testable with a poly(n=ffl) sample (in at most exponential time). TESTING MAY BE HARDER THAN LEARNING. In contrast to Proposition 2.1 and to Corollary 2.2, we show that there are classes which are efficiently learnable (though not by a proper learning algorithm) but are not efficiently testable. This is proven by observing that many hardness results for proper learning (cf. [31, 13, 32]) actually establish the hardness of testing (for the same classes). Furthermore, we believe that it is more natural to view these hardness results as referring to testing. Thus, the separation between efficient learning and efficient proper learning translates to a separation between efficient learning and efficient testing. 5 We remark that in case the function is F have an easy to recognize representation, one can easily guarantee that the algorithm never outputs a function not in F . Standard classes considered in works on proper learning typically have this feature. Proposition 2.3 If NP 6ae BPP then there exist function classes which are not poly(n=ffl)-time testable but are poly(n=ffl)-time (non-properly) learnable. We stress that while Proposition 2.1 generalizes to learning and testing under specific distributions, and to learning and testing with queries, the proof of Proposition 2.3 uses the premise that the testing (or proper learning) algorithm works for any distribution and does not make queries. TESTING MAY BE EASIER THAN LEARNING. Proposition 2.4 There exist function classes F such that F has a property testing algorithm whose sample complexity and running time are O(log(1=ffi)=ffl), yet any learning algorithm for F must have sample complexity exponential in n. The impossibility of learning the function class in Proposition 2.4 is due to its exponential VC-dimension, (i.e., it is a pure information theoretic consideration). We now turn to function classes of exponential (rather than double exponential) size. Such classes are always learnable with a polynomial sample, the question is whether they are learnable in polynomial-time. We present a function class which is easy to test but cannot be learned in polynomial-time (even under the uniform distri- bution), provided trapdoor one-way permutations exist (e.g., factoring is intractable). Proposition 2.5 If there exist trapdoor one-way permutations then there exists a family of functions which can be tested in poly(n=ffl)-time but can not be learned in poly(n=ffl)-time, even with respect to the uniform distribution. Furthermore, the functions can be computed by poly(n)-size circuits. The class presented in Proposition 2.5 consists of multi-valued functions. We leave it as an open problem whether a similar result holds for a class of Boolean functions. LEARNING AND TESTING WITH QUERIES (under the uniform distribution). Invoking known results on linearity testing [14, 7, 19, 10, 11, 8] we conclude that there is a class of 2 n functions which can be tested within query complexity O(log(1=ffi)=ffl), and yet learning it requires at least n queries. Similarly, using results on low-degree testing [7, 6, 21, 36], there is a class of which can be tested within query complexity O( log(1=ffi) ffl \Delta n), and yet learning it requires exp(n) many queries. AGNOSTICLEARNING AND TESTING. In a variant of PAC learn- ing, called Agnostic PAC learning [26], there is no promise concerning the target function f . Instead, the learner is required to output a hypothesis h from a certain hypothesis class H, such that h is ffl-close to the function in H which is closest to f . The absence of a promise makes agnostic learning closer in spirit to property testing than basic PAC learning. In particular, agnostic learning with respect to a hypothesis class H implies proper learning of the class H and thus property testing of H. LEARNING AND TESTING DISTRIBUTIONS. The context of learning (cf., [25]) and testing distributions offers a dramatic demonstration to the importance of a promise (i.e., the fact that the learning algorithm is required to work only when the target belongs to the class, whereas the testing algorithm needs to work for all targets which are either in the class or far away from it). Proposition 2.6 There exist distribution classes which are efficiently learnable (in both senses mentioned above) but cannot be tested with a subexponential sample (regardless of the running-time). 3. Testing Graph Properties We concentrate on testing graph properties using queries and with respect to the uniform distribution. We consider undirected, simple graphs (no multiple edges or self-loops). For a simple graph G, we denote by V(G) its vertex set and assume, without loss of generality, that jV(G)jg. The graph G is represented by the (symmetric) Boolean function where g(u; only if there is an edge between u and v in G. This brings us to associated undirected graphs with directed graphs, where each edge in the undirected graph is associated with a pair of anti-parallel edges. Specifically, for a graph G, we denote by E(G) the set of ordered pairs which correspond to edges in G (i.e., (u; v) 2 E(G) iff there is an edge between u and v in G). The distance between two N -vertex graphs, G 1 and G 2 , is defined as the number of entries which are in the symmetric difference of E(G 1 ) and E(G 2 ). We denote This notation is extended naturally to a set, C, of N -vertex graphs; that is, dist(G; C) )g. 3.1. Testing k-Colorability In this subsection we present an algorithm for testing the k-Colorability property for any given k. Namely, we are interested in determining if the vertices of a graph G can be colored by k colors so that no two adjacent vertices are colored by the same color, or if any k-partition of the graph has at least fflN 2 violating edges (i.e. edges between pairs of vertices which belong to the same side of the partition). The test itself is straightforward. We uniformly select a sample, denoted X, of O vertices of the graph, query all pairs of vertices in X to find which are edges in G, and check if the induced subgraph is k-Colorable. In lack of efficient algorithms for k-Colorability, for k - 3, we use the obvious exponential-time algorithm on the induced subgraph. The resulting algorithm is called the k-Colorability Testing Algorithm. Towards analyzing it, we define violating edges and good k-partitions. 6 Definition 3.1.1 (violating edges and good k-partitions): We say that an edge (u; v) 2 E(G) is a violating edge with respect to a k-partition say that a k-partition is ffl-good if it has at most fflN 2 violating edges (otherwise it is ffl-bad). The partition is perfect if it has no violating edges. Theorem 3.1 The k-Colorability Testing Algorithm is a property testing algorithm for the class of k-Colorable graphs whose query complexity is poly(k log(1=ffi)=ffl) and whose running time is exponential in its query complexity. If the tested graph G is k-Colorable, then it is accepted with probability 1, and with probability at least 1 \Gamma ffi (over the choice of the sampled vertices), it is possible to construct an ffl-good k-partition of V(G) in time poly(k log(1=ffi)=ffl) \Delta jV(G)j. Proof: If G is k-Colorable then every subgraph of G is k- Colorable, and hence G will always be accepted. The crux of the proof is to show that every G which is ffl-far from the class of k-Colorable graphs, denoted G k , is rejected with probability at least We establish this claim by proving its counter-positive. Namely, that every G which is accepted with probability greater than ffi , must have an ffl-good k-partition (and is thus ffl-close to G k ). This is done by giving a (construc- tive) proof of the existence of an ffl-good k-partition of V(G). Hence, in case G 2 G k , we also get an efficient probabilistic procedure for finding an ffl-good k-partition of V(G). Note that if the test rejects G then we have a certificate that in form of the (small) subgraph induced by X which is not k-colorable. We view the set of sampled vertices X as a union of two disjoint sets U and S, where U is a union of ' (disjoint) sets , each of size m. The size of S is m as well, where 4k=ffl. The set U (or rather a k-partition of U) is used to define a k-partition of V(G). The set S ensures that with high probability, the k-partition of U which is induced by the perfect k-partition of defines an ffl-good partition of V(G). In order to define a k-partition of V(G) given a k-partition of U, we first introduce the notion of a clustering of the vertices in V(G) with respect to this partition of U. More precisely, we define the clustering based on the k-partition of a subset U, where this partition, denoted (U 0 k ), is the one induced by the k-partition of U. The clustering is defined so that vertices in the same cluster have neighbors on the 6 k-partitions are associated with mappings of the vertex set into the canonical k-element set [k]. The partition associated with shall use the mapping notation -, and the explicit partition notation same sides of the partition of U 0 . For every A ' [k], the A- cluster, denoted CA , contains all vertices in V(G) which have neighbors in U 0 i for every i 2 A (and do not have neighbors in the other U 0 's). The clusters impose restrictions on possible extensions of the partition of U 0 to partitions of all V(G), which do not have violating edges incident to vertices in U 0 . Namely, vertices in CA should not be placed in any V i such that i 2 A. As a special case, C ; is the set of vertices that do not have any neighbors in U 0 (and hence can be put on any side of the partition). In the other extreme, C [k] is the set of vertices that in any extension of the partition of U 0 will cause violations. For each i, the vertices in C [k]nfig are forced to be put in V i , and thus are easy to handle. It is more difficult to deal with the the clusters CA where jAj Definition 3.1.2 (clusters): Let U 0 be a set of vertices, and let - 0 be a perfect k-partition of U 0 . Define U 0 ig. For each subset A ' [k] we define the A-cluster with respect to - 0 as follows: The relevance of the above clusters becomes clear given the following definitions of extending and consistent partitions. Definition 3.1.3 (consistent extensions): Let U 0 and - 0 be as above. We say that a k-partition - of V(G) extends a k-partition - 0 of U 0 if extended partition - is consistent with - 0 if -(v) 6= - 0 (u) for every is the [k]-cluster Thus, each vertex v in the cluster CA (w.r.t - 0 defined on forced to satisfy -(v) 2 - every k-partition - which extends - 0 in a consistent manner. There are no restrictions regarding vertices in C ; and vertices in C [k] (the latter is guaranteed artificially in the definition and the consequences will have to be treated separately). For the consistency condition forces We now focus on the main problem of the analysis. Given a k-partition of U, what is a good way to define a k-partition of V(G)? Our main idea is to claim that with high probability the set U contains a subset U 0 so that the clusters with respect to the induced k-partition of U 0 determine whatever needs to be determined. That is, if these clusters allow to place some vertex on a certain side of the partition, then doing so does not introduce too many violating edges. The first step in implementing this idea is the notion of a restricting vertex. Definition 3.1.4 (restricting vertex): A pair (v; i), where is said to be restricting with respect to a k-partition - 0 (of U 0 ) if v has at least ffl neighbors 7 In the Bipartite case, this is easy too (since C ; is likely to contain few vertices of high degree). in [B:i= 2B CB . Otherwise, (v; i) is non-restricting. A vertex restricting with respect to - 0 if for every A the pair (v; i) is restricting. Otherwise, v is non-restricting. As always, the clusters are with respect to - 0 . Thus, a vertex v 2 CA is restricting if for every adding v to U 0 (and thus to U 0 ) will cause may of its neighbors to move to a cluster corresponding to a bigger subset. That is, v's neighbors in the B-cluster (w.r.t (U 0 move to the (B [ fig)-cluster (w.r.t (U 0 Given a prefect k-partition of U, we construct U 0 in steps starting with the empty set. At step j we add to U 0 a vertex which is a restricting vertex with respect to the k-partition of the current set U 0 . If no such vertex exists, the procedure terminates. When the procedure terminates (and as we shall see it must terminate after at most ' steps), we will be able to define, based on the k-partition of the final U 0 , an ffl-good k-partition of V(G). The procedure defined below is viewed at this point as a mental experiment. Namely, it is provided in order to show that with high probability there exists a subset U 0 of U with certain desired properties (which we later exploit). Restriction Procedure (Construction of U 0 ) Input: a perfect k-partition of 1. U 0 ;. 2. For do the following. Consider the current set U 0 and its partition - 0 (induced by the perfect k-partition of U). ffl If there are less than (ffl=8)N restricting vertices with respect to - 0 then halt and output U 0 . ffl If there are at least (ffl=8)N restricting vertices but there is no restricting vertex in U j , then halt and output error. ffl Otherwise (there is a restricting vertex in U j ), add the first (by any fixed order) restricting vertex to U 0 . 3.1.5 For every U and a perfect k-partition of U, after at most iterations, the Restriction Procedure halts and outputs either U 0 or error. The proof of this claim, as well as all other missing proofs, can be found in our report [22]. Before we show how U 0 can be used to define a k-partition - of V(G), we need to ensure that with high probability, the restriction procedure in fact outputs a set U 0 and not error. To this end, we first define the notion of a covering set. Definition 3.1.6 (covering sets - for k-coloring): We say that U is a covering set for V(G), if for every perfect k-partition of U, the Restriction Procedure, given this partition as input, halts with an output U 0 ae U (rather than an error message). In other words, U is such that for every perfect k-partition of U and for each of the at most ' iterations of the procedure, if there exist at least (ffl=8)N restricting vertices with respect to the current partition of U 0 , then U j will include at least one such restricting vertex. Lemma 3.1.7 With probability at least 1 \Gamma ffi, a uniformly chosen set of size ' \Delta is a covering set. Definition 3.1.8 (closed partitions): Let U 0 be a set and - 0 a k-partition of it. We call (U closed if there are less than (ffl=8)N restricting vertices with respect to - 0 . Clearly, if the Restriction Procedure outputs a set U 0 then this set together with its (induced) partition are closed. If (U is closed, then most of the vertices in V(G) are non-restricting. Recall that a non-restricting vertex v, belonging to a cluster [k], has the following property. There exists at least one index A, such that (v; i) is non-restricting. It follows from Definition 3.1.4 that for every consistent extension of - 0 to - which satisfies there are at most fflN violating edges incident to v. 8 However, even if v is non-restricting there might be indices A such that (v; i) is restricting, and hence there may exist a consistent extensions of - 0 to - which in which there are more than ffl violating edges incident to v. Therefore, we need to define for each vertex its set of forbidden indices which will not allow to have restricting pair (v; i). Definition 3.1.9 (forbidden sets): Let (U closed and consider the clusters with respect to - 0 . For each v 2 V(G) n U 0 we define the forbidden set of v, denoted F v , as the smallest set satisfying ffl For every if v has at least (ffl=4)N neighbors in the clusters CB for which 2 B, then i is in F v . For Lemma 3.1.10 Let (U be an arbitrary closed pair and 's be as in Definition 3.1.9. Then: 8 N . 2. Let - be any k-partition of V(G) that Then, the number of edges (v; v 0 is at most (ffl=2)N 2 . First note that by definition of a consistent extension no vertex in cluster CB , where i 2 B, can have -value i. Thus, all violated edges incident to v are incident to vertices in clusters CB so that B. Since the pair (v; i) is non-restricting, there are at most fflN such edges. The lemma can be thought of as saying that any k-partition which respects the forbidden sets is good (i.e., does not have many violating edges). However, the partition applies only to vertices for which the forbidden set is not [k]. The first item tells us that there cannot be many such vertices which do not belong to the cluster C [k] . We next show that, with high probability over the choice of S, the k-partition - 0 of U 0 (induced by the k-partition of U[S) is such that C [k] is small. This implies that all the vertices in C [k] (which were left out of the partition in the previous lemma) can be placed in any side without contributing too many violating edges (which are incident to them). Definition 3.1.11 (useful k-partitions): We say that a pair 8 N . Otherwise it is ffl-unuseful. The next claim directly follows from our choice of m and the above definition. 3.1.12 Let U 0 be a fixed set of size ' and - 0 be a fixed k-partition of U 0 so that (U S be a uniformly chosen set of size m. Then, with probability at least ffik \Gamma' , there exists no perfect k-partition of U 0 [ S which extends - 0 . The following is a corollary to the above claim and to the fact that the number of possible closed pairs (U by all possible k-partitions of U is at most k ' . Corollary 3.1.13 If all closed pairs (U are determined by all possible k-partitions of U are unuseful, then with probability at least over the choice of S, there is no perfect k-partition of We can now wrap up the proof of Theorem 3.1. If G is accepted with probability greater than ffi , then by Lemma 3.1.7, the probability that it is accepted and U is a covering set is greater than ffi =2. In particular, there must exist at least one covering set U, such that if U is chosen then G is accepted with probability greater than ffi =2 (with respect to the choice of S). That is, (with probability greater than ffi =2) there exists a perfect partition of U [ S. But in such a case (by applying Corollary 3.1.13), there must be a useful closed pair (U (where U 0 ae U). If we now partition V(G) as described in Lemma 3.1.10, where vertices with forbidden set [k] are placed arbitrarily, then from the two items of Lemma 3.1.10 and the usefulness of (U that there are at most fflN 2 violating edges with respect to this partition. This completes the main part of the proof. (Theorem 3.1) 3.2. Testing Max-Clique Let !(G) denote the size of the largest clique in graph G, and C ae jV(G)jg be the set of graphs having cliques of density at least ae. The main result of this subsection is: Theorem 3.2 Let ' There exists a property testing algorithm, A, for the class C ae whose edge-query complexity is O(' 2 ae 2 =ffl 6 ) and whose running time is exp('ae=ffl 2 ). In particular, A uniformly selects O(' 2 ae 2 =ffl 4 ) vertices in G and queries the oracle only on the existence of edges between these vertices. In case G 2 C ae , one can also retrieve in time set of ae \Delta jV(G)j vertices in G which is almost a clique (in the sense that it lacks at most ffl \Delta jV(G)j 2 edges to being a clique). Theorem 3.2 is proven by presenting a seemingly unnatural algorithm/tester (see below). However, as a corollary, we observe that "the natural" algorithm, which uniformly selects poly(log(1=ffi)=ffl) many vertices and accepts iff they induce a subgraph with a clique of density ae \Gamma ffl 2 , is a valid C ae -tester as well. Corollary 3.3 Let R be a uniformly selected set of m vertices in V (G). Let GR be the subgraph (of G) induced by R. Then, 2In the rest of this subsection we provide a motivating discussion to the algorithm asserted in Theorem 3.2. Recall that jV(G)j denotes the number of vertices in G. Our first idea is to select at random a small sample U of V(G) and to consider all subsets U 0 of size ae\Delta jUj of U where poly(1=ffl). For each U 0 let T(U 0 ) be the set of all vertices which neighbor every vertex in U 0 (i.e., \Gamma(u)). In the subgraph induced by T(U 0 ), consider the set Y(U 0 ) of aeN vertices with highest degree in the induced subgraph. Clearly, if G is ffl-far from C ae , then at least fflN 2 edges to being a clique (for every choice of U and U 0 ). On the other hand, we show that if G has a clique C of size aeN then, with high probability over the choice of U, there exists a subset U 0 ae U such that Y(U 0 ) misses at most (ffl=3)N 2 to being a clique (in particular, U will do). Assume that for any fixed U 0 we could sample the vertices in Y(U 0 ) and perform edge queries on pairs of vertices in this sample. Then, a sample of O(t=ffl 2 ) vertices (where suffices for approximating the edge density in Y(U 0 ) to within an ffl=3 fraction with probability In particular a sample can distinguish between a set Y(U 0 ) which is far from being a clique and a set Y(U 0 ) which is almost a clique. The point is that we need only consider possible sets is only a polynomial in 1=ffl. The only problem which remains is how to sample from Certainly, we can sample sampling V(G) and testing membership in T, but how do we decide which vertex is among those of highest degree? The first idea is to estimate the degrees of vertices in T using an additional sample, denoted W. Thus, instead of considering the aeN vertices of highest degree in T, we consider the aeN vertices in T having the most neighbors in T " W. The second idea is that we can sample T, order vertices in this sample according to the number of neighbors in T " W, and take the ae fraction with the most such neighbors. 3.3. The General Partition Problem The following General Graph Partition property generalizes all properties considered in previous subsections. In particular, it captured any graph property which requires the existence of partitions satisfying certain fixed density constraints. These constraints may refer both to the number of vertices on each side of the partition and to the number of edges between each pair of sides. ae lb be a set of non-negative parameters so that ae lb GP \Phi be the class of graphs which have a k-way partition denotes the set of edges with one endpoint in V j and one in V j 0 . That is, Eq. (3) places lower and upper bounds on the relative sizes of the various parts; whereas Eq. (4) imposes lower and upper bounds on the density of edges among the various pairs of parts. For example, k-colorability is expressed by setting % ub setting ae lb ae ub similarly setting the % xx 's for j 0 6= j). Theorem 3.4 There exists an algorithm A such that for every given set of parameters \Phi, algorithm A is a property testing algorithm for the class GP \Phi with query complexity log(k=fflffi), and running time exp Recall that better complexities for Max-CUT and Bisection (as well as for k-Colorability and ae-Clique), are obtained by custom-made algorithms. Acknowledgments We wish to thank Noga Alon, Ravi Kannan, David Karger and Madhu Sudan for useful discussions. --R The algorithmic aspects of the regularity lemma. A new rounding procedure for the assignment problem with applications to dense graph arrangement problems. Polynomial time approximation schemes for dense instances of NP-hard problems Proof verification and intractability of approximation problems. Probabilistic checkable proofs: A new characterization of NP. Checking computations in polylogarithmic time. Linearity testing in characteristic two. Free bits Efficient probabilistically checkable proofs and applications to approximation. Improved non-approximability results Can finite samples detect singularities of real-valued functions? In 24th STOC Training a 3-node neural network is NP-complete Learnability and the Vapnik-Chervonenkis dimension On determining the rationality of the mean of a random variable. The complexity of colouring problems on dense graphs. Approximating clique is almost NP-complete The regularity lemma and approximation schemes for dense problems. Property testing and its connection to learning and approximation. Distinguishability of sets of distributions. On the learnability of discrete distributions. Toward efficient agnostic learning. On probably correct classification of concepts. Lecture notes on evasiveness of graph properties. The hardness of approximations: Gap location. Computational limitations on learning from examples. The minimum consistent DFA problem cannot be approximated within any polynomial. On recognizing graph properties from adjacency matrices. On the time required to recognize properties of graphs: A problem. Robust functional equations and their applications to program testing. Robust characterization of polynomials with applications to program testing. A theory of the learnable. On the uniform convergence of relative frequencies of events to their probabilities. Probably almost discriminative learning. Lower bounds to randomized algorithms for graph properties. A general classification rule for probability mea- sures --TR A theory of the learnable Using dual approximation algorithms for scheduling problems theoretical and practical results The complexity of colouring problems on dense graphs A polynomial approximation scheme for scheduling on uniform processors: Using the dual approximation approach Computational limitations on learning from examples Learnability and the Vapnik-Chervonenkis dimension Training a 3-node neural network in NP-complete Checking computations in polylogarithmic time Self-testing/correcting for polynomials and for approximate functions Approximating clique is almost NP-complete (preliminary version) Can finite samples detect singularities of real-valued functions? Toward efficient agnostic learning The minimum consistent DFA problem cannot be approximated within any polynomial Small-bias probability spaces Efficient probabilistically checkable proofs and applications to approximations On probably correct classification of concepts Self-testing/correcting with applications to numerical problems The algorithmic aspects of the regularity lemma Improved non-approximability results On the learnability of discrete distributions The hardness of approximation Probably Almost Discriminative Learning Polynomial time approximation schemes for dense instances of <italic>NP</italic>-hard problems MAX-CUT has a randomized approximation scheme in dense graphs Testing of the long code and hardness for clique Adaptively secure multi-party computation Some optimal inapproximability results Property testing in bounded degree graphs Spot-checkers A sublinear bipartiteness tester for bounded degree graphs Recycling queries in PCPs and in linearity tests (extended abstract) Testing problems with sub-learning sample complexity Fast Probabilistic Algorithms for Verification of Polynomial Identities Robust Characterizations of Polynomials withApplications to Program Testing Linearity testing in characteristic two Free bits, PCPs and non-approximability-towards tight results Clique is hard to approximate within n1- A new rounding procedure for the assignment problem with applications to dense graph arrangement problems The regularity lemma and approximation schemes for dense problems Probabilistically checkable proofs and the testing of hadamard-like codes --CTR Michal Parnas , Dana Ron , Ronitt Rubinfeld, Testing membership in parenthesis languages, Random Structures & Algorithms, v.22 n.1, p.98-138, January Oren Ben-Zwi , Oded Lachish , Ilan Newman, Lower bounds for testing Euclidean Minimum Spanning Trees, Information Processing Letters, v.102 n.6, p.219-225, June, 2007 Michal Parnas , Dana Ron, Testing the diameter of graphs, Random Structures & Algorithms, v.20 n.2, p.165-183, March 2002 Hana Chockler , Dan Gutfreund, A lower bound for testing juntas, Information Processing Letters, v.90 n.6, p.301-305, Uriel Feige , Gideon Schechtman, On the integrality ratio of semidefinite relaxations of MAX CUT, Proceedings of the thirty-third annual ACM symposium on Theory of computing, p.433-442, July 2001, Hersonissos, Greece J. 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Fernandez de la Vega , Ravi Kannan , Marek Karpinski, Random sampling and approximation of MAX-CSP problems, Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, May 19-21, 2002, Montreal, Quebec, Canada Eldar Fischer, On the strength of comparisons in property testing, Information and Computation, v.189 n.1, p.107-116, 25 February 2004 Alon , Asaf Shapira, Testing satisfiability, Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms, p.645-654, January 06-08, 2002, San Francisco, California Alon, Testing subgraphs in large graphs, Random Structures & Algorithms, v.21 n.3-4, p.359-370, October 2002 Eldar Fischer , Ilan Newman, Testing versus estimation of graph properties, Proceedings of the thirty-seventh annual ACM symposium on Theory of computing, May 22-24, 2005, Baltimore, MD, USA Hana Chockler , Orna Kupferman, -Regular languages are testable with a constant number of queries, Theoretical Computer Science, v.329 n.1-3, p.71-92, 13 December 2004 Eldar Fischer , Arie Matsliah, Testing graph isomorphism, Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm, p.299-308, January 22-26, 2006, Miami, Florida Artur Czumaj , Christian Sohler, Estimating the weight of metric minimum spanning trees in sublinear-time, Proceedings of the thirty-sixth annual ACM symposium on Theory of computing, June 13-16, 2004, Chicago, IL, USA Eldar Fischer , Ilan Newman, Testing of matrix properties, Proceedings of the thirty-third annual ACM symposium on Theory of computing, p.286-295, July 2001, Hersonissos, Greece Alon , Asaf Shapira, Testing satisfiability, Journal of Algorithms, v.47 n.2, p.87-103, July Ioannis Giotis , Venkatesan Guruswami, Correlation clustering with a fixed number of clusters, Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm, p.1167-1176, January 22-26, 2006, Miami, Florida Michal Parnas , Dana Ron, Testing metric properties, Information and Computation, v.187 n.2, p.155-195, 15 December Christian Borgs , Jennifer Chayes , Lszl Lovsz , Vera T. Ss , Balzs Szegedy , Katalin Vesztergombi, Graph limits and parameter testing, Proceedings of the thirty-eighth annual ACM symposium on Theory of computing, May 21-23, 2006, Seattle, WA, USA Ccile Germain-Renaud , Dephine Monnier-Ragaigne, Grid result checking, Proceedings of the 2nd conference on Computing frontiers, May 04-06, 2005, Ischia, Italy Robert Krauthgamer , Ori Sasson, Property testing of data dimensionality, Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms, January 12-14, 2003, Baltimore, Maryland Eldar Fischer , Guy Kindler , Dana Ron , Shmuel Safra , Alex Samorodnitsky, Testing juntas, Journal of Computer and System Sciences, v.68 n.4, p.753-787, June 2004 Beate Bollig, A large lower bound on the query complexity of a simple boolean function, Information Processing Letters, v.95 n.4, p.423-428, 31 August 2005 Alon , Asaf Shapira, Linear equations, arithmetic progressions and hypergraph property testing, Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms, January 23-25, 2005, Vancouver, British Columbia Michal Parnas , Dana Ron, Testing metric properties, Proceedings of the thirty-third annual ACM symposium on Theory of computing, p.276-285, July 2001, Hersonissos, Greece Y. Kohayakawa , V. Rdl , L. Thoma, An optimal algorithm for checking regularity: (extended abstract), Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms, p.277-286, January 06-08, 2002, San Francisco, California Artur Czumaj , Christian Sohler, Soft kinetic data structures, Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms, p.865-872, January 07-09, 2001, Washington, D.C., United States Eli Ben-Sasson , Oded Goldreich , Prahladh Harsha , Madhu Sudan , Salil Vadhan, Robust pcps of proximity, shorter pcps and applications to coding, Proceedings of the thirty-sixth annual ACM symposium on Theory of computing, June 13-16, 2004, Chicago, IL, USA Cristina Bazgan , W. Fernandez de la Vega , Marek Karpinski, Polynomial time approximation schemes for dense instances of minimum constraint satisfaction, Random Structures & Algorithms, v.23 n.1, p.73-91, August Eldar Fischer , Eric Lehman , Ilan Newman , Sofya Raskhodnikova , Ronitt Rubinfeld , Alex Samorodnitsky, Monotonicity testing over general poset domains, Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, May 19-21, 2002, Montreal, Quebec, Canada Eldar Fischer , Ilan Newman , Ji Sgall, Functions that have read-twice constant width branching programs are not necessarily testable, Random Structures & Algorithms, v.24 n.2, p.175-193, March 2004 P. Drineas , A. Frieze , R. Kannan , S. Vempala , V. Vinay, Clustering Large Graphs via the Singular Value Decomposition, Machine Learning, v.56 n.1-3, p.9-33 Harry Buhrman , Lance Fortnow , Ilan Newman , Hein Rhrig, Quantum property testing, Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms, January 12-14, 2003, Baltimore, Maryland Eli Ben-Sasson , Madhu Sudan , Salil Vadhan , Avi Wigderson, Randomness-efficient low degree tests and short PCPs via epsilon-biased sets, Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, June 09-11, 2003, San Diego, CA, USA Nikhil Bansal , Avrim Blum , Shuchi Chawla, Correlation Clustering, Machine Learning, v.56 n.1-3, p.89-113 Gereon Frahling , Christian Sohler, Coresets in dynamic geometric data streams, Proceedings of the thirty-seventh annual ACM symposium on Theory of computing, May 22-24, 2005, Baltimore, MD, USA Abraham D. Flaxman , Alan M. Frieze, The diameter of randomly perturbed digraphs and some applications, Random Structures & Algorithms, v.30 n.4, p.484-504, July 2007 Ravi Kumar , Ronitt Rubinfeld, Algorithms column: sublinear time algorithms, ACM SIGACT News, v.34 n.4, December W. Fernandez de la Vega , Marek Karpinski , Claire Kenyon , Yuval Rabani, Approximation schemes for clustering problems, Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, June 09-11, 2003, San Diego, CA, USA V. Rdl , M. Schacht, Property testing in hypergraphs and the removal lemma, Proceedings of the thirty-ninth annual ACM symposium on Theory of computing, June 11-13, 2007, San Diego, California, USA Viraj Kumar , Mahesh Viswanathan, Conformance testing in the presence of multiple faults, Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms, January 23-25, 2005, Vancouver, British Columbia Michael A. Bender , Dana Ron, Testing properties of directed graphs: acyclicity and connectivity, Random Structures & Algorithms, v.20 n.2, p.184-205, March 2002 Alon , Asaf Shapira, Every monotone graph property is testable, Proceedings of the thirty-seventh annual ACM symposium on Theory of computing, May 22-24, 2005, Baltimore, MD, USA Michal Parnas , Dana Ron , Ronitt Rubinfeld, Tolerant property testing and distance approximation, Journal of Computer and System Sciences, v.72 n.6, p.1012-1042, September 2006 Artur Czumaj , Funda Ergn , Lance Fortnow , Avner Magen , Ilan Newman , Ronitt Rubinfeld , Christian Sohler, Sublinear-time approximation of Euclidean minimum spanning tree, Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms, January 12-14, 2003, Baltimore, Maryland Ccile Germain-Renaud , Nathalie Playez, Result checking in global computing systems, Proceedings of the 17th annual international conference on Supercomputing, June 23-26, 2003, San Francisco, CA, USA Alon , Asaf Shapira, Testing subgraphs in directed graphs, Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, June 09-11, 2003, San Diego, CA, USA Oded Goldreich , Madhu Sudan, Locally testable codes and PCPs of almost-linear length, Journal of the ACM (JACM), v.53 n.4, p.558-655, July 2006 Oded Goldreich, Property testing in massive graphs, Handbook of massive data sets, Kluwer Academic Publishers, Norwell, MA, 2002 Fast approximate probabilistically checkable proofs, Information and Computation, v.189 n.2, p.135-159, March 15, 2004 Fast approximate PCPs, Proceedings of the thirty-first annual ACM symposium on Theory of computing, p.41-50, May 01-04, 1999, Atlanta, Georgia, United States M. Kiwi, Algebraic testing and weight distributions of codes, Theoretical Computer Science, v.299 n.1-3, p.81-106, Artur Czumaj , Christian Sohler, Testing hypergraph colorability, Theoretical Computer Science, v.331 n.1, p.37-52, 15 February 2005 Asaf Shapira, A combinatorial characterization of the testable graph properties: it's all about regularity, Proceedings of the thirty-eighth annual ACM symposium on Theory of computing, May 21-23, 2006, Seattle, WA, USA Nina Mishra , Dana Ron , Ram Swaminathan, A New Conceptual Clustering Framework, Machine Learning, v.56 n.1-3, p.115-151

approximation algorithms;computational learning theory;graph algorithms

285064

A Theory for Total Exchange in Multidimensional Interconnection Networks.

AbstractTotal exchange (or multiscattering) is one of the important collective communication problems in multiprocessor interconnection networks. It involves the dissemination of distinct messages from every node to every other node. We present a novel theory for solving the problem in any multidimensional (cartesian product) network. These networks have been adopted as cost-effective interconnection structures for distributed-memory multiprocessors. We construct a general algorithm for single-port networks and provide conditions under which it behaves optimally. It is seen that many of the popular topologies, including hypercubes, k-ary n-cubes, and general tori satisfy these conditions. The algorithm is also extended to homogeneous networks with 2k dimensions and with multiport capabilities. Optimality conditions are also given for this model. In the course of our analysis, we also derive a formula for the average distance of nodes in multidimensional networks; it can be used to obtain almost closed-form results for many interesting networks.

Introduction Multidimensional (or cartesian product) networks have prevailed the interconnection network design for distributed memory multiprocessors both in theory and in practice. Commercial machines like the Ncube, the Cray T3D, the Intel iPSC, Delta and Paragon, have a node interconnection structure based on multidimensional networks such as hypercubes, tori and meshes. These networks are based on simple basic dimensions: linear arrays in meshes [15], rings in k-ary n-cubes [6] and general tori, complete graphs in generalized hypercubes [4]. Structures with quite powerful dimensions have also been proposed, e.g. products of trees or products of graphs based on groups [21, 9]. One important issue related to multiprocessor interconnection networks is that of information dissemination. Collective communications for distributed-memory multiprocessors have recently received considerable attention, as for example is evident from their inclusion in the Message Passing Interface standard [18] and from their support of various constructs in High Performance Fortran [12, 16]. This is easily justified by their frequent appearance in parallel numerical algorithms [11, 13, 3]. Broadcasting, scattering, gathering, multinode broadcasting and total exchange constitute a set of representative collective communication problems that have to be efficiently solved in order to maximize the performance of message-passing parallel programs. A general survey regarding such communications was given in [10]. In total exchange, which is also known as multiscattering or all-to-all personalized communication, each node in a network has distinct messages to send to all the other nodes. Various data permutations occurring e.g. in parallel FFT and basic linear algebra algorithms can be viewed as instances of the total exchange problem [3]. The subject of this work is the development of a general theory for solving the total exchange problem in multidimensional networks. A multitude of quantities or properties in such networks can be decomposed to quantities and properties of the individual dimensions. For example, the degree of a node is the sum of the degrees in each of the dimensions. We show here that the total exchange problem can also be decomposed to the simpler problem of performing total exchange in single dimensions. This is a major simplification to an inherently complex problem for inherently complex networks. We provide general algorithms applicable to any multidimensional network A Theory for Total Exchange in Multidimensional Interconnection Networks given that we have total exchange algorithms for each dimension. Optimality conditions are given and it is seen that they are met for many popular networks, e.g. hypercubes, tori and generalized hypercubes to name a few. The results presented here apply to packet-switched networks that follow the so-called constant model [10]. The assumptions pertaining the model we will follow are: ffl communication links are bidirectional and fully duplex ffl a message requires one time unit (or step) to be transferred between two nodes ffl only adjacent nodes can exchange messages. Another parameter of the model is that of port capabilities. Depending on whether a node can communicate with one or all of its neighbors at the same time unit, two basic possibilities arise: Single-port: a node can send at most one message and receive at most one message at each step. Multiport: a node can send and receive messages from all its neighbors simultaneously. As discussed in [10], the above assumptions constitute the standard model when examining theoretical aspects of communications in packet-switched networks. Furthermore, results and conclusions under this model can form the basis of arguments for other models, such as the linear one which also quantifies the effect of message lengths. Many recent works focus exclusively on wormhole-routed networks (an excellent survey on collective communications for such machines was given in [17]). However, we believe that studies should not be limited to one particular type of architecture: "it is important to consider several types of communication models, given the broad variety of present and future communication hardware" [2]. In addition, since a circuit-switched or wormhole routed network can emulate a packet-switched network by performing only nearest-neighbor communications, the results also constitute a reference point for methods developed for the former type of networks. Algorithms to solve the total exchange problem for specific networks and under a variety of assumptions have appeared in many recent works, mostly concentrating in hypercubes and two-dimensional tori (e.g. [22, 14, 2, 23]). Under the single-port model we know of two optimal algorithms, in [3, pp. 81-83] for hypercubes, and in [19] for star graphs. In contrast, our results are applicable not only to one particular structure but rather provide a general procedure for solving the problem in any multidimensional network. 2. Multidimensional Networks 3 This paper is organized as follows. We introduce formally multidimensional networks in the next section and we give some of their properties related to our study. Section 3 gives lower bounds on the time required for solving the total exchange problem under both port assumptions. In the same section we derive a new formula for the single-port bound as applied to the networks of interest. The result has its own merit as it also provides almost closed-form formulas for the average distance in networks for which no such formula was known up to now. In Section 4 we concentrate on single-port networks. We develop a total exchange algorithm and we give conditions under which it behaves optimally. We also review known results about simple dimensions and conclude that our method can be optimally applied to hypercubes, k-ary n-cubes and other popular interconnects. In Section 5 we modify the algorithm and adapt it to the multiport model. The extension works for networks which have dimensions (homogeneous networks). Again, we provide optimality conditions and observe that they are satisfied for a number of interesting topologies. The results are summarized in Section 6. 2. Multidimensional Networks E) be an undirected graph 1 [5] with node (or vertex) set V and edge (or link) set E. This is the usual model of representing a multiprocessor interconnection network: processors correspond to nodes and communication links correspond to edges in the graph. The number of nodes in G is An edge in E between nodes v and u is written as the unordered pair (v; u) and v and u are said to be adjacent to each other, or just neighbors. A path in G from node v to node u, denoted as v ! u, is a sequence of nodes u, such that all vertices are distinct and for all We say that the length of a path is ' if it contains ' vertices apart from v. The distance, dist(v; u), between vertices v and u is the length of a shortest path between v and u. Finally, the eccentricity of v, e(v), is the distance to a node farthest from v, i.e. The maximum eccentricity in G is known as the diameter of G. 1 The terms 'graph' and `network' are considered synonymous here. 4 A Theory for Total Exchange in Multidimensional Interconnection Networks a b Figure 1. Cartesian product of two graphs Given k graphs G product is defined as the graph E) whose vertices are labeled by a k-tuple (v We will call such products of graphs multidimensional graphs and G i will be called the ith dimension of the product. The ith component of the address tuple of a node will be called the ith address digit or the ith coordinate. The definition of E above in simple words states that two nodes are adjacent if they differ in exactly one address digit. Their differing coordinates should be adjacent in the corresponding dimension. An example is given in Fig. 1. Dimension 1 is a graph consisting of a two-node path with consists of a three-node ring with 3g. Their product has node set (a; 1); (a; 2); (a; 3); (b; 1); (b; 2); (b; According to the definition, node (a; 1) has the following neighbors: since node a is adjacent to node b in the first dimension, node (a; 1) will be adjacent to node (b; 1); since node 1 is adjacent to both nodes 2 and 3 in the second dimension, node (a; 1) will also be adjacent to nodes (a; 2) and (a; 3). Hypercubes are products of two-node linear arrays (or rings), tori are products of rings. If all dimensions of the torus consist of the same ring, we obtain k-ary n-cubes [6]. Meshes are products of linear arrays [15]. Generalized hypercubes are products of complete graphs [4]. If all dimensions G i , are identical then the network is characterized as homogeneous. 3. Lower Bounds for Total Exchange 5 Multidimensional graphs have is the number of nodes in k. It is also known that if dist i (v is the distance between v i and u i in G i then the distance between dist It will be convenient to use the don't care symbol `\Lambda' as a shorthand notation for a set of addresses. An appearance of this symbol at an element of an address tuple represents all legal values of this element. In the previous example, (a; while denotes the whole node set of the graph. 3. Lower Bounds for Total Exchange In the total exchange problem, a node v has to send messages, one for each of the other nodes in an n-node network. Let us first assume that the single-port model is in effect. If there exist n d nodes in distance d from v, where then the messages sent by v must cross links in total. For all messages to be exchanged, the total number of link traversals must be The quantity s(v) is known as the total distance or the status [5] of node v. Every time a message is communicated between adjacent nodes one link traversal occurs. Under the single-port model nodes are allowed to transmit only one message per step, so that the maximum number of link traversals in a single step is at most n. Consequently, we can at best subtract n units from SG in each step, so that a lower bound on total exchange time is In other words, total exchange under the single-port assumption requires time bounded below by the average status, AS(G), of the vertices. 6 A Theory for Total Exchange in Multidimensional Interconnection Networks For multiport networks tighter bounds are obtained through cuts of the network. Partition the vertex set V in two disjoint sets V 1 and V 2 such that be the number of edges in E joining the two parts, i.e. edges from nodes in V 1 destined for nodes in V 2 must cross these C V1 V2 edges. The total number of such messages is jV 1 jjV 2 j. Since only C V1V2 messages are able to pass from V 1 to V 2 at a time, we obtain the following lower bound for total exchange time: We are of course interested in maximizing the fraction in the right-hand side by selecting V 1 and appropriately so that the tightest possible bound results. In many cases a bisection of the graph is the most appropriate choice, although any sensible partition will yield quite tight bounds. 3.1. Status in multidimensional networks In the course of our analysis on the single-port model we will need to compare the time needed for total exchange with the lower bound of (3). We present here a formula for the status and the average status of vertices in multidimensional graphs, as required by (3). The results are based on the status of vertices in individual dimensions. Theorem 1 Let is the status of v i in G i , the status of Proof. The status of node v can be calculated through (2) or by using the equivalent formula: where dist(v; u) is the distance between v and u. Hence, the status of v i in G i can be written as dist 3. Lower Bounds for Total Exchange 7 We know that in a multidimensional network the distance between two vertices is equal to the sum of distances between the corresponding coordinates (Eq. (1)). Consequently, from (5) we obtain dist dist ae X dist oe ae X oe ae n oe as claimed. The quantity s(v)=(n \Gamma 1) is known as the average distance of node v, giving the average number of links that have to be traversed by a message departing from v. It is an important performance measure of the network since under uniform addressing distributions it is directly linked with the average delay a message experiences before reaching its destination [20]. Hence, Theorem 1 can also be used to calculate the average distance of vertices in many graphs for which no closed-form formula was known up to now. As an example, in generalized hypercubes [4] each dimension is a complete graph with m i vertices, In a complete graph all nodes are adjacent to each other, so that s i (v Consequently, the average distance in generalized hypercubes is In [4] it was possible to derive a formula only for the case where all m i are equal to each other. In the context of the total exchange problem we are interested in the average status of the nodes in the network. Let AS(G i ) be the average status of G i , defined in (3) as We have the following corollary. 8 A Theory for Total Exchange in Multidimensional Interconnection Networks . If AS(G i ) is the average status of G i , then the average status of G is given by Proof. From Theorem 1 we obtain (v 1 ;:::;v k )2G which, divided by n, gives the required result. 4. Single-port Algorithm A \Theta B. A k-dimensional network G 1 \Theta \Delta \Delta \Delta \Theta G k can still be expressed as the product of two graphs by taking so we may consider two dimensions without loss of generality. Let Finally, let \Phi Graph G consists of n 2 (interconnected) copies of VA . Let A j be the jth copy of A with node set takes all values in VA . Similarly, G can be viewed as n 1 copies of B, and we let be the ith copy of B with node set (v i ; ). An example is shown in Fig. 2. 4. Single-port Algorithm 9 (a) (b) A A A A1 2 3 43 Figure 2. A 4 \Theta 3 torus as (a) four copies of a three-node ring or (b) three copies of a four-node ring We will develop the basic idea behind our algorithm through the example in Fig. 2. Consider the top node of A 1 . This node belongs to A 1 as well as B 1 . All nodes in A 1 have, among other messages, messages destined for the rest of the nodes in A 1 . These messages can be distributed by performing a total exchange within A 1 . In addition, nodes in A 1 have messages for all nodes in A 2 , A 3 and A 4 . Somehow, these messages have to travel to their appropriate destinations. What we will do is the following: all messages of the top node of A 1 meant for the nodes in A 2 will be transferred to the top node of A 2 . All messages of the middle node of A 1 destined for the nodes in A 2 will be transferred to the middle node of A 2 . Similar will be the case for the bottom node of A 1 . Once all these messages have arrived in A 2 , the only thing remaining is to perform a total exchange within A 2 and all these messages will be distributed to the correct destinations. Next, nodes of A 1 have to transfer their messages meant for A 3 to nodes of A 3 . The procedure will be identical to the procedure we followed for messages meant for A 2 . Finally, the remaining messages in A 1 are destined for A 4 and one more repetition of the above procedure will complete the task. Notice that what we did for messages originating at nodes of A 1 has to be done also for messages originating at the other copies of A, i.e. A 2 , A 3 and A 4 . We are now ready to formalize our arguments. We are going to adopt the following the message of node destined for node (v k ; u l ). We will furthermore introduce the ' ' symbol to denote a corresponding set of messages. For example, messages of node (v Theory for Total Exchange in Multidimensional Interconnection Networks 1 For every 2 For every 3 For every 5 For every 6 Do in parallel for all A j , 7 In A j perform total exchange of messages m ( ;u k (messages reside in node (v Figure 3. Algorithm A1 destined for the nodes of A l , and m (v destined for node messages of (v Notice that this last set normally includes nodes. Since no node sends messages to itself, it is always implied that from any set of messages, we have removed every message whose source and destination are the same. Consider the set of messages set represents our total exchange problem: every node has one message for every other node. Next consider the set m ( This is the set of messages of nodes in A j destined for the other nodes in A j : they can be distributed by a total exchange operation within A j . Finally, consider the set m (v for the nodes of A k . This set will be transferred to node (v Thus, after such transfers, node (v and so on. Notice that every node in A k will have received messages meant for every node in A k : these messages clearly can be distributed to the appropriate destinations through a total exchange operation within A k . To recapitulate, we can solve the total exchange problem in using Algorithm A1 shown in Fig. 3. First we perform all the transfers we described above and then we perform the total exchanges within each A j . The transfers correspond to lines 1-4 in Algorithm A1. After they are completed, every node (v i ; u j ), for every i, j, will have received all messages meant for the jth copy of A originating at nodes (v 4. Single-port Algorithm 11 Lines 5-7 of the algorithm distribute these messages to the correct vertices of A j in n 2 rounds. In the kth round a total exchange is performed and the exchanged messages have originated from A k . Algorithm A1 solves the total exchange problem but lines 1-4 do not show how the transfer of messages is exactly implemented. First of all, there may exist path collisions between transfers from (v and transfers from (v i we try to do them simultaneously. Let us consider again the example in Fig. 2. At some point all nodes in A 1 want to transfer their messages, say, for nodes in A 4 . We make the observation that these transfers can indeed be done in parallel. That is, the top node of A 1 can transfer its messages to the top node in A 4 , the middle node of A 1 can transfer its own messages to the middle node of A 4 and so on, without any interference between them. The trick is to use only paths in the second dimension (B). That is, all the transfers of the top node of A 1 use links in B 1 , all transfers from the bottom node of A 1 use links in B 3 , etc. Consequently, a straightforward way of parallelizing line 1 is the following: when transferring messages from (v allow use of links in the second dimension. In other words, the allowable paths (v paths (v have no node in common. Consequently, lines 1-4 can be rewritten in the improved form: 1 Do in parallel for all 2 For every 3 For every using links in B We may still improve matters by further parallelizing lines 1-3. Within B i we need to transfer messages from every vertex u j to every other vertex u k . In Table 1 we list the messages to be transferred by some vertex (v . Notice that we do not have to transfer messages meant for A j anywhere, so the jth column of the table is actually unused (it will only be used for a total exchange within A j ). Column k contains all messages of (v to be transferred first to node (v 12 A Theory for Total Exchange in Multidimensional Interconnection Networks For . . . . . . Table 1. Messages to be transferred from node s actually unused since messages of (v do not have to be transferred to any other copy of A. Instead of transferring the messages column by column (i.e. transfer all messages in column 1 to A 1 , then all messages in column 2 to A 2 , etc.) we transfer them horizontally (row by row). The batch R r of messages in row r contains all messages m (v We will transfer all of them, except of course for m (v which is meant for a node of A j . Let us consider again the network in Fig. 2 and assume that the bottom nodes of A 1 , A 2 , A 3 and A 4 want to transfer their first batch, R 1 . The batch of the bottom node of A 1 contains one message for each of the bottom nodes of A 2 , A 3 and A 4 . Similarly, batch R 1 for the bottom node of A 2 contains one message for the other three nodes in question. It should be immediately clear that these messages constitute an instance of the total exchange problem in every node has one message for every other node in B 1 . In general, when every node (v transfers its own batch R r of Table 1, a total exchange within B i can distribute the messages appropriately. Consequently, all rows of Table 1 of every node will be transferred where they should by performing n 1 total exchanges in at the rth exchange all nodes (v batch of messages (rth row of the corresponding tables). Based on the above discussion, and recalling that transfers within B i do not interfere with transfers within may express our total exchange algorithm in its final form, Algorithm A2, appearing in Fig. 4. Algorithm A2 is a general solution to the total exchange problem for any multidimensional network. If the network has k ? 2 dimensions, 4. Single-port Algorithm 13 2 Do in parallel for all In B i perform total exchange with node (v sending messages m (v 4 For every 5 Do in parallel for all A j , 6 In A j perform total exchange with node (v sending messages m (v Figure 4. Algorithm A2 Algorithm A2 can be used recursively, by taking . The total exchanges in A j (lines 4-6) can be performed by invoking the algorithm with The algorithm is in a highly desirable form: it only utilizes total exchange algorithms for each of the dimensions. The problem of total exchange in a complex network is now reduced to the simpler problem of devising total exchange algorithms for single dimensions. For example, we are in a position to systematically construct algorithms for tori, based on algorithms for rings. We now proceed to determine the time requirements of the algorithm and the conditions under which it behaves optimally. 4.1. Optimality conditions It is not very hard to calculate the time required for Algorithm A2. This is because it is written in a form suitable for the single-port model: every node participates in one total exchange operation at a time. When each total exchange is performed under the single-port model, in effect no node sends/receives more than one message at a time. Theorem 2 If single-port total exchange algorithms for graphs A and B take steps correspondingly then Algorithm A2 for 14 A Theory for Total Exchange in Multidimensional Interconnection Networks time units. Proof. The result is straightforward: lines 1-3 perform n 1 total exchanges within B i (for all parallel), each requiring steps. Similarly, lines 4-6 perform n 2 total exchanges within A j (for all in parallel), each requiring steps. Corollary 2 If and a single-port total exchange algorithm for G i takes total exchange in G under the single-port model can be performed in steps, where Proof. The proof is by induction. If we only have one dimension then the corollary is trivially true. Assume as an induction hypothesis that it holds for up to dimensions. Then we must have where T 0 is the time needed for total exchange in G j. If we let Theorem 2 gives as required. Theorem 3 If single-port total exchange for every dimension can be performed in time equal to the lower bound of (3) then the same is true for G. Proof. If in G i total exchange can be performed in time equal to the lower bound of (3) then From Corollary 2, we must have 5. Multiport Algorithm 15 which, combined with Corollary 1, shows that and the algorithm is thus optimal. The last theorem provides the main optimality condition for Algorithm A2. If we have total exchange algorithms for every dimension and these algorithms achieve the bound of (3) then Algorithm A2 also achieves this bound. For example, in hypercubes every dimension is a two-node graph. Trivially, in a two-node graph the time for total exchange is just one step, equal to the average status. Thus the optimality condition is met and the presented algorithm is an optimal algorithm for single-port hypercubes. More generally, we have shown elsewhere [8] that there exist algorithms that need time equal to (3) for any Cayley [1] network. Consequently, the optimality condition is met for arbitrary products of Cayley networks. Rings and complete graphs are examples of Cayley networks and thus Algorithm A2 solves optimally the total exchange problem in k-ary n-cubes, general tori and generalized hypercubes. 5. Multiport Algorithm In this section we will modify Algorithm A2 to work better under the multiport model. In its present form, Algorithm A2 is not particularly efficient under this model. This is because lines 4-6 are executed after lines 1-3 have finished. During execution of lines 1-3 only edges of the second dimension (B) are used while lines 4-6 use only edges of the first dimension (A). In the multiport model we try to keep as many edges busy as possible and the behavior of Algorithm A2 does not contribute to that effect. We seek, consequently, to transfer messages in both dimensions simultaneously. In other words we will reconstruct the algorithm such that lines 1-3 overlap in time as much as possible with lines 4-6. The theory we present here applies to homogeneous networks. We recall that a multidimensional network is homogeneous when all its dimensions are identical. Thus, H k for some graph H. We will only consider the two-dimensional case, i.e. also be seen that the algorithm we derive is applicable when the dimensionality of the graph is in general a power of 2, i.e. E) where is, G has n 2 nodes. A Theory for Total Exchange in Multidimensional Interconnection NetworksBBB Figure 5. A 3 \Theta 3 homogeneous mesh For A 1 For A 2 For A 3 Table 2. Messages to be transferred from node (1,1) The network in Fig. 5 will be used as an example for our arguments. For node (1; 1) we give the messages it will distribute in Table 2. The messages in the first column are meant for the other nodes in A 1 . A total exchange within A 1 may thus begin immediately to distribute such messages. Since this total exchange uses only links in the first dimension, node (1,1) is also available to participate in some total exchange in the second dimension (i.e. in B 1 ). In a general network, node (v can participate in a total exchange within B i as soon as the first total exchange in A j starts. Within A j the transferred messages are m (v in column j of Table 1. Let us see what messages will be involved in the first total exchange within B i . Our objective is the following: we want every node (v to receive messages so that after this total exchange in B i is done, another total exchange can be initiated within A j . Consequently, we seek to arrange the transfers so that (v receives one message for each node in A j , i.e. receive messages with destinations Notice that any node (v receive messages through a total exchange in B i : since A j has n nodes (including (v all the receptions of (v should be meant for nodes other than (v 5. Multiport Algorithm 17 In the network in Fig. 5, we let for example node (1,1) send m (1;1) (2; 2). This message will at some point be received by node (1,2) and it will provide one message for the forthcoming total exchange in A 2 . If (1,2) sends m (1;2) (2; 3) then node (1,3) will also be provided with one message for total exchange in A 3 . Similarly, needed by node (1,1). We define the following operators: \Theta These operators work like addition/subtraction modulo n but produce numbers ranging from 1 to n instead of 0 to are better suited for our purposes here. Based on this operator and the preceding discussion, we see that one effective scheduling is to let node (v and all Hence, this node will also receive will use for the next total exchange in A j . Let us see what other messages will be sent during this first total exchange in B i . In our example it is seen that since node (1,1) decided to send m (1;1) (2; 2), it cannot send another message to node (1,2). Thus it has to send a message to node (1,3). Since this node will receive which covers one destination in A 3 , the only choice for (1,1) is to send m (1;1) (3; 3). This message completes the set of messages needed by (1,3) for the next total exchange in A 3 since all other vertices in A 3 are now covered. Similarly, (1,2) and (1,3) must send m (1;2) (3; 1) and three nodes will have a complete set of messages, suitable for total exchanges within A 1 , A 2 and A 3 . In general, the second message that node (v provide node (v second message for the total exchange in A j \Phi . The pattern should now be clear: during the first total exchange in B i , every node (v sends the following messages: A Theory for Total Exchange in Multidimensional Interconnection Networks or, in a compact form: This node will provide node (v with the 'th message it needs (i.e. a message destined for node (v i\Phi Notice that the above set contains one message to be received by each node (v i.e. it is a perfect set for participation in the first total exchange in B i . Also, it should be clear that (v receive the following messages: Again notice that this set contains one message for each node (v . Thus we achieved our goal: every node in B i receives a full set of messages to be used for the subsequent total exchange in A j . B, the first total exchange in A j finishes exactly when the first total exchange in B i finishes. Thus the second total exchange in A j can start immediately, using the newly acquired (through the exchange in B i ) messages. Then the story repeats itself: a second total exchange in B i can be performed simultaneously with the second total exchange in A j . Our goal for this total exchange in B i remains the same: to distribute messages that can be used for a third total exchange in A j . The idea behind selecting a group of messages for this second total exchange in B i is similar to the one in the first total exchange we saw above. Now, we let (v The situation is repeated continuously. While the rth total exchange within A j is in progress, the rth total exchange in B i is also performed in order to provide nodes with messages for the next total exchange in A j . During the rth exchange in B i a node (v sends the 5. Multiport Algorithm 19 following messages: Observe that the destinations v i\Phi n ' are in the order given by That is, the natural sequence which we used in the first total exchange in B i is left-rotated by r positions. Based on this observation, it is easy to verify that the above set of messages can be given in the compact form: Similarly, it is seen that after the rth exchange in B i , node (v received messages which can be used during the (r + 1)th total exchange in A j . Let us recapitulate. During the first total exchange in A j , (v taneously, total exchanges in B i start. During the rth exchange in B i the same node sends the set of messages given in (7), and receives the set given in (8). This set will be used for the (r exchange in A j . This will occur for all total exchanges in B i are performed in parallel with the total exchanges in A j . The last (nth) total exchange in A j will involve the messages received during the (n \Gamma 1)th total exchange in B i . It can be noticed that (v i ; u j ) has sent all its messages meant for nodes in all other copies of A, A k (k 6= j), except for nodes (v In the example of Fig. 5, we saw that during the first two exchanges in B 1 , node (1,1) sent all its messages with the exception of messages m (1;1) (1; which are destined for node (1,2) and (1,3). The situation is similar for nodes (1,2) and (1,3). In conclusion, messages m (v are the 20 A Theory for Total Exchange in Multidimensional Interconnection Networks only messages remaining to be sent. Observe that this is a perfect set of messages for a (final) total exchange in B i . This nth exchange can be performed while the nth exchange in A j occurs. What we have described up to now is formulated as Algorithm A3 in Fig. 6. The total exchanges in the copies of A and B are completely parallelized, hence lines 1-3. Lines 4-8 perform the transfers we described above in B i . Lines 9-13 perform the total exchanges in A j . Notice how simple lines 11-13 are: whatever was sent through the rth exchange in B i is used during the (r 1)th exchange in A j . As it is, the algorithm works for any two-dimensional homogeneous network. Extension to more than two dimensions seems rather difficult because the homogeneity will be lost, in the sense that A could be different than B. For example, if can be written as only vice versa. However, it is easy to see that the algorithm is applicable if the dimensionality is a power of 2 . If then we let . The algorithm can then be applied recursively for A and B, by e.g. setting , and so on. We proceed now to determine the time requirements of Algorithm A3 and to give optimality conditions. 5.1. Optimality conditions Theorem 4 If H has n nodes and total exchange in H requires TH steps then Algorithm A3 in Proof. Procedure TEA() performs n total exchanges in A j (for all thus requiring nTH steps. Similarly, TEB() also requires nTH steps. The algorithm finishes when both procedures have finished, i.e. at time By the recursive application of the algorithm for networks where the dimensionality is a power of 2 we have the following corollary. 5. Multiport Algorithm 21 1 Do in parallel 4 For 5 Do in parallel for all 6 Perform total exchange in 7 Do in parallel for all 8 Perform total exchange in 9 Do in parallel for all A j , Perform total exchange in A j : node (v 12 Do in parallel for all A j , Perform total exchange in A j : node (v the messages received from the second dimension (B i ); Figure 6. Algorithm A3 for multiport homogeneous networks: 22 A Theory for Total Exchange in Multidimensional Interconnection Networks Corollary 3 Let . If total exchange in H requires TH time units, then total exchange in G can be performed in steps. Proof. The proof is by induction. The case of two dimensions was covered in Theorem 4. If, as an induction hypothesis, for G apply Theorem 4 with G 0 treated as H, T 0 treated as TH , and n d=2 treated as n. It is then seen that claimed. Theorem 5 Let . If total exchange in H can be performed in time equal to the lower bound of (4) then the same is true for G. Proof. From Corollary 3, total exchange in G requires If TH achieves the lower bound in (4) then there exists a partition VH1 , VH2 of the node set of H such that is the number of links separating the two parts. Consider the following partition of V , the node set of G: Then clearly, . Notice that G contains n d\Gamma1 copies of H and that in order to separate the two parts we only need to disconnect each copy of H, by removing links only in the first dimension. Since C VH 1 VH 2 links are needed to disconnect each copy of H, we obtain Thus, V 1 and V 2 is a partition of G such that 6. Discussion 23 which is equal to T , the time needed for total exchange in G. Thus the bound in (4) is tight for G, too. Summarizing, Algorithm A3 is a multiport total exchange algorithm for homogeneous networks whose dimensionality is a power of 2. If total exchange in H can be performed in time equal to the lower bound of (4) then Algorithm A3 optimally solves the problem in G. For example, in [7] we have given algorithms that achieve this lower bound in linear arrays and rings. Consequently, Algorithm A3 leads to an optimal total exchange algorithm for homogeneous meshes and tori with dimensions. 6. Discussion We have given a systematic procedure for performing total exchange in multidimensional net- works. The main contribution is probably the existence of a decomposition of the problem to simpler subproblems. Given that we have total exchange algorithms for single dimensions, we can synthesize an algorithm for the multidimensional structure. In contrast with all the other works on the problem, this approach is not limited to one particular network but to any graph that can be expressed as a cartesian product. Except for the structured nature of our method, we also showed that it is optimal with respect to the number of communication steps for many popular networks. Under the single-port assumption, Algorithm A2 provides optimal solutions for hypercubes, k-ary n-cubes, general tori and actually any product of Cayley graphs. For most of these networks, this is the first optimal algorithm to appear in the literature. Under the multiport assumption, we reached similar conclusions for homogeneous networks dimensions: Algorithm A3 solves the problem in any such network. Optimality is also guaranteed if the single-dimension algorithm achieves the bound of (4). In particular, based on known results for linear arrays and rings, meshes and k-ary n-cubes with 2 k dimensions can optimally take advantage of our algorithm. We are currently studying the behavior of the algorithm in the case where the number of dimensions is not a power of two. Some preliminary results indicate that the algorithm could still be applicable. A Theory for Total Exchange in Multidimensional Interconnection Networks --R "A group-theoretic model for symmetric interconnection networks," "Optimal communication algorithms for hypercubes," Parallel and Distributed Computation: Numerical Meth- ods "Generalized hypercube and hyperbus structures for a computer network," Distance in Graphs. "Deadlock-free message routing in multiprocessor interconnection networks," "Optimal total exchange in linear arrays and rings," "Optimal total exchange in Cayley graphs," "Methods and problems of communication in usual networks," "On the impact of communication complexity on the design of parallel numerical algorithms," "Compiling Fortran D for MIMD distributed-memory machines," "Communication efficient basic linear algebra computations on hypercube architectures," "Optimum broadcasting and personalized communication in hypercubes," Introduction to Parallel Algorithms and Architectures: Arrays "High Performance Fortran," "Collective communication in wormhole-routed massively parallel computers," "MPI: A message-passing interface standard," "Communication aspects of the star graph interconnection net- work," "The performance of multicomputer interconnection net- works," "Product-shuffle networks: towards reconciling shuffles and butterflies," "Data communications in hypercubes," "Communication algorithms for isotropic tasks in hypercubes and wraparound meshes," --TR --CTR Yu-Chee Tseng , Sze-Yao Ni , Jang-Ping Sheu, Toward Optimal Complete Exchange on Wormhole-Routed Tori, IEEE Transactions on Computers, v.48 n.10, p.1065-1082, October 1999 V. Dimakopoulos, All-port total exchange in cartesian product networks, Journal of Parallel and Distributed Computing, v.64 n.8, p.936-944, August 2004 Shan-Chyun Ku , Biing-Feng Wang , Ting-Kai Hung, Constructing Edge-Disjoint Spanning Trees in Product Networks, IEEE Transactions on Parallel and Distributed Systems, v.14 n.3, p.213-221, March V. Dimakopoulos , Nikitas J. Dimopoulos, Optimal Total Exchange in Cayley Graphs, IEEE Transactions on Parallel and Distributed Systems, v.12 n.11, p.1162-1168, November 2001

total exchange;collective communications;interconnection networks;packet-switched networks;multidimensional networks

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Capabilities-Based Query Rewriting in Mediator Systems.

Users today are struggling to integrate a broad range of information sources providing different levels of query capabilities. Currently, data sources with different and limited capabilities are accessed either by writing rich functional wrappers for the more primitive sources, or by dealing with all sources at a lowest common denominator. This paper explores a third approach, in which a mediator ensures that sources receive queries they can handle, while still taking advantage of all of the query power of the source. We propose an architecture that enables this, and identify a key component of that architecture, the Capabilities-Based Rewriter (CBR). The CBR takes as input a description of the capabilities of a data source, and a query targeted for that data source. From these, the CBR determines component queries to be sent to the sources, commensurate with their abilities, and computes a plan for combining their results using joins, unions, selections, and projections. We provide a language to describe the query capability of data sources and a plan generation algorithm. Our description language and plan generation algorithm are schema independent and handle SPJ queries. We also extend CBR with a cost-based optimizer. The net effect is that we prune without losing completeness. Finally we compare the implementation of a CBR for the Garlic project with the algorithms proposed in this paper.

Introduction Organizations today must integrate multiple heterogeneous information sources, many of which are not conventional SQL database management systems. Examples of such information sources include bibliographic databases, object repositories, chemical structure databases, WAIS servers, etc. Some of these systems provide powerful query capabilities, while others are much more limited. A new challenge for the database community is to allow users to query this data using a single powerful query language, with location transparency, despite the diverse capabilities of the underlying systems. Figure (1.a) shows one commonly proposed integration architecture [1, 2, 3, 4]. Each data source has a wrapper, which provides a view of the data in that source in a common data model. Each wrapper can translate queries expressed in the common language to the language of its underlying information source. The mediator provides an integrated view of the data Research partially supported by Wright Laboratories, Wright Patterson AFB, ARPA Contract F33615-93-C-1337. Wrapper i CBR component subqueries target query Mediator Query Decomposition description of queries supported by wrapper plan for (1.a) Client Mediator Wrapper 1 Wrapper 2 Wrapper n Information Information Information Source n Figure 1: (a) A typical integration architecture. (b) CBR-mediator interaction. exported by the wrappers. In particular, when the mediator receives a query from a client, it determines what data it needs from each underlying wrapper, sends the wrappers individual queries to collect the required data, and combines the responses to produce the query result. This scenario works well when all wrappers can support any query over their data. However, in the types of systems we consider, this assumption is unrealis- tic. It leads to extremely complex wrappers, needed to support a powerful query interface against possibly quite limited data sources. For example, in many systems the relational data model is taken as the common data model, and all wrappers must provide a full SQL interface, even if the underlying data source is a file system, or a hierarchical DBMS. Alternatively, this assumption may lead to a "lowest common denomina- tor" approach in which only simple queries are sent to the wrappers. In this case, the search capabilities of more sophisticated data sources are not exploited, and hence the mediator is forced to do most of the work, resulting in unnecessarily poor performance. We would like to have simple wrappers that accurately reflect the search capabilities of the underlying data source. To enable this, the mediator must recognize differences and limitations in capabilities, and ensure that wrappers receive only queries that they can handle. For Garlic [1], an integrator of heterogeneous multimedia data being developed at IBM's Almaden Re-search Center, such an understanding is essential. Garlic needs to deal efficiently with the disparate data types and querying capabilities needed by applications as diverse as medical, advertising, pharmaceutical research, and computer-aided design. In our model, a wrapper is capable of handling some set of queries, known as the supported queries for that wrap- per. When the mediator receives a query from a client, it decomposes it into a set of queries, each of which references data at a single wrapper. We call these individual queries target queries for the wrappers. A target query need not be a supported query; it may sometimes be necessary to further decompose it into simpler supported Component SubQueries (CSQs) in order to execute it. A plan combines the results of the CSQs to produce the answer to the target query. To obtain this functionality, we are exploring a Capabilities-Based Rewriter (CBR) module (Fig- ure 1.b) as part of the Garlic query engine (media- tor). The CBR uses a description of each wrapper's ability, expressed in a special purpose query capabilities description language, to develop a plan for the wrapper's target query. The mediator decomposes a user's query into target queries q for each wrapper w without considering whether q is supported by w. It then passes q to the CBR for "inspection." The CBR compares q against the description of the queries supported by wrapper and produces a plan p for q, if either (i) q is directly supported by w, or (ii) q is computable by the mediator through a plan that involves selection, projection and join of CSQs that are supported by w. The mediator then combines the individual plans p into a complete plan for the user's query. The CBR allows a clean separation of wrapper capabilities from mediator internals. Wrappers are "thin" modules that translate queries in the common model into source-specific queries. 2 Hence, wrappers reflect the actual capabilities of the underlying data sources, while the mediator has a general mechanism for interpreting those capabilities and forming execution strategies for queries. This paper focuses on the technology needed to enable the CBR approach. We first present a language for describing wrappers' query capabilities. The descriptions look like context-free grammars, modified to describe queries rather than arbitrary strings. The descriptions may be recursive, thus allowing the description of infinitely large supported queries. In addition, they may be schema- independent. For example, we may describe the capabilities of a relational database wrapper without re- In general, there is a one-to-one mapping and no optimization is involved in this translation. All optimization is done at the mediator. ferring to the schema of a specific relational database. An additional benefit of the grammar-like description language is that it can be appropriately augmented with actions to translate a target query to a query of the underlying information system. This feature has been described in [5] and we will not discuss it further in this paper. The second contribution of this paper is an architecture for the CBR and an algorithm to build plans for a target query using the CSQs supported by the relevant wrapper. This problem is a generalization of the problem of determining if a query can be answered using a set of materialized queries/views [6, 7]. However, the CBR uses a description of potentially infinite queries as opposed to a finite set of materialized views. The problem of identifying CSQs that compute the target query has many sources of exponentiality even for the restricted case discussed by [6, 7]. The CBR algorithm uses optimizations and heuristics to eliminate sources of exponentiality in many common cases. In the next section, we present the language used to describe a wrapper's query capabilities. In Section 3 we describe the basic architecture of the CBR, identifying three modules: Component SubQuery Discov- ery, Plan Construction, and Plan Refinement. These components are detailed in Sections 4, 5 and 6, re- spectively. Section 7 summarizes the run-time performance of the CBR, while Section 8 compares the CBR with related work. Finally, Section 9 concludes with some directions for future work in this area. 2 The Relational Query Description RQDL is the language we use to describe a wrap- per's supported queries. We discuss only Select- Project-Join queries in this paper. In section 2.1 we introduce the basic language features , followed in sections 2.2 and 2.3 by the extensions needed to describe infinite query sets and to support schema-independent descriptions. Section 2.4 introduces a normal form for queries and descriptors that increases the precision of the language. The complete language specification appears in [8]. The description language focuses on conjunctive queries. We have found that it is powerful enough to express the abilities of many wrappers and sources, such as lookup catalogs and object databases. Indeed, we believe that it is more expressive than context-free grammars (we are currently working on the proof). 2.1 Language Basics An RQDL specification contains a set of query tem- plates, each of which is essentially a parameterized query. Where an actual query might have a con- stant, the query template has a constant placeholder, allowing it to represent many queries of the same form. In addition, we allow the values assumed by the constant placeholders to be restricted by specifier- metapredicates. A query is described by a template (loosely speaking) if (1) each predicate in the query matches one predicate in the template, and vice versa, and (2) any metapredicates on the placeholders of the template evaluate to true for the matching constants in the query. The order of the predicates in query and template need not be the same, and different variable names are of course possible. For example, consider a "lookup" facility that provides information - such as name, department, office address, and so on - about the employees of a company. The "lookup" facility can either retrieve all employees, or retrieve employees whose last name has a specific prefix, or retrieve employees whose last name and first name have specific prefixes. 3 We integrate "lookup" into our heterogeneous system by creating a wrapper, called lookup, that exports a predicate emp(First-Name, Last-Name, Department, Office, Manager). ( The Manager field may be 'Y' or 'N'.) The wrapper also exports a predicate prefix(Full, Prefix) that is successful when its second argument is a prefix of its first argument. This second argument must be a string, consisting of letters only. We may write the following Datalog query to retrieve emp tuples for persons whose first name starts with 'Rak' and whose last name starts with 'Aggr': prefix(FN,'Rak'), prefix(LN,'Aggr') In this paper we use Datalog [9] as our query language because it is well-suited to handling SPJ queries and facilitates the discussion of our algorithms. 4 We use the following Datalog terms in this paper: Distinguished variables are the variables that appear in the target query head. A join variable is any variable that appears twice or more in the target query tail. In the query (Q1) the distinguished variables are FN, LN, D, O and M and the join variables are FN and LN. Description (D2) is an RQDL specification of lookup's query capabilities. The identifiers starting with and $LP) are constant placeholders. isalpha() is a metapredicate that returns true if its argument is a string that contains letters only. Metapredicates start with an underscore and a lowercase letter. Intuitively, template (QT2.3) describes query (Q1) because the predicates of the query match those of the template (despite differences in order and in variable names), and the metapredicates evaluate to true when $FP is mapped to 'Rak' and $LP to 'Aggr'. (D2) answer(F,L,D,O,M) :- (QT2.1) emp(F,L,D,O,M) answer(F,L,D,O,M) :- (QT2.2) emp(F,L,D,O,M), prefix(L, $LP), isalpha($LP) answer(F,L,D,O,M) :- (QT2.3) emp(F,L,D,O,M), prefix(L, $LP), prefix(F,$FP), 3 The "lookup" facility is very similar to a Stanford University facility. 4 We could have used SPJ SQL queries instead of Datalog. Then, we would use a description language that looks like SQL and not Datalog. The same notions, i.e., placeholders, nonter- minals, and so on, hold. The CBR algorithm is also the same. In general, a template describes any query that can be produced by the following steps: 1. Map each placeholder to a constant, e.g., map $LP to 'Aggr'. 2. Map each template variable to a query variable, e.g., map F to FN. 3. Evaluate the metapredicates and discard any template that contains at least one metapredicate that evaluates to false. 4. Permute the template's subgoals. 2.2 Descriptions of Large and Infinite Sets of Supported Queries RQDL can describe arbitrarily large sets of templates (and hence queries) when extended with non-terminals as in context-free grammars. Nonterminals are represented by identifiers that start with an underscore capital letter. They have zero or more parameters and they are associated with nonterminal templates. A query template t containing non-terminals describes a query q if there is an expansion of t that describes q. An expansion of t is obtained by replacing each nonterminal N of t with one of the nonterminal templates that define N until there is no nonterminal in t. For example, assume that lookup allows us to pose one or more substring conditions on one or more fields of emp. For example, we may pose query (Q3), which retrieves the data for employees whose office contains the strings 'alma' and 'B'. substring(O,'alma'), substring(O,'B') (D4) uses the nonterminal Cond to describe the supported queries. In this description the query template (QT4.1) is supported by nonterminal templates such as (NT4.1). (D4)answer(F,L,D,O,M) :- (QT4.1) emp(F,L,D,O,M), Cond(F,L,D,O,M) substring(F, $FS), Cond(F,L,D,O,M) substring(L, $LS), Cond(F,L,D,O,M) substring(D, $DS), Cond(F,L,D,O,M) substring(O,$OS), Cond(F,L,D,O,M) substring(M, $MS), Cond(F,L,D,O,M) To see that description (D4) describes query (Q3), we expand Cond(F,L,D,O,M) in (QT4.1) with the nonterminal template (NT4.4) and then again expand Cond with the same template. The Cond subgoal in the resulting expansion is expanded by the empty template (NT4.6) to obtain expansion (E5). substring(O,$OS), substring(O,$OS1) Before a template is used for expansion, all of its variables are renamed to be unique. Hence, the second occurrence of placeholder $OS of template (NT4.4) is renamed to $OS1 in (E5). (E5) describes query (Q3), i.e., the placeholders and variables of (E5) can be mapped to the constants and variables of (Q3). 2.3 Schema Independent Descriptions of Supported Queries Description (D4) assumes that the wrapper exports a fixed schema. However, the query capabilities of many sources (and thus wrappers) are independent of the schemas of the data that reside in them. For exam- ple, a relational database allows SPJ queries on all of its relations. To support schema independent descriptions RQDL allows the use of placeholders in place of the relation name. Furthermore, to allow tables of arbitrary arity and column names, RQDL provides special variables called vector variables, or simply vec- tors, that match lists of variables that appear in a query. We represent vectors in our examples by identifiers starting with an underscore ( ). In addition, we provide two built-in metapredicates to relate vectors and attributes: subset and in. subset( R, succeeds if each variable in the list that matches R appears in the list that matches A. in($Position, X, matches a variable list, and there is a query variable that matches X and appears at the position number that matches $Position. (For readability we will use italics for vectors and bold for metapredicates). For example, consider a wrapper called file-wrap that accesses tables residing in plain UNIX files. It may output any subset of any table's fields and may impose one or more substring conditions on any field. Such a wrapper may be easily implemented using the UNIX utility AWK. (D6) uses vectors and the built-in metapredicates to describe the queries supported by file-wrap. substring(X,$S), Cond( In general, to decide whether a query is described by a template containing vectors we must expand the nonterminals, map the variables, placeholders, and vectors, and finally, evaluate any metapredicates. To illustrate this, we show how to verify that query (Q7) is described by (D6). substring(O,'alma'), substring(O,'B') First, we expand (QT6.1) by replacing the non-terminal Cond with (NT6.1) twice, and then with (NT6.2), thus obtaining expansion (E8). in($Position,X, A),substring(X,$S), in($Position1,X1, A),substring(X1,$S1), subset( R, Expansion (E8) describes query (Q7) because there is a mapping of variables, vectors, and placeholders of (E8) that makes the metapredicates succeed and makes every predicate of the expansion identical to a predicate of the query. Namely, vector A is mapped to [F,L,D,O,M], vector R to [L,D], placeholders $Position and $Position1 to 4, $S to 'alma', $S1 to 'B', and the variables X and X1 to O. We must be careful with vector mappings; if the vector V that maps to appears in a metapredicate, we replace However, if the vector V appears in a predicate as p( V ) the mapping results in Finally, the metapredicate in(4, O, [F,L,D,O,M]) succeeds because O is the fourth variable of the list, and subset([L,D], [F,L,D,O,M]) succeeds because [L,D] is a "subset" of [F,L,D,O,M]. Vectors are useful even when the schema is known as the specification may otherwise be repetitious, as in description (D4). In our running example, even though we know the attributes of emp, we save effort by not having to explicitly mention all of the column names to say that a substring condition can be placed on any column. 2.4 Query and Description Normal Form If we allow templates' variables and vectors to map to arbitrary lists of constants and variables, descriptions may appear to support queries that the underlying wrapper does not support. This is because using the same variable name in different places in the query or description can cause an implicit join or selection that does not explicitly appear in the description. For example, consider query (Q9), which retrieves employees where the manager field is 'Y' and the first and last names are equal, as denoted by the double appearance of FL in emp. (D6) should not describe query (Q9). Nevertheless, we can construct expansion (E10), which erroneously matches query (Q9) if we map A to [FL,FL,D,O,'Y'] and R to [FL,D]: This section introduces a query and description normal form that avoids inadvertently describing joins and selections that were not intended. In the normal form both queries and descriptions have only explicit equalities. A query is normalized by replacing every constant c with a unique variable V and then by introducing the subgoal more, for every join variable V that appears n ? 1 times in the query we replace its instances with the Vn and introduce the subgoals We replace any appearance of V in the head with V 1 . For example, query (Q11) is the normal form of (Q9). FL1=FL2, M='Y' Description (D6) does not describe (Q11) because (D6) does not support the equality conditions that RQDL Plan Refinement Target Query Specification Plans (not fully optimized) Component SubQueries Component SubQuery Discovery Algebraically Optimal Plans Plan Construction Figure 2: The CBR's components appear in (Q11). Description (D12) supports equality conditions on any column and equalities between any two columns: (NT12.2) describes equalities with constants and (NT12.3) describes equalities between the columns of our table. (D12) answer( R) :- (QT12.1) Table in($Position,X, A), substring(X, $S), in($Position1,X, A), X=$C, Cond( in($Pos1,X, A), in($Pos2,Y, A), X=Y, Cond( For presentation purposes we use the more compact unnormalized form of queries and descriptions when there is no danger of introducing inadvertent selections and joins. However, the algorithms rely on the normal form. 3 The Capabilities-Based Rewriter The Capabilities-Based Rewriter (CBR) determines whether a target query q is directly supported by the appropriate wrapper, i.e., whether it matches the description d of the wrapper's capabilities. If not, the CBR determines whether q can be computed by combining a set of supported queries (using selections, projections and joins). In this case, the CBR will produce a set of plans for evaluating the query. The CBR consists of three modules, which are invoked serially (see Figure Component SubQuery (CSQ) Discovery: finds supported queries that involve one or more subgoals of q. The CSQs that are returned contain the largest possible number of selections and joins, and do no projection. All other CSQs are pruned. This prevents an exponential explosion in the number of CSQs. ffl plan construction: produces one or more plans that compute q by combining the CSQs exported by CSQ Discovery. The plan construction algorithm is based on query subsumption and has been tuned to perform efficiently in the cases typically arising in capabilities-based rewriting. ffl plan refinement: refines the plans constructed by the previous phase by pushing as many projections as possible to the wrapper. 3.1 Consider query (Q13), which retrieves the names of all managers that manage departments that have employees with offices in the 'B' wing, and the employees' office numbers. This query is not directly supported by the wrapper described in (D12). emp(F1,L1,D,O1,M1), substring(O1,'B') The CSQ detection module identifies and outputs the following CSQs: answer 14 answer 15 emp(F1,L1,D,O1,M1), substring(O1, 'B') Note, the CSQ discovery module does not output the 2 4 CSQs that have the tail of (Q14) but export a different subset of the variables F0, L0, D, and O0 (like- wise for (Q15). The CSQs that export fewer variables are pruned. The plan construction module detects that a join on D of answer 14 and answer 15 produces the required answer of (Q13). Consequently, it derives the plan (P16). answer 14 answer 15 Finally, the plan refinement module detects that variables O0, F1, L1, and M1 in answer 14 and answer 15 are unnecessary. Consequently, it generates the more efficient plan (P19). answer 17 answer emp(F1,L1,D,O1,M1), substring(O1, 'B') answer 17 (F0,L0,D), answer (D,O1)The CBR's goal is to produce all algebraically optimal plans for evaluating the query. An algebraically optimal plan is one in which any selection, projection or join that can be done in the wrapper is done there, and in which there are no unnecessary queries. More Definition 3.1 (Algebraically Optimal Plan P ) plan P is algebraically optimal if there is no other plan P 0 such that for every CSQ s of P there is a corresponding CSQ s 0 of P 0 such that the set of sub-goals of s 0 is a superset of the set of subgoals of s (i.e., s 0 has more selections and joins than s) and the set of exported variables of s is a superset of the set of exported variables of s 0 (i.e., s 0 has more projections than In the next three sections we describe each of the modules of the CBR in turn. The CSQ discovery module takes as input a target query and a description. It operates as a rule production system where the templates of the description are the production rules and the subgoals of the target query are the base facts. The CSQ discovery module uses bottom-up evaluation because it is guaranteed to terminate even for recursive descriptions [10]. How- ever, bottom-up derivation often derives unnecessary facts, unlike top-down. We use a variant of magic sets rewriting [10] to "focus" the bottom-up derivation. To further reduce the set of derived CSQs we develop two pruning techniques as decsribed in Sections 4.2 and 4.3. Reducing the number of derived CSQs makes the CSQ discovery more efficient and also reduces the size of the input to the plan construction module. The query templates derive answer facts that correspond to CSQs. In particular, a derived answer fact is the head of a produced CSQ whereas the underlying base facts, i.e., the facts that were used for deriving answer, are the subgoals of the CSQ. Nonterminal templates derive intermediate facts that may be used by other query or nonterminal templates. We keep track of the sets of facts underlying derived facts for pruning CSQs. The following example illustrates the bottom-up derivation of CSQs and the gains that we realize from the use of the magic-sets rewriting. The next subsection discusses issues pertaining to the derivation of facts containing vectors. EXAMPLE 4.1 Consider query (Q3) and description (D4) from page 3. The subgoals emp(F,L,D,O,M), substring(O, 'alma'), and substring(O,'B') are treated by the CSQ discovery module as base facts. To distinguish the variables in target query subgoals from the templates' variables we ``freeze'' the vari- ables, e.g. F,L,D,O, into similarly named constants, e.g. f,l,d,o. Actual constants like 'B' are in single quotes. In the first round of derivations template (NT4.6) derives fact Cond(F,L,D,O,M) without using any base fact (since the template has an empty body). Hence, the set of facts underlying the derived fact is empty. Variables are allowed in derived facts for nontermi- nals. The semantics is that the derived fact holds for any assignment of frozen constants to variables of the derived fact. In the second round many templates can fire. For example, derives the fact Cond(F,L,D,o,M) using Cond(F,L,D,O,M) and substring(o,'alma'), or using Cond(F,L,D,o,M) and substring(o,'B'). Thus, we generate two facts that, though identical, they have different underlying sets and hence we must retain both since they may generate different CSQs. In the second round we may also fire (NT4.6) again and produce Cond(F,L,D,O,M) but we do not retain it since its set of underlying facts is equal to the version of Cond(F,L,D,O,M) that we have already produced. Eventually, we generate answer(f,l,d,o,m) with set of underlying facts femp(f,l,d,o,m), substring(o, 'alma'), substring(o,'B')g. Hence we output the CSQ (Q3), which, incidentally, is the target query. The above process can produce an exponential number of facts. For example, we could have proved Cond(o,L,D,O,M), Cond(F,o,D,O,M), Cond(o,o,D,O,M), and so on. In general, assuming that emp has n columns and we apply m substrings on it we may derive n m facts. Magic-sets can remove this source of exponentiality by "focusing" the nontermi- nals. Applying magic-sets rewriting and the simplifications described in Chapter 13.4 of [10] we obtain the following equivalent description. We show only the rewriting of templates (NT4.4) and (NT4.6). The others are rewritten similarly. emp(F,L,D,O,M), Cond(F,L,D,O,M) mg Cond(F,L,D,Office,M), substring(Office, $OS), Cond(F,L,D,Office,M) mg Cond(F,L,D,O,M) emp(F,L,D,O,M) only Cond(f,l,d,o,m) facts (with different underlying sets) are produced. Note, the magic-sets rewritten program uses the available information in a way similar to a top-down strategy and thus derives only relevant facts. 2 4.1 Derivations Involving Vectors When the head of a nonterminal template contains a vector variable it may be possible that a derivation using this nonterminal may not be able either to bind the vector to a specific list of frozen variables or to allow the variable as is in the derived fact. The CSQ discovery module can not handle this situation. For most descriptions, magic-sets rewriting solves the problem. We demonstrate how and we formally define the set of non-problematic descriptions. For example, let us fire template (NT6.1) of (D6) on the base facts produced by query (Q3). Assume also that (NT6.2) already derived Cond( A). Then we derive that Cond( holds, with set of underlying facts fsubstring(o, 'alma')g, provided that the constraint " A contains o" holds. The constraint should follow the fact until A binds to some list of frozen variables. We avoid the mess of constraints using the following magic-sets rewriting of (D6). (D21) answer( R) :- (QT21.1) Table subset( R, mg Cond( A), in($Position,X, A), substring(X,$S), Cond( When rules (NT21.1) and (NT21.2) fire the first subgoal instantiates variable A to [f,l,d,o,m] and they derive only Cond([f,l,d,o,m]). Thus, magic- sets caused A to be bound to the only vector of inter- est, namely [f,l,d,o,m]. Note a program that derives facts with unbound vectors may not be problematic because no metapredicate may use the unbound vector variable. However we take a conservative approach and consider only those programs that produce facts with only bound vector variables. Magic-sets rewriting does not always ensure that derived facts have bound vectors. In the rest of this section we describe sufficient conditions for guaranteeing the derivation of facts with bound vectors only. First we provide a condition (Theorem 4.1) that guarantees that a program (that may be the result of magic rewriting) does not derive facts with unbound vectors. Then we describe a class of programs that after being magic rewriteen satisfy the condition of Theorem 4.1. Theorem 4.1 A program will always produce facts with bound vector variables if in all rules " \Gammatail" tail has a non-metapredicate subgoal that refers to V , or in general V can be assigned a binding if all non-metapredicate subgoals in tail are bound. 2 Intuitively, after we magic-rewrite a program it will deriving facts with unbound vectors only if a nonterminal of the initial program derives uninstan- tianted vectors and in the rules that is used it does not share variables with predicates or nonterminals s that bind their arguments (otherwise, the magic predicate will force the the rules that produce uninstan- tianted vectors to focus on bindings of s.) For ex- ample, specification (MS6) does not derive uninstan- tianted vectors because the nonterminal Cond, that may derive uninstantianted variables, shares variables with Table A). [8] provides a formal criterion for deciding whether the bottom-up evaluation derives facts that have vector variables. This criterion is used by the following algorithm that derives CSQs given a target query and a description. Algorithm 1 Input: Target query Q and Description D Output: A set of CSQs s Check if the program derives facts with vector variables (see [8]) Reorder each template R in D such that All predicate subgoals occur in the front of the rule A nonterminal N appears after M if N depends on M for grounding. Metapredicates appear at the end of the rule Rewrite D using Magic-sets Evaluate bottom-up the rewritten description D as described in [8] Note, template R can always be reordered. The proof appears in [8]. 4.2 Retaining Only "Representative" CSQs A large number of unneeded CSQs are generated by templates that use vectors and the subset metapred- icate. For example, template (QT12.1) describes for a particular A all CSQs that have in their head any subset of variables in A. It is not necessary to generate all possible CSQs. Instead, for all CSQs that are derived from the same expansion e, of some template t, where e has the form metapredicate listi, and V does not appear in the hpredicate and metapredicate listi we generate only the representative CSQ that is derived by mapping V to the same variable list as W . 5 All represented CSQs, i.e., CSQs that are derived from e by mapping V to a proper subset of W are not gener- ated. For example, the representative CSQ (Q15) and the represented CSQ (Q18) both are derived from the expansion (E22) of template (QT12.1). in($Position,X, A), substring(X,'B'), subset( R, The CSQ discovery module generates only (Q15) and not (Q18) because (Q15) has fewer attributes than (Q18) and is derived by by mapping the vector R to the same vector with A, i.e., to [F1,L1,D,O1,M1]. Representative CSQs often retain unneeded attributes and consequently Representative plans, i.e., plans containing representative CSQs, retrieve unneeded at- tributes. The unneeded attributes are projected out by the plan refinement module. Theorem 4.2 Retaining only representative CSQs does not lose any plan, i.e., if there is an algebraically optimal plan p s that involves a represented query s then p s will be discovered by the CBR. 2 The intuitive proof of this claim is that for every plan p s there is a corresponding representative plan r derived by replacing all CSQs of p s with their rep- resentatives. Then, given that the plan refinement component considers all plans represented by a representative plan, we can be sure that the CBR algorithm does not lose any plan. The complete proof appears in [8]. Evaluation: Retaining only a representative CSQ of head arity a eliminates 2 a \Gamma 1 represented CSQs thus 5 In general, the hlist of predicates and metapredicates i may contain metapredicates of the form in(hpositioni,hvariable i i, m. In this case, the template describes all CSQs that output a subset of W and a superset of hvariableimg. The CSQ discovery module out- puts, as usual, the representative CSQ and annotates it with the set S that provides the "minimum" set of variables that represented CSQs must export. In this paper we will not describe any further the extensions needed for the handling of this case. eliminating an exponential factor from the execution time and from the size of the output of the CSQ discovery module. Still, one might ask why the CSQ discovery phase does not remove the variables that can be projected out. The reason is that the "projection" step is better done after plans are formed because at that time information is available about the other CSQs in the plan and the way they interact (see Section 6). Thus, though postponing projection pushes part of the complexity to a later stage, it eliminates some complexity altogether. The eliminated complexity corresponds to those represented CSQs that in the end do not participate in any plan because they retain too few variables. 4.3 Pruning Non-Maximal CSQs Further efficiency can be gained by eliminating any CSQ Q that has fewer subgoals than some other CSQ checks fewer conditions than Q 0 . A CSQ is maximal if there is no CSQ with more subgoals and the same set of exported variables, modulo variable renaming. We formalize maximality in terms of subsumption [10]: Definition 4.1 (Maximal CSQs) A CSQ s m is a maximal CSQ if there is no other CSQ s that is subsumed by s m . 2 Evaluation: In general, the CSQ discovery module generates only maximal CSQs and prunes all others. This pruning technique is particularly effective when the CSQs contain a large number of conditions. For example, assume that g conditions are applied to the variables of a predicate. Consequently, there are 2 CSQs where each one of them contains a different proper subset of the conditions. By keeping "maximal CSQs only" we eliminate an exponential factor of 2 g from the output size of the CSQ discovery module. Theorem 4.3 Pruning non-maximal CSQs does not lose any algebraically optimal plan. 2 The reason is that for every plan p s involving a non-maximal CSQ s there is also a plan pm that involves the corresponding maximal CSQ s m such that pm pushes more selections and/or joins to the wrapper than p s , since s m by definition involves more selections and/or joins than s. 5 Plan Construction In this section we present the plan construction module (see Figure 2.) In order to generate a (rep- resentative) plan we have to select a subset S of the CSQs that provides all the information needed by the target query, i.e., (i) the CSQs in S check all the sub-goals of the target query, (ii) the results in S can be joined correctly, and (iii) each CSQ in S receives the constants necessary for its evaluation. Section 5.1 addresses (i) with the notion of "subgoal consumption." Section 5.2 checks (ii), i.e., checks join variables. Section 5.3 checks (iii) by ensuring bindings are avail- able. Finally, Section 5.4 summarizes the conditions required for constructing a plan and provides an efficient plan construction algorithm. 5.1 Set of Consumed Subgoals We associate with each CSQ a set of consumed sub-goals that describes the CSQs contribution to a plan. Loosely speaking, a component query consumes a sub-goal if it extracts all the required information from that subgoal. A CSQ does not necessarily consume all its subgoals. For example, consider a CSQ s e that semijoins the emp relation with the dept relation to output each emp tuple that is in some department in relation dept. Even though this CSQ has a subgoal that refers to the dept relation it may not always consume the dept subgoal. In particular, consider a target query Q that requires the names of all employees and the location of their departments. CSQ s e does not output the location attribute of table dept and thus does not consume the dept subgoal with respect to query Q. We formalize the above intuition by the following definition: Definition 5.1 (Set of Consumed Subgoals for a CSQ) A set S s of subgoals of a CSQ s constitutes a set of consumed subgoals of s if and only if 1. s exports every distinguished variable of the target query that appears in S s , and 2. s exports every join variable that appears in S s and also appears in a subgoal of the target query that is not in S s .Theorem 5.1 Each CSQ has a unique maximal set of consumed subgoals that is a superset of every other set of consumed subgoals. 2 The proof of the uniqueness of the maximal consumed set appears in [8]. Intuitively the maximal set describes the "largest" contribution that a CSQ may have in a plan. The following algorithm states how to compute the set of maximal consumed subgoals of a CSQ. We annotate every CSQ s with its set of maximal consumed subgoals, C s . Algorithm 2 Input: CSQ s and target query Q Output: CSQ s with computed annotation C s Insert in C s all subgoals of s Remove from C s subgoals that have a distinguished attribute of Q not exported by s Repeat until size of C s is unchanged Remove from C s subgoals that: Join on variable V with subgoal g of Q where g is not in C s , and Join variable V is not exported by s Discard CSQ s if C s is empty. This algorithm is polynomial in the number of the subgoals and variables of the CSQ. Also, the algorithm discards all CSQs that are not relevant to the target query: Definition 5.2 (Relevant CSQ) A CSQ s is called relevant if C s is non-empty. 2 Intuitively, irrelevant CSQs are pruned out because in most cases they do not contribute to a plan, since they do not consume any subgoal. Note, we decide the relevance of a CSQ "locally," i.e., without considering other CSQs that it may have to join with. By pruning non-relevant CSQs we can build an efficient plan construction algorithm that in most cases (Section 5.2) produces each plan in time polynomial in the number of CSQs produced by the CSQ discovery module. However, there are scenarios (see the extended version [8]) where the relevance criteria may erroneously prune out a CSQ that could be part of a plan. We may avoid the loss of such plans by not pruning irrelevant CSQs and thus sacrificing the polynomiality of the plan construction algorithm. In this paper we will not consider this option. 5.2 Join Variables Condition It is not always the case that if the union of consumed subgoals of some CSQs is equal to the set of the target query's subgoals then the CSQs together form a plan. In particular, it is possible that the join of the CSQs may not constitute a plan. For exam- ple, consider an online employee database that can be queries for the names of all employees in a given divi- sion. The database can also be queried for the names of all employees in a given location. Further, the name of an employee is not uniquely determined by their location and division. The employee database cannot be used to find employees in a given division and in a given location by joining the results of two queries - one on division and the other on location. To see this, consider a query that looks for employees in "CS" in "New York". Joining the results of two independent queries on division and location will incorectly return as answer a person named "John Smith" if there is a "John Smith" in "CS" in "San Jose" and a different "John Smith" in "Electrical" in "New York". Intuitively, the problem arises because the two independent queries do not export the information necessary to correctly join their results. We can avoid this problem by checking that CSQs are combined only if they export the join variables necessary for their correct combination. The theorem of Section 5.4 formally describes the conditions on join variables that guarantee the correct combination of CSQs. 5.3 Passing Required Bindings via Nested Loops Joins The CBR's plans may emulate joins that could not be pushed to the wrapper, with nested loops joins where one CSQ passes join variable bindings to the other. For example, we may compute (Q13) by the following steps: first we execute (Q23); then we collect the department names (i.e., the D bindings) and for each binding d of D, we replace the $D in (Q24) with d and send the instantianted query to the wrap- per. We use the notation /$D in the nested loops plan (P25) to denote that (Q24) receives values for the $D placeholder from D bindings of the other CSQs - (Q23) in this example. answer 23 answer 24 The introduction of nested loops and binding passing poses the following requirements on the CSQ discovery ffl CSQ discovery: A subgoal of a CSQ s may contain placeholders /$hvari, such as $D, in place of corresponding join variables (D in our example.) Whenever this is the case, we introduce the structure /$hvari next to the answer s that appears in the plan. All the variables of s that appear in such a structure are included in the set B s , called the set of bindings needed by s. For ex- ample, fg. CSQ discovery previously did not use bindings information while deriving facts. Thus, the algorithm derives useless CSQs that need bindings not exported by any other CSQ. The optimized derivation process uses two sets of attributes and proceeds iteratively. Each iteration derives only those facts that use bindings provided by existing facts. In addition, a fact is derived if it uses at least one binding that was made available only in the very last itera- tion. Thus, the first iteration derives facts that need no bindings, that is, for which B s is empty. The next iteration derives facts that use at least one binding provided by facts derived in iteration one. Thus, the second iteration does not derive any subgoal derived in the first iteration, and so on. The complete algorithm that appears in [8] formalizes this intuition. The bindings needed by each CSQ of a plan impose order constraints on the plan. For example, the existence of D in B 24 requires that a CSQ that exports D is executed before (Q24). It is the responsibility of the plan construction module to ensure that the produced plans satisfy the order constraints. Evaluation The pruning of CSQs with inappropriate bindings prunes an exponential number of CSQs in the following common scenario: Assume we can put an equality condition on any variable of a subgoal p. Consider a CSQ s that contains p and assume that n variables of p appear in subgoals of the target query that are not contained in s. Then we have to generate versions of s that describe different binding pat- terns. Assuming that no CSQ may provide any of the variables it is only one (out the 2 n ) CSQs useful. 5.4 A Plan Construction Algorithm In this section we summarize the conditions that are sufficient for construction of a plan. Then, we present an efficient algorithm that finds plans that satisfy the theorem's conditions. Finally, we evaluate the algo- rithm's performance. Theorem 5.2 Given CSQs s corresponding heads answer i (V i sets of maximal consumed subgoals C i and sets of needed bindings , the plan is correct if ffl consumed sets condition: The union of maximal consumed sets [ i=1;:::;n C i is equal to the target query's subgoal set. ffl join variables condition: If the set of maximal consumed subgoals of CSQ s i has a join variable V then every CSQ s j that contains V in its set of maximal consumed subgoals C j exports V . ffl bindings passing condition: If there must be a CSQ s exports V . 2 The proof is based on the theory of containment mappings appropriately extended to take into consideration nested loops [8]. The plan construction algorithm in the extended version of the paper [8] is based on Theorem 5.2. The algorithm takes as input a set of CSQs derived by the discovery process described later, and the target query Q. At each step the algorithm selects a CSQ s that consumes at least one subgoal that has not been consumed by any CSQ s 0 considered so far and for which all variables of B s have been exported by at least one s 0 . Assuming that the algorithm is given m CSQs (by the CSQ discovery module) it can construct a set that satisfies the consumed sets and the bindings passing conditions in time polynomial in m. Neverthe- less, if the join variables condition does not hold the algorithm takes time exponential in m because we may have to create exponentially many sets until we find one that satisfies the join variables condition. How- ever, the join variables condition evaluates to true for most wrappers we find in practice (see following dis- cussion) and thus we usually construct a plan in time polynomial in m. For every plan p there may be plans p 0 that are identical to p modulo a permutation of the CSQs of p. In the worst case there are is the number of CSQs in p. Since it is useless to generate permutations of the same plan, The algorithm creates a total order OE of the input CSQs and generates plans by considering CSQ s 1 before CSQ s 2 only the CSQs are considered in order by OE. Note, a query s 2 must always be considered after a query s 1 if s 1 provides bindings for s 2 . Hence, OE must respect the partial order OE OE provides bindings to s 2 . The plan construction algorithm first sorts the input CSQs in a total order that respects the PO b OE. Then it procedes by picking CSQs and testing the conditions of Theorem 5.2 until it consumes all subgoals of the target query. The algorithm capitalizes on the assumption that in most practical cases every CSQ consumes at least one subgoal and the join variables condition holds. In this case, one plan is developed in time polynomial in the number of input CSQs. The following lemma describes an important case where the join variables condition always holds. Lemma 5.1 The join variables condition holds for any set of CSQs such that 1. no two CSQs of the set have intersecting sets of maximal consumed subgoals, or 2. if two CSQs contain the subgoal g(V their sets of maximal consumed subgoals then they both export variables Condition (1) of Lemma 5.1 holds for typical wrappers of bibliographic information systems and lookup services (wrappers that have the structure of (D12)), relational databases and object oriented databases - wrapped in a relational model. In such systems it is typical that if two CSQs have common subgoals then they can be combined to form a single CSQ. Thus, we end up with a set of maximal CSQs that have non-intersecting consumed sets. Condition (2) further relaxes the condition (1). Condition (2) holds for all wrappers that can export all variables that appear in a CSQ. The two conditions of Lemma 5.1 cover essentially any wrapper of practical importance. 6 Plan Refinement The plan refinement module filters and refines constructed plans in two ways. First, it eliminates plans that are not algebraically optimal. The fact that CSQs of the representative plans have the maximum number of selections and joins and that plan refinement pushes the maximum number of projections down is not enough to guarantee that the plans produced are algebraically optimal. For example, assume that CSQs are interchangeable in all plans, and the set of subgoals of s 1 is a superset of the set of subgoals of exports a subset of the variables exported by s 2 . The plans in which s 2 participates are algebraically worse than the corresponding plans with s 1 . Nevertheless, they are produced by the plan construction module because s 1 and s 2 may both be maximal, and do not represent each other because they are produced by different template expansions. Plan refinement must therefore eliminate plans that include s 2 . Plan refinement must also project out unnecessary variables from representative CSQs. Intuitively, the necessary variables of a representative CSQ are those variables that allow the consumed set of the CSQ to "interface" with the consumed sets of other CSQs in the plan. We formalize this notion and its significance by the following definition (note, the definition is not restricted to maximal consumed sets): Definition 6.1 (Necessary Variables of a Set of Consumed Subgoals:) A variable V is a necessary variable of the consumed subgoals set S s of some CSQ s if, by not exporting V , S s is no longer a consumed set. 2 The set of necessary variables is easily computed: Given a set of consumed subgoals S, a variable V of S is a necessary variable if it is a distinguished variable, or if it is a join variable that appears in at least one subgoal that is not in S. Due to space limitations the complete plan refinement algorithm and its evaluation appear in [8]. Its main complication is due to the fact that unecessary variables cannot always be projected out when the maximal consumed sets of the CSQs intersect. 7 Evaluation The CBR algorithm employs many techniques to eliminate sources of exponentiality that would otherwise arise in many practical cases. The evaluation paragraphs of many sections in this paper describe the benefit we derive from using these techniques. Remember that our assumption that every CSQ consumes at least one subgoal led to a plan construction module that develops a plan in time polynomial to the number of CSQs produced by the CSQ detection mod- ule, provided that the join variables condition holds. This is an important result because the join variables condition holds for most wrappers in practice, as argued in Subsection 5.4. The CBR deals only with Select-Project-Join queries and their corresponding descriptions. It produces algebraically optimal plans involving CSQs, i.e., plans that push the maximum number of selections, projections and joins to the source. However, the CBR is not complete because it misses plans that contain irrelevant CSQs (see Definition 5.2 and the discussion of Section 5.1.) On the other hand, the techniques for eliminating exponentiality preserve completeness, in that we do not miss any plan through applying one of these techniques (see justifications in Sections 4.2, 4.3.) 8 Related Work Significant results have been developed for the resolution of semantic and schematic discrepancies while integrating heterogeneous information sources. How- ever, most of these systems [11, 12, 4, 13] do not address the problem of different and limited query capabilities in the underlying sources because they assume that those sources are full-fledged databases that can answer any query over their schema. 6 The recent interest in the integration of arbitrary information sources, including databases, file systems, the Web, and many legacy systems, invalidates the assumption that all underlying sources can answer any query over the data they export and forces us to re-solve the mismatch between the query capabilities provided by these sources. Only a few systems have addressed this problem. HERMES [11] proposes a rule language for the specification of mediators in which an explicit set of parameterized calls can be made to the sources. At run-time the parameters are instantiated by specific values and the corresponding calls are made. Thus, HERMES guarantees that all queries sent to the wrappers are supported. Unfortunately, this solution reduces the interface between wrappers and mediators to a very simple form (the particular parameterized 6 The work in query decomposition in distributed databases has also assumed that all underlying systems are relational and equally able to perform any SQL query. calls), and does not fully utilize the sources' query power. DISCO [14]describes the set of supported queries using context-free grammars. This technique reduces the efficiency of capabilities-based rewriting because it treats queries as "strings." The Information Manifold [15] develops a query capabilities description that is attached to the schema exported by the wrapper. The description states which and how many conditions may be applied on each attribute. RQDL provides greater expressive power by being able to express schema-independent descriptions and descriptions such as "exactly one condition is allowed." TSIMMIS suggests an explicit description of the wrapper's query capabilities [5], using the context-free grammar approach of the current paper. (The description is also used for query translation from the common query language to the language of the underlying source.) However, TSIMMIS considers a restricted form of the problem wherein descriptions consider relations of prespecified arities and the mediator can only select or project the results of a single CSQ. This paper enhances the query capability description language of [5] to describe queries over arbitrary schemas, namely, relations with unspecified arities and names, as well as capabilities such as "selections on the first attribute of any relation." The language also allows specification of required bindings, e.g., a bibliography database that returns "titles of books given author names." We provide algorithms for identifying for a target query Q the algebraically optimal CSQs from the given descriptions. Also, we provide algorithms for generating plans for Q by combining the results of these CSQs using selections, projections, and joins. The CBR problem is related to the problem of determining how to answer a query using a set of materialized views [16, 6, 7, 17]. However, there are significant differences. These papers consider a specification language that uses SPJ expressions over given relations specifying a finite number of views. They cannot express arbitrary relations, arbitrary arities, binding requirements (with the exception of [7]), or infinitely large queries/views. Also, they do not consider generating plans that require a particular evaluation order due to binding requirements. [6] shows that rewriting a conjunctive query is in general exponential in the total size of the query and views. [17] shows that if the query is acyclic we can rewrite it in time polynomial to the total size of the query and views. [6, 7] generate necessary and sufficient conditions for when a query can be answered by the available views. By contrast, our algorithms check only sufficient conditions and might miss a plan because of the heuristics used. Our algorithm can be viewed as a generalization of algorithms that decide the subsumption of a datalog query by a datalog program (i.e., the description). Recently [18] proposed Datalog for the description of supported queries. It also suggested an algorithm that essentially finds what we call maximal CSQs. 9 Conclusions and Future Work In this paper, we presented the Relational Query Description Language, RQDL, which provides powerful features for the description of wrappers' query capabilities. RQDL allows the description of infinite sets of arbitrarily large queries over arbitrary schemas. We also introduced the Capabilities-Based Rewriter, CBR, and presented an algorithm that discovers plans for computing a wrapper's target query using only queries supported by the wrapper. Despite the inherent exponentiality of the problem, the CBR uses optimizations and heuristics to produce plans in reasonable time in most practical situations. The output of the CBR algorithm, in terms of the number of derived plans, remains a major source of exponentiality. Though the CBR prunes the output plans by deriving a plan only if no other plan pushes more selections, projections or joins to the source, it may still be the case that the number of plans is exponential in the number of subgoals and/or join vari- ables. For example, consider the case where our query involves a chain of n joins and each one of them can be accomplished either by a left-to-right nested loops join, or a right-to-left nested loops join, or a local join. In this case, CBR has to output 3 n plans where each of the plans employs one of the three join methods. Then, the mediator's cost-based optimizer would have to estimate the cost of each one of the plans and choose the most efficient. We could modify the CBR to generate all of these plans or only some of them, depending on the time to be spent on optimization. Currently, we are looking at implementing a CBR for IBM's Garlic system [1]. We are also investigating tighter couplings between the mediator's cost-based optimizer and the CBR. Finally, we are investigating more powerful rewriting techniques that may replace a target query's subgoals with combinations of semantically equivalent subgoals that are supported by the wrapper. Acknowledgements We are grateful to Mike Carey, Hector Garcia- Molina, Anand Rajaraman, Anthony Tomasic, Jeff Ullman, Ed Wimmers, and Jennifer Widom for many fruitful discussions and comments. --R Towards heterogeneous multimedia information systems: The Garlic approach. Object exchange across heterogeneous information sources. Amalgame: a tool for creating interoperating persistent The Pegasus heterogeneous multi-database system A query translation scheme for the rapid implementation of wrappers. Answering queries using views. Answering queries using templates with binding patterns. Principles of Database and Knowledge-Base Systems Principles of Database and Knowledge-Base Systems HERMES: A heterogeneous reasoning and mediator system. An approach to resolving semantic heterogeneity in a federation of autonomous Integration of Information Systems: Bridging Heterogeneous Databases. Scaling heterogeneous databases and the design of DISCO. Query processing in the information manifold. Computing queries from derived relations. Query folding. Answering queries using limited external processors. --TR --CTR Stefan Huesemann, Information sharing across multiple humanitarian organizations--a web-based information exchange platform for project reporting, Information Technology and Management, v.7 n.4, p.277-291, December 2006 Yannis Papakonstantinou , Vinayak Borkar , Maxim Orgiyan , Kostas Stathatos , Lucian Suta , Vasilis Vassalos , Pavel Velikhov, XML queries and algebra in the Enosys integration platform, Data & Knowledge Engineering, v.44 n.3, p.299-322, March Alin Deutsch , Lucian Popa , Val Tannen, Physical Data Independence, Constraints, and Optimization with Universal Plans, Proceedings of the 25th International Conference on Very Large Data Bases, p.459-470, September 07-10, 1999 Andreas Koeller , Elke A. Rundensteiner, A history-driven approach at evolving views under meta data changes, Knowledge and Information Systems, v.8 n.1, p.34-67, July 2005

query containment;heterogeneous sources;query rewriting;cost optimization;mediator systems

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A Parallel Algorithm to Reconstruct Bounding Surfaces in 3D Images.

The growing size of 3D digital images causes sequential algorithms to be less and less usable on whole images and a parallelization of these algorithm is often required. We have developed an algorithm named Sewing Faces which synthesizes both geometrical and topological information on bounding surface of 6-connected3D objects. We call such combined information a skin. In this paper we present a parallelization of Sewing Faces. It is based on a splitting of 3D images into several sub-blocks. When all the sub-blocks are processed a gluing step consists of merging all the sub-skins to get the final skin. Moreover we propose a fine-grain approach where each sub-block is processed by several parallel processors.

Introduction Over the past decade, 3D digitalization techniques such as the Magnetic Resonance Imaging have been extensively developed. They have opened new research topics in 3D digital image processing and are of primary importance in many application domains such as medical imaging. The classical notions of 2D image processing have been extended to 3D (pixels into voxels, 4- connectivity into 6-connectivity, etc) and the 2D algorithms have to be adapted to 3D problems ([4], [2]). In this process the amount of data is increased by an order of magnitude (from n 2 pixels in a 2D image to n 3 voxels in a 3D image, where n is the size of the image edges) and in consequence the time complexity of 3D algorithms is also increased by an order of magnitude. In order to still get efficient algorithms in terms of running time and to deal with growing size images, these algorithms have to be parallelized. Among the many problems in 3D image processing, we focus in this paper on the problem of the reconstruction of bounding surfaces of 6-connected objects in 3D digital images. A 3D digital image is characterized by a 3D integer matrix called block; each integer I(v) of the block defines a value associated with a volume element or voxel v of the image. An image describes a set of objects such as organs in medical images. The contour of an object is composed of all the voxels which belong to the object but which have at least one of their adjacent voxels in the background. From this set of voxels we compute the bounding surface of the object. It is a set of closed surfaces enclosing the object. We have developed in [6] a sequential algorithm for bounding surfaces reconstruction. The objective of this paper is to present a parallel version of this algorithm. This parallelization is based on a decomposition of the 3D block into sub-blocks. On each sub-block a fragment of the bounding surface is computed. Once all the fragments have been determined a final step consists in merging them together in order to retrieve the complete bounding surface. The paper is organized as follows. Section 2 recalls the principles of bounding surfaces reconstruction. Section 3 presents some basic notions of 3D digital images while section 4 briefly recalls our sequential algorithm for bounding surfaces reconstruction. Section 5 discusses the sub-blocks decomposition, then sections 6 and 7 respectively present the coarse-grain and fine-grain parallelizations of the algorithm. In section 8 we briefly show how to transform a reconstructed surface into a 2D mesh. Finally section 9 presents experimental results and compares our approach with related works. Some of the notions introduced in this paper are illustrated with figures. Since they are not always easy to visualize in 3D they will be presented using the 2D analogy. Surface reconstruction The closed surfaces that bound an object can be determined in two different ways : ffl using a method by approximation, the surface is reconstructed by interpolating the discretized data. The Marching Cubes [5] developed by Lorensen and Cline is such a method it builds a triangulation of the surface. Various extensions of the method have been proposed, either by defining a heuristic to solve ambiguous cases [9] or by reducing the number of generated triangles. Faster reconstructions have been developed. Some are based on parallelized versions of the algorithm [7]. Others use the octree abstract data type [3] which reduces the number of scanned voxels. ffl using an exact method, the surface is composed of faces shared by a voxel of the object and a voxel of the background. Such a method has been proposed by Artzy et al. [1]. The efficiency of the various reconstruction algorithms is strongly related to the type of scan used to determine the surface. Hence the surface reconstruction can be realized either by a complete search among all the voxels of the block or by a contour following for which only the voxels of the object contour are scanned. The contour following approach yields more efficient algorithms whose time complexity is proportional to the number of voxels of the contour instead of the number of voxels of the whole block. The Marching Cubes algorithm is based on a whole-block scanning while the method proposed in [1] relies on a contour following. The determination of the bounding surface of an object is useful to visualize the object but also to manipulate it, using techniques such as a distortion of a surface, a transformation of a surface into a surface mesh, a derefinement of a surface by merging adjacent coplanar faces, a reversible polyhedrization of discretized volumes. In the former case, the surface needs only to be defined by geometrical information, i.e. the list of its triangles in case of approximation methods or the list of its faces in case of an exact method. In the latter case however, the surface must be defined not only by geometrical information but also by topological information, i.e. information stating how the faces are connected together. Note that it is of course possible to recover the topological information from the geometrical one. For each face, one have to scan all the other faces defining the surface in order to find its adjacent faces, i.e. the ones which share one edge with it. If the surface contains n faces then this topological reconstruction is O(n 2 ). To avoid this quadratic operation the topological information must be collected together with the geometrical information. Figure 1: A block The algorithm we have developed in [6] reconstructs the bounding surface of any 6-connected object of a 3D digital image. It is called Sewing Faces and its characteristics are the following ffl it is an exact method. It extracts faces belonging to a voxel of the object and a voxel of the background. ffl it is based on a contour following. Its time complexity is therefore proportional to the number of voxels of the contour. ffl it synthesizes both geometrical and topological information. In this case the reconstructed surface is named a skin. The topological information is synthesized using sews stating how two adjacent faces of the bounding surface are connected together. ffl its time and space complexity are both linear according to the number of faces of the skin, as proved in [6]. 3 Notions of 3D digital images A 3D block (see figure 1) can be seen as a stack of adjacent voxel slices pushed together according to any one of the three axes x, y or z. A voxel is made of six faces (whose types can be numbered as shown in figure 2), and twelve edges. Each face has an opposite face in a voxel; for instance face of type 2 is opposite to face of type 5 (cf. figure 2). In the following we call face i a face of type i. Two faces that share one edge are adjacent. Two voxels that share one face are 6-adjacent; if they share only one edge they are 18-adjacent (see figure 3). In the following we call object in a block a set of 6-connected voxels. be any two voxels of set '. If there exists a path x are 6-adjacent, then ' is 6-connected. If block B contains more than one object, that is to say if B is made of several 6-connected components ' i , we call this set of objects a composed object and we denote it by \Theta : \Theta = S The voxels which are not in object \Theta are in the background. A boolean function is defined on block B. It is denoted by \Theta B (v) and states whether or not voxel v belongs to object \Theta. There are several ways to define function \Theta B (v), depending on the z z y Figure 2: Type of voxel faces (b) face shared by u and v edge shared by u and v (a) Figure 3: (a) : u and v are 6-adjacent, (b) : u and v are 18-adjacent type of digitalized data. If the block is already thresholded, we may define \Theta B user-defined. If the block is not segmented, we may use a function \Theta B true . On the contrary, if we want to define the object as the complement of the background, we may define \Theta B Other more sophisticated definitions are possible. If an object contains n holes, its bounding surface is made of (1 n) borders that are not connected together. Each border is a closed surface made of adjacent faces sewed together. Figure 4 illustrates this notion with a 1-hole object using the 2D analogy. Since objects are 6-connected sets of voxels, there exist three different types of sews between two adjacent faces of a border. These three types of relations have been presented by Rosenfeld [8] and are named 1-sew, 2-sew and 3-sew (see figure 5). They depend on the adjacency relation between the voxel(s) supporting the two faces. Figure 4: A 2D object with one hole and its two borders 3-sew 2-sew 1-sew Figure 5: Three types of sews be a 6-connected object with n holes. Each border \Upsilon ' i of ' i is a pair ffl F is the set of faces which separate 6-connected component ' i from the background by a closed ffl R is the sewing relation. It is a set of 4-tuples (f expressing that faces f 1 and f 2 are sewed together through their common edge e using a sew of type s or 3). Using this definition the notions of skin are introduced as follows. Definition 3 The skin of object ' i characterized by n holes is the union of all its borders and is denoted by S ' i ;j . The skin of composed object ' i is the union of the skins S ' i and is denoted by S \Theta : S \Theta Since the skin of a composed object is a union of borders, it is also defined as a pair (F; R) as introduced by definition 2. 4 Sewing Faces : an algorithm for skin reconstruction For each border \Upsilon to be reconstructed, the starting point of Sewing Faces is a pair (v; i) where v is a voxel of the object contour and i is a face of v belonging to border \Upsilon. Such a pair is called a starting-voxel and can be either given by the user or determined by a dichotomous search algorithm that depends on the type of the 3D image. The principles of the sequential version of Sewing Faces are the following. From the starting voxel, the algorithm first computes its faces that belong to the bounding surface and then detects among its adjacent voxels (either 6 or 18-adjacent) the ones that belong to the contour. For each of these voxels it determines the faces that are also included in the bounding surface and realizes their sews with adjacent faces of the skin. The process is then iterated for all the adjacent voxels that also belong to the contour. Step-by-step the bounding surface is reconstructed based on a contour following. The algorithm uses a hash table to memorize the faces that have already been added to the skin, and a stack to store the voxels to be examined. It uses two main functions GetF aces (v; i) and T reat Seq: (v) that are now described. They rely on the notion of neighbor of a voxel according to a given face defined by its type (cf. figure 2). Definition 4 Let v be a voxel whose face of type i is shared with voxel u. Voxel u is called the neighbor of v by its face i and is denoted by n(v; Function GetF aces (v; i) determines the faces of v belonging to the skin when its face of does. It runs as follows : (1) add 1 face of type i of v to F . (2) add to F any of the four faces of type j of v adjacent to face of type i such that n(v; 2 \Theta or n(v; B, i.e. such that the neighbor of v by face j is not in the object. (3) if one face has been added to the border during steps (1) or (2) then : (a) add to F face of type k of v which is opposite to i if n(v; 2 \Theta or n(v; (b) add to R the 1-sews (i; e that have been added to F during steps (1), (2) or (3)(a). (c) push v in the stack of voxels to be treated. Hence voxel v is not entirely treated at this step. In particular its 2-sews ans 3-sews with adjacent voxels have not been detected yet. (d) add to the hash table the faces of v added to the skin. Function T reat Seq: (v) determines all the 6-adjacent or 18-adjacent voxels of v such that one or more of their faces belong to the skin, and sews these faces to those of v when they share one edge, i.e. realizes the 2 and 3\Gammasews. The data structure used by T reat Seq: (v) is an array which indicates for each face type, the reference in F of the corresponding face of voxel v if this face has already been added to the skin. The corresponding entry is empty if the face is not in F or if the face has not already been added to the skin. T reat Seq: (v) is defined as follows operation add does not add a face twice in F , i.e. it checks whether the given face already belongs to F and actually adds it only if it does not belong to F . and one of its adjacent faces in v (call it j) belong to the border, face k opposite to i and adjacent to j also belongs to the border if the neighbor of v by face k is not in the object. reat Seq: (v) get from the hash table for each edge e of v shared by two faces such that faces v do /* w is not in object \Theta B */ /* in this case face j of voxel u */ /* belongs to the skin */ else /* w is in object \Theta B */ /* in this case the opposite face to face */ /* i in voxel w belongs to the skin */ endif endfor Using these two functions the sequential version of Sewing Faces, denoted by SF Seq: ( ) is defined by: for each starting voxel (v; i) do GetF aces (v; i) while not empty (stack) do reat Seq: (top (stack)) 5 Sub-blocks decomposition Let us suppose that the blocks we deal with contain one byte long integers. The memory size required for blocks of size 128 3 , 512 3 or 1024 3 is respectively 2MB, 128MB or 1GB. To allow any computer to run Sewing Faces on such large-size data, the whole block must be decomposed into sub-blocks and the algorithm must be processed on these sub-blocks. The size of the sub-blocks depends on the available amount of memory. In the general case, each voxel intensity being g bytes long, a c bytes memory space available for the sub-block storage can hold sub-blocks up to size (l; l; l) with l = (g=c) 1=3 . Using a parallel machine, such a decomposition into sub-blocks is also very useful since the bounding surfaces reconstruction algorithm may be run simultaneously on the different sub-blocks assigned to different processors of the architecture. 5.1 Sub-blocking Splitting the block into sub-blocks is the sub-blocking operation. A sub-block has six faces, each face is a voxel slice. It is fully defined by the coordinates of its origin in the block and by its size according to axes x, y and z. sub-block block gluing faces GFGF GF GF GF GF GF GF GF GFGFGFGFGFGFGF Figure The gluing faces and forbidden faces of a sub-block On each sub-block, Sewing Faces builds a sub-skin. All the sub-skins must finally be merged to get the whole skin. In order to be able to glue all the sub-skins together, the overlap between two adjacent sub-blocks must be two-slices wide. be two sub-blocks whose origins in block B are (B:x 1 whose sizes are (b 1 :z). If there exist two axes ff; fi 2 x; z such that B:ff if the third axis fl is such that or such that are said to be adjacent. On any sub-block two particular types of faces are emphasized: the gluing faces and the forbidden faces, as illustrated in figure 6 using the 2D analogy. They correspond to the overlap between sub-blocks and are therefore defined only "inside" the whole block and not on its own faces. Definition 6 Let b be a sub-block and f b be one of its faces such that f b is not included into a face of block B. Face f b is called a forbidden face. A slice which is adjacent to a forbidden face is called a gluing face. The execution of Sewing Faces on sub-block b reconstructs the sub-skin associated with b. This sub-skin is similar to a skin except on the "border" of sub-block b where some sews are missing. The missing sews connect a face of a voxel lying on one of its gluing faces with a face of a voxel belonging to a sub-block adjacent to b. Notice that since the sews of type 1 involve only one voxel, they can always be detected even if they are on a gluing face. The missing sews are therefore only of type 2 or 3. They must be detected and memorized to be treated during the final gluing phase. To do so the complete neighboring of the gluing faces voxels must be examined. This explains the presence of the forbidden faces adjacent to the gluing faces. Since the voxels of the forbidden faces are not treated in sub-block b, any forbidden face must be shared by an adjacent sub-block where it is considered as a gluing face. These adjacency relations between sub-blocks guarantee that the gluing process of the sub-skins will be possible. They are characterized as follows. GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF mB.size.y B.size.y -n GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF GF Figure 7: A valid sub-blocking Definition 7 A set fb of sub-blocks such that : does not belong to a forbidden face of b i , and ffl 8v such that are adjacent is a valid sub-blocking (with respect to the object). Notice that a valid set of sub-blocks need not to cover the whole block. If part of the image contains only background voxels, it is unnecessary to process it. Figure 7 shows a valid sub- blocking using the 2D analogy. In the general case it is easy to automatically split a block into a valid sub-blocking whose sub-blocks can be separately processed. 5.2 Gluing phase Once the sub-skins on the different sub-blocks have been reconstructed, they must be merged together to get the final skin. This process requires the realization of the missing sews. We call the whole process (merging and sewing) the gluing phase. For any sub-block b i the sub-skin is characterized by pair is the set of faces belonging to the sub-skin of b i and R b i is the corresponding set of fully realized sews. During the sub-skin computation the missing sews, called half-sews, are memorized. They are defined as follows. Definition 8 A half-sew is a quintuple (e; s; f; coord; t) where ffl e is the edge to be sewed; ffl s is the type of sew to realize; ffl f is the face of set F sharing edge e; ffl coord are the coordinates of voxel v that owns f ; ffl t is the type of the unknown face that share edge e with face f . For each sub-block this information is memorized using either a hash table or a list ordered according to field coord. The gluing process is realized in two steps. The first one consists in merging together the different sets F b i computed on sub-blocks . The second step concerns set RB defining the sews between all the faces of FB . This step is realized by building full-sews from pairs of half-sews describing the two parts of a sew. Definition 9 A full-sew is composed of two half-sews characterized by : define the coordinates of the two adjacent voxels involved in the sew; At the end of the gluing process, the skin describes the geometry and the topology of all the borders that were pointed out by the starting voxels given at the beginning. 5.3 Fragmentation of the objects The following problem occurs when decomposing a block into sub-blocks. A 6-connected object contained in the block is fragmented on the different sub-blocks and on a given sub-block the fragment of the object may not be 6-connected. If this problem is not carefully taken into account the skin reconstruction will be incorrect. Figure 8 illustrates such a situation using the 2D analogy. The initial block is splitted into two sub-blocks with an overlap of two slices : the gluing and forbidden faces. The 6-connected object is distributed on the two sub-blocks in such a way that on sub-block 2 there are two non-connected fragments of the object. Therefore if the algorithm on sub-block 2 runs with only one starting voxel, then only one of the two fragments will be reconstructed. Since it is not realistic to expect the user to give as many starting voxels as there are fragments of the objects on the different sub-blocks, this problem must be automatically solved by the method. This is realized on the gluing faces. When a voxel of the contour lies on a gluing face, its adjacent voxels belonging to the forbidden face may also belong to the contour and are therefore considered as new starting voxels for the adjacent sub-block. In figure 8 let the two voxels drawn starting voxel starting voxel new starting voxel Figure 8: Fragmentation of a 6\Gammaconnected object into non 6\Gammaconnected fragments in dark grey be the starting voxels on each sub-block. During its processing, sub-block 1 detects that the sub-skin it is reconstructing gets out in sub-block 2. A new starting voxel (drawn in black) is then pushed by sub-block 1 on the stack of sub-block 2 and no part of the skin is lost. also send to sub-block 2 as other starting voxel the voxel mentioned on the figure. Sub-block 2 will discard this voxel since it has already been treated while constructing the small fragment on top of figure 8 from it own starting voxel. Vice versa, sub-block 2 will send two new starting voxels to sub-block 1. They will also be discarded by sub-block 1 since they have already been treated. This mecanism guarantees that the whole skin will finally be reconstructed. As a consequence only one starting-voxel (v; i) (as described in section 4) per sub-block is required at the beginning of the algorithm. 6 Coarse-grain parallelization The coarse-grain parallelization of Sewing Faces is based on the notions of sub-blocking and gluing phase. The idea is to decompose the 3D block of data into sub-blocks as defined in section 5. The sub-skins on each sub-block are computed simultaneously on different processors. Once all the sub-skins have been determined the final gluing phase is realized. The computation of a sub-skin is similar to the sequential version, except for the voxels of the gluing faces that may induce missing sews. For any such voxel the following steps must be realized (1) store the corresponding half-sew (as defined in definition 8) in order to later compute the full-sew; (2) determine which voxel v 0 of the forbidden face should be treated to realize this sew; (3) determine which sub-block b j owns v 0 in a slice that is not a forbidden on the stack of voxels of sub-block b j as a new starting voxel. This allows to solve the fragmentation problem introduced in section 5.3. These four points are realized by function HalfSews ( ). Notice that in order to realize point (3), each sub-block has to know the origin and the size of all the other sub-blocks. Point (4) can be realized either with a message passing mecanism on a distributed memory processor or using shared memory on a share memory machine. The parallel version of the algorithm, denoted by SF Par: (), is obtained by changing function reat Seq: (v) into T reat Par: (v) as follows. reat Par: (v) get from the hash table for each edge e of v shared by two faces such that faces v do if e defines a missing sew then HalfSews else /* w is not in object \Theta B */ /* in this case face j of voxel u */ /* belongs to the skin */ else /* w is in object \Theta B */ /* in this case the opposite face to face */ /* i in voxel w belongs to the skin */ endif endif endfor When all the sub-blocks have terminated their execution, each sub-skin is characterized by a and the gluing process can begin. As explained in section 5.2 the half sews associated with each sub-block are memorized into ordered lists. The gluing phase consists therefore in scanning these lists in order to complete the sews. Due to the ordering of the lists according to the coordinates of the supporting voxels, this process is linear relatively to the length of these lists. The number of voxels to be treated in each sub-block obviously depends on the objects that are considered and also on the sub-blocking. Sub-blocking may be data-driven : the whole image is scanned in order to detect the objects before splitting the block. This method is used by [7] to achieve load-balancing in Marching Cubes parallelization. Such a whole block scanning is terribly cost-effective compared to the contour following used by Sewing Faces to get good time performance. Moreover we believe that a good load balancing is strongly related to the domain of application. For a given type of images, for instance brain images obtained from magnetic resonance imaging techniques, all the images to be treated are rather similar relatively to the question of load-balancing. Therefore we think that before using Sewing Faces in a given field, a preliminary study must be conducted on a set of standard images in order to detect the appropriate sub-block decomposition that will result in effficent load-balancing. Such a decomposition may be for example to split the block either into cubic sub-blocks or into slices. 7 Fine-grain parallelization The fine-grain parallelization of Sewing Faces consists in allowing several processors to deal with the same sub-block. Obviously input data and data structures must be shared by all the processors running Sewing Faces on the same sub-block. In consequence the main difficulty of such a parallelization is to prevent data from corruption. Let us see in detail where corruption problems may arise : possible data-corruption because it is read-only; ffl hash table : when array faces new is written in the hash table, if its previous version old currently stored in the hash table contains values that are not stored in new then the two arrays must be merged. Moreover 1-sews between faces that (faces new v [i] and faces old new v [j] and faces old (faces new v [j] and old new v [i] and faces old must be added to R. During this merging step, access in hash table to array faces v [ ] by other processors must be forbidden using a semaphore-like method; ffl stack of voxels : no possible data-corruption since the voxels are just pushed to or poped from the common stack. Depending on the stack implementation, semaphore-like operations may be used to guarantee the correctness of the stack information (such as its set F of faces : it is simply implemented using an array. When a new skin face f is detected during one GetF aces ( ) execution, the next available face number in F must be read and incremented using indivisible instruction or any other semaphore-like operation; ffl set R of sews : no data curruption is possible because R is a write-only file; ffl set of half-sews : it is implemented as a write-only disk file, therefore no data corruption is possible. Moreover, the gluing process can easily deal with duplicated half-sews by omitting them when the case arise. The fine-grain parallelization of Sewing Faces solves the load balancing problem in the general case. All the processors executing the algorithm on the same sub-block share one stack of voxels, one hash table, etc. As a result they all finish their execution at the same time when the stack of voxels is empty. When the fine-grain approach is used alone without any coarse-grain parallelization, the load-balancing is always optimal. The fine-grain parallelization may be combined with a coarse-grain decomposition using a cluster of share-memory processors. The different sub-blocks are assigned to the different machines of the cluster. On each share memory machine the different processors may compute the sub-skin associated with one sub-block through the shared stack of voxels. Embedding So far we have only realized topological operations (adding faces into set F , adding sews into set R), without considering real coordinates of the vertices. Therefore the extracted surface is a topological surface. In order to visualize it, it must be transformed it into a geometrical surface, i.e. into a 2D mesh. Such a process is called an embedding. A face embedded in the 3D space becomes a facet. To convert all the faces into facets, a starting point is required : the real Table 1: Execution time related to the sequential version SF Size of the 3D block Number of faces Time (s.) 260 316656 6.0 300 421968 8.0 coordinates of the four vertices of a given face f of each border. From the type of face f and from its sews types, it is easy to deduce from which face arises each of the four faces sewed with f . We can compute the coordinates of the four faces adjacent to f . And so on. We thus obtain the real coordinates of all the faces of FB and we get all the facets. If the three dimensions of the basic parallelepiped representing one voxel are integer values, embedding of the skin does not require any computation with real values. The embedding process is O(f) where f is the number of faces of the skin. 9 Results We have already proved [6] that the sequential version of Sewing Faces is linear in time and space according to the number of faces in the skin. Linearity is still achieved by the coarse-grain and the fine-grain approach. In tables 1, 2 and 3 we focus on the execution time of the coarse-grain version versus the sequential version. The input data consist of digital balls of growing size (from 230 3 to 300 3 ). Each image contains only one object whose skin is made of only one border. The starting voxels are automatically detected by a dichotomous algorithm. Sub-blockings are obtained by splitting each block in 8 sub-blocks b i=0;7 with identical sizes. Tests summarized in tables 1, 2, and 3 were realized on a Intel Pentium 133 MHz computer running under Linux. Indicated times are user + system times. Table results obtained with the sequential version of Sewing Faces. The number of faces and the elapsed time (in seconds) are indicated. Indicated times include the block loading, the research of the starting voxel and the skin building. Table 2 shows the elapsed time of the coarse-grain version of Sewing Faces on any elementary sub-block. The measured time includes the sub-block loading, the research of the starting voxel, the sub-skin building and the half-sews detection. The third column indicates the elapsed time related to the gluing process. Finally the last column shows the theoretical time obtained on a multi-processor architecture where all sub-blocks are processed altogether at the same time. It is obtained by adding the gluing time and the elementary sub-block time. Table 3 points out the time saved by the coarse-grain approach and shows it as a percentage. The last column indicates the speed-up factor due to the coarse-grain approach and underlines the fact that using 8 processors we get an speed-up factor of about 5. Let us now study the influence of the sub-blocking on the parallel computation time. The efficiency obviously depends on the number and the structure of the sub-blocks. If the number of Table 2: Execution time related to the coarse-grain parallel version of SF Par: ( ) Size of the 3D block Time to compute on a sub-block Time of the gluing phase Total time (s.) 300 1.1 0.5 1.6 Table 3: Speed-up obtained with the coarse-grain parallelization Size of the 3D block time saved (%) Speed-up 260 80.00 5.0 300 80.00 5.0 sub-blocks is too large, the gluing phase will become more time-consuming due to the increasing number of missing sews. This problem has been studied with a synthetic block of size 225 \Theta 300 \Theta 225 representing a bar of size 210 \Theta 300 \Theta 210. The block is decomposed into vertical sub-blocks of equal width. In order to analyze the influence of the number of sub-blocks on the efficiency of the parallel version of Sewing Faces, we have executed it on the above-mentioned block, increasing the number of sub-blocks. This experiment has been realized on a SGI R4400. Figure 9 shows the experimental results. It indicates that the computation time decreases with the number of sub-blocks until we reach 12 sub-blocks. With more than 12 sub-blocks the computation time remains almost the same. We may take advantage of the lack of penality in terms of computation time when increasing the number of sub-blocks, to execute Sewing Faces on more than 12 sub-blocks in case of memory limitation. The experiment has also been conducted on brain images (cf. figure 11) using cubic sub-blocks of equal size. The previous results are confirmed : above a certain number of sub-blocks, 27 in this case as indicated in figure 10, the computation time does not decrease anymore due to the increasing number of missing sews to glue. Figure 12 shows the result for a decomposition into two sub-blocks. Conclusions We have proposed in this paper a parallel version of a reconstruction surface algorithm. Our goal is not only to increase time performances but also to deal with large 3D images that are now time # of sub-blocks Figure 9: Parallel time computation for the bar currently available in practical fields such as medical imaging. This parallelization is based on a sub-block decomposition. In order to compute the skin using a contour following approach, a two-slices wide overlapping between adjacent sub-blocks is required. Moreover sub-blocks must communicate with their neighbors to guarantee the computation of the whole skin. The initial choice of the data structures used by the sequential version of Sewing Faces (one stack of voxels and one hash table for voxel faces) allows to easily and fully parallelize it, using a coarse-grain and/or a fine-grain approach. Moreover the notions of border, composed object and sub-block overlapping cause the parallel version of Sewing Faces to be very flexible. Notice moreover that the sub-block based algorithm may also be useful on a mono processor machine to deal with blocks of data that are too large to fit in memory. Experimental results have shown the efficiency of the parallel version of the algorithm, both on synthetic balls data and on brain images. --R Octrees for Faster Isosurface Generation. Digital Topology Marching Cubes Sewing Faces Wu Digital Surfaces. Allen Van Gelder and Jane Wilhelms Topological Considerations in Isosurface Generation. --TR

computer graphics;parallel applications;3D digital images;coars and fine-grain parallelization;bounding surfaces reconstruction

285191

Linearly Derived Steiner Triple Systems.

We attach a graph to every Steiner triple system. The chromatic number of this graph is related to the possibility of extending the triple system to a quadruple system. For example, the triple systems with chromatic number one are precisely the classical systems of points and lines of a projective geometry over the two-element field, the Hall triple systems have chromatic number three (and, as is well-known, are extendable) and all Steiner triple systems whose graph has chromatic number two are extendable. We also give a configurational characterization of the Hall triple systems in terms of mitres.

Introduction In November of 1852 Steiner [32] posed an infinite series of questions concerning what are now known as Steiner triple systems. The second of these questions asked whether or not it was always possible, given a Steiner triple system on n points, to introduce n(n 4-subsets of the underlying set of the given triple system with the property that no one of these 4-subsets contained a triple and, on the other hand, each of the 3-subsets of the underlying set not a triple were in one of the chosen 4-subsets (necessarily unique). In other words he asked whether or not every Steiner triple system extends to a Steiner quadruple system. Triple systems that do extend are now known as derived triple systems. 1 Despite an enormous effort, especially in the last twenty years, not a great deal of progress has been made in answering this second question, but it is known that all Steiner triple systems on fifteen or fewer points are derived [10, 25] - a result obtained by computer attack. We here introduce a rather restricted notion of "derived" for Steiner triple systems. The ideas were inspired, indeed flowed naturally from, the binary view of Steiner triple systems taken in [2]. We call a triple system possessing such an extension to a Steiner quadruple system linearly derived . Although The author wishes especially to thank Paul Camion and Pascale Charpin. The research atmosphere that they have created at Projet Codes, INRIA surely contributed to this investigation, the bulk of which took place during the Spring of 1995 while the author was a visitor there. Apparently Pl-ucker in 1835 had not only defined triple systems but had even asked about extending them to quadruple systems and Pl-ucker's work may have been the inspiration for Steiner's paper. See [37]. not all Steiner triple systems are linearly derived, all the geometric 2 systems are - and many, many others too. If a Steiner triple system is linearly derived so are any subsystems it may possess. Hence whenever one finds a Steiner triple system that is not linearly derived one knows that any system of which it is a subsystem cannot be linearly derived. To find a "linear extension" of a Steiner triple system one restricts one's attention to those 4-subsets of the underlying set of the system that are symmetric differences of two triples - that is, those 4-subsets that are the support of the binary sum of the incidence vectors of two triples. We call such 4-subsets linear 4-subsets. Being linearly derived is related to the existence of quadrilaterals (the unique four-line configurations on six points) and non-Fano planes (the unique six-line configurations on seven points) in the triple system. In par- ticular, the non-Fano planes of the system are in one-to-one correspondence with those linear 4-subsets of the system that can be seen in three distinct ways as the symmetric difference of two triples. In order to more easily express and understand the results we present, we attach a graph to each Steiner triple system. The vertices of the graph are the linear 4-subsets of the system with two joined by an edge if the two 4-subsets intersect in three points. The geometric systems of points and lines of projective spaces over F 2 are characterized as those Steiner triple systems whose graph is monochromatic (that is, there are no edges). Among the quadrilateral-free Steiner triple systems one finds the Hall triple systems (which can be viewed as generalizations of the geometric systems of points and lines of the affine spaces over F 3 We show that these systems have graphs with chromatic number three and that they are linearly derived. In fact, any quadrilateral-free Steiner triple system whose graph has chromatic number three will be linearly derived, but it may be that the Hall triple systems are characterized among the quadrilateral-free systems as those whose graph has chromatic number three. There are Steiner triple systems whose graphs have chromatic number two; they are linearly derived and seem to form an interesting class of systems, which, as far as we know, have never been considered. A linear extension Following [5] we call a Steiner triple system geometric if it is the system of lines of a projective geometry over the binary field or an affine geometry over the ternary field. of such a system can be constructed from the quadrilaterals and non-Fano planes of the systems - in fact, from the quadrilaterals alone. Indeed, one of the most surprising outcomes - to the author at least - of this investigation was the realization that if a Steiner triple system contains enough Fano planes and proper non-Fano planes, then it is linearly derived. This result, Theorem 4.3, may be viewed as a generalization of the Stinson-Wei [33] result characterizing the geometric binary Steiner triple systems in terms of the number of quadrilaterals. Probably the vast majority of systems have graphs with chromatic number greater than three. Since the maximum possible degree of the graph of a Steiner triply system is eight, it follows from Brooks's theorem [6, Theorem 8.4] that the chromatic number of the graph is at most eight. We do not have an example of a linearly derived Steiner triple system whose graph has chromatic number greater than three. The cyclic system of order 5 (on 13 points) has chromatic number greater than three and is not linearly derived; the quadrilateral-free system of order 6 (on 15 points) also has chromatic number greater than three. Mitres, the unique five-line configurations on seven points with two "par- allel" lines, play a role in the discussion to follow. We also give a configurational characterization of the Hall triple systems couched in terms of mitres. Just as the geometric binary systems are characterized as those systems with the maximum number of quadrilaterals, the Hall triple systems are characterized as those systems with the maximum number of mitres. Only an elementary knowledge of design theory (such as can be found, for example, in Chapter 1 of [4]) and graph theory are necessary in order to read this paper. Any of the standard references on design theory and graph theory will contain the necessary background. Moreover, we define most notions as they are introduced. The only departure from the nomenclature of the vast existing literature on Steiner triple systems is that we use "order" in the design-theory sense; it does not denote the number of points n of the triple system, but rather (n \Gamma 3)=2. Linear 4-subsets and their graph Recall that a 4-subset of the underlying set of a Steiner triple system is linear if it is the symmetric difference of two triples. The name comes from the binary view of Steiner triple systems: a linear 4-subset is the support of a weight-four vector in the binary code of the triple system that is the sum of the incidence vectors of two triples. Linear 4-subsets come in three flavors according to whether they arise just one way, two ways, or three ways as a symmetric difference (or sum). Clearly three is the maximum number, since a 4-set splits in exactly three ways as the disjoint union of two 2-subsets. We will distinguish these linear 4-subsets by calling them singly-linear, doubly-linear and triply-linear, respectively. A triangle in a Steiner triple system is simply a 3-subset of the underlying set which is not a triple of the system. A Steiner triple system on n points triangles. A triangle can be expanded to a linear 4-subset in three ways, clearly, but the resulting linear 4-subsets might not differ. In fact, a triangle is a subset of one, two, or three linear 4-subsets so triangles, too, come in three flavors: a Type I triangle is contained in exactly one linear 4-subset; a Type II triangle is in exactly two linear 4-subsets; a Type III triangle is in exactly three linear 4-subsets. It is a simple matter to check that all four 3-subsets of a linear 4-subset are triangles, that all four 3-subsets of a triply-linear 4-subset are of Type I, and that all four 3-subsets of a doubly-linear 4-subset are of Type II. But a singly-linear 4-subset could contain both Type II and Type III triangles. Sometimes a singly-linear 4-subset contains only Type II triangles. In fact a singly-linear 4-subset contains only Type II triangles if and only if its valency in the graph is four. Of course, in a quadrilateral-free Steiner triple system all linear 4-subsets are singly-linear and contain only Type III triangles. In such systems, the valency of every vertex is eight. denote the set of singly-linear 4-subsets, L 2 denote the set of doubly-linear 4-subsets, and L 3 denote the set of triply-linear 4-subsets. Sim- ilarly, \Delta I will denote the set of Type I triangles, \Delta II will denote the set of Type II triangles, and \Delta III will denote the set of Type III triangles. The number of linear 4-subsets of a Steiner triple system on n points depends, in general, not only on n but on the system. There are, however, ways to form the symmetric difference of two triples. The following lemma summarizes the easily proven relationships among the cardinalities of the sets just described and some simple consequences of these relationships. Lemma 2.1 1. jL 2. 3. 4. 5. A quadrilateral of a Steiner triple system is the four-line configuration 3 containing exactly six points. Here is the configuration: s s s s Doubly-even 4-subsets are obviously related to quadrilaterals but the notions are different: a quadrilateral gives rise to three doubly-linear 4-subsets and the triply-linear 4-subsets complicate the relationship still further. On the other hand a quadrilateral-free Steiner triple system is the same as a Steiner triple system all of whose linear 4-subsets are singly-linear. In fact, "linearity" is a somewhat finer notion. Here is the easily proven relationship between quadrilaterals and linearity: Proposition 2.2 The number of quadrilaterals in a Steiner triple system is3 jL 2 and, in particular, the number of doubly-linear 4-subsets of a Steiner triple system is divisible by 3. 3 We are using "line" rather than "triple" here to conform with common usage. The four-line configurations of a Steiner triple system have been catalogued and counted by Grannell, Griggs and Mendelsohn [13] and the number of quadrilaterals determines the number of all other four-line configurations. Corollary 2.3 The number of singly-linear 4-subsets of a Steiner triple system is divisible by 3. Proof: By Lemma 2.1 we have that and the result is obvious. 2 A triply-linear 4-subset of a Steiner triple system corresponds precisely to a configuration isomorphic to a non-Fano plane. Such a configuration is the six-line configuration on exactly seven points and can be viewed as the usual Fano plane with a line removed; as a linear space, it plays an important role in coordinatization problems - see Delandtsheer, [9, Page 197]. Here is the picture, the triply-linear 4-subset being the complement of the missing line (the vertices of the large triangle plus the central point): s s s s Observe that the triply-linear 4-subset given by the non-Fano plane is the support of the binary sum of the incidence vectors of the six lines of the configuration. In order to easily state some of the results we wish to discuss we attach a graph 4 to each Steiner triple system. Since no other graphs will appear in the paper we simply call it the graph of the Steiner triple system. The vertices of the graph are the linear 4-subsets and two vertices will be connected by an edge if the two linear 4-subsets intersect in three points. The vertices of the graph corresponding to the elements of L 3 are clearly isolated vertices and there are no edges between vertices corresponding to elements of L 2 There will be edges between elements of L 2 and elements of L 1 , however. 4 There are at least two other ways to attach a graph to a Steiner triple system. Perhaps the most interesting of these other graphs is the "block graph", which is a strongly- regular graph. For an account of this graph the reader may wish to consult Peeters, [28, Section 3.1]. The graph we are introducing is quite different and should not be confused with the others. In fact, the valency of an element of L 2 is clearly four. Because every Type III triangle of a Steiner triple system gives rise to a 3-clique of the graph, implies, by Lemma 2.1, that the graph is at least 3-chromatic; in particular, the graph of any quadrilateral-free Steiner triple system is at least 3-chromatic. As we shall see, the Hall triple systems do have 3-chromatic graphs. Since every triangle is contained in a linear 4-subset and no linear 4- subset contains a triple of the system, it is at least conceivable that one might find among the linear 4-subsets the means to extend the triple system to a quadruple system. In terms of the graph of the system, an extension of a given Steiner triple system found among the linear 4-subsets, is the same thing as a stable 5 subset of the vertices of its graph with cardinality precisely24 Because a collection of linear 4-subsets will form an extension provided there are enough with no two intersecting in three points, the graph of a Steiner triple system can have a stable subset of vertices of cardinality at most 624 We call a Steiner triple system that posesses such an extension linearly derived and the collection of 1n(n linear 4-subsets a linear extension. Since L 2 [L 3 is always a stable subset of the graph of a Steiner triple system, we have the following easy Proposition 2.4 For a Steiner triple system on n points In the case of equality L 2 [L 3 is a linear extension and the system is linearly derived. Remarks: 1) In fact, equality does occur for some Steiner triple systems. As we shall see, Corollary 4.2, they are precisely those systems whose graph has 5 A stable subset of a graph is a set of vertices with no two connected by an edge; such a subset is also sometimes called an independent set of vertices. A stable subset of a graph is the same as a clique of the complementary graph. 6 In general it is "hard" - in the technical, computer-science sense - to compute the maximum number of vertices in a stable subset of a graph and hard to compute the chromatic number also. For an expository discussion of these matters see Knuth [18]. chromatic number one or two. 2) Since the triangles of Type I are in a unique linear 4-subset - which is, in fact, triply-linear - any linear extension of a Steiner triple system must contain all triply-linear subsets; that is, L 3 must be a subset of any linear extension. Observe that a linear 4-subset is always among the points of the subsystem generated by any one of its triangles. It follows that a linear extension of any Steiner triple system automatically yields linear extensions of all of its subsystems. Since the cyclic Steiner triple system of order 5 (on 13 points) is, as we shall see, not linearly derived, there will be many triple systems without linear extensions. In the next section we investigate the geometric Steiner triple systems and show not only that they are linearly derived but that their graphs have the smallest possible chromatic numbers. 3 Linear extensions of geometric systems The Steiner triple systems we will deal with in this section are either the systems of points and lines of a projective geometry over the binary field or Hall triple systems - i.e., Steiner triple systems each of whose triangles is contained in a subsystem isomorphic to the affine plane of order 3. We begin with a characterization, in terms of the graph, of the binary geometric systems. This result, although different, is equivalent to Theorems 3.1 and 3.2 in [33]. Proposition 3.1 If a Steiner triple system has no singly-linear 4-subsets then it has precisely 1n(n\Gamma1)(n\Gamma3) linear 4-subsets and these 4-subsets form a linear extension. Moreover, the triple system is the system of points and lines of a projective geometry over the binary field and its graph is monochromatic Proof: By Lemma 2.1 jL is empty so is L 2 . It follows that all linear subsets are triply-linear and that all triangles are of Type I. Hence each triangle is contained in a unique linear 4-subset and we have the necessary linear extension. The extension is simply given by L 3 and the graph has no edges, i.e., is monochromatic. That the system must be a geometric binary system follows rather easily. One can simply apply the Theorem 3.2] or see below - since the number of quardilaterals is maximal by Proposition 2.2, or use [15, 16].2 Corollary 3.2 The graph of a Steiner triple system is monochromatic if and only if it is a geometric binary system. Proof: It follows easily from the discussion presented with the description of the graph that, if the graph is monochromatic, then jL and we have the geometric system. On the other hand, in a geometric binary system of points and lines of a projective geometry every triangle generates a Fano plane and every linear 4-subset is triply-linear. In particular the graph is monochromatic.2 The characterization of the binary geometric systems due to Stinson and Wei [33, Theorems 3.1 and 3.2] is the following: A Steiner triple system on n points has at quadrilaterals with equality if and only if it is the geometric system of points and lines of a projective geometry over the binary field. From the point of view of [2], such a characterization should have a "ternary" analogue - and indeed it does. Since the characterization is via local properties, however, it is not the systems of points and lines of affine geometries over the ternary field that are characterized but, instead, the Hall triple systems - that is, systems that are locally like the affine plane of order 3. In the ternary analogue, quadrilaterals are replaced by mitres. 7 A mitre is the five-line configuration with exactly seven points and two disjoint triples: s s Here is the characterization: 7 The terminology is due to Andries Brouwer. Theorem 3.3 A Steiner triple system on n points has at most12 mitres with equality if and only if it is a Hall triple system. Proof: We first describe a pre-mitre - which can be viewed as a mitre with one of the disjoint triples removed. Here is the picture: s s s s Now, the four-line configuration above may or may not complete to a mitre. The important point is that every mitre gives rise to exactly two such configurations and that the number of such configurations is easily counted (see [13] where pre-mitres are simply the four-line configurations of type C 15 There are precisely 1n(n pre-mitres in every Steiner triple system on n points. This immediately gives the inequality with equality if and only if every pre-mitre completes to a mitre. Since every triangle embeds in a pre-mitre - in at least three ways, in fact - starting with a triangle and using the fact that all pre-mitres complete to a mitre, it is an easy exercise to show that every triangle is contained in a subsystem isomorphic to an affine plane of order 3, that is, that the triple system is a Hall triple system. On the other hand, given a Hall triple system every pre-mitre must be in an affine plane of order 3 and hence must complete to a mitre, since the three points that should be collinear must be on one of the two lines parallel to the base of the pre-mitre.2 The proof above also gives the following Corollary 3.4 A Steiner triple system in which every pre-mitre completes to a mitre must be a Hall triple system. We next want to show that the geometric systems of points and lines of an affine geometry over the ternary field are linearly derived. We show more: that Hall triple systems are linearly derived and, in fact, we show that the graph of a Hall triple system is 3-chromatic, which is an apparently stronger property. First note that a Hall triple system cannot contain a quadrilateral since affine planes of order 3 do not contain quadrilaterals. Thus, in this case we have that both L 2 and L 3 are empty, with all linear 4-subsets singly-linear - in contrast to the binary geometric systems where all linear 4-subsets were triply-linear. That is, Hall triple systems are quadrilateral-free. There are certainly many, many other quadrilateral-free Steiner triple systems, the first non-Hall system occuring at order system #80 of the eighty triple systems on 15 points. Andries Brouwer proved that quadrilateral-free Steiner triple systems existed for all n j 3 (mod 6); this was independently shown by Griggs, Murphy and Phelan in [14], where a brief history of the problem is given. Griggs has constructed systems for many and, moreover, conjectures that quadrilateral-free Steiner triple systems exist for all admissible orders except 2 and 5. Observe that the graph of a quadrilateral-free Steiner triple system is regular of valency 8, contains 3-cliques, but does not contain a 4-clique. Its chromatic number is, of course, at least 3. We first show that, if it is 3, then it is linearly derived. Proposition 3.5 In a quadrilateral-free Steiner triple system whose graph has chromatic number 3, the linear 4-subsets split into three disjoint subsets, each of which yields a linear extension of the triple system. Proof: It follows from Lemma 2.1 that a quadrilateral-free Steiner triple system on n points has precisely8 n(n linear 4-subsets. Since the cardinality of a stable subset of its graph is at whenever the graph is 3-chromatic the linear 4-subsets must split into three stable subsets of cardinality each corresponding to one of the three colors and each yielding a linear extension.2 We next show that the graphs of Hall triple systems have chromatic number three by splitting their linear 4-subsets into three disjoint linear exten- sions. We do not have an example of a quadrilateral-free Steiner triple system whose graph is 3-chromatic that is not a Hall triple system. Theorem 3.6 The linear 4-subsets of a Hall triple system on n points split into three disjoint subsets, each of cardinality 1n(n \Gamma 1)(n \Gamma 3), with each yielding a linear extension. In particular, the graph of a Hall triple system is 3-chromatic. Proof: Simple counting shows that there are precisely fifty-four 4-subsets of an affine plane of order 3 that do not contain a line. All of these must be linear 4-subsets since an affine plane of order 3 is quadrilateral-free and hence contains fifty-four linear 4-subsets. It is well-known (see, for example, [22, Section 7] for a direct proof from first principles or use the small Mathieu design to get the desired splitting) that these fifty-four 4-subsets split into three extensions - which are therefore necessarily linear extensions. We now apply an idea first used in [3]. It can be applied to our present situation either through [1] or [17] to split the linear 4-subsets of any Hall triple system into the required three disjoint pieces. Briefly stated, the idea is that the linear 4-subsets all occur in subsystems of affine planes of order 3 and that one can piece together the extensions of these planes to get linear extensions of the Hall triple system.2 Thus we have shown that the geometric binary system of points and lines of a projective geometry over the binary field has a linear extension as does the geometric ternary system of points and lines of an affine space over the ternary field. Moreover, we have characterized the binary geometric system in terms of the graph and shown that, in this context, the Hall systems are a natural generalization of the system of points and lines of the affine geometry in the sense that they have linear extensions also and that their graphs have chromatic number three. We do not have any other examples of quadrilateral-free Steiner triple system whose graph has chromatic number 3. We are indebted to Wendy Myrvold for the following computer-generated, but easily hand-checked, proof that the graph of the unique quadrilateral-free Steiner triple system of order 6 (on 15 points) has chromatic number greater than 3. Example: The unique quadrilateral-free Steiner triple system on 15 points can be taken to be the cycles of the following three triples: There are twenty-one cycle classes of (necessarily singly-linear) 4-subsets. The reader can check, step-by-step, that at least four colors are necessary by examining the vertices in the following order: f1; 2; 8; 12g, f2; 4; 8; 12g, Of course, one must first check that the above 4-subsets are linear, but that too is easily done by hand. 4 Chromatic number two There are Steiner triple systems whose graphs have chromatic number 2. We characterize these in the following result. It is a case in which the inequality of Lemma 2.1 is an equality - thus also an extremal case. Theorem 4.1 The graph of a Steiner triple system has chromatic number 2 if and only if it has singly-linear 4-subsets and the number of singly-linear 4- subsets equals the number of doubly-linear 4-subsets. Moreover, in this case, the graph, when its isolated points are removed, is a regular bipartite graph of valency 4 and the Steiner triple system has at least two linear extensions, one consisting of the singly-linear and triply-linear 4-subsets and the other of the doubly-linear and triply-linear 4-subsets. implies the existence of 3-cliques in the graph of a Steiner triple system, for the graph to be 2-chromatic we must have it is a simple matter to check directly from the definitions that, putting L 3 aside, the graph on is bipartite with parts L 1 and L 2 and is regular of valency 4. It follows immediately from Lemma 2.1 that and since there are no edges among the vertices in L 2 this set forms a linear extension. Since there are no edges among the vertices in L 1 either, also yields a linear extension. It is also clear that when jL is empty and all the above obtains.2 Remark: If the bipartite subgraph of the graph of the Steiner triple system is disconnected, then there will definitely be more than two linear extensions since, for each connected component, one can choose either of the parts. Examples: 1) The Steiner triple systems #2, #3 and #16 in the usual ordering ([23]) of the 80 on 15 points satisfy the conditions of the Theorem and hence have the announced linear extensions. 8 We are indebted to Vladimir Tonchev for making the necessary computations for both this example and the one that follows (which is easily done by hand). In fact, he computed the data listed in Section 5 giving, for each of the 80 systems, not only the number of linear 4-subsets of each type but also, by Proposition 2.2, the number of quadrilaterals. The number of triply-linear 4-subsets is the number of non-Fano planes contained in the Steiner triple system as six-line configurations on seven points - as we have already remarked. For the systems of full 2-rank (#23 9 through #80 except for #61) none of these non-Fano planes complete to a Fano plane. For the systems #1 through #22 and #61 one can use the 2-rank to compute easily the number of non-Fano planes that do not complete to a Fano plane - since the number of Fano planes is is the corank, a Fano plane being a maximal subsystem in this case. For example, #61 has no such non-Fano planes while #2, which is of 2-rank 12 and hence has seven Fano planes, has eight non-Fano planes which do not complete to a Fano plane. The general results presented above show that four of the eighty Steiner triple systems of order 6 are linearly derived and explicitly give the linear extensions. But all of them are known to be derived and the history of the proof of that fact is interesting. As recently as 1980 the possibility of a computer attack on the eighty systems was 8 These are precisely those systems which are both mitre-free and possess singly-linear 4-subsets, [7, page 215]. 9 In [23, Page 42] it is incorrectly reported (see [24], where this error and several misprints are recorded) that system #23 has a subsystem; in fact, it has none, but it does have four non-Fano planes. The same reference missed the occurrence - apparently by a misreading of the beautiful classification done by hand over seventy-five years ago by White, Cole and Cummings [38] - of non-Fano planes in nine of the systems, each of which has a single non-Fano plane. This omission was also observed by Grannell and Griggs, [12, Page 187], who give the nine systems and the single non-Fano plane in each one. deemed "not feasible" (see [29, Page 113]) but Grannell and Griggs [12] rather quickly made progress via computer and then the remaining fourteen cases were despatched with a computer attack by Diener, Schmitt and de Vries [10]. See the Introduction to [10] for a slightly more detailed history and the body of that paper for a short description of the methods used. 2) The two Steiner triple systems of order 5 (on 13 points) have the following "linear" structure: System Singly \Gamma Doubly \Gamma T riply \Gamma Quadrilaterals linear linear linear The cyclic system is mitre-free [7, page 221]. Wendy Myrvold generated, by computer, the following easily hand-checked proof that the chromatic number of the cyclic system is at least 4. The system can be taken to be the cycles of and, by examining the vertices of the graph in the following order, one sees that at least four colors are needed: f2; 3; 4; 8g, f2; 3; 4; 6g, f3; 4; 6; 8g, These systems are known to be derived (see [25]) but not linearly. In fact, Myrvold ran her clique program on the cyclic system which gave 52 as the cardinality of the maximal stable subset. It follows from this computer result that no Steiner triple system containing a subsystem isomorphic to the cyclic system of order 5 can be linearly derived. Corollary 4.2 For a Steiner triple system on n points we have that with equality if and only its graph has chromatic number 1 or 2. Moreover, in the case of equality both linear extensions. Remark: It is at least theoretically possible for a Steiner triple system to have lots of quadrilaterals, but be non-Fano-free - that is, have no non- Fano planes. It seems unlikely that such systems exist since the non-Fano planes of a system produce many of its quadrilaterals in known examples. We do, however, have the following, perhaps empty, result, which, in any case, does give a sharper upper bound on the number of quadrilaterals in the non-Fano-free case than that given, in general, by the Stinson-Wei result. Suppose a Steiner triple system on n points contains no non-Fano planes. Then the number of quadrilaterals is at most72 with equality if and only if the chromatic number of its graph is 2. Moreover, in the case of equality, both L 1 and L 2 yield linear extensions. The proof is simple: Since L 3 is empty and jL Proposition 2.2 implies the inequality. We have equality if and only if Proposition 2.2 and Lemma 2.1. The graph is bipartite, the parts being L 1 and L 2 and either part yields a linear extension. One can phrase this characterization entirely in terms of quadri- laterals. Observe first that in any Steiner triple system each quadrilateral has three 4-subsets of its six points that do not contain a line of the quadrilateral. These subsets will be either doubly-linear or triply-linear. In the following result we call these subsets the 4-subsets given by the quadrilateral. A Steiner triple system without non-Fano planes has a graph with chromatic number 2 if and only if the collection of those 4-subsets given by the quadrilaterals of the system forms an extension. We now suppose given an arbitrary Steiner triple system on n points. Let F denote the number of Fano planes contained in the system and N the number of proper non-Fano planes, that is, non-Fano planes which do not complete to a Fano plane of the system. Then the number of triply-linear 4-subsets is 7F + N . Since every non-Fano plane which does not complete to a Fano plane gives rise to 6 doubly-linear 4-subsets, we have the following consequence of Lemma 2.1, Proposition 2.2 and Proposition 2.4: Theorem 4.3 Suppose given a Steiner triple system on n points containing F Fano planes and N proper non-Fano planes Then and the number of quadrilaterals of the system is at least 7F +3N . Moreover, equality implies that the Steiner triple system has a graph with chromatic number 1 or 2 and precisely 7F the geometric system of points and lines of a projective geometry over the binary field or else N 6= 0, jL and L 1 [L 3 providing linear extensions. Proof: Since jL 1 and hence the inequality. Proposition 2.2 underbounds the number of quadri- laterals. Equality forces jL and the rest of the Theorem Remarks: 1) Observe that each of the three systems on 15 points that have graphs with chromatic number two - #2, #3 and #16 - satisfy the bound of the Theorem, that is F system #1 does also - as do all the geometric binary systems. 2) Systems of of order 8 (on 19 points) cannot furnish further examples of equality since the right-hand-side of the inequality is not an integer, but, for systems of order 9 (systems on 21 points), the right-hand-side is an integer presumably further examples do exist in this case. It would be very interesting to have a suitably easy construction of Steiner triple systems on n points, where n j 0; with F in any Steiner triple system on n points, it is, at least in principle, possible that its graph might be highly disconnected consisting of the isolated vertices comprising L 3 , a bipartite subgraph with one part L 2 and the other part jL 2 j vertices from L 1 , with the other components of the graph, in the case in which of chromatic number 3. Should that happen, then, as in the Proposition 3.5 and the result above, we could extract at least six linear extensions consisting of one-third of the vertices from the 3-chromatic components, either one of the two parts of the bipartite subgraph and the isolated vertices. The cyclic system of order 5 shows that this is not always LINEAR STRUCTURE OF SYSTEMS OF ORDER SIX possible since, as we have seen, the chromatic number of its graph is at least 4. In fact, for such a highly disconnected graph to arise one must have singly-linear 4-subsets all of whose 3-subsets are of Type II since each vertex corresponding to elements of L 2 is of valency four and therefore those vertices from L 1 making up the other part of the bipartite subgraph must also have valency four. Moreover, it is impossible to have more than jL 2 j such singly- linear 4-subsets and, when there is such a set of singly-linear 4-subsets, all remaining singly-linear 4-subsets must have all their 3-subsets of Type III. Usually, this makes it easy to check that such a decomposition of the graph cannot occur. Neither of the two triple systems of order 5 have graphs with such a decomposition for just this reason. We have no examples of such a decomposition with jL it is possible that none exist. 5 Linear structure of systems of order six We record below results of computations made by Vladimir Tonchev and Robert Weishaar; for further information see [36]. The corank is simply is the 2-rank of the given system (the rank over the binary field of the incidence matrix of the system). The number of Fano planes contained in a system is simply c is the corank. Since the number of non-Fano planes is the same as the number of triply-linear 4- subsets, one has that the number of proper non-Fano planes is where T is the number of triply-linear 4-subsets and c is the corank. The number of quadrilaterals - which we have recorded - is simply Proposition 2.2, where D is the number of doubly-linear 4-subsets. LINEAR STRUCTURE OF SYSTEMS OF ORDER SIX 19 Number linear linear linear 6 QUASI-LINEARLY DERIVED SYSTEMS? 21 6 Quasi-linearly derived systems? It is widely believed that every Steiner triple system is derived. Since the problem can be reduced to those systems of full 2-rank (see [2, Corollary 6.2]) it follows from Corollary 6.3 of [2] that, if all Steiner triple systems are derived, then a suitable set of quadruples always can be found among the binary linear span of incidence vectors of the triples. In this work we have demanded that the quadruples be found among the binary sums of the incidence vectors of two triples. One could investigate what further systems can be shown to be derived by looking at the binary sums of the incidence vectors of either two or four triples (using only those yielding, of course, vectors of weight four). In fact, one could envisage a measure of how difficult it is to find a derivation - using six or fewer triples, eight or fewer triples, etc. We have not tried to do this. In the case of four triples we do have the study of Grannell, Griggs and Mendelsohn [13] of four-line configurations to aid us. Of the sixteen possible four-line configurations only three, C and C 15 give weight-four vectors appears to be the most interesting candidate, being a pre- mitre. The weight-four vector produced by such a pre-mitre will have all four of its 3-subsets triangles if and only if it does not complete to a mitre. Only those, of course, could be used in trying to find what might be called a quasi-linear extension of the triple system. Since we know, Corollary 3.4, that whenever each pre-mitre complete to a mitre we have a Hall triple system - which has a linear extension - in any other case there will be 4-subsets coming from pre-mitres that could be used to find a quasi-linear extension. We do not have an example of a Steiner triple system that has no linear extension but does have a quasi-linear extension, but one presumes that such systems exist. Although it is believed that all Steiner triple systems are derived, one still does not have a suitably easy construction of derived triple systems. It might conceivably be true that for all admissible n ? 13 there is a linearly derived Steiner triple system and, if one could find an easy construction, one would have an alternative to Hanani's recursive techniques for providing derived 10 The support of such a weight-four vector might, in fact, be linear. We have already noted that a triply-linear 4-subset is the support of the binary sum of the incidence vectors of the six lines of the non-Fano plane to which it corresponds. 7 AN HISTORICAL ASIDE 22 Steiner triple systems for all admissible orders. 7 An historical aside In [13, Page 52] Grannell, Griggs and Mendelsohn wonder about the fact that certain results on four-line configurations had been neglected in the literature of Steiner triple systems. The same question could be asked about the results we have here described since the ideas seem so natural, flowing easily from a either a geometric or coding-theoretic view of the subject, and the results so easily discovered. We would like to suggest a possible explanation different from that proposed by Grannell, Griggs and Mendlesohn: an insularity - and unreceptiveness to ideas coming from disparate disciplines - that seems to be particularly acute among some workers investigating Steiner triple systems. Thoralf Skolem complained [31, Page 274] - perhaps because of the announcement by Hanani at the 1958 International Conference of Mathematicians - about the inattention paid by workers in the field to his appendices to Netto's book. Indeed, his ideas on constructing triple systems 11 seem to have been ignored for over a half century - except by members of the Belgian school. When Skolem, probably best known for his work during the 20s and 30s on the foundations of set theory, complained in 1958, his ideas had been in print for well over a quarter of a century in the Second Edition, published in 1927, of the Lehrbuch der Combinatorik by Eugen Netto, [26, Note 16]. It was nearly a quarter of a century after his complaint that Lindner, [21, Page 184], observed that Skolem's ideas led to an "incredibly simple" proof of the existence of Steiner triple systems on 6n in 1987 that proof entered the English monograph literature: Steeet and Street, [34, Chapter 5]. 12 Ever since its very infancy - in the work of Pl-ucker on cubic curves in see de Vries [37] - Steiner systems have been intimately related to 11 A direct and easy proof of existence of Steiner triple systems of all admissible orders can be constructed from [27, 30, 31]. See, for example, [34, Chapter 5] for an efficient and easily understood direct construction. I am indebted to D. R. Stinson for pointing out these two sources to me. Neither of them took account of the work of O'Keefe; the Streets became aware of O'Keefe's work only after the publication of their book. geometry and its configurational aspects. 13 Yet, in [29, Pages 105-106] - the same paper that asserted the infeasibility of the use of computers in determining whether or not the eighty systems of order six were derived - the contributions made by finite geometry to the study of Steiner triple systems are summarily dismissed as "of minor consequence" for the question of deriv- ability. This despite not only the history of the subject but also the fact that the fundamental work of Teirlinck [35] - and its elaboration by Doyen, Hubaut and Vandensavel [11] - had already shown the importance the geometric systems had for Steiner systems of orders other than the geometric ones. Insularity may be more prevalent in combinatorial mathematics (see, for example, the historical essay by Crapo, [8]) but it clearly can also appear in the most developed fields with even world-class mathematicians being unreceptive to new ideas (see Lang, [19, 20]). --R On the binary codes of Steiner triple systems. Graph Theory with Applications. Dimensional linear spaces. Hans Ludwig de Vries. Ranks of incidence matrices of Steiner triple systems. Derived Steiner triple systems of order 15. A small basis for four- line configurations in Steiner triple systems On an infinite class of Steiner systems constructed from affine spaces. The sandwich theorem. Mordell's review Mordell's review A survey of embedding theorems for Steiner systems. Transitive Erweiterungen endlicher Permutationsgrup- pen Small Steiner triple systems and their properties. Small Steiner triple systems and their properties - Errata On the Steiner systems S(3 Lehrbuch der Combinatorik. Verification of a conjecture of Ranks and structure of graphs. A survey of derived triple systems. On certain distributions of integers in pairs with given differences. Some remarks on the triple systems of Steiner. Combinatorische Aufgabe. Some results on quadrilaterals in Steiner triple systems. Combinatorics of Experimental Design. On projective and affine hyperplanes. Steiner triple systems of order 15 and their codes. Hans Ludwig de Vries. Complete classification of the triad systems on fifteen elements. --TR --CTR J. D. Key , H. F. Mattson, Jr., Edward F. Assmus, Jr. (19311998), Designs, Codes and Cryptography, v.17 n.1-3, p.7-11, Sept. 1999

graphs;chromatic number;derived triple systems;steiner triple systems

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Achieving bounded fairness for multicast and TCP traffic in the Internet.

There is an urgent need for effective multicast congestion control algorithms which enable reasonably fair share of network resources between multicast and unicast TCP traffic under the current Internet infrastructure. In this paper, we propose a quantitative definition of a type of bounded fairness between multicast and unicast best-effort traffic, termed "essentially fair". We also propose a window-based Random Listening Algorithm (RLA) for multicast congestion control. The algorithm is proven to be essentially fair to TCP connections under a restricted topology with equal round-trip times and with phase effects eliminated. The algorithm is also fair to multiple multicast sessions. This paper provides the theoretical proofs and some simulation results to demonstrate that the RLA achieves good performance under various network topologies. These include the performance of a generalization of the RLA algorithm for topologies with different round-trip times.

Introduction Given the ubiquitous presence of TCP traffic in the Internet, one of the major barriers for the wide-range deployment of reliable multicast is the lack of an effective congestion control mechanism which enables multicast traffic to share network resources reasonably fairly with TCP. Because it is crucial for the success of providing multicast services over the Internet, this problem has drawn great attention in the reliable multicast and Internet community. It was a central topic in the recent Reliable Multicast meetings [8], and many proposals have emerged recently [7, 15, 13, 1, 18, 3, 19]. In this introductory section, we first give an overview of the previous work and then we discuss our experience with the problem and introduce our approach. The basic problem can be described as the following. Consider the transport layer of a multicast connection with This material is based upon work supported by the U.S. Army Research Office under grant number DAAH04-95-1-0188. one sender and multiple receivers over the Internet. The sender has to control its transmission rate based on the loss information obtained from all the receivers. We assume that there is TCP background traffic; the end-to-end loss information is the only mechanism to indicate congestion; the participating receivers have time-varying capacities; and different receivers can be losing different information at different times. The objective of the control algorithm is to avoid congestion and to be able to share network resources reasonably fairly with the competing TCP connections. The previously proposed multicast flow/congestion control algorithms for the Internet can be broadly classified into two categories: 1) Use multiple multicast groups. 2) A single group with rate-based feedback control algorithms. The first category includes those proposals using forward error correction or layered coding [18, 19]. They require setting up multiple multicast groups and require co-ordination between receivers, which are not always possible. Some limitations of this type of control are identified in [13]. Many of the proposed rate-based schemes share a common framework: The sender updates its rate from time to time (normally at a relatively large interval on the order of a second) based on the loss information obtained. It reduces its rate multiplicatively (usually by half, the same as TCP does) if the loss information indicates congestion, otherwise it increases its rate linearly. Different proposals differ in their ways of determining the length of the update interval and acquisition of loss information; they have different criteria to determine congestion, etc. It is largely agreed that, with no congestion, the rate should be increased linearly with approximately one packet per round-trip time, which is the same as TCP does. A critical aspect of these rate-based algorithms is when to reduce the rate to half, or how congestion is determined from the loss information from all the receivers. Most of the proposed algorithms are designed with the objective of identifying the bottleneck branches 1 of the multicast session and reacting only to the losses on the bottleneck links [13, 7]. Some also try to be fair to TCP [1, 15]. The algorithms have to be adaptive as well, i.e., be able to migrate to new bottlenecks once they come up and persist for a long time. In the following, we discuss two examples in detail. The loss-tolerant rate controller (LTRC) proposal is based on checking an average loss rate against a threshold [13]. The algorithm tries to react only to the most congested paths and ignore other loss information. The algorithm 1 By bottleneck branches, we refer to the branches where the band-width share of multicast traffic is the smallest among all multicast paths, assuming equal share of all connections going through the path. identifies congestion and reduces the sender rate if the reported loss rate (an exponentially-weighted moving average) from some receiver is larger than a certain threshold. The rate is not reduced further within a certain period of time after the last reduction. It is not clear how to choose the loss threshold values for an arbitrary topology with any number of receivers to drive the system to the desired operating region. The monitor-based flow control (MBFC) is a double- threshold-check scheme [15]. That is, a receiver is considered congested if its average loss rate during a monitor period is larger than a certain threshold (loss-rate threshold), and the sender recognizes congestion only if the fraction of the receiver population considered congested is larger than a certain threshold (loss-population threshold). Using the loss- population threshold to determine whether to reduce the rate or not is a means to average the QoS over all receivers, and is not aimed to work with the slowest receiver. As a special case, with the loss-population threshold set to minimum (one congested receiver is counted as congestion), the MBFC reduces to the case of tracing the slowest receiver, but, again, it is difficult to derive a meaningful threshold value to be able to single out the most congested receiver. If the threshold value is too small, there could be excessive congestion signals because different receivers could experience congestion at different times. There are many other proposals which we cannot cover in this introduction. Although many of the proposals are claimed to be TCP-friendly based on the simulation results for certain network topologies, none have provided a quantitative description of fairness of their algorithms to TCP and a proof of their algorithms' ability to guarantee fairness. We have carried out extensive simulations to study the interaction of TCP traffic with other forms of rate-controlled traffic in both unicast and multicast settings, with both drop-tail and RED (random early drop) gateways 2 . We summarize our major observations here and the details are discussed in the rest of this paper. First of all, there is no consensus on the fairness issue between reliable multicast and unicast traffic, let alone a useful quantitative definition. Should a multicast session be treated as a single session which deserves no more band-width than a single TCP session when they share network should the multicast session be given more bandwidth than TCP connections because it is intended to serve more receivers? If the latter argument is creditable, how much more bandwidth should be given to the multicast session and how do we define "fairness" in this case? This paper addresses this problem and proposes an algorithm which allows a multicast session to obtain a larger share of resources when only a few of the multicast receivers are much more congested than others. However, we believe that a consensus on the definition of relative fairness between multicast and unicast traffic is achievable once an algorithm shown to be "reasonably fair" to TCP is accepted by the Internet community. The toughest barrier to designing a fair multicast congestion control algorithm is that most of the current Internet routers are still of drop-tail type. A drop-tail router uses a first in first out buffer to store arriving packets when the outgoing link is busy. The FIFO buffer has a finite size and the arriving packet is dropped if the buffer is already full. Since drop-tail routers do not distinguish packets from different traffic flows, they do not enforce any fairness for the connections Routers and gateways are used interchangeably in this paper. See [5] for definitions for drop-tail and RED gateways. sharing resources through them. Also with drop-tail routers, the packet loss pattern is very sensitive to the way packets arrive at the router and is difficult to control in general. Since TCP packets tend to arrive at the router in clusters [21], any rate-based algorithm with an evenly-spaced packet arrival pattern may experience a very different loss rate from that of the competing TCP connections through a drop-tail gateway. Therefore, rate-based algorithms adjusting source transmission rate based on average loss rate cannot be fair to TCP in general. But, it has been pointed out that rate-based schemes are better suited for multicast flow/congestion control than window-based schemes [15]. This is true in general in terms of scalability and ease of design. However, if our design objective is to be fair to window-based TCP, rate-based schemes have difficulty, if not an impossibility, in achieving the goal without help from the networks. For the algorithms assuming the same loss rate for the competing connections, RED gateways can be used to achieve the goal. The RED gateway is proposed as an active router management scheme which enables the routers to protect themselves from congestion collapse [5]. A RED gateway detects incipient congestion by computing the average queue size at the gateway buffer. If the average queue size exceeds a preset minimum threshold but below a maximum thresh- old, the gateway drops each incoming packet with a certain if the maximum threshold is exceeded, all arriving packets are dropped. RED gateways are designed so that, during congestion, the probability that the gateway notifies a particular connection to reduce its window (or rate) is roughly proportional to that connection's share of the bandwidth through the gateway. Therefore, RED gateways not only keep the average queue length low but ensure fairness and avoid synchronization effects [6]. For our work, the most important fact about the RED gateway is that all connections going through it see the same loss probability. RED gateways also make fair allocation of network resources for connections using different forms of congestion control possible. Adoption of RED gateways will greatly ease the multicast congestion control problem, but the current Internet still uses mostly drop-tail gateways. Therefore, it is important to design an algorithm which works for drop-tail gateways and might work better for RED gateways. However, even with RED gateways, it is still very difficult to locate the bottlenecks of a multicast session based on loss information alone (refer to section 3.2 for details). Many proposals for reliable multicast flow control do try to locate the bottleneck links using some threshold-based mechanism, such as the LTRC (loss-tolerant rate controller) discussed above, but it is very difficult to choose a universal threshold which works for all kinds of network topolo- gies. [16] has shown that a loss-threshold-based additive- increase-multiplicative-decrease multicast control algorithm is not fair to TCP with RED gateways. Based on the above observations, we decided to choose a window-based approach to design a usable mechanism to do multicast congestion control in the current Internet in- frastructure. Specifically, we propose a random listening algorithm which does not require locating the bottleneck link. The algorithm is simple and possesses great similarity to TCP; it ensures some reasonable fairness, defined later as "essential fairness", to TCP with RED gateways or drop-tail gateways in a restricted topology to be defined in the next section. Although the scheme inherits many of the identified drawbacks of TCP (some of them are alleviated in our multicast scheme), it might be the only way that the multicast sessions can potentially share bandwidth reasonably fairly with TCP connections with drop-tail routers. The rest of the paper is organized as follows: We propose a quantitative definition for fairness between multicast and unicast traffic in section 2. Our algorithm is presented in section 3, and we prove that it is essentially fair to TCP in section 4. In section 5 we present some simulation results indicating the performance of our algorithm sharing resources with TCP. We also briefly discuss a generalization of the algorithm which works for topologies with different round-trip times and its performance. Section 6 concludes the paper by addressing some possible future work. Our design of the multicast congestion control algorithm is motivated by the design of the TCP congestion control scheme [9, 12]. We summarize the basic properties of the TCP scheme in the following: Probing extra bandwidth: increase the congestion window by one packet per round-trip time until a loss is seen. ffl Responsive to congestion: reduce the congestion window to half upon a detection of congestion (i.e., a packet loss). ffl Fair: by using the same protocol, the TCP connections between the same source and destination pair (along the same route) share the bottleneck bandwidth equally in the steady state. 3 Similarly we list our design objectives for the multicast congestion control algorithm to be: ffl Able to probe and grab extra bandwidth. ffl Responsive to congestion. Multicast Fairness: multiple multicast sessions between the same sender and receiver groups should share the bandwidth equally on average over the long run. ffl Fair to TCP: the multicast traffic has to be able to share the bandwidth reasonably fairly with TCP in order to be accepted by the Internet community. This is a complex issue to be addressed in the rest of this section. Note that our performance goals, including definitions of fairness, are focused on the average behavior in the steady state, assuming connections last for a long time. Our work in this paper is based on this assumption. We do not try to guarantee fairness to short-lived connections, but our algorithm does provide opportunities for them to be set up and to transmit data. This is a reasonable decision because the multicast session is presumably cumbersome with many links involved and thus it is impossible to react optimally to every disturbance, especially short-lived ones. 2.1 Observations We observe that TCP fairness is defined and achieved only for the connections between the same sender and receiver, that is, the paths have to have the same round-trip times Generally speaking, the TCP connections share bandwidth equally as long as they have equal round-trip times and the same number of congested gateways on their path. But a slight difference in the round-trip times could result in very different outcome in bandwidth share due to the phase effect discussed in [5]; therefore, we restrict the fairness definition to the connections between the same source and destination pair along the same route which is the best way of ensuring equal round-trip times. and the same number of congested gateways. It is well-recognized that the unfairness of the TCP congestion algo- rithm, such as biases against connections with multiple congested gateways, or against connections with longer round-trip times and against bursty traffic, is exhibited in networks with drop-tail gateways [5]. This observation leads us to define the relative fairness between multicast and TCP traffic on a restricted topology only, where the sender has the same round-trip time to all the receivers in the multicast group. As we mentioned in the introductory section, there is no consensus on the issue of fairness between multicast and unicast traffic. But the following is obvious: An ideal situation is to be able to design a multicast algorithm which can control and adjust the bandwidth share of the multi-cast connection to be equal to some constant c times that of the competing TCP connection, with c being controllable by tuning some parameters of the algorithm. On the other extreme, the minimum requirements of reasonable fairness should include the following: 1) Do not shut out TCP com- pletely. 2) The throughput of the multicast session does not diminish to zero as the number of receivers increases. Anything in between the ideal and the minimum could be reasonable provided that the cost to achieve it is justifiable. Loosely speaking, a useful definition for "essentially fair" could be the following: when sharing a link with TCP, the multicast session should get neither too much nor too little bandwidth; that is, some kind of bounded fairness. We quantify this definition next. 2.2 Concepts First we introduce a restricted topology, referred to through-out this paper, on which the fairness concepts are defined. We also introduce the notation used throughout the paper. Consider a multicast session fS one sender S and N receivers . The sender also has connections to each R i along the same path (a branch of the multicast tree) [see figure 1], where could be zero (no competing TCP connection). Imagine a virtual link (or a logical connection), L i , between S and R i . Note that the virtual links might share common physical paths. We assume that the round-trip times, RTT i , of are equal on the average. 4 Denote the minimum link capacity (or available bandwidth) along L i by - i (pkt/sec). We define the "soft bottleneck " of a multicast session, denoted by Lsb , as the branch with the smallest - i That is, g. We say that the multicast is "absolutely fair " to TCP if the multicast session operates with an average throughput equal to min i f- i the steady state. In other words, "absolute fairness" requires that the multicast session be treated as a single session and equally share the bottleneck bandwidth with competing TCP connections on its soft bottleneck paths. As we mentioned before, absolute fairness is difficult to achieve based on loss information alone. By relaxing the definition somewhat, we introduce an important concept called "essential fairness ". We say that a multicast session is "es- sentially fair " to TCP if its average throughput, denoted by 4 Notice that the round-trip time includes both queueing delay and propagation delay. Therefore, it is time varying. In our analysis in this paper, we assume a nice property of round-trip time: it is uniformly distributed between pure propagation delay and propagation delay plus maximum queueing delay. It is based on the single bottle-neck queue model. 5 In contrast, a "hard bottleneck " would be the link with minimum . Also, there could be multiple soft bottlenecks with equal Networks R R R R between S and R L has bottleneck link capacity - , connections. Dashed line : virtual link L L 's might share common physical path in the network. multicast connection, and m TCP connections. Figure 1: A restricted topology. - RLA , in the long run is bounded by a - b - TCP , where - TCP is the average throughput of the competing TCP connections on the soft bottleneck path, and a; b are functions of N such that a - b ! N . Absolute fairness is a special case of essential fairness with g. b=a can serve as an indication of the tightness of the fairness measure. The flexibility of allowing an interval of fairness is necessary because absolute fairness might not be achievable in some networks whereas a fairness measure is needed. It is a reasonable representation of the vague term "reasonably fair", and would appear to be acceptable by many applications. The merit of the essential fairness concept lies in its boundedness, so the networks and applications can have some idea of what they can expect. Our definition can be used to measure and compare the fairness of existing multicast algorithms. We will prove later that the random listening algorithm we propose in this paper is essentially fair to TCP and it achieves more tightly bounded fairness with RED gateways than with drop-tail gateways. In summary, we have defined three key concepts for multicast sharing with unicast traffic on a restricted topology: soft bottleneck, absolute fairness and essential fairness. The definitions can be easily extended to the case with multiple multicast sessions between the same sender and receiver group. In the next section, we present a random listening algorithm which achieves the design objectives described in the beginning of this section. 3 Random Listening Algorithm (RLA) In this paper, we focus on the congestion control problem. We assume that the sender has infinite data to send and the receivers are infinitely fast, so that the network is always the bottleneck. Hereafter, we refer to a congested receiver, meaning the path between the sender and the receiver is experiences packet drops. We also define a congestion signal as an indication of congestion according to the algorithm; congestion probability as the ratio of the number of congestion signals the sender detected to the number of packets the sender sent; congestion frequency as the average number of congestion signals the sender detected per time unit. TCP considers packet losses as indications for congestion. In particular, one or multiple packet drops from one window of packets in TCP are considered as one congestion signal since they usually cause one window cut (or cause retransmission timeout) [4]. The number of window cuts is equal to the number of congestion signals in TCP in the ideal case without timeout event. Our simulation experience convinced us that, with drop-tail gateways, algorithms might have to be "TCP-like" in order to be TCP-friendly. By TCP-like, we refer to the essential feature of the congestion window adjustment policy: increasing by one every round-trip time with no congestion and reducing by half upon congestion. But TCP-like alone is not enough to ensure TCP-friendliness. More importantly, to be fair to TCP, we have to make sure that the multi-cast sender and the competing TCP senders are consistent in their way of measuring congestion. RED gateways ensure that the competing connections sharing the same bottleneck link experience the same loss probability no matter what type of congestion control algorithms are used. However, the situation with drop-tail gateways is more complicated. We will show that, with some added random processing time to eliminate phase effects, we can design our algorithm to ensure that the competing connections see the same congestion frequency. This problem is examined in detail next. 3.1 TCP's Macro-effect with Drop-tail Gateways With drop-tail gateways, a basic feature of TCP traffic is that the sender increases its transmission rate to fill up the bottleneck buffer until it sees a packet loss. Then the sender's transmission rate is sharply reduced to allow the buffer to be drained. The TCP congestion control policy results in a typical behavior of the buffer in front of the bottleneck router: the buffer occupancy periodically oscillates between empty (or almost empty) and full. Although this periodicity is neither necessarily of equal interval nor deter- ministic, depending upon the behavior of the cross traffic other than TCP, it is certainly the macroscopic behavior of the network routers carrying TCP traffic. We call the period starting from a low occupancy to a full buffer and then dropping back to a low occupancy, a "buffer period ". Through simulation, we find that the buffer period normally lasts much longer than two round-trip times, 2RTT , and the buffer-full period 6 , during which the buffer is full or nearly full, normally lasts around 2RTT or less in the steady state. During the buffer-full period within each buffer pe- riod, a sender could lose more than one packet. In our algorithm to be presented, we group the losses within 2RTT as one congestion signal. This way we approximately make sure of one congestion signal per buffer period if any packet is dropped. The reason for doing so is that it is not desirable to reduce a window multiple times due to closely spaced packet drops. TCP actually considers multiple packet drops within one window as one congestion signal. Another phenomenon is that, with drop-tail gateways, the packet drop pattern is very sensitive to the packet arrival pattern. In particular, we find that the packet drop pattern is very sensitive to the interval between two consecutive packet arrivals at the bottleneck buffer. If the interval is slightly smaller than the service time of the bottleneck server, the next packet is more likely to be dropped when the buffer is nearly full. Otherwise, if it is slightly larger than the bottleneck server service time, the next packet is less likely to be dropped because one packet will leave the buffer in between. This is one type of phase effect identified in [5]. Phase effects do not take place in competing TCP connections when the round-trip times are exactly the same. But in a multicast session which consists of multiple links with different instantaneous round-trip times, adding a random 6 This time interval roughly corresponds to the "drop period" defined in [5]. amount of processing time is necessary to avoid the phase effect. Therefore, a uniformly distributed random processing time up to the bottleneck server service time is added in our simulation with drop-tail gateways. The phase effect might not be significant in the real Internet because of mixing of different packet sizes, in which case our algorithm is expected to work well, sharing bandwidth reasonably fairly with TCP traffic. With drop-tail gateways and added randomness to eliminate the phase effect, our TCP-like multicast algorithm is designed to make sure that the multicast sender sends packets in a fashion similar to the TCP senders. Then all senders have a similar chance to encounter packet drops in a restricted topology with equal round-trip times, provided that the congestion window sizes are large enough. That is, both the multicast and TCP senders see roughly the same number of congestion signals over a long period of time, or the congestion frequencies should be the same on the average over a large number of simulations. For the connections with smaller window sizes, they might experience fewer packet drops, which results in a desirable situation for our problem of bandwidth allocation between multicast and unicast traffic. The reason is explained in section 5. 3.2 Rationale for Random Listening If the objective is to achieve absolute fairness between multi-cast and TCP traffic, we have to locate the soft bottlenecks of the multicast session and react only to the congestion signals from the soft bottleneck paths. However, it is diffi- cult, if not impossible, to locate the soft bottlenecks based on the loss information alone. For the TCP connections to achieve the same average throughput, the larger the round-trip time is, the larger the window and hence smaller loss probability required. Therefore, for a multicast session with different round-trip times between the sender and receivers, it is not reasonable to expect the bottleneck would be the branch with the largest loss probability. Although, for the restricted topology with equal round-trip times, the soft bottlenecks are the branches with the largest loss probability, it is still difficult to locate them based on loss information alone. This is because either the sender or the receiver has to calculate a moving average of the loss probability for each receiver and the sender has to react to only the loss reports from the bottlenecks which have the largest loss probability. But, since losses are rare and stochastic events, a certain interval of time and enough samples are needed to make the loss probability estimate significant. It would take too long to locate the soft bottlenecks correctly; the wrong action based on the non-bottleneck branches could cause undesirable performance results. Based on these observations, we decided to trade off the absolute fairness (requiring locating the soft bottlenecks) with fast response. Now examine figure 1. The multicast sender is receiving congestion signals from all congested receivers. Obviously the sender does not want to reduce its window upon each congestion signal. Otherwise, as the number of receivers increases, the number of congestion signals will increase, and the throughput of the multicast session will decrease as the number of receivers increases. Suppose that the sender knows how many receivers, say n, are reporting congestion. An appealing solution would be to reduce the window every n congestion signals. To see why, consider a simple flat tree topology as in figure 1 with all the receivers, links and background TCP connections identical and independent, and all connections starting at the same time. Then the buffer periods are synchronized and the sender receives n congestion signals in each buffer period. Obviously it is desired that the sender only reduce its window once every buffer period. This deterministic approach is certainly a possible solution here. But in a more realistic network with not everything identical, where buffer periods are asynchronous and congestion signals come at different times, the sequence of congestion signals arriving at the sender could be very irregular, and thus the deterministic approach would not work well. On the other hand, in such a statistical environment, a random approach could be a good candidate to produce good average performance. This is the rationale we use to propose a random listening scheme to handle a complex stochastic stream of congestion feedback signals. The basic idea is that, upon receiving a congestion signal, the sender reduces its window with probability 1=n, where n is the number of receivers reporting frequent losses. There- fore, on the average, the sender reduces its window every n congestion signals. If all the receivers experience the same average congestion, the sender reacts as if listening to one representative of them. If the sender detects one receiver experiencing the worst congestion (on the soft bottleneck) and the others in better condition with less frequent congestion signals, it reduces the window less frequently than the TCPs on the soft bottleneck branch, resulting in a larger average window size of the multicast sender than that of the TCP connections on this branch. But we can prove that the multicast bandwidth share is bounded in terms of the bandwidth share. Based on this idea, we propose a random listening algorithm to be presented next, with its performance to be discussed in the rest of this paper. 3.3 The Algorithm The design closely follows the TCP selective acknowledgment procedure (SACK) [12]. We focus on the congestion control part of the algorithm. Here we only outline the essential part of the algorithm. The complete algorithm is implemented using Network Simulator (NS2) [17], and more information is available at [20]. The important variables are summarized below. Their meaning and maintenance are the same as in TCP unless specified differently here. The items preceded by a bullet are new to our algorithm. smoothed round-trip times between the sender and receiver i. moving average of the window size. ffl num trouble rcvr : a dynamic count of the number of receivers which are reporting losses frequently. dynamically adjusted threshold to determine the probability of reducing the window upon a congestion signal. For a restricted topology, 1=num trouble rcvr. number uniformly distributed in (0, 1), generated when a decision as to whether to reduce the window or not is needed. ffl last window cut : the time when the cwnd was reduced to half last time. ffl cperiod start i : the starting time of a congestion period (i.e., the period in which packets are dropped) at receiver i. This is used to group the losses within two round-trip times into one congestion signal. ffl min last ack : the minimum value of the cumulative ACK sequence number from all receivers. All packets up to this sequence number are received by all receivers reach all : the maximum packet number which is correctly received by all receivers. It could be different from min last ack because selective acknowledgment is used. ffl rexmit thresh : if the number of receivers requesting a retransmission of a lost packet is larger than rexmit thresh, the retransmission is multicasted. Otherwise the retransmission is unicasted. The skeleton of the RLA is the following: 1. Loss detection method. Our multicast receivers use selective acknowledgments using the same format as receivers [12]. A loss is detected by the sender via identifying discontinuous ack sequence numbers or timeout. To accommodate out-of-order delivery of data, the sender considers a packet P is lost if a packet with a sequence number at least three higher than P is selectively ACKed. 2. Congestion detection method. A congestion period starts when a loss is detected and the cperiod start i is beyond is reset to the current time. The losses within 2\Lambdasrtt i of cperiod start i are ignored. 3. Window adjustment upon a congestion detection. Upon a congestion detected from receiver i by the above method: ffl update num trouble rcvr. If it is a rare loss from a receiver not considered as a troubled receiver (see rule 6 below), skip the following steps. ffl if last window cut is beyond 2 awnd srtt i , 7 cwnd / cwnd=2. forced-cut. ffl else, generate a uniform random number -, it. else cwnd / cwnd=2. randomized-cut. 4. Window adjustment upon ACKs. Once a packet is ACKed by all the receivers, cwnd / cwnd+ 1=cwnd. 5. The window lower bound moves when max reach all increases, but the window upper bound should never exceed min last ack plus available receiver buffer size. 6. Update of num trouble rcvr: a congested receiver is considered as a troubled receiver only if the receiver's congestion probability is larger than a certain thresh- old, which is set to 1=(j min congestion interval). here is a constant, and is recommended to be set to 20. min congestion interval is the smallest of the exponentially-weighted moving average of the interval lengths between congestion signals from all receivers. This setting is justified in the proof in the next section. trouble rcvr is the dynamic count of the number of troubled receivers. A detailed implementation instruction is available at [20]. 7 In ideal TCP with deterministic losses, cwnd has a maximum size of W and a minimum of W=2. cwnd is halved every W=2 round trips, or RTT W=2 seconds. Here for the multicast connection using random listening approach, to avoid ignoring too many consecutive congestion signals due to the randomness of the algorithm, we choose to force the reduction of the congestion window if the previous window cut happens at least 2 awnd round trips ago. The threshold value is ad hoc but works well from our simulation experience: Basically we don't want cwnd to grow too large or the forced-cut to happen too often. Note that there are two different treatments to a congestion signal: forced-cut and randomized-cut. The forced actions are intended to protect the system by damping the randomness. Without the forced-cut step, the algorithm can possibly result in too long of a continuous increment of cwnd, which is not desirable. We also implemented a retransmission scheme 8 to recover loss packets and a fast-recovery mechanism to prevent a suddenly widely-open window which is undesirable because it can cause congestion and a burst of packet losses. Many details in the implementation are not described here and can be found in [20]. Many of them are just straight-forward extensions from the TCP algorithm. We believe it is beneficial to keep it as similar to TCP as possible. Then any changes to TCP or in networks to improve TCP performance can be easily incorporated and are likely to improve the performance of our algorithm as well. 4 Fairness of the RLA In this section, we prove that our RLA is essentially fair to TCP. That is, with the restricted topology where a multicast session is sharing resources with unicast TCP connections, the multicast session gets a bandwidth share which is c times the share of a competing TCP connection on the soft bottleneck branch, where c is a bounded constant. We present a simple proof based on some gross simplifications of the system and the algorithm. Although a sophisticated proof based on advanced stochastic processes, similar to the proof for the TCP case [14], is possible, we choose a simple approach which is easier to understand and better illustrates our idea. We also prove the multicast fairness property of the RLA, one of the design objectives mentioned in section 2, using a simple two-session model. We first present a simple estimation of TCP's performance adopted from [14], then we use the same idea and result to prove our theorems. The key part of the proofs is to show that the RLA results in an average window size bounded from above and below by functions of the congestion probability (the ratio of the number of congestion signals to the number of packets sent, see section 3) on the soft bottleneck branch. Since on each common link, the RLA sender and the competing TCP senders see the same loss probability with RED gateways [6], or the same congestion frequency with drop-tail gateways with phase effects eliminated (see section 3.1), a relation between congestion probabilities of the two types of traffic can be derived, based on which the bandwidth shares can be calculated. 4.1 Estimation of TCP Throughput We consider TCP SACK here and use the approximation technique introduced in [14]. Although there are many subtleties in the implementation of fast retransmission and fast recovery, etc., we list the most important parts of the algorithm relevant to congestion control here. The sender maintains cwnd and ssthresh, with the same meaning as defined in section 3. The sender also estimates the round-trip time and calculates the timeout timer based 8 There could be different ways of doing retransmission as long as the retransmission traffic does not interfere too much with the normal transmission. In our implementation, the sender waits until it hears from all the receivers and it retransmits a lost packet by multicast if the number of receivers requesting it is larger than a threshold (rexmit thresh) and by unicast otherwise. The receiver can also trigger an immediate retransmission of a lost packet by unicast if it sets a field in the packet. on the estimation. The TCP congestion window cwnd evolves in the following way: (1) Upon receiving a new ACK: cwnd else cwnd / cwnd congestion avoidance phase. (2) Upon a loss detection: set ssthresh / cwnd=2, and cwnd / cwnd=2: (3) Upon timeout, set ssthresh / cwnd=2, and cwnd / 1, denotes the integer part of x. We consider cwnd as a random process and are interested in its average value in the steady state since it is roughly proportional to the average throughput of the TCP connection. Assume perfect detection of packet losses, and that the slow-start and the timeout events can be ignored in the steady state analysis [10]. Suppose we run the algorithm for a long time and the resulting congestion probability (the number of window cuts over the number of packets sent) is p. The resulting average window size can be approximated in the following way [14]. Denote the cwnd right after receiving the acknowledgment with sequence number t by W t . Then in the steady state, the random process W t evolves as fol- lows: given W t , with probability and with probability p, W between jumps upon acknowledgment arrivals. Now considering the average drift of W t if W denoted by D(w), we have . The drift is positive if w ! w and negative otherwise. Then the stationary distribution of must have most of its probability in the region around w . This gives an ad hoc approximation of the average window The unit is in terms of packet. Throughout this paper, we call this approximation "proportional average (PA) window size". It can be shown that W is a good approximation to the time average of the random process W t and in fact is proportional to it. We adopt this simple approximation approach in our analysis since it is adequate for our purpose. Also note that the above simple derivation gives a result similar to the popular formula for TCP throughput estima- (packets), as in [11], with a slightly different constant. Comparison of the two formulas shows that the average throughput is roughly proportional to the ratio of the average window size to the average round-trip time. Both formulas only work for the cases with small loss probability. Therefore, in the rest of our paper, we only consider the cases with (used in [11]), called moderate congestion. The performance of TCP (and TCP-like algorithms) deteriorates in heavy congestion because of frequent timeout events. Maintaining fairness is then not as important an issue as long as no one is completely shut out. 4.2 Estimation of RLA Throughput Applying the above drift analysis technique to our RLA algorithm proposed in section 3, we can derive the following proposition. Proposition: Consider a restricted topology (with TCP background traffic and the RLA used by the multicast sender) and n receivers persistently reporting congestion. The congestion probabilities seen by the multicast sender from the n receivers are p i , congestion is moderate so that Denote the proportional average of the congestion window size by W . Then W satisfies the following: Due to space limitations, we cannot show the complete proof which is available in reference [20]. The basic idea and methodology are illustrated using the following simple case of two receivers with independent loss path (see figure 2(a)). (a) independent losses G (b) common losses Figure 2: Two simple cases with two receivers only. In figure 2(a), the sender sees independent congestion signals from receivers 1 and 2, denoted by R1 and R2, respec- tively. We assume that all traffic is persistent and thus, in the steady state, Therefore, if the sender detects a congestion signal at time t0 , it cuts its window by half with probability 1; if the outcome turns out to be ignoring the congestion signal, the lost packet will be ACKed later (at time t1 ) and will cause the congestion window to increase by 1 W , based on the fourth rule of the RLA (see section 3.3). In our proof, we ignore the possible difference between the congestion window sizes at time t0 and t1 since 1 is small for a relatively large window size. Now for each packet sent by the sender, the possible outcomes are (w.p. stands for with probability): 1. No congestion signal from both receivers, w.p. then W W . 2. Cause one congestion signal from R1, w.p. then W 2 , or W / Ww.p. 13. Cause one congestion signal from R2, w.p. then W W w.p. 1, or Cause two congestion signals from both receivers w.p. then W W w.p. 1, or 2 , or 4 . The positive drift of W isW and the negative drift of W is The neutral point gives the approximation for the average window size to be It is easy to check out by simple algebraic manipulation that, for any pmax ? 0, W ? holds. To prove the upper bound in equation 2 with loss of generality assume the following holds: that f(p1) is an increasing function of p1 for 1. Therefore, for p1 ! 5%, x larger than 0.03 is sufficient for x - f(p1) to hold. This condition is ensured in the RLA algorithm by controlling the way the variable "num trouble rcvr" is dynamically counted. The RLA algorithm counts a receiver as a "trou- bled receiver" only if the interval lengths between the congestion signals are smaller than j min congestion interval, that is, its average congestion probability is larger thanj pmax . We recommend in our algorithm to take or 1 which leaves more room than the above 0.03 bound. Protocol designers can choose a proper value for j based on the above analysis. Note that we did not consider the forced-cut action in the RLA algorithm, which is rarely invoked (as shown in the simulation results). The effect of the forced-cut could be a slightly smaller average window size which does not affect our results in any significant way. With more complex algebraic manipulation involved, the results can be extended to a case with n receivers with independent loss paths. Using the same approach, for a topology with common losses only (see figure 2(b) for an illustration of the case with two receivers), we can prove equation 2 holds. The general case stated in the Proposition can be proved using the above results and the following Lemma: Lemma: A higher degree of correlation in loss due to common path results in a larger average congestion window size if the RLA is used. The proof is omitted due to space limitations. Intuitively, for the same congestion probability, correlation in the congestion signals results in more window increments and less window cuts on the average. This is because congestion signals come in groups in the correlated case. This has the potential of causing a deep cut in cwnd at once, while the independent congestion signals come one at a time but more frequently, and cause potentially more window cuts. We deliberately choose the bounds in the form of equation because is related to the average window size of the competing TCP connections. Now we are ready to proceed to show that the RLA is essentially fair to TCP. Theorem I: Consider a restricted topology with RED gate- ways. If there are n receivers persistently reporting congestion and the largest congestion probability is less than 5%, the RLA algorithm is essentially fair to TCP with 3n. 9 Due to space limitations, here we only outline the major steps of the proof. First, since RED gateways ensure that on each link, all competing connections see the same loss 9 See section 2 for the definition of a and b. probability [6], we denote the largest loss probability, occurring at the soft bottleneck branches, by p l . We can derive the relations between p l and the corresponding congestion probabilities, p RLA c for the multicast connection, and p TCP c for the TCP connection. Then, using these relations and the Proposition, we can derive the following inequality:3 Second, we have to consider the round-trip times in order to estimate the throughput. Here we have to notice that in the multicast RLA, a packet is considered acknowledged only if the sender has received ACKs from all the receivers. Then the round-trip time for each packet in the RLA is always the largest among the round-trip times on all the links when the packet is sent. Denote the average round-trip time for RLA by RTT RLA and that for TCP by RTT (recall each of the branches in the restricted topology (figure 1) has equal average round-trip times). Using the approximation that the round-trip time is equal to a fixed propagation delay plus a varying queueing delay, we can derive the following: Finally, combining the bounds for average window size and average round-trip times, we have the following: RTT RLA RTT That is, the long term average throughput of the multi-cast sender is no less than a third of the TCP throughput on the soft bottleneck branch, and no more than 3n times that. Therefore, the RLA is essentially fair to TCP according to the definition of essential fairness in section 2. Theorem II: Consider a restricted topology with drop-tail gateways and the phase effect eliminated. If there are n receivers persistently reporting congestion and the largest congestion probability is less than 5%, the RLA is essentially fair to TCP with This theorem can be proved similarly to theorem I, using the fact that, with drop-tail gateways and the phase effect eliminated, the competing RLA and TCP traffic see the same congestion frequency. The proof is omitted due to space limitations. 4.3 Remarks In the above two theorems, we proved that the RLA is essentially fair to TCP with the restricted topology with equal round-trip times. Note that the bounds in the theorems are widely separated for sizable n; this is because they work for all situations including the cases with extremely unbalanced congestion branches. The algorithm actually delivers desirable performance in the following way: if all the troubled receivers have the same degree of congestion, the RLA results in a throughput no larger than four times that of the competing TCP throughput for any n (this can be proved [20]); on the other extreme, if there is one most congested receiver and the other receivers experience only minor congestion just enough to be counted as troubled receivers, the actual throughput of the RLA is close to the upper bound which is in the order of n for drop-tail gateways. That is, the multicast connection on the soft bottleneck branch gets times the smallest throughput among the competing TCP connections. This might be desirable because this single bottleneck is slowing down the other receivers. If this is not desirable, the RLA can implement an option to drop this slow receiver. For the situations in between the above two extreme cases, the RLA gives reasonable per- this is demonstrated in the simulation results in section 5. In summary, the RLA achieves a higher share of band-width than the TCPs on the soft bottleneck branches when only a few receivers in the multicast session are much more congested than others. This is reasonable because the multicast session serves more receivers and it should suffer less on a single highly congested bottleneck. 4.4 Multicast Fairness of RLA The RLA is fair in the sense that the senders of competing multicast sessions between the same sender and receiver group will have the same average cwnd in the steady state. Consider a simple case with two competing sessions with n receivers in each session on the same topology of the form in figure 1. The cwnd's of the two senders are correlated random processes. The problem can be modeled as a randomly moving particle on a plane, with x and y axes being the cwnd of sender 1 and 2, respectively (see figure 3). This model is a generalization of the deterministic model for unicast congestion control used in [2], where the authors proved that the linear increase/multiplicative decrease scheme converges to the fair operating point. Our algorithm is a generalization of the unicast algorithm to multiple receivers and introduces randomness. We will show that although the cwnd does not converge to a single point, the desired operating point (equal share of the bottleneck bandwidth, see figure 3) is a recurrent point and most of the probability mass would be focused on the general area of this point. fairness line desired operating point pipe 3 pipe 3 Y Figure 3: Fairness of RLA to each other. In our analysis below, we assume there is no feedback delay. 10 Since the two sessions share exactly the same path, the two senders get the same congestion signals. The senders are informed of congestion by receiver i if cwnd1 exceeds the pipe size of virtual link L i which is the largest RTT value times the available bandwidth of the link (see figure 3). Otherwise the sender is informed of no congestion. Focus on the troubled receivers, recalling there are n of them per session. We order the pipe sizes of these troubled links as pipek and there are n i receivers with pipe size pipe i and In figure 3, This assumption is necessary to allow us to use a simple and neat analysis. Our simulation results indicated that the fairness we claimed here still holds when propagation delay is involved. in the white region with no congestion in the lightly shaded region and the senders receive n1 congestion signals once they enter this region; and pipe2 pipe3 in the dark shaded region and the senders receive once they enter this region, and so on. If there is no congestion, both cwnd's increase linearly which results in an upward movement of the particle along the line (see the movement of x0 to x1 in figure 3). In the congested region with a certain number of congestion signals fed to the senders, each sender independently generates a random number to decide whether to increase or cut the window. In our model, the particle randomly chooses one of the moving directions which are combinations of increasing or cutting of each window. For example, in figure 3, assuming after a round-trip time, can move to (cwnd1 +1; cwnd2 +1), or (cwnd1=2; cwnd2=2), or (cwnd1 + 1; cwnd2=2), or (cwnd1=2; cwnd2 1). Obviously the movement of the particle is Markovian since the next movement only depends on the current location of the particle. From this Markovian model, we can draw several conclusions about the system. First, the desired operating point (see figure 3) is a recurrent point. This is because from any starting point, there exists at least one convergent path to the desired point (e.g., along the dotted line in figure 3) with positive probability. Note that, in our model, there is a positive probability for the cwnd to grow to infinity, but this does not happen in the real system because we incorporated a forced-cut mechanism in the algorithm. Secondly, the average cwnd's of the two senders are the same. This is obvious because the two senders get the same congestion signals and react randomly but identically and independently, that is, the roles of the two senders are inter- changeable. In other words, if we switch the x and y axes, the moving particle follows the same stochastic process, and the marginal distributions along the axes are the same which gives the same mean value. Finally, most of the probability mass would be focused on the general area of the desired operating point in figure 3. To illustrate the idea, we consider a simple case where all of the n links have the same pipe size pipe. Then the plane is divided into two regions: a non-congested region with pipe and the rest a congested region with arriving to the sender upon each packet loss. Since in the RLA, the losses within two RTTs are grouped into one congestion signal, we consider a discrete-time version of the system with the time unit being two round-trip times, i.e., \Deltat = 2RTT . Denote cwnd of the kth multicast session by Wk , 2. If there is no congestion, congestion with n congestion signals, 2. W1 and W2 control the movement of the particle along x and y axes, respectively. The average drift along the x axis is 2 if W1 or 2 pipe. The time unit is 2RTT . The average drift along the y axis is symmetric and can be obtained by replacing W1 with W2 in the above equation. The drift diagram with is drawn in figure 4; the drift is scaled down by a factor of 5 to make the picture clear. The drift diagram shows that the particle controlled by the two congestion window sizes along the two axes has a desired operating point average drift of cwnd1 average drift of Figure 4: Average drift diagram of two competing cwnd's. trend to move towards the desired operating point. Figure 5 is the density plot of the occurrence of the point during one simulation run; 11 the higher numbers of occurrence, the darker the area. It shows that most of the probability mass is in an area centered around the desired operating point which is (20, 20) in this case. Figure 5: Density plot of the occurrence of (cwnd1 ; cwnd2 ). Performance Evaluation A version of the RLA is implemented in Network Simulator simulation purposes to test the RLA performance under various network topologies. Here we present some of the simulation results in a four-level tertiary tree network topology (see figure 6), where the links and nodes are labeled with the first number index indicating their level and the second indicating an order in each level. We describe most of the simulation parameters used in the simulations shown below. In figure 6, all senders (RLA or TCP) are located at the root node S, all receivers at leaf nodes R27. The nodes in between are gateways; they could be either drop-tail or RED type. All nodes have a buffer of size 20 packets. In the case of RED gateways, 11 The simulation setup consists of two multicast sessions with 27 receivers in each in a topology of the form shown in figure 1. There is one TCP session from the sender node to each of the receiver nodes. All receivers have the same capability. Each path has a delay band-width product of 60 shared by 2 multicast and 1 TCP sessions. There- fore, each session is supposed to get an average cwnd of 20. R Figure Four-level tertiary tree. the minimum threshold is 5 and the maximum threshold is 15. (Other parameters are the default values used in the standard NS2.0 RED gateway). The one-way propagation delays of the first three level links are all 5 ms, and those of the last level links are 100 ms. We tested the situations with bottleneck links at different levels to study the effect of independent or correlated losses. All the non-bottleneck link speeds are set to 100 Mbps. Data packet size is set to 1000 bytes. All simulations have 27 receivers (except for the case with different round-trip times), and all receivers are troubled receivers. In the simulation results presented here, rexmit thresh is set to 0 (i.e., all retransmissions are multicasted). All simulations are run for 3000 seconds and statistics are collected after the first 100 seconds. The simulation results are briefly summarized in the following. 5.1 Multicast Sharing with TCP The results for drop-tail gateways are shown in figure 7 and figure 9 for RED gateways. There are 5 cases with different soft bottleneck locations. The second row (most congested links) in the figures indicates the soft bottleneck location. The corresponding link speeds are set so that the soft bottleneck bandwidth share is min i second (recall m i is the number of background TCP connections between the sender and receiver i). We list the performance of the RLA, including the average throughput in packets per second, average congestion window size, average round-trip time (for those packets correctly received without retransmissions), the number of congestion signals the multicast sender detected from all receivers, the number of window cuts and the number of forced window cuts, over the entire simulation period (after the first 100 seconds). We also list the worst and the best case TCP performance (WTCP and BTCP rows in the figures) among the competing flows. As we can see from figure 7, the RLA achieves reasonable fairness with TCP even with drop-tail gateways. Comparing cases 1, 2 and 3, we can see that a higher correlation among the packet losses results in a larger average window size and a higher throughput. This agrees with our Lemma in section 4.2. The throughputs of the RLA and TCP connections in all cases satisfy the essential fairness requirement with In fact, the bounds are quite loose for these cases. The actual performance of the RLA algorithm in most cases is much more "reasonable" than the bounds indicated, in the sense that in most cases the RLA can achieve a tighter bounded fairness. With the simulation setup, the measured essential fairness has bounds these cases, which is very reasonable performance and acceptable for many applications. cwnd signals # wnd cut forced cut cwnd # wnd cut cwnd # wnd cut RTT RTT RTT Links Congested Most Case A T144.123247081.889.621.981827.219797022.072223.36880.27065179.20.26980.30.27040.012759017.984243.8405117540.23118.953.700.238570.7(sec) (sec) (sec) thrput (pkt/sec) (pkt/sec) thrput thrput (pkt/sec) Figure 7: Simulation results with drop-tail gateways. We can also see from figure 7 that the number of window cuts taken by the RLA sender is roughly 1of the congestion signals the multicast sender detected, as desired. In figure 8, we consider the congestion signals from each receiver separately and list the worst, best and average number of congestion signals the sender detected from each of the receivers on the links with the same level of congestion. The number is over the entire simulation period (2900 seconds). We also list the results for the competing TCP connections. It demonstrates that the TCP sender and the RLA sender see roughly the same number of congestion signals on each branch on average. Therefore, they see the same congestion frequency, as we argued in section 3.1. Note that the discrepancies between the RLA and TCP congestion frequencies are larger in cases 4 and 5. This is because their congestion window sizes are very different (refer to figure 7 for window sizes). In these cases, a larger window likely incurs more losses. Although it breaks the assumption of equal congestion frequencies, it creates a desired balance: the larger the window, the more losses and then the window is more likely to be reduced to half and vice versa. This balance actually helps to achieve a tighter bounded fairness as we observed in the simulations.7629521082 Worst Average Best Worst Best Average RLA Branch more all links all links all links Case 861 861 879 8188319256464057228428997131082less congested more Figure 8: Statistics of the number of congestion signals. Figure 9 shows the corresponding results for the RED gateways. All simulation setups are the same except that the gateway type is changed to RED, and we do not use random overhead in these simulations because RED gateways eliminate the phase effect. The results show that with RED gateways, the fairness between multicast and TCP is closer to absolute, especially in case 1. This is expected as suggested by the bounds derived in section 4 and is also intuitive because RED gateways are designed to enforce fairness. cwnd signals # wnd cut forced cut cwnd # wnd cut cwnd # wnd RTT RTT RTT Links Congested Most Case A (sec) cut (sec) (sec)thrput (pkt/sec) thrput thrput (pkt/sec) (pkt/sec) Figure 9: Simulation results with RED gateways. 5.2 Multiple Multicast Sessions To test the multicast fairness property of the RLA algo- rithm, we have simulated the above scenarios with two overlapping multicast sessions from the sender to the same re- ceivers. In all cases, the two multicast sessions share band-width almost equally and have roughly the same average window size. In particular, in the topology of case 3 mentioned above, the two multicast senders achieve throughputs of 65.1 and 65.9 pkt/sec respectively, and average window sizes of 19.9 and 20.1 packets respectively. 5.3 Different Round-Trip Times This paper has focused on the restricted topology with equal round-trip times where fairness is meaningfully defined. But, in reality, most multicast sessions comprise receivers located at different distances from the sender. These cases have to be addressed properly in order for an algorithm to be ac- cepted. We have a generalized version of the RLA algorithm presented in section 3 to work for the cases with different round-trip times. The basic idea is to set )=num trouble rcvr. In our experiment, we are using the function of the form because it has been shown that, for TCP-like window adjustment policy, the average throughput is proportional to (RTT there are no queueing delays [5, 10]. Note that in the case of equal round-trip times, the above pthresh is the same as in the original RLA. In the case of different round-trip times, the receiver with a smaller round-trip time has a much smaller pthresh, that is, a much larger fraction of the congestion signals is ignored. In the case of different round-trip times, we are not able to provide any theoretical proofs of bounded fairness. But our initial experimental results show the generalized algorithm is promising in providing a reasonable share of band-width among multicast and TCP traffic. Here we present a set of simulation results with the same topology as in the above simulations but adding the nodes G31 through G39 also as receivers which are of significantly different round-trip times from the leaf nodes since the level four links have a one-way propagation delay of 100 ms. Here we show simulation results for two cases with the bottlenecks at level 2 links or level 3 links respectively. Both cases have a total of 36 receivers, all troubled receivers. The results are summarized in figure 10 and they show a reasonable share of bandwidth between the multicast and TCP traffic. cwnd # cong signals cut forced cut cwnd # wnd cut cwnd # wnd cut Congested Most Links thrput (pkt/sec) thrput thrput (pkt/sec) (pkt/sec) RTT (sec) RTT RTT (sec) (sec) A R Figure 10: Results with different round-trip times. 6 Conclusions and Future work In this paper, we introduced a quantitative definition for essential fairness between multicast and unicast traffic. We also proposed a random listening algorithm (RLA) to achieve essential fairness for multicast and TCP traffic over the Internet with drop-tail or RED gateways. RLA is simple and achieves bounded fairness without requiring locating the soft bottleneck links. Although our RLA is based on the TCP congestion control mechanism, it is worth noting that the idea of "random listening" can be used in conjunction with other forms of congestion control mechanism, such as rate-based control. The key idea is to randomly react to the congestion signals from all receivers and to achieve a reasonable reaction to congestion on the average over a long run. There are many interesting possibilities which are worth exploring in this direction. Due to space limitations, many details of the algorithm and simulation results are not shown in this paper. Our on-going work is to carry out more and larger scale simulations and to refine the algorithm based on the experience gained from the simulations. Acknowledgement This work was inspired by a summer project the first author worked on at Lucent Bell Labs. She would like to thank Dr. Zheng Wang for the basic inspiration. Many thanks are also due to Dr. Sanjoy Paul, Dr. Ramachandran Ramjee, Dr. Jamal Golestani, all of Bell Labs, for useful discussions. --R Notes on FEC supported congestion control for one to many reliable multicast. Analysis of the increase and decrease algorithms for congestion avoidance in computer networks. A congestion control mechanism for reliable multicast. On traffic phase effects in packet-switched gateways Random early detection gateways for congestion avoidance. A congestion control architecture for bulk data transfer Congestion avoidance and control. The performance of tcp/ip for networks with high bandwidth-delay products and random loss Tcp selective acknowledgement op- tions A loss tolerant rate controller for reliable multicast. The stationary behavior of ideal tcp congestion avoidance. The direct adjustment algorithm UCB/LBNL/VINT. One to many reliable bulk-data transfer in the mbone Achieving bounded fairness for multicast and TCP traffic in the Internet. Observations on the dynamics of a congestion control algorithm: The effects of two-way traffic --TR Congestion avoidance and control Analysis of the increase and decrease algorithms for congestion avoidance in computer networks Observations on the dynamics of a congestion control algorithm Random early detection gateways for congestion avoidance Simulation-based comparisons of Tahoe, Reno and SACK TCP The performance of TCP/IP for networks with high bandwidth-delay products and random loss --CTR Dan Rubenstein , Jim Kurose , Don Towsley, The impact of multicast layering on network fairness, ACM SIGCOMM Computer Communication Review, v.29 n.4, p.27-38, Oct. 1999 Dan Rubenstein , Jim Kurose , Don Towsley, The impact of multicast layering on network fairness, IEEE/ACM Transactions on Networking (TON), v.10 n.2, April 2002 Yair Amir , Baruch Awerbuch , Claudiu Danilov , Jonathan Stanton, A cost-benefit flow control for reliable multicast and unicast in overlay networks, IEEE/ACM Transactions on Networking (TON), v.13 n.5, p.1094-1106, October 2005 Homayoun Yousefi'zadeh , Hamid Jafarkhani , Amir Habibi, Layered media multicast control (LMMC): rate allocation and partitioning, IEEE/ACM Transactions on Networking (TON), v.13 n.3, p.540-553, June 2005 Anca Dracinschi Sailer , Serge Fdida, Generic congestion control, Computer Networks: The International Journal of Computer and Telecommunications Networking, v.41 n.2, p.211-225, 5 February B. Baurens, Groupware, Cooperative environments for distributed: the distributed systems environment report, Springer-Verlag New York, Inc., New York, NY, 2002

internet;phase effect;RED and drop-tail gateways;flow and congestion control;multicast

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Efficient Distributed Detection of Conjunctions of Local Predicates.

AbstractGlobal predicate detection is a fundamental problem in distributed systems and finds applications in many domains such as testing and debugging distributed programs. This paper presents an efficient distributed algorithm to detect conjunctive form global predicates in distributed systems. The algorithm detects the first consistent global state that satisfies the predicate even if the predicate is unstable. Unlike previously proposed run-time predicate detection algorithms, our algorithm does not require exchange of control messages during the normal computation. All the necessary information to detect predicates is piggybacked on computation messages of application programs. The algorithm is distributed because the predicate detection efforts as well as the necessary information are equally distributed among the processes. We prove the correctness of the algorithm and compare its performance with respect to message, storage, and computational complexities with that of the previously proposed run-time predicate detection algorithms.

Introduction Development of distributed applications requires the ability to analyze their behavior at run time whether to debug or control the execution. In particular, it is sometimes essential to know if a property is satisfied (or not) by a distributed computation. Properties of the computation, which specify desired (or undesired) evolutions of the program's execution state, are described by means of predicates over local variables of component processes. A basic predicate refers to the program's execution state at a given time. These predicates are divided into two classes called local predicates and global predicates. A local predicate is a general boolean expression defined over the local state of a single process, whereas a global predicate is a boolean expression involving variables managed by several processes. Due to the asynchronous nature of a distributed computation, it is impossible for a process to determine the total order in which the events occurred in the physical time. Consequently, it is often impossible to determine the global states through which a distributed computation passed through, complicating the task of ascertaining if a global predicate became true during a computation. Basic predicates are used as building blocks to form more complex class of predicates such as linked predicates [14], simple sequences [5, 9, 1], interval-constrained sequences [1], regular patterns [4] or atomic sequences [8, 9]. The above class of properties are useful in characterizing the evolution of the program's execution state, and protocols exist for detecting these properties at run time by way of language recognition techniques [2]. When the property (i.e., a combination of the basic properties) contains no global predicate, the detection can be done locally without introducing any delays, without defining a centralized process and without exchanging any control messages. Control information is just piggybacked to the existing message of the application. However, if the property refers at least to one global predicate, then all possible observations of the computation must be considered. In other words, the detection of the property requires the construction and the traversal of the lattice of consistent global states representing all observations of the computation. When the property reduces to one global predicate, the construction of the lattice can be avoided in some cases. If the property is expressed as a disjunction of local predicates, then obviously no cooperation between processes is needed in order to detect the property during a computation. A form of global predicate, namely, the conjunction of local predicates, has been the focus of research [5, 6, 7, 12, 17] during the recent years. In such predicates, the number of global states of interest in the lattice is considerably RR 4 M. Hurfin, M. Mizuno, M. Raynal, M. Singhal reduced because all global states that includes a local state where the local predicate is false need not be examined. Previous Work The problem of global predicate detection has attracted considerable attention lately and a number of global predicate detection algorithms have been proposed in the recent past. In the centralized algorithm of Cooper and Marzullo [3], every process reports each of its local states to a process, which builds a lattice of the global computation and checks if a state in the computation satisfies the global predicate. The power of this algorithm lies in generality of the global predicates it can detect; however, the algorithm has a very high overhead. If a computation has n processes and if m is the maximum number of events in any process, then the lattice consists of O(m n ) states in the worst case. Thus, the worst case time complexity of this algorithm is O(m n ). The algorithm in [10] has linear space complexity; however, the worst case time complexity is still linear in the number of states in the lattice. Since the detection of generalized global predicates by building and searching the entire state space of a computation is utterly prohibitive, researchers have developed faster, more efficient global predicate detection algorithms by restricting themselves to special classes of predicates. For example, a form of global predicate that is expressed as the conjunction of several local predicates has been the focus of research [5, 6, 7, 12, 17] recently. Detection of such predicates can be done during a replay of the computation [12, 17] or during the initial computation [5, 6, 7].This paper focus on the second kind of solution which allows one to detect the predicate even before the end of the computation. In the Garg-Waldecker centralized algorithm to detect such predicates [6], a process gathers information about the local states of the processes, builds only those global states that satisfy the global predicate, and checks if a constructed global state is consistent. In the distributed algorithm of Garg and Chase [7], a token is used that carries information about the latest global consistent state (cut) such that the local predicates hold at all the respective local states. The worst case time complexity of both these algorithms is O(mn 2 ) which is linear in m and is much smaller than the worst case time complexity of the methods that require searching the entire lattice. However, the price paid is that not all properties can be expressed as the conjunction of local predicates. Recently, Stoller and Schneider [16] proposed an algorithm that combines the Garg-Waldecker approach [6] with any approach that constructs a lattice to detect Possibly(\Phi). distributed computation satisfies Possibly(\Phi) iff predicate \Phi holds in a state in the corresponding lattice.) This algorithm has the best features of both INRIA Efficient Distributed Detection of Conjunctions of Local Predicates 5 the approaches - it can detect Possibly(\Phi) for any predicate \Phi and it detects a global predicate expressed as the conjunction of local predicates in time linear in m (the maximum number of events in any process). Paper Objectives This paper presents an efficient distributed algorithm to detect conjunctive form global predicates in distributed systems. We prove the correctness of the algorithm and compare its performance with that of the previous algorithms to detect conjunctive form global predicates. The rest of the paper is organized as follows: In the next section, we define system model and introduce necessary definitions and notations. Section 3 presents the first global predicate detection algorithm and gives a correctness proof. The second algorithm is presented in Section 4. In Section 5, we compare the performance of the proposed algorithms with the existing algorithms for detecting conjunctive form global predicates. Finally, Section 6 contains the concluding remarks. System Model, Definitions, and Notations 2.1 Distributed Computations A distributed program consists of n sequential processes denoted by P 1 The concurrent execution of all the processes on a network of processors is called a distributed computation. The processes do not share a global memory or a global clock. Message passing is the only way for processes to communicate with one another. The computation is asynchronous: each process evolves at his own speed and messages are exchanged through communication channels, whose transmission delays are finite but arbitrary. We assume that no messages are altered or spuriously introduced. No assumption is made about the FIFO nature of the channels. 2.2 Events 2.2.1 Definition and Notations Activity of each process is modeled by a sequence of events (i.e., executed action). Three kinds of events are considered: internal, send, and receive events. Let e x the x th event which occurs at process P i . Figure 1 shows an example of distributed computation involving two processes P 1 and P 2 . In this example, event e 2 1 is a send event and event e 1 2 is the corresponding receive event. Event e 1 1 is an internal event. RR 6 M. Hurfin, M. Mizuno, M. Raynal, M. Singhal Figure 1: A distributed computation For each process P i , we define an additional internal event denoted as e 0 i that occurred at process P i at the beginning of the computation. So, during a given computation, execution of process P i is characterized by a sequence of events: Furthermore, if the computation terminates, the last action executed at process (denoted as e m i by an imaginary internal event denoted as e m i +1 2.2.2 Causal Precedence Relation Between Events The "happened-before" causal precedence relation of Lamport induces a partial order on the events of a distributed computation. This transitive relation, denoted by OE, is defined as follows: 8e x or There exists a message m such that i is a send event (sending of m to P j e y j is a receive event (receiving of m from or There exists an event e z such k and e z This relation is extended to a reflexive relation denoted . INRIA Efficient Distributed Detection of Conjunctions of Local Predicates 7 2.3 Local states 2.3.1 Definition and Notations At a given time, the local state of a process P i is defined by the values of the local variables managed by this process. Although occurrence of an event does not necessarily cause a change of the local state, we identify the local state of a process at a given time with regard to the last occurrence of an event at this process. We use oe x i to denote the local state of P i during the period between event e x i and event e x+1 . The local state oe 0 i is called the initial state of process P i . Figure 2: Local states of processes 2.3.2 Causal Precedence Relation Between Local States The definition of the causal precedence relation between states (denoted by \Gamma!) is based on the happened-before relation between events. This relation is defined as follows: 8oe x Two local states oe x i and oe y are said to be concurrent if there is no causal dependency between them (i.e., oe x j and oe y A set of local states is consistent if any pair of elements are concurrent. In the distributed computation shown in Figure 2, foe 0 2 g and foe 2,oe 4 2 g are three examples of consistent sets of local states. RR 8 M. Hurfin, M. Mizuno, M. Raynal, M. Singhal 2.4 Intervals 2.4.1 Definition and Notations Since causal relations among local states of different processes are caused by send and receive events, we introduce the notion of intervals to identify concurrent sequences of states of a computation. An interval is defined to be a segment of time on a process that begins with a send or receive event (called a communication event) and ends with the next send or receive event. Thus, a process execution can be viewed as a consecutive sequence of intervals. In order to formally define intervals, we first introduce a new notation to identify communication events. We use " x i to denote the x th send or receive event at P i . Thus, for each " x i , there exists exactly one e y i that denotes the same event. Furthermore, the imaginary event e 0 are renamed as " 0 . If the computation terminates, imaginary event i at process P i is renamed as " l i +1 (l i is the number of communication events that occurred at process P i ). The x th interval of process P i , denoted by ' i , is a segment of the computation that begins at " and ends at " x . Thus, the first interval at P i is denoted by ' 0 . If the computation terminates, the last interval of process P i is identified by ' l i Figure 3: The corresponding set of intervals We say that interval ' x i contains the local state oe y (or oe y i is contained in ' x the following property holds: (" x This relation is denoted by oe y . If this relation does not hold, it is denoted by oe y . By definition, any interval contains at least one local state. INRIA Efficient Distributed Detection of Conjunctions of Local Predicates 9 2.4.2 Causal Precedence Relation Between Intervals The relation that expresses causal dependencies among intervals is denoted as !. This relation induces a partial order on the intervals of distributed computation and is defined as follows: A set of intervals is consistent if for any pair of intervals in the set, say ' x 2.5 Global States A global state (or cut) is a collection of n local states containing exactly one local state from each process P i . A global state is denoted by f oe x 1 g. If a global state f oe x 1 is consistent, it is identified by The set of all consistent global states of a distributed computation form a lattice whose minimal element is the initial global state \Sigma(0; 0; from the distributed computation can reach the latter from the former when process P i executes its next event i . Each path of the lattice starting at the minimal element corresponds to a possible observation of the distributed computation. Each observation is identified by a sequence of events where all events of the computation appear in an order consistent with the "happen before" relation of Lamport. The maximal element called the final global state and exists only if all the processes of the distributed computation have terminated. Given a computation and a predicate on a global state \Phi, we can use the two modal operators proposed by Cooper and Marzullo [3] to obtain two different pro- perties, namely, Possibly(\Phi) and Definitely(\Phi). A distributed computation satisfies if and only if the lattice has a consistent global state verifying the predicate \Phi, whereas Definitely(\Phi) is satisfied by the computation if and only if each observation (i.e., each path in the lattice) passes through a consistent global state verifying \Phi. In this paper, we focus on the class of global predicates formed as the conjunction of local predicates and we consider only the first satisfaction rule: Possibly(\Phi). This rule is particularly attractive to test and debug distributed executions. RR M. Hurfin, M. Mizuno, M. Raynal, M. Singhal 2.6 Conjunctions of Local Predicates A local predicate defined over the local state of process P i is denoted as L i . Notation oe x indicates that the local predicate L i is satisfied when process P i is in the local state oe x . Due to its definition, a local predicate L i can be evaluated by process P i at any time without communicating with any other process. We extend the meaning of symbol j= to intervals as follows: i such that (oe y Let \Phi denote a conjunction of p local predicates. Without loss of generality, we assume that the p processes involved in the conjunction \Phi are . In this paper, we write either \Phi or to denote the conjunction. A set of p local states f oe x 1 p g is called a solution if and only if:? ! p g is a consistent set. A global state \Sigma(x 1 called a complete solution if this set of local states includes a solution. By definition, Possibly(\Phi) is verified if there exists a complete solution. A consistent set of local states containing less than n local states may be completed to form a consistent global state (i.e., a consistent set of n elements), and thus, a solution may be extensible to one or more complete solutions. The goal of a detection algorithm is not to calculate the whole set of complete solutions but only to determine a solution. This approach is not restrictive. In order to deal with complete solutions rather than with solutions, a programmer can simply add to the conjunction, local predicates L are true in any local state. Consequently, we will no longer speak about complete solution. Due to the link between local states and intervals, the following definition of a solution is obviously consistent with the first one. A set of p local states f oe y 1 p g is a solution iff there exists a set of p g is a consistent set. INRIA Efficient Distributed Detection of Conjunctions of Local Predicates 11 Let S denote the set of all solutions. If S is not empty, the first solution is the unique element of S denoted by f oe f 1 p g such that every element p g of S satisfies the following property: As the property to detect is expressed as a conjunction of local predicates, this particular solution, if it exists, is well defined in the computation. The set of intervals that includes this solution is also well defined. We denote this set of intervals and we say that this set of intervals is the first one which verifies \Phi. 3 Detection Algorithms for Conjunction of Local Predicate 3.1 Overview As mentioned in the previous section, Possibly(\Phi) is verified by detecting a set of concurrent intervals, each of which verifies its local predicate. We have developed the following two approaches to resolve this problem: 1. In the first approach, processes always keep track of sets of concurrent intervals. For each such set, each process checks whether its interval in the set verifies its local predicate. 2. In the second approach, a process always keeps track of a set of intervals, each of which verifies its local predicate. For each such set, the process checks whether all intervals in the set are concurrent. Thus, algorithms designed for those complementary approaches are dual of each other. This section described the algorithm corresponding to the first approach in detail, including its correctness proof. The next section describes an algorithm corresponding to the second approach. 3.2 The First Algorithm 3.2.1 Dependency Vectors To identify a set of p concurrent intervals, the algorithm keeps track of causal dependencies among intervals by using a vector clock mechanism similar to that described RR M. Hurfin, M. Mizuno, M. Raynal, M. Singhal in [13]. Each process P i (1 i n) maintains an integer vector D i [1.p], called the dependency vector. Since causal relations between two intervals at different processes are created by communication events (and their transitive relation), values in D i are advanced only when a communication event takes place at P i . We use D x i to denote the value of vector D i when process P i is in interval ' x i . This value is computed at the time " x i is executed at process P i . Each process P i executes the following protocol: 1. When process P i is in interval ' 0 all the components of vector D i are zero. 2. When P i (1 i p) executes a send event, D i is advanced by setting D i [i] := 1. The message carries the updated D i value. 3. When P i executes a receive event, where the message contains Dm , D i is advanced by setting D i [k] := max(D i [k]; Dm [k]) for 1 k p. Moreover, the dependency vector is advanced by setting D i [i] := D belongs to the set of p processes directly implicated in the conjunction (i.e., When a process P i (1 i p) is in interval ' x i , the following properties are observed [15]: 1. D x it represents the number of intervals at P i that precede interval 2. D x represents the number of intervals at process P j that causally precede the interval ' x 3. The set of intervals f' D x p g is consistent. 4. None of the intervals ' y j such that y ! D x (i.e., intervals at P j that causally precede ' D x can be concurrent with ' x . Therefore, none of them can form a set of intervals with ' x i that verifies \Phi. Let D a and D b be two dependency vector clock values. We use the following notations. ffl D a = D b iff 8i; D a ffl D a D b iff 8i; D a [i] D b [i] INRIA Efficient Distributed Detection of Conjunctions of Local ffl D a ! D b iff (D a D b The following result holds: 3.2.2 Logs Each process maintains a log, denoted by Log i , that is a queue of vector clock entries. Log i is used to store the value of the dependency vector associated with the past local intervals at P i that may be a part of a solution (i.e., intervals that verify L i and have not been tested globally yet). Informations about causal relations of a stored interval with intervals at other processes will be used in the future when the stored interval will be examined. When P i is in a local state oe y contained in an interval ' x , such that oe y it enqueues vector clock value D x i in Log i if this value has not already been stored. Even if there exists more than one local state in the same interval ' x i that verifies the value D x i needs to be logged only once since we are interested in intervals instead of states. 3.2.3 Cuts In addition to the vector clock, each process P i (1 i n) maintains an integer vector C i [1.p], called a cut and a boolean vector B i [1.p]. Vector C i defines the first consistent global state which could verify \Phi. In others words, all previous global states dont satisfy \Phi. By definition, C denotes a set of p intervals that may be the first solution. If some informations received by P i show that this set is certainly not a solution, the cut is immediately updated to a new potential solution. At any time, C i [j] denotes the number of interval of P j already discarded and indicate the j of the first interval of P j not yet eliminated. If the conjunction is satisfied during the computation, the cut C will evolve until it denotes the first solution. Let C x denotes the value of C i [j] after the communication event " x i has been executed at P i . The value of C i remains unchanged in the interval. Each P i maintains the values C i in such a way that none of the intervals that precede event " C i [j] j at can form a set of intervals that verifies \Phi. Therefore, each process P i (1 i p) may discard any values D i in Log i such that D i [i] ! C i [i]. Each P i (1 i n) also maintains the vector B i in such a way that B i [j] holds if the interval ' C i [j] j at P j is certain to verify its local predicate. Thus, if the system is not certain whether the RR 14 M. Hurfin, M. Mizuno, M. Raynal, M. Singhal interval verifies its local predicate, B i [j] is set to false. To maintain this condition, the cut C and the B vector must be exchanged among processes. When P i sends a message, it includes vectors C in the message. 3.3 Descriptions of the Algorithm A formal description of the algorithm is given in Section 3.4. The algorithm consists of the following three procedures that are executed at a process ffl A procedure A that is executed each time local predicate L i associated to P i ffl A procedure B that is executed when P i (1 i n) sends a message. ffl A procedure C that is executed when P i (1 i n) receives a message. In addition to vector clock D i , cut C i , boolean vector B i , and log Log i , each process maintains a boolean variable not logged yet i . Variable not logged yet i is true iff the vector clock value that is associated with the current interval has not been logged in Log i . This variable helps avoid logging the same vector clock value more than once in Log i . A: When the local predicate L i becomes true : Let oe y i be the local state that satisfies the local predicate. If ' x i is the interval that includes state oe y then the vector clock value D x i , which is associated with interval i , is logged in Log i if it has not been logged yet. To indicate that the vector clock for this interval has already been logged in, process P i sets variable not logged yet i to false. Furthermore, if the current interval (denoted by ' x or ' D x also the oldest interval of P i not yet discarded (denoted by ' C x i [i] is set to true to indicate that ' C x sends a message: Since it is the beginning of a new interval, a process P i (1 i p) advances the vector clock by setting D i resets variable not logged yet i to true. If the log is empty, none of the intervals that precedes the new interval can form a set of intervals that verifies \Phi. In particular, this remark holds for the last interval that ends just when the current execution of procedure B (i.e., the sending action) occurs. Consequently, the last interval (and also all the intervals of P i that causally INRIA Efficient Distributed Detection of Conjunctions of Local Predicates 15 precede this one) can be discarded by setting C i [i] to the current value of D i [i]. (At this step of the computation, " D i [i] i is the identity of the current send event). Finally, sends a message along with C receives a message from P j that contains D Based on the definition of vector clocks, it advances D i and resets variable not logged yet i to true. From the definition of a cut, at any process P k (1 k p), none of the intervals that precedes interval ' C i [k] k or ' C j [k] k can form a set of concurrent intervals that verifies \Phi. Thus, C i is advanced to the componentwise maximum of C i and C j . B i is updated so that it contains more up-to-date of the information in B i and B j . process P i deletes log values for intervals that precede ' C i [i] since these intervals do not belong to sets of concurrent intervals that verify \Phi. After this operation, there are two possibilities: ffl Case 1 Log i becomes empty i.e. Log i does not contain any interval that occurs after ' C i [i] i and before ' D i [i] In this case, none of the intervals at P i represented by ' y i i such that y i ! D i [i] can form a set of concurrent intervals that verify \Phi. The algorithm needs to consider only future intervals, denoted by ' z i , such that D i [i] z. Since none of the intervals ' y k k such that y k ! D i [k] at other processes P k can form a set of concurrent intervals with such future intervals ' z C i is advanced to D i . When process P i executes the receive action, it has no informations about intervals ' D i [k] p). Therefore, all components of vector B i are set to false. contains at least one entry that was logged after the occurrence of event " C i [i] the oldest such logged entry be D log . From the properties of vector D and the definition of a cut, at any process P k , 1kp, none of the intervals preceding ' D log k or ' C i [k] k can form a set of concurrent intervals with ' D log i that verifies \Phi. Thus, C i is advanced to the componentwise maximum of D log . Similar to Case 1, if the value C i [k] is modified (i.e., it takes its value from D log not certain whether P k 's local predicate held in the interval ' D log k . Thus, B i [k] is set to false. If the value C i [k] remains unchanged, the value B i [k] will also remain unchanged. RR M. Hurfin, M. Mizuno, M. Raynal, M. Singhal Furthermore, since ' D log verified its local predicate, B i [i] is set to true. At this point, P i checks whether B i [k] is true for all k. If so, this indicates that each interval in the concurrent set of intervals 1 f' C i [1] verifies its local predicate and thus, \Phi is verified. 3.4 A Formal Description of the Algorithm Initialization procedure executed by any process P i if (i p) then Create(Log i ); not yet logged i := true; endif Procedure A executed by process when the local predicate L i becomes true if (not yet logged i ) then logged i := false; endif Procedure B executed by any process P i when it sends a message if (i p) then logged i := true; endif Append vectors D i , C i , and B i to the message; Send the message; Procedure C, executed by any process P i when it receives a message from P j We will prove in subsection 3.7 that a set of intervals numbered by C i values always are concurrent. INRIA Efficient Distributed Detection of Conjunctions of Local Predicates 17 Extract vectors D j , C j and B j from the message; if (i p) then logged i := true; while ((not (Empty(Log i ))) and (Head(Log i do /* Delete all those logged intervals that from the current /* knowledge do not lie on a solution. /* Construct a solution that passes through the next local interval. else /* Construct a solution that passes through the logged interval. endif endif Deliver the message; Function Combine Maxima ((C1,B1), (C2,B2)) B: vector [1.p] of boolean; C: vector [1.p] of integers; for to p do case endcase 3.5 A Simple Example Since the algorithm is quite involved, we illustrate the operation of the algorithm with the help of an example. In Figure 4, a local state contained in an interval is RR M. Hurfin, M. Mizuno, M. Raynal, M. Singhal represented by a grey area if it satisfies the associated local predicate. At different step of the computation, we indicate the values of the main variables used to detect items with square brackets next to a process interval, respectively, depict the contents of vectors D and C. Those values remain unchanged during the entire interval. The value of vector B after execution of a communication event is indicated between round brackets. Initial value of interval number at two processes is 0 and C vector is (0 0) at both processes. When the local predicate holds in interval ' 0 enqueues D 1 vector into Log 1 . Process P 1 also set B 1 [1] to true because it is certain that ' C 1 [1] sends message m1, it increments D 1 [1] to 1 and sends vectors B 1 , C 1 , and D 1 in the message. When receives message m1, it increments D 2 [2] to 1 and then updates its B, C, and D vectors. P 2 finds its Log empty and constructs a potential solution using its D vector and stores it into its C vector. When the local predicate becomes true in state oe 1 to Log 2 . As the variable not yet logged 2 is false when process P 2 is in local state oe 2 2 , the vector clock D is not logged twice during the same interval. When P 2 sends message m2, it increments D 2 [2] to 2 and sends vectors B 2 , in the message. When P 1 receives message m2, it increments D 1 [1] to 2 and then updates its B, C, and D vectors. After merging with the vectors received in the message, P 1 finds that C 1 [1] (=1)?Head(Log 1 )[1] (=0) and discards this entry from Log 1 . Since Log 1 is empty, P 1 constructs a potential solution using its D vector and stores it into its C vector. When the local predicate becomes true in interval oe 2 to Log 1 . When P 1 sends message m3, it increments D 1 [1] to 3 and sends vectors B 1 , in the message. In the meantime, local predicate holds in state oe 3 2 and consequently, P 2 logs vector D 2 to Log 2 . When receives message m3, it increments D 2 [2] to 3 and then updates its B, C, and D vectors. After merging with the vectors received in the message, P 2 finds that C 2 [2] (=2)?Head(Log 2 )[2] (=1) and discards this entry. Since the next entry in Log 2 cannot be discarded, P 2 constructs a potential solution using Head(Log 2 )[2] vector and stores it into its C and B vectors. The potential solution goes through 1 . The fact that this interval satisfies L 1 is known by process P 2 (B 2 [1] is true). After P 2 sets B 2 [2] to true, it finds that all entries of vector B 2 are true and declares the verification of the global predicate. INRIA Efficient Distributed Detection of Conjunctions of Local Predicates 19 Figure 4: An Example to Illustrate Algorithm 1. 3.6 Extra messages The algorithm is able to detect if a solution exists without adding extra-message during the computation and without defining a centralized process. The algorithm depends on the exchange of computation message between processes to detect the predicate. As a consequence, not only the detection may be delayed, but also in some cases the computation may terminate and the existing solution may go unde- tected. For example, if the first solution is the set consisting of the p last intervals, g, the algorithm will not detect it. To solve this problem, if a solution has not been found when the computation terminates, messages containing vector D, C, and B are exchanged between the p processes until the first solution is found. To guarantee the existence of at least one solution, we assume that the set of intervals f' l 1 +1 p g is always a solution. Extra messages are exchanged only after the computation ends and the first solution has not been detected yet. To reduce the overhead due to these extra messages between processes, one can use a privilege (token) owned by only one process at a time. The token circulates around a ring consisting of the p processes, disseminating the information about the three vectors. Another solution consists of sending the token to a process who may know the relevant information (i.e., a process P j such that B i [j] is false). RR M. Hurfin, M. Mizuno, M. Raynal, M. Singhal 3.7 Correctness of the algorithm be two vector timestamps such that the sets of intervals represented by f' V 1 [1] are both concur- rent. Then, the set of intervals represented by f' V 3 [1] p g is concurrent, Proof: We show that a pair of intervals (' V 3 [i] any combination of i and concurrent. Renumbering the vectors V 1 and V 2 if necessary, suppose V 3 There are two cases to consider: 1. This case is obvious because, from the assumption, ' V 1 [i] are concurrent. 2. Suppose on the contrary that ' V 3 [i] j are not concurrent. There are two cases to consider: In this case, " i . This contradicts the assumption that ' V 1 [j] are concurrent. By applying the same argument as the case (a), this leads to a contradiction to the assumption that ' V 2 [i] are concurrent. are concurrent. 2 The following lemma guarantees that a cut C i always keeps track of a set of concurrent intervals. Lemma 2: At any process P i , at any given time, f' C i [1] p g is a set of concurrent intervals. Proof: C i is updated only in one of the following three ways: 1. When a receive event is executed, by executing C:= D. 2. When a receive event is executed by taking maximum of C i and C j (the cut contained in the message sent by process P j ), and then by taking maximum of C i and D log (The oldest value of the dependency vector still in the log). INRIA Efficient Distributed Detection of Conjunctions of Local Predicates 21 3. When a send event occurs, entry C i [i] is set to D i [i]. Let update(C x the update on cut C i at communication event " x giving the value C x i . We define a partial order relation (represented by ";") on all updates on all cuts C i (1 i p) as follows: 1. If " x are two consecutive communication events that occur at P i (i.e., 2. If there exists a message m such that " x i is the sending of m to P j and " y i is the receiving of m from P i , then update(C x Let SEQ be a topological sort of the partial order of update events. We prove the lemma by induction on the number of updates in SEQ. Induction Base: Since the set of intervals f' 0 p g is concurrent, initially the lemma holds for any C i (i.e., after update(C 0 Induction Hypothesis: Assume that the lemma holds up to t applications of the updates. Induction Steps: Suppose st update occurs at process P i and let C x the cut value after the st update operation denoted as update(C x i is a receive event and C x From the definition of vector clocks, for any vector clock value D, the set of intervals f' D[1] p g is concurrent. Thus, C x represents a set of concurrent intervals. Case 2: " x i is a receive event and C x As " y j is the corresponding send event, update(C y Clearly, the relation update(C holds. Since from induction hypo- thesis, both C represent sets of concurrent intervals, max(C represents a set of concurrent intervals (from Lemma 1). Since from the definition of vector clocks, D log (the oldest entry still in the log) represents a set of concurrent intervals, C x represents a set of concurrent intervals (from Lemma 1). Case 3: " x i is a send event, C x i [i], and 8j such that j 6= i, C x RR 22 M. Hurfin, M. Mizuno, M. Raynal, M. Singhal Suppose no receive event occurs at process P i before the send event " x . All the entries of vector D x i and vector C x i [i], are zero. Therefore, i and the proof is the same as in Case 1. Suppose now that " y i is the last receive event who occurs before the sending event " x in the log has been discarded since this event occurred. As the Log is empty, we can conclude that the log was also empty when this receive event occurred and hence, C y . As this event is the last receive event that occurs before " x i , we conclude that 8j such that j 6= i, D y and C y . Then, the proof is the same as in case 1.The following lemma shows that if there is a solution, a cut C i will not miss and pass beyond the solution. Lemma 3: Consider a particular cut (identified by an integer vector S) such that p g is a set of concurrent intervals that verifies \Phi. i denote any communication event. If, for all communication events " y j such that " y Proof: Proof is by contradiction. Suppose there exists a communication event " x such that :(C x and for any communication event " y j such that " y i is the first event that advances C i beyond S. There are two cases to consider: 1. C x From hypothesis, C holds. Therefore, entry C i [j] is modified during execution of event " x i . This event is necessarily a receive event (" z k is the corresponding send event). Note that ' C x i denote the same interval. S[j] ! D x holds. So from the definition of dependency vectors, ' S[j] i . However, either (' C x i . This contradicts the hypothesis that f' S[1] p g is a set of concurrent intervals. INRIA Efficient Distributed Detection of Conjunctions of Local Predicates 23 From the assumption, C k S. Therefore, C x log holds. So from the definition of dependency vec- tors, ' S[j] . Because of the max operation, D log so ' S[j] . Then the proof is the same as in Case 1. 2. C x i is either a send or a receive event. From the algorithm, it is clear that this case occurs only if none of the intervals that occurred between ' C (including this) and ' D x verifies This contradicts the fact that ' S[i] verifies L i since C k [i]); D log i is a receive event and " z k is the corresponding send event. From the assumption, C and C z k [i] S[i]. Assume that C k [i]). Then C S[i] and therefore C x From the algorithm, it is clear that this case occurs only if none of the intervals that occurred between ' C (including this) and ' D log verifies This contradicts the fact that ' S[i] verifies L i since C (= D log [i]).The following lemma proves that the algorithm keeps making progress if it has not encountered a solution. Lemma 4: Suppose process P i has executed the algorithm at the x th communication event " x is in the interval ' x that the set f' C x does not verify \Phi. Then, there exists " y j such that C x . Proof: There are two reasons for f' C x p g not verifying \Phi: 1. ' C x i does not verify RR M. Hurfin, M. Mizuno, M. Raynal, M. Singhal In this case, at " x could not find an interval that verifies L i , and therefore, set to x (i.e., the value of D x [i]). At the next communication event updates at least the i th entry of C i by setting C x+1 i [i] to D x+1 2. There exists at least one process P k (1 k p) such that ' C x k does not verify In this case, P k will eventually advance C k [k] to a value greater than C x (refer to Case 1). This new value computed when event " z occurs will propagate to other processes. Extra messages eventually exchanged at the end of the computation guarantee that there will eventually be a communication event j at a process P j such that that " z and C x .Finally, the following theorem shows that \Phi is verified in a computation iff the algorithm detects a solution. Theorem: [1] If there exists an interval ' x i on a P i such that, during this interval, holds for all k (1 k p), then f' C x verifies \Phi. Conversely, if f' C x verifies \Phi for an event " x on some processor there exists a communication event " y j such that for all k (1 k p), holds. Proof: [1] Proof is by contradiction. Suppose ' C x k does not verify L k for some k. We show that as long as C i [k] is not changed, B i [k] is false. There are two cases to consider: 1. i does not verify L i is updated to D i [i] when communication events " x occurs. If " x i is a receive i [i] is set to false at the same time and remains false during the interval necessarily empty for the entire duration of interval ' C x modified only when event " x+1 occurs. i is a send event, the value of B i [i] is unchanged since the last receive event or since the beginning of the computation if no receive event occurs at process before the send event " x i . In both cases, Log i remains empty during this entire period and B i [i] remains false. INRIA Efficient Distributed Detection of Conjunctions of Local Predicates 25 2. i 6= k: updated C i [k] to C x there existed a process P j that advanced to C x [k], and the value was propagated to P i . P j must have set B j [k] to false and this information must have propagated to P i . This value was propagated to P i without going through P k (else P k would have been advanced to a value greater than C x i [k]). It is easy to see that B x i [k] is false: Since k is the only process that can change B k [k] to true, P i will never see B true together with C i [2] Assume that f' C x verifies \Phi. Message exchanges guarantee that there will eventually be a communication event " y j such that for all k, 1 k p, . When process P k is in interval ' C x [k] is set to true. From Lemma 3, once a process P h sets C h [k] to C x does not change this value in the future. This implies that all the processes P h that are on the path of the message exchange from " C x k to " y to C x i [k] and B h [k] to true - none of such processes P h sets B h [k] to false by advancing C h [k] beyond C x information is eventually propagated to P j , and so B y holds for all k, 1 k p.4 The Second Algorithm In the second algorithm, every process always keeps track of a set of intervals for all the processes such that each of the intervals verifies its local predicate. For each such set, the process checks whether all the intervals in the set are concurrent. 4.1 Overview of the Algorithm 4.1.1 Verified Intervals In this algorithm, only the intervals that verify their associated local predicates are of interest. We call such intervals verified intervals. A new i is used to identify the x th verified interval of process P i . Thus, for i , there exists exactly one ' y i that denotes the same interval. RR 26 M. Hurfin, M. Mizuno, M. Raynal, M. Singhal \Omega 0\Omega 1\Omega 0\Omega 1" 0 Figure 5: The corresponding set of verified intervals. 4.1.2 Dependency Vectors As in the first algorithm, each process P i (1 i n) maintains a dependency vector track of the identity of the next verified interval that P i will encounter. Even though P i does not know which interval it will be, it knows that the next verified interval will be denoted When has encountered the x th verified interval denoted, by \Omega detailed description of Log i is given later.) At this moment, D i increments D i [i] by one to x to look for the next verified . Note that the existence of the verified i , is not guaranteed at this moment. The local predicate may not be satisfied anymore during the computation. In the first algorithm, vector D remains the same for the entire duration of an interval. In the second algorithm, on the contrary, vector D may change once during an interval if this interval is a verified interval. In order to capture causal relation among verified intervals at different processes, the following protocol is executed on D i by a process P i (1 i n): 1. Initially, all the components of vector D i are zero. 2. When P i executes a send event, it sends D i along with the message. 3. When P i executes a receive event, where a message contains Dm , D i is advanced by setting D i [k] := max(D i [k]; Dm [k]) for 1 k p. INRIA Efficient Distributed Detection of Conjunctions of Local Predicates 27 Clearly, the following properties hold: 1. D i [i] represents the number of verified intervals at P i whose existence can be confirmed by P i (i.e., P i has already passed through its (D i [i]) st verified interval). 2. D i [j](j 6= i) represents the number of verified intervals that have occurred at process P j and causally precede the current interval of P i . 3. The set of verified intervals p g is not necessarily consistent. Yet, by definition, if exists, it satisfies the associated local predicate. 4. None of the j such that y ! D i [j] (i.e., verified intervals at P j that causally precede\Omega D i [j] can be concurrent . Therefore, none of them can form a set of intervals i that verifies \Phi. 4.1.3 Logs Each process P i maintains a log, denoted by Log i , in the same manner as in the first algorithm. When P i verifies its local predicate L i , it enqueues the current D i before incrementing D i [i] by one. When Log i is not empty, notation D log i is used to denote the value of the vector clock at the head of Log i . Necessarily, when the log is not empty, the existence of \Omega D log i has already been confirmed by P i . 4.1.4 Cuts Like the first algorithm, each process P i maintains an integer vector C i and a boolean vector . The meaning of C i is similar to that of the first algorithm; that is the next possible interval at P j that may be in a solution and none of the verified intervals that precedes\Omega C i [j] j can be in a solution. Therefore, each process P i may discard any values D i in Log i such that D i The meaning of B i is also similar to that in the Algorithm 1. Each P i maintains in such a way that B i [j] is true if process P i is certain that verified interval j has been confirmed by P j . Furthermore, process P i is certain that none of the verified causally precede\Omega C i [j] . Thus, if P i is not certain RR 28 M. Hurfin, M. Mizuno, M. Raynal, M. Singhal whether the verified j has already been confirmed by P j , B i [j] is set to false. 4.2 Descriptions of the Algorithm A formal description of the algorithm is given in Section 4.3. As the first algorithm, the second algorithm consists of three procedures that are executed at a process P i . Again, we assume that the set of intervals f' l 1 +1 p g is a solution. Extra messages are exchanged after the computation ends only if the first solution has not been discovered yet. When the local predicate L i becomes true: Let oe y i be a local state that satisfies the local predicate. P i has entered the verified i that includes state oe y i . It logs D i in Log i if D i has not been logged yet since the beginning of this interval. In order to indicate that the vector clock for this interval has already been logged, it sets variable not logged yet i to false. The counter of verified interval D i [i] is incremented by one to reflect that the current interval is a verified interval. Furthermore, if the current verified interval is also the oldest verified interval of discarded (denoted [i] is set to true to confirm the existence When P i sends a message: Since it marks the beginning of a new interval, P i resets variable not logged yet i to true and then it sends the message along with C When receives a message from P j that contains D Since a new interval begins, it resets variable not logged yet i to true. As in the first algorithm, none of the intervals at any process P k that precede\Omega C i [k] can form a set of concurrent intervals that verifies \Phi. Thus, C i is advanced to the componentwise maximum of C i and C j . At this moment, B i is also updated. "B i [k] is true" means that P i is certain that the existence k has been confirmed. Thus, if C i [k] and at least one of B i [k] and B j [k] is true, B i [k] is set to true. deletes all entries from Log i that precede\Omega C i [i] i since all those verified intervals are no more potential components of a solution. After this operation, there are two cases to consider: INRIA Efficient Distributed Detection of Conjunctions of Local Predicates 29 ffl Case 1 Log i becomes empty: In this case, none of the verified intervals at P i up to this moment forms a set of concurrent verified intervals. The algorithm needs to consider only verified intervals that will occur in the future. If such intervals exist,\Omega D i [i] i will be the first one. Since all of the verified k such that y ! D i [k] at other processes P k causally precede\Omega D i [i] none of such intervals can be in a solution. Thus, cut C i is advanced to D i . When process P i executes the receive action, it is not certain whether P k (1 p) has encountered the verified k . Therefore, all components of vector B i are set to false. contains at least one logged interval: Let the oldest of such logged entry be D log . From the properties of vector D and the definition of a cut, all of the verified intervals at any process P k preceding\Omega C i [k] or\Omega D log causally precede\Omega D log none of such intervals can be in a solution. Thus, C i is advanced to the componentwise maximum of D log Similar to Case 1, if the value C i [k] is modified (i.e., it takes its value from log not certain whether P k 's will encounter the verified interval ' D log k . Thus, B i [k] is set to false. If C i [k] remains unchanged, B i [k] also remains unchanged to follow other processes' decision. Furthermore, B i [i] is set to true since P i has confirmed the existence of\Omega D log At this moment, with the new information (C may be able to detect a solution. Thus, P i checks whether B i [k] is true for all k. If so, this indicates that all the verified intervals in set have been confirmed and concurrent with one another since no process has detected causal relations between any pair of intervals in the set. 4.3 Formal Description of the Algorithm Initialization procedure executed by any process P i if (i p) then Create(Log i ); not yet logged i := true; endif RR M. Hurfin, M. Mizuno, M. Raynal, M. Singhal Procedure A executed by process when the local predicate L i becomes true if (not yet logged i ) then logged i := false; endif Procedure B executed by any process P i when it sends a message if (i p) then not yet logged i := true; endif Append vectors D i , C i , and B i to the message; Send the message; Procedure C executed by any process P i when it receives a message from P j Extract vectors D j , C j , and B j from the message; if (i p) then not yet logged i := true; while ((not (Empty(Log i ))) and (Head(Log i do /* Delete all those logged intervals that from the current /* knowledge do not lie on a solution. /* Construct a solution that passes through the next local verified interval. else /* Construct a solution that passes through the logged verified interval. endif endif Deliver the message; INRIA Efficient Distributed Detection of Conjunctions of Local Predicates 31 4.4 Discussion 4.4.1 An Example To help the readers understand the algorithm, in Figure 5, we illustrate the operation of the second algorithm for a computation similar to that one used in Figure 4. In Figure 5, the contents of vector D and C that are indicated next to a process interval in square brackets, are the values of the vectors just after evaluation of the last local state of the interval (i.e., just before execution of the communication event). F FA@ FA@ F FA@ FA@ FA@ F FA@ F FA@ F TA@ TA@ Figure An Example to Illustrate Algorithm 2. 4.4.2 Difference Between Both Approaches The second algorithm can be considered an optimization of the first one. Interval counters D and C evolve more slowly in the second algorithm and updates of both vectors occur less often. For example, vector C is not modified on a send action. Each algorithm finds the first solution in a different way. In the first algorithm, each interval of the solution is located via a number of communication events that occur before the process encounters this interval. In the second algorithm, the delivered information is the number of validated interval that precede the solution. The difference between both algorithms is much more on the semantics and the properties of the control variables rather than on the way they are updated. For example, update of vector C is made in a similar way in both algorithms. Yet, each RR M. Hurfin, M. Mizuno, M. Raynal, M. Singhal component is managed as a counter of interval (in the first algorithm) or as a counter of verified interval (in the second algorithm). Both algorithms employ complementary approaches to find the first solution. In the first algorithm, the corresponding set of interval is always concurrent (i.e., it satisfies the first criterion of the solution). In the second algorithm, the elements of the set are always verified intervals (i.e., the set satisfies the second criterion of the solution). A correctness proof of the second algorithm is similar to the proof of the first algorithm. However, Lemma 1 and Lemma 2 become irrelevant in the second algo- rithm. Instead, the following lemma becomes useful: Lemma 5: At any time during execution of P i , if B i [j] (1 j p) is true then Proof: Process P j has necessarily updated B j [j] to true to account for the fact that j has been encountered and is the oldest interval not discarded yet. At the same time, P j used the value at the head of the Log, D log j , to update all the other components C j [k] to a value greater or equal to D log [k] in order to invalidate all the verified intervals that are in the causal past . Therefore, as the cut C never decreases (due to the merge operation made at the beginning of each receive share with P j the same vision of the values of B[j] and C[j] and at the same time keep an older value for some component C i [k]. 2 The rest of the proof is the same as that of the first algorithm with the definition of interval appropriately modified. 5 A Comparison with Existing Work Previous work in detecting conjunctive form global predicates has been mainly by Garg and Waldecker [6] and Garg and Chase [7]. Garg-Waldecker algorithm is centralized [6], where each process reports all its local states satisfying its local predicate to a checker process. The checker process gathers this information, builds only those global states that satisfy the global predicate, and checks if a constructed global state is consistent. This algorithm has a message, storage, and computation complexities of O(Mp 2 ) where M is the number of messages sent by any process and p is the number of processes over which the global predicate is defined. INRIA Efficient Distributed Detection of Conjunctions of Local Predicates 33 In [7], Garg and Chase present two distributed algorithms for detection of conjunctive form predicates. In these algorithm, all processes participate in the global predicate detection on equal basis. The first distributed algorithm requires vector clocks and employs a token that carries information about the latest global consistent cut such that the local predicates hold at all the respective local states. The message, storage, and computation complexities of this algorithm is the same as of Garg- Waldecker [6] algorithm, namely, O(Mp 2 ). However, the worst case message, sto- rage, and computation complexities for a process in this algorithm is O(Mp); thus, the distribution of work is more equitable than in the centralized algorithm. The second distributed algorithm does not use vector clocks and uses direct dependencies instead. The message, storage, and computation complexities of this algorithm are O(Mn) and the worst case message, storage, and computation complexities for a process in this algorithm are O(M ); thus, this algorithm is desirable when p 2 is greater than n. The proposed predicate detection algorithm does not cause transfer of any additional messages (except in the end provided the predicate is not detected when the computation terminates). The control information needed for predicate detection is piggybacked on computation messages. On the contrary, the distributed algorithms of Garg and Chase may require exchange of as many as Mp and Mn control mes- sages, respectively. Although the worst case volume of control information exchanged is identical, namely, O(Mp 2 ), in the first Garg and Chase algorithm and in the proposed algorithm, the latter results in no or few additional message exchanges. A study by Lazowska et al. [11] showed that message send and receive overhead can be considerable (due to context switching and execution of multiple communication protocol layers) and it is desirable to send few bigger from performance point of view. 6 Concluding Remarks Global predicate detection is a fundamental problem in the design, coding, testing and debugging, and implementation of distributed programs. In addition, it finds applications in many other domains in distributed systems such as deadlock detection and termination detection. This paper presented two efficient distributed algorithms to detect conjunctive form global predicates in distributed systems. The algorithms detect the first consistent global state that satisfies the predicate and work even if the predicate is uns- RR 34 M. Hurfin, M. Mizuno, M. Raynal, M. Singhal table. The algorithms are based on complementary approaches and the second algorithm can be considered an optimization of the first one, where the vectors D and C increase at a lower rate. We proved the correctness of the algorithms. The algorithms are distributed because the predicate detection efforts as well as the necessary information are equally distributed among the processes. Unlike previous algorithms to detect conjunctive form global predicates, the algorithms do not require transfer of any additional messages during the normal computation; instead, they piggyback the control information on computation messages. Additional messages are exchanged only if the predicate remains undetected when the computation terminates. --R Consistent Detection of Global Predicates. Detection of Unstable Predicates in Distributed Pro- grams Detection of Weak Unstable Predicates in Distributed Programs. Distributed Algorithms for Detecting Conjunctive Predi- cates Global Events and Global Breakpoints in Distributed Sys- tems Detecting Atomic Sequences of Predicates in Distributed Computations. Linear Space Algorithm for On-line Detection of Global States File Access Performance of Diskless Workstations Global conditions in debugging distributed programs. Virtual Time and Global States of Distributed Systems. Breakpoints and Halting in Distributed Programs. A Way to Capture Causality in Distributed Systems. Faster Possibility Detection by Combining Two Ap- proaches "le de Nancy-Brabois, Campus scientifique, 615 rue du Jardin Botanique, BP 101, 54600 VILLERS LE S NANCY Unite de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unite de recherche INRIA Rho" --TR --CTR Punit Chandra , Ajay D. Kshemkalyani, Distributed algorithm to detect strong conjunctive predicates, Information Processing Letters, v.87 n.5, p.243-249, 15 September Loon-Been Chen , I-Chen Wu, An Efficient Distributed Online Algorithm to Detect Strong Conjunctive Predicates, IEEE Transactions on Software Engineering, v.28 n.11, p.1077-1084, November 2002 Ajay D. Kshemkalyani, A Fine-Grained Modality Classification for Global Predicates, IEEE Transactions on Parallel and Distributed Systems, v.14 n.8, p.807-816, August Guy Dumais , Hon F. Li, Distributed Predicate Detection in Series-Parallel Systems, IEEE Transactions on Parallel and Distributed Systems, v.13 n.4, p.373-387, April 2002 Emmanuelle Anceaume , Jean-Michel Hlary , Michel Raynal, Tracking immediate predecessors in distributed computations, Proceedings of the fourteenth annual ACM symposium on Parallel algorithms and architectures, August 10-13, 2002, Winnipeg, Manitoba, Canada Neeraj Mittal , Vijay K. Garg, Techniques and applications of computation slicing, Distributed Computing, v.17 n.3, p.251-277, March 2005 Scott D. Stoller, Detecting global predicates in distributed systems with clocks, Distributed Computing, v.13 n.2, p.85-98, April 2000 Punit Chandra , Ajay D. Kshemkalyani, Causality-Based Predicate Detection across Space and Time, IEEE Transactions on Computers, v.54 n.11, p.1438-1453, November 2005

on-the-fly global predicate detection;distributed systems

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Safe metaclass programming.

In a system where classes are treated as first class objects, classes are defined as instances of other classes called metaclasses. An important benefit of using metaclasses is the ability to assign properties to classes (e.g. being abstract, being final, tracing particular messages, supporting multiple inheritance), independently from the base-level code. However, when both inheritance and instantiation are explicitly and simultaneously involved, communication between classes and their instances raises the metaclass compatibility issue. Some languages (such as SMALLTALK) address this issue but do not easily allow the assignment of specific properties to classes. In contrast, other languages (such as CLOS) allow the assignment of specific properties to classes but do not tackle the compatibility issue well.In this paper, we describe a new model of metalevel organization, called the compatibility model, which overcomes this difficulty. It allows safe metaclass programming since it makes it possible to assign specific properties to classes while ensuring metaclass compatibility. Therefore, we can take advantage of the expressive power of metaclasses to build reliable software. We extend this compatibility model in order to enable safe reuse and composition of class specific properties. This extension is implemented in NEOCLASSTALK, a fully reflective SMALLTALK.

Introduction It has been shown that programming with metaclasses is of great benefit [KAJ interesting use of metaclasses is the assignment of specific properties to classes. For example, a class can be abstract, have a unique instance, trace messages received by its instances, define pre-post conditions on its methods, forbid redefinition of some particular methods. These properties can be implemented using metaclasses, allowing thereby the customization of the classes behavior [LC96]. From an architectural point of view, using metaclasses organizes applications into abstraction levels. Each level describes and controls the level immediately below to which it is causally connected [Mae87]. Reified classes communicate with other objects including their own instances. Thus, classes can send messages to their instances and instances can send messages to their classes. Such message sending is named inter-level communication [MMC95]. However, careless inheritance at one level may break inter-level communication resulting in an issue called the compatibility issue [BSLR96]. We have identified two symmetrical kinds of compatibility issues. The first one is the upward compatibility issue, which was named metaclass compatibility by Nicolas Graube [Gra89], and the second one is the downward compatibility issue. Both kinds of compatibility issues are important impediments to metaclass programming that one should always be aware of. Currently, none of the existing languages dealing with metaclasses allow the assignment of specific properties to classes while ensuring compatibility. Clos [KdRB91] allows one to assign any property to classes, but it does not ensure compatibility. On the other hand, both SOM [SOM93] and Smalltalk [GR83] address the compatibility issue but they introduce a class property propagation problem. Indeed, a property assigned to a class is automatically propagated to its subclasses. Therefore, in SOM and Small- talk, a class cannot have a specific property. For example, when assigning the abstractness property to a given Smalltalk class, subclasses become abstract too [BC89]. It follows that users face a dilemma: using a language that allows the assignment of specific class properties without ensuring compatibility, or using a language that ensures compatibility but suffers from the class property propagation problem. In this paper, we present a model - the compatibility model - which allows safe metaclass program- ming, i.e. it makes it possible to assign specific properties to classes without compromising compatibility. In addition to ensuring compatibility, the compatibility model avoids class property propagation: a class can be assigned specific properties without any side-effect on its subclasses. We implemented the compatibility model in NeoClasstalk, a Smalltalk extension which introduces many features including explicit metaclasses [Riv96]. Our experiments [Led98][Riv97] showed that the compatibility model allows programmers to fully take advantage of the expressive power of metaclasses. This effort has resulted (i) in a tool that permits a programmer unfamiliar with metaclasses to transparently deal with class specific properties, and (ii) in an approach allowing reuse and composition of class properties. This paper is organized as follows. Section 2 presents the compatibility issue. We give some examples to show its significance. Section 3 shows how existing programming languages address the compatibility issue, and how they deal with the property propagation problem. Section 4 describes our solution and illustrates it with an example. In section 5, we deal with reuse and composition of class specific properties within the compatibility model. Then, we sketch out the use of the compatibility model for both base-level and meta-level programmers. The last section contains a concluding summary. Inter-level communication and compatibility We define inter-level communication as any message sending between classes and their instances (see Figure 1). Indeed, class objects can interact with other objects by sending and receiving messages. In particular, an instance can send a message to its class and a class can send a message to some of its instances. We use Smalltalk as an example to illustrate this issue 1 . Message Sending A Class level Instance level Figure 1: Inter-level communication Two methods allow inter-level communication in Smalltalk: new and class. When one of them is used, the involved objects belong to different levels of abstraction ffl An object receiving the class message returns its class. Then, the class method makes it possible to go one level up. The following two instance methods - excerpted from Visual Works Smalltalk include message sending to the receiver's class. message name is sent to the class: ObjectAEprintOn: aStream title := self class name. message daysInYear: is sent to the class: DateAEdaysInYear "Answer the number of days in the year represented by the receiver." " self class daysInYear: self year ffl A class receiving the new message returns a new instance. Therefore, the new method makes it possible to go one level down. The following two class methods include message sending to the newly created instances. message at:put: is sent to a new instance: ArrayedCollection anObject newCollection j newCollection := self new: 1. newCollection at: 1 put: anObject. "newCollection message on: is sent to a new instance: Browser classAEopenOn: anOrganizer self openOn: (self new on: anOrganizer) with- TextState: nil Thus, inter-level communication in Smalltalk is materialized by sending the messages new and class. Other languages where classes are reified (such as Clos and SOM) also allow similar message sending. Since these inter-level communication messages are embedded in methods, they are inherited whenever methods are inherited. Ensuring compatibility means making sure that these methods will not induce any failure in subclasses, i.e. all sent messages will always be understood. We have identified two kinds of compatibility: upward compatibility 3 and downward compatibility. We use the Smalltalk syntax and terminology throughout this paper. measures we made over a Visual Works Smalltalk image show that inter-level communication is very frequent. 25% of classes include instance methods referencing the class and 24% of metaclasses define methods referencing an instance. 3 Nicolas Graube named this issue metaclass compatibility [Gra89]. <i-foo> <c-bar> inheritance instantiation self class c-bar Figure 2: Compatibility need to be ensured at a higher level 2.1 Upward compatibility Suppose A implements a method i-foo that sends the c-bar message to the class of the receiver (see Figure 2). B is a subclass of A. When i-foo is sent to an instance of B, the B class receives the c-bar message. In order to avoid any failure, B should understand the c-bar message (i.e. MetaB should implement or inherit a method c-bar). Definition of upward compatibility: Let B be a subclass of the class A, MetaB the metaclass of B, and MetaA the metaclass of A. Upward compatibility is ensured for MetaB and MetaA iff: every possible message that does not lead to an error for any instance of A, will not lead to an error for any instance of B. 2.2 Downward compatibility Suppose implements a method c-foo that sends the i-bar message to a newly created instance (see Figure 3). MetaB is created as a subclass of MetaA. When c-foo is sent to B (an instance of MetaB), B will create an instance which will receive the i-bar message. In order to avoid any failure, instances of B should understand the i-bar message (i.e. B should implement or inherit the i-bar method). <i-bar> <c-foo> A inheritance instantiation self new i-bar Figure 3: Compatibility need to be ensured at a lower level Definition of downward compatibility: Let MetaB be a subclass of the metaclass MetaA. Downward compatibility is ensured for two classes B instance of MetaB and A instance of every possible message that does not lead to an error for A, will not lead to an error for B. 3 Existing models We will now show why none of the known models allow the assignment of specific properties to classes while ensuring compatibility. 3.1 When (re)defining a class in Clos, the validate-superclass generic function is called, before the direct are stored [KdRB91]. As a default, validate-superclass returns true if the metaclass of the new class is the same as the metaclass of the superclass 4 , i.e. classes and their subclasses must have the same metaclass. Therefore, incompatibilities are avoided but metaclass programming is very constrained. inheritance instantiation B A Figure 4: By default in Clos, subclasses must share the same metaclass as their superclass Figure 4 shows a hierarchy of two classes that illustrates the Clos default compatibility management policy. Since class B inherits from A, B and A must have the same metaclass. In order to allow the definition of classes with different behaviors, programmers usually redefine the validate-superclass method to make it always return true. Thus, Clos programmers can have total freedom to use a specific metaclass for each class. So, they can assign specific properties to classes, but the trade-off is that they need to be always aware of the compatibility issue. 3.2 SOM SOM is an IBM CORBA compliant product which is based on a metaclass architecture [DF94b]. The SOM kernel follows the ObjVlisp model [Coi87]. SOM metaclasses are explicit and can have many instances. Therefore, users have complete freedom to organize their metaclass hierarchies. 3.2.1 Compatibility issue in SOM SOM encourages the definition and the use of explicit metaclasses by introducing a unique concept named derived metaclasses which deals with the upward compatibility issue [DF94a]. At compile-time, SOM automatically determines an appropriate metaclass that ensures upward compatibility. If needed, SOM automatically creates a new metaclass named a derived metaclass to ensure upward compatibility. instantiation inheritance self class c-bar A>>i-foo class: parent: A; metaclass: MetaB; A Derived <c-bar> <i-foo> Figure 5: SOM ensures upward compatibility using derived metaclasses In fact, it also returns true if the superclass is the class named t, or if the metaclass of one argument is standard-class and the metaclass of the other is funcallable-standard-class. Suppose that we want to create a class B, instance of MetaB and subclass of A. SOM will detect an upward compatibility problem, since MetaB does not inherit from the metaclass of A (MetaA). Therefore, automatically creates a derived metaclass (Derived), using multiple inheritance in order to support all necessary class methods and variables 5 . Figure 5 shows the resulting construction. When an instance of B receives i-foo, it goes one level higher and sends c-bar to the B class. B understands the c-bar message since its metaclass (i.e. Derived) is a derived metaclass which inherits from both MetaB and MetaA. inheritance instantiation self new i-bar SOMObject A <i-bar> <c-foo> SOMClass Figure example SOM does not provide any policy or mechanism to handle downward compatibility. Suppose that MetaB is created as a subclass of MetaA (see Figure 6). The c-foo method which is inherited by MetaB sends the i-bar message to a new instance. When the B class receives the c-foo message, a run-time error will occur because its instances do not understand the i-bar message. 3.2.2 Class property propagation in SOM SOM does not allow the assignment of a property to a given class, without making its subclasses be assigned the same property. We name this defect the class property propagation problem. In the following example, we illustrate how derived metaclasses implicitly cause undesirable propagation of class properties. inheritance instantiation class: parent: A; metaclass: SoleInstance; Derived SoleInstance Released A Figure 7: Class property propagation in SOM Suppose that the A class of Figure 7 is a released class, i.e. it should not be modified any more. This property is useful in multi-programmer development environments for versionning purposes. In order to avoid any change, A is an instance of the Released metaclass. Let B a class that has a unique instance: B is an instance of the SoleInstance metaclass. But as B is a subclass of A, SOM creates B as instance of an automatically created derived metaclass which inherits from both SoleInstance and Released. Thus, as soon as B is created, it is automatically "locked" and acts like a released class. So, we cannot define any new method on it! 5 The semantics of derived metaclasses guarantees that the declared metaclass takes precedence in the resolution of multiple inheritance ambiguities (i.e. MetaB before MetaA). Besides, it ensures the instance variables of the class are correctly initialized by the use of a complex mechanism. 3.3 Smalltalk-80 In Smalltalk, metaclasses are partially hidden and automatically created by the system. Each metaclass is non-sharable and strongly coupled with its sole instance. So, the metaclass hierarchy is parallel to the class hierarchy and is implicitly generated when classes are created. 3.3.1 Compatibility issue in Smalltalk-80 Using parallel inheritance hierarchies, the Smalltalk model ensures both upward and downward com- patibility. Indeed, any code dealing with new or class methods, is inherited and works properly. instantiation inheritance class <c-bar> <c-foo> <i-foo> A class>>c-foo self new i-bar self class c-bar A>>i-foo A class <i-bar> Figure 8: Smalltalk ensures both upward and downward compatibilities When one creates the B class, a subclass of A (see Figure 8), Smalltalk automatically generates the metaclass of B ("B class" 6 ), as a subclass of "A class", the metaclass of A. Suppose A implements a method i-foo that sends c-bar to the class of the receiver. If i-foo is sent to an instance of B, the B class receives the c-bar message. Thanks to the parallel hierarchies, the B class understands the c-bar message, and upward compatibility is ensured. In a similar manner, downward compatibility is ensured thanks to the parallel hierarchy. 3.3.2 Class property propagation in Smalltalk-80 Since metaclasses are automatically and implicitly managed by the system, Smalltalk drastically reduces the opportunity to change class behaviors, making metaclass programming "anecdotal". As with SOM, Smalltalk does not allow the assignment of a property to a class without propagating it to its subclasses. inheritance A class A class A class>>new self error: 'I am Abstract' instantiation Figure 9: Class property propagation in Smalltalk In Figure 9, the A class is abstract since its subclasses must implement some methods to complete the instance behavior. B is a concrete class as it implements the whole set of these methods. Suppose that we want to enforce the property of abstractness of A. In order to forbid instantiating A, we define the class method A classAEnew which raises an error. Unfortunately, "B class" inherits the method new from "A class". As a result, attempting to create an instance of B raises an error 7 6 The name of a Smalltalk metaclass is the name of its unique instance postfixed by the word 'class'. 7 This example is deliberately simple, and one could avoid this problem by redefining new in "B class". But, this solution 4 The compatibility model Among the previous models, only the Smalltalk one with its parallel hierarchies ensures full compati- bility. However, it does not allow the assignment of specific properties to classes. On the other hand, only the Clos model allows the assignment of specific properties to classes. Unfortunately, it does not ensure compatibility. We believe that these two goals can both be achieved by a new model which makes a clear separation between compatibility and class specific properties. inheritance instantiation Abstract class>>new self error: 'I am Abstract' class A A class Abstract class Figure 10: Avoiding the propagation of abstractness We illustrate this idea of separation of concerns by refactoring the example of Figure 9. We create a new metaclass named "Abstract class" as a subclass of "A class" (see Figure 10). The A class is redefined as an instance of this new metaclass. As "Abstract + A class" redefines the new method to raise an error, A cannot have any instance. However, since "B class" is not a subclass of "Abstract the B class remains concrete. The generalization of this scheme is our solution, named the compatibility model. In the remainder of this paper, names of metaclasses defining some class property are denoted with the concatenation of the property name, the + symbol and the superclass name. For example, "Abstract class" is a subclass of "A class" that defines the property of abstractness named Abstract. 4.1 Description of the compatibility model The compatibility model extends the Smalltalk model by separating two concerns: compatibility and specific class properties. A metaclass hierarchy parallel to the class hierarchy ensures both upward and downward compatibility like in Smalltalk. An extra metaclass "layer" is introduced in order to locally assign any property to classes. Classes are instances of metaclasses belonging to this layer. So, the compatibility model is based on two "layers" of metaclasses, each one addressing a unique concern: Compatibility concern: This issue is addressed by the metaclasses organized in a hierarchy parallel to the class hierarchy. We name such metaclasses: compatibility metaclasses. They define all the behavior that must be propagated to all (sub)classes. All class methods which send messages to instances should be defined in these metaclasses. Besides, all messages sent to classes by their instances should be defined in these metaclasses too. Specific class properties concern: This issue is addressed by metaclasses that define the class specific properties. We name such metaclasses: property metaclasses. A class with a specific property is instance of a property metaclass which inherits from the corresponding compatibility metaclass. The property metaclass is not joined to other property metaclasses, since it defines a property specific to the class. is a kind of inheritance anomaly [MY93] that increases maintenance costs. AClass BClass metaclasses Compatibility metaclasses Property <i-foo> A <i-bar> A>>i-foo self new i-bar AClass>>c-foo self class c-bar inheritance instantiation <c-foo> BProperty <c-bar> AProperty BClass Figure 11: The compatibility model Figure shows 8 the compatibility model applied to a hierarchy consisting of two classes: A and B. They are respectively instances of the "AProperty metaclasses. "AProperty properties specific to class A, while "BProperty properties specific to class B. As "AProperty + AClass" and "BProperty + BClass" are not joined by any link, class property propagation does not occur. Thus, A and B can have distinct properties. are subclasses of the AClass and BClass meta- classes, both upward and downward compatibility are guaranteed. Suppose that A defines two instance methods i-foo and i-bar. The i-foo method sends the c-bar message to the class of the receiver. The i-bar method is sent to a new instance by the c-foo method. Because the AClass and BClass metaclass hierarchy is parallel to the A and B class hierarchy, inter-level communication failure is avoided. 4.2 Example: Refactoring the Smalltalk-80 Boolean hierarchy The Boolean hierarchy of Smalltalk 9 is depicted in Figure 12. Boolean is an abstract class which defines a protocol shared by True and False. True and False are concrete classes that cannot have more than one instance. These properties (i.e. abstractness and having a sole instance) are implicit in Smalltalk. Using the compatibility model, we refactor the Boolean hierarchy to emphasize them. instantiation inheritance Boolean class False class Boolean True class True False Figure 12: The Boolean hierarchy of Smalltalk We first create "Boolean class", which is a compatibility metaclass. The second step consists of creating the property metaclass "Abstract Boolean class", which enforces the Boolean class to be abstract. Finally, we build the Boolean class by instantiating the "Abstract metaclass. To refactor the False class, we first create the "False class" metaclass, as a subclass of "Boolean class" to ensure the compatibility. The second step consists of creating the property metaclass "SoleInstance Compatibility metaclasses are surrounded with a dashed line and property metaclasses are drawn inside a grey shape. 9 We prefer this academic example to emphasize our ideas rather than a more complex example which should require a more detailed presentation. False class", which enforces the False class to have at most one instance. At last, we create the False class by instantiating the "SoleInstance + False class" metaclass. The True class is refactored in the same way. The result of rebuilding the whole hierarchy of Boolean is shown in Figure 13. Abstract Boolean class SoleInstance False class instantiation inheritance False class Boolean Boolean class False True class SoleInstance class True Figure 13: The Boolean hierarchy after refactoring 5 Reuse and composition within the compatibility model We have experimented the compatibility model in NeoClasstalk 10 [Riv97], a fully reflective Smalltalk. We quickly faced the need of class property reuse and composition. Indeed, unrelated classes belonging to different hierarchies can have the same properties, and a given class can have many properties. In the previous section, both the True class and the False class have the same property: having a unique instance. Besides, we assigned only one property to each class of the Boolean hierarchy. But, a class need to be assigned many properties. For example, the False class must not only have a unique instance, but it also should not be subclassed (such a class is final in Java terminology). So, we have to reuse and compose these class properties with respect to our compatibility model. In this section, we propose an extension of our compatibility model that deals with reuse and composition of class properties. Any language where classes are treated as regular objects may integrate our extended compatibility model. NeoClasstalk has been used as a first experimentation platform. 5.1 Reuse of class properties In Smalltalk, since metaclasses behave in a different way than classes, they are defined as instances of a particular class, a meta-metaclass, called Metaclass. Metaclass defines the behavior of all metaclasses in Smalltalk. For example, the name of a metaclass is the name of its sole instance postfixed by the word 'class'. MetaclassAEname "thisClass name, ' class' We take advantage of this concept of meta-metaclasses to reuse class properties. Since metaclasses implementing different properties have different behaviors, we need one meta-metaclass for each class property. Property metaclasses defining the same class property are instances of the same meta-metaclass. When a property metaclass is created, the meta-metaclass initializes it with the definition of the corresponding class property. Thus, the code (instance variables, methods, . ) corresponding to the definition NeoClasstalk and all related papers can be downloaded from http://www.emn.fr/cs/object/tools/neoclasstalk/neoclasstalk.html of the class property, is automatically generated. Reuse is achieved by creating property metaclasses defining the same class property as instances of the same meta-metaclass, i.e. they are initialized with the same class property definition (an example of such an initialization is given in section 5.4.2). The root of the meta-metaclass hierarchy named PropertyMetaclass describes the default structure and behavior of property metaclasses. For example, the name of a property metaclass is built from the property name and the superclass name: PropertyMetaclassAEname "self class name, '+', self superclass name In the refactored Boolean hierarchy of section 4.2, both "SoleInstance False class" and "SoleInstance True class" define the property of having a unique instance. Reuse is achieved by defining both "SoleInstance False class" and "SoleInstance True class" as instances of SoleInstance, a subclass of PropertyMetaclass (see Figure 14). SoleInstance False class Abstract Boolean class metaclass level class level meta-metaclass level False class Boolean instantiation inheritance Boolean class False True class SoleInstance class Abstract SoleInstance True PropertyMetaclass Figure 14: Reuse properties in the Boolean hierarchy 5.2 Composition of class properties Since a given class can have many properties, the model must support the composition of class proper- ties. We chose to use many property metaclasses organized in a single inheritance hierarchy, where each metaclass implements one specific class property. To illustrate this idea, we modify the instantiation link for the False class (see Figure 15). We define two property metaclasses, one for each property. The first property metaclass is "SoleInstance which inherits from the compatibility metaclass "False class". The second one is "Final False class", which is the class of False. It is defined as a subclass of "SoleInstance False class". The resulting scheme respects the compatibility model: it allows the assignment of two specific properties to the False class and still ensures compatibility. 5.2.1 Conflict management The solution of the property metaclasses composition issue is not trivial. Indeed, it is necessary to deal with conflicts that arise when composing different property metaclasses. When using inheritance to compose property metaclasses, two kinds of conflicts can arise: name conflicts and value conflicts [DHH SoleInstance False class Boolean class Abstract Boolean class False class SoleInstance False class Final False Boolean instantiation inheritance Figure 15: Assigning two properties to False Name conflicts happen when orthogonal property metaclasses define instance variables or methods which have the same name. Two property metaclasses are orthogonal when they define unrelated class properties. Name conflicts for both instance variables and methods are avoided by adapting the definition of a new property metaclass according to its superclasses. For example, although the two property metaclasses "SoleInstance False class" and "SoleInstance + True class" define the same property for their respective instances (classes False and True), they may use different instance variable names or method names. Value conflicts happen when non-orthogonal property metaclasses define methods which have the same name. Most of these conflicts are avoided by making the property metaclass hierarchy act as a cooperation chain, i.e. a property metaclass explicitly refer to the overridden methods defined in its superclasses 11 . Therefore, each property metaclass acts like a mixin [BC90]. 5.2.2 Example of cooperation between property metaclasses Suppose that we want to assign two specific properties to the False class of Figure 16: (i) tracing all messages (Trace) and (ii) having breakpoints on particular methods (BreakPoint). These two properties deal with the message handling which is based in NeoClasstalk on the technique of the "method wrappers" described in [Duc98] and [BFJR98]. The executeMethod:receiver:arguments: method is the entry point to handle messages in NeoClasstalk, i.e. customizing executeMethod:receiver:arguments: allows a specialization of the message sending 12 . Thus, when the object false receives a message, the class False receives the message executeMethod:receiver:arguments:. According to the inheritance hierarchy, (1) the trace is first done, then (2), by the use of super, the breakpoint is performed, and (3) a regular method application is finally executed (again called using super). ffl (3) StandardClassAEexecuteMethod: method receiver: rec arguments: args ffl (2) BreakPoint+False classAEexecuteMethod: method receiver: rec arguments: args method selector == stopSelector ifTrue: [self halt: 'Breakpoint for ', stopSelector]. "super executeMethod: method receiver: rec arguments: args ffl (1) Trace+BreakPoint+False classAEexecuteMethod: In NeoClasstalk, as in Smalltalk, this is achieved using the pseudo-variable super. executeMethod:receiver:arguments: method is provided by StandardClass (the root of all metaclasses in Neo- which just applies the method on the receiver with the arguments. False class instantiation inheritance False Trace BreakPoint BreakPoint False class Trace BreakPoint False class PropertyMetaclass metaclass level class level meta-metaclass level Figure Composition of non-orthogonal properties method receiver: rec arguments: args self transcript show: method selector; cr. "super executeMethod: method receiver: rec arguments: args 5.3 The extended compatibility model Generalizing previous examples allows us to define the extended compatibility model (see Figure 17) which enables reusing and composing class properties. Each property metaclass defines the instance variables and methods involved in a unique property. Property metaclasses specific to a given class are organized in a single hierarchy. The root of this hierarchy is a subclass of a compatibility metaclass 13 . Each property metaclass is an instance of a meta-metaclass which describes a specific class property, allowing its reuse. metaclasses Property BClass AClass metaclasses Compatibility BClass AClass BClass BClass AClass AClass PropertyMetaclass Figure 17: The Extended Compatibility Model 13 This single hierarchy may be compared to an explicit linearization of property metaclasses composed using multiple inheritance [DHHM94]. Metaclass creation, composition and deletion are managed automatically with respect to the extended compatibility model. Each time a new class is created, a new compatibility metaclass is automatically created. This can be done in the same way that Smalltalk builds its parallel metaclass hierarchy. The assignment of a property to this class results in the insertion of a new metaclass into its property metaclass hierarchy. This insertion is made in two steps 1. first, the new property metaclass becomes a subclass of the last metaclass of the property metaclass 2. then, the class becomes instance of this new property metaclass. NeoClasstalk provides protocols for dynamically changing the class of an object (changeClass:) and the superclass of a class (superclass:) [Riv97]. Thus, the implementation of these two steps is immediate in NeoClasstalk, and is provided by the composeWithPropertiesOf: method. aClass self superclass: aClass class. aClass changeClass: self. 5.4 Programming within the extended compatibility model We distinguish two kinds of programmers: (i) "base level programmers" who implement applications using the language and development tools, and (ii) "meta level programmers" for whom the language itself is the application. 5.4.1 Base Level Programming To make our model easy to use for a "base-level programmer", the NeoClasstalk programming environment includes a tool that allows one to assign different properties to a given class using a Smalltalk-like browser (see Figure 18). These properties can be added and removed at run-time. The metaclass level is automatically built according to the selection of the "base-level programmer". 5.4.2 Meta Level Programming In order to introduce new class properties, "meta-level programmers" must create a subclass of the Prop- ertyMetaclass meta-metaclass. This new meta-metaclass stores the instance variables and the methods that should be defined by its instances (property metaclasses). When this new meta-metaclass is instan- tiated, the previous instance variables are added to the resulting property metaclass and the methods are compiled 15 at initialization time For example, the evaluation of the following expression creates a property metaclass - instance of the meta-metaclass Trace - that assigns the trace property to the True class. Trace new composeWithPropertiesOf: True 14 The removal of a property metaclass is done in a symmetrical way. solution consists of doing the compilation only once, resulting in proto-methods [Riv97]. Thus, when the property metaclass gets initialized, proto-methods are "copied" into the method dictionary of the property metaclass, allowing a fast instantiation of meta-metaclasses. This assumes that initialization is part of the creation process, which is true in almost every language. This is tradition- nally achieved in Smalltalk by the redefinition of new into super new initialize [SKT96]. Figure properties assigned to a class using a browser In order to achieve the trace, messages must be captured and then logged in a text collector. Therefore, property metaclasses instances of Trace must define an instance variable (named transcript) corresponding to a text collector and a method that handles messages. Message handling is achieved using the executeMethod:receiver:arguments: method which source code was already presented in 5.2.2. These definitions are generated when the property metaclasses are initialized, i.e. using the initialize method of the Trace meta-metaclass: TraceAEinitialize super initialize. self instanceVariableNames:' transcript '. self generateExecuteMethodReceiverArguments. 6 Conclusion Considering classes as first class objects organizes applications in different abstraction levels, which inevitably raises upward and downward compatibility issues. Existing solutions addressing the compatibility issues (such as Smalltalk) do not allow the assignment of specific properties to a given class without propagating them to its subclasses. The compatibility model proposed in this paper addresses the compatibility issue and allows the assignment of specific properties to classes without propagating them to subclasses. This is achieved thanks to the separation of the two involved concerns: compatibility and class properties. Upward and downward compatibilities are ensured using the compatibility metaclass hierarchy that is parallel to the class hierar- chy. The property metaclasses, allowing the assignment of specific properties to classes, are subclasses of these compatibility metaclasses. Therefore, we can take advantage of the expressive power of metaclasses to define, reuse and compose class properties in a environment which supports safe metaclass programming. Class properties improve readability, reusability and quality of code by increasing separation of concerns allow a better organization of class libraries and frameworks for designing reliable software. We are strongly convinced that our compatibility model enables separation of concerns based on the metaclass paradigm. Therefore, it promotes building reliable software which is easy to reuse and maintain. Acknowledgments The authors are grateful to Mathias Braux, Pierre Cointe, St'ephane Ducasse, Nick Edgar, Philippe Mulet, Jacques Noy'e, Nicolas Revault, and Mario S-udholt for their valuable comments and suggestions. Special thanks to the anonymous referees who provided detailed and thought-provoking comments. --R Programming with Explicit Metaclasses in Smalltalk. Wrappers to the Rescue. Concurrency and Distribution in Object Oriented Programming. Noury Bouraqadi-Sa-adani are First Class: the ObjVlisp Model. Derived Metaclasses in SOM. Reflections on Metaclass Programming in SOM. Le point sur l'h'eritage multiple. Proposal for a Monotonic Multiple Inheritance Linearization. Evaluating Message Passing Control Techniques in Smalltalk. Metaclass Compatibility. Separation of Concerns. "Object-Oriented Programming: The CLOS Perspective" The Art of the Metaobject Protocol. Explicit Metaclasses As a Tool for Improving the Design of Class Libraries. Reflection and Distributed Systems Adaptive Object-Oriented Software: The Demeter Method with Propagation Patterns Concepts and Experiments in Computational Reflection. Towards a Methodology for Explicit Composition of MetaObjects. Research Directions in Concurrent Object-Oriented Programming A New Smalltalk Kernel Allowing Both Explicit and Implicit Metalclass Pro- gramming Object Behavioral Evolution Within Class Based Reflective Languages. Smalltalk with Style. Advances in Object-Oriented Metalevel Architectures and Reflec- tion --TR Smalltalk-80: the language and its implementation Concepts and experiments in computational reflection are first class: The ObjVlisp Model Metaclass compatibility Programming with explicit metaclasses in Smalltalk-80 Mixin-based inheritance The art of metaobject protocol Metaobject protocols Analysis of inheritance anomaly in object-oriented concurrent programming languages Proposal for a monotonic multiple inheritance linearization Reflections on metaclass programming in SOM Smalltalk with style Towards a methodology for explicit composition of metaobjects Concurrency and distribution in object-oriented programming Advances in Object-Oriented Metalevel Architectures and Reflection Adaptive Object-Oriented Software Wrappers to the Rescue Explicit Metaclasses as a Tool for Improving the Design of Class Libraries --CTR Stphane Ducasse , Oscar Nierstrasz , Nathanael Schrli , Roel Wuyts , Andrew P. Black, Traits: A mechanism for fine-grained reuse, ACM Transactions on Programming Languages and Systems (TOPLAS), v.28 n.2, p.331-388, March 2006

compatibility;class property propagation;metaclasses;class specific properties

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Visualizing dynamic software system information through high-level models.

Dynamic information collected as a software system executes can help software engineers perform some tasks on a system more effectively. To interpret the sizable amount of data generated from a system's execution, engineers require tool support. We have developed an off-line, flexible approach for visualizing the operation of an object-oriented system at the architectural level. This approach complements and extends existing profiling and visualization approaches available to engineers attempting to utilize dynamic information. In this paper, we describe the technique and discuss preliminary qualitative studies into its usefulness and usability. These studies were undertaken in the context of performance tuning tasks.

INTRODUCTION Effective performance of many software engineering tasks requires knowledge of how the system works. Gaining the desired knowledge by studying or statically analyzing the source code can be difficult. Static analysis, for instance, can help a software engineer determine if two classes can interact, but it does not help the engineer determine how many objects of a class might exist at run-time, nor how many method calls might occur between particular objects. Determining answers to these questions requires an investigation of dynamic information collected as the software system executes. This Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers, or to redistribute to lists, requires specific permission and/or a fee. OOPSLA '98 10/98 Vancouver, B.C. c dynamic information helps bridge "the dichotomy between the code structure as hierarchies of classes and the execution structure as networks of objects" [1, p. 326]. Software engineers require tool support to effectively access and interpret dynamic system information, because the quantity, level of detail and complex structure of this information would otherwise be overwhelming. In creating a tool to help an engineer access this information, two goals must be paramount: the tool must be usable and it must be useful for the task it is designed to address. Usability is defined in terms of practicality and simplicity of interface; usefulness is defined in terms of easing the performance or completion of a task of importance, especially in comparison to alternative methods. Many tools have been developed to provide engineers access to dynamic information. Profilers, for instance, provide numerical summaries of dynamic information, such as the length of time spent executing a method. This information can be helpful when trying to tackle some system performance problems. Other tasks, however, such as verifying that objects are interacting appropriately according to defined roles [6], require additional structural information. The usefulness of profilers for these types of task degrades because the relevant dynamic information is not evident from a summary numeric value produced on a per method or per class basis. When structural dynamic information is needed, an engineer may attempt to use an object-level visualizer (e.g., [1, 6, 7]). These visualizers provide such displays as the interactions between objects (or classes) and the number of objects created of each class. When the task requires views involving many classes in a large system, the usability of these tools de- grades, as they tend to display complex interactions between multiple objects in a haze of extraneous, overlain information. In part to overcome this complexity problem, Sefika et al. introduced an architectural-oriented visualization approach [14] that allows an engineer to investigate the operation of the system at both coarse- and fine-grained levels. Some of the design choices made in their approach limit its applicability. Their approach is on-line, limiting its usefulness for some kinds of tasks. Their approach requires hard-wired instrument classes to be attached to the system, limiting its flexibility and reducing its usability. We have developed an off-line, flexible approach for visualizing the operation of an object-oriented system at an architectural level. Our approach abstracts two fundamental pieces of dynamic information: the number of objects involved in the execution, and the interactions between the objects. We visualize these two pieces of information in terms of a high-level view of the system that is selected by the engineer as useful for the task being performed. To represent the information collected across a system's ex- ecution, we use a sequence of cels. Each cel displays abstracted dynamic information representing both a particular point in the system's execution, and the history of the execution to that point. The integration of "current" and "histori- cal" information is intended to ease the interpretation of the display by the engineer. Using our prototype, a software engineer can navigate both forwards and backwards through the cels comprising views on the execution. Our approach complements and extends existing approaches to accessing dynamic system information. Our approach ffl allows an unfamiliar system to be studied without alteration of source code, ffl permits lightweight changes to the abstraction used for condensing the dynamic information, ffl supplies a visualization independent of the speed of execution of the system being studied, and ffl allows a user to investigate the abstracted information in a detailed manner by supporting both forwards and backwards navigation across the visualizations. To investigate the usefulness and usability of the approach, we have performed preliminary, qualitative studies of the use of the technique to aid performance-tuning tasks on Smalltalk programs. These studies show that the technique can help software engineers make better use of dynamic system information when performing tasks such as performance enhancement We begin in Section 2 by describing our visualization tech- nique; Section 3 discusses the creation of a visualization. In Section 4, we discuss our initial evaluation efforts intended to assess the usability and usefulness of the approach. In Section 5, we consider the design choices we made in our visualization technique. Section 6 describes related work and Section 7 concludes with a description of directions for future work. Our visualization technique abstracts information that has been previously collected during a system's execution and uses concepts from the field of computer animation to display that information to a user. We begin the description of our technique by focusing on the visualization itself, and then describe how a software engineer can construct such a visualization Figures show different views withina visualization produced during one of our case studies that was investigating a performance problem in a reverse engineering program (Section 4.1). The two windows in Figures 1 and 2 each provide one view-a cel showing events that occurred within a particular interval of the system's execution, defined as a set of n events where n is adjustable. The view in Figure 3 shows a summary view of all events occurring in the trace, and Figure 4 gives a detailed, textual view of some of the information within the summary view. Sections 2.1, 2.2, and 2.3 describe these views in more detail. Full details of the runningexample we use are provided in Section 4.1. Our prototype permits a software engineer to easily switch from a particular cel to the summary view. A user may also move through the sequence of cels sequentially or via random access; animation controls, such as play, stop, step forward and step backward, allow a user to review the execution trace and pause or return to points of interest. We discuss the navigation capabilities of our visualization in Section 2.3. 2.1 Cels A cel consists of a canvas upon which are drawn a set of wid- gets. These widgets consist of: ffl boxes, each representing a set of objects abstracted within the high-level model (Section 3.2) defined by the engineer, ffl a directed hyperarc between and through some of the boxes, ffl a set of directed arcs between pairs of boxes, each of which indicates that a method on an object in the destination box has been invoked from a method on an object in the source box, ffl a bar-chart style of histogram associated with each box, indicating the ages of and garbage collection information about the objects associated with that box, ffl annotations and bars within each box, and ffl annotations associated with each directed arc. Each box drawn identically within each cel represents a particular abstract entity specified by the engineer, and thus, does not change through the animation. The grey rectangles in Figures labelled Clustering, SimFunc, Module- sAndSuch, and Rest are boxes corresponding to abstract entities of the same names. Two of these entities, Clustering and SimFunc, each correspond to a class in the reverse engineering tool source; the other two entities represent collections of classes. ModulesAndSuch Stop Summary Step >> Play << Step Stop Data Visualization1037 39166360123Cel#: 13 Rest Clustering Rest ModulesAndSuch Rest Figure 1: A window showing an example cel in the visualization technique. ModulesAndSuch Stop Summary Play << Step Step >> Options000000111111111111 Data Visualization Stack: Clustering - Rest - SimFunc Rest Rest Clustering Clustering ModulesAndSuch Figure 2: A window showing the next cel after that in Figure 1 for the same system and execution trace. Step >> << Step Stop Play Cel: 4 Stop << Step Step >> Play Options Data Visualization Summary Rest ModulesAndSuch Clustering Clustering Rest Rest ModulesAndSuch ModulesAndSuch Deallocation Age Deallocation Age Deallocation Age Deallocation Age Allocation Pattern Allocation Pattern Allocation Pattern Allocation Pattern9992 Figure 3: A window showing the Summary View for the same system and execution trace as shown in Figures 1 and 2. OK Cancel 37228 -> 40604 age: 3376: #MethodContext 37229 -> 40604 age: 3375: #BlockContextTemplate 37229 -> 40604 age: 3375: #MethodContext 37232 -> 40604 age: 3372: #MethodContext 37206 -> 40604 age: 3398: #Array 37217 -> 40604 age: 3387: #MethodContext 37196 -> 40604 age: 3408: #Array 37215 -> 40604 age: 3389: #EsCompactBlockContextTemplate 37214 -> 40604 age: 3390: #BlockContextTemplate 37214 -> 40604 age: 3390: #MethodContext 37213 -> 40604 age: 3391: #EsCompactBlockContextTemplate 37212 -> 40604 age: 3392: #MethodContext 37211 -> 40604 age: 3393: #EsCompactBlockContextTemplate Data Visualization 37217 -> 40604 age: 3387: #BlockContextTemplate Allocations over sample time Figure 4: A pop-up window produced by clicking on the Allocation Pattern histogram for the Clustering entity of Figure 3. The path of the hyperarc represents the call stack at the end of the current interval being displayed. In Figure 1, the current call stack only travels between the Clustering and Rest boxes-the hyperarc is marked in red (shown as a dashed black line herein); in Figure 2, the call stack has extended to SimFunc as well. The set of directed arcs represents the total set of calls between boxes up to the current interval; they are displayed in blue (shown as solid black herein). Because the total number of pairs of boxes is manageable, this set does not obscure the rest of the cel significantly. Multiple instances of interaction between two boxes are shown as a number annotating the directed arc. The same two arcs are shown in Figures 1 and 2 from Clustering to Rest, with 123 calls, and from Rest to Sim- Func, with 122 calls. Object creation, age, and destruction are a particular focus within the visualization. Each box is annotated with numbers and bars indicating the total number of objects that map to the box that have been allocated and deallocated until the current interval. The length of a bar for a given box is proportional to the maximum number of objects represented by that box over the course of the visualization. The Clustering box of Figure 1 shows that a total of 1127 objects associated with it had been created to this point in the execution, and that 1037 of these had been garbage collected. The histogram associated with each box shows this information as well, but in a more refined form. An object that was created in the interval being displayed has survived for a single interval; stepping ahead one cel, if it still exists, the object has survived for two intervals, and so on. The kth bin of the histogram shows the total number of objects mapped to the box that are of age k; to limit the number of bins in the histogram, any objects older than some threshold age T are shown in the rightmost bin of the histogram. The histogram attached to the Clustering box in Figure 1 indicates that all of its 1127 objects were created relatively far in the past, more before the one being shown here. Colour is used to differentiate those objects that still exist from those that have been garbage collected; each bar of the histogram is divided into a lower, green part (marked in a vertical-line pattern herein) for the living objects and an up- per, red part (marked in a diagonal-line pattern herein) for the deleted objects. In Figure 1, the upper part of the bar in Clus- tering's histogram shows that roughly 80% of the old objects have been deallocated. Yellow (shown as light grey herein) is used both within the box annotations and within histograms to indicate a change that just occurred during the interval. More specifically, it is used to show objects that have just been created or deleted. In Figure 2, which shows the interval immediately after that of Figure 1, an additional 324 objects had just been allocated that were related to Clustering. This allocation is shown both by the yellow (light grey) portion of the upper bar, and the yellow (light grey) bar in the first bin of the histogram No complex graph layout algorithms are currently used to produce the views. The drawing package used in the prototype supports interactive rearrangement of the widgets by its user. Each cel is intended to represent a combination of information not present in its predecessor (in terms of the original ex- ecution) and a summary-to-date of the information in its predecessors and itself. The new information is difficult to interpret in isolation; the context provided by the summary-to-date eases this interpretation. See Section 5.3 for further discussion 2.2 Summary View In addition to the individual cels, a summary view is provided to display the overall execution of the system being studied. This view shows the same boxes, directed arcs, arc annota- tions, and box annotations as the final cel of the animated view. In addition, it displays two histograms per box; these are different from the histograms in the animated view. One, the allocation pattern, shows the entire execution trace divided into a set of ten equal-length intervals; if the trace consists of 10n events then each interval consists of n contiguous events. The height of each bar represents the number of objects allocated in that interval that map to that box. The other histogram, the deallocation age, shows the age of every object associated with the box when the object was garbage col- lected; if an object had not been garbage collected when tracing ended, it is displayed in the rightmost bar. For example in Figure 3, Clustering's deallocation-age histogram shows that most of its objects were deallocated at a very young age while the rest still existed when tracing stopped-this is the case for all of the boxes in this example except ModulesAnd- Such, whose associated objects were always deallocated at a young age. Clustering's allocation pattern is fairly uniform, showing only a slight increase in allocations halfway through execution; on the other hand, SimFunc stopped allocating objects after the halfway point. 2.3 Navigating the Visualization There are three forms of interaction with the visualization: view selection, animation control, and detail querying. View selection simply entails choosing between a summary view, or the detailed, cel-based view. It would be reasonable to allow multiple, simultaneous views, both summary and cel-based, but this is not provided by the current imple- mentation. However, the off-line nature of this technique (Section 3.1) allows multiple instances of the tool to be run simultaneously. Animation control is provided by several buttons, a slider, and the textual entry of particular values. The buttons are Step Backwards and Forwards (by step-size number of cels), Play, and Stop. The slider is used for random access to a cel in a drag-and-drop fashion. Textual entry is used to specify step- and interval sizes, and animation speed; altering the step size allows the engineer to move through the animation more quickly by not showing some cels-this allows the animation to proceed more quickly when the redisplay rate of the graphics software and hardware is slower than the desired rate of animation. By default, an interval ends upon an event that caused a frame to be added or removed from the execution stack; these events include making or returning from a method call, and generally, each object allocation or deallocation. This granularity is generally too fine to be usable-with tens of thousands of method calls occurring and similar numbers of objects being created and destroyed, not much changes between two adjacent cels, and histograms tend to have empty bins except for the rightmost. Therefore, we allow the engineer to re-set the interval size; a size of ten, for example, indicates that each cel should represent the changes to the system produced by ten events. Detail querying allows the engineer to connect observations made via the abstract visualization to the actual classes, object allocations and deallocations, and method calls that are being abstracted. This is done by clicking on the appropriate widget for which details are sought. Arcs, hyperarcs, and histograms can be clicked on in this way; all cause a textual dialog window to pop-up (Figure 4). This pop-up window contains a list of the dynamic entities that were associated with the widget of interest. For exam- ple, the pop-up for an arc contains a list of all the calls between the boxes connected by that arc; the pop-up for the al- location/deallocation histogram of the animated view gives a list of the objects that mapped to that box, when they were created, how old they were when garbage collected, and what method caused them to be created. Selection of an entry within these pop-up windows could be used to automatically position the view in a textual code browser in a future version of the tool. 3 CONSTRUCTING THE VISUALIZATION A software engineer employs a four-stage process when using our visualization technique (Figure 5). 1. Data is collected from the execution of the system being studied, and is stored to disk. 2. The software engineer designs a high-level model of the system as a set of abstract entities that are selected to investigate the structural aspects of interest. For example, in Figure 5, "utilities" and "database" are specified as entities 3. The engineer describes a mapping between the raw dynamic information collected and the abstract entities. Figure 5, for instance, shows that any dynamic information whose identifier begins with "foo" (such as objects whose class name starts with "foo") is to be mapped to the "utilities" entity. This mapping is applied to the raw information collected by the tool, producing an abstract visualization of the dynamic data. 4. The software engineer interacts with the visualization, and interprets it to investigate the dynamic behaviour of the system. The process is deliberately divided into multiple stages to increase its usability. Rather than having to complete the entire process every time any change is required for the task of discovery, iteration can occur over any suffix of the pro- cess. For example, the software engineer might begin with any extremely coarse view of the program, knowing very little about its performance; after interacting with the resulting visualization and gaining a partial understanding of the studied system's operation, the software engineer need only alter the high-level model and corresponding mapping to generate a new visualization-there is no need to re-collect the identical dynamic information. This process is based on the work of Murphy et al. [12]. We compare our visualization technique to this previous work in Section 6. 3.1 Stage 1: Gathering Dynamic Information Dynamic system information is collected for every method call, object creation, and object deletion performed by the system being investigated. In other words, trace information is collected. This information currently consists of an ordered list of: ffl the class of the object calling the method or having the object created, and ffl the class of the object and method being called or returning from, or the class of object being created or deleted. Since the tool currently uses complete trace information, the complete call stack for any given moment can be reconstructed Stage 3 Stage 4 database utilities foo* -> utilities *db* -> database Figure 5: The process. Because our implementation is in Smalltalk, this information is collected by instrumenting the Smalltalk virtual machine (VM) to log these events as they occur. There is nothing inherent in the tool in its use of Smalltalk; it could be as easily implemented in any language in which the execution was instrumented to collect the required information. Because a software engineer often needs to understand dynamic problems that only occur after significant initialization of the studied system, the collection of the trace information needs to be performed only during portions of the execution. This eliminates extraneous information not of interest, and speeds up the process of collection. In our implementation, VM methods are available to dynamically activate and deactivate tracing. We used these methods to collect data for Figures through 4 that included only the main iteration loop of the algorithm, excluding execution pertaining to initialization and the output of results. 3.2 Stage 2: Choosing a High-level View The software engineer typically begins an investigation with some idea of the static structure of the system being studied. Even when this is not the case, the naming conventions and organization of the source code itself often allow some guess as to the system's structure. The engineer chooses a high-level structural view to use as the basis for visualization by stating the names of the abstract entities. These entities may correspond to actual system com- ponents, be aggregates or subdivisions thereof, or have little connection to reality. In Figures 1 through 4, for instance, the investigator chose two entities representing specific classes in the program (Clustering and SimFunc), and two entities representing collections of classes (Rest and ModulesAndSuch ). 3.3 Stage 3: Specifying a Mapping For the tool to indicate the dynamic interactions between the abstract entities, it needs to have a map relating dynamic system entities to the abstract ones. A map indicates that specific system entities, such as objects of a given class or methods matching a particular pattern, are to be represented by a specific abstract entity (and thus, by a box in the visualization). An engineer states this mapping using a declarative mapping language. To be usable, a mapping language must allow an engineer to easily express the relationships between entities. The mapping language's constructs are based on the standard Smalltalk notion of structural hierarchy: methods are grouped into classes, classes into categories, categories into subapplications, and subapplications into applications. Amap consists of an ordered set of entries, each of which has three parts: 1. a name indicating the level of the Smalltalk structural hierarchy being mapped, 2. a regular expression indicating the set of names to map for the particular structural hierarchy level being mapped, and 3. the name of the abstract entity to which these dynamic system entities are to be mapped. Methods are provided for mapping a class regular expression plus method regular expression simultaneously, and subapplication, class, category, and method simultaneously. For example, say the engineer has defined an abstract entity named foo, and every message foo passed to classes named *bar* within the subapplications dog* should be mapped there; this would be indicated by a map entry: matchingSubApplication: 'dog*' class: '*bar*' category: '*' method: 'foo' mapTo: 'foo'. The example in Figures 1 through 4 used the following map: matchingClass: 'ArchClusteringAnalysis' mapTo: 'Clustering' matchingClass: 'ArchModuleGroup' mapTo: 'ModulesAndSuch' matchingClass: 'ArchProcedure' mapTo: 'ModulesAndSuch' matchingClass: 'ArchSymbol' mapTo: 'ModulesAndSuch' matchingClass: 'ArchSimFunc' mapTo: 'SimFunc' matchingSubApplication: 'Schwanke*' mapTo: 'Rest' Because we are interested in visualizing the interactions between system components, the tool takes note both of the method being called and the method being executed when it was called. The same set of map entries is used to map both; the visualization itself will differentiate between incoming and outgoing calls. Individual objects are also mapped in this way. Because it is often important where an object was created, we track objects not simply based on their class, but also in terms of the call stack that was present when it was created. Such an object will typically be mapped to a particular abstract entity through the mechanism described above; the object is treated as belonging to that abstract entity and is represented through its visualization (i.e., through its representation as a box). The mapping possesses two important properties: it is partial and it is ordered. The ordering means that each system entity is mapped to a single abstract entity, the first one for which the map entry is a valid match. The mapping is partial because a software engineer does not need to express the structure of the entire system before investigating it. If a system entity fails to match every entry in the map, it is not represented in the resulting visualization. This feature both decreases the overhead for the tool and removes unwanted information from the visualization. If the engineer wants every dynamic entity to appear in the visualization, a final entry in the map of the form: matchingAnything: '*' mapTo: 'default' will act as a default abstract entity for all dynamic entities that "fall through" the other map entries. Three fundamental questions that must be answered about any software visualization are: ffl Is the technique useful to software engineers trying to perform a task on a system? ffl Is the technique usable by software engineers? ffl For what kinds of software engineering tasks is the visualization helpful? Evaluating a technique against each of these questions requires a number of careful, in-depth studies. These studies are warranted only after an initial determination of the coarse-grained utility of a technique. In this paper, we report on results from our preliminary investigations into the utility of our visualization technique. In our preliminary investigations, we chose to fix the kind of software engineering task studied to be performance tun- ing. This task was chosen because it is heavily reliant on dynamic system information and because it tends to be delegated to "expert" developers. A visualization technique that can aid a non-expert developer in tackling performance problems would thus be beneficial in increasing the use of dynamic system information by engineers, which was one of our initial goals. We also chose to focus on the usefulness of the technique, rather than its usability. This decision was reasonable because the main features of the technique affecting its usability have been investigated in other related domains. The iterative selection of the high-level entities and designation of the mapping by the software, for instance, are also characteristic of the software reflexion model approach from which this visualization technique is derived. Users of the software reflexion model approach have not had difficulties performing these steps [10, 11]. Our preliminary studies, then, focus on investigating the usefulness of the visualization. We report on two case stud- ies. The first case study (Section 4.1) discusses the use of the visualization technique by one of the authors to determine why a Smalltalk implementation of a reverse engineering algorithm [13] was running slower than expected. In this sce- nario, we focus on the differences in information provided by the visualization technique compared to a profiler. In the second case study (Section 4.2), we had both an expert and a non-expert Smalltalk developer use the visualization to attempt to discover the cause of a performance problem with the visualization technique itself. We report on both qualitative and quantitative data collected about the use of the visualization. 4.1 Case Study #1 A hierarchical agglomerative reverse engineering algorithm attempts to automatically cluster entities, such as procedures in a C program, comprising a software system into subsystems (modules) based on a similarity function. One of the authors wanted to determine why a Smalltalk implementation of a particular algorithm [13] executed significantly more slowly than a C++ implementation. The algorithm starts by placing each procedure in a separate module. It then iteratively computes the similarity function between each possible pair of modules; in each iteration, the most similar pair of modules is combined. The algorithm terminates when a specified number of modules are left or when no modules are similar enough to be combined. The performance investigator had knowledge of the design of each program, but had not implemented either program. To examine the performance of the Smalltalk implementation, the investigator first used the IBM VisualAge for Smalltalk execution profiler. With this tool, a user can either sample or trace the execution of an application, and then view collected statistics, such as the amount of execution time spent in particular methods or the number of garbage collection scav- enges. After perusing several of these views, the investigator determined about 16% of the execution time was spent in methods of the ArchClusteringAnalysis class that contains the main iteration loop, 5.5% was spent in methods of the ArchCache class that acts as a cache for already computed similarity values, and 4.6% was spent in computing new similarity values. This result was not surprising. The information confirmed the investigator's understanding of how the program works, but did not provide any hints as to whether the performance could be enhanced. The investigator next applied the visualization tech- nique, choosing a high-level model consisting of four entities. One entity, Clustering, represented the ArchClusteringAnalysis class. Another, SimFunc, represented the class that had methods for computing the similarity function. A third, ModulesAndSuch, represented the functions and modules whose similarity was to be compared, and a fourth, Rest, represented all other classes comprising the program. The mapping associated the appropriate classes (and sub-applications) with these boxes. The investigator collected trace information for the main iteration loop of the program and then began interacting with the visualization. Playing throughthe abstracted information, the investigator noted the large number of objects (over 4500) associated with the SimFunc entity. The investigator viewed the summary and queried it for the objects associated with SimFunc's box. The object list contained many Set and MethodContext objects (Figure 4). These results confirmed that the cost of computing the similarity between two modules was high and should be minimized. Returning to a "play" through the vi- sualization, the investigator noted that the ratio of calls from Clustering to Rest and from Rest to SimFunc was lower than expected. Prior investigation had shown that the majority of the calls between Clustering and Rest were due to calls on the ArchCache object; calls from Rest to SimFunc represent new computations of similarity. 1 This insight led the investigator to study the ArchCache class. The investigator found that the "key" value used to store and access similarity values in the cache was not causing as many hits as it could. A slight modification to the formation of keys resulted in an increase of just over 25% in the speed of the program. The visualization technique aided this performance-tuning task by presenting information that caused the investigator to ask, and answer, the "right" questions about the implementa- tion. Insight into structural interactions in the system helped the investigator narrow in on the algorithmic problem. The investigator made use of the both the interaction and object allocation and deallocation information, the summary view, and the ability to play, and re-play, through the traced execution. 4.2 Case Study #2 In the second case study, the tool was used to investigate its own performance problems; specifically, due to a structural design flaw, it was faster to step forward than to step backward in the visualization tool. This flaw centered on the fact that the implementor had chosen to generate cels on the fly and often used simple linked lists to hold the required information for the arc annotations; as a result, adding to these lists via the method better design for the program would havebeen to hide the cache behind the ArchSimFunc interface. addInteractionsFrom:to:between: was fast, but removing from the lists via removeInteractionsFrom:to:between: required a linear-order search through each list. The implementor of the tool had discovered this flaw and informed the experimenters of its existence and its cause. To prepare for the studies, the experimenters gathered a trace consisting of stepping forwards and backwards in the visualization tool. 2 An initial high-level model and mapping were also prepared for the participants as the short study periods were intended to focus on the visualization itself, rather than the process of creating a visualization. The high-level model was very simple, and can be seen in the visualization shown in Figure 6; the classes used by the tool all had names that began with a two- or three-letter prefix, and thus were mapped to abstract entities with these prefixes as names. In a separate session each, a previously collected trace was given to two experimental participants: an expert at solving performance problems in Smalltalk applications, and a non-expert in solving performance problems in any language. Each participant was given an introduction to the tool and a short training session in which each had the opportunity to use the tool on a toy problem. Then, the symptom of the flaw in the tool was explained, and the parameters and interaction that we had traced were described. Each was asked to determine three or fewer points of interest within the source code for the tool that they saw as being good candidates for more detailed analysis; they were also asked to answer a set of questions periodically in regards to their perceptions of the tool and progress in their task. We audio-taped these question and answer sessions. We also captured automatically a log of the navigation pattern through the visualization using instrumentation built into the prototype. 4.2.1 The Expert Participant The expert participant began with a ten-minute inspection of the summary view: the Gp box was seen to have the most objects allocated, and most of these were immediately deal- located. Querying the attendant Allocation Pattern histogram showed that many of these objects were of the classes Point, MethodContext, and BlockContext. The animated view was then used, both in step forward and backward mode and in play mode, to examine the range of cels where many of these objects were being allocated; a repetitive call pattern was observed between the Gp and Cdf boxes. The arcs and hyperarcs between these boxes were queried for de- tails, and the methods involved in this pattern were found by the participant. A separate code browser was then used to investigate the details causing this behaviour. After studyingthe 2 The visualization tool had to be run on a different, pre-existing execution trace. A toy example was used for this purpose, but choice of input was not a factor in the tool's symptoms. The second participant actually received a trace of only a step backwards. system for an hour, the participant decided that the likeliest cause was in the methods ffl removeInteractionsFrom:to:between:, and ffl addInteractionsFrom:to:between:. The participant noted the similarity of code in these two meth- ods. This observation made sense because the fundamental problem was due to the data structure. The participant was thus able to indicate a useful point to continue the investiga- tion, as had been requested at the start of the study. The expert participant liked two features of the tool in particular ffl the summary view, although the participant stated: "in this case [the effect] was slightly obvious [in the summary view]-it may not be so obvious in other cases"; and ffl the animation of the hyperarc resulting from pressing "play", because of the way one can watch "how things go into loops or circles or watch the communication back and forth between different things, or specific things." The expert participant felt the tool lacked two desirable features ffl integration between it and a traditional code browser, so one could, for example, select a method in a pop-up detail window and have the code browser display that method; and ffl the lack of ability to view a detailed stack dump, comparable to that available from a Smalltalk debugger, particularly so that the parameter types being passed could be seen (this cannot be seen from the static code because Smalltalk is dynamically typed). The actual values being passed were deemed desired in some instances. Code browser integration is a desired feature that has not yet been implemented; the tool has been designed to accommodate this change. The tool did allow the participant to narrow the search to particular points of interest that could then be investigated via a debugger or similar means. The desire for greater, integrated informationfrom the tool is understand- able, but runs contrary to its design philosophy of complementing existing techniques-it is not intended to supercede the use of a debugger. This desire also highlights the tension between off-line and on-line approaches to accessing dynamic information. 4.2.2 The Non-Expert Participant The non-expert participant made extensive use of both the object histograms and the allocation/deallocation bars in the detailed view to investigate the performance problem. Specif- ically, the participant would find cels in which object deallocation was not keeping pace with object allocation (i.e., Cw << Step Stop Play Step >> Summary Stop Options00011111100000011111111111100000000000011111111111111111111111100110011 Data Visualization Cdf Gp Cg Cg72010Cw Figure Case study #2 visualization. the green bar-shown herein via a vertical-line pattern-was longer than the red bar-shown herein via a diagonal-line pattern-within a box) and would then step forward to see when objects were being allocated. Queries on the associated histograms were then used to determine the classes of the allocated objects. Less frequently, the participant would investigate the calls involved with the allocations. For the first forty minutes, the participant worked solely with the visualization tool. After that, the participant began to use the Smalltalk code browsers to study the associated code. After approximately an hour with the tool, the participant had identified two methods, including the removeInteractionsFrom method, as a point in the code at which to continue the investigation. This determination was based, in part, on noticing a correlation between an increase in message sends between the Gp and Cg boxes and the number of objects allocated by Cg. Similar to the case of the expert participant then, the non-expert found the correct area of code to investigate, which was the task that had been posed. The non-expert found the deallocation age histograms and the ability to determine the correlation between abstract information to method and object names by clicking on histograms and interactions in the visualization particularly help- ful. However, the non-expert indicated a desire for different displays of this information, finding the "screen with all the methods [was] too cluttered." Similar to the expert, the non-expert desired more integration with other Smalltalk tools, such as the code browser. For instance, the participant wanted to be able to select a call from a list of interactions and visit that call site in the code. During an interview part of the way through the study pe- riod, the participant noted that it was difficult to attack the task because of a lack of knowledge of what could cause performance problems. The visualization tool provided some clue as to how to proceed because of its emphasis on particular dynamic information. The applicability of the dynamic information chosen for other tasks requires further research. Key features of our technique include off-line operation, a navigable visualization of the collected data, cels based on a running summary, and the use of a declarative mapping to abstract fine-grained information about a system's execution. We discuss each of these features and our use of trace information 5.1 Off-line Operation Using an on-line visualization technique can be a slow, unidirectional procedure. Taking the technique off-line and separating the visualization from the system execution can achieve two benefits. First, it allows the information to be preprocessed as a whole prior to visualization, enabling the generation of summary information about the entire execution. For the performance tuning tasks described in the case studies, summary information was used to provide clues about which parts of the system to investigate as potential sources of the problem. After accessing summary information, the users returned to investigate detailed parts of the execution. Second, it allows any partial trace of an execution to be reviewed without having to re-run the entire execution. This review capability permits the visualization to be navigable in a way that is not possible for an on-line technique. Not only may the trace be replayed from any arbitrary point, but also it may be played backwards, or at a rate that is independent of the speed of the original execution of the system being studied 5.2 Navigable Visualization One advantage of an off-line visualization approach is the navigation capability provided to the software engineer. The user can unfold the execution in a forward, "play", mode, but then can perform detailed investigations of particular parts of the execution by moving the visualization both forward and backward. In our current prototype, we do not associate any information about the actual execution time with the off-line navigation. Each step forward or backward in our visualization takes time proportional to the display time of the next cel, rather than representing the length of time required by an associated method call, allocation or garbage collection. For some tasks, including performance tuning, it would sometimes be helpful to have steps between cels represent the system running time. 5.3 Running Summary We believe that separately displaying individual events, or small groups of contiguous events, makes for an insufficient visualization of a system execution because of a lack of connection to the greater context of that execution. Some sort of summary information is also needed. We considered two means of providing such summary in- formation: a single summary picture, such as that in Figure 3, and a set of pictures showing the change to the state of the system over individual intervals of its execution ("delta" in- formation), which is not provided by our tool. But neither alone would be sufficient to illustrate the dynamic nature of the information we are attempting to visualize. The summary picture clearly does not contain any temporal ordering of events-it is difficult to look at one and mentally reconstruct the sequence of events that produced it. Furthermore, this summary alone cannot contain enough detail about the execution to be useful without becoming so cluttered that it is rendered unusable. Delta pictures address the concern of visualization of the temporal nature of the information; however, it is difficult to understand the relationship between a delta picture and the execution in toto. To reach a compromise between these alternatives, we chose to provide a running summary of the execution within the individual cels. This implicitly provides the temporal component of the summary information while maintaining context for the delta information within a cel. Two other alternatives to maintain context are possible. In the first, we could begin with a summary view such as that provided by our tool. But rather than being a single, static picture, it could also be divided into a sequence of cels each of which would show the same summary information while highlightingin a different colour, say, the information that was changed or added over the represented interval, such as the directed arcs that were traversed, or the subset of objects that were deallocated. The second alternative is similar, but instead of highlighting only the information that is different for that interval, a running summary of all the information that had changed from the start of execution of the system to the current interval would be highlighted. Both can suffer from the fact that a complete summary view can quickly become too detailed, leading to information overload. However, both these schemes could be used to complement the delta plus running-summary combination currently used in our cels; we have not yet investigated this possibility. 5.4 Mapping Objects Each cel maps objects to abstraction units. Associating an object with an abstraction unit using our declarative mapping approach requires a means of "naming" objects. We chose to name-more precisely, identify-an object based on where it is created in the code: a software engineer identifies objects mapping to a particular abstraction unit by describing a part of the call stack that exists when one of the objects is created. This approach has the advantage that an engineer can identify collections of objects by perusing the source code and describing the locations where relevant allocations occur. Another possible choice would be to name objects based on their class. However, this approach to naming would not allow objects of the same class to be mapped to different abstraction units, limiting the ability of the engineer to differentiate distinct uses of classes. Currently, the mapping provided by the engineer is applied uniformly to all dynamic information collected as the system executes. A ramification of this decision is that once an object is associated with an abstraction unit, it remains associated with that unit for the duration of the visualization. Some- times, though, it may be useful to modify the association of objects to abstraction units over the course of the execution. For instance, if an object is created in one subsystem, but is then immediately passed as an argument to another subsys- tem, it may be useful to capture the "migration" of the object. Supporting this migration would require not only a means to allow the engineer to describe when and how the migration would occur, but also would require updates to the use of histograms for object allocation and deallocation. Further understanding of how this capability might help in the performance of tasks is required before support is added. 5.5 Dynamic Information Our current prototype visualizes trace information collected about a system's execution. Trace information has the benefit that it is complete: all object interactions, allocations, and deallocations are included in the trace. Complete information is easy for the engineer to reason about. However, trace information has the often cited problem of being voluminous [9, 2, 8]. Tracing even small pieces of a system's execution can result in a huge amounts of data. Althoughwe have been able to successfully use trace data to investigate some performance problems, the use of trace information limits the flexibility and usability of our current prototype. We plan to investigate the use of sampled information as a basis for our prototype to overcome some of these limitations. 6 RELATED WORK De Pauw et al. have developed a number of visualizations to describe the execution of an object-oriented system, including inter-class call cluster diagrams, inter-class call matrices, a histogram of instances, and an allocation matrix [1]. All of these visualizations show fine-grained execution information about individual classes and objects. The utility of these visualizations degrades as the size, measured in the number of classes, of a system grows. Several other similar object- and class-level visualization approaches have been developed (e.g., [6, 5]); these techniques share the same scalability problem Lange and Nakamura in the Program Explorer tool allow the developer to integrate, off-line, static and dynamic information about a program to aid comprehension activities [7, 8]. For instance, they show how this combination of information can help a developer find and investigate instances of design patterns in a system. The visualizations they produce are also at a fine-grained level. Vlissides et al. use a different notion of pattern, which they refer to as execution patterns, to help developers investigate the large amount of fine-grained execution information available about a system [3]. Specifically, they allow a developer to query an on-line animation for patterns appearing in a dynamic execution stream. In both the Program Explorer and execution pattern approaches, the developer must apply detailed knowledge about a system to formulate appropriate queries. Jerding et al. have applied the information mural approach to create a scalable visualization of fine-grained program events [4]. The result, an execution mural, places classes vertically on the screen and uses single pixel vertical bars, with various colouring approaches, to indicate calls between classes. The interactions occurring in the system are then shown across the screen. Using this approach, thousands of interactions occurring between objects can be visualized on one screen. The authors extend these ideas to a Pattern Mural that provides an information mural display of automatically detected common occurring sequences of calls (patterns) in the execution. Although this approach may help a developer find unexpected patterns, or verify existing patterns in the code, it still visualizes only fine-grained informationabout the system. The approach taken by Sefika et al. differs in allowing a developer to utilize coarse-grained system information to produce visualizations [14]. Using their technique, a developer may introduce various abstractions into the system instrumentation process, including subsystem, framework and pattern-level abstractions. The abstractions can then be used as a basis for several visualizations including affinity and ternary dia- grams. The coarser-grained visualizations produced with this technique make it easier for developers to investigate inter-component interactions in large systems than previous approaches Some of the design decisions Sefika et al. made in developing their technique limit its flexibility. Choosing an on-line approach permits a link between the speed shown in the visualization and the execution speed. However, as we have discussed, an on-line approach limits the modes of investigation available to an engineer. Choosing an approach that hard-wires the abstractions of interest into the instrumentation process provides an effective data gathering mechanism; how- ever, it decreases the usability of the technique by making it more difficult for an engineer to apply it to a new system. We have been able to easily apply our technique to different systems because of the separation in our process between data gathering and visualization. Our visualization technique buildson the software reflexion model technique developed by Murphy et al. [12, 10]. The reflexion model technique helps an engineer access both static and dynamic information about a system by enabling a comparison between a posited high-level model and a model representing informationextracted from either the static source or from a system's execution. Similar to our visualization tech- nique, the software reflexion model depends on a declarative mapping language. Our visualization technique extends the reflexion model work in three fundamental ways: by applying the abstraction approach across discrete intervals of the execution with animation controls, by providing support to map dynamic entities rather than only static entities, and by mapping memory aspects of an execution in addition to interac- tions. Our visualization technique also uses the running summary model rather than the complete summary model used in the reflexion model approach. AND FUTURE WORK Condensing dynamic information collected during a system's execution in terms of abstractions that represent coarse system structure, such as frameworks and subsystems, can help software engineers investigate the behaviour of a system. We have developed a visualization technique that allows engineers to flexibly define the coarse structure of interest, and to flexibly navigate through the resulting abstracted views of the system's execution. Our approach complements and extends existing visualization techniques. Our preliminary investigations into the usefulness and usability of the visualization indicate it shows promise for enhancing a software engineer's ability to utilize dynamic information when performing tasks on a system. To date, we have focused on the use of dynamic information to aid one particular software engineering task-performance tuning. We intend to continue our investigations into the utility of the entire technique through more extensive case studies on a wider range of tasks on larger systems. Although there is evidence elsewhere [10, 11] that the iterative mapping approach is usable for static information, our further studies will investigate if this remains true for dynamic information. ACKNOWLEDGMENTS This work was funded by a British Columbia Advanced Systems Institute Industrial Partnership Program grant, by OTI, Inc., and by an NSERC research grant. We thank Edith Law for participating in the case study, and we thank the anonymous reviewers for their comments. --R Visualizing the behavior of object-oriented systems Modeling object-oriented program execution Execution patterns in object-oriented visualization Visualizing interactions in program executions. Interactive visualization of design patterns can help in framework understanding. Efficient program tracing. Lightweight Structural Summarization as an Aid to Software Evolution. Reengineering with reflexion models: A case study. Software reflexion models: Bridging the gap between source and high-level models An intelligent tool for re-engineering software modularity --TR Visualizing the behavior of object-oriented systems Interactive visualization of design patterns can help in framework understanding Software reflexion models Architecture-oriented visualization Visualizing interactions in program executions An intelligent tool for re-engineering software modularity Efficient program tracing Object-Oriented Program Tracing and Visualization Reengineering with Reflexion Models Modeling Object-Oriented Program Execution Lightweight structural summarization as an aid to software evolution --CTR George Yee, Visualization for privacy compliance, Proceedings of the 3rd international workshop on Visualization for computer security, November 03-03, 2006, Alexandria, Virginia, USA Steven P. Reiss, Visual representations of executing programs, Journal of Visual Languages and Computing, v.18 n.2, p.126-148, April, 2007 Andrs Moreno , Mike S. Joy, Jeliot 3 in a Demanding Educational Setting, Electronic Notes in Theoretical Computer Science (ENTCS), 178, p.51-59, July, 2007 Johan Moe , David A. Carr, Using execution trace data to improve distribute systems, SoftwarePractice & Experience, v.32 n.9, p.889-906, July 2002 Lei Wu , Houari Sahraoui , Petko Valtchev, Program comprehension with dynamic recovery of code collaboration patterns and roles, Proceedings of the 2004 conference of the Centre for Advanced Studies on Collaborative research, p.56-67, October 04-07, 2004, Markham, Ontario, Canada Robert J. Walker , Gail C. Murphy, Implicit context: easing software evolution and reuse, ACM SIGSOFT Software Engineering Notes, v.25 n.6, p.69-78, Nov. 2000 Bradley Schmerl , David Garlan , Hong Yan, Dynamically discovering architectures with DiscoTect, ACM SIGSOFT Software Engineering Notes, v.30 n.5, September 2005 Davor ubrani , Gail C. Murphy, Hipikat: recommending pertinent software development artifacts, Proceedings of the 25th International Conference on Software Engineering, May 03-10, 2003, Portland, Oregon Robert J. Walker , Gail C. Murphy , Jeffrey Steinbok , Martin P. Robillard, Efficient mapping of software system traces to architectural views, Proceedings of the 2000 conference of the Centre for Advanced Studies on Collaborative research, p.12, November 13-16, 2000, Mississauga, Ontario, Canada Iain Milne , Glenn Rowe, OGRE: Three-Dimensional Program Visualization for Novice Programmers, Education and Information Technologies, v.9 n.3, p.219-237, September 2004 Avi Bryant , Andrew Catton , Kris De Volder , Gail C. Murphy, Explicit programming, Proceedings of the 1st international conference on Aspect-oriented software development, April 22-26, 2002, Enschede, The Netherlands Giuseppe Pappalardo , Emiliano Tramontana, Automatically discovering design patterns and assessing concern separations for applications, Proceedings of the 2006 ACM symposium on Applied computing, April 23-27, 2006, Dijon, France Hong Yan , David Garlan , Bradley Schmerl , Jonathan Aldrich , Rick Kazman, DiscoTect: A System for Discovering Architectures from Running Systems, Proceedings of the 26th International Conference on Software Engineering, p.470-479, May 23-28, 2004 Abdelwahab Hamou-Lhadj , Timothy C. Lethbridge, A survey of trace exploration tools and techniques, Proceedings of the 2004 conference of the Centre for Advanced Studies on Collaborative research, p.42-55, October 04-07, 2004, Markham, Ontario, Canada Paul Gestwicki , Bharat Jayaraman, Methodology and architecture of JIVE, Proceedings of the 2005 ACM symposium on Software visualization, May 14-15, 2005, St. Louis, Missouri Raimondas Lencevicius , Urs Hlzle , Ambuj K. Singh, Dynamic Query-Based Debugging of Object-Oriented Programs, Automated Software Engineering, v.10 n.1, p.39-74, January Eleni Stroulia , Tarja Syst, Dynamic analysis for reverse engineering and program understanding, ACM SIGAPP Applied Computing Review, v.10 n.1, p.8-17, Spring 2002 Martin P. Robillard , Gail C. Murphy, Representing concerns in source code, ACM Transactions on Software Engineering and Methodology (TOSEM), v.16 n.1, p.3-es, February 2007

software visualization;performance;software structure;program comprehension;execution trace;programming environments

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Multiple dispatch as dispatch on Tuples.

Many popular object-oriented programming languages, such as C++, Smalltalk-80, Java, and Eiffel, do not support multiple dispatch. Yet without multiple dispatch, programmers find it difficult to express binary methods and design patterns such as the "visitor" pattern. We describe a new, simple, and orthogonal way to add multimethods to single-dispatch object-oriented languages, without affecting existing code. The new mechanism also clarifies many differences between single and multiple dispatch.

INTRODUCTION Single dispatch, as found in C++ [Stroustrup 97], Java [Arnold & Gosling 98, Gosling et al. 96], Smalltalk-80 [Goldberg & Robson 83], and Eiffel [Meyer 92, Meyer 97], selects a method using the dynamic class of one object, the message's receiver. Multiple dispatch, as found in CLOS [Chapter 28, Steele 90] [Paepcke 93], Dylan [Shalit 97, Feinberg et al. 97], and Cecil [Chambers 92, Chambers 95], generalizes this idea, selecting a method based on the dynamic class of any subset of the message's arguments. Multiple dispatch is in many ways more expressive and flexible than single dispatch in object-oriented (OO) programming [Bobrow et al. 86, Chambers 92, Castagna 97, Moon 86]. In this paper we propose a new, simple, and orthogonal way of adding multiple dispatch to existing languages with single dispatch. The idea is to add tuples as primitive expressions and to allow messages to be sent to tuples. Selecting a method based on the dynamic classes of the elements of the tuple gives multiple dispatch. To illustrate the idea, we have designed a simple class-based OO language called Tuple * . While perhaps not as elegant as a language built directly with multiple dispatch, we claim the following advantages for our mechanism: 1. It can be added to existing single dispatch languages, such as C++ and Java, without affecting either (a) the semantics or (b) the typing of existing programs written in these languages. 2. It retains the encapsulation mechanisms of single- dispatch languages. 3. It is simple enough to be easily understood and remembered, we believe, by programmers familiar with standard single dispatch. 4. It is more uniform than previous approaches of incorporating multiple dispatch within a single- dispatching framework. To argue for the first two claims, we present the semantics of the language in two layers. The first layer is a small, class-based single-dispatching language, SDCore, of no interest in itself. The second layer, Tuple proper, includes SDCore and adds multiple dispatch by allowing messages to be sent to tuples. Our support for the third claim is the simplicity of the mechanism itself. If, even now, you think the idea of sending messages to tuples is clearly equivalent to multiple dispatch, then you agree. To support the fourth claim, we argue below that our mechanism has advantages over others that solve the same problem of incorporating multiple dispatch into existing single-dispatching languages. The rest of this paper is organized as follows. In Section 2 we describe the single-dispatching core of Tuple; this is needed only to show how our mechanism can be added to such a language. In Section 3 we describe the additions to the single- dispatching core that make up our mechanism and support our first three claims. Section 4 supports the fourth claim by comparison with related work. In Section 5 we offer Appears in the OOPSLA '98 Conference Proceedings, Conference on Object-Oriented Programming, Systems, and Applications, Vancouver, British Columbia, Canada, October 18- 22, 1998, pp. 374-387. * With apologies to Bruce et al., whose TOOPLE language [Bruce et al. 93] is pronounced the same way. some conclusions. In Appendix A we give the concrete syntax of Tuple, and in Appendix B we give the language's formal typing rules. In this section, we introduce the syntax and semantics of SDCore by example. 2.1 Syntax and Semantics The single-dispatching core of Tuple is a class-based language that is similar in spirit to conventional OO languages like C++ and Java. Integer and boolean values, along with standard operations on these values, are built into the language. Figure 1 illustrates SDCore with the well-known Point/ColorPoint example. The Point class contains two fields, representing an x and y coordinate respectively. Each point instance contains its own values for these fields, supplied when the instance is created. For example, the expression new Point(3,4) returns a fresh point instance with xval and yval set to 3 and 4 respectively. The Point class also contains methods for retrieving the values of these two fields and for calculating the distance from the point to some line. (We assume the Line class is defined elsewhere.) For ease of presentation, SDCore's encapsulation model is extremely simple. An instance's fields are only accessible in methods where the instance is the receiver argument. An instance may contain inherited fields, which this rule allows to be accessed directly; this is similar to the protected notion in C++ and Java. The ColorPoint class is a simple subclass of the Point class, augmenting that definition with a colorval field and a method to retrieve the color. (We assume that a class Color is defined elsewhere.) The ColorPoint class inherits all of the Point class's fields and methods. To create an instance of a subclass, one gives initial values for inherited fields first and then the fields declared in the subclass. For example, one would write new ColorPoint(3,5,red). As usual, subclasses can also override inherited methods. To simplify the language, SDCore has only single inheritance. (However, multiple inheritance causes no problems with respect to the new ideas we are advocating.) We use a standard syntax for message sends. For example, if p1 is an instance of Point or a subclass, then the expression p1.distanceFrom(ln2) invokes the instance's distanceFrom method with the argument ln2. Within the method body, the keyword self refers to the receiver of the message, in this case p1. Dynamic dispatching is used to determine which method to invoke. In particular, the receiver's class is first checked for a method implementation of the right name and number of arguments. If no such method exists, then that class's immediate superclass is checked, and so on up the inheritance hierarchy until a valid implementation is found (or none is found and a message-not-understood error occurs). For simplicity, and to ease theoretical analysis, we have not included assignment and mutation in SDCore. Again, these could be easily added. 2.2 Type Checking for SDCore An overview of the static type system for SDCore is included here for completeness. The details (see Appendix B) are intended to be completely standard. For ease of presentation SDCore's type system is simpler than that found in more realistic languages, as this is peripheral to our contributions. There are four kinds of type attributes. The types int and bool are the types of the built-in integer and boolean values, respectively. The function type (T 1 ,.,T n )-T n+1 is the type of functions that accept as input n argument values of types respectively and return a result of type T n+1 . The class type, CN, where CN is a class name, is the type of instances of the class CN. Types are related by a simple subtyping relation. The types int and bool are only subtypes of themselves. The ordinary contravariant subtyping rule is used for function types [Cardelli 88]. A class type CN 1 is a subtype of another class type CN 2 if the class CN 1 is a subclass of the class CN 2 . To ensure the safety of this rule, the type system checks that, for every method name m in class CN 2 , m's type in CN 2 is a supertype of m's type in CN 1 . * Classes that do not meet this check will be flagged as errors. Thus every subclass that passes the type checker implements a subtype of its superclass. To statically check a message send expressionof the form check that the static type of E 0 is a subtype of a class type CN whose associated class contains a method I of type (T 1 ,.,T n )-T n+1 , where the types of the expressions are subtypes of the T 1 ,.,T n class Point { fields (xval:int, yval:int) method x():int { xval } method y():int { yval } method distanceFrom(l:Line):int { . } class ColorPoint inherits Point { fields (colorval:Color) method color():Color { colorval } Figure 1: The classes Point and ColorPoint. * Unlike C++ and Java, SDCore does not allow static overloading of method names. However, since we add multiple dispatch to SDCore by a separate mechanism, and since dynamic dispatch can be seen as dynamic overloading, there is little reason to do so. respectively; in this case, E . For example, p1.distanceFrom(ln2) has type int, assuming that p1 has type Point and ln2 has type Line. 2.3 Problems with Single Dispatching Single dispatching does not easily support some programming idioms. The best-known problem of this sort is the "binary method problem" [Bruce et al. 95]. For example, consider adding an equality test to the Point and ColorPoint classes above as follows. (For simplicity, in SDCore we have not included super, which would allow ColorPoint's equal method to call Point's equal method.) class Point { method equal(p:Point):bool { class ColorPoint inherits Point { method equal(p:ColorPoint):bool - a type error in SDCore { and As is well-known, this makes it impossible for ColorPoint to be considered a subtype of Point [Cook et al. 90]. In other words, ColorPoint instances cannot be safely used wherever a Point is expected, so polymorphism on the point hierarchy is lost. (For this reason the example is ill-typed in SDCore.) The problem is semantic, and not a fault of the SDCore type system. It stems from the asymmetry in the treatment of the two Point instances being tested for equality. In particular, one instance is the message receiver and is thereby dynamically dispatched upon, while the other is an ordinary argument to the message and plays no role in method selection [Castagna 95]. Multiple dispatch avoids this asymmetry by dynamically dispatching based on the run-time class of both arguments. A more general problem is the "visitor" design pattern [Pages 331-344, Gamma et al. 95]. This pattern consists of a hierarchy of classes, typically representing the elements of a structure. In order to define new operations on these elements without modifying the element classes, a separate hierarchy of visitors is created, one per operation. The code to invoke when a "visit" occurs is dependent both on which visitor and which element is used. Multimethods easily express the required semantics [Section 7, Baumgartner et al. 96], while a singly-dispatched implementation must rely on unwieldy simulations of multiple dispatching. OF SDCORE Tuple extends SDCore with tuple expressions, tuple classes, tuple types, and the ability to declare and send messages to tuples, which gives multiple dispatch. Nothing in the semantics or typing of SDCore is changed by this extension. 3.1 Syntax and Semantics In Tuple the expression creates a tuple of size n with components v 1 ,.,v n , where each v i is the value of the corresponding Figure shows how one would solve the Point/ColorPoint problem in Tuple. Rather than defining equal methods within the Point and ColorPoint classes, we create two new tuple classes for the methods. In the first tuple class, a tuple of two Point instances is the receiver. The names p1 and p2 can be used within all methods of the tuple class to refer to the tuple components. However, tuple classes are client code and as such have no privileged access to the fields of such components. The second tuple class is similar, defining equality for a tuple of two ColorPoint instances. (We assume that there is a tuple class for the tuple (Color, Color) with an equal method.) There can be more than one tuple class for a given tuple of classes. Since no changes are made to the Point or ColorPoint classes when adding equal methods to tuple classes, the subtype relationship between ColorPoint and Point is unchanged. That is, by adding the equal method to a tuple class instead of to the original classes of Figure 1, ColorPoint remains a safe subtype of Point. The syntax for sending a message to a tuple is analogous to that for sending a message to an instance. For example, (p1,p2).equal() sends the message "equal()" to the tuple (p1,p2), which will invoke one of the two * As in ML [Milner et al. 90], we do not allow tuples of length one. This prevents ambiguity in the syntax and semantics. For example, an expression such as (x).g(y) is interpreted as a message sent to an instance, not to a tuple. Tuples have either zero, or two or more elements. We allow the built-in types int and boolean to appear as components of a tuple class as well. Conceptually, one can think of corresponding built-in classes int and boolean, each of which has no non-trivial subclasses or superclasses. tuple class (p1:Point, p2:Point) { method equal():bool { and tuple class (cp1:ColorPoint, { method equal():bool { and and (cp1.color(), Figure 2: Two tuple classes holding methods for testing equality of points. equal methods. Just as method lookup in SDCore relies on the dynamic class of the receiver, method lookup in Tuple relies on the dynamic classes of the tuple components. Therefore, the appropriate equal method is selected from either the (Point, Point) or the (ColorPoint, tuple class based on the dynamic classes of p1 and p2. In particular, the method from the (ColorPoint, ColorPoint) tuple class is only invoked if both arguments are ColorPoint instances at run-time. The use of dynamic classes distinguishes multiple dispatch from static overloading (as found, for example, in Ada 83 [Ada 83]). The semantics of sending messages to tuples, multiple dispatch, is similar to that in Cecil [Chambers 95]. Consider the expression value v i , and where C d,i is the minimal dynamic class of v i . A method in a tuple class (C 1 ,.,C n ) is applicable to this expression if the method is named I, if for each 1-i-n the dynamic class C d,i is a subclass of C i , and if the method takes m-n additional arguments. (The classes of the additional arguments are not involved in determining applicability, but their number does matter.) Among the applicable methods (from various tuple classes), a unique most-specific method is chosen. A method M 1 in a tuple class (C 1,1 ,.,C 1,n ) is more specific than a method M 2 in a tuple class (C 2,1 ,.,C 2,n ) if for each 1-i-n, C 1,i is a subclass of C 2,i . (The other arguments in the methods are not involved in determining specificity.) If no applicable methods exist, a message-not-understood error occurs. If there are applicable methods but no most- specific one, a message-ambiguous error occurs. Algorithms for efficiently implementing multiple dispatch exist (see, e.g., [Kiczales & Rodriguez 93]). This semantics immediately justifies part 1(a) of our claim for the orthogonality of the multiple dispatch mechanism. An SDCore expression cannot send a message to a tuple. Furthermore, the semantics of message sends has two cases: one for sending messages to instances and one for sending messages to tuples. Hence the meaning of an expression in SDCore is unaffected by the presence of multiple dispatch. The semantics for tuple classes also justifies our second claim. That is, since tuple classes have no special privileges to access the fields of their component instances, the encapsulation properties of classes are unaffected. However, because of this property, Tuple, like other multimethod languages, does not solve the "privileged access" aspect of the binary methods problem [Bruce et al. 95]. It may be that a mechanism such as C++ friendship grants would solve most of this in practice. We avoided giving methods in tuple classes default privileged access to the fields of the instances in a tuple because that would violate information hiding. In particular, any client could access the fields of instances of a class C simply by creating a tuple class with C as a component. 3.2 Multiple Dispatch is not just for Binary Methods Multimethods are useful in many common situations other than binary methods [Chambers 92, Baumgarter et al. 96]. In particular, any time the piece of code to invoke depends on more than one argument to the message, a multimethod easily provides the desired semantics. For example, suppose one wants to print points to output devices. Consider a class Output with three subclasses: Terminal, an ordinary Printer, and a ColorPrinter. We assume that ColorPrinter is a subclass of Printer. Printing a point to the terminal requires different code than printing a point to either kind of printer. In addition, color printing requires different code than black-and-white printing. Figure 3 shows how this situation is programmed in Tuple. In this example, there is no binary method problem. In particular, the addition of print methods to the Point and ColorPoint classes will not upset the fact that ColorPoint is a subtype of Point. The problem is that we need to invoke the appropriate method based on both whether the first argument is a Point or ColorPoint and whether the second argument is a Terminal, Printer, or ColorPrinter. In a singly-dispatched language, an unnatural work-around such as Ingalls's "double dispatching" technique [Ingalls 86, Bruce et al. 95] is required to encode the desired behavior. 3.3 Tuples vs. Classes The ability to express multiple dispatching via dispatching on tuples is not easy to simulate in a single-dispatching language, as is well-known [Bruce et al. 95]. The Ingalls double-dispatching technique mentioned above is a faithful simulation but often requires exponentially (in the size of the more methods than a multimethod-based solution. A second attempt to simulate multiple dispatch in single- dispatching languages is based on product classes [Section tuple class (p:Point, out:Terminal) { method print():() { - prints Points to the terminal . } tuple class (p:Point, out:Printer) { method print():() { - prints Points to the printer . } tuple class (cp:ColorPoint, out:ColorPrinter) { method print():() { - print ColorPoints to the printer in color . } Figure 3: Multimethods in tuple classes for printing. The unit tuple type, (), is like C's void type. 3.2, Bruce et al. 95]. This simulation is not faithful, as it loses dynamic dispatch. However, it is instructive to look at how this simulation fails, since it reveals the essential capability that Tuple adds to SDCore. Consider the following classes in SDCore (adapted from the Bruce et al. paper). class TwoPoints { fields(p1:Point, p2:Point) method { and class TwoColorPoints { fields(cp1:ColorPoint, cp2:ColorPoint) method { and and (new TwoColors( cp1.color(), With these classes, one could create instances that simulate tuples via the new expression of SDCore. For example, an instance that simulates a tuple containing two Point instances is created by the expression new TwoPoints(my_p1,my_p2). However, this loses dynamic dispatching. The problem is that the new expression requires the name of the associated class to be given statically. In particular, when the following message send expression is executed (new TwoPoints(my_p1,my_p2)).equal() the method in the class TwoPoints will always be invoked, even if both my_p1 and my_p2 denote ColorPoint instances at run-time. By contrast, a tuple expression does not statically determine what tuple classes are applicable. This is because messages sent to tuples use the dynamic classes of the values in the tuple instead of the static classes of the expressions used to construct the tuple. For example, even if the static classes of my_p1 and my_p2 are both Point, if my_p1 and my_p2 denote ColorPoint instances, then the message send expression (my_p1,my_p2).equal() will invoke the method in the tuple class for (ColorPoint, ColorPoint) given in Figure 2. Hence sending messages to tuples is not static overloading but dynamic overloading. It is precisely multiple dispatch. Of course, one can also simulate multiple dispatch by using a variant of the typecase statement to determine the dynamic types of the arguments and then dispatching appropriately. (For example, in Java one can use the getClass method provided by the class Object.) However, writing such code by hand will be more error-prone than automatic dispatch by the language. Such dispatch code will also need to be duplicated in each method that requires multiple dispatch, causing code maintenance problems. Every time a new class enters the system, the dispatch code will need to be appropriately rewritten in each of these places, while in Tuple no existing code need be modified. Further problems can arise if the intent of the dispatch code (to do dispatch) is not clear to maintenance programmers. By contrast, when writing tuple classes it is clear that multiple dispatch is desired. The semantics of Tuple ensures that each dispatch is handled consistently, and the static type system ensures that this dispatching is complete and unambiguous. 3.4 Type Checking for Tuple We add to the type attributes of SDCore product types of the these are the types of tuples containing elements of types T 1 ,.,T n . A product type (T 1 -,.,T n -) is a subtype of (T 1 ,.,T n ) when each T i - is a subtype of T i . Because of the multiple dispatching involved, type checking messages sent to tuples is a bit more complex than checking messages sent to instances (see Appendix B for the formal typing rules). We divide the additions to SDCore's type system into client-side and implementation-side rules [Chambers & Leavens 95]. The client-side rules check messages sent to tuples, while the implementation-side rules check tuple class declarations and their methods. The aim of these rules is to ensure statically that at run-time no type mismatches occur and that no message-not-understood or message-ambiguous error will occur in the execution of messages sent to tuples. The client-side rule is the analog of the method application rule for ordinary classes described above. In particular, given an application that there is some tuple class for the product type (T 1 ,.,T n ) such that the static type of Further, that tuple class must contain a method implementation named I with m-n additional arguments such that the static types of E are subtypes of the method's additional argument types. Because the rule explicitly checks for the existence of an appropriate method implementation, this eliminates the possibility of message- not-understood errors. However, the generalization to multiple dispatching can easily cause method-lookup ambiguities. For example, consider again the Point/ColorPoint example from Section 2. Suppose that, rather than defining equality for the tuple classes (Point,Point) and (ColorPoint,ColorPoint), we had defined equality instead for the tuple classes (Point,ColorPoint) and (ColorPoint,Point). According to the client-side rule above, an equal message sent to two ColorPoint expressions is legal, since there exists a tuple class of the right type that contains an appropriate method implementation. The problem is that there exist two such tuple classes, and neither is more specific than the other. Therefore, at run-time such a method invocation will cause a method-ambiguous error to occur. Our solution is based on prior work on type checking for multimethods [Castagna et al. 92, Castagna 95]. For each pair of tuple classes that have a method named I that accepts k additional arguments, we check two conditions. The first check ensures monotonicity [Castagna et al. 92, Castagna 95, Goguen & Meseguer 87, Reynolds 80]. Let -U- be the types of the methods named I in the tuple classes respectively. Suppose that (T 1 ,.,T n ) is a subtype of must be a subtype of -U-. By the contravariant rule for function types, this means that for each j, S j - must be a subtype of S j , and U must be a subtype of U-. The second check ensures that the two methods are not ambiguous. We define two types S and T to be related if either S subtypes T or vice versa. In this case, min(S,T) denotes the one of S and T that subtypes the other. It must be the case that (T 1 ,.,T n ) and (T 1 -,.,T n -) are not the same tuple. Further, if for each j, T j and T j - are related, then there must be a tuple class (min(T 1 ,T 1 -),.,min(T n ,T n -)) that has a method named I with k additional arguments. The existence of this method is necessary and sufficient to disambiguate between the two methods being checked. The type rules for tuple classes and message sends to tuples validate part (b) of our first claim. That is, Tuple's extensions to the SDCore type system are orthogonal. The typing rules in Tuple are a superset of the typing rules in SDCore. Hence, if an SDCore program or expression has type T, it will also have type T in Tuple. In Tuple we chose a by-name typing discipline, whereby there is a one-to-one correspondence between classes and types. This unification of classes with types and subclasses with subtypes allows for a very simple static type system. It also reflects the common practice in popular object-oriented languages. Indeed, this approach is a variant of that used by C++ and Java. (Java's interfaces allow a form of separation of types and classes.) Although the type system is simplistic, the addition of multimethods to the language greatly increases its expressiveness, allowing safe covariant overriding while preserving the equivalence between subclassing and subtyping. There are several other ways in which we could design the type system. For example, a purely structural subtyping paradigm could be used, with classes being assigned to record types based on the types of their methods. Another possibility would be to maintain by-name typing but keep this typing and the associated subtyping relation completely orthogonal to the class and inheritance mechanisms. This is the approach taken in Cecil [Chambers 95]. We ruled out these designs for the sake of clarity and simplicity. Another design choice is whether to dispatch on classes or on types. In Tuple, this choice does not arise because of the strong correlation between classes and types. In particular, the Tuple dispatching semantics can be viewed equivalently as dispatching on classes or on types. However, in the two alternate designs presented above, the dispatching semantics could be designed either way. Although both options are feasible, it is conceptually simpler to dispatch on classes, as this nicely generalizes the single-dispatching semantics and keeps the dynamic semantics of the language completely independent of the static type system. The names of the tuple formals in a tuple class are, in a sense, a generalization of self for a tuple. They also allow a very simple form of the pattern matching found in functional languages such as ML [Milner et al. 90]. Having the tuple formals be bound to the elements of the tuple allows Tuple, like ML, to include tuple values without needing to build into the language primitive operations to extract the components of a tuple. It is interesting to speculate about the advantages that might be obtained by adding algebraic datatypes and more extensive pattern-matching features to object-oriented languages (see also [Bourdoncle & Merz 97, Ernst et al. 98]). In this section we discuss two kinds of related work. The first concerns generic-function languages; while these do not solve the problem we address in this paper, using such a language is a way to obtain multiple dispatch. The second, more closely-related work, addresses the same problem that we do: how to add support for multiple dispatch to languages with single dispatch. An inspirational idea for our work is the technique for avoiding binary methods by using product classes described by Bruce et al. [Section 3.2, Bruce et al. 95]. We discussed this in detail in Section 3.3 above. Another source of inspiration for this work was Castagna's paper on covariance and contravariance [Castagna 95]. This makes clear the key idea that covariance is used for all arguments that are involved in method lookup and contravariance for all arguments that are not involved in lookup. In Tuple these two sets of arguments are cleanly separated, since in a tuple class the arguments in the tuple are used in method lookup, and any additional arguments are not used in method lookup. The covariance and contravariance conditions are reflected in the type rules for Tuple by the monotonicity condition. 4.1 Generic-Function Languages Our approach provides a subset of the expressiveness of CLOS, Dylan, and Cecil multimethods, which are based on generic functions. Methods in tuple classes do not allow generic functions to have methods that dynamically dispatch on different subsets of their arguments. That is, in Tuple the arguments that may be dynamically dispatched upon must be decided on in advance, since the distinction is made by client code when sending messages to tuples. In CLOS, Dylan, and Cecil, this information is not visible to clients. On the other hand, a Tuple programmer can hide this information by always including all arguments as part of the tuple in a tuple class. (This suggests that a useful syntactic sugar for Tuple might be to use f(E 1 ,.,E n ) as sugar for is at least 2.) Second, generic function languages are more uniform, since they only have one dispatching mechanism and can treat single dispatching as a degenerate case of multiple dispatching rather than differentiating between them. Although we believe that these advantages make CLOS-style multimethods a better design for a new language, the approach illustrated by Tuple has some key advantages for adapting existing singly-dispatched languages to multimethods. First, our design can be used by languages like C++ and Java simply by adding tuple expressions, tuple types, tuple classes, and the ability to send messages to tuples. As we have shown, existing code need not be modified and will execute and type check as before. This is in contrast to the generic function model, which causes a major shift in the way programs are structured. Second, our model maintains class-based encapsulation, keeping the semantics of objects as self-interpreting records. The generic function model gives this up and must base encapsulation on scoping constructs, such as packages [Chapter 11, Steele 90] or local declarations [Chambers & Leavens 97]. 4.2 Encapsulated and Parasitic Multimethods Encapsulated multimethods [Section 4.2.2, Bruce et al. 95] [Section 3.1.11, Castagna 97] are similar in spirit to our work in their attempt to integrate multimethods into existing singly-dispatched languages. The following example uses this technique to program equality tests for points in an extension to SDCore. class Point { method equal(p:Point):bool { class ColorPoint inherits Point { method equal(p:Point):bool { method equal(p:ColorPoint):bool { and With encapsulated multi-methods, each message send results in two dispatches (in general). The first is the usual dispatch on the class of the receiving instance (messages cannot be sent to tuples). This dispatch is followed by a second, multimethod dispatch, to select a multimethod from within the class found by the first dispatch. In the example above, the message p1.equal(p2) first finds the dynamic class of the object denoted by p1. If p1 denotes a ColorPoint, then a second multimethod dispatch is used to select between the two multimethods for equal in the class ColorPoint. In essence, the first dispatch selects a generic function formed from the multimethods in the class of the receiver, and the second dispatch is the usual generic function dispatch on the remaining arguments in the message. One seeming advantage of encapsulated multimethods is that they have access to the private data of the receiver object, whereas in Tuple, a method in a tuple class has no privileged access to any of the elements in the tuple. In languages like C++ and Java, where private data of the instances of a class are accessible by any method in the class, this privileged access will be useful for binary methods. However, this advantage is illusory for multimethods in general, as no special access is granted to private data of classes other than that of the receiver. This means that access must be provided to all clients, in the form of accessor methods, or that some other mechanism, such as C++ friendship grants, must provide such access to the other arguments' private data. Two problems with encapsulated multimethods arise because the multimethod dispatch is preceded by a standard dispatch to a class. The first problem is the common need to duplicate methods or to use stub methods that simply forward the message to a more appropriate method. For example, since ColorPoint overrides the equal generic function in Point, it must duplicate the equal method declared within the Point class. As observed in the Bruce et al. paper, this is akin to the Ingalls technique for multiple polymorphism [Ingalls 86]. Parasitic multimethods [Boyland & Castagna 97], a variant of encapsulated multimethods, remove this disadvantage by allowing parasitic methods to be inherited. The second problem caused by the two dispatches is that existing classes sometimes need to be modified when new are added to the system. For example, in order to program special behavior for the equality method accepting one Point and one ColorPoint (in that order), it is necessary to modify the Point class, adding the new encapsulated multimethod. This kind of change to existing code is not needed in Tuple, as the programmer simply creates a new tuple class. Indeed, Tuple even allows more than one tuple class with the same tuple of component classes, allowing new multimethods that dispatch on the same tuple as existing multimethods to enter the system without requiring the modification of existing code. Encapsulated and parasitic multimethods have an advantage in terms of modularity over both generic-function languages and Tuple. The modularity problem of generic-function languages, noted by Cook [Cook 90], is that independently-developed program modules, each of which is free of the possibility of message-ambiguous errors, may cause message-ambiguous errors at run-time. For example, consider defining the method equal in three modules: module A defines it in a tuple class (Point, Point), module B in a tuple class (Point, ColorPoint), and module C in a tuple class (ColorPoint, Point). By themselves these do not cause any possibility of message- ambiguous errors, and a program that uses either A and B or A and C will type check. However, a program that includes all three modules may have message-ambiguous errors, since a message sent to a tuple of two ColorPoint instances will not be able to find a unique most-specific method. Therefore, a link-time check is necessary to ensure type safety. Research is underway to resolve this problem for generic function languages [Chambers & Leavens 95], which would also resolve it for Tuple. However, to date no completely satisfactory solution has emerged. The design choices of encapsulated and parasitic multimethods were largely motivated by the goal of avoiding this loss of modularity. Encapsulated multimethods do not suffer from this problem because they essentially define generic functions within classes, and each class must override such a generic function as a whole. (However, this causes the duplication described above.) Parasitic multimethods do not have this problem because they use textual ordering within a class to resolve ambiguities in the inheritance ordering. However, this ordering is hard to understand and reason about. In particular, if there is no single, most-specific parasite for a function call, control of the call gets passed among the applicable parasites in a manner dependent on both the specificity and the textual ordering of the parasites, and determining at which parasite this ping-ponging of control terminates is difficult. Boyland and Castagna also say that, compared with their textual ordering methods by specificity as we do in Tuple "is very intuitive and clear" [Page 73, Boyland & Castagna 97]. Finally, they note that textual ordering causes a small run-time penalty in dispatching, since the dispatch takes linear instead of logarithmic time, on the average. The key contribution of this work is that it describes a simple, orthogonal way to add multiple dispatch to existing single-dispatch languages. We showed that the introduction of tuples, tuple types, tuple classes for the declaration of multimethods, and the ability to send messages to tuples is orthogonal to the base language. This is true in both execution and typing. All that tuple classes do is allow the programmer to group several multimethods together and send a tuple a message using multimethod dispatching rules. Since existing code in single-dispatching languages cannot send messages to tuples, its execution is unaffected by this new capability. Hence our mechanism provides an extra layer, superimposed on top of a singly-dispatched core. Design decisions in this core do not affect the role or functionality of tuples and tuple classes. Tuple also compares well against related attempts to add multiple dispatch to singly-dispatched languages. We have shown that Tuple's uniform dispatching semantics avoids several problems with these approaches, notably the need to "plan ahead" for multimethods or be forced to modify existing code as new classes enter the system. On the other hand, this uniformity also causes Tuple to suffer from the modularity problem of generic-function languages, which currently precludes the important software engineering benefits of separate type checking. The Tuple language itself is simply a vehicle for illustrating the idea of multiple dispatching via dispatching on tuples. Although it would complicate the theoretical analysis of the mechanism, C++ or Java could be used as the singly- dispatched core language. ACKNOWLEDGMENTS Thanks to John Boyland for discussion and clarification about parasitic multimethods. Thanks to the anonymous referees for helpful comments. Thanks to Olga Antropova, John Boyland, Giuseppe Castagna, Sevtap Karakoy, Clyde Ruby, and R. C. Sekar for comments on an earlier draft. Thanks to Craig Chambers for many discussions about multimethods. Thanks to Olga Antropova for the syntactic sugar idea mentioned in Section 4.1. Thanks to Vassily Litvinov for pointing us to [Baumgartner et al. 96] and to Craig Kaplan for an idea for an example. Leavens's work was supported in part by NSF Grants CCR 9593168 and CCR-9803843. --R American National Standards Institute. Subtyping recursive types. The Java Programming Language. On the Interaction of Object-Oriented Design patterns and Programming Languages Merging Lisp and Object-Oriented Programming Type Checking Higher-Order Polymorphic Multi- Methods Parasitic Methods: An Implementation of Multi-Methods for Java Safe and decidable type checking in an object-oriented language The Hopkins Object Group A Semantics of Multiple Inheritance. A Calculus for Overloaded Functions with Subtyping. Covariance and contravariance: conflict without a cause. The Cecil Language: Specification and Rationale: Version 2.0. Typechecking and Modules for Multi-Methods BeCecil, A Core Object-Oriented Language with Block Structure and Multimethods: Semantics and Typing Inheritance is not Subtyping. Predicate Dispatching: A Unified Theory of Dispatch. The Dylan Programming Book. Design Patterns: Elements of Reusable Object-Oriented Software Steele</Author>, <Author>Guy L.</Author> Steele The Java Language Specification. A Simple Technique for Handling Multiple Polymorphism. Rodriguez Jr. The Language. The Definition of Standard ML. Using Category Theory to Design Implicit Conversions and Generic Operators. The Structure of Typed Programming Languages. The Dylan Reference Manual: The Definitive Guide to the New Object-Oriented Dynamic Language Steele Jr. --TR Smalltalk-80: the language and its implementation Object-oriented programming with flavors CommonLoops: merging Lisp and object-oriented programming A simple technique for handling multiple polymorphism A semantics of multiple inheritance The definition of Standard ML Common LISP: the language (2nd ed.) Inheritance is not subtyping The C++ programming language (2nd ed.) Eiffel: the language A calculus for overloaded functions with subtyping Subtyping recursive types Safe and decidable type checking in an object-oriented language Object-oriented programming Efficient method dispatch in PCL The structure of typed programming languages Design patterns Covariance and contravariance Typechecking and modules for multimethods On binary methods The Dylan reference manual Object-oriented programming Object-oriented software construction (2nd ed.) Parasitic methods Type checking higher-order polymorphic multi-methods The Java programming language (2nd ed.) The Java Language Specification Object-Oriented Multi-Methods in Cecil Dispatching Using category theory to design implicit conversions and generic operators Object-Oriented Programming Versus Abstract Data Types --CTR Rajeev Kumar , Vikram Agrawal, Multiple dispatch in reflective runtime environment, Computer Languages, Systems and Structures, v.33 n.2, p.60-78, July, 2007 Timmy Douglas, Making generic functions useable in Smalltalk, Proceedings of the 45th annual southeast regional conference, March 23-24, 2007, Winston-Salem, North Carolina Ran Rinat, Type-safe convariant specialization with generalized matching, Information and Computation, v.177 n.1, p.90-120, 25 August 2002 Curtis Clifton , Gary T. Leavens , Craig Chambers , Todd Millstein, MultiJava: modular open classes and symmetric multiple dispatch for Java, ACM SIGPLAN Notices, v.35 n.10, p.130-145, Oct. 2000 Todd Millstein , Mark Reay , Craig Chambers, Relaxed MultiJava: balancing extensibility and modular typechecking, ACM SIGPLAN Notices, v.38 n.11, November Antonio Cunei , Jan Vitek, PolyD: a flexible dispatching framework, ACM SIGPLAN Notices, v.40 n.10, October 2005 Paolo Ferragina , S. Muthukrishnan , Mark de Berg, Multi-method dispatching: a geometric approach with applications to string matching problems, Proceedings of the thirty-first annual ACM symposium on Theory of computing, p.483-491, May 01-04, 1999, Atlanta, Georgia, United States Curtis Clifton , Todd Millstein , Gary T. Leavens , Craig Chambers, MultiJava: Design rationale, compiler implementation, and applications, ACM Transactions on Programming Languages and Systems (TOPLAS), v.28 n.3, p.517-575, May 2006 Yuri Leontiev , M. Tamer zsu , Duane Szafron, On type systems for object-oriented database programming languages, ACM Computing Surveys (CSUR), v.34 n.4, p.409-449, December 2002

semantics;tuple;multimethods;generic functions;typing;binary methods;single dispatch;multiple dispatch;language design

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Similarity and Symmetry Measures for Convex Shapes Using Minkowski Addition.

AbstractThis paper is devoted to similarity and symmetry measures for convex shapes whose definition is based on Minkowski addition and the Brunn-Minkowski inequality. This means, in particular, that these measures are region-based, in contrast to most of the literature, where one considers contour-based measures. All measures considered in this paper are invariant under translations; furthermore, they can be chosen to be invariant under rotations, multiplications, reflections, or the class of affine transformations. It is shown that the mixed volume of a convex polygon and a rotation of another convex polygon over an angle is a piecewise concave function of . This and other results of a similar nature form the basis for the development of efficient algorithms for the computation of the given measures. Various results obtained in this paper are illustrated by experimental data. Although the paper deals exclusively with the two-dimensional case, many of the theoretical results carry over almost directly to higher-dimensional spaces.

Introduction The problem of shape similarity has been extensively investigated in both machine vision and biological vision. Although for human perception, different features such as shape, color, reflectance, functional information play an important role while comparing objects, in machine vision usually only geometric properties of shapes are used to introduce shape similarity. In the literature, one finds two concepts for expressing the similarity of shapes: distance functions measuring dissimilarity and similarity measures expressing how similar two shapes are. In this paper we shall work with similarity measures. In practice, similarity approaches have to be invariant under certain classes of transfor- mations, e.g. similitudes (i.e., translations, rotations, change of scale). Affine transformations are also of great practical value as these can approximate shape distortions arising when an object is observed by a camera under arbitrary orientations with respect to the image plane [1]. A well-known method to develop a similarity approach which is invariant under a given class of transformations is to perform a shape normalization first [2], [3], [4]. In Subsection IV-C of this paper we discuss one particular method based on the ellipse of inertia. In the literature, one finds several different methods for comparing shapes. Among the best known ones are matching techniques [4]. We mention here also contour matching [5], [6], structural matching [7] (which is based on specific structural features), and point set matching [8]. In several approaches, one uses the Hausdorff distance for point sets to describe similarity [9]. An interesting construction of a similitude invariant distance function for polygonal shapes is given in [5]; here one computes the L 2 -distance of the so-called turning functions representing the boundary of the polygons. Several authors use the concept of a scale space to develop a multiresolution similarity approach [10], [11]. Finally, Fourier descriptors derived from contour representations have been used by various authors to describe shape similarity and symmetry [12], [13], [14], [15]. See [16] for a comprehensive discussion. Similarity measures can be used to compute "how symmetric a given shape is", e.g. with respect to reflection in a given line. For many objects, presence or absence of symmetry is a major feature, and therefore the problem of object symmetry identification is of great interest in image analysis and recognition, computer vision and computational geometry. Unfortunately, in many practical cases, exact symmetry does not occur or, if it does, is disturbed by noise. In such circumstances it is useful to define measures of symmetry which give quantitative information about the amount of symmetry of a shape. There exists a vast literature dealing with all kinds of symmetry of shapes and (grey-scale) images: central symmetry [17], [18], [19], reflection symmetry [20], [21], [22], rotation symmetry [23], [24], skew symmetry [25], [26]; see also [27], [28]. The Brunn-Minkowski theory [29] allows us to introduce a general framework for comparing convex shapes of arbitrary dimension. In this paper we introduce and investigate a class of similarity and symmetry measures for convex sets which are based on Minkowski addition, the Brunn-Minkowski inequality, and the theory of mixed volumes. Although we deal with the 2D case, most of results can be extended to higher dimensions. The similarity measures examined in this paper are translation invariant by definition. In ad- dition, they can be defined in such a way that they are also invariant with respect to other transformation groups such as rotations and reflections. We propose efficient algorithms for the computation of similarity measures for convex polygons which are invariant under November 4, 1997 similitude transformations. These algorithms are based on the observation that the given measures are piecewise concave functionals; thus to find their maximal values it is sufficient to compute them only for a finite number of points. Moreover this number is bounded by are the number of vertices of the polygons son. We also propose efficient algorithms for the computation of affine invariant similarity measures. In this case the calculation is preceded by a normalization of the polygonal shapes into so-called canonical shapes. Now the computation of the affine invariant similarity measure is reduced to the computation of the rotation invariant similarity measure for their respective normalizations. In this paper we investigate symmetry measures for convex shapes which are invariant under line reflections and rotations. We introduce symmetry measures using two different approaches. The first approach uses similarity measures, the second is a direct approach. We propose efficient algorithms for the computation of rotation and reflection invariant symmetry measures for convex polygons. The normalization technique makes it possible to compute skew symmetry measures as well. We conclude with an overview of this paper. We start with some notations and recall some basic concepts in Section II. In Section III we give short treatment of the theory of mixed volumes, the Brunn-Minkowski inequality, and some derived inequalities. A formal definition of similarity measures can be found in Subsection IV-A, where we also present some examples based on Minkowski addition. In Subsection IV-B we investigate similarity measures for convex polygons which are invariant under rotations and multiplications, and we present an algorithm to compute such measures efficiently. An affine invariant similarity measure is presented in Subsection IV-C. To define it, we introduce an image normalization (canonical form) based on the ellipse of inertia known from classical mechanics. Symmetry measures are introduced in Section V; there we also give several examples, some of them based on similarity measures. In Section VI we illustrate our theoretical findings with some experimental results, and we end with some conclusions in Section VII. In this paper, most results are given without proof. Readers interested in such proofs, as well as some additional results, may refer to our report [30] from which this paper has been extracted. II. Preliminaries In this section we present some basic notation and other prerequisites needed in the sequel of the paper. By K(IR 2 ), or briefly K, we denote the family of all nonempty compact subsets of IR 2 . Provided with the Hausdorff distance [29] this is a metric space. The compact, convex subsets of IR 2 are denoted by and the convex polygons by just P. In this paper, we are not interested in the location of a shape A ' other words, two shapes A and B are said to be equivalent if they differ only by translation. We denote this as A j B. The Minkowski sum of two sets It is well-known [29] that every element A of C is uniquely determined by its support function given by: Here ha; ui is the inner product of vectors a and u, 'sup' denotes the supremum, and S 1 denotes the unit circle. It is also known that [29]: for C. The support set F (A; u) of A at u 2 S 1 consists of all points a 2 A for which A polygon P ' IR 2 can be represented uniquely by specifying the position of one of its vertices, the lengths and directions of all its edges, and the order of the edges. Below, p i will denote the length of edge i and u i is the vector orthogonal to this edge: see Fig. 1. By we denote the angle between the positive x-axis and u i . Since we are not interested in the location of P , it is sufficient to give the sequence is the number of vertices of P . We will call this sequence the perimetric representation of P . In Fig. 1 we give an illustration. We denote this sequence by M(P ). If the polygon is convex, then the order of does not have to be specified in advance, since in this case the normal vectors are ordered counter-clockwise. In this case we can think of M(P ) as a set. But we can also use the November 4, 1997 Fig. 1. Perimetric representation of a polygon. so-called perimetric measure M(P; \Delta) as an alternative representation [19]: 0; otherwise. We point out that the perimetric measure is a special case of the concept of area measure [29]. It is evident that for every convex polygon P , we have the identity where the sum is taken over all u for which M(P; u) 6= 0. This relation expresses that the contour of P is closed. Moreover, any discrete function that satisfies this relation is the perimetric measure of a convex polygon. It is well-known that the Minkowski addition of two convex polygons can be computed by merging both perimetric representations; see e.g. [31], [32]. Mathematically, this amounts to the following relation: for In the second part of this section we consider affine transformations on IR 2 . The reader may refer to [33] for a comprehensive discussion. The group of all affine transformations on IR 2 is denoted by G 0 . If g 2 G 0 and A 2 K, then Ag. We write for every A 2 K. This is equivalent to saying that g \Gamma g 0 is a translation. We denote by G the subgroup of G 0 containing all linear transformations, i.e., transformations g with Lemma 1: For any two sets A; B ' IR d and for every G, we have We introduce the following notations for subsets of G: ffl I: isometries (distance preserving transformations); ffl R: rotations about the origin; multiplications with respect to the origin by a positive ffl L: (line) reflections (lines passing through the origin); ffl S: similitudes (rotations, reflections, multiplications). Observe that I; R; M and S are subgroups of G. For every transformation g 2 G we can compute its determinant 'det g' which is, in fact, the determinant of the matrix corresponding with g. Note that this value is independent of the choice of the coordinate system. If g is an isometry then j det the converse is not true, however. If H is a subgroup of G, then H+ denotes the subgroup of H containing all transformations with positive determinant. For example, I multiplications and rotations. If H is a subgroup of G, then the set fmh j h 2 H; m 2 Mg is also a subgroup, which will be denoted by MH. Denote by r ' the rotation in IR 2 around the origin over an angle ' in a counter-clockwise direction, and by ' ff the reflection in IR 2 with respect to the line passing through the origin which makes an angle ff with the positive x-axis. The following relations hold: In what follows, the topology on K is the one induced by the Hausdorff metric, also called myopic topology [34]. At several instances in this paper we shall need the following concept. Definition 1: Let H ' G and J ' K. We say that H is J -compact if, for every A 2 J and every sequence fh n g in H, the sequence fh n (A)g has a limit point of the form h(A), It is easy to verify that R is K-compact. However, the subcollection fr R is a rotation with '=- irrational, is not K-compact. The following result is easy to prove. Lemma 2: Assume that H is J -compact and let f : J ! IR be a continuous function. If A 2 J and f 0 := sup h2H f(h(A)) is finite, then there exists an element h 0 2 H such that f(h 0 III. Mixed volumes and the Brunn-Minkowski inequality In this section we present a brief account of the theory of volumes and mixed volumes of compact sets (also called 'mixed areas' in the 2-dimensional case). For a comprehensive treatment the reader may consult the book of Schneider [29]. The volume (or area) of a compact set A will be denoted by V (A). It is well-known that for every affine transformation g the following relation holds: The mixed volume V (A; B) of two compact, convex sets A; B ' IR 2 is implicitly defined by the following formula for the volume of A \Phi B: Fig. 2 for an illustration. Fig. 2. The right figure is P \Phi Q. The sum of the volumes of the light grey regions equals 2V (P; Q), the sum of the volumes of the dark grey regions equals V (P ). The mixed volume has the following properties being arbitrary compact, convex for every affine transformation is continuous in A and B (11) with respect to the Hausdorff metric: Note for example that (9) is a straightforward consequence of (3) and (4)-(5). In this paper the following well-known inequality plays a central role. See Hadwiger [35] or Schneider [29] for a comprehensive discussion. Theorem 1 (Brunn-Minkowski inequality) For two arbitrary compact sets the following inequality holds: with equality if and only if A and B are convex and homothetic modulo translation, i.e., The Brunn-Minkowski inequality (12) in combination with (5) yields the following inequality for mixed volumes: and as before equality holds iff A and B are convex and B j -A for some - ? 0. This latter inequality is called Minkowski's inequality. Using the fact that for two arbitrary real numbers x; y one has equality y, one derives from (12) that: with equality iff A j B and both sets are convex. The mixed volume of two convex polygons can be easily computed using support functions and perimetric representations. Assume that the perimetric representation of Q is given by the sequence (v Furthermore, if h(P; \Delta) is the support function of P , then Fig. 2 for an illustration of this formula. Note that with this formula the additivity of V (P; Q) as stated in (10) follows immediately from the additivity of the support function; see (1). Furthermore, (15) in combination with (6) shows that V (P; Q) is increasing in both arguments. In fact, this observation holds for arbitrary compact, convex sets, i.e., We conclude this section with a formula for the computation of the volume of a 2- dimensional polygon (not necessarily convex) using its perimetric representation. Several formulas for calculating volumes of polyhedra are known [36]. Let the vertices (ordered counter-clockwise) of a polygon P be given by Refer to [36] for some further information. If P is a polygon with perimetric representation then the vertices are given by (putting x 6 u j . Here P needs not be convex. Substitution into (17) gives This is the formula which we will use in the sequel of this paper. IV. Similarity measures This section, which is concerned with similarity measures, falls apart into three subsec- tions. In Subsection IV-A we give a formal definition and present some basic properties. In the next two subsections we treat, respectively, similarity measures that are invariant under rotations and multiplications (Subsection IV-B) and similarity measures that are invariant under arbitrary affine transformations (Subsection IV-C). A. Definition and basic properties One of the goals of this paper is to find a tool which enables us to compare different shapes, but in such a way that this comparison is invariant under a given group H of transformations and can be computed efficiently. For example, if we take for H all rota- tions, then our comparison should return the same outcome for A and B as for A and r(B), where r is some rotation. Towards this goal one could try to find a distance function (or metric) d(A; B) which equals zero if and only if B j h(A) for some h 2 H. Many authors, however, rather work with so-called similarity measures than with distance functions. In this paper we will follow this convention. Definition 2: Let H be a subgroup of G and J ' K. A function oe : J \Theta J ! [0; 1] is called an H-invariant similarity measure on J if 1. 2. 3. 4. 5. oe is continuous in both arguments with respect to the Hausdorff metric. When H contains only the identity mapping, then oe will be called a similarity measure. Although not stated explicitly in the definition above, it is also required that J is invariant under H, that is, If oe is a similarity measure on J and H is a J -compact subgroup of G, then defines an H-invariant similarity measure on J . Unfortunately, oe 0 is difficult to compute in many practical situations. Below, however, we present two cases (with which this can be done efficiently if one restricts attention to convex polygons. Let H be a given subgroup of G, and define and Proposition 1: If H is a C-compact subgroup of G, then (a) oe 1 is an H-invariant similarity measure on C; (b) oe 2 is an MH-invariant similarity measure on C. In [30] we present a simple example which shows that compactness is essential. We conclude this section with the following simple but useful result. Recall that ' 0 is the line reflection with respect to the x-axis. Proposition 2: Let oe be a similarity measure on J , and define (a) If oe is R-invariant, then oe 0 is an I-invariant similarity measure. (b) If oe is G+ -invariant, then oe 0 is a G-invariant similarity measure. To give the reader an idea of the flavour of the proof, we show, for the result in (b), property 3 of the definition of an H-invariant similarity measure (Definition 2). G. There are two possibilities: g 2 G+ or g 2 G n G+ . We consider the second case. We can write Now which had to be demonstrated. B. Rotations and multiplications In this section we consider similarity measures on P which are S+ -invariant, i.e., invariant under rotations and multiplications. We use the similarity measures defined in (20)- In these expressions, the terms V (P \Phi r ' (Q)) and play an important role. Let the perimetric representations of the convex November 4, 1997 polygons P and Q be given by respectively. To compute The support set F (P; r ' (v j )) consists of a vertex of P unless ' is a solution of r ' (v j g. Angles ' for which this holds (i.e., r ' (v are called critical angles. The set of all critical angles for P and Q is given by f( where 6 u denotes the angle of vector u with the positive x-axis. We denote the critical angles by 0 - ' 2-. It is evident that N - n P nQ . Now, fix a vertex We have seen that the support set F (P; r ' (v j )) consists of a vertex C of P ; see Fig. 3. d a O Fig. 3. The support set F (P; r ' (v j consists of the vertex C. be the angle between the line through C with normal vector r ' (v j ) and the line through O and C. It follows that h(P; r ' (v j Taking the second derivative with respect to ' we find we find a similar result for ' 7! Thus we arrive at the following result. Proposition 3: The volume V (P \Phi r ' (Q)) and the mixed volume are functions of ' which are piecewise concave on (' This result is illustrated in Fig. 4.525660 Fig. 4. Left: convex polygons P and Q. Right: the function ' 7! V (P \Phi r ' (Q)) is piecewise concave. The ffi's indicate the location of the critical angles. Consider the S+ -invariant similarity measure obtained from (20) by choosing Then sup -?0; '2[0;2-) =h =h =h Thus, in order to compute oe 1 (P; Q) we have to minimize two expressions, one in - and one in '. The first expression achieves its minimum at 2 , the second at one of the critical angles associated with The similarity measure oe 2 given by (21) results in From Proposition 1 we know that oe 2 is S+ -invariant, too. As above, the maximum is attained at one of the critical angles associated with Q, and we get In Section III we have given some formulas for the computation of (mixed) volumes of convex polygons. The expression in (19) uses the perimetric representation, and we use it to get the following result. Proposition 4: Given the perimetric representation of the convex polygons P and Q, the time complexity of computing oe 1 and oe 2 is O(n P nQ (n P are the number of vertices of P and Q, respectively. If we choose Using that ' min where ~ To find the minimum, we need to consider the critical angles of well as those of Q. C. Affine invariant similarity measure G, then the similarity measures defined in (20) and (21), respectively, are affine invariant (that is, invariant under arbitrary affine transformations). Unfortunately, we do not have efficient algorithms to compute them. However, using the approach of Hong and Tan in [2], we are able to define similarity measures which can be computed efficiently, and which are invariant under a large group of affine transformations, namely G+ , the collection of all linear transformations which have a determinant which is positive. In combination with Proposition 2, this leads to similarity measures which are G-invariant. The basic idea is to transform a set A to its so-called canonical form A ffl in such a way that two sets A and B are equivalent modulo a transformation in G+ if and only if A ffl and are equivalent modulo rotation. The definition of the canonical form, as discussed by Hong and Tan [2], is based on the concept of the ellipse of inertia known from classical mechanics [37]. Note, however, that Hong and Tan [2] use a slightly different approach; they introduce the moment curve which is closely related to the ellipse of inertia. Throughout this section we restrict ourselves to the family of compact sets with positive area, in the sequel denoted by K+ . Consider an axis through the centroid of A, and denote, for a point (x; y) 2 A, by r(x; y) its distance to this axis. The moment of inertia with respect to the axis is given by: Z Z A Here ' denotes the angle between the axis and the x-axis in some fixed coordinate system. An easy calculation shows that (m xx +m yy (see also [38, p.48-53]). Here m A xydxdy. The point q sin ') on the axis traces an ellipse when ' varies between and 2-, the so-called ellipse of inertia. ellipse of inertia a yy Fig. 5. The ellipse of inertia of a shape. This ellipse, depicted in Fig. 5, has its long axis at angle ' 0 , which is the unique solution in [0; -) of the equations sin 2m xy cos Let 2a and 2b be the lengths of these axes, respectively. One easily finds that a = which yields that xy xy A simple calculation shows thata 2 The following definition is due to Hong and Tan [2]. Definition 3: A shape is said to be in canonical form if its centroid is positioned at the origin and its ellipse of inertia is a unit circle. Proposition 5: Every compact set can be transformed into its canonical form by means of a transformation in G+ , namely by a stretching along the long axis of the ellipse of inertia by a factor b=(ab) 1=4 and along the short axis by a factor a=(ab) 1=4 . The proof of this result is based on the observation that, under the transformation (x; y) 7! (-x; -y), the second moments scale as follows: Our next result shows how the canonical form of a shape is affected by affine transformations Proposition (a) For every A 2 K+ we have (b) If In [30] we use the notion of covariance matrix to prove (b); see also [3]. With this result it is easy to construct G+ -invariant similarity measures from R-invariant ones. Proposition 7: Let oe : K+ \Theta K+ ! [0; 1] be an R-invariant similarity measure, and define then oe ffl is a G+ -invariant similarity measure. As the map A 7! A ffl preserves convexity, we get the same result for shapes in as well as for shapes in P. To apply these results for convex polygons, there are at least two possibilities. We can compute the canonical shape of the polygon itself or of the set given by its vertices considered as point masses. In the latter case, which is the one considered below, the previous findings remain valid, albeit that integrals have to be replaced by summations. Furthermore, the stretching factors in Proposition 5 become b (along the long axis) and a (along the short axis), respectively. Suppose we are given the perimetric representation of a convex polygon P . The computation of M(P ffl consists of the following steps (putting 1. Fixing the origin at the first vertex of P , we can find the coordinates of the other vertices; see (18). 2. The centroid of P is given by 3. The second moments m are given by 4. Compute b from (23)-(24). 5. Define OE 6 in such a way that \Gamma-=2 - OE i from 6 6 a tan OE i a tan and Example 1: Consider the rotation invariant similarity measure given by (21), i.e., oe ffl on P as in Proposition 7, then oe ffl is a G+ -invariant similarity measure. Using Proposition 2(b) we obtain a G-invariant similarity measure. V. Symmetry measures Exact symmetry only exists in the mathematician's mind. It is never achieved in the real world, neither in nature nor in man-made objects [39]. Thus, in order to access symmetry of objects (convex 2-dimensional polygons in our case), we need a tool to measure the amount of symmetry. Towards that goal Gr-unbaum [18] introduced the concept of a symmetry measure; refer to [23], [21] for some other references. Below we give a formal definition of this concept. But first we recall some basic terminology. We will restrict attention to the 2-dimensional case, but most of what we say carries over immediately to higher dimensions. The symmetry group of a set A ' IR 2 consists of all g 2 G such that g(A) j A. The use of the word 'group' is justified by the observation that these transformations constitute a subgroup of G. An element g in this subgroup is called a symmetry of A and A is said to be g-symmetric. An element g 2 G for which denoting the identity transformation) for some finite m - 1 is called a cyclic transformation of order m. Sometimes we to denote the dependence on g. It is evident that j det if g is cyclic. In this paper we are mostly interested in symmetries of a given shape which are cyclic. However, as shown in Example 2(b), there may also exist symmetries which are not cyclic. Example 2: (a) Not every cyclic transformation is an isometry. For example, (x; y) 7! (2y; x=2) is cyclic of order 2, but it is not an isometry. (b) Not every symmetry is cyclic. Let B be the unit disk in IR 2 and let A := g(B) for some 2 G. It is obvious that gr ' g \Gamma1 is a symmetry of A for every ' 2 [0; 2-], since r ' is a symmetry of B. In most cases, however, this symmetry is not cyclic. Let, for example, g be the transformation (x; y) 7! (2x; y). Then A is the ellipse x 1. For every ' with '=- irrational, the transformation gr ' g \Gamma1 is a non-cyclic, non-isometric symmetry of the ellipse. If H is a subgroup of G, then the set of cyclic transformations in H is denoted by C(H). It is easy to see that In general, C(H) is not a subgroup. Let e 2 G be a cyclic transformation of order m. We define the mapping e : K ! K by Here the denominator m represents a scaling with factor 1=m. It is easy to see that e (A) is e-symmetric, and we call this set the e-symmetrization of A. Observe that e is not an affine transformation. As a matter of fact, e is defined for shapes rather than for points. Every line reflection ' ff is a cyclic transformation of order 2. The corresponding symmetrization of a set A, that is (A))=2, is called Blaschke symmetrization of A [29]. Proposition 8: Let A 2 C. If e is a cyclic transformation, then V (e (A)) - V (A). Furthermore, the following statements are equivalent: e is a symmetry of A; if the greatest common divisor of k and m equals 1. If e is a cyclic transformation of order m and k / m, then e k is a cyclic transformation of order m, and (e k ) . It is easy to see that every cyclic rotation of order m is of the form Symmetry measures were introduced by Gr-unbaum [18] for point symmetries. Here we will generalize this definition to arbitrary families of cyclic transformations. Definition 4: Let E be a given collection of cyclic transformations and J ' K. A is called an E-symmetry measure on J if, for every e 2 E, the function -(\Delta; e) is continuous on J with respect to the Hausdorff topology, and if Suppose that, in addition, the following property holds: if e has order m and k / m, then -(A; then - is called a consistent E-symmetry measure. G be such that heh we say that - is H-invariant if Note that in this definition we have restricted ourselves to cyclic transformations. Example 3: It is easy to show that defines a symmetry measure for all cyclic rotations r ' (i.e., '=- rational). This symmetry measure is invariant under similitudes. It is not consistent, however. The consistency condition (4) has the following intuitive interpretation. Suppose that a shape A is (nearly) symmetric with respect to rotation over 2-=m, then it is also (nearly) symmetric with respect to rotation over an angle 2k-=m, where 1 - k - m. Moreover, if k / m, then the converse also holds. There are at least two different ways to make an E-symmetry measure consistent. Our next result, the proof of which is straightforward, shows how this can be done. Proposition 9: If - is an E-symmetry measure, then k/me Y k/me both define a consistent E-symmetry measure. If - is H-invariant, then - min and - are H-invariant as well. It is easy to see that - consistent. The next result shows how one can obtain symmetry measures from similarity measures. Proposition 10: Let H be a subgroup of G and E ' C(H) such that If oe is an H-invariant similarity measure, then - given by is a consistent H-invariant E-symmetry measure. Remarks. (a) If we do not assume the conditions in (26), the equality e (A) j h(A) yields that This implies that h(A) is e-symmetric. (b) It is tempting to replace (27) by: -(A; However, such a definition does not allow us to consider invariance under groups H which contain e. For, oe(A; if oe is H-invariant and e 2 H. The following example is based on the similarity measure oe 2 given by (21). Example 4: Let E consist of the rotations e is a positive integer. Furthermore, let . It is clear that condition (26) in Proposition 10 holds, hence defines an S+ -invariant E-symmetry measure. There are other construction methods for symmetry measures besides those based on similarity measures. Below we present several examples of symmetry measures based on Minkowski addition. In Proposition 8 we have seen that V (e e is a cyclic transformation. Let E be a collection of cyclic transformations; we define Proposition 11 below shows that - 1 defines a consistent E-symmetry measure. There is alternative way to define a symmetry measure using mixed volumes. It is based upon the observation (see (13)) that if e is a cyclic transformation. We define Note that in Example 3 we have discussed the case where E comprises all cyclic rotations. At first sight, it seems possible to define yet another symmetry measure by replacing in (29). However, a simple calculation using properties shows that V (A; e using (7), one gets that Therefore, such a definition would coincide with - 1 in (28). Proposition 11: Let E be a given collection of cyclic transformations, then - 1 and - 2 given by (28) and (29), respectively, are E-symmetry measures. The measure - 1 is consistent Suppose, furthermore, that H ' G is such that heh and - 2 are H-invariant. If e is a finite-order rotation or a reflection, and if P is a convex polygon whose perimetric representation is given, then it is easy to compute the perimetric representation of e (P ) by merging the perimetric representations of e i (P ); see Section II. This also leads to an efficient computation of the symmetry measure - 1 . Example 5 (Rotations) Let E consist of all cyclic rotations. Then - 1 given by (28) is a consistent S-invariant E-symmetry measure, S being the group of similitudes. Because of the consistency of - 1 , it suffices to consider :g. Given a polygon P and a rotation r over the angle 2-=m, for some m - 1; the r-symmetrization r (P ) is a polygon which is symmetric under rotations of order m. If M(P; u) is the perimetric measure of P , then we can use (2) to find the perimetric measure of r (P M(r (P ); u) =m =m It is obvious that mod 2-: Using formula (19), we can compute - 1 directly. Table II in Section VI contains the outcomes for a given collection of convex polygons. Example 6 (Line reflections) In this example we restrict ourselves to convex polygons. If E consists of all line reflections, then - 1 given by (28) defines an S-invariant E-symmetry measure. For a line reflection ' ff we find Like in the previous example, we can compute the perimetric measure M(' ff (P ); u) if the perimetric measure of P is given: and 6 u) mod 2-: In Table III in Section VI we compute the symmetry measure - 1 for several convex polygons for the angles The symmetry measure - 2 given by (29) amounts to Thus we get that In most of the literature, one does not compute the symmetry measure for specific line reflections ' ff , but rather the maximum over all lines. In our setting this leads to the following definition. A function ' : K ! [0; 1] is called an index of reflection symmetry if ' is continuous, and only if A is reflection symmetric with respect to some line. If - is a measure of reflection symmetry, such as - 1 in (30), then is an index of reflection symmetry. The computation of this index can be done efficiently because of the following observa- tions. Since V we conclude from Proposition 3 that ff 7! V concave on (ff the angles 2ff k are the critical angles of lying between 0 and 2-. Thus every ff k is of the form 1( 6 where P has perimetric representation f(u g. This yields that the minimum of ff 7! V is achieved at one of the angles ff k . Using the same argument as in Proposition 4, one finds that the index can be computed in O(n 3 In Table III we also give the index as well as the angle of the reflection axis for which the index (maximum) is attained. Example 7 (Skew-symmetry) A shape A is said to be skew-symmetric if there exists an affine transformation g 2 G+ such that g(A) is reflection symmetric with respect to some line. In this example we show that one can use the notion of canonical shapes (Subsection IV-C) to find 'how skew-symmetric' a given shape is. Suppose that A is skew-symmetric; then g(A) is reflection symmetric for some g 2 G+ . The symmetry line of g(A) coincides with one of the axes of inertia, and therefore it is also a symmetry axis of (g(A)) ffl . As this latter shape is a rotation of A ffl (see Proposition 6(b)), we conclude that A ffl is reflection symmetric, too. Conversely, if A ffl is reflection symmetric, then A is skew-symmetric (for, A ffl is the result of two stretchings along the principal axes of the ellipse of inertia of A). Thus we find that A is skew-symmetric if and only if A ffl is reflection symmetric. This yields immediately that we obtain an index of skew symmetry from any index of reflection symmetry applied to the canonical shapes; see Example 6. VI. Experimental results In this section the results obtained previously will be applied to some concrete examples. We consider four, more or less regular, shapes, namely: a triangle, a square, a tetragon with one reflection axis, and a regular octagon. These shapes, along with their canonical forms, are depicted in Fig. 6. In this figure, we depict four other convex polygons (and their canonical forms), namely: P , a reflection of P denoted by P refl , a distortion of P denoted by Q (the lower three points have been shifted in the x-direction), and an affine transformation of Q denoted by Q aff . In Table I we compute the similarity measure oe 2 given by (22) which is S+ -invariant. In the first row we compute oe 2 (Q; R), where R is one of the other polygons depicted in Fig. 6. In the third row we compute the values oe 2 (Q aff ; R). The second row contains the values oe ffl 2 is the G+ -invariant similarity measure obtained from Proposition 7. Observe that we do not compute oe ffl since these values are identical to R). Note for example that using oe 2 , which is invariant under rotations and multi- plications, Q is more similar to the square than to the tetragon (values 0.724 and 0.692, respectively), whereas oe ffl 2 , which is G+ -invariant, gives opposite results (values 0.907 and 0.920, respectively). In Table I we also give the angle at which the maximum in expression (22) is achieved. Often, this 'optimal angle' depends, to a large extent, on the similarity measure that is being employed. Table II and Table III are concerned with symmetry measures for rotations and re- flections, respectively. In Table II we illustrate the measure - 1 discussed in Exam- 6triangle triangle ffl square square ffl tetragon tetragon ffl octagon octagon ffl refl aff Fig. 6. Polygons used in the experiments described in this section. Note that Q ffl aff is a rotation of Q ffl . ple 5 for corresponding with rotations over 360 ffi =m. Observe that In fact, it is easy to see that both - 1 and - 2 defined in (28) and (29), respectively, satisfy for every shape A and every affine transformation g. Table II shows the rotation symmetries of the square ( 90 ffi and 180 ffi degrees) and the degrees). The triangle is not rotation symmetric, but the measure is maximal at an angel of 120 ffi degrees (value 0.696). Note also that P is almost -rotation invariant (value 0.995). Table III shows the reflection symmetry measure of Example 6 for five different reflection axes. Furthermore, the two bottom rows capture the maximum over all axes (i.e., the index November 4, 1997 of reflection symmetry; see Example 6) and the angle at which this maximum is attained. Table III shows that the triangle and tetragon have one reflection axis ( 90 ffi ), and the square and octagon have three reflection axes Furthermore, P is almost reflection symmetric with respect to the axes at 0 ffi and 90 ffi (value 0.995). The angle at which the index of reflection is attained is almost the same for P and Q and 95:3 ffi respectively). VII. Conclusions The objectives of this paper are twofold: on the one hand we wanted to give a formal definition of similarity and symmetry measures that are invariant under a given group of transformations, and to derive some general properties of such measures; see for example Propositions 2, 7, 9, and 10. But, on the other hand, we have introduced some new examples of such measures based on Minkowski addition and the Brunn-Minkowski inequality. We believe that our analysis shows that such measures can be useful in certain applica- tions. By no means, however, do we claim that our approach can be usefully applied in every shape analysis problem. It is clear that our restriction to convex sets is a very severe one: we will come back to this issue below. As our approach is based on the area of shapes, it will be difficult to compare it with boundary-oriented approaches, such as boundary matching. Van Otterloo [16, p.143] points out that the quality of a similarity measure is a subjective matter: "it is usually not possible to make general statements about the quality of a similarity measure on the basis of results in a particular application: a measure that performs well in character recognition does not necessarily perform well in industrial inspection." Our similarity measures do not possess the triangle inequality. Moreover, the approach applied in the paper is limited to convex shapes. In particular this second limitation is a major drawback. To use our approach with non-convex shapes there are at least two options. Firstly, one might still use the perimetric measure of a nonconvex shape, even though it characterizes only a convex shape. Alternatively, one can choose to work with the convex hull of non-convex shapes. In both cases, one has to give up some properties of a similarity measure as given in Definition 2, in particular 4. Although our exposition is mainly restricted to the 2D case, the approach has a straight-forward 8extension to 3D (and higher dimensional) shapes. For example, in the case of 3D shapes, instead of using the perimetric representation, we must use the so-called slope diagram representation [40]. Note that from the computational point of view the 3D case becomes much more difficult, however. We will study such problems in our future work. Acknowledgments A. Tuzikov is grateful to the Centrum voor Wiskunde en Informatica (CWI, Amsterdam) for its hospitality. He also would like to thank G. Margolin and S. Sheynin for discussions on some results in this paper. --R "Recognize the similarity between shapes under affine transformation," "Image normalization for pattern recognition," "An efficiently computable metric for comparing polygonal shapes," "Identification of partially obscured objects in two and three dimensions by matching noisy characteristic curves," Computer Vision "Similarity and affine invariant distances between 2D point sets," "Comparing images using the Hausdorff distance," "A multiresolution algorithm for rotation-invariant matching of planar shapes," "Scale-based description and recognition of planar curves and two-dimensional shapes," "Elliptic Fourier features of a closed contour," "Classification of partial 2-D shapes using Fourier descriptors," "Shape discrimination using Fourier descriptors," "Fourier descriptors for plane closed curves," A Contour-Oriented Approach to Digital Shape Analysis "A determinition of the center of an object by autoconvolution," "Measures of symmetry for convex sets," "Convexity and symmetry: Part 2," "On symmetry detection," "Measures of axial symmetry for ovals," "On the detection of the axes of symmetry of symmetric and almost symmetric planar images," "Measures of N-fold symmetry for convex sets," "Detection of generalized principal axes in rotationally symmetric shapes," "Symmetry detection through local skewed symmetries," "Finding axes of skewed symmetry," "Detection of partial symmetry using correlation with rotated- reflected images," "Symmetry as a continuous feature," The Brunn-Minkowski Theory "Similarity and symmetry measures for convex sets based on Minkowski addition," "A unified computational framework for Minkowski operations," Metric Affine Geometry "Computing volumes of polyhedra," Symmetry in Science and Art "Mathematical morphological operations of boundary-represented geometric objects," --TR --CTR Jos B. T. M. Roerdink , Henk Bekker, Similarity measure computation of convex polyhedra revisited, Digital and image geometry: advanced lectures, Springer-Verlag New York, Inc., New York, NY, 2001 Alexander V. Tuzikov , Stanislav A. Sheynin, Symmetry Measure Computation for Convex Polyhedra, Journal of Mathematical Imaging and Vision, v.16 n.1, p.41-56, January 2002 Hamid Zouaki, Representation and geometric computation using the extended Gaussian image, Pattern Recognition Letters, v.24 n.9-10, p.1489-1501, 01 June Antonio Chella , Marcello Frixione , Salvatore Gaglio, Conceptual Spaces for Computer Vision Representations, Artificial Intelligence Review, v.16 n.2, p.137-152, October 2001 Andrew B. Kahng, Classical floorplanning harmful?, Proceedings of the 2000 international symposium on Physical design, p.207-213, May 2000, San Diego, California, United States Bertrand Zavidovique , Vito Di Ges, The S-kernel: A measure of symmetry of objects, Pattern Recognition, v.40 n.3, p.839-852, March, 2007

symmetry measure;minkowski addition;convex set;similarity measure;brunn-minkowski inequality

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Location- and Density-Based Hierarchical Clustering Using Similarity Analysis.

AbstractThis paper presents a new approach to hierarchical clustering of point patterns. Two algorithms for hierarchical location- and density-based clustering are developed. Each method groups points such that maximum intracluster similarity and intercluster dissimilarity are achieved for point locations or point separations. Performance of the clustering methods is compared with four other methods. The approach is applied to a two-step texture analysis, where points represent centroid and average color of the regions in image segmentation.

Introduction Clustering explores the inherent tendency of a point pattern to form sets of points (clusters) in multidimensional space. Most of the previous clustering methods assume tacitly that points having similar locations or constant density create a single cluster (location- or density-based clustering). Two ideal cases of these clusters are shown in Figure 1. Location or density becomes a characteristic property of a cluster. Other properties of clusters are proposed based on human perception [1, 2, 3] (Figure 2 left) or specific tasks (texture discrimination from perspective distortion [4]), e.g., points having constant directional change of density in Figure 2 right. The properties of clusters have to be specified before the clustering is performed and are usually a priori unknown. This work presents a new approach to hierarchical clustering of point patterns. Two hierarchical clustering algorithms are developed based on the new approach. The first algorithm detects clusters with similar locations of points (location-based clustering). This method achieves identical results as centroid clustering [5, 6, 7] with slight improvement in uniqueness of solutions. The second algorithm detects clusters with similar point separations (density-based clustering). This method can create clusters with points being spatially interleaved and having dissimilar densities called transparent clusters. Figure 3 shows two transparent clusters. The detection orig. data20x2 Figure 1: Ideal three (left) and two (right) clusters for location (left) and density based clusterings. orig.data Orig data Figure 2: Illustration of other possible properties of points creating a cluster. Left - two clusters with smoothly varying nonhomogeneous densities taken from [1]. Right - two clusters with constant directional change of density. Figure 3: Two transparent clusters, C of transparent clusters is a unique feature of the method among all existing clustering methods. The two methods are developed using similarity analysis. The similarity analysis relates intra-cluster dissimilarity with inter-cluster dissimilarity. The dissimilarity of point locations or point separations is considered for clustering and is denoted in general as a dissimilarity of elements e i . Each method can be described as follows. First, every element e i gives rise to one cluster C e i having elements dissimilar to e i by no more than a fixed number '. Second, a sample mean - of all elements in C e i is cal- culated. Third, clusters would be formed by grouping pairs of elements if the sample means computed at the two elements are similar. Fourth, the degree of dissimilarity ' is used to form several (multiscale) partitions of a given point pattern. A hierarchical organization of clusters within multiscale partitions is built by agglomerating clusters for increasing degree of dissimilarity. Lastly, the clusters that do not change for a large interval of ' are selected into the final partition. Experimental evaluation is conducted for synthetic point patterns, standard point patterns (8Ox - handwritten character recognition, IRIS - flower recognition) and point patterns obtained from image texture analysis. Performance of the clustering methods is compared with four other methods (partitional - FORGY, CLUSTER, hierarchical - single link, complete link [5]). Detection of clusters perceived by humans (Gestalt clusters [1]) is shown. Location- and density-based clusterings are suitable for texture analysis. A texture is modeled as a set of uniformly distributed identical primitives [8] (see Figure 4). A primitive is described by a set of photometric, geometrical or topological features (e.g., color or shape). Spatial coordinates of a primitive are described by another set of features. Thus the point pattern obtained from texture analysis consists of two sets of features (e.g., centroid location and average color of primitives) and has to be decomposed first. Location-based clustering is used to form clusters corresponding to identical primitives in one subspace (color of primitives). Density-based clustering creates clusters corresponding to uniformly distributed primitives in the other subspace (centroid locations of primitives). The resulting texture is identified by combining clustering results in the two subspaces. This decomposition approach is also demonstrated on point patterns obtained in other application domains. In general, it is unknown how to determine the choice of subspaces. Thus an exhaustive search for the best division in terms of classification error is used in the experimental part for handwritten character recognition and taxonomy applications. The salient features of this work are the following. First, a decomposition of the Figure 4: Example of textures. Top - original image. Bottom - the resulting five textures (delineated by black line) obtained by location- and density-based clusterings. clustering problem into two lower-dimensional problems is addressed. Second, a new clustering approach is proposed for detecting clusters having any constant property of points (location or density). Third, a density-based clustering method using the proposed approach separates spatially interleaved clusters having various densities, thus is unique among all existing clustering methods. The methods can be related to the graph-theoretical algorithms. This paper is organized as follows. Section 2 provides a short overview of previous clustering methods. Theoretical development of the proposed clustering method is presented in Section 3. Analytical, numerical and experimental performance evaluations of the clustering method follow in Section 4. Section 5 presents concluding remarks. Previous Work Clustering is understood as a low-level unsupervised classification of point patterns [5, 9]. A classification method assigns every point into only one cluster without a priori knowledge. All methods are divided into partitional and hierarchical methods. Partitional methods create a single partition of points while hierarchical methods give rise to several partitions of points that are nested. Partitional clustering methods can be subdivided roughly into (1) error-square clusterings [5, 6], (2) clustering by graph theory [10, 2, 5, 3]) and (3) density estimation clusterings [7, 5, 11, 6]. Error-square clusterings minimize the square error for a fixed number of clusters. These methods require to input the number of sought clusters as well as the seeds for initial cluster centroids. Comparative analysis in this work is performed using the implementations of error-square clusterings called FORGY and CLUSTER [5]. FORGY uses only one K-means pass, where K is a given number of clusters in the final partition. CLUSTER uses the K-means pass followed by a forcing pass. During the forcing pass all mergers of the clusters obtained from the K-means pass are performed until the minimum square error is achieved. Clustering by graph theory uses geometric structures such as, minimum spanning tree (MST), relative neighborhood graph, Gabriel graph and Delaunay triangulation. The methods using these geometric structures construct the graph first, followed by removal of inconsistent edges of the graph. Inconsistent edges and how to remove edges are specified for each method. Due to computational difficulties, only methods using MST are used for higher than three dimensional point patterns. Density estimation clusterings have used the two approaches: (a) count the number of points within a fixed volume (mode analysis, Parsen window), (b) calculate the volume for a fixed number of points (k-nearest neighbors). These methods vary in their estimations (Parsen, Rosenblatt, Loftsgaarden and Quesenberry [11]). Two most commonly used hierarchical clusterings are single-link and complete-link methods [12, 5]. Both methods are based on graph theory. Every point represents a node in a graph and two nodes are connected with a link. A length of a link is computed as the Euclidean distance between two points. Single- and complete-link clusterings begin with individual points in separate clusters. Next, all links having smaller length than a fixed threshold create a threshold graph. The single link method redefines current clustering if the number of maximally connected subgraphs of the threshold graph is less than the number of current clusters. The complete-link method does the same for maximally complete subgraphs. A connected subgraph is defined as any graph that connects all nodes (corresponding to points). A complete subgraph is any connected subgraph that has at least one link for all pairs of nodes (points). The implementations of single-link and complete-link clusterings based on Hubert's and Johnson's algorithms [5] are used for the comparative analysis in this work. The use of clustering methods can be found in many applications related to remote sensing [13, 14, 15], image texture detection [16], taxonomy [12, 17], geography [18, 19] and so on. The objective of this work is to contribute to (1) the theoretical development of non-existing clustering methods and (2) the use of clustering for texture detection. 3 Location- and Density-Based Clusterings First, a mathematical framework is established in Section 3.1. The clustering method is proposed in Section 3.2. The algorithms for hierarchical location- and density-based clusterings are outlined in Section 3.3 and related methods to the proposed ones are compared in Section 3.4. 3.1 Mathematical formulation An n-dimensional (nD) point pattern is defined as a set of points I = fp i g P with coordinates general goal of unsupervised clustering is to partition a set of points I into non-overlapping subsets of points fC j g N and , where W j is an index set from all integer numbers in the interval The subsets of points are called clusters and are characterized in this work by the similarity (dissimilarity) of point locations or point separations. A notion of an element e is introduced to refer either to a point location or a point separation d(l p1 ;p 2 (the Euclidean distance between two points also called length of a link d(l p1 ;p 2 In general, every cluster of elements can be characterized by its maximum intra-cluster dissimilarity intra-cluster similarity ') and minimum inter-cluster dissimilarity the dissimilarity value of any two elements defined as the Euclidean distance Figures 5 and 6 show a cluster of points CF j characterized by and a cluster of point separations (links) CL j characterized by One would like to obtain clusters with a minimum intra-cluster and maximum inter-cluster dissimilarity (maximum intra-cluster and minimum inter-cluster similarity) in order to decrease the probability of misclassification. Thus our goal is to partition a point pattern I into nonoverlapping clusters fC j g N having a minimum intra-cluster and maximum inter-cluster dissimilarity of elements. If clusters of elements are not clusters of points as in the case point separations then a mapping from the clusters obtained to clusters of points is performed. The mapping from clusters of point separations (links) to clusters of points takes two steps: (1) Construct a minimum spanning tree from the average values of individual clusters of links. (2) Form clusters of points sequentially from clusters of links in the order given by the minimum spanning tree (from smaller to larger average values of clusters). d d CF CF d d a F Figure 5: Characteristics of clusters of points. Clusters of two-dimensional points are illustrated. All points from one cluster are within a sphere having the center at p midp and radius ffik d midp a s a s e e a s e e e e l k0 e 0.5e 0.5e e Figure Characteristics of clusters of links. Top: Three clusters of links partitioning two-dimensional points into three clusters of points. Bottom: Characteristics of the three clusters of links. The horizontal axis represents values of Euclidean distances between pairs of points d(l k ). Links in each cluster of links differ in length by no more than ". 3.2 The clustering method Given a set of elements I and the goal, the unknown parameters of the classification problem are the values ' and ff for each final cluster, as well as, the number of final clusters N. Two steps are taken to partition the input elements into clusters. First, a value of intra-cluster dissimilarity ' is fixed and clusters characterized by ' are formed by grouping pairs of elements. The result of the first step is a set of clusters denoted as fCE ' m=1 since they are only characterized by '. Second, a value of ' is estimated. A cluster CE ' j with the estimated value ' is selected into the final partition fCE j g N . The choice of CE ' j is driven by a maximization of inter-cluster dissimilarity ff and a minimization of intra-cluster dissimilarity '. The final partition is aimed to be identical with a ground truth partition fC j g N j=1 , which is assumed to exist for the purpose of evaluating the classification accuracy (number of misclassified elements). The development of the proposed classification method is described next by addressing the following issues. (1) Given a fixed value of intra-cluster dissimilarity ', how to estimate an unknown cluster at a single element? (2) How to group pairs of elements based on estimates calculated at each element? (3) How to estimate a value of intra-cluster dissimilarity ' of an unknown cluster of (1) Estimate of an unknown cluster derived from a single element In order to create an unknown cluster C j , every pair of elements in C j should be grouped together. The grouping is based on a certain estimate of the cluster C j computed at each element. The best estimate of an unknown cluster C j is obtained at a single element e i if the element e i gives rise to a cluster C e i identical with the unknown one C j . It would be possible to create the cluster C e i if the unknown cluster C j of elements is characterized by a value of inter-cluster dissimilarity ff larger d d d a F d d CF d> d1p Figure 7: A cluster of points with e ee a s > e e e Figure 8: A cluster of links with than a value of intra-cluster dissimilarity '. (see Figures 7 and 8). Under the assumption ff ? ', the cluster C e i is obtained from any element e grouping together all other elements e k satisfying the inequality k e Thus for any two elements e 1 and e 2 from the cluster C e i , their pairwise dissimilarity is always less than 2'; if e then 2'. The last fact about 2' intra-cluster dissimilarity leads to a notation C e i (2) Grouping elements into clusters using similarity analysis The final clusters fCE ' characterized by ' are obtained in the following way: (a) Create clusters C 2' g, such that k e (b) Compare all pairs of clusters C 2' (c) Assign elements into the final clusters of elements based on the comparisons in (b). Steps (b) and (c) are performed using similarity analysis. The similarity analysis relates intra-cluster dissimilarity ' and inter-cluster dissimilarity ff of an unknown cluster C ';ff . The relationship between ff and ' breaks down into two cases; ff ? ' and ff - '. For ff ? ', an unknown cluster C ';ff j has a value of inter-cluster dissimilarity larger than a value of intra-cluster dissimilarity. In this case, clusters C 2' are either identical to or totally different from an unknown cluster C ';ff . Thus two elements and e 2 would belong to the same final cluster CE ' e2 . For ff - ', an unknown cluster C ';ff j has a value of inter-cluster dissimilarity smaller or equal to a value of intra-cluster dissimilarity. In this case, clusters C 2' are not identical to an unknown cluster C ';ff . A cluster C 2' is a superset of C ';ff because the cluster C 2' also contains some exterior elements of C ';ff j due to ff - '. For ff - ', it is not known how to group elements into clusters and the analysis of this case proceeds. Our analysis assumes that the case ff - ' occurs due to a random noise. This assumption about random noise leads to a statistical analysis of similarity of clusters C 2' . Two issues are investigated next: (i) a statistical parameter of a cluster C ';ff j that would be invariant in the presence of noise and (ii) a maximum deviation of two statistically invariant parameters computed from clusters C ';ff . First, let us assume that deterministic values of elements are corrupted by a zero mean random noise with a symmetric central distribution. Then a sample mean (average) of elements would be a statistically invariant parameter because the mean of noise is zero. Although the sample mean of noise corrupted elements varies depending on each realization of noise, it is a fixed number for a given set of noise corrupted elements. Thus the sample mean - - j of noise corrupted elements in C ';ff j is a statistically invariant parameter under the aforementioned assumptions about noise. Second, a sample mean is computed from each cluster C 2' and is denoted as . The deviation of - from - - j is under investigation. If C 2' is a subset of C ' PROBABILITY DISTRIBUTION e e Figure 9: Confidence interval for 1D case of e i . then the sample mean - would not deviate by more than ' from - This statement is always true. If there are two arbitrary subsets e l then their sample means would not be more than ' apart, as is a superset of C ' oe C ' then the same deviation of - e i from either - is assumed as before for e '. The validity of the previous if statement depends on the ratio of elements from the true cluster C ' j and other clusters exterior to C ' . Thus for the second issue, the sample mean - is not expected to deviated from - by more than Figure and any two elements would be grouped together if their corresponding sample means - - e1 and - - e2 are not more than ' apart; . The inequality k - used for ff - ' can be applied to the case ff ? '. There would be no classification error in the final partition fCE ' m=1 for the case the inequality was used. For ff - ', the classification error is evaluated in a statistical framework as a probability P r(k - '). The complement probability corresponds to a confidence interval of the mean estimator with a confidence coefficient - and upper and lower confidence limits \Sigma'. (3) Estimation of intra-cluster dissimilarity ' The value of intra-cluster dissimilarity ' is a priori unknown for an unknown cluster . An estimation of the value ' is based on the assumption that an unknown cluster C j with a maximum inter-cluster dissimilarity ff does not change the elements in C j for a large interval of values '. Thus clusters CE ' j that do not change their elements for a large interval of ' are selected into the final partition fCE j g N . The set fCE j g N j=1 is an estimate of the ground truth partition fC j g N . The procedure for an automatic selection of ' uses an analysis of hierarchical classification results and consists of four steps: (i) produce multiple sets of clusters by varying the value ' called multiscale classification, (ii) organize multiscale sets of clusters into a hierarchy of clusters, (iii) detect clusters that do not change their elements for a large interval of ' and (iv) select the value ' based on the analysis in Step (iii). Hierarchical organization of the output is defined as a nested sequence of sets of clusters along the scale axis. The nested sequence is understood as follows: a cluster obtained at scale ' cannot split at scale ' cannot merge at scale with other clusters. The hierarchy of multiscale classification results is guaranteed by modifying elements within the final cluster CE ' m created at each scale ' to the sample mean of elements of the cluster. This implementation of hierarchical organization can be supported by the following fact. Two elements e 1 and e 2 which have identical values belong to the same cluster CE ' m for all scales ' - 0. 3.3 Clustering algorithms Proposed density-based clustering, where elements are links, requires (1) to map clusters of links into clusters of points and (2) to process a large number of links. These two issues are tackled before the final algorithms for location- and density-based clusterings are provided. Specifics of density-based clustering In order to obtain clusters of points, a mapping from clusters of links to clusters of points is designed. The mapping consists of three steps. (1) Compute an average link length of each cluster of links. (2) Construct a minimum spanning tree from the average values of individual clusters of links. (3) Form clusters of points sequentially from clusters of links in the order given by the minimum spanning tree (from smaller to larger average values). Knowing the mapping, the number of processed links is decreased by merging links in the order of the link distances d(l k ) (from the shortest links to the longest links). Clusters CL " are created and the corresponding clusters of points CS " are derived immediately. No other links, which contain already merged points will be processed afterwards. When the union of all clusters of points includes all given points ([CS " more links are processed. Clustering algorithm for location-based clustering (2) Create a cluster CF 2ffi at each point (3) Calculate sample means of CF 2ffi Group together any two points p 1 and p 2 into a cluster CF ffi (5) Assign the sample mean of a cluster CF ffi m to all points in CF ffi m (for all m). Increase ffi and repeat from Step 2 until all points are clustered into one cluster. Select those clusters CF ffi m into the final partition fCF j g N j=1 that do not change over a large interval of ffi values. Clustering algorithm for density-based clustering calculate point separations d(l k ) (length of links) for all pairs of points p i . (2) Order d(l k ) from the shortest to the longest; d(l 1 (3) Create clusters of links CL 2" l k for each individual link l k , such that d(l k Calculate sample means - l k (5) Group together pairs of links l 1 and l 2 sharing one point p i into a cluster of links ". Assign those unassigned points to clusters CS " which belong to links creating clusters . Remove all links from the ordered set, which contain already assigned points. Perform calculations from Step 3 for increased upper limit there exist unassigned points. Assign the link average of a cluster CL " m to all links from the cluster CL " (for all m). Increase " and repeat from Step 2 until all points are partitioned into one cluster. Select those CL " clusters into the final partition fCS j g N j=1 that do not change over a large interval of " values. 3.4 Related clustering methods Location-based clustering is related to centroid clustering [5] and density-based clustering is related to Zahn's method [1]. Centroid clustering achieves results identical to the proposed location-based clustering although the algorithms are different (see [5]). The only difference in performance is in the case of equidistant points, when the proposed method gives a unique solution, while the centroid clustering method does not, due to sequential merging and updating of point coordinates. Zahn's method consists of the followings steps: (1) Construct the minimum spanning tree (MST) for a given point pattern. (2) inconsistent links in the MST. (3) Remove inconsistent links to form connected components (clusters). A link is called inconsistent if the link distance is significantly larger than the average of nearby link distances on both sides of the link. The proposed density-based clustering differs from the Zahn's clustering in the following ways: (1) We use the average of the largest set of link distances (descriptors of CL 2" l k rather than nearby link distances for defining inconsistent link and this leads to more accurate estimates of inconsistent links. (2) We replace the threshold for removing inconsistent links ("significantly larger" in the definition of inconsistent links) with a simple statistical rule. (3) We work with all links from a complete graph 1 rather than a few links selected by MST (this is crucial for detecting transparent clusters). Performance Evaluation The problem of image texture analysis is introduced in Section 4.1. This problem statement explains our motivation for pattern decomposition followed by using both location- and density-based clusterings. Theoretical and experimental evaluations of the methods follow next. The evaluation focuses on (1) clustering accuracy in Section 4.2, (2) detection of Gestalt clusters in Section 4.3, and (3) performance on real applications in Section 4.4. 4.1 Image texture analysis An image texture is modeled as a set of uniformly distributed identical primitives shown in Figure 4. Each primitive in Figure 4 is characterized by its color and size. All primitives having similar colors and shapes are uniformly distributed therefore the centroid coordinates of all texture interior primitives have similar inter-neighbor distances. The goal of image texture analysis is to (1) obtain primitives, (2) partition the primitives into sets of primitives called texture and (3) describe each texture using interior primitives and their distribution. In this work, all primitives are found Links between all pairs of points create a complete graph according to the notation in graph theory. based exclusively on their color. A homogeneity based segmentation [20] is applied to an image. The segmentation partitions an image into homogeneous regions called primitives (similar colors are within a region). A point pattern is obtained from all primitives (regions) by measuring an average color and centroid coordinates of each primitive. Given the pattern, a decomposition of features is performed first. One set of features corresponds to the centroid measurements and the other to the color measurements of primitives. Two lower dimensional patterns are created from these features. The location-based clustering is applied to the pattern consisting of the color feature and the density-based clustering is applied to the pattern consisting of the centroid coordinate features. Clustering results are combined and shown in Figure 4 (bottom). The cluster similarity in each subspace provides a texture description characterized by similarity of primitives and uniformity of distribution. The density-based clustering was applied to the pattern shown in Figure 10 (top). The points from the pattern are numbered from zero to the maximum number of points. The output in Figure 10 (bottom) shows three clusters labeled by the number of a point from each cluster that has the minimum value of its number. Partial spatial occlusions of blobs in the original image gave rise to a corrupted set of features corresponding to the centroid coordinates of primitives. From this follows that the lower dimensional points are not absolutely uniformly distributed in the corresponding subspace. The value of similarity " was selected manually. The method demonstrates its exceptional property of separating spatially interleaved clusters which is a unique property of the clustering methods described here. dist 26 26 26 26 2626 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 Figure 10: Spatially interleaved clusters. Top - original point pattern. Bottom - density-based clustering at 4.2 Accuracy and computational requirements Experimental analysis of clustering accuracy is evaluated by measuring the number of misclassified points with respect to the ground truth. Clustering accuracy is tested for (1) synthetic point patterns generated using location and density models of clusters and (2) standard test point patterns (80x, IRIS), which have been used by several other researches to illustrate properties of clusters (80x is used in [5] and IRIS in [12, 5, 1]). Computational requirements are stated. Experimental results are compared with four other clustering methods, two hierarchical methods - single link and complete link, and two partitional - FORGY and CLUSTER [5]. Synthetic and standard point patterns A point pattern is generated and the points are numbered. Detected clusters are shown pictorially as sets of points labeled by same number. The common number for a cluster corresponds to the number of a point that has the minimum value of its number. Two models were used to generate synthetic point pattern. First, three locations in a two-dimensional space gave rise to a synthetic pattern with three clusters. These three locations were perturbed by Gaussian noise (zero mean, variation oe) with various values of the standard deviation oe. The number of points derived by perturbations of each location varied as well. Figure 11 shows two realizations of synthetic patterns (left column). Results obtained from the location-based clustering are shown in Figure 11, right column. Second, a 2D synthetic point pattern (64 points) was generated with four clusters (30, 10, 12, 12 points) of different densities. The point pattern is shown in Figure 12 (left). Points from the pattern were corrupted by uniform noise \Delta \Sigma 0:5 and by Gaussian noise Figure 12 middle and right). Results obtained from density-based clustering method for the point patterns are shown in Figure 13. input data2040 Figure 11: Clusters detected by location-based clustering. Left column - three clusters, points (top) and 60 points (bottom), Gaussian noise location-based clustering corresponding to the left column point patterns 2.5 orig. data7.512.517.5 data uniform noise10x2 data Gaussian noise15 Figure 12: Synthetic pattern with four clusters of different densities. Internal link distances between points from the four clusters are equal to 1, 2, 3 and (first left). Locations of points are corrupted by noise with uniform (middle) and Gaussian (right) distributions. 1.301122 dist 52 52 52 52 52 52 52 52 dist 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 dist 38 53 53 53 53 53 53 53 53 53 53 53 dist -55150. dist 000000 000000 000000 000000 0000003030303042 42 52 52 52 52 52 52 52 52 52 52 52 52 -55152. dist 000000 000000 000000 000000 000000 Figure 13: Clusters detected by density-based clustering. Clustering results for the patterns shown in Figure 12. Cluster labels for points without noise (top row), with uniform noise (middle row) and with Gaussian noise (bottom row). The number above each plot refers to the value of ". We selected two standard point patterns obtained from (1) handwritten character recognition problem (recognition of 80X with 8 features) and (2) flower recognition problem (Fisher-Anderson iris data denoted as IRIS; recognition of iris setosa, iris versicolor and iris virginica with 4 features). The features are shown in Figure 14. The data set 80x contains 45 points with 3 categories each of size 15 points. The data set IRIS contains 150 points with 3 categories each of size 50. Results expressed in terms of misclassified points are in Table 3. Decomposition of features followed by location- and density-based clusterings is explored for each point pattern (80X and IRIS). It is unknown how to determine the choice of features for the decomposition. The goal is to create lower dimensional point patterns showing inherent tendency to form sets of points with similar locations or approximately constant density. An exhaustive search for the best division in terms of classification error was used. Comparisons with other clustering methods Two partitional clustering methods (FORGY, CLUSTER) and two hierarchical clus- 8petal width and length sepal width and length Figure 14: Features for standard point patterns. Features are shown for the 80x (top) and the IRIS (bottom) standard data. tering methods (single and complete link) were compared with the proposed methods. Compared four methods are fully described in [5]. The comparison is based on the number of misclassified points with respect to the ground truth. The two hierarchical methods were selected because they cluster points using links (clustering by graph theory) that is similar to the proposed density-based clustering. The other two meth- ods, FORGY and CLUSTER, cluster points using their coordinates that is similar to the proposed location-based clustering. The misclassified points for hierarchical methods were counted from the best possible non-overlapping point pattern partition (the closest to the ground truth) with dominant labels within correct clusters. The misclassified points for partitional methods were counted from the closest partition to the ground truth for the two input values (variables), (1) a random seed location for the initial clustering and (2) a number of expected clusters in the result. A summary of clustering results in terms of misclassified points is provided in Tables 1, 2 and 3 for synthetic and standard data shown in Figures 11, 12 and 14. The order of methods based on their performance is shown in the most right column of each table. The performance criterion is the sum of misclassified points for several point patterns with known ground truth clusters (shown in the second column from right in each table). The best method for a class of point patterns shown in Figure 11 is the proposed location-based clustering (see Table 1). A class of point patterns shown in Figure 12 was clustered the most accurately by the proposed density-based clustering (see Table 2). A combination of location- and density-based clusterings applied to 80X and IRIS data led to the best clustering results (see Table 3). The eight-dimensional point pattern 80X was decomposed experimentally into two lower-dimensional spaces; one 4-dimensional subspace (features 1,2,7,8) and one 4-dimensional subspace (features 3,4,5,6) in order to achieve the result stated in Table 3. By applying the location-based clustering to n 4-dimensional points followed by the density-based clustering applied to n 4-dimensional points we could separate 0 from 8X and then 8 from X. The four-dimensional point pattern IRIS was decomposed experimentally as well, but the clustering results were not better than the results from location-based clustering applied alone. All six methods used for the comparison were applied to a class of point patterns with spatially interleaved clusters, e.g., Figures 3 and 10. Proposed density-based clustering outperforms all other methods because it is the only method that is able to separate spatially interleaved clusters. Time and memory requirements time requirement for running each method is linearly proportional to the number of processed elements (N point points, N link links) and to the number of used elements for a sample mean calculation at each element (N CF 2ffi and N CL 2" l k ). The number of processed links N link was reduced by sequential mapping of clusters of links to clusters of points therefore the time requirement was lowered. Time measurements were taken for various patterns. For example, the user time needed for clustering a point pattern having points (similar to one in Figure 11 bottom) takes in average 0:06s Table 1: Number of misclassified points resulting from clustering data in Figure 11. method / data pts order locat. clus. dens. clus. 1 11 8 11 31 6. single link 1 2 7 9 19 5. complete link 1 2. Table 2: Number of misclassified points resulting from clustering data in Figure 12. method / data no noise order locat. clus. dens. clus. 0 3 1 4 1. single link 9 14 13 36 6. complete link 0 11 4 15 2. Table 3: Number of misclassified points resulting from clustering 80X and IRIS data. method / data 80x IRIS perform. pts 150 pts P order locat. dens. clus. 7 14 21 1. locat. clus. 24 14 38 4. dens. clus. single link 24 25 49 7. complete link 12 34 46 6. 2. per one location-based clustering and 1:33s per one density-based clustering on Sparc machine. The size of memory required is linearly proportional to the number of processed elements (N point points, N link links). 4.3 Detection of Gestalt clusters Gestalt clusters are two-dimensional clusters of points that are perceived by humans as point groupings [1, 10]. The goal of this Section is to test the properties of the proposed methods for detecting and describing Gestalt clusters. Properties of the location- and density-based methods are demonstrated using the data of sample cluster problems from [1] and [2]. The sample problems [1] are (1) composite cluster problem, (2) particle track description, (3) touching clusters, (4) touching Gaussian clusters and (5) density gradient detection. Each of the problems tackles one or more configurations of clusters in a given point pattern. The configurations of clusters refer to the properties of individual clusters in a point pattern by which the clusters are detected. The properties are, for exam- ple, location and density of clusters, distribution of points within a cluster (Gaussian clusters), spatial shape of clusters (lines in particle tracks problems), mutual spatial position of clusters (touching clusters, surrounding clusters), density gradient of clusters. Point patterns containing clusters with the abovementioned properties are in Figures 15, 16, 17 and 18. Results corresponding to Gestalt clusters are shown as well. All results were obtained using the proposed methods. There are two acceptable results in Figure 16 for the case of touching clusters with identical densities. The choice of the method and the similarity parameter of a shown result are made manually. The proposed approach to clustering of elements (points or links) gave rise to location- and density-based clusterings. These clusterings can detect and describe Gestalt clusters equally well as graph-theoretical methods using minimum spanning tree [1]. Cases of point patterns similar to Figure require special treatment using the graph-theoretical methods (detecting and removing denser cluster followed by clustering the rest of the points). This drawback is not present in the proposed methods. 4.4 Experimental results on real data. Experimental results on real data are reported for image texture analysis and syn- thesis. The analysis is conducted by (1) segmenting image into regions, (2) creating a point pattern from regions, (3) clustering point pattern and (4) presenting application dependent results. In the following application, the goal is to represent image texture in a very concise way suitable for storage or coding of texture. In order to achieve this goal, a density of texture primitives (homogeneous regions) is assumed to be constant over the whole texture. Thus a concise representation of textures will consist of description of primitives and spatial distributions. Figure 19 (left) shows an image with a regular texture (tablecloth) having approximately constant density of dark, bright and gray rectangles. A decomposition of the tablecloth into sets of dark, bright and gray rectangles was performed by (1) creating a point pattern with three features (centroid locations and average intensity value of segmented regions), (2) using the density-based clustering (see the result of clustering in Figure 20) and (3) separating those regions into one image that gave rise to the points grouped into one cluster during the clustering. The decomposition is shown in Figure (top). A possible synthesis of the image is shown in Figure 21 (bottom). The synthesis starts with painting the background (intensity of the largest region) followed by laying a region representative from each cluster at all centroid locations from the cluster. In this way, a textured image is represented more efficiently than 1.789437 dist00000000000 00 0262626 26 26 26 26 26262626 26 26 26 26 26 2626262626266060606060 Figure 15: Point pattern showing a composite cluster problem and a problem of the particle track description. Top - original data, bottom - result of density-based clustering at dist dist 34 34 34 34 34 34 34 34 3434 34 34 34 34 34 3434 34 3434 3434 34 34 Figure A problem of touching clusters. Top - original data, bottom - results of density- (left) and location- (right) based clusterings at orig.data Figure 17: A problem of touching Gaussian clusters. original data, right - result of location-based clustering at dist 72 72 7272 72 72 72 72 72 72 72 72 72 72 72 72 72 Figure A problem of density gradient detection. original data, right - result of density-based clustering at any single pixel or region based description. 5 Conclusion We have presented a new clustering approach using similarity analysis. Two hierarchical clustering algorithms, location- and density-based clusterings, were developed based on this approach. The two methods process locations or point separations denoted as elements e i . The methods start with grouping elements into clusters C e i for every element e i . All elements in C e i are dissimilar to e i by no more than a fixed value '. The dissimilarity of two elements is defined as their Euclidean distance. A sample mean - of all elements in C e i is calculated. Clusters are formed by grouping elements having similar - . The resulting set of clusters is identified among all clusters obtained by varying '. Those clusters that do not change over a large interval of are selected into the final partition in order to minimize intra-cluster dissimilarity and maximize inter-cluster dissimilarity. Figure 19: Image texture analysis: Image "Tablecloth". Left to right: original image, segmented image, contours of segmented im- age, centroids of segmented regions overlapped with the original image. 7.946564 dist0222222 Figure 20: Result of density-based clustering. A point pattern obtained from Figure 19 is clustered. Numerical labels denote the clusters corresponding to dark blobs of the tablecloth (label 2), white blobs of the tablecloth (label 3), a piece of banana shown in the left corner (label 78) and background with left top triangle and shading of the banana (label 0). Figure 21: Texture analysis and synthesis. The image shown in Figure 19 is decomposed and reconstructed. Top row - image decomposition (analysis) based on obtained clusters shown in Figure 20, bottom row image reconstruction (synthesis). Location-based clustering achieves results identical to centroid clustering. Density-based clustering can create clusters with points being spatially interleaved and having dissimilar densities. The separation of spatially interleaved clusters is a unique feature of the density-based clustering among all existing methods. The accuracy and computational requirements of the proposed methods were evaluated thoroughly. Synthetic point patterns and standard point patterns (8Ox - handwritten character recognition, were used for quantitative experimental evaluation of accu- racy. Performance of the clustering methods was compared with four other methods. Correct detections of various Gestalt clusters were shown. Location- and density-based clusterings were used for image texture analysis. A texture was defined as a set of uniformly distributed identical primitives. Primitives were found by segmenting an image into color homogeneous regions. A point pattern was obtained from textured images by measuring centroid locations and average colors of primitives. Features of this point pattern were divided into two sets, because each set of features required a different clustering model. The centroid locations of primitives, were hypothesized to have uniform distribution therefore the density-based clustering was applied to form clusters in this lower-dimensional subspace corresponding to features of the centroid locations. Properties of primitives, such as color, were modeled to be identical within a texture, therefore the location-based clustering was applied to form clusters in the second lower-dimensional subspace corresponding to the color feature. Resulting texture was identified by combining clustering results in the two subspaces. In a nutshell, this clustering problem required (1) a point pattern decomposition into two lower-dimensional point patterns, (2) location- and density-based clusterings to form clusters from the two point patterns and (3) texture identification using both clustering results. The decomposition approach motivated by image texture analysis was explored for point patterns that originated from hand-written character recognition and taxonomy problems. The contributions of this work can be summarized as (1) addressing a decomposition of the clustering problem into two lower-dimensional problems, (2) proposing a new clustering approach for detecting clusters having a constant property of interior points, such as location or density, and (3) developing a density-based clustering method that separates spatially interleaved clusters having various densities. Acknowledgments The authors greatfully acknowledge all people who provided the data for exper- iments. Point patterns from [1] and [2, 21] - Mihran Tuceryan, Texas Instruments; Standard point patterns 80X and IRIS - Chitra Dorai with permission of Anil Jain; The authors thank for providing the four clustering methods to Chitra Dorai and Professor Anil Jain from the Pattern Recognition and Image Processing Laboratory, Michigan State University. This research was supported in part by Advanced Research Projects Agency under grant N00014-93-1-1167 and National Science Foundation under grant IRI 93-19038. --R "Graph-theoretical methods for detecting and describing gestalt clusters," Extraction of Perceptual Structure in Dot Patterns. "Dot pattern processing using voronoi neighborhoods," "Shape from texture: Integrating texture-element extraction and surface estimation," Algorithms for Clustering Data. analysis. John Wiley and sons inc. "Uniformity and homogeneity based hierarchical cluster- ing," Pattern Classification and Scene Analysis. John Wiley and sons inc. Freeman and com- pany "A binary division algorithm for clustering remotely sensed multispectral images," "A new clustering algorithm applicable to multi-scale and polarimetric sar images," "Texture segmentation using voronoi polygons," The Advanced Theory of Statistics "A comparison of three exploratory methods for cluster detection in spatial point patterns," Models of Spatial Processes. "Segmentation of multidimensional images," "Extraction of early perceptual structure in dot pat- terns: Integrating region, boundary and component gestalt," --TR --CTR Jos J. Amador, Sequential clustering by statistical methodology, Pattern Recognition Letters, v.26 n.14, p.2152-2163, 15 October 2005 Qing Song, A Robust Information Clustering Algorithm, Neural Computation, v.17 n.12, p.2672-2698, December 2005 Chaolin Zhang , Xuegong Zhang , Michael Q. Zhang , Yanda Li, Neighbor number, valley seeking and clustering, Pattern Recognition Letters, v.28 n.2, p.173-180, January, 2007 Kuo-Liang Chung , Jhin-Sian Lin, Faster and more robust point symmetry-based K-means algorithm, Pattern Recognition, v.40 n.2, p.410-422, February, 2007 Hichem Frigui , Cheul Hwang , Frank Chung-Hoon Rhee, Clustering and aggregation of relational data with applications to image database categorization, Pattern Recognition, v.40 n.11, p.3053-3068, November, 2007 Ana L. N. Fred , Jos M. N. Leito, A New Cluster Isolation Criterion Based on Dissimilarity Increments, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.25 n.8, p.944-958, August Ana L. N. Fred , Anil K. Jain, Combining Multiple Clusterings Using Evidence Accumulation, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.27 n.6, p.835-850, June 2005

point patterns;density-based clustering;location-based clustering;hierarchy of clusters;spatially interleaved clusters;clustering

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Scalable S-To-P Broadcasting on Message-Passing MPPs.

AbstractIn s-to-p broadcasting, s processors in a p-processor machine contain a message to be broadcast to all the processors, 1 sp. We present a number of different broadcasting algorithms that handle all ranges of s. We show how the performance of each algorithm is influenced by the distribution of the s source processors and by the relationships between the distribution and the characteristics of the interconnection network. For the Intel Paragon we show that for each algorithm and machine dimension there exist ideal distributions and distributions on which the performance degrades. For the Cray T3D we also demonstrate dependencies between distributions and machine sizes. To reduce the dependence of the performance on the distribution of sources, we propose a repositioning approach. In this approach, the initial distribution is turned into an ideal distribution of the target broadcasting algorithm. We report experimental results for the Intel Paragon and Cray T3D and discuss scalability and performance.

Introduction The broadcasting of messages is a basic communication operation on coarse-grained, message passing massively parallel processors (MPPs). In the standard broadcast operation, one processor broadcasts a message to every other processor. Various implementations of this operation for architectures with different machine characteristics have been proposed [5, 9, 12, 13, 14]. Another well-studied broadcasting operation is the all-to-all broadcast in which every processor broadcasts a message to every other processor [3, 7, 8, 15]. Let p be the number of proces- sors. Assume that s of the p processors, which we call source processors, contain a message to be broadcast to every other processor, 1 - s - p. In this paper we present broadcasting algorithms that handle all ranges of s. We report experimental results for s-to- Research supported in part by ARPA under contract DABT63-92-C-0022ONR. The views and conclusions contained in this paper are those of the authors and should not be interpreted as representing official policies, expressed or implied, of the U.S. government. broadcasting algorithms on the Intel Paragon and discuss their scalability and performance. In general, quantities influencing scalability, and thus the choice of which algorithm gives the best per- formance, include the number of processors, the message sizes, and the number of source processors [10]. Our algorithms are scalable with respect to p, s, and the message sizes; i.e., they maintain their speedup as these parameters change. For s-to-p broadcast- ing, other factors influence scalability as well. For any fixed s, a particular algorithm exhibits a different behavior depending where the s source processors are located. Each algorithm has ideal distribution patterns and distribution patterns giving poor performance. Poor distribution patterns for one algorithm can be ideal for another. Thus, the location of the source processors and the relationship of these locations to the size and dimensions of the architecture effect the scalability of an algorithm. In order to study these relationships to the fullest extent, we assume that every processor knows the position of the source processors and the size of the messages. This implies synchronization occurs before the broadcasting In this paper we describe a number of different broadcasting algorithms and investigate for each algorithm its good and bad distribution patterns. We characterize features of s-to-p broadcast algorithms that perform well on a wide variety of source dis- tributions. Some of our algorithms are tailored towards meshes, others are based on architecture-independent approaches. We show that algorithms that ffl exhibit a fast increase in the number of processors actively involved in the broadcasting process and ffl increase the message length at these processors as slowly as possible give the best performance. We show that achieving these two goals can be difficult for regular machine sizes (i.e., machines whose dimensions are a power of 2). This, in turn, implies that good or bad input distributions cannot be characterized by the pattern alone. The dimension of the machine plays a crucial role as well. The performance obtained on ideal distributions can vary greatly from that obtained on poor distributions. We propose the approach of repositioning sources to guarantee a good performance. The basic idea is to perform a permutation to transform the given distribution into an ideal distribution for a particular algorithm which is then invoked to perform the actual broadcast. The paper is organized as follows. In Section 2 we describe the algorithms that do not reposition their sources. In Section 3 we discuss different repositioning approaches. Section 4 describes the different source distributions we consider. In Section 5 we discuss performance and scalability of the proposed algorithms on the Intel Paragon. Section 6 concludes. Algorithms without Reposi- tioning In this section we describe s-to-p broadcasting algorithms which do not reposition the sources. Our first class of broadcast algorithms generalizes an efficient 1-to-p broadcasting approach. S-to-p broadcasting could be done by having each one of the s source processor initiate a 1-to-p broadcast. How- ever, having the s broadcasting processes take place without interaction is inefficient. Our approach is to let each processor initiate a broadcast, but whenever messages from different sources meet at a proces- sor, messages are combined. Further broadcasting steps proceed thus with larger messages. We use a Binomial heap broadcasting tree [6, 9] to guide the broadcasts. In Algorithm Br Lin, we view the processors of the mesh as forming a linear array (by using a snake-like row-major indexing). The existence of a linear array is not required and the approach is architecture-independent. If processors P i and both contain a message to be broadcast, they exchange their messages and form a larger message consisting of the original and the received message. If only one of the processors contains a message, it sends it to the other one. Then, Algorithm Br Lin proceeds recursively on the first p=2 and the last p=2 processors. Algorithms Br Lin behaves differently for different machine sizes. Whether the number of processors actively involved in the broadcasting process increases, depends on where the source processors are located. For example, when the input distribution consists of columns, the first log p=2 iterations introduce no new sources. For meshes with an odd number of rows, new sources are introduced in the case of column distribution. In order to study the use of only column links or row links during a single iteration for arbitrary mesh sizes, we introduce Algorithm Br xy. In Algorithm Br xy, we first select either rows or columns. Assume the rows were selected. We then view each row as a linear array and invoke Algorithm Br Lin within each row. After this, we invoke Algorithm Br Lin within each column. We consider two versions of Algorithm Br xy which differ on how dimensions are selected. In Algorithm Br xy source, the number of sources in the rows and columns determine the order of the dimen- sions. Recall that every processor knows the positions of the sources. Every processor determines , the maximum number of sources in a row, and , the maximum number of sources in a col- umn. If the rows are selected and Algorithm Br Lin is invoked on the rows. Otherwise, the columns are selected first. A reason for choosing the dimensions in this order is the following. When the rows contain fewer elements, the broadcasting done within the rows is likely to generate messages of smaller size to be broadcast within the columns. Assume sources are located in a few, say ff columns. where r is the number of rows of the mesh. First broadcasting in the rows results in every processors containing ff messages at the time the column broadcast starts. For the sake of comparison, we also consider a version of Algorithm Br xy which compares the dimensions and broadcasts first along longer dimension. Assume the mesh consists of r rows and c columns. Algorithm Br xy dim selects the rows if r - c and the columns if r ! c. In the algorithms described so far processors issue sends and receives to facilitate communication. We do not make use of existing communication operations generally available in communication libraries [1, 2, 7]. S-to-p broadcasting can easily be stated in terms of known communication operations. We considered two such approaches. The first one, Algorithm Xor, invokes an all-to-all personalized exchange communication [7]. The second such approach results in Algorithm 2-Step. This algorithm performs the broadcast by invoking two regular communication operations, one s-to-one followed by an one-to-all operation. In the s-to-one communication, processor receives the s messages from the source processors. combines the s messages and initiates an one-to-all broadcast. 3 Algorithms with Reposi- tioning On coarse-grained machines like the Paragon, sending relatively short messages is cheap compared to the cost of an entire s-to-p operation. At the same time, experimental results show that the performance of our s-to-p algorithms can differ by a factor up to 2 for the same number of sources, depending on where the sources are positioned. Each algorithm has its own ideal source distribution. In this section we consider the approach of repositioning the sources and then invoking an s-to-p algorithm on its ideal input distribution. Algorithm Repos is invoked with one of the algorithms described in the previous section. For the sake of an example, assume it is Algorithm Br Lin. The first step generates Br Lin's ideal input distribution for s sources on the given machine size and machine dimension. This is achieved by each source processor sending its message to a processor determined by the ideal distribution. We refer to the next section for a discussion on ideal distributions. Whether it pays to perform the redistribution depends on the quality of the initial distribution of sources. We point out that our current implementation of Algorithm Repos does not check whether the initial distribution is actually close enough to an ideal distribution. We simply perform the repositioning. Our second class of repositioning algorithms not only repositions the sources, but also makes use of the observation that the time for broadcasting s=2 sources on a p=2-processor machine is less than half of the time for broadcasting s sources on a p-processor machine. Assume we partition the p processors into a group G 1 consisting of p 1 processors and into a group G 2 consisting of p 2 processors. The partition of the processors into two groups is independent of the position of the sources, and may depend on the choice of the broadcasting algorithm invoked on each processor group. After the broadcasting within G 1 and G 2 is completed, every processor in G 1 (resp. G 2 ) exchanges its data with a processor in G 2 (resp. G 1 ). This communication step corresponds to a permutation between the processors in G 1 and G 2 . We refer to Algorithm Part Lin as the algorithm based on this principle and using Algorithm Br Lin within the sub-machines. We refer to Algorithm Part xy source as the algorithm based on this principle and using Algorithm Br xy-source within the sub-machines. 4 Source Distributions In this section we discuss different patterns of source distribution used in our experiments. Some of these distributions exploit the strengths while other highlight the weaknesses of the proposed algorithms. Some are chosen because we expect them to be difficult distributions for all algorithms. For the sake of brevity, distributions may only be explained at an intuitive level. Assume the machine is a mesh of size with r - c and that processors are indexed in row-major order. Let c e. ffl Row and Column Distributions: In row distribution source processors. These i rows are spaced evenly. Every row, with the exception of the last one, contains c source processors. For mesh, R(30) has the source processors positioned as shown in Figure 1. Column distribution C(s) is defined analogously. ffl Equal Distribution: In equal distribution E(s), processor (1; 1) is a source processor and every dp=se- th or bp=sc-th processor is a source processor. For particular values of s, r, and c, E(s) can turn into a row, column, or diagonal distribution, or exhibit a rather irregular position of sources. ffl Right and Left Diagonal Distributions: The right diagonal distribution, Dr(s), has the s source processors positioned on i diagonals. We include the diagonal from (1; 1) to (r; r). The remaining are spaced evenly (using modulo arithmetic), with the last diagonal not necessarily filled with sources. The left diagonal distribution Dl(s) has source processors on the diagonal from (1; c) to (r; 1) and spaces the remaining accordingly. ffl Band Distribution: The band distribution B(s) is similar to the right diagonal distribution. The right diagonal distribution contains i diagonals, each having width 1. The band distribution B(s) contains r e evenly distributed bands, each having width d s br e. ffl Cross Distribution: The cross distribution C(s) corresponds to the union of a row and a column dis- tribution, with the number of source processors in the row distribution being roughly equal to the number of processors in the column distribution. ffl Square Block Distribution: In the square block distribution, SB(s), the source processors are contained in a square mesh of size d se \Theta d se. Figure shows three of the above distributions for 30. The remainder of this section describes how the algorithms handle different distributions. The Row l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l Cross l m l m l l l l l l l l l l l l l l l l l l l l l l l l l l Diagonal l l l l l l l l l l l l Figure 1: Placement of source processors in row, cross, and right diagonal distributions on a machine. performance of the algorithms on these distributions is discussed in the next section. Consider first Algorithm Br xy source. One expects row and column distributions to be ideal source distributions. Algorithm Br xy source will choose the first dimension so that the number of source processors is increased as fast as possible, while the message length increases as slowly as possible. How- ever, not all row and column distributions are equally good. For example, in R(20) on a mesh of size 10\Theta10, the first and the sixth row contain the source processors and thus the first iteration does not increase the number of source processors. Having the 20 sources positioned in the first and the seventh row eliminates this. This is an important observation for the algorithms generating ideal distributions. It shows that the machine dimension effects the ideal distribution of sources. The diagonal distribution places the same number of sources in each row and col- umn. One would expect Algorithm Br xy source to perform quite well on diagonal distributions. The performance of Algorithm Br xy source on the equal distribution will vary. Cross, square block, and band distributions should be considerably more expensive since the source positions may not allow a fast increase in the number of sources. The behavior of Algorithm Br Lin on these source distributions is quite different. First, neither row or column distribution are ideal distributions for Br Lin. For an even number of rows, an iteration achieves no increase in the number of sources on the column distribution. Consider the row distri- bution. When the number of rows is a power of 2, Algorithm Br Lin is actually identical to Algorithm Br xy source. When the number of rows is odd, communication in an iteration occurs between processors not in the same column and congestion will increase. The equal distribution can turn into a row or a column distribution and will thus not be ideal either. The behavior of the algorithm on the left and the right diagonal distribution can differ (no such difference exists for Algorithm Br xy source). On a machine of size 10 \Theta 10, Dr(10) experiences no increase in the number of sources in the first iteration (since processor P 50 lies on the diagonal). For other machine sizes, the right diagonal distribution may not experience such disadvantage. The left diagonal distribution is least sensitive towards the size of the machine and it achieves the desired properties of an efficient broadcasting algorithm. The remaining distributions appear to be difficult distributions. Finally, Algorithm Br xy dim suffers the obvious drawbacks when the selection of the dimension is done according to the size of the dimensions and not according to the number of sources. The ideal distribution for Algorithm Br xy dim will either be a row or a column distribution, depending on the dimensions 5 Experimental Results In this section we report performance results for the broadcasting algorithms on the Intel Paragon. We consider machine sizes from 4 to 256 processors and message sizes from 32 bytes to 16K bytes. We study the performance over a whole spectrum of source numbers ranging from 1 to p and a representative selection of source distributions. In this paper we report only the performance for the case when all source processors broadcast messages of the same length. In our experiments, using different length messages did not influence the performance of the algorithms. In particular, for a given algorithm a good distribution remains a good distribution when the length of messages varies. Throughout this sec- tion, we use L to denote the size of the messages at source processors. Most implementation issues follow in a straightforward way from the descriptions given in the previous sections. We point out that we do not synchronize globally after each iteration or after one dimension has been handled. In all our algorithms, as soon as a processor has all relevant data, it continues. 5.1 Performance of Algorithms without Repositioning In the following we first study the scalability of the five algorithms described in Section 2 for standard scalability parameters such as machine size, number of source processors, and message length. We then consider other relevant parameters, including the distribution of the source processors, the dimension of the machine, and the interaction of the dimension of the machine and the source processor distribution with respect to a particular algorithm. We show that these parameters have a significant impact on the performance. Xor Br_Lin Br_xy_source 100103050Number of Sources Time (msec) Figure 2: Performance of algorithms when the number of sources varies from 1 to 100, assuming and equal distribution on a The communication operations invoked in Algorithms Xor and 2-Step use the implementations described in [7]. In particular, the all-to-all exchange algorithm views the exchanges as consisting of p permutations and it uses the exclusive-or on processor indices to generate the permutations. The most efficient Paragon implementation of an one-to-all communication views the mesh as a linear array and applies the communication pattern used in Algorithm Br Lin; i.e., processor P i exchanges a message with and then the one-to-all communication is performed within each machine half. We did not expect Algorithms Xor and 2-Step to give good per- formance. Xor simply exchanges too many messages and Algorithm 2-Step creates unnecessary communication bottlenecks. However, we did want to see their performance against the other proposed algorithms to show the disadvantage of using existing communication routines in a brute-force way. Figure 2 shows the performance of the five algo- rithms. From this figure it is apparent that Algorithms 2-Step and Xor are not efficient. In particu- lar, for more than 4 sources, Algorithm 2-Step suffers congestion at the node which receives all the messages. Algorithm Xor is inherently inefficient because of the large number of sends issued by the source processors. For Algorithm 2-Step, the rate of increase in the execution time is steeper than the increase in number of sources. This is due to the fact that as the number of source processors increases, the bottleneck processor in Algorithm 2-Step receives more messages in the first step and sends out more data in the second step. However, in the case of Algorithm Xor, with the increase in number of source processors, the increase in the number of sources is more distributed among all processors. Xor Br_Lin Br_xy_source Message Length (in bytes) Time (msec) Figure 3: Performance of algorithms when L varies from 32 bytes to 16K keeping s =30 on a machine with right diagonal distribution. Xor Br_Lin Br_xy_source Machine Size Time (msec) Figure 4: Performance of algorithms when the machine size varies, assuming having approximately sources in a right diagonal distribution The bandwidth of the network is high enough to handle this type of increased communication volume better. The performance of the other three algo- rithms, Br Lin, Br xy source, and Br xy dim scales linearly with the increase in number of sources. Depending on the number of sources and how the equal distribution places sources in the machine, the performance of these algorithms differs slightly. Figure 3 shows the performance for a right diagonal distribution with when the message size changes. As already stated, the diagonal distributions place the same number of sources in the rows and columns. Once again, regardless of how small a message size, Algorithms 2-Step and Xor perform poorly. The almost flat curve up to a message size of 1K for Algorithm Xor further supports our observation related to Figure 2. The other three algorithms experience little increase in the time until bytes. Then we see a linear increase. Figure 4 shows the behavior of the five algorithms when the machine size varies from 4 to 256 processor. Algorithm Xor is as good as any other algorithm for small machine sizes (4 to 16 processors). This feature is also observed when the number of sources is close to p for small machine sizes. The first three figures give the impression that algorithms Br Lin, Br xy source, and Br xy dim give the same performance. However, this is not true. In the following we show that different distributions and different machine sizes effect these algorithms in different ways. Br_Lin Br_xy_source row dia blk cro6789 Distributions Time (msec) Figure 5: Performance of three algorithms on a 10x10 machine with assuming different source distributions with Figure 5 shows the performance for using different distribution patterns. The figure confirms the discussion given in Section 4 with respect to ideal and difficult distributions. Algorithm Br xy source gives roughly the same performance on the first 4 distributions, but for the square block and cross distribution we see a considerable increase in time. We point out that the same performance on the first 4 distributions for Br xy source is not true in general. However, the row and the column distribution show up as ideal distributions. Square block and cross distributions require more time for all three al- gorithms. As expected, Algorithm Br Lin performs best on them. This is due to the fact that in Algorithm Br Lin sources can spread to different rows and columns in the first few iterations, thus utilizing the links more efficiently. On the other hand, for the square block distribution, Algorithms Br xy source, Br xy dim have only few columns and rows available to generate new sources. The big increase in Algorithm Br xy dim for the row distribution indicates the importance of choosing the right dimension first. Br_Lin Br_xy_source 407.58.59.510.511.5Number of Sources Time (msec) Figure Performance of three algorithms on a 10x10 machine with a right diagonal distribution. The total message size is kept at 80K and the number of sources varies. Figure 6 shows the performance of the three algorithms when the total message size (i.e., the sum of the message sizes in the source processors) is fixed. An interesting aspect of the performance curves is that if the data is spread among a larger number of sources, the broadcast operation is accomplished faster. For example the 80K size, data spread among 5 sources takes approximately 11.4 msec using Algorithm Br xy source. However, the same amount of data spread among 40 sources to begin with takes only 7.3 msec. This plot highlights our claim made earlier that for a given amount of data more number of sources involved in broadcasting yield faster execution times. Time Figure 7: Performance of Algorithm Br Lin when and the dimensions vary, assuming equal distribution. Three source sizes are shown and 4K in all the cases. Figure 7 shows the performance of three algorithms when the dimensions of the machine vary. It demonstrates that performance is related to the size the dimensions. For the same number of sources, message size, and number of pro- cessors, a distribution gives different performance (hence is considered good or bad) depending on the dimension of machine. For a small number of sources (for example the machine dimensions may not affect the performance. For a large number of sources, machine dimensions impact the performance considerably more. It seems like an anomaly to have faster performance for The reason lies in the distribution and the number of rows. When the equal distribution tends to place the source processors within columns. This does not allow a fast increase in the number of sources. On the other hand, for the source processors are, with the exception of size 4 \Theta 30, positioned along diagonals. 5.2 Performance of Algorithms with Repositioning Algorithms Br xy source and Br Lin exhibit good performance for a variety of source distributions and machine dimensions. However, each algorithm has source distributions which exhibit the algorithm's weaknesses. In addition, the relationship between source distribution and machine dimension can result in a performance loss. The algorithm cannot change the machine dimension, but the problems arising from the source distribution can be avoided by performing a repositioning. In Section 3 we described a repositioning and a partitioning approach. We next discuss the performance of the repositioning approach using Algorithm Br xy source. We use the row distribution as the ideal source distribution for Br xy source. Similar results hold for the repositioning algorithm using Br Lin with the left diagonal distribution as the ideal source distribution. Let Algorithm Repos xy source be the repositioning algorithm invoking Br xy source. In this algorithm we first perform a permutation to redistribute source processors according to the row distribution. We generate a row distribution that positions the rows so that the number of new sources increases as fast as possible (the exact position of the rows depends on the number of rows of the mesh). The cost of the permutation depends on s, where the s source processors are located, and the message length. In Figure 8 we show the percentage difference between Algorithms Repos xy source and Br xy source on four input distributions when the number of sources increases from 16 to 192. The machine size is and the message size is 6K bytes. Repositioning pays for all distributions except the band distribution. Repositioning for the band distribution costs up to 6:5% more. Translating this into actual time, when s is less than 150, Repos xy source -10103050Number of Sources Percentage Difference o- equal +- cross *- band x- blk Figure 8: The difference between Repos xy source and Br xy source on different input distributions for a machine with varying the number of sources. costs between 1 and 2 msec more. For repositioning costs 7.5 msec more, indicating that repositioning is expensive for large source numbers. The repositioning approach results in a significant gain for the cross and square block distributions. In terms of actual time, the gain for repositioning on the cross distributions lies between 13 and 31 msec. A gain of 13 msec is observed when for all other source numbers the gain lies between 20 and 31 msec. The conclusion of our experimental study is that repositioning pays (i.e., the cost of the initial permutation is less than the gain resulting from working with an ideal source distribution) unless: ffl the number of sources is large (s ? p=2 appears to be the breakpoint), or ffl the number of processors is small (for p - 16, there appears to be little difference between the algorithms and different source distributions), or ffl the message length is small (i.e., less that 1K). Clearly, if the input distribution is close to an ideal distribution, it does not pay to reposition. However, none of our algorithms tries to analyze the input dis- tribution. The effect of the message length on the repositioning is illustrated in Figure 9. The figure shows the percentage difference for the same four distributions on a 16 \Theta 16 machine and 75 sources when the message length increases. For a message size of less than 1K, repositioning pays only for the cross distribution. As the message size increases, the benefit of repositioning increases for all distributions. Not surprisingly, for large message length, the gain tapers off and decreases for some distributions. In Section 3 we also proposed to combine the repositioning with a partitioning approach. We first generate an ideal source distribution and then create two Message Length Percentage Difference o- equal +- cross *- band x- blk Figure 9: The difference between Repos xy source and Br xy source on different input distributions for a machine with varying the message length. broadcasting problems, each on one half of the ma- chine. Let Algorithm Part xy source be such a partitioning algorithm using Br xy source within each machine half. We have compared Part xy source against the performance of Repos xy source and Br xy source. Our results showed that for the Intel Paragon the partitioning approach hardly ever gives a better performance than repositioning alone. The reason lies in the cost of the final permutation. The exchange of long messages done in the final step dominates the performance and eliminates the gain obtained from broadcasting on smaller machines. The performance of partitioning algorithms could be different for machines of other characteristics, but we conjecture that for other currently available MPPs the outcome will be the same. 6 Conclusions We described different s-to-p broadcasting algorithms and analyzed their scalability and performance on the Intel Paragon. We showed that the performance of each algorithm is influenced by the distribution of the source processors and a relationship between the distribution and the dimension of the machine. Each algorithm has ideal distributions and distributions on which the performance degrades. To reduce the dependence of the performance on the input distribution we proposed a repositioning approach. In this approach the given distribution is turned into an ideal distribution of the target broadcasting algorithm. We have also compiled and run our programs under MPI environment, using MPI point-to-point message passing primitives. We have observed a performance loss of 2 to 5% in every implementation. --R "CCL: A Portable and Tunable Collective Communication Library for Scalable Parallel Computers," "Interprocessor Collective Communication Library," "Multiphase Complete Exchange on a Circuit Switched Hypercube," "Benchmarking the CM-5 Multicomputer," "On the Design and Implementation of Broadcast and Global Combine Operations Using the Postal Model," Introduction to Algorithms "Communication Operations on Coarse-Grained Mesh Architectures," "An Architecture for Optimal All-to-All Personalized Communication," "Opti- mal Broadcast and Summation in the LogP Model," Introduction to Parallel Computing "Multicast in Hypercube Multiprocessors," "Performance Evaluation of Multicast Wormhole Routing in 2D- Mesh Multicomputers," "Optimum Broadcasting and Personalized Communication in Hypercubes," "Many-to- Many Communication With Bounded Traffic," "All-to-all Communication on Meshes with Wormhole Routing," --TR --CTR Yuh-Shyan Chen , Chao-Yu Chiang , Che-Yi Chen, Multi-node broadcasting in all-ported 3-D wormhole-routed torus using an aggregation-then-distribution strategy, Journal of Systems Architecture: the EUROMICRO Journal, v.50 n.9, p.575-589, September 2004

scalability;message-passing MPPs;broadcasting;communication operations

287341

Deterministic Voting in Distributed Systems Using Error-Correcting Codes.

AbstractDistributed voting is an important problem in reliable computing. In an N Modular Redundant (NMR) system, the N computational modules execute identical tasks and they need to periodically vote on their current states. In this paper, we propose a deterministic majority voting algorithm for NMR systems. Our voting algorithm uses error-correcting codes to drastically reduce the average case communication complexity. In particular, we show that the efficiency of our voting algorithm can be improved by choosing the parameters of the error-correcting code to match the probability of the computational faults. For example, consider an NMR system with 31 modules, each with a state of m bits, where each module has an independent computational error probability of 103. In this NMR system, our algorithm can reduce the average case communication complexity to approximately 1.0825m compared with the communication complexity of 31m of the naive algorithm in which every module broadcasts its local result to all other modules. We have also implemented the voting algorithm over a network of workstations. The experimental performance results match well the theoretical predictions.

Introduction Distributed voting is an important problem in the creation of fault-tolerant computing systems, e.g., it can be used to keep distributed data consistent, to provide mutual exclusion in distributed systems. In an N Modular Redundant (NMR) system, when the N computational modules execute identical tasks, they need to be synchronized periodically by voting on the current computation state (or result, and they will be used interchangeably hereafter), and then all modules set their current computation state to the majority one. If there is no majority result, then other computations are needed, e.g., all modules recompute from the previous result. This technique is also an essential tool for task-duplication-based checkpointing [12]. In distributed storage systems, voting can also used to keep replicated data consistent. Many aspects of voting algorithms have been studied, e.g., data approximation, reconfigurable voting and dynamic modification of vote weights, metadata-based dynamic voting[3][5][9]. In this paper, we focus on the communication complexity of the voting problem. Several voting algorithms have been proposed to reduce the communication complexity [4][7]. These algorithms are nondeterministic because they perform voting on signatures of local computation results. Recently, Noubir et. al. [8] proposed a majority voting scheme based on error-control codes: each module first encodes its local result into a codeword of a designed error-detecting code and sends part of the codeword. By using the error-detecting code, discrepancies of the local results can be detected with some probability, and then by a retransmission of full local results, a majority voting decision can be made. Though the scheme drastically reduces the average case communication complexity, it can still fail to detect some discrepancies of the local results and might reach a false voting result, i.e., the algorithm is still a probabilistic one. In addition, this scheme is only using the error detecting capabilities of codes. As this paper will show, in general, using only error-detecting codes ( EDC ) does not help to reduce communication complexity of a deterministic voting algorithm. Though they have been used in many applications such as reliable distributed data replication [1], error-correcting codes ( ECC ) have not been applied to the voting problem. For many applications[12], deterministic voting schemes are needed to provide more accurate voting results. In this paper, we propose a novel deterministic voting scheme that uses error- correcting/detecting codes. The voting scheme generalizes known simple deterministic voting algorithms. Our main contributions related to the voting scheme include: (i) using the correcting in addition to the detecting capability of codes ( only the detection was used in known schemes ) to drastically reduce the chances of retransmission of the whole local result of each node thus the communication complexity of the voting, (ii) a proof that our scheme provably reaches the same voting result as the naive voting algorithm in which every module broadcasts its local result to all other modules, and (iii) the tuning of the scheme for optimal average case communication complexity by choosing the parameters of the error-correcting/detecting code, thus making the voting scheme adaptive to various application environments with different error rates. The paper is organized as follows: in Section 2, we describe the majority voting problem in NMR systems. Our voting algorithm together with its correctness proof are described in Section 3. Section 4 analyzes both the worst case and the average case communication complexity of the algorithm. Section 5 presents experimental results of performances of the proposed voting algorithm, as well as two other simple voting algorithms for comparison. Section 6 concludes the paper. 2 The Problem Definition In this section, we define the model of the NMR system and its communication complexity. Then we address the voting problem in terms of the communication complexity. 2.1 NMR System Model An NMR system consists of N computational modules which are connected via a communication medium. For a given computational task, each module executes a same set of instructions with independent computational error probability p. The communication medium could be a bus, a shared memory, a point-to-point network or a broadcast network. Here we consider the communication medium as a reliable broadcast network, i.e., each module can send its computation result to all other modules with only one error-free communication operation. The system evolution is considered to be synchronous, i.e., the voting process is round-based. 2.2 Communication Complexity The communication complexity of a task in an NMR system is defined as the total number of bits that are sent through the communication medium in the whole execution procedure of the task. In a broadcast network, let m ij be the number of the bits that the ith module sends at the jth round of the execution of a task, then the communication complexity of the task is is the number of the modules in the system, and K is the number of rounds needed to complete the task. 2.3 The Voting Problem Now consider the voting function in an NMR system. In an NMR system, in order to get a final result for a given task, after each module completes its own computation separately, it needs to be synchronized with other modules by voting on the result. Denote X i as the local computational result of the ith module, the majority function is defined as follows: OE otherwise where in general, N is an odd natural number, and OE is any predefined value different from all possible computing results. Example 1 If changes to 0010, and other X 0 then The result of voting in an NMR system is that each module gets Majority(X final result , where X is the local computation result of the ith module. One obvious algorithm for the voting problem is that after each module computes the task, it broadcasts its own result to all the other modules. When a module receives all other modules' results, it simply performs the majority voting locally to get the result. The algorithm can be described as follows: Algorithm 1 Send-All Voting For Module Wait Until Receive all This algorithm is simple: each module only needs one communication (i.e., broadcast) oper- ation, but apparently its communication complexity is too high. If the result for the task has bits, then the communication complexity of the algorithm is Nm bits. In most cases, the probability of a module to have a computational error is rather low, namely at most times all modules shall have the same result, thus each module only needs to broadcast part of its result. If all the results are identical, then each module shall agree with that result. If not, then modules can use Algorithm 1. Namely, we can get another simple improved voting algorithm as follows: Algorithm 2 Simple Send-Part Voting For Module Partition the local result X i into N symbols: X Wait Until Receive all If Else Wait Until Receive all In the above algorithm, * is a concatenation operation of strings, e.g., is an equality evaluation: FALSE otherwise Some padding may be needed if the local result is not an exact multiple of N. The following example demonstrates a rough comparison of the two algorithms. round of communication is needed, and in total 25 bits are transmitted. On the other hand, with Algorithm broadcast 0, and is the majority voting result. In this case, 2 rounds of communication are done, and 10 bits ( 5 bits for X and 5 bits for F ) are transmitted. results in and the Else part of the algorithm is executed, finally the majority voting result is obtained by voting on all the X i 's, i.e., rounds of communication are needed, and in total, bits ( 25 bits for X i 's and 5 bits for F) are transmitted. 2 From the above example, it can be observed: 1. Algorithm 1 always requires only 1 round of communication, and Algorithm 2 requires 2 or 3 rounds of communication; 2. The Else part of Algorithm 2 is actually Algorithm 3. The communication complexity of Algorithm 1 is always Nm, but the communication complexity of Algorithm 2 may be m+N or Nm+N , depending on the X i 's; 4. In Algorithm 2, by broadcasting and voting on the voting flags, i.e., F i 's, the chance for getting a false voting result is eliminated. If the Else part of Algorithm 2, i.e., Algorithm 1, is not executed too often, then the communication complexity can be reduced to m+N from Nm, and in most cases, m AE N , thus m. So the key idea used to reduce the communication complexity is to reduce the chance to execute Algorithm 1. In most computing environments, each module has low computational error probability p, thus most probably all modules either (1) get the same result or (2) only few of them get different results from others. In case (1), Algorithm 2 has low communication complexity, but in case (2), Algorithm 1 is actually used and the communication complexity is high (i.e., Nm+N) , but if we can detect and correct these discrepancies of the minor modules' results, then the Else part of the Algorithm 2 does not need to be executed, the communication complexity can still be low. This detecting and correcting capability can be achieved by using error-correcting codes. 3 The Solution Based on Error-Correcting Codes Error-correcting codes ( ECC ) can be used in the voting problem to reduce the communication complexity. The basic idea is that instead of broadcasting its own computation result X i di- rectly, P i , the ith module, first encodes its result X i into a codeword Y i of some code, and then broadcasts one symbol of the codeword to all other modules. After receiving all other symbols of the codeword, it reassembles them into a vector again. If all modules have the same result, i.e., all are equal, then the received vector is the codeword of the result, thus it can be decoded to the result again. If the majority result exists, i.e., Majority(X OE, and there are t c) modules whose results are different from the majority result X, then the symbols from all these modules can be regarded as error symbols with respect to the majority result. As long as the code is designed to correct up to t errors, these error symbols can be corrected to get the codeword corresponding to the majority result, thus Algorithm 1 does not need to be executed. When the code length is less than Nm, the communication complexity is reduced compared to Algorithm 1. On the other hand, if only error-detecting codes are used, once error results are detected, Algorithm 1 still needs to be executed, and thus increases the whole communication complexity of the voting. Thus error-correcting codes are preferable to error-detecting codes for voting. By properly choosing the error-correcting codes, the communication complexity can always be lowered than that of Algorithm 1. But it is possible that the majority result does not exist, i.e., Majority(X the vector that each module gets can still be decoded to a result. As observed from the above example, introduction of the voting flags can avoid this false result. 3.1 A Voting Algorithm with ECC With a properly designed error-correcting code which can detect up to d and correct up to t error voting algorithm using this code is as follows: Algorithm 3 ECC Voting For Module Wait Until Receive all Y If Y is undecodable Execute Algorithm Else If Else Execute Algorithm Notice that to execute Algorithm 1, each module P i does not need to send its whole result It only needs to send additional of its codeword Y i . Since the code is designed to detect up to d and correct up to t symbols, it can correct up to d the unsent d+t symbols of Y i can be regarded as erasures and recovered, hence the original X i can be decoded from Y i . To see the algorithm more clearly, the flow chart of the algorithm is given in Fig. 1, and the following example shows how the algorithm works. Example 3 There are 5 modules in an NMR system, and the task result has 6 bits, i.e., Here the EVENODD code [2] is used which divides 6-bit information into 3 symbols the reassembled vector Y into N symbols Broadcast(Y (i)), wait until get Execute "Send-All Voting" Majority(F , . , F Figure 1: Flow Chart of Algorithm 3 and encode information symbols into a 5-symbol codeword. This code can correct 1 error symbol, Now if then with the EVENODD code, after each module broadcasts 1 symbol (i.e., 2 bits) of the codewords, the reassembled vector is Y=0000000001. Since Y has only 1 error symbol, it can be decoded into X=000000. From the flow chart of the algorithm, we can see that is the majority voting result. In this case, there are 2 rounds of communication, and the communication complexity is 15 bits. As a comparison, algorithm 1 needs 1 round of communication, and its communication complexity is 30 bits; on the other hand, algorithm needs 3 rounds of communication, and the communication complexity in this case is 35 bits. In this example, the EVENODD code is used, but actually the code itself does not affect the communication complexity as long as it has same properties as the EVENODD code, namely, an MDS code with From the flow chart of the algorithm, the introduction of voting on F i 's ensures not to reach a false voting result, and going to the Send-All Voting in worst case guarantees not to fail to reach the majority result if it exists. Thus the algorithm can give a correct majority voting result. A rigorous correctness proof of the algorithm is as follows. 3.2 Correctness of the Algorithm Theorem 1 Algorithm 3 gives Majority(X set of local computational results Proof: From the flow chart of the algorithm, it is easy to see that the algorithm terminates in the following two cases: 1. Executing the Send-All Voting algorithm: the correct majority voting result is certainly reached; 2. Returning a X: in this case, since Majority(F are equal to X, X is the majority result. 2 To see how the algorithm works with various cases of the local results X i 's give two stronger observations about the algorithm, which also help to analyze the communication complexity of the algorithm. outputs OE, i.e., Algorithm 3 never gives a false voting result. Proof: It is easy to see from the flow chart that after the first round of communication, each module gets a same vote vector Y . According to the decodability of Y, there are two cases: 1. If Y is undecodable, then the Send-All Voting algorithm is executed, and the output will be OE; 2. If Y is decodable, the decoded result X now can be used as a reference result. But since there does not exist a majority voting result, majority of the X i 's are not equal to the X, i.e., so the Send-All Voting algorithm is executed, and the output again will be OE. 2 output is exactly the X, i.e., Algorithm 3 will not miss the majority voting result. Proof: Suppose there are e modules whose local results are different from the majority result X, then e - 1. If e - t, then there are e error symbols in the vote vector Y with respect to the corresponding codeword of the majority result X, so Y can be correctly decoded into X, and majority of are equal to X, i.e., majority of F i 's is 1, hence the correct majority result X is outputted. 2. If e ? t, then Y is either undecodable or incorrectly decoded into another X 0 , where X 0 6= X. In either case, the Send-All voting algorithm is executed and the correct majority result X is reached. 2 3.3 Proper Code Design In order to reduce the communication complexity, we need an error-correcting code which can be used in practice for Algorithm 3. Consider a block code with length M. Because of the symmetry among the N modules, M needs to be a multiple of N, i.e., each codeword consists of N symbols, and each symbol has k bits, thus Nk. If the minimum distance of the code is d min , then 2 c, since the code should be able to detect up to d error symbols and correct up to t error symbols[6]. Recall that the final voting result has m bits, the code to design is a (Nk; m; (d To get the smallest value for k, by the Singleton Bound in coding theory[6], we get (2) Equality holds for all MDS Codes[6]. So given a designed (d; t), the smallest value for k is d m e. If m is an integer, all MDS Codes can achieve this lower bound of k. One class of commonly used MDS codes for arbitrary distances is the Reed-Solomon code[6]. If m is not an integer, then any (Nk; m; (d+ t)k +1) block code can be used, where e, one of the examples is the BCH code, which can also have arbitrary distances[6]. The exact parameters of (k; d; t) can be obtained by shortening setting some information symbols to zeros ) or puncturing ( deleting some parity symbols Notice that In most applications, N - m, thus the N bits of F i 's can be neglected, and k is approximately the number of the bits that each module needs to send to get final voting result, so the communication complexity of Algorithm 3 is always lower than that of Algorithm 1. In this paper, only the communication complexity of voting is considered, since in many systems, computations for encoding and decoding on individual nodes are much faster than reliable communications among these nodes, which need rather complicated data management in different communication stacks, retransmission of packets between distributed nodes when packet loss happens. However, in real applications, design of proper codes should also make the encoding and decoding of the codes as computationally efficient as possible. When the distances of codes are relatively small, which is the case for most applications when the error probability p is relatively low, more computation-efficient MDS codes exist, such as codes in [2], [10] and [11], all of which require only bitwise exclusive OR operations. Communication Complexity Analysis 4.1 Main Results From the flow chart of Algorithm 3, we can see if the algorithm terminates in branch 1, i.e., the algorithm gets a majority result, then the communication complexity is N(k+1); if it terminates in branch 2, then the communication complexity is N(m+1); finally if the algorithm terminates in branch 3, the communication complexity is Nm, thus the worst case communication complexity Cw is N(m Denote C a as the average case communication complexity of Algorithm 3, and define the average reduction factor ff as the ratio of C a over the communication complexity of the Send-All Voting algorithm (i.e., Nm), namely Nm , then the following theorem gives the relation between ff and the parameters of an NMR system and the corresponding code: Theorem 2 For an NMR system with N modules each of which executes an identical task of m-bit result and has computational error with probability p independent of other modules' activities, if Algorithm 3 uses an ECC which can detect up to d and correct up to t error symbols, and holds for the average reduction factor of Algorithm 3: where Proof: To get the average case communication complexity C a of Algorithm 3, we need to analyze the probability P i of the algorithm terminating in the branch i, 3. First assume that if a module has an erroneous result X i , then it contributes an error symbol to the voting vector Y . From the proof of Observation 2, if the algorithm terminates in the branch 1, then at most modules have computational errors, thus the probability of this event is exactly P 1 . The event that the algorithm reaches the branch 2 corresponds to the decoder error event of a code with minimum distance of d+t+1, thus [6] b d+t where fA i g is the weight distribution of the code being used, and P ik is the probability that a received vector Y is exactly Hamming distance k from a weight-i (binary) codeword of the code. More precisely, !/ If the weight distribution of the code is unknown, P 2 can be approximately bounded by b d+t Since the second term in the right side of the inequality above is just the probability of event that correctable errors happen. Finally P 3 is the probability of the decoder failure event, Now notice the fact that a module has an erroneous result can also contribute a correct symbol to the voting vector, the average case communication complexity is and the average reduction factor is Notice that e, and we get the result of ff as in Eq. (3). 2 Remarks on the theorem : From Eq. (3), we can see the relation between the average reduction factor ff and each branch of Algorithm 3. The first term relates to the first branch whose reduction factor is k , or 1 when m is large enough relative to N , the round-off error of partition can be neglected, and P 1 is the probability of that branch. One would expect this term to be the dominant one for the ff, since with a properly designed code tuned to the system, the algorithm is supposed to terminate at Branch 1 in most cases. The second term is simply the probability that the algorithm terminates at either Branch 2 or Branch 3, where the reduction factor is 1 ( i.e., there is no communication reduction since all the local results are transmitted considering the 1 bit for F i 's in Branch 2. The last term is due to the 1 bit for voting on F i 's. When the local result size is large enough, i.e., m AE 1, this 1 bit can be neglected in our model. Thus in most applications, the result in the theorem can be simplified as since the assumption that m AE 1 is quite reasonable. From the above theorem and its proof, it can be seen that for a given NMR system (i.e., N and p), P 1 is only a function of t, so if t is chosen, from Eq. (3) or Eq. (11) it is easy to see that ff monotonically decreases as d decreases. Recall that 0 - t - d, thus for a chosen t, setting can make ff minimum when m AE 1. Even though it is not straight forward to get a closed form of t which minimize ff, it is almost trivial to get such an optimal t by numerical calculation. Fig. 2 shows relations between ff and (t; p; N ). Fig. 2a and Fig. 2b show how ff (using Eq. change with t for some setup of (N; p) when t. It is easy to see that for small p and reasonable N , a small t (e.g., t - 2, for N - 51 with can achieve minimal ff. These results show that for a quite good NMR system (e.g., p - 0:01), only by putting a small amount of redundancy of the local results and apply error-correcting codes on them, the communication complexity of the majority voting can be drastically reduced. Since the majority result is of m bits, and each module shall get an identical result after the voting, the communication complexity of the voting problem is at least m bits, thus ff - 1 is the lower bound of ff. Fig. 2c shows the closeness of the theoretical lower bound of ff and the minimum ff that Algorithm 3 can achieve for some setup of NMR systems. 4.2 More Observations From the above results, we can see that the communication complexity of the Algorithm 3 is determined by the code design parameters (d; t). In an NMR system with N modules, we only need to consider the case where at most b N modules have different local results with the majority (a) ff vs. t for different p with fixed vs. t for different N with fixed p=.1 alpha alpha lower bound Figure 2: Relations between ff and (t; p; N) result, thus the only constraints of (d,t) is 2 c. For some specific values of (d,t), the algorithm reduces to the following cases: 1. When repetition code is used, the algorithm becomes Algorithm 1. Since a repetition code is always the worst code in terms of redundancy, it should always be avoided for reducing the communication complexity of voting. On the other hand, when d=t=0: the algorithm becomes Algorithm 2, and from Fig. 2, we can see that for a small enough p and reasonable N , e.g., actually is a best solution of the majority voting problem in terms of the communication complexity. Besides, Algorithm 2 has low computational complexity since it does not need any complex encoding and decoding operations. Thus the ECC voting algorithm is a generalized voting algorithm, and its communication complexity is determined by the code chosen. 2. then the code only has detecting capability, but if m AE N , then from the analysis above, increasing d actually makes ff increasing. Thus it is not good to put some redundancy to the local results only for detecting capability when m AE N , i.e., using only EDC ( error detecting code ) does not help to reduce the communication complexity of voting. The scheme proposed in [8] is in this class with 3. 2 c: as analyzed above, in general, it is not good to have d ? t in terms of ff, since increase of d will increase ff when t is fixed. But in this case, Algorithm 3 has a special property: branch 2 of the algorithm can directly come to declare there is no majority result without executing the Send-All Voting algorithm, simply because the code now can detect up to b Nc errors, so if there was a majority result, then Y (refer to the Fig. 1) can have at most b N erroneous modules, and since Y is decodable, the majority of the local results should agree with the decoded result X, i.e., Majority(F contradicts with the actual there is no majority result. By setting d to b N 3 always has 2 rounds of communication and the worst case communication complexity is thus Nm instead of N(m + 1) for the general case, and this achieves the lower bound of the worst case communication complexity of the distributed majority voting problem [8]. 5 Experimental Results In this section, we show some experimental results of the three voting algorithms discussed above. The experiments are performed over a cluster of Intel Pentium/Linux-2.0.7 nodes connected via a 100 Mbps Ethernet. Reliable communication is implemented by a simple improved UDP scheme: whenever there is a packet loss, the voting operation is considered as a failure and redone from beginning. By choosing suitable packet size, there is virtually no packet loss using UDP. To examine real performances of the above three voting algorithms, N nodes vote on a result of length m using all the three voting algorithms. For the ECC Voting algorithm, an EVENODD Code is used, which corrects 1 error symbol, i.e., for the ECC Voting algorithm. Random errors are added to local computing results with a preassigned error probability p, independent of results at other nodes in the NMR system. Performances are evaluated by two parameters for each algorithm: the total time to complete the voting operation T and the communication time for the voting operation C. Among all the local T 's and C's, the maximum T and C are chosen to be the T and C of the whole NMR system, since if the voting operation is considered as a collective operation, the system's performance is determined by the worst local performance in the system. For each set of the NMR system parameters ( N nodes and error probability voting operation is done 200 times and random computation errors in each run are independent of those in other runs, and the arithmetic average of C's and T 's are regarded as the performance parameters for the tested NMR system. Experimental results are shown in figures 3 through 5. Fig. 3 compares the experimental average reduction factors of the voting algorithms with the theoretical results as analyzed in the previous section, when they are applied in an NMR system of 5 nodes. Fig. 4 shows the performances ( T and C ) of the voting algorithms. Detailed communication patterns of the voting algorithms are shown in Fig. 5 to provide some deeper insight into the voting algorithms. Fig. 3a and Fig. 3b show the experimental average reduction factors of the voting communication for the Simple Send-Part Voting algorithm and the ECC Voting algorithm. Fig. 3a and Fig. 3b also show the theoretical average reduction factors of the algorithm 2 and 3 as computed from the Eq. (11). Notice that the average communication time reduction factors ff of both algorithm 2 and algorithm 3 are below 1, and as the computing result size increases, the reduction factor approaches the theoretical bound, with the exception of the smallest computing result size of 1 Kbyte. Fig. 4 shows the performances of each voting algorithm applied in an NMR system of 5 nodes. Fig. 4ab show the total voting time T and Fig. 4cd show the communication time C for voting. The only different parameter of the NMR systems related to the figures a and symmetrically c and d ) is the error probability p: in the figures a and c, while in the figures b and d. It is easy to see from the figures that for the voting algorithm 1 Voting ), T and C are the same, since besides communication, there is no additional local computation. Fig. 4ab show that the algorithm 2 ( Simple Send-Part Voting ) and 3 ( ECC Voting ) perform better than the algorithm 1 ( Send-All Voting ) in terms of the total voting time T . On the other hand, Fig. 4cd show, in terms of C, i.e., the communication complexity, the ECC Voting algorithm is better than the Simple Send-Part Voting algorithm when the error (a) error probability computing result size m ( Kbyte ) average reduction factor a (b) error probability computing result size m ( Kbtye ) average reduction factor a Figure 3: Average Reduction Factors (C(i) is the experimental average reduction factor of communication time for voting using algorithm i, and ff(i) is the theoretical bound of the average communication reduction factor using algorithm i, probability is relatively large ( Fig. 4c ) and worse than the Simple Send-Part Voting algorithm when the error probability is relatively small ( Fig. 4d ), which is consistent with the analysis results in the previous section. In the analysis in the previous section, the size of local computing result m does not show up as a variable in the average reduction factor function ff, since the communication complexity is only considered as proportional to the size of the messages that need to be broadcasted. But practically, communication time is not proportional to the message size, since startup time of communication also needs to be included. More specially, in the Ethernet environment, since the maximum packet size of each physical send ( broadcast ) operation is also limited by the physical ethernet, communication completion time becomes a more complicated function of the message size. Thus from the experimental results, it can be seen that for a computing result of small size, e.g., 1 Kbyte, the Send-All Voting algorithm actually performs best in terms of both C and T , since the startup time dominates the performance of communication. Also, the communication time for broadcasting the 1-bit voting flags cannot be neglected as analyzed in the previous section, particularly for a small size computing result. This can also be seen from the detailed voting communication time pattern in Fig. 5ab: round 2 of the communication is for the 1-bit voting flag, even though it finishes in much more shorter time than round 1, but is still not negligibly small. This explains the fact that for small size computing results, the average communication time reduction factors of algorithm 2 and algorithm 3 are quite apart from their theoretical bound. Further examination of the detailed communication time pattern of voting provides a deeper insight into algorithm 3. From Fig. 5cd, it is easy to see that in the first round of communication, algorithm 2 needs less time than algorithm 3 since the size of the message to be broadcasted is smaller for algorithm 2. Besides, the first round of communication time does not vary as the error probability p varies for the both algorithms. The real difference between the two algorithms lies in the third round of communication. From Fig. 5c, this time is small for the both algorithms since the error probability p is small ( 0.01 ). But as the error probability p increases to 0.1, as shown in Fig. 5d, for algorithm 2, this time also increases to be bigger than the first round time, since it has no error-correcting capability and once full message needs to be broadcasted, its size is much bigger than in the first round. On the other hand, for algorithm 3, though it also increases, the communication time for the third round is still much smaller than in the first round, this comes from the error-correcting codes that algorithm 3 uses, since the code can correct errors at one computing node, which is the most frequent error pattern that happens. Thus even though the error probability is high, in most cases, the most expensive third round of communication can still be avoided, and algorithm 3 performs better ( in terms of communication complexity or time ) than algorithm 2 in high error probability systems, just as the predicted analysis in the previous section. 6 Conclusions We have proposed a deterministic distributed voting algorithm using error-correcting codes to reduce the communication complexity of the voting problem in NMR systems. We also have given a detailed theoretical analysis of the algorithm. By choosing the design parameters of the error-correcting code, i.e., (d; t), the algorithm can achieve a low communication complexity which is quite close to its theoretical lower bound. We have also implemented the voting algorithm over a network of workstations, and the experimental performance results match well the theoretical analysis. The algorithm proposed here needs 2 or 3 rounds of communication. It is left as an open problem whether there is an algorithm for the distributed majority voting problem with its average case communication complexity less than Nm using only 1 round of communication. --R "An Optimal Strategy for Computing File Copies" "EVENODD: An Efficient Scheme for Tolerating Double Disk Failures in RAID Architectures" "Voting Using Predispositions" "Fault-Masking with Reduced Redundant Communication" "Voting Without Version Numbers," The Theory of Error Correcting Codes "Parallel Data Compression for Fault Tolerance" "Using Codes to Reduce the communication Complexity of Voting in NMR Systems" "Voting Algorithms" "X-Code: MDS Array Codes with Optimal Encoding" "Checkpointing in Parallel and Distributed Systems" --TR

MDS code;error-correcting codes;NMR system;majority voting;communication complexity

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A comparison of reliable multicast protocols.

We analyze the maximum throughput that known classes of reliable multicast transport protocols can attain. A new taxonomy of reliable multicast transport protocols is introduced based on the premise that the mechanisms used to release data at the source after correct delivery should be decoupled from the mechanisms used to pace the transmission of data and to effect error recovery. Receiver-initiated protocols, which are based entirely on negative acknowledgments (NAKS) sent from the receivers to the sender, have been proposed to avoid the implosion of acknowledgements (ACKS) to the source. However, these protocols are shown to require infinite buffers in order to prevent deadlocks. Two other solutions to the ACK-implosion problem are tree-based protocols and ring-based protocols. The first organize the receivers in a tree and send ACKS along the tree; the latter send ACKS to the sender along a ring of receivers. These two classes of protocols are shown to operate correctly with finite buffers. It is shown that tree-based protocols constitute the most scalable class of all reliable multicast protocols proposed to date.

Introduction The increasing popularity of real-time applications supporting either group collaboration or the reliable dissemination of multimedia information over the Internet is making the provision of reliable and unreliable end-to-end multicast services an integral part of its architecture. Minimally, an end- to-end multicast service ensures that all packets from each source are delivered to each receiver in the session within a finite amount of time and free of errors and that packets are safely deleted within a finite time. Additionally, the service may ensure that each packet is delivered only once and in the # Supported in part by the Office of Naval Research under Grant N00014- 94-1-0688, and by the Defense Advanced Research Projects Agency (DARPA) under grant F19628-96-C-0038 Correspondence to: B.N. Levine order sent by the source. Although reliable broadcast protocols have existed for quite some time [3], viable approaches on the provision of end-to-end reliable multicasting over the Internet are just emerging. The end-to-end reliable multicast problem facing the future Internet is compounded by its current size and continuing growth, which makes the handling of acknowledgements a major challenge commonly referred to as the acknowledgement (ack) implosion problem. The two most popular approaches to end-to-end reliable multicasting proposed to date are called sender-initiated and receiver-initiated. In the sender-initiated approach, the sender maintains the state of all the receivers to whom it has to send information and from whom it has to receive acknowledgments (acks). Each sender's transmission or re-transmission is multicast to all receivers; for each packet that each receiver obtains correctly, it sends a unicast ack to the sender. In contrast, in the receiver-initiated approach, each receiver informs the sender of the information that is in error or missing; the sender multicasts all packets, giving priority to retransmissions, and a receiver sends a negative acknowledgement (nak) when it detects an error or a lost packet. The first comparative analysis of ideal sender-initiated and receiver-initiated reliable multicast protocols was presented by Pingali et al. [17, 18]. This analysis showed that receiver-initiated protocols are far more scalable than sender-initiated protocols, because the maximum through-put of sender-initiated protocols is dependent on the number of receivers, while the maximum throughput of receiver-initiated protocols becomes independent of the number of receivers as the probability of packet loss becomes negligi- ble. However, as this paper demonstrates, the ideal receiver-initiated protocols cannot prevent deadlocks when they operate with finite memory, i.e., when the applications using the protocol services cannot retransmit any data themselves, and existing implementations of receiver-initiated protocols have inherent scaling limitations that stem from the use of messages multicast to all group members and used to set timers needed for nak avoidance, the need to multicast naks to all hosts in a session, and to a lesser extent, the need to store all messages sent in a session. This paper addresses the question of whether a reliable multicast transport protocol (reliable multicast protocol, for short) can be designed that enjoys all the scaling properties of the ideal receiver-initiated protocols, while still being able to operate correctly with finite memory. To address this question, the previous analysis by Pingali et al. [17, 18, 22] is extended to consider the maximum throughput of generic ring-based protocols, which organize receivers into a ring, and two classes of tree-based protocols, which organize receivers into ack trees. These classes are the other three known approaches that can be used to solve the ack implosion problem. Our analysis shows that tree- and ring-based protocols can work correctly with finite memory, and that tree-based protocols are the best choice in terms of processing and memory requirements. The results presented in this paper are theoretical in nature and apply to generic protocols, rather than to specific implementations; however, we believe that they provide valuable architectural insight for the design of future reliable multicast protocols. Section 2 presents a new taxonomy of reliable multicast protocols that organizes known approaches into four protocol classes and discusses how many key papers in the literature fit within this taxonomy. This taxonomy is based on the premise that the analysis of the mechanisms used to release data from memory after their correct reception by all receivers can be decoupled from the study of the mechanisms used to pace the transmission of data within the session and the detection of transmission errors. Using this taxonomy, we argue that all reliable unicast and multicast protocols proposed to date that use naks and work correctly with finite memory (i.e., without requiring the application level to store all data sent in a session) use acks to release memory and naks to improve throughput. Section 3 addresses the correctness of the various classes of reliable multicast protocols introduced in our taxonomy. Section 4 extends the analysis by Pingali et al. [17, 18, 22] by analyzing the maximum throughput of three protocol classes: tree-based, tree-based with local nak avoidance and periodic polling (tree-NAPP), and ring-based protocols. Section 5 provides numerical results on the performance of the protocol classes under different scenarios, and discusses the implications of our results in light of recent work on reliable multicasting. Section 6 provides concluding remarks. new taxonomy of reliable multicast protocols We now describe the four generic approaches known to date for reliable multicasting. Well-known protocols (for unicast and multicast purposes) are mapped into each class. Our taxonomy differs from prior work [8, 17, 18, 22] addressing receiver-initiated strategies for reliable multicasting in that we decouple the definition of the mechanisms needed for pacing of data transmission from the mechanisms needed for the allocation of memory at the source. Using this approach, the protocol can be thought as using two windows: a congestion window (cw ) that advances based on feedback from receivers regarding the pacing of transmissions and detection of errors, and a memory allocation window (mw ) that advances based on feedback from receivers as to whether the sender can erase data from memory. In practice, proto-010101000000000000000000000000000000000000000011111111111111111111111111111111111111111111111111000000000000000000111111111111111111111111111000000000000000000111111111111111111111111111000000000000000000000000111111111111111111111111000000000000000000000000000000000000000011111111111111111111111111111111111111111111111111 Ack Source Receiver Nak Fig. 1. A basic diagram of a sender-initiated protocol cols may use a single window for pacing and memory (e.g., TCP [10]) or separate windows (e.g., NETBLT [4]). Each reliable protocol assumes the existence of multi-cast routing trees provided by underlying multicast routing protocols. In the Internet, these trees will be built using such protocols as DVMRP [6], Core-Based Trees (CBT) [1], Ordered Core-Based Trees (OCBT) [20], Protocol-Independent Multicast (PIM) [7], or the Multicast Internet Protocol (MIP) [14]. 2.1 Sender-initiated protocols In the past [17, 18], sender-initiated protocols have been characterized as placing the responsibility of reliable delivery at the sender. However, this characterization is overly restrictive and does not reflect the way in which several reliable multicast protocols that rely on positive acknowledgements from the receivers to the source have been designed. In our taxonomy, a sender-initiated reliable multicast protocol is one that requires the source to receive acks from all the receivers, before it is allowed to release memory for the data associated with the acks. Receivers are not restricted from directly contacting the source. It is clear that the source is required to know the constituency of the receiver set, and that the scheme suffers from the ack implosion problem. However, this characterization leaves unspecified the mechanism used for pacing of transmissions and for the detection of transmission errors. Either the source or the receivers can be in charge of the retransmission timeouts! The traditional approach to pacing and transmission error detection (e.g., TCP in the context of reliable unicasting) is for the source to be in charge of the retransmission timeout. However, as suggested by the results reported by Floyd et al. [8], a better approach for pacing a multicast session is for each receiver to set its own timeout. A receiver sends acks to the source at a rate that it can accept, and sends a nak to the source after not receiving a correct packet from the source for an amount of time that exceeds its retransmission timeout. An ack can refer to a specific packet or a window of packets, depending on the specific retransmission strategy. A simple illustration of a sender-initiated protocol is presented in Fig. 1. Notice that, regardless of whether a sender-driven or receiver-driven retransmission strategy is used, the source is still in charge of deallocating memory after receiving all the acks for a given packet or set of packets. The source keeps packets in memory until every receiver node has positively acknowledged receipt of the data. For a sender-initiated pro- tocol, if a sender-driven retransmission strategy is used, the sender "polls" the receivers for acks by retransmitting after a timeout. If a receiver-driven retransmission strategy is used, the receivers "poll" the source (with an ack) after they time out. 1 It is important to note that, just because a reliable multi-cast protocol uses naks, it does not mean that it is receiver- initiated, i.e., that naks can be the basis for the source to ascertain when it can release data from memory. The combination of acks and naks has been used extensively in the past for reliable unicast and multicast protocols. For exam- ple, NETBLT is a unicast protocol that uses a nak scheme for retransmission, but only on small partitions of the data (i.e., its cw ). In between the partitions, called "buffers", are acks for all the data in the buffer (i.e., the mw ). Only upon receipt of this ack does the source release data from mem- therefore, NETBLT is really sender-initiated. In fact, naks are unnecessary in NETBLT for its correctness, i.e., a buffer can be considered one large packet that eventually must be acked, and are important only as a mechanism to improve throughput by allowing the source to know sooner when it should retransmit some data. A protocol similar to NETBLT is the "Negative Acknowledgments with Periodic Polling" (NAPP) protocol [19]. This protocol is a broadcast protocol for local area networks (LANs). Like NETBLT, NAPP groups together large partitions of the data that are periodically acked, while lost packets within the partition are naked. NAPP advances the cw by naks and periodically advances the mw by acks. Because the use of naks can cause a nak implosion at the source, NAPP uses a nak avoidance scheme. As in NET- BLT, naks increase NAPP's throughput, but are not necessary for its correct operation, albeit slow. The use of periodic polling limits NAPP to LANs, because the source can still suffer from an ack implosion problem even if acks occur less often. Other sender-initiated protocols, like the Xpress Transfer Protocol (XTP) [21], were created for use on an internet, but still suffer from the ack implosion problem. The main limitation of sender-initiated protocols is not that acks are used, but the need for the source to process all of the acks and to know the receiver set. The two known methods that address this limitation are: (a) using naks instead of acks, and (b) delegating retransmission responsibility to members of the receiver set by organizing the receivers into a ring or a tree. We discuss both approaches subsequently. 2.2 Receiver-initiated protocols Previous work [17, 18] characterizes receiver-initiated protocols as placing the responsibility for ensuring reliable packet delivery at each receiver. The critical aspect of these protocols for our taxonomy is that no acks are used. The receivers send naks back to the source when a retransmission Of course, the source still needs a timer to ascertain when its connection with a receiver has failed.010011010000000000000000000000000000000000000000111111111111111111111111111111111111111111111111110000000000000000001111111111111111111111111110000000000000000001111111111111111111111111110000000000000000000000000000000000000000000000000011111111111111111111111111111111111111111111111111 Nak Source Receiver Fig. 2. A basic diagram of a receiver-initiated protocol is needed, detected by either an error, a skip in the sequence numbers used, or a timeout. Receivers are not restricted from directly contacting the source. Because the source receives feedback from receivers only when packets are lost and not when they are delivered, the source is unable to ascertain when it can safely release data from memory. There is no explicit mechanism in a receiver-initiated protocol for the source to release data from memory (i.e., advance the mw ), even though its pacing and retransmission mechanisms are scalable and efficient (i.e., advancing the cw ). Figure 2 is a simple illustration of a receiver-initiated protocol. Because receivers communicate naks back to the source, receiver-initiated protocols have the possibility of experiencing a nak implosion problem at the source if many receivers detect transmission errors. To remedy this problem, previous work on receiver-initiated protocols [8, 17, 18] adopts the nak avoidance scheme first proposed for NAPP, which is a sender-initiated protocol. Receiver-initiated with nak avoidance (RINA) protocols have been shown [17, 18, 22] to have better performance than the basic receiver-initiated protocol. The resulting generic RINA protocol is as follows [17, 18]. The sender multicasts all packets and state information, giving priority to retransmissions. Whenever a receiver detects a packet loss, it waits for a random time period and then multicasts a nak to the sender and all other receivers. When a receiver obtains a nak for a packet that it has not received and for which it has started a timer to send a nak, the receiver sets a timer and behaves as if it had sent a nak. The expiration of a timer without the reception of the corresponding packet is the signal used to detect a lost packet. With this scheme, it is hoped that only one nak is sent back to the source for a lost transmission for an entire receiver set. Nodes farther away from the source might not even get a chance to request a retransmission. The generic protocol does not describe how timers are set accurately. The generic RINA protocol we have just described constitutes the basis for the operation of the Scalable Reliable Multicasting (SRM) algorithm [8]. SRM has been embedded into an internet collaborative whiteboard application called wb. SRM sets timers based on low-rate, periodic "session messages" multicast by every member of the group. The messages specify a time stamp used by the receivers to estimate the delay from the source, and the highest sequence number generated by the node as a source. 2 The average Multiple sources are supported in SRM, we focus on the single-source case for simplicity. bandwidth consumed by session messages is kept small (e.g., by keeping the frequency of session messages low). SRM's implementation requires that every node stores all packets, or that the application layer store all relevant data. We note that naks from receivers are used to advance the cw , which is controlled by the receivers, and the sequence number in each multicast session message is used to "poll" the receiver set, i.e., to ensure that each receiver is aware of missing packets. Although session messages implement a "polling" function [19], they cannot be used to advance the mw , as in a sender-initiated protocol, because a sender specifies its highest sequence number as a source, not the highest sequence number heard from the source. 3 In practice, the persistence of session messages forces the source to process the same number of messages that would be needed for the source to know the receiver set over time (one periodic message from every receiver). Accordingly, as defined, the basic dissemination of session messages in SRM does not scale, because it defeats one of the goals of the receiver-initiated paradigm, i.e., to keep the receiver set anonymous from the source for scaling purposes. There are other issues that limit the use of RINA protocols for reliable multicasting. First, as we show in the next section, a RINA protocol requires that data needed for re-transmission be rebuilt from the application. This approach is reasonable only for applications in which the immediate state of the data is exclusively desired, which is the case of a distributed whiteboard. However, the approach does not apply for multimedia applications that have no current state, but only a stream of transition states. Second, naks and retransmissions must be multicast to the entire multicast group to allow suppression of naks. The nak avoidance scheme was designed for a limited scope, such as a LAN, or a small number of Internet nodes (as it is used in tree-NAPP protocols, described in the next sec- tion). This is because the basic nak avoidance algorithm requires that timers be set based on updates multicast by every node. As the number of nodes increases, each node must do increasing amount of work! Furthermore, nodes that are on congested links, LANs or regions may constantly bother the rest of the multicast group by multicasting naks. Approaches to limit the scope of naks and retransmission are still evolving [8]. However, current proposals still rely on session messages that reach all group members. Another example of a receiver-initiated protocol is the "log-based receiver-reliable multicast" (LBRM) [9], which uses a hierarchy of log servers that store information indefinitely and receivers recover by contacting a log server. Using log servers is feasible only for applications that can afford the servers and leaves many issues unresolved. If a single server is used, performance can degrade due to the load at the server; if multiple servers are used, mechanisms must still be implemented to ensure that such servers have consistent information. The ideal receiver-initiated protocol has three main advantages over sender-initiated protocols, namely: (a) the source does not know the receiver set, (b) the source does 3 Our prior description of SRM [11, 12] incorrectly assumed that session messages contained the highest sequence number heard from the source. We thank Steve McCanne for pointing this out. not have to process acks from each receiver, and (c) the receivers pace the source. The limitation of this protocol is that it has no mechanism for the source to know when it can safely release data from memory. Furthermore, as we have argued, the practical implementations of the receiver-initiated approach fail to provide advantages (a) and (b). The following two protocol classes organize the receiver set in ways that permit the strengths of receiver-initiated protocols to be applied on a local scale, while providing explicit mechanisms for the source to release memory safely (i.e., efficient management of the mw ). 2.3 Tree-based protocols Tree-based protocols are characterized by dividing the receiver set into groups, distributing retransmission responsibility over an acknowledgement tree (ack tree) structure built from the set of groups, with the source as the root of the tree. A simple illustration of a tree-based protocol is presented in Fig. 3. The ack tree structure prevents receivers from directly contacting the source, in order to maintain scalability with a large receiver set. The ack tree consists of the receivers and the source organized into local groups , with each such group having a group leader in charge of retransmissions within the local group. The source is the group leader in charge of retransmissions to its own local group. Each group leader other than the source communicates with another local group (to either a child or the group leader) closer to the source to request retransmissions of packets that are not received correctly. Group leaders may be children of another local group, or minimally, may just be in contact with another local group. Each local group may have more than one group leader to handle multiple sources. Group leaders could also be chosen dynamically, e.g., through token passing within the local group. Hosts that are only children are at the bottom of the ack tree, and are termed leaves. Obviously, an ack tree consisting of the source as the only leader and leaf nodes corresponds to the sender-initiated scheme. Acknowledgments from children in a group, including the source's own group, are sent only to the group leader. The children of a group send their acknowledgements to the group leader as soon as they receive correct packets, advancing the cw ; we refer to such acknowledgements as local acks or local naks, i.e., retransmissions are triggered by local acks and local naks unicast to group leaders by their children. Similar to sender-initiated schemes, the use of local naks is unnecessary for correct operation of the protocol. Tree-based protocols can also delegate to leaders of sub-trees the decision of when to delete packets from memory (i.e., advance the mw ), which is conditional upon receipt of aggregate acks from the children of the group. Aggregate acks start from the leaves of the ack tree, and propagate toward the source, one local group at a time. A group leader cannot send an aggregate ack until all its children have sent an aggregate ack. Using aggregate acks is necessary to ensure that the protocol operates correctly even if group leaders fail, or if the ack tree is partitioned for long periods Group Leaf Source Fig. 3. A basic diagram of a tree-based protocol of time [12]. If aggregate acks are not used, i.e., if a group leader only waits for all its children to send local acks before advancing the mw , then correct operation after group leaders fail can only be guaranteed by not allowing nodes to delete packets; this is the approach used in all tree-based protocols [13, 16, 24] other than Lorax [12]. The Lorax protocol [12] is the first tree-based protocol to build a single shared ack tree for use by multiple sources in a single ses- sion, and to use aggregate acks to ensure correct operation after hosts in the ack tree fail. The use of local acks and local naks for requesting retransmissions is important for throughput. If the source scheduled retransmissions based on aggregate acks, it would have to be paced based on the slowest path in the ack tree. Instead, retransmissions are scheduled independently in each local group. Tree-based protocols eliminate the ack-implosion prob- the source from having to know the receiver set, and operate solely on messages exchanged in local groups (between a group leader and its children in the ack tree). Furthermore, if aggregate acks are used, a tree-based protocol can work correctly with finite memory even in the presence of receiver failures and network partitions. To simplify our analysis and description of this protocol, we assume that the group leaders control the retransmission timeouts; however, such timeouts can be controlled by the children of the source and group leaders. Accordingly, when the source sends a packet, it sets a timer, and each group leader sets a timer as it becomes aware of a new packet. If there is a timeout before all local acks have been received, the packet is assumed to be lost and is retransmitted by the source or group leader to its children. The first application of tree-based protocols to reliable multicasting over an internet was reported by Paul et al. [15], who compared three basic schemes for reliable point-to- multipoint multicasting using hierarchical structures. Their results have been fully developed as the reliable multicast transport protocol (RMTP) [13, 16]. While our generic protocol sends a local ack for every packet sent by the source, RMTP sends local acks only periodically, so as to conserve bandwidth and to reduce processing at each group leader, increasing attainable throughput. We define a tree-NAPP protocol as a tree-based protocol that uses nak avoidance and periodic polling [19] in the local groups. Naks alone are not sufficient to guarantee reliability with finite memory, so receivers send a periodic positive local ack to their parents to advance the cw . Note that messages sent for the setting of timers needed for nak000000111111001111001110000000000001111111111111111 Receiver Set Source Nak Ack Fig. 4. A basic diagram of a ring-based protocol avoidance are limited to the local group, which is scalable. The tree-based multicast transport protocol (TMTP) [24] is an example of a tree-NAPP protocol. 2.4 Ring-based protocols Ring-based protocols for reliable multicast were originally developed to provide support for applications that require an atomic and total ordering of transmissions at all receivers. One of the first proposals for reliable multicasting is the token ring protocol (TRP) [3]; its aim was to combine the throughput advantages of naks with the reliability of acks. The Reliable Multicast Protocol (RMP) [23] discussed an updated WAN version of TRP. Although multiple rings are used in a naming hierarchy, the same class of protocol is used for the actual rings. Therefore, RMP has the same throughput bounds as TRP. We base our description of generic ring-based protocols on the LAN protocol TRP and the WAN protocol RMP. A simple illustration of a ring-based protocol is presented in Fig. 4. The basic premise is to have only one token site responsible for acking packets back to the source. The source times out and retransmits packets if it does not receive an ack from the token site within a timeout period. The ack also serves to timestamp packets, so that all receiver nodes have a global ordering of the packets for delivery to the application layer. The protocol does not allow receivers to deliver packets until the token site has multicast its ack. Receivers send naks to the token site for selective repeat of lost packets that were originally multicast from the source. The ack sent back to the source also serves as a token passing mechanism. If no transmissions from the source are available to piggyback the token, then a separate unicast message is sent. Since we are interested in the maximum throughput, we will not consider the latter case in this pa- per. The token is not passed to the next member of the ring of receivers until the new site has correctly received all packets that the former site has received. Once the token is passed, a site may clear packets from memory; accordingly, the final deletion of packets from the collective memory of the receiver set is decided by the token site, and is conditional on passing the token. The source deletes packets only when an ack/token is received. Note that both TRP and RMP specify that retransmissions are sent unicast from the token site. Because our analysis focuses on maximum attainable throughput of protocol classes, we will assume that the token is passed exactly once per message. 3 Protocol correctness A protocol is considered correct if it is shown to be both safe and live [2]. Given the minimum definition of reliable service we have assumed, for any reliable multicast protocol to be live, no deadlock should occur at any receiver or at the source. For the protocol to be safe, all data sent by the source must be delivered to a higher layer within a finite time. To address the correctness of protocol classes, we assume that nodes never fail during the duration of a reliable multicast session and that a multicast session is established correctly and is permanent. Therefore, our analysis of correctness focuses only on the ability of the protocol classes to sustain packet losses or errors. We assume that there exists some non-zero probability that a packet is received error-free, and that all senders and receivers have finite memory. The proof of correctness for ring-based protocols is given by Chang and Maxemchuk [3]. The proof that sender-initiated unicast protocols are safe and live is available from many sources (e.g., Bertsekas and Gallager [2]). The proof does not change significantly for the sender-initiated class of reliable multicast protocols and is omitted for brevity. The liveness property at each receiver is not violated, because each node can store a counter of the sequence number of the next packet to be delivered to a higher layer. The safety property proof is also essentially the same, because the source waits for acks from all members in the receiver set before sliding the cw and mw forward. Theorems 1 and 2 below demonstrate that the generic tree-based reliable multi-cast protocol (TRMP for short) is correct, and that receiver-initiated reliable multicast protocols are not live. Theorem 1: TRMP is safe and live. Proof. Let R be the set of all the nodes that belong to the reliable multicast session, including a source s. The receivers in the set are organized into a B-ary tree of height h. The proof proceeds by induction on h. For the case in which reduces to a non-hierarchical sender-initiated scheme of with each of the B receivers practicing a given retransmission strategy with the source. Therefore, the proof follows from the correctness proof of unicast retransmission protocols presented by Bertsekas and Gallager [2]. For h > 1, assume the theorem holds for any t such that must prove the theorem holds for some Liveness. We must prove that each member of a tree of height t is live. Consider a subset of the tree that starts at the source and includes all nodes of the tree up to a height of (t - 1); the leaves of this subtree are also group leaders in the larger tree, i.e., group leaders of the nodes at the bottom of the larger tree. By the inductive hypothesis, the liveness property is true in this subtree. We must only show that TRMP is live for a second subset of nodes consisting of leaves of the larger tree and their group leader parents. Each group in this second subset follows the same protocol, and it suffices to prove that an arbitrary group are live. The arbitrary group in the second subset of the tree constitutes a case of sender-initiated reliable multicast, with the only difference that the original transmission is sent from the source (external to the group), not the group leader. Since leaves can only contact the group leader, we must prove this relationship is live. The inductive hypothesis guarantees that the group leader and its parent is live. Assume the source transmits a packet i at time c 1 , and that it is received correctly and delivered at all leaves of the arbitrary group at time c 2 . Let c 3 be the time at which the group leader deletes the packet and advances the mw . The protocol is live and will not enter into deadlock if finite. The rest of the proof follows from the proof by Bertsekas and Gallager [2] for unicast protocols, where the group leader takes the place of the source. Therefore, TRMP is live. Safety. The safety of TRMP follows directly, because our proof of liveness shows that any arbitrary packet i is delivered at each receiver within a finite time. QED Theorem 2. A receiver-initiated reliable protocol is not live. Proof. The proof is by example focusing on the sender and an arbitrary member of the receiver set R (where R # 1). - Sender node, X , has enough memory to store up to M packets. Each packet takes 1 unit of time to reach a receiver node Y . Naks take a finite amount of time to reach the sender. - Let p i denote the i th packet, i beginning from zero. p 0 is sent at start time 0, but it is lost in the network. sends the next (M - 1) packets to Y successfully. sends a nak stating that p 0 was not received. The nak is either lost or reaches the sender after time M when the sender decides to send out packet pM . store up to M packets, and it has not received any naks for p 0 by time M , it must clear assuming that it has been received correctly. receives the nak for p 0 at time M deadlocked, unable to retransmit p 0 . QED The above indicates that the ideal receiver-initiated protocol requires an infinite memory to work correctly. In prac- tice, this requirement implies that the source must keep in memory every packet that it sends during the lifetime of a session. Theorem 1 assumes that no node failures or network traffic occur. However, node failures do happen in practice, which changes the operational requirements of practical tree-based protocols. For tree-based protocols, it can be shown that deleting packets from memory after a node receives local acks from its children is not live. Aggregate acks are necessary to ensure correct operation of tree-based protocols in the presence of failures. Lorax [12] is the only tree-based protocol that uses aggregate acks and can operate with finite memory in the presence of node failures or network partitions. 4 Maximum throughput analysis 4.1 Assumptions To analyze the maximum throughput that each of the generic reliable multicast protocols introduced in Sect. 2 can achieve, we use the same model as Pingali et al. [17, 18], which focuses on the processing requirements of generic reliable multicast protocols, rather than the communication band-width requirements. Accordingly, the maximum throughput of a generic protocol is a function of the per-packet processing rate at the sender and receivers, and the analysis focuses on obtaining the processing times per packet at a given node. We assume a single sender, X , multicasting to R identical receivers. The probability of packet loss is p for any node. Figure 5 summarizes all the notation used in this sec- tion. For clarity, we assume a single ack tree rooted at a single source in the analysis of tree-based protocols. A selective repeat retransmission strategy is assumed in all the protocol classes since it is well known to be the retransmission strategy with the highest throughput [2], and its requirement of keeping buffers at the receivers is a non-issue given the small cost of memory. Assumptions specific to each protocol are listed in Sect. 2, and are in the interest of modeling maximum throughput. We make two additional assumptions: (1) no acknowledgements are ever lost, and (2) all loss events at any node in the multicast of a packet are mutually independent. Such multicast routing protocols as CBT, OCBT, PIM, MIP, and DVMRP [1, 5, 7, 14, 20] organize routers into trees, which means that there is a correlation between packet loss at each receiver. Our first assumption benefits all classes, but especially favors protocols that multicast acknowledge- ments. In fact, this assumption is essential for RINA proto- cols, in order to analyze their maximum attainable through- put, because nak avoidance is most effective if all receivers are guaranteed to receive the first nak multicast to the receiver set. As the number of nodes involved in nak avoidance increases, the task of successful delivery of a nak to all receivers becomes less probable. Both RINA and tree-NAPP protocols are favored by the assumption, but RINA protocols much more so, because the probability of delivering naks successfully to all receivers is exaggerated. Our second assumption is equivalent to a scenario in which there is no correlation among packet losses at receivers and the location of those receivers in the underlying multicast routing tree of the source. Protocols that can take advantage of the relative position of receivers in the multicast routing tree for the transmission of acks, naks, or retransmissions would possibly attain higher throughput than predicted by this model. However, no class is given any relative advantage with this assumption. Table 1 summarizes the bounds on maximum throughput for all the known classes of reliable multicast protocols. Our results clearly show that tree-NAPP protocols constitute the most scalable alternative. 4.2 Sender- and receiver-initiated protocols Following the notation introduced by Pingali et al. [17, 18], we place a superscript A on any variable related to the sender-initiated protocol, and N1 and N2 on variables related to the receiver-initiated and RINA protocols, respec- tively. The maximum throughput of the protocols for a constant stream of packets to R receivers is [17, 18] 1/# A Table 1. Analytical bounds Protocol Processor requirements p as a con- stant Sender-initiated [17, O Receiver- initiated nak avoidance [17, O Ring-based (uni- cast retrans.) O Tree-based O(B(1 - p) Tree-NAPP O # . (3) Even as the probability of packet loss goes to zero, the throughput of the sender-initiated protocol is inversely dependent on R, the size of the receiver set, because an ack must be sent by every receiver to the source once a transmission is correctly received. In contrast, as p goes to zero, the throughput of receiver-initiated protocols becomes independent of the number of receivers. Notice, however, that the throughput of a receiver-initiated protocol is inversely dependent with R, the number of receivers, or with ln R, when the probability of error is not negligible. We note that this result assumes perfect setting of the timers used in a RINA protocol without cost and that a single nak reaches the source, because we are only interested in the maximum attainable throughput of protocols. 4.3 Tree-based protocols We denote this class of protocols simply by H1, and use that superscript in all variables related to the protocol class. In the following, we derive and bound the expected cost at each type of node and then consider the overall system throughput. To make use of symmetry, we assume, without loss of generality, that there are enough receivers to form a full tree at each level. Without loss of generality, we assume that each local group in the ack tree consists of B children and a group leader. This allows us to make use of symmetry in our throughput calculations. We also assume that local acks advance the mw rather than aggregate acks, because by assumption no receiver fails in the system. We assume perfect setting of timers without cost and that a single nak reaches the source, because we are only interested in the maximum attainable throughput of protocols. 4.3.1 Source node We consider first X H1 , the processing costs required by the source to successfully multicast an arbitrarily chosen packet Branching factor of a tree, the group size. R - Size of the receiver set. Time to feed in a new packet from the higher protocol layer. Xp - Time to process the transmission of a packet. Xa , Times to process transmission of an ack, nak, or local ack, respectively. Time to process a timeout at a sender or receiver node, respectively. Yp - Time to process a newly received packet. Time to deliver a correctly received packet to a higher layer. Ya , Yn , Y h - Times to process and transmit an ack, nak, or local ack, respectively. Probability of loss at a receiver; losses at different receivers are assumed to be independent events. r - Number of local acks sent by receiver r per packet using a tree-based protocol. r - Number of acks sent by a receiver r per packet using a unicast protocol. Total number of local acks received from all receivers per packet. Mr - Number of transmissions necessary for receiver r to successfully receive a packet. - Number of transmissions for all receivers to receive the packet correctly (for protocols A, N1 and N2); Number of transmissions for all receivers to receive the packet correctly for protocols H1 and H2. Processing time per packet at sender and receiver, respectively, in protocol w # {A, N1, N2, H1, H2, R}. Processing time per packet at a group leader in tree-based and tree- NAPP protocols, respectively. per packet at the token-site in ring-based protocols. # w x - Throughput for protocol w # {A, N1, N2, H1, H2, R} where x is one of the source s, receiver (leaf) r, group leader h, or token-site t. No subscript denotes overall system throughput. X# , Y# - Times to process the reception and transmission, respectively, of a periodic local ack. Fig. 5. Notation to all receivers using the H1 protocol. The processing requirement for an arbitrary packet can be expressed as a sum of costs: +(receiving acks) where X f is the time to get a packet from a higher layer, is the time taken on attempt m at successful transmission of the packet, X t (m) is the time to process a timeout interrupt for transmission attempt m, X h (i) is the time to process local ack i, M H1 is the number of transmissions that the source will have to make for this packet using the H1 protocol, and L H1 is the number of local acks received using the H1 protocol. Taking expectations, we have What we have derived so far is extremely similar to Eqs. 1 and 2 in the analysis by Pingali et al. [17, 18]. In fact, we can use all of their analysis, with the understanding that B is the size of the receiver subset from which the source collects local acks. Therefore, the expected number of local acks received at the sender is Substituting Eq. 6 into Eq. 5, we can rewrite the expected cost at the source node as Pingali et al. [17, 18] have shown that the expected number of transmissions per packet in A, N1, and N2 equals # . (8) Because in H1 the number of receivers R = B, the expected number of transmissions per packet in the H1 protocol is which can be simplified to [17, 18, 19] Pingali et al. [17, 18] provide a bound of E[M ] that we apply to E[M H1 ] with Using Eq. 11, we can bound Eq. 7 as follows It then follows that, when p is a constant, E[X H1 O(B 4.3.2 Leaf nodes Let Y H1 denote the requirement on nodes that do not have to forward packets (leaves). Notice that leaf nodes in the H1 protocol will process fewer retransmissions and thus send fewer acknowledgements than receivers in the A protocol. We can again use an analysis similar to the one by Pingali et al. [17, 18] for receivers using a sender-initiated protocol. (receiving transmissions) (sending local acks) where Y p (i) is the time it takes to process (re)transmission i, Y h (i) is the time it takes to send local ack i, Y f is the time to deliver a packet to a higher layer, and L H1 is the number of local acks generated by this node h (i.e., the number of transmissions correctly received). Since each receiver is sent transmissions with probability p that a packet will be lost, we obtain Taking expectations of Eq. 13 and substituting Eq. 14, we have Again, noting the bound of E[M H1 ] given in Eq. 11, When p is treated as a constant, E[Y H1 4.3.3 Group leaders To evaluate the processing requirement at a group leader, h, we note that a node caught between the source and a node with no children has a two jobs: to receive and to retransmit packets. Because it is convenient, and because a group leader is both a sender and receiver, we will express the costs in terms of X and Y . Our sum of costs is (receiving transmissions) (sending local acks) (collecting local acks) Just as in the case for the source node, L H1 is the expected number of local acks received from node h's children for this packet, and L H1 h is the number of local acks generated by node h. We can substitute Eqs. 6 and 14 into Eq. to obtain The first two terms are equivalent to the processing requirements of a leaf node. The last two are almost the cost for a source node. Substituting and subtracting the difference yields In other words, the cost on a group leader is the same as a source and a leaf, without the cost of receiving the data from higher layers and one less transmission (the original one). Substituting Eqs. 12 and 16 into Eq. 20, we have When p is a constant, E[H H1 which is the dominant term in the throughput analysis of the overall system 4.3.4 Overall system analysis Let the throughput at the sender # H1 s be 1/E[X H1 ], at the group leaders # H1 be 1/E[H H1 ], at the leaf nodes # H1 r be 1/E[Y H1 ]. The throughput of the overall system is r }. (22) From Eqs. 12, 16, and 21 it follows that If p is a constant and if p # 0, we obtain Therefore, the maximum throughput of this protocol, as well as the throughput with non-negligible packet loss, is independent of the number of receivers. This is the only class of reliable multicast protocols that exhibits such degree of scalability with respect to the number of receivers. 4.4 Tree-based protocols with local nak avoidance and periodic polling To bound the overall system throughput in the generic Tree- NAPP protocol, we repeat the method used for the tree-based class; we first derive and bound the expected cost at the source, group leaders, and leaves. As we did for the case of tree-based protocols, we assume that there are enough receivers to form a full tree at each level. We place a superscript H2 on any variables relating to the generic Tree-NAPP protocol. 4.4.1 Source node We consider first X H2 , the processing costs required by the source to successfully multicast an arbitrarily chosen packet to all receivers using the H2 protocol. The processing requirement for an arbitrary packet can be expressed as a sum of costs: +(receiving local naks) +(receiving periodic local acks) where X f is the time to get a packet from a higher layer, (i) is the time for (re)transmission attempt i, X n (m) is the time for receiving local nak m from the receiver set, # is the amortized time to process the periodic local ack associated with the current congestion window, and M H2 is the number of transmission attempts the source will have to make for this packet. Taking expectations, where we have Using Eq. 11, the bound of E[M H1 ], we can bound Eq. 28 as follows It then follows that, when p is a constant, E[X H2 4.4.2 Leaf nodes Let Y H2 denote the processing requirement on nodes that do not have to forward packets (leaves). The sum of cost can be expressed as (receiving transmissions) (sending periodic local acks) (sending local naks) (receiving local naks) Let Y p (i) be the time it takes to process the (re)transmission r be the number of transmissions required for the packet to be received by receiver r, Y n (j) be the time it takes to send local nak j, X n (j) be the time it takes to receiver local nak j (from another receiver), Y t (k) be the time to set timer k, Y f be the time to deliver a packet to a higher layer, and Y # be the amortized cost of sending a periodic local ack for a group of packets of which this packet is a member. Taking expectations of Eq. 30, It follows from the distribution of M r that [17, 18] . (32) Therefore, noting Eq. 32 and that Prob{M r > derive from Eq. 31 the expected cost as Again, using the bound of E[M H1 ] given in Eq. 11, we can bound Eq. 33 by # . (34) When p is treated as a constant, E[Y H2 4.4.3 Group leaders The sum of costs for group leaders, which have the job of both sender and receiver, is (receiving transmissions) (sending periodic local acks) (receiving periodic local acks) (receiving local naks) (sending local naks) (retransmissions to children) Taking expectations and substituting Eq. 32, we obtain Similar to group leaders in the H1 protocol, the processing cost at a group leader is the same as a source and a leaf, without the cost of receiving the data from a higher layer and one less transmission. Substituting Eq. 28 and Eq. 33 into Eq. 36 and subtracting the difference, the expected cost can be expressed as Therefore, Eq. 36 can be bounded by # . (38) When p is a constant, E[H H2 ] # O(1). Therefore, all nodes in the Tree-NAPP protocol have a constant amount of work to do with regard to the number of receivers. 4.4.4 Overall system analysis The overall system throughput for the H2 protocol is the minimum throughput attainable at each type of node in the tree, that is, r }. (39) From Eqs. 29, 34, and 38, it follows that Accordingly, if either p is constant or p # 0, we obtain from Eq. 40 that # O(1). (41) Therefore, the maximum throughput of the Tree-NAPP pro- tocol, as well as the throughput with non-negligible packet loss, is independent of the number of receivers. 4.5 Ring-based protocols In this section, we analyze the throughput of ring-based pro- tocols, which we denote by a superscript R, using the same assumptions as in Sects. 4.3 and 4.4. Because we are interested in the maximum attainable throughput, we are assuming a constant stream of packets, which means we can ignore the overhead that occurs when there are no acks on which to piggyback token-passing messages. 4.5.1 Source Source nodes practice a special form of unicast with a roaming token site. The sum of costs incurred is (processing acks) r X a (i) Mr where M r is the number of transmissions required for the packet to be received by the token site, and has a mean of r be the number of acks from a receiver r (in this case the token site) sent unicast, i.e., the number of packets correctly received at r. This number is always 1, accordingly: Taking expectations of Eq. 42, we obtain r If we again assume constant costs for all operations, it can be shown that which, when p is a constant, is O(1) with regard to the size of the receiver set. 4.5.2 Token site The current token site has the following costs: (note both TRP and RMP specify that retransmissions are sent unicast to other R - 1 receivers.) (multicasting ack/token ) (processing naks) (unicasting retransmissions) r (R - 1)Prob{M r >1} Mr where L R is the number of naks received at the token site when using a ring protocol. To derive L R , consider M r , the number of transmissions necessary for receiver r to successfully receive a packet. M r has an expected value of 1/(1-p), and the last transmission is not naked. Because there are (R - 1) other receivers sending naks to the token site, we obtain (R - 1)p Therefore, the mean processing time at the token site is (R (R - 1)p The expected cost at the token site can be bounded by (R - 1)p with regard to the number of receivers. When p is a constant, 4.5.3 Receivers Receivers practice a receiver-initiated protocol with the current token site. We assume there is only one packet for the ack, token, and time stamp multicast from the token site per data packet. The cost associated with an arbitrary packet are therefore (receiving first transmission) (sending naks) (receiving retransmissions) r Mr Mr The first term in the above equation is the cost of receiving the ack/token/time stamp packet from the token site; the second is the cost of receiving the first transmission sent from the sender, assuming it is received error free; the third is the cost of delivering an error-free transmission to a higher layer; the fourth is the cost of receiving the retransmissions from the token site, assuming that the first failed; and the last two terms consider that a nak is sent only if the first transmission attempt fails and that an interrupt occurs only if a nak was sent. Taking expectations, we obtain As shown previously [17, 18], . (52) Substituting Eqs. 43, 52, and 32 into Eq. 51, we have Assuming all operations have constant costs, it can be shown that with regard to the size of the receiver set. If we consider p as a constant, then E[Y R 4.5.4 Overall system analysis The overall system throughput of R, the generic token ring protocol, is equal to the minimum attainable throughput at each of its parts: r }. (55) From Eqs. 45, 49 and 54 it follows that, if p is a constant and for p # 0, we obtain (R - 1)p 5 Numerical results To compare the relative performance of the various classes of protocols, all mean processing times are set equal to 1, except for the periodic costs X # and Y # which are set to 0.1. Figure 6 compares the relative throughputs of the protocols A, N1, N2, H1, H2, and R as defined in Sect. 2. The graph represents the inverse of Eqs. 19, 36, and 48, re- spectively, which are the throughputs for the tree-based, tree- NAPP, and ring-based protocols, as well as the inverse of the throughput equations derived previously [17, 18] for sender- and receiver-initiated protocols. The top, middle and bottom graphs correspond to increasing probabilities of packet loss, 10%, and 25%, respectively. Exact values of E[M H1 were calculated using a finite version of Eq. 9; Exact values of E[M ] were similarly calculated [22]. The performance of nak avoidance protocols, especially tree-NAPP protocols, is clearly superior. However, our assumptions place these two subclasses at an advantage over their base classes. First, we assume that no acknowledgements are lost or are received in error. The effectiveness of nak avoidance is dependent on the probability of naks reaching all receivers, and thus, without our assumption, the effectiveness of nak avoidance decreases as the number of receivers involved increases. Accordingly, tree-NAPP protocols have an advantage that is limited by the branching factor of the ack tree, while RINA protocols have an advantage that increases with the size of the entire receiver set. Second, we assume that the timers used for nak avoidance are set perfectly. In reality, the messages used to set timers would be subject to end-to-end delays that exhibit no regularity and can become arbitrarily large. 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000.2Throughput H2 A 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000.2Number ofReceivers Fig. 6. The throughput graph from the exact equations for each protocol. The probability of packet loss is respectively. The branching factor for trees is set at 10 We conjecture that the relative performance of nak avoidance would actually lie closer to their respective base classes, depending on the effectiveness of the nak avoidance scheme; in other words, the curves shown are upper bounds of nak avoidance performance. Our results show that, when considering only the base classes (since not one has an advantage over another), the tree-based class performs better than all the other classes. When considering only the subclasses that use nak avoidance, tree-NAPP protocols perform better than RINA protocols, even though our model provides an unfair advantage to RINA protocols. It is the hierarchical structure organization of the receiver set in tree-based protocols that guarantees scalability and improves performance over other protocols. Using nak avoidance on a small scale increases performance even further. In addition, if nak avoidance failed for a tree-NAPP protocol (e.g., due to incorrect setting of timers), the performance would still be independent of the size of the receiver set. RINA protocols do not have this property. Failure of the nak avoidance for RINA protocols would result in unscalable performance like that of a receiver-initiated protocol, which degrades quickly with increasing packet loss. Any increase in processor speed, or a smaller branching factor would also increase throughput for all tree-based pro- tocols. However, for the same number of receivers, a smaller branching factor implies that some retransmissions must traverse a larger number of tree-hops towards receivers expecting them further down the tree. For example, if a packet is lost immediately at the source, the retransmission is multi-cast only to its children and all other nodes in the tree must wait until the retransmission trickles down the tree struc- ture. This poses a latency problem that can be addressed by taking advantage of the dependencies in the underlying multicast routing tree. Retransmissions could be multicast only toward all receivers attached to routers on the subtree of the router attached to the receiver which has requested the missing data. The number of tree-hops from the receiver to the source is also a factor in how quickly the source can release data from memory in the presence of node failures, as discussed by Levine et al. [12]. Supportable receivers H2 N2 R A Fig. 7. Number of supportable receivers for each protocol. The probability of packet loss is respectively. The branching factor for trees is set at 10 Figure 7 shows the number of supportable receivers by each of the different classes, relative to processor speed requirements. This number is obtained by normalizing all classes to a baseline processor, as described by Pingali et al. [17, 18]. The baseline uses protocol A and can support exactly one is the speed of the processor that can support at most R receivers under protocol #, we set - A 1. The baseline cost is equal to [17, 18] =- A [1] . (58) Using Eqs. 18, 36, 48, and 58, we can derive the following -s for tree-based, tree-NAPP, and ring-based protocols, respectively: (R - 1)p # . (61) The number of supportable receivers derived for sender- and receiver-initiated protocols are shown to be [17, 18] (2E[M ]) . (64) From Fig. 6 and 7, it is clear that tree-based protocols can support any number of receivers for the same processor speed bound at each node, and that tree-NAPP protocols attain the highest maximum throughput. It is also important to note that the maximum throughput that RINA protocols can attain becomes more and more insensitive to the size of the receiver set as the probability of error decreases. Because we have assumed that a single nak reaches the source, that naks are never lost, and that session messages incur no processing load, we implicitly assume the optimum behavior of RINA protocols. The simulation results reported for SRM by Floyd et al. [8] agree with our model and result from assuming no nak losses and a single packet loss in the ex- periments. Figure 7 shows that tree-NAPP protocols can be made to perform better than the best possible RINA protocol by limiting the size of the local groups. Because of the unicast nature of retransmissions in ring-based protocols, these protocols approach sender-initiated protocols; this indicates that allowing only multicast retransmissions would improve performance greatly. 6 Conclusions We have compared and analyzed the four known classes of reliable multicast protocols. Of course, our model constitutes only a crude approximation of the actual behavior of reliable multicast protocols. In the Internet, an ack or a nak is simply another packet, and the probability of an ack or nak being lost or received in error is much the same as the error probability of a data packet. This assumption gives protocols that use nak avoidance an advantage over other classes. Therefore, it is more reasonable to compare them separately: our results show that tree-based protocols without nak avoidance perform better than other classes that do not use nak avoidance, and that tree-NAPP protocols perform better than RINA protocols, even though RINA protocols have an artificial advantage over every other class. We conjecture that, once the effects of ack or nak failure, and the correlation of failures along the underlying multicast routing trees are accounted for, the same relative performance of protocols will be observed. The results are summarized in Table 1. It is already known that sender-initiated protocols are not scalable because the source must account for every receiver listening. Receiver-initiated protocols are far more scalable, unless nak avoidance schemes are used to avoid overloading the source with retransmission requests. However, because of the unbounded-memory requirement, this protocol class can only be used efficiently with application-layer support, and only for a limited set of applications. Furthermore, to set the timers needed for nak avoidance, existing instantiations of RINA protocols require all group members to transmit session messages periodically, which makes them unscalable. Ring-based protocols were designed for atomic and total ordering of packets. TRP and RMP limit their throughput by requiring retransmissions to be unicast. It would be possible to reduce the cost bound to O(ln R), assuming p to be a constant, if the nak avoidance techniques presented by Ramakrishnan and Jain [19] were used. Our analysis shows that ack trees are a good answer to the scalability problem for reliable multicasting. Practical implementations of tree-based protocols maintain the anonymity of the receiver set, and only the tree-based and tree-NAPP classes have throughputs that are constant with respect to the number of receivers, even when the probability of packet loss is not negligible (which would preclude accurate setting of nak avoidance timers). Because tree-based protocols delegate responsibility for retransmission to receivers and because they employ techniques applicable to either sender- or receiver-initiated protocols within local groups (i.e., a node and its children in the tree) of the ack tree only, any mechanism that can be used with all the receivers of a session in a receiver-initiated protocol can be adopted in a tree-based protocol, with the added benefit that the throughput and number of supportable receivers is completely independent of the size of the receiver set, regardless of the likelihood with which packets, acks, and naks are received correctly. On the other hand, while the scope of naks and retransmissions can be reduced without establishing a structure in the receiver set [8], limiting the scope of the session messages needed to set nak avoidance timers and to contain the scope of naks and retransmissions require the aggregation of these messages. This leads to organizing receivers into local groups that must aggregate sessions messages sent to the source (and local groups). Doing this efficiently, how- ever, leads to a hierarchical structure of local groups much like what tree-based protocols require. Hence, it appears that organizing the receivers hierarchically (in ack trees or oth- erwise) is a necessity for the scaling of a reliable multicast protocol. --R Core based trees (CBT): An architecture for scalable inter-domain multicast routing Data Networks Reliable broadcast protocols. NETBLT: A high-throughput transport protocol Multicast routing in a datagram internetwork. Multicast routing in datagram internetworks and extended lans. An architecture for wide-area multicast routing A reliable multicast framework for light-weight sessions and application level framing Transmission control protocol. RMTP: A reliable multicast transport protocol. Multicast transport protocols for high-speed networks Reliable multicast transport protocol (RMTP). Protocol and Real-Time Scheduling Issues for Multi-media Applications A comparison of sender-initiated and receiver-initiated reliable multicast protocols A negative acknowledgement with periodic polling protocol for multicast over lan. XTP: The Xpress Transfer Protocol. A reliable dissemination protocol for interactive collaborative applications. He received the B. His current research interest is the analysis and design of algorithms and protocols for computer communication. --TR Data networks NETBLT: a high throughput transport protocol Multicast routing in datagram internetworks and extended LANs XTP: the Xpress Transfer Protocol Multicast routing in a datagram internetwork Core based trees (CBT) A comparison of sender-initiated and receiver-initiated reliable multicast protocols An architecture for wide-area multicast routing A reliable dissemination protocol for interactive collaborative applications Log-based receiver-reliable multicast for distributed interactive simulation A reliable multicast framework for light-weight sessions and application level framing Protocol and real-time scheduling issues for multimedia applications The case for reliable concurrent multicasting using shared ACK trees Reliable broadcast protocols A High Performance Totally Ordered Multicast Protocol The Ordered Core Based Tree Protocol A Comparison of Known Classes of Reliable Multicast Protocols --CTR V. Ramakrishna , Max Robinson , Kevin Eustice , Peter Reiher, An Active Self-Optimizing Multiplayer Gaming Architecture, Cluster Computing, v.9 n.2, p.201-215, April 2006 Christian Maihfer, A bandwidth analysis of reliable multicast transport protocols, Proceedings of NGC 2000 on Networked group communication, p.15-26, November 08-10, 2000, Palo Alto, California, United States Shuju Wu , Sujata Banerjee , Xiaobing Hou, A Comparison of Multicast Feedback Control Mechanisms, Proceedings of the 38th annual Symposium on Simulation, p.80-87, April 04-06, 2005 Shuju Wu , Sujata Banerjee , Xiaobing Hou, Performance Evaluation and Comparison of Multicast Feedback Control Mechanisms, Simulation, v.82 n.5, p.345-362, May 2006 Brian Neil Levine , Sanjoy Paul , J. J. Garcia-Luna-Aceves, Organizing multicast receivers deterministically by packet-loss correlation, Proceedings of the sixth ACM international conference on Multimedia, p.201-210, September 13-16, 1998, Bristol, United Kingdom Maurice Herlihy , Srikanta Tirthapura , Roger Wattenhofer, Ordered Multicast and Distributed Swap, ACM SIGOPS Operating Systems Review, v.35 n.1, p.85-96, January 1, 2001 Athina P. Markopoulou , Fouad A. Tobagi, Hierarchical reliable multicast: performance analysis and placement of proxies, Proceedings of NGC 2000 on Networked group communication, p.27-35, November 08-10, 2000, Palo Alto, California, United States Ryan G. Lane , Scott Daniels , Xin Yuan, An empirical study of reliable multicast protocols over Ethernet-connected networks, Performance Evaluation, v.64 n.3, p.210-228, March, 2007 Suman Banerjee , Seungjoon Lee , Ryan Braud , Bobby Bhattacharjee , Aravind Srinivasan, Scalable resilient media streaming, Proceedings of the 14th international workshop on Network and operating systems support for digital audio and video, June 16-18, 2004, Cork, Ireland Suman Banerjee , Seungjoon Lee , Bobby Bhattacharjee , Aravind Srinivasan, Resilient multicast using overlays, ACM SIGMETRICS Performance Evaluation Review, v.31 n.1, June Philip K. McKinley , Ravi T. Rao , Robin F. Wright, H-RMC: a hybrid reliable multicast protocol for the Linux kernel, Proceedings of the 1999 ACM/IEEE conference on Supercomputing (CDROM), p.8-es, November 14-19, 1999, Portland, Oregon, United States Brian Neil Levine , Sanjoy Paul , J. J. Garcia-Luna-Aceves, Organizing multicast receivers deterministically by packet-loss correlation, Multimedia Systems, v.9 n.1, p.3-14, July Suman Banerjee , Seungjoon Lee , Bobby Bhattacharjee , Aravind Srinivasan, Resilient multicast using overlays, IEEE/ACM Transactions on Networking (TON), v.14 n.2, p.237-248, April 2006 Carlos A. S. Oliveira , Panos M. Pardalos, A survey of combinatorial optimization problems in multicast routing, Computers and Operations Research, v.32 n.8, p.1953-1981, August 2005

ACK implosion;multicast transport protocols;reliable multicast;tree-based protocols

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Using the Matrix Sign Function to Compute Invariant Subspaces.

The matrix sign function has several applications in system theory and matrix computations. However, the numerical behavior of the matrix sign function, and its associated divide-and-conquer algorithm for computing invariant subspaces, are still not completely understood. In this paper, we present a new perturbation theory for the matrix sign function, the conditioning of its computation, the numerical stability of the divide-and-conquer algorithm, and iterative refinement schemes. Numerical examples are also presented. An extension of the matrix-sign-function-based algorithm to compute left and right deflating subspaces for a regular pair of matrices is also described.

Introduction . Since the matrix sign function was introduced in early 1970s, it has been the subject of numerous studies and used in many applications. For example, see [30, 31, 11, 26, 23] and references therein. Our main interest here is to use the matrix sign function to build parallel algorithms for computing invariant subspaces of nonsymmetric matrices, as well as their associated eigenvalues. It is a challenge to design a parallel algorithm for the nonsymmetric eigenproblem that uses coarse grain parallelism effectively, scales for larger problems on larger machines, does not waste time dealing with the parts of the spectrum in which the user is not interested, and deals with highly nonnormal matrices and strongly clustered spectra. In the work of [2], after reviewing the existing approaches, we proposed a design of a parallel nonsymmetric eigenroutine toolbox, which includes the basic building blocks (such as LU factorization, matrix inversion and the matrix sign function), standard eigensolver routines (such as the QR algorithm) and new algorithms (such spectral divide-and-conquer using the sign function). We discussed how these tools could be used in different combinations on different problems and architectures, for extracting all or some of the eigenvalues of a nonsymmetric matrix, and/or their corresponding invariant subspaces. Rather than using "black box" eigenroutines such as provided by EISPACK [32, 21] and LAPACK [1], we expect the toolbox approach to allow us more flexibility in developing efficient problem-oriented eigenproblem solvers on high performance machines, especially on parallel distributed memory machines. However, the numerical accuracy and stability of the the matrix sign function and divide-and-conquer algorithms based on it are poorly understood. In this paper, we will address these issues. Much of this work also appears in [3]. Let us first restate some of basic definitions and ideas to establish notation. The matrix sign function of a matrix A is defined as follows [30]: Let be the Jordan canonical form of a matrix A 2 C n\Thetan , where the eigenvalues of J+ lie in the open right half plane (C+ ) and those of J \Gamma lie in the open left half plane (C \Gamma ). Published at SIAM J. Mat. Anal. Appl., Vol.19, pp.205-225, 1998 y Department of Mathematics, University of Kentucky, Lexington, KY 40506 (bai@ms.uky.edu). z Computer Science Division and Mathematics Department, University of California, Berkeley, Z. BAI AND J. DEMMEL Then the matrix sign function of A is: We assume that no eigenvalue of A lies on the imaginary axis; otherwise, sign(A) is not defined. It is easy to show that the spectral projection corresponding to the eigenvalues of A in the open right and left half planes are P respectively. Let the leading columns of an orthogonal matrix Q span the range space of P+ (for example, Q may be computed by the rank-revealing QR decomposition of the spectral decomposition (1) where are the eigenvalues of A in C+ , and -(A 22 ) are the eigenvalues of A in C \Gamma . The algorithm proceeds in a divide-and-conquer fashion, by computing the eigenvalues of A 11 and A 22 . Rather than using the Jordan canonical form to compute sign(A), it can be shown that sign(A) is the limit of the following Newton iteration A k+1 =2 (2) The iteration is globally and ultimately quadratic convergent. There exist different scaling schemes to speedup the convergence of the iteration, and make it more suitable for parallel computation. By computing the matrix sign function of a M-obius transformation of A, the spectrum can be divided along arbitrary lines and circles, rather than just along the imaginary axis. See the report [2] and the references therein for more details. Unfortunately, in finite precision arithmetic, the ill conditioning of a matrix A k with respect to inversion and rounding errors, may destroy the convergence of the Newton iteration (2), or cause convergence to the wrong answer. Consequently, the left bottom corner block of the matrix Q T AQ in (1) may be much larger than where u denotes machine precision. This means that it is not numerically stable to approximate the eigenvalues of A by the eigenvalues of A 11 and A 22 , as we would like. In this paper, we will first study the perturbation theory of the matrix sign func- tion, its conditioning, and the numerical stability of the overall divide-and-conquer algorithm based on the matrix-sign function. We realize that it is very difficult to give a complete and clear analysis. We only have a partial understanding of when we can expect the Newtion iteration to converge, and how accurate it is. In a coarse analysis, we can also bound the condition numbers of intermediate matrices in the Newton iter- ation. Artificial and possibly very pathological test matrices are constructed to verify our theoretical analysis. Besides these artificial tests, we also test a large number of eigenvalue problems of random matrices, and a few eigenvalue problems from appli- cations, such as electrical power system analysis, numerical simulation of chemical reactions, and areodynamics stability analysis. Through these examples, we conclude that the most bounds for numerical sensitivity and stability of matrix sign function computation and its based algorithms are reachable for some very pathological cases, but they are often very pessimistic. The worst cases happen rarely. In addition, we discuss iterative refinement of an approximate invariant subspace, and outline an extension of the matrix sign function based algorithms to compute both left and right deflating subspaces for a regular matrix pencil A \Gamma -B. MATRIX SIGN FUNCTION FOR COMPUTING INVARIANT SUBSPACES 3 The rest of this paper is organized as the following: Section 2 presents a new perturbation bound for the matrix sign function. Section 3 discusses the numerical conditioning of the matrix sign function. The backward error analysis of computed invariant subspace and remarks on the matrix sign function based algorithm versus the QR algorithm are presented in section 4. Section 5 presents some numerical examples for the analysis of sections 2, 3 and 4. Section 6 describes the iteration refinement scheme to improve an approximate invariant subspace. Section 7 outlines an extension of the matrix sign function based algorithms for the generalized eigenvalue problem. Concluding remarks are presented in section 8. 2. A perturbation bound for the matrix sign function. When a matrix A has eigenvalues on pure imaginary axis, its matrix sign function is not defined. In other words, the set of ill-posed problems for the matrix sign function, is the set of matrices with at least one pure-imaginary eigenvalue. Computationally, we have observed that when there are the eigenvalues of A close the pure imaginary axis, the Newton iteration and its variations are very slowly convergent, and may be misconvergent. Moreover, even when the iteration converges, the error in the computed matrix sign function could be too large to use. It is desirable to have a perturbation analysis of the matrix sign function related to the distance from A to the nearest ill-posed problem. Perturbation theory and condition number estimation of the matrix sign function are discussed in [25, 23, 29]. However, none of the existing error bounds explicitly reveals the relationship between the sensitivity of the matrix sign function and the distance to the nearest ill-posed problem. In this section, we will derive a new perturbation bound which explicitly reveals such relationship. We will denote all the eigen-values of A with positive real part by -+ (A), i.e., denotes the smallest singular value of A. In addition, we recall the well-known inequality is the matrix 2-norm. Theorem 2.1. Suppose A has no pure imaginary and zero eigenvalues, is a perturbation of A and ffl j kffiAk. Let Then 3: Furthermore, if then 4 Z. BAI AND J. DEMMEL O Re Im r -r r Fig. 1. The semi-circle \Gamma Proof. We only prove the bound (7). The bound (5) can be proved by using a similar technique. Following Roberts [30] (or Kato [24]), the matrix sign function can also be defined using Cauchy integral representation: where Z \Gamma is any simple closed curve with positive direction enclosing -+ (A). sign the spectral projector for -+ (A). Here, without loss of generality, we take \Gamma to be a semi-circle with radius Figure 1). From the definition (8) of sign(A), it is seen that to study the stability of the matrix sign function of A to the perturbation ffi A, it is sufficient to just study the sensitivity of the projection be the projection corresponding to -+ A), from the condition (6), no eigenvalues of A are perturbed across or on the pure imaginary axis, and the semi-circle \Gamma also encloses -+ Therefore we have Z Z r \Gammar Z -=2 \Gamma-=2 [(re where the first integral, denoted I 1 , is the integral over the straight line of the semi-circle \Gamma, the second integral, denoted I 2 , is the integral over the curved part of the semi-circle \Gamma. Now, by taking the spectral norm of the first integral term, and noting the definition of !, the condition (6) and the inequality (3), we have Z r \Gammar MATRIX SIGN FUNCTION FOR COMPUTING INVARIANT SUBSPACES 5 Z r \Gammar Z r \Gammar Z r \Gammar By taking the spectral norm of the second integral term I 2 , we have Z -=2 \Gamma-=2 k(re Z -=2 \Gamma-=2 re i' re i' r r where the third inequality follows from (3) and the fourth from the choice of the radius r of the semi-circle \Gamma. The desired bound (7) follows from the bounds on kI 1 k and kI 2 k and the identity A few remarks are in order: 1. In the language of pseudospectra [35], the condition (6) means that the kffiAk- pseudospectra of A do not cross the pure imaginary axis. 2. From the perturbation bound (7), we see that the stability of the matrix sign function to the perturbation requires not only the kffiAk-pseudospectra of the A to be bounded away from the pure imaginary axis, but also A to be small (recall that dA is the distance from A to the nearest matrix with a pure-imaginary eigenvalue). 3. It is natural to take A as the condition number of the matrix sign function. Algorithms for computing dA and related problems can be found in [14, 9, 8, 12]. 4. The bound (7) is similar to the bound of the norm of the Fr'echet derivative of the matrix sign function of A at X given by Roberts [30]: is the length of the closed contour \Gamma. Recently, an asymptotic perturbation bound of sign(A) was given by Byers, He and Mehrmann [13]. They show that to first order in ffi A, kffiAk; 6 Z. BAI AND J. DEMMEL where A is assumed to have the form of (1), kffiAk is sufficiently small and the separation of the matrices A 11 and A 22 [33].\Omega is the Kronecker product. Comparing the bounds (7) and (9), we note that first the bound (7) is a global bound and is an asymptotic bound. Second, the assumption (6) for the bound (7) has a simple geometric interpretation (see Remark 2 above). It is unspecified how to interpret the assumption on sufficient small kffiAk for the bound (9). 3. Conditioning of matrix sign function computation. In [2], we point out that it may be much more efficient to compute to half machine precision only, i.e., to compute S with an absolute error bounded by u 1=2 kSk. To avoid ill conditioning in the Newton iteration and achieve the half machine precision, we believe that the matrix A must have condition number less than u \Gamma1=2 . If A is ill conditioned, say having singular values less than u 1=2 kAk, we need to use a preprocessing step to deflate small singular values by a unitary similarity transformation, and obtain a submatrix having condition number less than u \Gamma1=2 , and then compute the matrix sign function of this submatrix. Such a deflation procedure may be also needed for the intermediate matrices in the Newton iteration in the worst case. We now look more closely at the situation of near convergence of the Newton iteration, and relate the error to the distance to the nearest ill-posed problem [18]. As before, the ill-posed problems are those matrices with pure-imaginary eigenvalues. Without loss of generality, let us assume A is of the form A 11 A 12 where Otherwise, for any matrix B, by the Schur decomposition, we can where A has the above form, and then R be the solution of the Sylvester equation A which must exist and be unique since A 11 and A 22 have no common eigenvalues. Then it is known that the spectral projector P corresponding to the eigenvalues of A 11 is ' I R and . The following lemma relates R and the norm of the projection P to sign(A) and its condition number. Lemma 3.1. Let A and R be as above. Let ae 1. I \Gamma2R 2. ae, and therefore Proof. 1. Let ' I R I . It is easy to verify that if R satisfies (12), then I \Gamma2R MATRIX SIGN FUNCTION FOR COMPUTING INVARIANT SUBSPACES 7 2. Using the singular value decomposition (SVD) of R: URV one can reduce computing the SVD of S to computing the SVD of I \Gamma2\Sigma which, by permutations, is equivalent to computing the SVDs of the 2 by 2 matrices . This is in turn a simple calculation. We note that for the solution R of the Sylvester equation (12), we have where the equality is attainable [33]. From Lemma 3.1, we see that the conditioning of the matrix sign function computation is closely related to the norm of the projection P , therefore the norm of R, which in turn is closely related to the quantity Specifically, when kRk is large, and If kA 12 k is moderate, an ill conditioned matrix sign function means large kRk, which in turn means small Following Stewart [33], it means that it is harder to separate the invariant subspaces corresponding to the matrices A 11 and A 22 . The following theorem discusses the conditioning of the eigenvalues of sign(A), and the distance from sign(A) to the nearest ill-posed problem. Theorem 3.2. Let A and R be as in Lemma 3.1. Then 1. Let ffi S have the property that S + ffi S has a pure imaginary eigenvalue. Then may be chosen with no smaller. In the language of [35], the ffl-pseudospectrum of S excludes the imaginary axis for ffl ! 1=kSk, and intersects it for ffl - 1=kSk. 2. The condition number of the eigenvalues of S is kPk. In other words, perturbing S by a small ffi S perturbs the eigenvalues by at most kPk kffiSk+O(kffiSk 2 ). 3. If A is close to S and -(S) ! u \Gamma1=2 , then Newton iteration (2) in floating point arithmetic will compute S with an absolute error bounded by u 1=2 kSk. Proof. 1. The problem is to minimize oe min (S \Gamma iiI) over all real i, where oe min is the smallest singular value of S \Gamma iiI. Using the same unitary similarity transformation and permutation as in the part 1 of Lemma 3.1, we see that this is equivalent to minimizing oe min over all oe j and real i. This is a straightforward calculation, with the minimum being obtained for 8 Z. BAI AND J. DEMMEL 2. The condition number of a semi-simple eigenvalue is equal to the secant of the acute angle between its left and right eigenvectors [24, 17]. Using the above reduction to 2 by 2 subproblems (this unitary transformation of coordinates does not changes angles between vectors), this is again a straightforward calculation. 3. Since the absolute error ffi S in computing 1 essentially by the error in computing S For the Newton iteration to converge, ffi S cannot be so large that S + ffi S has pure imaginary eigenvalues; from the result 1, this means kffiSk Therefore, if u iteration (2) will compute S with an absolute error bounded by u 1=2 kSk. It is naturally desired to have an analysis from which we know the conditioning of the intermediate matrices A k in the Newton iteration. It will help us in addressing the question of how to detect possible appearance of pure imaginary eigenvalues and to modify or terminate the iteration early if necessary. Unfortunately, it is difficult to make a clean analysis far from convergence, because we are unable to relate the error in each step of the iteration to the conditioning of the problem. We can do a coarse analysis, however, in the case that the matrix is diagonalizable. Theorem 3.3. Let A have eigenvalues - j (none pure imaginary or zero), right eigenvectors x j and left eigenvectors y j , normalized so Let A k be the matrix obtained at the kth Newton iteration (2). Then for all k, oe oe min Proof. We may express the eigen-decomposition of A as We wish to bound j- j;k j from above and below for all k. This is easily done by noting that so that all - j;k lie inside a disk defined by This disk is symmetric about the real axis, so its points of minimum and maximum absolute value are both real. Solving for these extreme points yields MATRIX SIGN FUNCTION FOR COMPUTING INVARIANT SUBSPACES 9 This means oe Similarly oe which proves the bound (15). As we know, the error introduced at each step of the iteration is mainly caused by the computation of matrix inverse, which is approximately bounded in norm by when oe - 1. If uoe \Gamma3 ! oe min error cannot make an intermediate A k become singular and so cause the iteration to fail. Our analysis shows that if uoe \Gamma3 ! oe, or oe ? u 1=4 , then the iteration will not fail. This very coarse bound generalizes result 3 of Theorem 2. We note that if A is symmetric, by the orthonormal eigendecomposition of then from Theorem 3, we have Therefore, if It shows that when A is symmetric, the condition number of the intermediate matrices A k , which affects the numerical stability of the Newton iteration, is essentially determined by the square of the distance of the eigenvalues to the imaginary axis. 1 When A is nonsymmetric and diagonalizable, from Theorem 3.3, we also see that the condition number of the intermediate matrices A k is related to the norms of the spectral projectors corresponding to the eigenvalues - j and the quantities of the form ~ by a simple algebraic manipulation, we have referee predicted that in the symmetric case, the condition number of A k might be determined only by the distance, not the square of the distance. We were not able to prove such prediction. Z. BAI AND J. DEMMEL From this expression, we see that if there is an eigenvalue - j of A very near to the pure imaginary axis, i.e, ff j is small, then by the first order Taylor expansion of ~ oe j in term of ff j , we have Therefore, to first order in ff j , the condition numbers of the intermediate matrices A k O This implies that even if the eigenvalues of A are well-conditioned (i.e., the kP j k are not too large), if there are also eigenvalues of A closer to the imaginary axis than u 1=2 , then the condition number of A k could be large, so the Newton iteration could fail to converge. 4. Backward Stability of Computed Invariant Subspace. As discussed in the previous section, because of possible ill conditioning of a matrix with respect to inversion and rounding errors during the Newton iteration, we generally only expect to be able to compute the matrix sign function to the square root of the machine precision, provided that the initial matrix A has condition number smaller than u \Gamma1=2 . This means that when Newtion iteration converges, the computed matrix sign function u)kSk: Under this assumption, b is an approximate spectral projection corresponding to -+ (A). Therefore, if P ), the first ' columns b in the rank revealing QR decomposition of b span an approximate invariant subspace. b Q has the form A 11 A 12 A 22 with -( b being the approximate eigenvalues of A in C+ , and -( b A 22 ) being the approximate eigenvalues of A in C \Gamma . Since we expect the computed matrix sign function to be of half machine precision, it is reasonable to expect computing the invariant subspace to half precision too. This in turn means that the backward error in the computed decomposition b Q is bounded by O( that the problem is not very ill conditioned. In this section, we will try to justify such expectation. To this end, we first need to bound the error in the space spanned by the leading columns of the transformation matrix Q, i.e., we need to know how much a right singular subspace of the exact projection matrix perturbed when P is perturbed by a matrix of norm j. Since P is a projector, the subspace is spanned by the right singular vectors corresponding to all nonzero singular values of P (call the set of these singular values S). In practice, of course, this is a question of rank determination. From the well-known perturbation theory of the singular value MATRIX SIGN FUNCTION FOR COMPUTING INVARIANT SUBSPACES 11 decomposition [34, page 260], the space spanned by the corresponding singular vectors is perturbed by at most O(j)=gap S , where gap S is defined by To compute gap S , we note that there is always a unitary change of basis in which a projector is of the form I \Sigma , where diagonal with straightforward calculation, we find that the singular values of the projector are f where the number of ones in the set of singular values is equal to maxf2' \Gamma n; 0g. Since f Thus, the error ffi Q in Q is bounded by O( Hence, the backward error in the computed spectral decomposition is bounded by is the second order perturbation term of kffiQk. Therefore, if 2' - n, we have the following first order bound on the backward stability of computed invariant subspace: O( u)kSk O( u)kSk If we use the bound (5) of the matrix sign function S, then from (21), we have O( dA where dA , defined in (4), is the distance to the ill-posed problem. On the other hand, if we use the bound (13) for the matrix sign function S, then from (21) again, we have O( 22 ) is the separation of the matrices A 11 and A 22 , if A is assumed to have the form (11). We note that the error bound (23) is essentially the same as the error bound given by Byers, He and Mehrmann [13], although we use a different approach. In [13], it is assumed that in (19), where F 21 is the (2,1) block of the matrix F . Therefore, O( u) term in (23) is replaced by O(u). Z. BAI AND J. DEMMEL The bounds (22) and (23) reveal two important features of the matrix sign function based algorithm for computing the invariant subspace. First, they indicate that the backward error in the computed approximate invariant subspace appears no larger than the absolute error in the computed matrix sign function, provided that the spectral decomposition problem is not very ill conditioned (i.e., dA or ffi is not tiny). Second, if 2' - n, the backward error is a decreasing function of oe l . If oe ' is large, this means oe 1 and so are large, and this in turn means the eigenvalues close the imaginary axis are ill conditioned. It is harder to divide these eigenvalues. Of course as they become ill conditioned, dA decreases at the same time, which must counterbalance the increase in oe ' in a certain range. It is interesting to ask which error bound (22) and (23) is sharper, i.e., which one of the quantities dA and 22 ) is larger. In [13], an example of a 2 by 2 matrix is given to show that the quantity ffi is larger than the quantity dA . However, we can also devise simple examples to show that dA can be larger than More generally, by choosing A 11 to be a large Jordan block with a tiny eigenvalue, and dA to be close to the square root of ffi . dA is computed using "numerical brute force" to plot the function dA (- ) on a wide the range of - 2 IR, and search for the minimal value. Note that by modifying A to be A \Gamma oeI, where oe is a (sufficiently small) real number, dA will change, but ffi will not. Thus dA and ffi are not completely comparable quantities. We believe dA to be a more natural quantity to use than ffi , since ffi does not always depend on the distance to the nearest ill-posed problem. This is reminiscent of the difference between the quantities In practice, we will use the a posteriori bound kE 21 k=kAk anyway, since if we block upper-triangularize b Q by setting the (2; 1) block to zero, kE 21 k=kAk is precisely the backward error we introduce. Before ending of this section, let us comment on stability of the matrix sign function based algorithm versus the QR algorithm. The QR algorithm is a numerical backward stable method for computing the Schur decomposition of a general non-symmetric matrix A. The computed Schur form b T and Schur vectors b Q by the QR algorithm satisfy where E is of the order of ukAk. Numerical software for the QR algorithm is available in EISPACK [32] and LAPACK [1]. Although nonconvergent examples have been found, they are quite rare in practice [6, 16]. We note that the eigenvalues on the (block)-diagonal of b may appear in any order. Therefore, if an application requires an invariant subspace corresponding to the eigenvalues in a specific region in complex plane, a second step of reordering eigenvalues on the diagonal of b T is necessary. A guaranteed stable implementation of this reordering is described in [7]. The matrix sign function based algorithm can be regarded as an algorithm to combine these two steps into one. If the matrix sign function can be computed MATRIX SIGN FUNCTION FOR COMPUTING INVARIANT SUBSPACES 13 within the order of ukSk, then the analysis in this section shows that the matrix sign function based algorithm could be as stable as the QR algorithm plus reordering. Unfortunately, if the matrix is ill conditioned with respect to matrix inversion (which does not affect the QR algorithm), numerical unstable is anticipated in the computed matrix sign function. Therefore, in general, the matrix sign function is less stable than the QR algorithm plus reordering. 5. Numerical Experiments. In this section, we will present numerical examples to verify the above analysis. We will see that the numerical stability of the Newton iteration (2) and the backward accuracy of computed spectral decomposition (1) under the influence of the conditioning of the matrix A with respect to inversion, the condition number -(S) of and the distance \Delta(A) of the eigenvalues of A to the pure imaginary axis, where We use the easily computed quantity \Delta(A) as an surrogate of the quantity dA in (4). Let us recall that the analysis of sections 3 and 4 essentially claims that (1) If \Delta(A) ! u 1=2 , then the Newton iteration may fail to converge or fail to compute the matrix sign function within the absolute error u 1=2 kSk, even when the matrix sign function is well-conditioned. See (18). (2) If -(S) ? u \Gamma1=2 , then even the distance \Delta(A) is not small, the Newton iteration may still fail to compute the matrix sign function in the absolute error of O(u 1=2 kSk). See the part 3 of Theorem 3.2. (3) In general, the backward error in the computed spectral decomposition will be smaller than the absolute error in the computed matrix sign function. See (21). The following numerical examples will illustrate these claims. Our numerical experiments were performed on a SUN workstation 10 with machine precision " u. All the algorithms are implemented in Matlab 4.0a. We use the simple Newtion iteration (2) to compute the matrix sign function with the stopping criterion The maximal number of iterations is set to be 70. At the convergence, we have S, the computed matrix sign function. We use the QR decomposition with column pivoting as the rank revealing scheme: 1( b R\Pi, and finally compute A 11 A 12 A 22 where the first R) columns of b Q spans the invariant subspaces corresponding to -( b A 11 ), which are the approximate eigenvalues of A in C+ . kE 21 k=kAk is the backward error committed by the algorithm. All our test matrices are constructed of the form where U is an orthogonal matrix generated from the QR decomposition a random matrix with normal distribution having mean 0.0 and variance 1.0. We will choose different submatrices A 11 , A 22 and A 12 so that the generated matrices A have different specific features in order to observe our theoretical results in practice. 14 Z. BAI AND J. DEMMEL Table Numerical Results for Example 1 The exact matrix sign function of A and the condition number of S are computed as described in Lemma 3.1. The condition number of A is computed by Matlab function cond. In the following tables, iter is the number of iterations of the Newton iteration. number 10 ff in parenthesis next to an iteration number iter indicates that the convergence of the Newton iteration was stationary about O(10 ff ) from the iter th iteration forward, and failed to satisfy the stopping criterion even after the allowed maximal number of iterations. We have experimented numerous matrices with different pathologically ill conditioning in terms of the distance to the pure imaginary axis, the condition numbers of -(A) and -(S), and the different values of sep(A 11 ; A 22 ) and so on. Two selected examples presented here are of typical behaviors we observed. Example 1. In this example, the matrices A are of the form (24) with and A is a random 2 by 2 matrix with normal distribution multiplying by a parameter c. The generated matrix A have two complex conjugate eigenpairs s \Sigma i and \Gammas \Sigma i. As s ! 0, the distance too. The size of the parameter c will adjust the conditioning of the resulted matrix A and its matrix sign function. Table 1 reports the computed results for different values of From the table, we see that when the matrices are well conditioned and the corresponding the matrix sign function is also well-conditioned, as stated in the claim (1), the convergence rate and accuracy of the Newton iteration is clearly determined by the distance \Delta(A). When the distance becomes smaller, there is a steady increase in the number of the Newton iteration required to convergence and the loss of the accuracy in the computed matrix sign function and therefore the desired invariant subspace. From the table, we also see that when both \Delta(A) and -(S) are moderate, the Newton iteration fails to compute the matrix sign function in half machine precision. Nevertheless, the computed invariant subspace seems still have half machine precision, see the claim MATRIX SIGN FUNCTION FOR COMPUTING INVARIANT SUBSPACES 15 Table Numerical Results of Example 2 Example 2. In this example, the test matrices A are of the form (24). A 12 are 5 by 5 0)-normally distributed random matrices. The submatrices A 11 and A 22 are first set by 5 by 5 (1; 0)-normally distributed random upper tridiagonal matrices, and then the diagonal elements of A 11 and A 22 are replaced by dja ii j and \Gammadja ii j, respectively, where a ii (1 - i - n) are random numbers with normal distribution (0; 1), d is a positive parameter. A 12 are 5 by 5 (1; 0)-normally distributed random matrices. The numerical results are reported in Table 2. For the given parameter d, the eigenvalues are well-separated away from the pure imaginary axis (\Delta(A) is not small), however, as stated in the claim (2), we see the influence of the condition numbers -(S) to the convergence of the Newton iteration and therefore the accuracy of the computed matrix sign function and the invariant subspace. 6. Refining Estimates of Approximate Invariant Subspaces. When we use the matrix sign function based algorithm to deflate an invariant subspace of matrix A, we end up with the form A 11 A 12 A 22 where the size of kE 21 k=kAk reveals the accuracy and backward stability of computed invariant subspace spanning by b of A. If higher accuracy is desired, we may use iterative refinement techniques to improve the accuracy of computed invariant sub- space. The methods are due to Stewart [33], Dongarra, Moler and Wilkinson [20], and Chatelin [15]. Even though these methods all apparently solve different equations, as shown by Demmel [19], after changing variables, they all solve the same Riccati equation in the inner loop. Let us follow Stewart's approach to present the first class of methods. From (25), we know that b spans an approximate invariant subspaces and b spans an orthogonal complementary subspace. If let the true invariant subspace is represented as b its orthogonal complementary subspace as b Then Y is derived as follows: b will be an invariant subspace if and only if the lower left block of is zero, i.e. if the lower left corner of I \GammaY H Y I A 11 A 12 A 22 \GammaY I Z. BAI AND J. DEMMEL is zero. Thus, Y must satisfy the equation A 12 Y; which is the well-known algebraic Riccati equation. We may use the following two iterative methods to solve it: 1. The simple Newton iteration: A 22 Y with 2. The modified Newtion iteration: with Therefore, we only need to solve a Sylvester equation in the inner loop of the iterative refinement. In following numerical example, we only use the simple Newton iteration (26) to refine the approximate invariant subspace computed by matrix sign function based algorithm, with the following stopping criterion: Example 3 We continue the Example 2. Table 3 lists the 22 ), the number of iterative refinement steps and the backward accuracy of improved invariant subspace. As shown in the convergence analysis for the iterative solvers (26) and (27) of the Riccati equation by Stewart [33] and Demmel [18], if we let A 22 ); then under the assumptions k ! 1=4 and k ! 1=12, the iterations (26) and (27) converge, respectively. Therefore, sep( b A 22 ) is a key factor to the convergence of the iterative refinement schemes. The above examples verify such analysis. From the analysis of section 3, we recall that sep( b A 22 ) also affects the backward stability of the computed invariant subspace by the matrix sign function based algorithm in the first place (before iterative refinement). 7. Extension to the Generalized Eigenproblem. In this section, we outline a scheme to extend the matrix sign function based algorithm to solve the generalized eigenvalue problem of a regular matrix pencil A \Gamma -B. A matrix pencil A \Gamma -B is regular if A\Gamma-B is square and det(A\Gamma-B) is not identically zero. In [22], Gardiner and Laub have considered an extension of the Newton iteration for computing the matrix sign function to a matrix pencil for solving generalized algebraic Riccati equations. Here we discuss another possible approach, which includes the computation of both left and right deflating subspaces. For the given matrix pencil A \Gamma -B, the problem of the spectral decomposition is to seek a pair of left and right deflating subspaces L and R corresponding to the eigenvalues of the pencil in a specified region D in complex plane. In other words, we want to find a pair of unitary matrices QL and QR so that if MATRIX SIGN FUNCTION FOR COMPUTING INVARIANT SUBSPACES 17 Table Iterative Refinement Reresults of Example 2 d 28 where the eigenvalues of A 11 \Gamma -B 11 are the eigenvalues of A \Gamma -B in a selected region D in complex plane. Here, we will only discuss the region D to be the open right half complex plane. As the same treatment in the standard eigenproblem, by employing M-obius transformations (ffA divide-and-conquer, D can be the union of intersections of arbitrary half planes and (complemented) disks, and so a rather general region. To this end, by directly applying the Newton iteration to AB \Gamma1 , we have At convergence, In practice, we do not want to invert B if it is ill conditioned. Hence, by letting Z then the above iteration becomes This leads to the following iteration: converges quadratically to a matrix Z1 . Then Next, to find the desired deflating subspace, we use the rank revealing QR decomposition to calculate the range space of the projection corresponding to the spectral in the open right half plane, which has the same range space as Thus by computing the rank revealing QR decomposition of Z1 we obtain the invariant subspace of AB \Gamma1 without inverting B, i.e., where -(CR ) are the eigenvalues of the pencil A \Gamma -B in the open right half plane, -(CL ) are the ones of A \Gamma -B in the open left half plane. Therefore, we have obtained the left deflating subspace of A \Gamma -B. Z. BAI AND J. DEMMEL To compute the right deflating subspace of A\Gamma-B, we can applying the above idea to A H \Gamma -B H , since transposing swaps right and left spaces. The Newton iteration implicitly applying to A H B \GammaH turns out to be ~ converges quadratically to a matrix ~ Z1 . Using the same arguments as above, after computing the rank revealing QR decomposition of ~ QRRR \Pi R , we have ~ R A H B \GammaH ~ where -(DL ) are the eigenvalues of the pencil A\Gamma-B in the open left half plane, -(DR ) are the ones of A \Gamma -B in the open right half plane. Note that for the desired spectral decomposition, after transposing, we need to first compute the deflating subspace corresponding to the eigenvalues in the open left half plane. Let \Pi, where ~ \Pi is anti-diagonal identity matrix 2 , then we have From (29) and (30), we immediately have R D H0 D H L AQR and Q H BQR have the partitions A 21 A 22 we have R D H0 D H Note that -(CL ) are the eigenvalues of the pencil A \Gamma -B in the open left half plane, -(DR ) are the eigenvalues of the pencil A \Gamma -B in the open right half plane. Therefore, the above homogeneous Sylvester equation has only solution B From (31) or (32), we have A The computed unitary orthogonal matrices QL and QR give the desired spectral decomposition (28). 2 The permutation ~ \Pi can be avoided if we use the rank revealing QL decomposition. MATRIX SIGN FUNCTION FOR COMPUTING INVARIANT SUBSPACES 19 8. Closing Remarks. In this paper, we have presented a number of new results and approaches to further analyze the numerical behavior of the matrix sign function and algorithms using it to compute spectral decompositions of nonsymmetric matri- ces. From this analysis and numerical experiments, we conclude that if the spectral decomposition problem is not ill conditioned, the algorithm is a practical approach to solve the nonsymmetric eigenvalue problem. Performance evaluation of the matrix sign function based algorithm on parallel distributed memory machines, such as the Intel Delta and CM-5, is reported in [4]. During the course of this work, we have discovered a new approach which essentially computes the same spectral projection matrix as the matrix sign function approach does, and also uses basic matrix operations, namely, matrix multiplication and the QR decomposition. However, it avoids the matrix inverse. From the point of view of accuracy, this is a more promising approach. The new approach is based on the work of Bulgakov, Godunov and Malyshev [10, 27, 28]. In [5], we have improved their results in several important ways, and made it a truly practical and inverse free highly parallel algorithm for both the standard and generalized spectral decomposition problems. In brief, the difference between the matrix sign function and inverse methods is as follows. The matrix sign function method is significantly faster than it converges, but there are some very difficult problems where the inverse free algorithm gives a more accurate answer than the matrix sign function algorithm. The interested reader may see the paper [5] for details. Acknowledgements . The first author was supported in part by ARPA grant DM28E04120 and P-95006 via a subcontract from Argonne National Laboratory and by an NSF grant ASC-9313958 and in part by an DOE grant DE-FG03-94ER25219 via subcontracts from University of California at Berkeley. The second author was funded in part by the ARPA contract DAAH04-95-1-0077 through University of Tennessee subcontract ORA7453.02, ARPA contract DAAL03-91-C-0047 through University of Tennessee subcontract ORA4466.02, NSF contracts ASC-9313958 and ASC-9404748, contracts DE-FG03-94ER25219, DE-FG03-94ER25206, DOE contract No. W- through subcontract No. 951322401 with Argonne National Labora- tory, and NSF Infrastructure Grant Nos. CDA-8722788 and CDA-9401156. The information presented here does not necessarily reflect the position or the policy of the Government and no official endorsement should be inferred. The authors would like to acknowledge Ralph Byers, Chunyang He, Nick Higham and Volker Mehrmann for fruitful discussions on the subject. We would also like to thank the referees for their valuable comments on the manuscript. --R Design of a parallel nonsymmetric eigenroutine toolbox Design of a parallel nonsymmetric eigenroutine toolbox The spectral decomposition of nonsymmetric matrices on distributed memory parallel computers. Inverse free parallel spectral divide and conquer algorithms for nonsymmetric eigenproblems. Convergence of the shifted QR algorithm on 3 by 3 normal matrices. Reordering diagonal blocks in real schur form. A regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its L1-norm A bisection method for computing the H1 norm of a transfer matrix and related problems. Circular dichotomy of the spectrum of a matrix. Solving the algebraic Riccati equation with the matrix sign function. A bisection method for measuring the distance of a stable matrix to the unstable matrices. The matrix sign function method and the computation of invariant subspaces. On the stability radius of a generalized state-space system Simultaneous Newton's iteration for the eigenproblem. How the QR algorithm fails to converge and how to fix it. The condition number of equivalence transformations that block diagonalize matrix pencils. On condition numbers and the distance to the nearest ill-posed problem Three methods for refining estimates of invariant subspaces. Improving the accuracy of computed eigenvalues and eigenvectors. Matrix Eigensystem Routines - EISPACK Guide Extension A generalization of the matrix-sign function solution for algebraic Riccati equations The matrix sign decomposition and its relation to the polar decomposition. Perturbation Theory for Linear Operators. Polar decomposition and matrix sign function condition estimates. Matrix sign function algorithms for Riccati equa- tions Guaranteed accuracy in spectral problems of linear algebra Parallel algorithm for solving some spectral problems of linear algebra. Condition estimation for the matrix function via the Schur decomposition. Linear model reduction and solution of the algebraic Riccati equation. Separation of matrix eigenvalues and structural decomposition of large-scale systems Matrix Eigensystem Routines - EISPACK Guide and perturbation bounds for subspaces associated with certain eigenvalue problems. Matrix Perturbation Theory. Pseudospectra of matrices. --TR --CTR Daniel Kressner, Block algorithms for reordering standard and generalized Schur forms, ACM Transactions on Mathematical Software (TOMS), v.32 n.4, p.521-532, December 2006

newton's method;matrix sign function;eigenvalue problem;deflating subspaces;invariant subspace

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Parameter Estimation in the Presence of Bounded Data Uncertainties.

We formulate and solve a new parameter estimation problem in the presence of data uncertainties. The new method is suitable when a priori bounds on the uncertain data are available, and its solution leads to more meaningful results, especially when compared with other methods such as total least-squares and robust estimation. Its superior performance is due to the fact that the new method guarantees that the effect of the uncertainties will never be unnecessarily over-estimated, beyond what is reasonably assumed by the a priori bounds. A geometric interpretation of the solution is provided, along with a closed form expression for it. We also consider the case in which only selected columns of the coefficient matrix are subject to perturbations.

Introduction . The central problem in estimation is to recover, to good ac- curacy, a set of unobservable parameters from corrupted data. Several optimization criteria have been used for estimation purposes over the years, but the most im- portant, at least in the sense of having had the most applications, are criteria that are based on quadratic cost functions. The most striking among these is the linear least-squares criterion, which was first developed by Gauss (ca. 1795) in his work on celestial mechanics. Since then, it has enjoyed widespread popularity in many diverse areas as a result of its attractive computational and statistical properties (see, e.g., [4, 8, 10, 13]). Among these attractive properties, the most notable are the facts that least-squares solutions can be explicitly evaluated in closed forms, they can be recursively updated as more input data is made available, and they are also maximum likelihood estimators in the presence of normally distributed measurement noise. Alternative optimization criteria, however, have been proposed over the years including, among others, regularized least-squares [4], ridge regression [4, 10], total addresses: shiv@ece.ucsb.edu, golub@sccm.stanford.edu, mgu@math.ucla.edu, and sayed@ee.ucla.edu. least-squares [2, 3, 4, 7], and robust estimation [6, 9, 12, 14]. These different formulations allow, in one way or another, incorporation of further a priori information about the unknown parameter into the problem statement. They are also more effective in the presence of data errors and incomplete statistical information about the exogenous signals (or measurement errors). Among the most notable variations is the total least-squares (TLS) method, also known as orthogonal regression or errors-in-variables method in statistics and system identification [11]. In contrast to the standard least-squares problem, the TLS formulation allows for errors in the data matrix. But it still exhibits certain drawbacks that degrade its performance in practical situations. In particular, it may unnecessarily over-emphasize the effect of noise and uncertainties and can, therefore, lead to overly conservative results. More specifically, assume A 2 R m\Thetan is a given full rank matrix with m n, is a given vector, and consider the problem of solving the inconsistent linear system A"x b in the least-squares sense. The TLS solution assumes data uncertainties in A and proceeds to correct A and b by replacing them by their projections, " A and " b, onto a specific subspace and by solving the consistent linear system of equations b. The spectral norm of the correction in the TLS solution is bounded by the smallest singular value of \Theta A b . While this norm might be small for vectors b that are close enough to the range space of A, it need not always be so. In other words, the TLS solution may lead to situations in which the correction term is unnecessarily large. Consider, for example, a situation in which the uncertainties in A are very small, say A is almost known exactly. Assume further that b is far from the column space of A. In this case, it is not difficult to visualize that the TLS solution will need to rotate may therefore end up with an overly corrected approximant for A, despite the fact that A is almost exact. These facts motivate us to introduce a new parameter estimation formulation with prior bounds on the size of the allowable corrections to the data. More specifically, we formulate and solve a new estimation problem that is more suitable for scenarios in which a-priori bounds on the uncertain data are known. The solution leads to more meaningful results in the sense that it guarantees that the effect of the uncertainties will never be unnecessarily over-estimated, beyond what is reasonably assumed by the a-priori bounds. We note that while preparing this paper, the related work [1] has come to our attention, where the authors have independently formulated and solved a similar estimation problem by using (convex) semidefinite programming techniques and interior-point methods. The resulting computational complexity of the proposed solution is is the smaller matrix dimension. The solution proposed in this paper proceeds by first providing a geometric formulation of the problem, followed by an algebraic derivation that establishes that the optimal solution can in fact be obtained by solving a related regularized problem. The parameter of the regularization step is further shown to be obtained as the unique positive root of a secular equation and as a function of the given data. In this sense, the new formulation turns out to provide automatic regularization and, hence, has some useful regularization properties: the regularization parameter is not selected by the user but rather determined by the algorithm. Our solution involves an SVD step and its computational complexity amounts to O(mn 2 is again the smaller matrix dimension. A summary of the problem and its solution is provided in A NEW METHOD FOR PARAMETER ESTIMATION WITH UNCERTAIN Sec. 3.4 at the end of this paper. [Other problem formulations are studied in [15].] 2. Problem Formulation. Let A 2 R m\Thetan be a given matrix with m n and be a given vector, which are assumed to be linearly related via an unknown vector of parameters x 2 R n , The vector v 2 R m denotes measurement noise and it explains the mismatch between Ax and the given vector (or observation) b. We assume that the "true" coefficient matrix is A + ffiA, and that we only know an upper bound on the 2\Gammainduced norm of the perturbation ffiA: with j being known. Likewise, we assume that the "true" observation vector is b and that we know an upper bound j b on the Euclidean norm of the perturbation ffib: We then pose the problem of finding an estimate that performs "well" for any allowed perturbation (ffiA; ffib). More specifically, we pose the following min-max problem: Problem 1. Given A 2 R m\Thetan , with m n, nonnegative real numbers (j; j b ). Determine, if possible, an " x that solves x The situation is depicted in Fig. 2.1. Any particular choice for " x would lead to many residual norms, one for each possible choice of A in the disc in the disc (b second choice for " x would lead to other residual norms, the maximum value of which need not be the same as the first choice. We want to choose an estimate " x that minimizes the maximum possible residual norm. This is depicted in Fig. 2.2 for two choices, say " . The curves show the values of the residual norms as a function of A Fig. 2.1. Geometric interpretation of the new least-squares formulation. We note that if problem (2.4) reduces to a standard least squares problem. Therefore we shall assume throughout that j ? 0. [It will turn out that the solution to the above min-max problem is independent of j b ]. kresidualk Fig. 2.2. Two illustrative residual-norm curves. 2.1. A Geometric Interpretation. The min-max problem admits an interesting geometric formulation that highlights some of the issues involved in its solution. For this purpose, and for the sake of illustration, assume we have a unit-norm vector b, kbk uncertainties in it (j Assume further that A is simply a column vector, say a, with j 6= 0. That is, only A is assumed to be uncertain with perturbations that are bounded by j in magnitude (as in (2.2)). Now consider problem (2.4) in this context, which reads as follows: x This situation is depicted in Fig. 2.3. The vectors a and b are indicated in thick black lines. The vector a is shown in the horizontal direction and a circle of radius j around its vertex indicates the set of all possible vertices for a a Fig. 2.3. Geometric construction of the solution for a simple example. For any " x that we pick, the set f(a+ ffia)"xg describes a disc of center a"x and radius j"x. This is indicated in the figure by the largest rightmost circle, which corresponds to a choice of a positive " x that is larger than one. The vector in f(a + ffia)"xg that is furthest away from b is the one obtained by drawing a line from b through the center of the rightmost circle. The intersection of this line with the circle defines a A NEW METHOD FOR PARAMETER ESTIMATION WITH UNCERTAIN residual vector r 3 whose norm is the largest among all possible residual vectors in the set f(a ffia)"xg. Likewise, if we draw a line from b that passes through the vertex of a, it will intersect the circle at a point that defines a residual vector r 2 . This residual will have the largest norm among all residuals that correspond to the particular choice " More generally, any " x that we pick will determine a circle, and the corresponding largest residual is obtained by finding the furthest point on the circle from b. This is the point where the line that passes through b and the center of the circle intersects the circle on the other side of b. We need to pick an " x that minimizes the largest residual. For example, it is clear from the figure that the norm of r 3 is larger than the norm of r 2 . The claim is that in order to minimize the largest residual we need to proceed as follows: we drop a perpendicular from b to the lower tangent line denoted by ' 1 . This perpendicular intersects the horizontal line in a point where we draw a new circle (the leftmost circle) that is tangent to both ' 1 and ' 2 . This circle corresponds to a choice of " x such that the furthest point on it from b is the foot of the perpendicular from b to ' 1 . The residual indicated by r 1 corresponds to the desired solution (it has the minimum norm among the largest residuals). To verify this claim, we refer to Fig. 2.4 where we have only indicated two circles; the circle that leads to a largest residual that is orthogonal to ' 1 and a second circle to its left. For this second leftmost circle, we denote its largest residual by r 4 . We also denote the segment that connects b to the point of tangency of this circle with ' 1 by r. It is clear that r is larger than r 1 since r and r 1 are the sides of a right triangle. It is also clear that r 4 is larger than r, by construction. Hence, r 4 is larger than r 1 . A similar argument will show that r 1 is smaller than residuals that result from circles to its right. The above argument shows that the minimizing solution can be obtained as fol- lows: drop a perpendicular from b to ' 1 . Pick the point where the perpendicular meets the horizontal line and draw a circle that is tangent to both ' 1 and ' 2 . Its radius will be j"x, where " x is the optimal solution. Also, the foot of the perpendicular on ' 1 will be the optimal " b. The projection " b (and consequently the solution " x) will be nonzero as long as b is not orthogonal to the direction ' 1 . This imposes a condition on j. Indeed, the direction ' 1 will be orthogonal to b only when j is large enough. This requires that the circle centered around a has radius a T b, which is the length of the projection of a onto the unit norm vector b. This is depicted in Fig. 2.5. Hence, the largest value that can be allowed for j in order to have a nonzero solution " x is Indeed, if j were larger than or equal to this value, then the vector in the set (a that would always lead to the maximum residual norm is the one that is orthogonal to b, in which case the solution will be zero again. The same geometric argument will lead to a similar conclusion had we allowed for uncertainties in b as well. For a non-unity b, the upper bound on j would take the form We shall see that in the general case a similar bound holds, for nonzero solutions, and 6 CHANDRASEKARAN, GOLUB, GU, AND SAYED r Fig. 2.4. Geometric construction of the solution for a simple example. a Fig. 2.5. Geometric condition for a nonzero solution. is given by We now proceed to an algebraic solution of the min-max problem. A final statement of the form of the solution is given further ahead in Sec. 3.4. 3. Reducing the Minimax Problem to a Minimization Problem. We start by showing how to reduce the min-max problem (2.4) to a standard minimization A NEW METHOD FOR PARAMETER ESTIMATION WITH UNCERTAIN problem. To begin with, we note that which provides an upper bound for k . But this upper bound is in fact achievable, i.e., there exist (ffiA; ffib) for which To see that this is indeed the case, choose ffiA as the rank one matrix and choose ffib as the vector For these choices of perturbations in A and b, it follows that are collinear vectors that point in the same direction. Hence, which is the desired upper bound. We therefore conclude that which establishes the following result. Lemma 3.1. The min-max problem (2.4) is equivalent to the following minimization problem. Given A 2 R m\Thetan , with m n, nonnegative real numbers possible, an " x that solves 3.1. Solving the Minimization Problem. To solve (3.2), we define the cost function It is easy to check that L("x) is a convex continuous function in " x and hence any local minimum of L("x) is also a global minimum. But at any local minimum of L("x), it either holds that L("x) is not differentiable or its gradient 5L("x) is 0. In particular, note that L("x) is not differentiable only at " and at any " x that satisfies A"x 8 CHANDRASEKARAN, GOLUB, GU, AND SAYED We first consider the case in which L("x) is differentiable and, hence, the gradient of L("x) exists and is given by A T A+ ffI where we have introduced the positive real number By setting we obtain that any stationary solution " x of L("x) is given by We still need to determine the parameter ff that corresponds to " x, and which is defined in (3.3). To solve for ff, we introduce the singular value decomposition (SVD) of A: where U 2 R m\Thetam and V 2 R n\Thetan are orthogonal, and nal, with being the singular values of A. We further partition the vector U T b into m\Gamman . In this case, the expression (3.4) for " x can be rewritten in the equivalent form and hence, Likewise, ff A NEW METHOD FOR PARAMETER ESTIMATION WITH UNCERTAIN which shows that Therefore, equation (3.3) for ff reduces to the following nonlinear equation that is only a function of ff and the given data (A; b; j), Note that only the norm of b 2 , and not b 2 itself, is needed in the above expression. Remark. We have assumed in the derivation so far that A is full rank. If this were not the case, i.e., if A (and hence \Sigma) were singular, then equation (3.8) can be reduced to an equation of the same form but with a non-singular \Sigma of smaller dimension. Indeed, if we partition k be the first k components of b 1 ; n\Gammak be the last components of b 1 ; and let Then equation (3.8) reduces to r the same form as (3.8). From now on, we assume that A is full rank and, hence, \Sigma is A full rank is a standing assumption in the sequel : 3.2. The Secular Equation. Define the nonlinear function in ff, It is clear that ff is a positive solution to (3.8) if, and only if, it is a positive root of G(ff). Following [4], we refer to the equation as a secular equation. The function G(ff) has several useful properties that will allow us to provide conditions for the existence of a unique positive root ff. We start with the following result. Lemma 3.2. The function G(ff) in (3.10) can have at most one positive root. In ff ? 0 is a root of G(ff), then " ff is a simple root and G 0 ("ff) ? 0. Proof. We prove the second conclusion first. Partition where the diagonal entries of \Sigma 1 2 R k\Thetak are those of \Sigma that are larger than j, and the diagonal entries of \Sigma 2 2 R (n+1\Gammak)\Theta(n+1\Gammak) are the remaining diagonal entries of \Sigma and one 0. It follows that (in terms of the 2\Gammainduced norm for the diagonal matrices for all ff ? 0. Let u 2 R k be the first k components of the last components of It follows that we can rewrite G(ff) as the difference and, consequently, ff ? 0 be a root of G(ff). This means that ffI ffI which leads to the following sequence of inequalities: ffI ffI ffI ffI ffI ffI ffI Combining this relation with the expression for G 0 (ff), it immediately follows that ff must be a simple root of G(ff). Furthermore, we note that G(ff) is a sum of n rational functions in ff and hence can have only a finite number of positive roots. In the following we show by contradiction that G(ff) can have no positive roots other than " ff. Assume to the contrary that " ff 1 is another positive root of G(ff). Without loss of generality, we further assume that " ff 1 and that G(ff) does not have any root within the open It follows from the above proof that A NEW METHOD FOR PARAMETER ESTIMATION WITH UNCERTAIN But this implies that G(ff) ? 0 for ff slightly larger than " ff and G(ff) ! 0 for ff slightly smaller than " consequently, G(ff) must have a root in the interval ("ff; " contradiction to our assumptions. Hence G(ff) can have at most one positive root. Now we provide conditions for G(ff) to have a positive root. [The next result was in fact suggested earlier by the geometric argument of Fig. 2.3]. Note that A"x can be written as Therefore solving possible, is equivalent to solving This shows that a necessary and sufficient condition for b to belong to the column span of A is b Lemma 3.3. Assume j ? 0 (a standing assumption) and b 2 6= 0, i.e., b does not belong to the column span of A. Then the function G(ff) in (3.10) has a unique positive root if, and only if, Proof. We note that lim and that lim First we assume that condition (3.13) holds. It follows then that G(ff) changes sign on the interval (0; +1) and therefore has to have a positive root. By Lemma 3.2 this positive root must also be unique. On the other hand, assume that This condition implies, in view of (3.14), that G(ff) ! 0 for sufficiently large ff. We now show by contradiction that G(ff) does not have a positive root. Assume to the contrary that " ff is a positive root of G(ff). It then follows from Lemma 3.2 that G(ff) is positive for ff slightly larger than " ff since G 0 ("ff) ? 0, and hence G(ff) must have a root in ("ff; +1), which is a contradiction according to Lemma 3.2. Hence G(ff) does not have a positive root in this case. Finally, we consider the case We also show by contradiction that G(ff) does not have a positive root. Assume to the contrary that " ff is a positive root of G(ff). It then follows from Lemma 3.2 that ff must be a simple root, and a continuous function of the coefficients in G(ff). In ff is a continuous function of j. Now we slightly increase the value of j so that By continuity, G(ff) has a positive root for such values of j, but we have just shown that this is not possible. Hence, G(ff) does not have a positive root in this case either. We now consider the case b lies in the column span of A. This case arises, for example, when A is a square invertible matrix and It follows from b Now note that Therefore, by using the Cauchy-Schwarz inequality, we have and we obtain, after applying the Cauchy-Schawrtz inequality one more time, that Lemma 3.4. Assume j ? 0 (a standing assumption) and b lies in the column span of A. Then the function G(ff) in (3.10) has a positive root if, and only, if Proof. It is easy to check that lim A NEW METHOD FOR PARAMETER ESTIMATION WITH UNCERTAIN and that lim lim Arguments similar to those in the proof of Lemma 3.3 show that G(ff) does not have a positive root. Similarly G(ff) does not have a positive root if arguments similar to those in the proof of Lemma 3.3 show that G(ff) does not have a positive root if However, if lim Hence G(ff) must have a positive root. By Lemma 3.2 this positive root is unique. 3.3. Finding the Global Minimum. We now show that whenever G(ff) has a positive root " ff, the corresponding vector " x in (3.4) must be the global minimizer of L("x). Lemma 3.5. Let " ff be a positive root of G(ff) and let " x be defined by (3.4) for ff. x is the global minimum of L("x). Proof. We first show that where 4L("x) is the Hessian of L at " x. We take the gradient of L, Consequently, A T A \GammakA"x \Gamma bk 3\Gamma k"xk 3" We now simplify this expression. It follows from (3.4) that ffI and hence Substituting this relation into the expression for the Hessian matrix 4L("x), and simplifying the resulting expression using equation (3.3), we obtain ffI x 14 CHANDRASEKARAN, GOLUB, GU, AND SAYED Observe that the matrix ffI is positive definite since " can have at most one non-positive eigenvalue. This implies that 4L("x) is positive definite if and only if det (4L("x)) ? 0. Indeed, det ffI x x ffI x The last expression can be further rewritten using the SVD of A and (3.8): det x ffI x ffI =" x ffI x ffI ff x ffI Comparing the last expression with the function G(ff) in (3.10), we finally have det ff By Lemma 3.2, we have that G 0 ("ff) ? 0. Consequently, 4L("x) must be positive definite, and hence " x must be a local minimizer of L("x). Since L("x) is a convex function, this also means that " x is a global minimizer of L("x). We still need to consider the points at which L("x) is not differentiable. These include " any solution of Consider first the case b 2 6= 0. This means that b does not belong to the column span of A and, hence, we only need to check " follows from Lemma 3.3 that G(ff) has a unique positive root " ff and it follows from Lemma 3.5 that ffI is the global minimum. On the other hand, if condition (3.13) does not hold, then it follows from Lemma 3.3 that G(ff) does not have any positive root and hence is the global minimum. Now consider the case b which means that b lies in the column span of A. In this case L("x) is not differentiable at both " A NEW METHOD FOR PARAMETER ESTIMATION WITH UNCERTAIN condition (3.16) holds, then it follows from Lemma 3.4 that G(ff) has a unique positive ff and it follows from Lemma 3.5 that ffI is the global minimum. On the other hand, if where we have used the Cauchy-Schwartz inequality. It follows that is the global minimum in this case. Similarly, if j 2 , then is the global minimum. We finally consider the degenerate case j. Under this condition, it follows from (3.15) that Hence, This shows that L L(0). But since L("x) is a convex function in " x, we conclude that for any " x that is a convex linear combination of 0 and we also obtain Therefore, when there are many solutions " x and these are all scaled multiples of V \Sigma as in (3.17). 3.4. Statement of the Solution of the Constrained Min-Max Problem. We collect in the form of a theorem the conclusions of our earlier analysis. Theorem 3.6. Given A 2 R m\Thetan , with m n and A full rank, b nonnegative real numbers (j; j b ). The following optimization problem: min x always has a solution " x. The solution(s) can be constructed as follows. ffl Introduce the SVD of A, where U 2 R m\Thetam and V 2 R n\Thetan are orthogonal, and is diagonal, with being the singular values of A. ffl Partition the vector U T b into m\Gamman . ffl Introduce the secular function and First case: b does not belong to the column span of A. 1. If j 2 then the unique solution is " 2. If j ! 2 then the unique solution is " is the unique positive root of the secular equation Second case: b belongs to the column span of A. 1. If j 2 then the unique solution is " 2. If then the unique solution is " ff is the unique positive root of the secular equation 3. If j 1 then the unique solution is " 4. If then there are infinitely many solutions that are given by The above solution is suitable when the computation of the SVD of A is feasible. For large sparse matrices A, it is better to reformulate the secular equation as follows. Squaring both sides of (3.3) we obtain After some manipulation, we are led to \Theta where we have defined Therefore, finding ff reduces to finding the positive-root of \Theta In this form, one can consider techniques similar to those suggested in [5] to find ff efficiently. A NEW METHOD FOR PARAMETER ESTIMATION WITH UNCERTAIN 4. Restricted Perturbations. We have so far considered the case in which all the columns of the A matrix are subject to perturbations. It may happen in practice, however, that only selected columns are uncertain, while the remaining columns are known precisely. This situation can be handled by the approach of this paper as we now clarify. Given A 2 R m\Thetan , we partition it into block columns, \Theta and assume, without loss of generality, that only the columns of A 2 are subject to perturbations while the columns of A 1 are known exactly. We then pose the following Given A 2 R m\Thetan , with m n and A full rank, b 2 R m , and nonnegative real numbers (j x \Theta If we partition " x accordingly with A 1 and A 2 , say then we can write \Theta Therefore, following the argument at the beginning of Sec. 3, we conclude that the maximum over (ffiA 2 ; ffib) is achievable and is equal to In this way, statement (4.1) reduces to the minimization problem min \Theta This statement can be further reduced to the problem treated in Theorem 3.6 as follows. Introduce the QR decomposition of A, say R 11 R 12 where we have partitioned R accordingly with the sizes of A 1 and A 2 . Define4 Then (4.2) is equivalent to min R 11 R 12 \Gamma4 which can be further rewritten as min This shows that once the optimal " x 2 has been determined, the optimal choice for " is necessarily the one that annihilates the entry R That is, The optimal " x 2 is the solution of R 22 This optimization is of the same form as the problem stated earlier in Lemma 3.1 with " x replaced by " replaced by j 2 , A replaced by R 22 , and b replaced by Therefore, the optimal " x 2 can be obtained by applying the result of Theorem 3.6. Once " x 2 has been determined, the corresponding " x 1 follows from (4.5). 5. Conclusion. In this paper we have proposed a new formulation for parameter estimation in the presence of data uncertainties. The problem incorporates a-priori bounds on the size of the perturbations and admits a nice geometric interpretation. It also has a closed form solution that is obtained by solving a regularized least-squares problem with a regression parameter that is the unique positive root of a secular equation. Several other interesting issues remain to be addressed. Among these, we state the following: 1. A study of the statistical properties of the min-max solution is valuable for a better understanding of its performance in stochastic settings. 2. The numerical properties of the algorithm proposed in this paper need also be addressed. 3. Extensions of the algorithm to deal with perturbations in submatrices of A are of interest and will be studied elsewhere. We can also extend the approach of this paper to other variations that include uncertainties in a weighting matrix, multiplicatives uncertainties, etc (see, e.g., [15]). --R Robust solutions to least-squares problems with uncertain data Some modified matrix eigenvalue problems An analysis of the total least squares problem Generalized cross-validation for large scale problems Linear estimation in Krein spaces - Part I: Theory The Total Least Squares Problem: Computational Aspects and Analysis Fundamentals of Filtering and smoothing in an H 1 Society for Industrial and Applied Mathematics Fundamental Inertia Conditions for the Minimization of Quadratic Forms in Indefinite Metric Spaces "Parameter estimation in the presence of bounded modeling errors," --TR --CTR Arvind Nayak , Emanuele Trucco , Neil A. Thacker, When are Simple LS Estimators Enough? An Empirical Study of LS, TLS, and GTLS, International Journal of Computer Vision, v.68 n.2, p.203-216, June 2006 Mohit Kumar , Regina Stoll , Norbert Stoll, Robust Solution to Fuzzy Identification Problem with Uncertain Data by Regularization, Fuzzy Optimization and Decision Making, v.3 n.1, p.63-82, March 2004 Pannagadatta K. Shivaswamy , Chiranjib Bhattacharyya , Alexander J. Smola, Second Order Cone Programming Approaches for Handling Missing and Uncertain Data, The Journal of Machine Learning Research, 7, p.1283-1314, 12/1/2006 Ivan Markovsky , Sabine Van Huffel, Overview of total least-squares methods, Signal Processing, v.87 n.10, p.2283-2302, October, 2007

least-squares estimation;total least-squares;regularized least-squares;ridge regression;secular equation;modeling errors;robust estimation

287637

Computing rank-revealing QR factorizations of dense matrices.

We develop algorithms and implementations for computing rank-revealing QR (RRQR) factorizations of dense matrices. First, we develop an efficient block algorithm for approximating an RRQR factorization, employing a windowed version of the commonly used Golub pivoting strategy, aided by incremental condition estimation. Second, we develop efficiently implementable variants of guaranteed reliable RRQR algorithms for triangular matrices originally suggested by Chandrasekaran and Ipsen and by Pan and Tang. We suggest algorithmic improvements with respect to condition estimation, termination criteria, and Givens updating. By combining the block algorithm with one of the triangular postprocessing steps, we arrive at an efficient and reliable algorithm for computing an RRQR factorization of a dense matrix. Experimental results on IBM RS/6000 SGI R8000 platforms show that this approach performs up to three times faster that the less reliable QR factorization with column pivoting as it is currently implemented in LAPACK, and comes within 15% of the performance of the LAPACK block algorithm for computing a QR factorization without any column exchanges. Thus, we expect this routine to be useful in may circumstances where numerical rank deficiency cannot be ruled out, but currently has been ignored because of the computational cost of dealing with it.

INTRODUCTION We briefly summarize the properties of a rank-revealing QR factorization (RRQR factorization). Let A be an m \Theta n matrix (w.l.o.g. m - n) with singular values and define the numerical rank r of A with respect to a threshold - as follows: oe r oe r+1 Also, let A have a QR factorization of the form R 11 R 12 where P is a permutation matrix, Q has orthonormal columns, R is upper triangular and R 11 is of order r. Further, let -(A) denote the two-norm condition number of a matrix A. We then say that (2) is an RRQR factorization of A if the following properties are satisfied: Whenever there is a well-determined gap in the singular value spectrum between oe r and oe r+1 , and hence the numerical rank r is well defined, the RRQR factorization (2) reveals the numerical rank of A by having a well-conditioned leading submatrix R 11 and a trailing submatrix R 22 of small norm. We also note that the matrix \GammaI which can be easily computed from (2), is usually a good approximation of the nullvectors, and a few steps of subspace iteration suffice to compute nullvectors that are correct to working precision [Chan and Hansen 1992]. The RRQR factorization is a valuable tool in numerical linear algebra because it provides accurate information about rank and numerical nullspace. Its main use arises in the solution of rank-deficient least-squares problems, for example, in geodesy [Golub et al. 1986], computer-aided design [Grandine 1989], nonlinear least-squares problems [Mor'e 1978], the solution of integral equations [Eld'en and Schreiber 1986], and the calculation of splines [Grandine 1987]. Other applications arise in beamforming [Bischof and Shroff 1992], spectral estimation [Hsieh et al. 1991], and regularization [Hansen 1990; Hansen et al. 1992; Wald'en 1991]. Stewart [1990] suggested another alternative to the singular value decomposition (SVD), a complete orthogonal decomposition called URV decomposition. This factorization decomposes R 11 R 12 where U and V are orthogonal and both kR 12 k 2 and kR 22 k 2 are of the order oe r+1 . In particular, compared with RRQR factorizations, URV decompositions employ a general orthogonal matrix V instead of the permutation matrix P . URV decompositions are more expensive to compute, but they are well suited for nullspace updat- ing. RRQR factorizations, on the other hand, are more suited for the least-squares setting, since one need not store the orthogonal matrix V (the other orthogonal matrix is usually applied to the right-hand side "on the fly"). Of course, RRQR factorizations can be used to compute an initial URV decomposition, where We briefly review the history of RRQR algorithms. From the interlacing theorem for singular values [Golub and Loan 1983, Corollary 8.3.3], we have oe Hence, to satisfy condition (3), we need to pursue two tasks: Task 1. Find a permutation P that maximizes oe min (R 11 ). Task 2. Find a permutation P that minimizes oe max (R 22 ). Businger and Golub [1965] suggested what is commonly called the "QR factorization with column pivoting." Given a set of already selected columns, this algorithm chooses as the next pivot column the one that is "farthest away" in the Euclidean norm from the subspace spanned by the columns already chosen [Golub and Loan 1983, p.168, P.6.4-5]. This intuitive strategy addresses task 1. While this greedy algorithm is known to fail on the so-called Kahan matrices [Golub and Loan 1989, p. 245, Example 5.5.1], it works well in practice and forms the basis of the LINPACK [Dongarra et al. 1979] and LAPACK [Anderson et al. 1992a; Anderson et al. 1994b] implementations. Recently, Quintana-Ort'i, Sun, and Bischof [1995] developed an implementation of the Businger/Golub algorithm that allows half of the work to be performed with BLAS-3 kernels. Bischof also had developed restricted-pivoting variants of the Businger/Golub strategy to enable the use of BLAS-3 type kernels [1989] for almost all of the work and to reduce communication cost on distributed-memory machines [1991]. One approach to task-2 is based, in essence, on the following fact, which is proved in [Chan and Hansen 1992]. Lemma 1. For any R 2 IR n\Thetan and any W = n\Thetap with a nonsingular This means that if we can determine a matrix W with p linearly independent columns, all of which lie approximately in the nullspace of R (i.e., kRWk 2 is small), and if W 2 is well conditioned such that (oe min (W is not large, then we are guaranteed that the elements of the bottom right p \Theta p block of R will be small. Algorithms based on computing well-conditioned nullspace bases for A include these by Golub, Klema, and Stewart [1976], Chan [1987], and Foster [1986]. Other algorithms addressing task-2 are these by Stewart [1984] and Gragg and Stewart [1976]. Algorithms addressing task 1 include those of Chan and Hansen [1994] and Golub, Klema, and Stewart [1976]. In fact, the latter achieves both task 1 and task 2 and, therefore, reveals the rank, but it is too expensive in comparison with the others. Bischof and Hansen combined a restricted-pivoting strategy with Chan's algorithm [Chan 1987] to arrive at an algorithm for sparse matrices [Bischof and Hansen 1991] and also developed a block-variant of Chan's algorithm [Bischof and Hansen 1992]. A Fortran 77 implementation of Chan's algorithm was provided by Reichel and Gragg [1990]. Chan's algorithm [Chan 1987] guaranteed and That is, as long as the rank of the matrix is close to n, the algorithm is guaranteed to produce reliable bounds, but reliability may decrease with the rank of the matrix. Hong and Pan [1992] then showed that there exists a permutation matrix P such that for the triangular factor R partitioned as in (2) we have and oe min (R 11 are low-order polynomials in n and r (versus an exponential factor in Chan's algorithm). Chandrasekaran and Ipsen [1994] were the first to develop RRQR algorithms that satisfy (8) and (9). Their paper also reviews and provides a common framework for the previously devised strategies. In particular, they introduce the so-called unification principle, which says that running a task-1 algorithm on the rows of the inverse of the matrix yields a task-2 algorithm. They suggest hybrid algorithms that alternate between task-1 and task-2 steps to refine the separation of the singular values of R. Pan and Tang [1992] and Gu and Eisenstat [1992] presented different classes of algorithms for achieving (8) and (9), addressing the possibility of nontermination of the algorithms because of floating-point inaccuracies. The goal of our work was to develop an efficient and reliable RRQR algorithm and implementation suitable for inclusion in a numerical library such as LAPACK. Specifically, we wished to develop an implementation that was both reliable and close in performance to the QR factorization without any pivoting. Such an implementation would provide algorithm developers with an efficient tool for addressing potential numerical rank deficiency by minimizing the computational penalty for addressing potential rank deficiency. Our strategy involves the following ingredients: -An efficient block-algorithm for computing an approximate RRQR factorization based on the work by Bischof [1989], and -efficient implementations of RRQR algorithms well suited for triangular matrices based on the work by Chandrasekaran and Ipsen [1994] and Pan and Tang [1992]. These algorithms seemed better suited for triangular matrices than those suggested by Gu and Eisenstat [1992]. We expect that 1. 2. foreach i ng do res do 3. for to min(m; n) do 4. Let i - pvt - n be such that respvt is maximal 5. 7. 8. foreach ng do 9. res j := res 2 10. end foreach 11. end for Fig. 1. The QR Factorization Algorithm with Traditional Column Pivoting -in most cases the approximate RRQR factorization computed by the block algorithm is very close to the desired RRQR factorization, requiring little postpro- cessing, and -the almost entirely BLAS-3 based preprocessing algorithm performs considerably faster than the QR factorization with column pivoting and close to the performance of the QR factorization without pivoting. The paper is structured as follows. In the next section, we review the block algorithm for computing an approximate RRQR factorization based on a restricted- pivoting approach. In Section 3, we describe our modifications to Chandrasekaran and Ipsen's ``Hybrid-III'' algorithm and Pan and Tang's "Algorithm 3." Section 4 presents our experimental results on IBM RS/6000, and SGI R8000 platforms. In Section 5, we summarize our results. 2. A BLOCK QR FACTORIZATION WITH RESTRICTED PIVOTING In this section, we describe a block QR factorization algorithm which employs a restricted pivoting strategy to approximately compute a RRQR factorization, employing the ideas described in Bischof [1989]. We compute Q by a sequence of Householder matrices For any given vector x, we can choose a vector u so that the first canonical unit vector and j ff (see, for example,[Golub and Loan 1989, p. 196]). The application of a Householder matrix B := H(u)A involves a matrix-vector product z := A T u and a rank-one update B := A \Gamma 2uz T . Figure 1 describes the Businger/Golub Householder QR factorization algorithm with traditional column pivoting [Businger and Golub 1965] for computing the QR decomposition of an m \Theta n matrix A. The primitive operation [u; y] := genhh(x) computes u such that y = H(u)x is a multiple of e 1 , while the primitive operation After step i is completed, the values res are the length of the projections of the j th column of the currently permuted AP onto the orthogonal complement of the subspace spanned by the first i columns of AP . The values res j can be updated easily and do not have to be recomputed at every step, although e e e e e e @ @ Fig. 2. Restricting Pivoting for a Block Algorithm roundoff errors may make it necessary to recompute res periodically [Dongarra et al. 1979, p. 9.17] (we suppressed this detail in line 9 of Figure 1). The bulk of the computationalwork in this algorithm is performed in the apphh ker- nel, which relies on matrix-vector operations. However, on today's cache-based architectures (ranging from workstations to supercomputers) matrix-matrix operations perform much better. Matrix-matrix operations are exploited by using so-called block algorithms, whose top-level unit of computation is matrix blocks instead of vectors. Such algorithms play a central role, for example, in the LAPACK implementations [Anderson et al. 1992a; Anderson et al. 1994b]. LAPACK employs the so-called compact WY representation of products of Householder matrices [Schreiber and Van Loan 1989], which expresses the product of a series of m \Theta m Householder matrices (10) as where Y is an m \Theta nb matrix and T is an nb \Theta nb upper triangular matrix. Stable implementations for generating Householder vectors as well as forming and applying compact WY factors are provided in LAPACK. To arrive at a block QR factorization algorithm, we would like to avoid updating part of A until several Householder transformations have been computed. This strategy is impossible with traditional pivoting, since we must update res j before we can choose the next pivot column. While we can modify the traditional approach to do half of the work using block transformations, this is the best we can do (these issues are discussed in detail in [Quintana-Ort'i et al. 1995]). Therefore, we instead limit the scope of pivoting as suggested in [Bischof 1989], Thus, we do not have to update the remaining columns until we have computed enough Householder transformations to make a block update worthwhile. The idea is graphically depicted in Figure 2. At a given stage we are done with the columns to the left of the pivot window. We then try to select the next pivot columns exclusively from the columns in the pivot window, not touching the part of A to the right of the pivot window. Only when we have combined the Householder vectors defined by the next batch of pivot columns into a compact WY factor, do we apply this block update to the columns on the right. Since the leading block of R is supposed to approximate the large singular values of A, we must be able to guard against pivot columns that are close to the span of columns already selected. That is, given the upper triangular matrix R i defined by the first i columns of Q T AP and a new column determined by the new candidate pivot column, we must determine whether has a condition number that is larger than a threshold - , which defines what we consider a rank-deficient matrix. We approximate oe (R which is easy to compute. To cheaply estimate oe min (R i+1 ), we employ incremental condition estimation (ICE) [Bischof 1990; Bischof and Tang 1991]. Given a good estimate b oe min (R i defined by a large norm solution x to R T 1 and a new column , incremental condition estimation, with only 3k flops, computes s and c, s oe min (R oe min (R sx c A stable implementation of ICE based on the formulation in [Bischof and Tang 1991] is provided by the LAPACK routine xLAIC1. 1 ICE is an order of magnitude cheaper than other condition estimators (see, for example, [Higham 1986]). More- over, it is considerably more reliable than simply using j fl j as an estimate for oe min (R i+1 ) (see, for example, [Bischof 1991]). We also define b oe min (R The restricted block pivoting algorithm proceeds in four phases: Phase 1: Pivoting of largest column into first position. This step is motivated by the fact that the norm of the largest column of A is usually a good estimate for Phase 2: Block QR factorization with restricted pivoting. Given a desired block size nb and a window size ws, ws - nb, we try to generate nb Householder transformations by applying the Businger/Golub pivoting strategy only to the columns in the pivot window, using ICE to assess the impact of a column selection on the condition number via ICE. When the pivot column chosen from the pivot window would lead to a leading triangular factor whose condition number exceeds - , we mark all remaining columns in the pivot window (k, say) as "rejected," pivot them to the end of the matrix, generate a block transformation (of width not more than nb), apply it to the remainder of the matrix, and then reposition the pivot window 1 Here as in the sequel we use the conventionthat the prefix "x" generically refers to the appropriate one of the four different precision instantiations: SLAIC1, DLAIC1, CLAIC1, or ZLAIC1. a to encompass the next ws not yet rejected columns. When all columns have been either accepted as part of the leading triangular factor or rejected at some stage of the algorithm, this phase stops. Assuming we have included r 2 columns in the leading triangular factor, we have at this point computed an r 2 \Theta r 2 upper triangular matrix R that satisfies That is, r 2 is our estimate of the numerical rank with respect to the threshold - at this point. In our experiments, we chose This choice tries to ensure a suitable pivot window and "loosens up" a bit as matrix size increases. A pivot window that is too large will reduce performance because of the overhead in generating block orthogonal transformations and the larger number of unblocked operations. On the other hand, a pivot window that is too small will reduce the pivoting flexibility and thus increase the likelihood that the restricted pivoting strategy will fail to produce a good approximate RRQR factorization. In our experiments, the choice of w had only a small impact (not more than 5%) on overall performance and negligible impact on the numerical behavior. Phase 3: Traditional pivoting strategy among "rejected" columns. Since phase 2 rejects all remaining columns in the pivot window when the pivot candidate is rejected, a column may have been pivoted to the end that should not have been rejected. Hence, we now continue with the traditional Businger/Golub pivoting strategy on the remaining updating (14) as an estimate of the condition number. This phase ends at column r 3 , say, where and the inclusion of the next pivot column would have pushed the condition number beyond the threshold. We do not expect many columns (if any) to be selected in this phase. It is mainly intended as a cheap safeguard against possible failure of the initial restricted-pivoting strategy. Phase 4: Block QR factorization without pivoting on remaining columns. The columns not yet factored (columns r 3 are with great probability linearly dependent on the previous ones, since they have been rejected in both phase 2 and phase 3. Hence, it is unlikely that any kind of column exchanges among the remaining columns would change our rank estimate, and the standard BLAS-3 block QR factorization as implemented in the LAPACK routine xGEQRF is the fastest way to complete the triangularization. After the completion of phase 4, we have computed a QR factorization Kthat satisfies and for any column y in R(:; r 3 n) we have R r3' This result suggests that this QR factorization is a good approximation to a RRQR factorization and r 3 is a good estimate of the rank. However, this QR factorization does not guarantee to reveal the numerical rank correctly. Thus, we back up this algorithm with the guaranteed reliable RRQR implementations introduced in the next two sections. 3. POSTPROCESSING ALGORITHMS FOR AN APPROXIMATE RRQR FACTOR- IZATION In 1991, Chandrasekaran and Ipsen [1994] introduced a unified framework for RRQR algorithms and developed an algorithm guaranteed to satisfy (8) and and thus to properly reveal the rank. Their algorithm assumes that the initial matrix is triangular and thus is well suited as a postprocessing step to the algorithm presented in the prexeding section. Shortly thereafter, Pan and Tang [1992] introduced another guaranteed reliable RRQR algorithm for triangular matrices. In the following subsections, we describe our improvements and implementations of these algorithms. 3.1 The RRQR Algorithm by Pan and Tang We implement a variant of what Pan and Tang [1992] call "Algorithm 3." Pseudocode for our algorithm is shown in Figure 3. It assumes as input an upper triangular matrix R. \Pi R (i; denotes a right cyclic permutation that exchanges columns i and j, e.g., denotes a left cyclic permutation that exchanges columns i and j, i.e., j / j. In the algorithm, triu(A) denotes the upper triangular factor R in a QR factorization A = QR of A. As can be seen from Figure 3, we use this notation as shorthand for retriangularizations of R after column exchanges. Given a value for k, and a so-called f-factor 1, the algorithm is guaranteed to halt and produce a triangular factorization that satisfies oe min (R 11 oe (R 22 f Our implementation incorporates the following features: (1) Incremental condition estimation (ICE) is used to arrive at estimates for smallest singular values and vectors. Thus, oe (line 5) and v (line of Figure 3 can be computed inexpensively from u (line 2). The use of ICE significantly reduces implementation cost. (2) The QR factorization update (line 4) must be performed only when the if-test (line Thus, we delay it if possible. (3) For the algorithm to terminate, all columns need to be checked, and no new permutations must occur. In Pan and Tang's algorithm, rechecking of columns Algorithm 1. 2. u := left singular vector corresponding to oe min (R(1: k; 1: k)) 3. while ( accepted col - 4. R := triu(R \Delta \Pi R 5. Compute 7. accepted col := accepted col 8. else 9. v := right singular vector corresponding to oe 10. Find index q, 1 - q 11. R := triu(R \Delta \Pi L 12. u := left singular vector corresponding to oe min (R(1:k; 1: k)) 13. end if 14. if (i == n) then i 15. end while Fig. 3. Variant of Pan/Tang RRQR Algorithm after a permutation always starts at column k + 1. We instead begin checking at the column right after the one that just caused a permutation. Thus, we first concentrate on the columns that have not just been "worked over." (4) The left cyclic shift permutes the triangular matrix into an upper Hessenberg form, which is then retriangularized with Givens rotations. Applying Givens rotations to rows of R in the obvious fashion (as done, for example, in [Re- ichel and Gragg 1990]) is expensive in terms of data movement, because of the column-oriented nature of Fortran data layout. Thus, we apply Givens rotations in an aggregated fashion, updating matrix strips (R(1 : jb; (j \Gamma1)b+1 : jb)) of width b with all previously computed Givens rotations. Similarly, the right cyclic shift introduces a "spike" in column j, which is eliminated with Givens rotations in a bottom-up fashion. To aggregate Givens rotations, we first compute all rotations only touching the "spike" and the diagonal of R, and then apply all of them one block column at a time. In our experiments, we choose the width b of the matrix strips to be the same as the blocksize nb of the preprocessing. Compared with a straightforward implementation of Pan and Tang's "Algorithm 3," improvements (1) through (3) on average decreased runtime by a factor of five on 200 \Theta 200 matrices on an Alliant FX/80. When retriangularizations were frequent, improvement (4) had the most noticeable impact, resulting in a twofold to fourfold performance gain on matrices of order 500 and 1000 on an IBM RS/6000-370. Pan and Tang introduced the f-factor to prevent cycling of the algorithm. The higher f is, the tighter the bounds in (18) and (19), and the better the approximations to the k and k 1st singular values of R. However, if f is too large, it introduces more column exchanges and therefore more iterations, and, because of round-off errors, it might present convergence problems. We used in our work. Algorithm 1. 2. repeat 3. Golub-I-sf(f,k) 4. Golub-I-sf(f,k+1) 5. Chan-II-sf(f,k+1) 6. 7. until none of the four subalgorithms modified the column ordering Fig. 4. Variant of Chandrasekaran/Ipsen Hybrid-III algorithm Algorithm Golub-I-sf(f,k) 1. Find smallest index j, k - j - n, such that 2. kR(k: j; j)k 3. 4. R := triu(R \Delta \Pi R 5. end if Fig. 5. "f-factor" Variant of Golub-I Algorithm 3.2 The RRQR Algorithm by Chandrasekaran and Ipsen Chandrasekaran and Ipsen introduced algorithms that achieve bounds (18) and (19) with We implemented a variant of the so-called Hybrid-III algorithm, pseudocode for which is shown in Figures 4 - 6. Compared with the original Hybrid-III algorithm, our implementation incorporates the following features: (1) We employ the Chan-II strategy (an O(n 2 ) algorithm) instead of the so-called Stewart-II strategy (an O(n 3 ) algorithm because of the need for the inversion of that Ipsen and Chandrasekaran employed in their experiments. (2) The original Hybrid-III algorithm contained two subloops, with the first one looping over Golub-I(k) and Chan-II(k) till convergence, the second one looping over Golub-I(k+1) and Chan-II(k+1). We present a different loop ordering in our variant, one that is simpler and seems to enhance convergence. On matrices that required considerable postprocessing, the new loop ordering required about 7% less steps for 1000 \Theta 1000 matrices (one step being a call to Golub-I or Chan- II) than Chandrasekaran and Ipsen's original algorithm. In addition, the new ordering speeds up detection of convergence, as shown below. Algorithm 1. v := right singular vector corresponding to oe min (R(1:k; 1: k)). 2. Find largest index 3. if f \Delta jv 4. R := triu(R \Delta \Pi L 5. end if Fig. 6. "f-factor" Variant of Chan-II Algorithm (3) As in our implementation of the Pan/Tang algorithm, we use ICE for estimating singular values and vectors, and the efficient "aggregated" Givens scheme for the retriangularizations. We employ a generalization of the f-factor technique to guarantee termination in the presence of rounding errors. The pivoting method assigns to every column a "weight," namely, kR(k: in Golub-I(k) and v i in Chan-II(k), where v is the right singular vector corresponding to the smallest singular value of To ensure termination, Chandrasekaran and Ipsen suggested pivoting a column only when its weight exceeded that of the current column by at least n 2 ffl, where ffl is the computer precision; they did not analyze the impact of this change on the bounds obtained by the algorithm. In contrast, we use a multiplicative tolerance factor f like Pan and Tang; the analysis in [Quintana-Ort'i and Quintana-Ort'i 1996] proves that our algorithm achieves the bounds oe min (R 11 oe k (A); and (20) oe (R 22 These bounds are identical to (18) and (19), except that an f 2 instead of an f enters into the equation and that now 0 ! f - 1. We used our implementation. We claimed before that the new loop ordering can avoid unnecessary steps when the algorithm is about to terminate. To illustrate, consider the situation where we apply Chandrasekaran and Ipsen's original ordering to a matrix that almost reveals the rank: 1. Golub-I(k) Final permutation occurs here. Now the rank is revealed. 2. Chan-II(k) 3. Golub-I(k) Another iteration of inner k-loop since permutation occurred. 4. Chan-II(k) 5. Golub-I(k+1) Inner loop for 7. Golub-I(k) Another iteration of the main loop since permutation occurred in last pass. 8. 9. Golub-I(k+1) 10. Chan-II(k+1) Termination In contrast, the Hybrid-III-sf algorithm terminates in four steps: 1. Golub-I-sf(k) Final permutation 2. Golub-I-sf(k+1) 3. Chan-II-sf(k+1) 4. Chan-II-sf(k) Termination Algorithm RRQR(f,k) repeat call Hybrid-III-sf(f,k) or PT3M(f,k) ff if rank := k; stop else if ( ( ff - ) and (fi - ) )then else if ( ( ff - ) and ( fi - ) )then Fig. 7. Algorithm for Computing Rank-Revealing QR Factorization 3.3 Determining the Numerical Rank As Stewart [1993] pointed out, both the Chandrasekaran/Ipsen and Pan/Tang al- gorithms, as well as our versions of those algorithms, do not reveal the rank of a matrix per se. Rather, given an integer k, they compute tight estimates for To obtain the numerical rank with respect to a given threshold - , given an initial estimate for the rank (as provided, for example, by the algorithm described in Section 2), we employ the algorithm shown in Figure 7. In our actual implementation, ff and fi are computed in Hybrid-III-sf or PT3M. 4. EXPERIMENTAL RESULTS We report in this section experimental results with the double-precision implementations of the algorithms presented in the preceding section. We consider the following codes: DGEQPF. The implementation of the QR factorization with column pivoting currently provided in LAPACK. DGEQPB. An implementation of the "windowed" QR factorization scheme described in Section 2. DGEQPX. DGEQPB followed by an implementation of the variant of the Chan- drasekaran/Ipsen algorithm described in subsections 3.2 and 3.3. DGEQPY. DGEQPB followed by an implementation of the variant of the Pan/Tang algorithm described in subsections 3.1 and 3.3. DGEQRF. The block QR factorization without any pivoting provided in LAPACK In the implementation of our algorithms, we make heavy use of available LAPACK infrastructure. The code used in our experiments, including test and timing drivers and test matrix generators, is available as rrqr.tar.gz in pub/prism on ftp.super.org. We tested matrices of size 100; 150; 250; 500, and 1000 on an IBM RS/6000 Model 370 and SGI R8000. In each case, we employed the vendor-supplied BLAS in the ESSL and SGIMATH libraries, respectively. 4.1 Numerical Reliability We employed different matrix types to test the algorithms, with various singular value distributions and numerical rank ranging from 3 to full rank. Details of the test matrix generation are beyond the scope of this paper, and we give only a brief synopsis here. For details, the reader is referred to the code. Test were designed to exercise column pivoting. Matrix 6 was designed to test the behavior of the condition estimation in the presence of clusters for the smallest singular value. For the other cases, we employed the LAPACK matrix generator xLATMS, which generates random symmetric matrices by multiplying a diagonal matrix with prescribed singular values by random orthogonal matrices from the left and right. For the break1 distribution, all singular values are 1.0 except for one. In the arithmetic and geometric distributions, they decay from 1.0 to a specified smallest singular value in an arithmetic and geometric fashion, respectively. In the "reversed" distributions, the order of the diagonal entries was reversed. For test cases 7 though 12, we used xLATMS to generate a matrix of smallest singular value 5.0e-4, and then interspersed random linear combinations of these "full-rank" columns to pad the matrix to order n. For test cases 13 through 18, we used xLATMS to generate matrices of order n with the smallest singular value being 2.0e-7. We believe this set to be representative of matrices that can be encountered in practice. We report in this section on results for matrices of size noting that identical qualitative behavior was observed for smaller matrix sizes. We decided to report on the largest matrix sizes because the possibility for failure in general increases with the number of numerical steps involved. Numerical results obtained on the three platforms agreed to machine precision. For this case, we list in Table 1 the numerical rank r with respect to a condition threshold of 1:0e5, the largest singular value oe max , the r-th singular value oe r , the (r 1)st singular value oe r+1 , and the smallest singular value oe min for our test cases. Figures 8 and 9 display the ratio \Theta := (oe 1 =oe r ) where b-(R) as defined in (14) is the computed estimate of the condition number of R after DGEQPB (Figure 8) and DGEQPX and DGEQPY (Figure 9). Thus, \Theta is the ratio between the ideal condition number and the estimate of the condition number of the leading triangular factor identified in the RRQR factorization. If this ratio is close to 1, and b- is a good condition estimate, our RRQR factorizations do a good job of capturing the "large" singular values of A. Since the pivoting strategy and hence the numerical behavior of DGEQPB is potentially affected by the block size chosen, Figures 8 and 9 contain seven panels, each of which shows the results obtained with the test matrices and a block size ranging from 1 to 24 (shown in the top of each panel). We see that except for matrix type 1 in Figure 8, the block size does not play much of a rule numerically, although close inspection reveals subtle variations in both Figure 8 and 9. With block size 1, DGEQPB just becomes the standard Businger/Golub pivoting strategy. Thus, the first panel in Figure 8 corroborates the experimentally robust behavior of this algorithm. We also see that except for Table 1. Test Matrix Types Description r oe max oe r oe r+1 oe min Matrix with rank min(m;n) has full rank 3 Full rank 1000 1.0e0 5.0e-4 5.0e-4 5.0e-4 small in norm n) of full rank small in norm smallest sing. values clustered 1000 1.0e0 7.0e-4 7.0e-4-3 7.0e-4 7 Break1 distribution 501 1.0e0 5.0e-4 1.7e-15 1.0e-26 Reversed break1 distribution 501 1.0e0 5.0e-4 1.7e-15 1.2e-27 9 Geometric distribution 501 1.0e0 5.0e-4 3.3e-16 1.9e-35 Reversed geometric distribution 501 1.0e0 5.0e-4 3.2e-16 5.4e-35 11 Arithmetic distribution 501 1.0e0 5.0e-4 9.7e-16 1.4e-34 Reversed arithmetic distribution 501 1.0e0 5.0e-4 9.7e-16 1.2e-34 13 Break1 distribution 999 1.0e0 1.0e0 2.0e-7 2.0e-7 14 Reversed break1 distribution 999 1.0e0 1.0e0 2.0e-7 2.0e-7 Geometric distribution 746 1.0e0 5.0e-5 9.9e-6 2.0e-7 Reversed geometric distribution 746 1.0e0 5.0e-5 9.9e-6 2.0e-7 17 Arithmetic distribution 999 1.0e0 1.0e-1 2.0e-7 2.0e-7 Reversed arithmetic distribution 999 1.0e0 1.0e-1 2.0e-7 2.0e-7 Tests Optimal cond_no. Estimated cond_no. Fig. 8. Ratio between Optimal and Estimated Condition Number for s Optimal cond_no. Estimated cond_no. . QPY Fig. 9. Ratio between Optimal and Estimated Condition Number for DGEQPX (solid line) and DGEQPY (dashed) type 1, the restricted pivoting strategy employed in DGEQPB does not have much impact on numerical behavior. For matrix type 1, however, it performs much worse. Matrix 1 is constructed by generating n\Gamma 1 independent columns and generating the leading n+1 as random linear combinations of those columns, scaled by ffl 1 4 , where ffl is the machine precision. Thus, the restricted pivoting strategy, in its myopic view of the matrix, gets stuck, so to speak, in these columns. The postprocessing of these approximate RRQR factorizations, on the other hand, remedies potential shortcomings in the preprocessing step. As can be seen from Figure 9, the inaccurate factorization of matrix 1 is corrected, while the other, in essence correct, factorizations get improved only slightly. Except for small vari- ations, DGEQPX and DGEQPY deliver identical results. We also computed the exact condition number of the leading triangular submatrices identified in the triangularizations by DGEQPB, DGEQPX, and DGEQPY, and compared it with our condition estimate. Figure 10 shows the ratio of the exact condition number to the estimated condition number of the leading triangular factor. We observe excellent agreement, within an order of magnitude in all cases. Hence, the "spikes" for test matrices 13 and 14 in Figures 8 and 9 are not due to errors in our estimators. Rather, they show that all algorithms have difficulties when confronted with dense clusters of singular values. We also note that in this context, the notion of rank is numerically illdefined, since there is no sensible place to draw the line. The "rank" derived via the SVD is 746 in both cases, and our algorithms deliver estimates between 680 and 710, with minimal changes in the condition number of their corresponding leading triangular factors. In summary, these results show that DGEQPX and DGEQPY are reliable algorithms for revealing numerical rank. They produce RRQR factorizations whose s Exact Estimated -. QPX . QPY Fig. 10. Ratio between Exact and Estimated Condition Number of Leading Triangular Factor for DGEQPB (dashed), DGEQPX (dashed-dotted) and DGEQPY (dotted) leading triangular factors accurately capture the desired part of the spectrum of A, and thus reliable and numerically sensible rank estimates. Thus, the RRQR factorization takes advantage of the efficiency and simplicity of the QR factorization, yet it produces information that is almost as reliable as that computed by means of the more expensive singular value decomposition. 4.2 Computing Performance In this section we report on the performance of the LAPACK codes DGEQPF and DGEQRF as well as the new DGEQPB, DGEQPX, and DGEQPY codes. For these codes, as well as all others presented in this section, the Mflop rate was obtained by dividing the number of operations required for the unblocked version of DGEQRF by the runtime. This normalized Mflop rate readily allows for timing comparisons. We report on matrix sizes 100, 250, 500, and 1000, using block sizes (nb) of 1, 5, Figures show the Mflop performance (averaged over the versus block size on the IBM and SGI platforms. The dotted line denotes the performance of DGEQPF, the solid one that of DGEQRF and the dashed one that of DGEQPB; the 'x' and '+' symbols indicate DGEQPX and DGEQPY, respectively. On all three machines, the performance of the two new algorithms for computing RRQR is robust with respect to variations in the block size. The two new block algorithms for computing RRQR factorization are, except for small matrices on the SGI, faster than LAPACK's DGEQPF for all matrix sizes. We note that the SGI has a data cache of 4 MB, while the IBM platform has only a data cache. Thus, matrices up to order 500 fit into the SGI cache, but matrices of order 1000 do not. Therefore, for matrices of size 500 or less we observe limited benefits from the Block size Performance (in Performance (in Block size Performance (in Performance (in Fig. 11. Performance versus Block Size on IBM RS/6000-370: DGEQPF (\Delta \Delta \Delta), DGEQRF (-), Block size Performance (in Block size Performance (in Performance (in Block size Performance (in Fig. 12. Performance versus Block Size on SGI R8000: DGEQPF (\Delta \Delta \Delta), DGEQRF (-), DGE- Performance (in Fig. 13. Performance versus Matrix Type on an IBM RS/6000-370 for better inherent data locality of the BLAS 3 implementation in this computer. These results also show that DGEQPX and DGEQPY exhibit comparable performance. Figures 13 through 14 offer a closer look at the performance of the various test matrices. We chose nb = 16 and as a representative example. Similar behavior was observed in the other cases. We see that on the IBM platforms (Figure 13), the performance of DGEQRF and DGEQPF does not depend on the matrix type. We also see that, except for matrix types 1, 5, 15, and 16, the postprocessing of the initial approximate RRQR factorization takes up very little time, with DGEQPX and DGEQPY performing similarly. For matrix type 1, considerable work is required to improve the initial QR factorization. For matrix types 5 and 15, the performance of DGEQPX and DGEQPY differ noticeably on the IBM platform, but there is no clear winner. We also note that matrix type 5 is suitable for DGEQPB, since the independent columns are up front and thus are revealed quickly, and the rest of the matrix is factored with DGEQRF. The SGI platform (Figure 14) offers a different picture. The performance of all algorithms shows more dependence on the matrix type, and DGEQPB performs worse on matrix type 5 than on all others. Nonetheless, except for matrix 1, DGEQPX and DGEQPY do not require much postprocessing effort. The pictures for other matrix sizes are similar. The cost for DGEQPX and DGEQPY decreases as the matrix size increases, except for matrix type 1, where it increases as expected. We also note that Figures 11 though 12 would have looked even more favorable for our algorithm had we omitted matrix 1 or chosen the median (instead of the average) performance. Figure 15 shows the percentage of the actual amount of flops spent in monitoring e Performance (in Fig. 14. Performance versus Matrix Type on an SGI R8000 for the rank in DGEQPB and in postprocessing the initial QR factorization for different matrix sizes on the IBM RS/6000. We show only matrix types 2 through 18, since the behavior of matrix type 1 is rather different: in this special case, roughly 50% of the overall flops is expended in the postprocessing. Note that the actual performance penalty due to these operations is, while small, still considerably higher than the flop count suggests. This is not surprising given the relatively fine-grained nature of the condition estimation and postprocessing operations. Lastly, one may wonder whether the use of DGEQRF to compute the initial QR factorization would lead to better results, since DGEQRF is the fastest QR factorization algorithm. This is not the case, since DGEQRF does not provide any rank preordering, and thus performance gains from DGEQRF are annihilated in the postprocessing steps. For example, for matrices of order 250 on an IBM RS/6000-370, the average Mflop rate, excluding matrix 5, was 4.5, with a standard deviation of 1.4. The percentage of flops spent in postprocessing in these cases was on average 76.8 %, with a standard deviation of 6.7. For matrix 5, we are lucky, since the matrix is of low rank and all independent columns are at the front of the matrix. Thus, we spend only 3% in postprocessing, obtaining a performance of 49.1 Mflops overall. In all other cases, though, considerable effort is expended in the postprocessing phase, leading to overall disappointing performance. These results show that the preordering done by DGEQPB is essential for the efficiency of the overall algorithm. 5. CONCLUSIONS In this paper, we presented rank-revealing QR factorization (RRQR) algorithms that combine an initial QR factorization employing a restricted pivoting scheme in flops of pivoting in flops of pivoting Fig. 15. Cost of Pivoting (in % of flops) versus Matrix Types of Algorithms DGEQPX and DGEQPY on an IBM RS/6000-370 for Matrix Sizes 100 (+), 250 (x), 500 (*) and 1000 (o). with postprocessing steps based on variants of algorithms suggested by Chandrasekaran and Ipsen and Pan and Tang. The restricted-pivoting strategy results in an initial QR factorization that is almost entirely based on BLAS-3 kernels, yet still achieves at a good approximation of an RRQR factorization most of the time. To guarantee the reliability of the initial RRQR factorization and improve it if need be, we improved an algorithm suggested by Pan and Tang, relying heavily on incremental condition estimation and "blocked" Givens rotation updates for computational efficiency. As an alternative, we implemented a version of an algorithm by Chandrasekaran and Ipsen, which among other improvements uses the f-factor technique suggested by Pan and Tang to avoid cycling in the presence of roundoff errors. Numerical experiments on eighteen different matrix types with matrices ranging in size from 100 to 1000 on IBM RS/6000 and SGI R8000 platforms show that this approach produces reliable rank estimates while outperforming the (less reliable) QR factorization with column pivoting, the currently most common approach for computing an RRQR factorization of a dense matrix. ACKNOWLEDGMENTS We thank Xiaobai Sun, Peter Tang and Enrique S. Quintana-Ort'i for stimulating discussions on the subject. --R Incremental condition estimation. A parallel QR factorization algorithm with controlled local pivoting. SIAM Journal on Scientific and Statistical Computing A block algorithm for computing rank-revealing QR factorizations On updating signal subspaces. Robust incremental condition estimation. Preprint MCS-P225-0391 Linear least squares solution by Householder transformation. Rank revealing QR factorizations. Some applications of the rank revealing QR factor- ization On rank-revealing QR factorizations An application of systolic arrays to linear discrete ill-posed problems Rank and null space calculations using matrix decomposition without column interchanges. Rank degeneracy and least squares problems. Matrix Computations. Matrix Computations (2nd A comparison between some direct and iterative methods for certain large scale geodetic least-squares problem A stable variant of the secant method for solving nonlinear equations. An iterative method for computing multivariate C 1 piecewise polynomial interpolants. Rank deficient interpolation and optimal design: An example. Technical Report SCA-TR-113 A stable and efficient algorithm for the rank-one modification of the symmmetric eigenproblem Truncated SVD solutions to discrete ill-posed problems with ill- determined numerical rank Efficient algorithms for computing the condition number of a tridiagonal matrix. The rank revealing QR decomposition and SVD. Mathematics of Computation Comparisons of truncated QR and SVD methods for AR spectral estimations. The Levenberg-Marquardt algorithm: Implementationand theory Bounds on singular values revealed by QR factor- izaton Guaranteeing termination of Chandrasekaran Fortran subroutines for updating the QR factorization. ACM Transactions on Mathematical Software A storage efficient WY representation for products of Householder transformations. Rank degeneracy. An updating algorithm for subspace tracking. Determining rank in the presence of error. Using a fast signal processor to solve the inverse kinematic problem with special emphasis on the singularity problem. --TR Efficient algorithms for computing the condition number of a tridagonal matrix A comparison between some direct and iterative methods for certian large scale godetic least squares problems An aplicaiton of systolic arrays to linear discrete Ill posed problems A storage-efficient WY representation for products of householder transformations A block QR factorization algorithm using restricted pivoting Truncated singular value decomposition solutions to discrete ill-posed problems with ill-determined numerical rank Incremental condition estimation Algorithm 686: FORTRAN subroutines for updating the QR decomposition An updating algorithm for subspace tracking A parallel QR factorization algorithm with controlled local pivoting Structure-preserving and rank-revealing QR-factorizations LAPACK''s user''s guide Some applications of the rank revealing QR factorization Determining rank in the presence of error The modified truncated SVD method for regularization in general form On Rank-Revealing Factorisations Matrix computations (3rd ed.) A BLAS-3 Version of the QR Factorization with Column Pivoting Rank degeneracy and least squares problems Working Note 33: Robust Incremental Condition Estimation --CTR C. H. Bischof , G. Quintana-Ort, Algorithm 782: codes for rank-revealing QR factorizations of dense matrices, ACM Transactions on Mathematical Software (TOMS), v.24 n.2, p.254-257, June 1998 Enrique S. Quintana-Ort , Gregorio Quintana-Ort , Maribel Castillo , Vicente Hernndez, Efficient Algorithms for the Block Hessenberg Form, The Journal of Supercomputing, v.20 n.1, p.55-66, August 2001 Peter Benner , Maribel Castillo , Enrique S. Quintana-Ort , Vicente Hernndez, Parallel Partial Stabilizing Algorithms for Large Linear Control Systems, The Journal of Supercomputing, v.15 n.2, p.193-206, Feb.1.2000

rank-revealing orthogonal factorization;block algorithm;QR factorization;least-squares systems;numerical rank

287639

An object-oriented framework for block preconditioning.

General software for preconditioning the iterative solution of linear systems is greatly lagging behind the literature. This is partly because specific problems and specific matrix and preconditioner data structures in order to be solved efficiently, i.e., multiple implementations of a preconditioner with specialized data structures are required. This article presents a framework to support preconditioning with various, possibly user-defined, data structures for matrices that are partitioned into blocks. The main idea is to define data structures for the blocks, and an upper layer of software which uses these blocks transparently of their data structure. This transparency can be accomplished by using an object-oriented language. Thus, various preconditioners, such as block relaxations and block-incomplete factorizations, only need to be defined once and will work with any block type. In addition, it is possible to transparently interchange various approximate or exact techniques for inverting pivot blocks, or solving systems whose coefficient matrices are diagonal blocks. This leads to a rich variety of preconditioners that can be selected. Operations with the blocks are performed with optimized libraries or fundamental data types. Comparisons with an optimized Fortran 77 code on both workstations and Cray supercomputers show that this framework can approach the efficiency of Fortran 77, as long as suitable block sized and block types are chosen.

INTRODUCTION In the iterative solution of the linear system a preconditioner M is often used to transform the system into one which has better convergence properties, for example, in the left-preconditioned case, M \Gamma1 is referred to as the preconditioning operator for the matrix A and, in general, is a sequence of operations that somehow approximates the effect of A \Gamma1 on a vector. Unfortunately, general software for preconditioning is seriously lagging behind methods being published in the literature. Part of the reason is that many methods do not have general applicability: they are not robust on general problems, or they are specialized and need specific information (e.g., general direction of flow in a fluids simulation) that cannot be provided in a general setting. Another reason, one that we will deal with in this article, is that specific linear systems need specific matrix and preconditioner data structures in order to be solved efficiently; i.e., there need to be multiple implementations of a preconditioner with specialized data structures. For example, in some finite element applications, diagonal blocks have a particular but fixed sparse structure. A block SSOR preconditioner that needs to invert these diagonal blocks should use an algorithm suited to this structure. A block SSOR code that treats these diagonal blocks in a general way is not ideal for this problem. When we encounter linear systems from different applications, we need to determine suitable preconditioning strategies for their iterative solution. Rather than code preconditioners individually to take advantage of the structure in each appli- cation, it is better to have a framework for software reuse. Also, a wide range of preconditionings should be available so that we can choose a method that matches the difficulty of the problem and the computer resources available. This article presents a framework to support preconditioning with various, possibly user-defined, data structures for matrices that are partitioned into blocks. The main idea is to define data structures (called block types) for the blocks, and an upper layer of software which uses these blocks transparently of their data struc- ture. Thus various preconditioners, such as block relaxations and block incomplete only need to be defined once, and will work with any block type. These preconditioners are called global preconditioners for reasons that will soon become apparent. The code for these preconditioners is almost as readable as the code for their pointwise counterparts. New global preconditioners can be added in the same fashion. Global preconditioners need methods (called local preconditioners) to approximately or exactly invert pivot blocks, or solve systems whose coefficient matrices are diagonal blocks. For example, a block stored in a sparse format might be in- Object-Oriented Block Preconditioning \Delta 3 verted exactly, or an approximate inverse might be computed. Our design permits a variety of these inversion or solution techniques to be defined for each block type. The transparency of the block types and local preconditioners can be implemented through polymorphism in an object-oriented language. Our framework, called currently implements block incomplete factorization and block relaxation global preconditioners, a dense and a sparse block type, and a variety of local preconditioners for both block types. Users of BPKIT will either use the block types that are available, or add block types and local preconditioners that are appropriate for their applications. Users may also define new global preconditioners that take advantage of the existing block types and local preconditioners. Thus BPKIT is not intended to be complete library software; rather it is a framework under which software can be specialized from relatively generic components. It is appropriate to make some comments about why we use block preconditioning. Many linear systems from engineering applications arise from the discretization of coupled partial differential equations. The blocking in these systems may be imposed by ordering together the equations and unknowns at a single grid point, or those of a subdomain. In the first case, the blocks are usually dense; in the latter case, they are usually sparse. Experimental tests suggest it is very advantageous for preconditionings to exploit this block structure in a matrix [Chow and Saad 1997; Fan et al. 1996; Jones and Plassmann 1995; Kolotilina et al. 1991]. The relative robustness of block preconditioning comes partly from being able to solve accurately for the strong coupling within these blocks. From a computational point of view, these block matrix techniques can be more efficient on cached and hierarchical memory architectures because of better data locality. In the dense block case, block matrix data structures also require less storage. Block data structures are also amenable to graph-based reorderings and block scalings. When approximations are also used for the diagonal or pivot blocks (i.e., approximations with local preconditioners are used), these techniques are specifically called two-level preconditioners [Kolotilina and Yeremin 1986], and offer a middle-ground between accuracy and simpler computations. Beginning with [Underwood] in 1976 and then [Axelsson et al. 1984] and [Concus et al. 1985] more than a decade ago, these preconditioners have been motivated and analyzed in the case of block tridiagonal incomplete factorizations combined with several types of approximate inverses, and have recently reached a certain maturity. Most implementations of these methods, however, are not flexible: they are often coded for a particular block size and inversion technique, and further, they are almost always coded for dense blocks. The software framework presented here derives its flexibility from the use of an object-oriented language. We chose to use C++ [Stroustrup 1991] in real, 64-bit arithmetic. Other object-oriented languages are also appropriate. The framework is computationally efficient, since all operations involving blocks are performed with code that employs fundamental types, or with optimized Fortran 77 libraries such as the Level 3 BLAS [Dongarra et al. 1990], LAPACK [Demmel 1989], and the sparse BLAS toolkit [Carney et al. 1994]. By the same token, users implementing block types and local preconditioners may do so in practically any language, as long as the language can be linked with C++ by their compilers. BPKIT also has an interface for Fortran 77 users. Chow and M. A. Heroux BPKIT is available at http://www.cs.umn.edu/~chow/bpkit.html. Other C++ efforts in the numerical solution of linear equations include LAPACK++ [Dongarra et al. 1993] for dense systems, and Diffpack [Bruaset and Langtangen 1997], ISIS++ [Clay 1997], SparseLib++ and IML++ [Dongarra et al. 1994] for sparse systems. It is also possible to use an object-oriented style in other languages [Eijkhout 1996; Machiels and Deville 1997; Smith et al. 1995]. In Section 2, we discuss various issues that arise when designing interfaces for block preconditioning and for preconditioned iterative methods in general. We describe the specification of the block matrix, the global and local preconditioners, the interface with iterative methods, and the Fortran 77 interface. In Section 3, we describe the internal design of BPKIT, including the polymorphic operations on blocks that are needed by global preconditioners. In Section 4, we present the results of some numerical tests, including a comparison with an optimized Fortran 77 code. Section 5 contains concluding remarks. 2. INTERFACES FOR BLOCK PRECONDITIONING We have attempted to be general when defining interfaces (to allow for extensions of functionality), and we have attempted to accept precedents where we overlap with related software (particularly in the interface with iterative methods). For concreteness, we describe several methods which will be used in the numerical tests. Section 2 brings to light various issues in the software design of preconditioned iterative methods. 2.1 Block matrices A matrix that is partitioned into blocks is called a block matrix. Although with BPKIT any storage scheme may be used to store the blocks that are not zero, the locations of these blocks within the block matrix must still be defined. The block matrix class (data type) that is available in BPKIT, called BlockMat, contains a pointer to each block in the block matrix. The pointers for each row of blocks (block row) are stored contiguously, with additional pointers to the first pointer for each block row. This is the analogy to the compressed sparse row data structure [Saad 1990] for pointwise matrices; pointers point to blocks instead of scalar entries. The global preconditioners in BPKIT assume that the BlockMat class is being used. It is possible for users to design new block matrix classes and to code new global preconditioners for their problems, and still use the block types and local preconditioners in BPKIT. For the block matrix data structure described above, BPKIT provides conversion routines to that data structure from the Harwell-Boeing format [Duff et al. 1989]. There is one conversion routine for each block type (e.g., one routine will convert a Harwell-Boeing matrix into a block matrix whose blocks are dense). However, these routines are provided for illustration purposes only. In practice, a user's matrix that is already in block form (i.e., the nonzero entries in each block are stored contiguously) can usually be easily converted by the user directly into the BlockMat form. To be general, the conversion routines allow two levels of blocking. In many prob- lems, particularly linear systems arising from the discretization of coupled partial differential equations, the blockings may be imposed by ordering together the equa- Object-Oriented Block Preconditioning \Delta 5 tions and unknowns at a single grid point and those of a subdomain. The latter blocking produces coarse-grain blocks, and the smaller, nested blocks are called fine-grain blocks. Figure 1 shows a block matrix of dimension 24 with coarse blocks of dimension 6 and fine blocks of dimension 2. Fig. 1. Block matrix with coarse and fine blocks. The blocks in BPKIT are the coarse blocks. Information about the fine blocks should also be provided to the conversion routines because it may be desirable to store blocks such that the coarse blocks themselves have block structure. For ex- ample, the variable block row (VBR) [Saad 1990] storage scheme can store coarse blocks with dense fine blocks in reduced space. Optimized matrix-vector product and triangular solve kernels for the VBR and other block data structures are provided in the sparse BLAS toolkit [Carney et al. 1994; Remington and Pozo 1996]. local preconditioners or block operations, however, are defined for fine blocks (i.e., there are not two levels of local preconditioners). It is apparent that the use of very small coarse blocks will degrade computing performance due to the overhead of procedure calls. Larger blocks can give better computational efficiency and convergence rate in preconditioned iterative methods, and computations with large dense blocks can be vectorized. In this article, we will rarely have need to mention fine blocks; thus, when we refer to "blocks" with no distinction, we normally mean coarse blocks. To be concrete, we give an example of how a conversion routine is called when a block matrix is defined. The statement BlockMat B("HBfile", 6, DENSE); 6 \Delta E. Chow and M. A. Heroux defines B to be a square block matrix where the blocks have dimension 6, and the blocks are stored in a format indicated by DENSE (which is of a C++ enumerated type). The other block type that is implemented is CSR, which stores blocks in the compressed sparse row format. The matrix is read from the file HBfile, which must be encoded in the standard Harwell-Boeing format [Duff et al. 1989]. (The dimension of the matrix does not need to be specified in the declaration since it is stored within the file.) To specify a variable block partitioning (with blocks with different sizes), other interfaces are available which use vectors to define the coarse and fine partitionings. 2.2 Specifying the preconditioning A preconditioning for a block matrix is specified by choosing (1) a global preconditioner, and (2) a local preconditioner for each diagonal or pivot block to exactly or approximately invert the block or solve the corresponding set of equations. For example, to fully define the conventional block Jacobi preconditioning, one must specify the global preconditioner to be block Jacobi and the local preconditioner to be LU factorization. In addition, the block size of the matrix has a role in determining the effect of the preconditioning. At one extreme, if the block size is one, then the preconditioning is entirely determined by the global preconditioner. At the other extreme, if there is only one block, then the preconditioning is entirely determined by the local preconditioner. The block size parameterizes the effect and cost between the selected local and global preconditioners. The best method is likely to be somewhere between the two extremes. For example, suppose symmetric successive overrelaxation (SSOR) is used as the global preconditioner, and complete LU factorization is used as the local precondi- tioner. For linear systems that are not too difficult to solve, SSOR may be used with a small block size. For more challenging systems, larger block sizes may be used, giving a better approximation to the original matrix. In the extreme, the matrix may be treated as a single block, and the method is equivalent to LU factorization. A global preconditioner M is specified with a very simple form of declaration. In the case of block SSOR, the declaration is Two functions are used to specify the local preconditioner and to provide parameters to the global preconditioner: factorization for the blocks M.setup(B, 0.5, 3); // BSSOR(omega=0.5, iterations=3) Here B is the block matrix defined as in Section 2.1. The setup function provides the real data to the preconditioner, and performs all the computations necessary for setting up the global preconditioner, for example, the computation of the LU factors in this case. Therefore, localprecon must be called before setup. The setup function must be called again if the local preconditioner is changed. In these interfaces, the same local preconditioner is specified for all the diagonal blocks. Object-Oriented Block Preconditioning \Delta 7 In general, however, the local preconditioners are not required to be the same. In some applications, different variables (e.g., velocity and pressure variables in a fluids simulation) may be blocked together. It may then make sense to write a specialized global preconditioner with an interface that allows different local preconditioners to be specified for each block. 2.2.1 Global preconditioners. The global preconditioners that we have implemented in BPKIT are listed in Table 1, along with the arguments of the setup function, and any default argument values. General reference works describing these global preconditioners and many of the local preconditioners described later are [Axelsson 1994; Barrett et al. 1994; Saad 1995]. See also the BPKIT Reference Manual [Chow and Heroux 1996]. Here we briefly specify these preconditioners and make a few comments on how they may be applied. Table 1. Global preconditioners. setup arguments none level BTIF none BJacobi, BSOR and BSSOR are block versions of the diagonal, successive overrelax- ation, and symmetric successive overrelaxation preconditioners. BILUK is a block version of level-based incomplete LU (ILU) factorization. BTIF is an incomplete factorization for block tridiagonal matrices. A preconditioner for a matrix A is often expressed as another matrix M which is somehow an approximation to A. However, M does not need to be explicitly formed, but instead, only the operation of M \Gamma1 on a vector is required. This operation is called the preconditioning operation, or the application of the preconditioner. For iterative methods based on biorthogonalization, the transposed preconditioning operator M \GammaT is also needed. It is also possible to apply the preconditioner in a split fashion when the preconditioner has a factored form. For example, if M is factored as LU , then the preconditioned matrix is L and the operations of L \Gamma1 and U \Gamma1 on a vector are required. Many preconditioners M can be expressed in factored form. Consider the splitting of a block matrix A, where DA is the block diagonal of A, \GammaL A is the strictly lower block triangular part, and \GammaU A is the strictly upper part. The block SSOR preconditioner in the case of one iteration is defined by Chow and M. A. Heroux The scale factor 1=!(2\Gamma!) is important if the iterative method is not scale invariant. When used as a preconditioner, the relaxation parameter ! is usually chosen to be 1, since selecting a value is difficult. However, if more than one iteration is used and the matrix is far from being symmetric and positive definite, underrelaxation may be necessary to prevent divergence. Also, the simpler block SOR preconditioner (with one iteration) may be preferable over block SSOR if A is nonsymmetric. If k iterations of block are used, the preconditioner has the form although it is not implemented this way. Instead, the preconditioner is applied to a vector v by performing k SOR iterations on the system starting from the zero vector. The level-0 block ILU preconditioner for certain structured matrices including block 5-point matrices can be written in a very similar form called the generalized block SSOR form. Here, D is the block diagonal matrix resulting from the incomplete factorization. In general, however, a level-based block ILU preconditioner is computed by performing Gaussian elimination and neglecting elements in the factors that fall out of a predetermined sparsity pattern. Level-based ILU preconditioners are much more accurate than relaxation preconditioners, but for general sparse matrices, have storage costs at least that of the original matrix. Incomplete factorization of block tridiagonal matrices is popular for certain structured matrices where the blocks have banded structure. It is a special case of the generalized block SSOR form, and thus only a sequence of diagonal blocks needs to be computed and stored. The block partitioning may be along lines of a 2-D grid, or along planes of a 3-D grid. In general, any "striped" partitioning will yield a block tridiagonal matrix. The inverse-free form of block tridiagonal factorization is where D is a block diagonal matrix whose blocks D i are defined by the recurrence starting with D This inverse-free form only requires matrix-vector multiplications in the preconditioning operation. However, the blocks are typically very large, and an approximate inverse is used in place of the exact inverse in the above equation to make the factorization incomplete. Many techniques for computing approximate inverses are available [Chow and Saad 1998]. 2.2.2 Local preconditioners. Local preconditioners are either explicit or implicit depending on whether (approximate) inverses of blocks are explicitly formed. An example of an implicit local preconditioner is LU factorization. Object-Oriented Block Preconditioning \Delta 9 The global preconditioners that involve incomplete factorization require the inverses of pivot blocks. For large block sizes, the use of approximate or exact dense inverses usually requires large amounts of storage and computation. Thus sparse approximate inverses should be used in these cases. Implicit local preconditioners produce inverses that are usually dense, and are therefore usually not computationally useful for block incomplete factorizations. This use of implicit local preconditioners is disallowed within BPKIT. We also apply this rule for small block sizes, since dense exact inverses are usually most efficient in these cases. (Note that the explicit local preconditioner LP INVERSE for the CSR block type is meant to be used for testing purposes only. Also, if an exact factorization is sought, it is usually most efficient to use an LU factorization on the whole matrix.) The global preconditioners that involve block relaxation may use either explicit or implicit local preconditioners, but usually the implicit ones are used. Explicit local preconditioners can be appropriate for block relaxation when the blocks are small. Local preconditioners are also differentiated by the type of the blocks on which they operate. Not all local preconditioners exist for all block types; incomplete factorization, for example, is only meaningful for sparse types. Thus, a local preconditioner must be chosen that matches the type of the block. BPKIT requires the user to be aware of the restrictions in the above two paragraphs when selecting a local preconditioner. Due to the dynamic binding of C++ virtual functions, violations of these restrictions will only be detected at run-time. Table 2 lists the local preconditioners that we have implemented, along with their localprecon arguments, their block types, and whether the local preconditioner is explicit or implicit. In contrast to the setup function, localprecon takes no default arguments. We have included an explicit exact inverse local preconditioner for the CSR format for comparison purposes (it would be inefficient to use it in block tridiagonal incomplete factorizations, for example). Table 2. Local preconditioners. localprecon arguments Block type Expl./Impl. LP LU none DENSE implicit LP INVERSE none DENSE explicit LP SVD alpha1, alpha2 DENSE explicit LP LU none CSR implicit LP INVERSE none CSR explicit LP ILUT lfil, threshold CSR implicit LP APINV TRUNC semibw CSR explicit LP APINV BANDED semibw CSR explicit LP APINV0 none CSR explicit LP APINVS lfil CSR explicit LP DIAG none CSR explicit LP TRIDIAG none CSR implicit iterations CSR implicit iterations CSR implicit LP GMRES restart, tolerance CSR implicit LP LU is an LU factorization with pivoting. LP INVERSE is an exact inverse com- Chow and M. A. Heroux puted via LU factorization with pivoting. LP RILUK is level-based relaxed incomplete LU factorization. LP ILUT is a threshold-based ILU with control over the number of fill-ins [Saad 1994], which may be better for indefinite blocks. The local preconditions prefixed with LP APINV are new approximate inverse techniques; see [Chow and Saad 1998] and [Chow and Heroux 1996] for details. LP DIAG is a diagonal approximation to the inverse, using the diagonal of the original block, and LP TRIDIAG is a tridiagonal implicit approximation, ignoring all elements outside the tridiagonal band of the original block. LP SVD uses the singular value decomposition to produce a dense approximate inverse \Sigma is \Sigma with its singular values thresholded by ff 1 a constant ff 2 plus a factor ff 1 of the largest singular value oe 1 . This may produce a more stable incomplete factorization if there are many blocks to be inverted that are close to being singular [Yeremin 1995]. LP SOR, LP SSOR and LP GMRES are iterative methods used as local preconditioners. 2.3 Interface with iterative methods An object-oriented preconditioned iterative method requires that matrix and preconditioner objects define a small number of operations. In BPKIT, these operations are defined polymorphically, and are listed in Table 3. For left and right preconditionings, the functions apply and applyt may be used to apply the preconditioning operator (M \Gamma1 , or its transpose) on a vector. Split (also called two-sided, or symmetric) preconditionings use applyl and applyr to apply the left and right parts of the split preconditioner, respectively. For an incomplete factorization A - LU , applyl is the L \Gamma1 operation, and applyr is the To anticipate all possible functionality, the applyc function defines a combined matrix-preconditioner operator to be used, for example, to implement the Eisenstat trick [Eisenstat 1981]. If the Eisenstat trick is used with flexible preconditionings (described at the end of this section), the right preconditioner apply also needs to be used. Two functions not listed here are matrix member functions that return the row and column dimensions of the matrix, which are useful for the iterative method code to help preallocate any work-space that is needed. Not all the operations in Table 3 may be defined for all matrix and preconditioner objects, and many iterative methods do not require all these operations. The GMRES iterative method, for example, does not require the transposed operations, and the relaxation preconditioners usually do not define the split operations. This is a case where we violate an object-oriented programming paradigm, and give the parent classes all the specializations of their children (e.g., a specific preconditioner may not define applyl although the generic preconditioner does). This will be seen again in Section 3.2. The argument lists for the functions in Table 3 use fundamental data types so that iterative methods codes are not forced to adopt any particular data structure for vectors. The interfaces use blocks of vectors to support iterative methods that use multiple right-hand sides. The implementation of these operations use Level 3 BLAS whenever possible. All the interfaces have the following form: void mult(int nr, int nc, const double *u, int ldu, double* v, int ldv) const; Object-Oriented Block Preconditioning \Delta 11 Table 3. Operations required by iterative methods. Matrix operations mult matrix-vector product trans mult transposed matrix-vector product Preconditioner operations apply apply preconditioner applyt apply transposed preconditioner applyl apply left part of a split preconditioner applyr apply right part of a split preconditioner applyc apply a combined matrix-preconditioner operator applyct above, transposed where nr and nc are the row and column dimensions of the (input) blocks of vectors, u and v are arrays containing the values of the input and output vectors, respec- tively, and ldu and ldv are the leading dimensions of these respective arrays. The preconditioner operations are not defined as const functions, in case the preconditioner objects need to change their state as the iterations progress (and spectral information is revealed, for example). When a non-constant operator is used in the preconditioning, a flexible iterative method such as FGMRES [Saad 1993] must be used. In BPKIT, this arises whenever GMRES is used as a local preconditioner. Users may wish to write advanced preconditioners that work with the iterative methods, and which change, for example, when there is a lack of convergence. This is a simple way of enhancing the robustness of iterative methods. In this case, the iterative method should be written as a class function whose class also provides information about convergence history and possibly approximate spectral information [Wu and Li 1995]. 2.4 Fortran 77 interface Many scientific computing users are unfamiliar with C++. It is usually possible, however, to provide an interface which is callable from any other language. BPKIT provides an object-oriented type of Fortran 77 interface. Objects can be created, and pointers to them are passed through functions as Fortran 77 integers. Consider the following code excerpt (most of the parameters are not important to this call blockmatrix(bmat, n, a, ja, ia, num-block-rows, partit, btype) call preconditioner(precon, bmat, BJacobi, 0.d0, 0.d0, LP-LU, 0.d0, 0.d0) call flexgmres(bmat, sol, rhs, precon, 20, 600, 1.d-8) The call to blockmatrix above creates a block matrix from the compressed sparse row data structure, given a number of arguments. This "wrapper" function is actually written in C++, but all its arguments are available to a Fortran 77 pro- gram. The integer bmat is actually a pointer to a block matrix object in C++. The Fortran 77 program is not meant to interpret this variable, but to pass it to Chow and M. A. Heroux other functions, such as preconditioner which defines a block preconditioner with a number of arguments, or flexgmres which solves a linear system using flexible GMRES. Similarly, precon is a pointer to a preconditioner object. The constant parameters BJacobi and LP LU are used to specify a block Jacobi preconditioner, using LU factorization to solve with the diagonal blocks. The matrix-vector product and preconditioner operations of Table 3 also have "wrapper" functions. This makes it possible to use BPKIT from an iterative solver written in Fortran 77. This was also another motivation to use fundamental types to specify vectors in the interface for operations such as mult (see Section 2.3). Calling Fortran 77 from C++ is also possible, and this is done in BPKIT when it calls underlying libraries such as the BLAS. BPKIT illustrates how we were able to mix the use of different languages. 3. LOCAL MATRIX OBJECTS A block matrix may contain blocks of more than one type. The best choice for the types of the blocks depends mostly on the structure of the matrix, but may also depend on the proposed algorithms and the computer architecture. For example, if a matrix has been reordered so that its diagonal blocks are all diagonal, then a diagonal storage scheme for the diagonal blocks is best. Inversion of these blocks would automatically use the appropriate algorithm. (The diagonal block type and the local preconditioners for it would have to be added by the user.) To handle different block types the same way, instances of each type are implemented as C++ polymorphic objects (i.e., a set of related objects whose functions can be called without knowing the exact type of the object). The block types are derived from a local matrix class called LocalMat, a class that defines the common interface for all the block types. The global preconditioners refer to LocalMat objects. When LocalMat functions are called, the appropriate code is executed, depending on the actual type of the LocalMat object (e.g., DENSE or CSR). In addition, each block type has a variety of local preconditioners. The explicitness or implicitness of local preconditioners need to be transparent, since, for example, either can be used in block SSOR. Thus both types of preconditioners are derived from the same base class. In particular, local preconditioners for a given block type are derived from the base class which is that block type (e.g., the LP SVD local preconditioner for the DENSE type is derived from the DENSE block type). This gives the user the flexibility to treat explicit local preconditioners as regular blocks. Implicit local preconditioners are not derived separately because logically they are related to explicit local preconditioners. All block operations that apply to explicit preconditioners also apply to local preconditioners; however, many of these operations are inefficient for local preconditioners, and their use has been disallowed to prevent improper usage. Implicit preconditioners cannot be derived separately from explicit preconditioners because of their similarity from the point of view of global preconditioners. The LocalMat hierarchy is illustrated in Figure 2, showing the derivation of block types and the subsequent derivation of local preconditioners. These LocalMat classes form the "kernel" of BPKIT, and allow global preconditioners to be implemented without knowledge of the type of blocks or local preconditioners that are used. Users may also add to the kernel by deriving their own specific classes. Object-Oriented Block Preconditioning \Delta 13 CSR Fig. 2. LocalMat hierarchy. The challenge of designing the LocalMat class was to determine what operations are required to implement block preconditioners and to give these operations semantics that allow an efficient implementation for all possible block types. The operations are implemented as C++ virtual functions. The following subsections describe these operations. 3.1 Allocating storage An important difference between dense and sparse blocks is that the storage requirement for sparse blocks is not always known beforehand. Thus, in order to treat dense and sparse blocks the same way, storage is allocated for a block when it is required. As an optimization, if it is known that dense blocks are used (e.g., conversion of a sparse matrix to a block matrix with dense blocks), storage may be allocated beforehand by the user. Functions are provided to set the data pointers of the block objects. Thus it is possible to allocate contiguous storage for an array of dense blocks. 3.2 Local matrix functions Table 4. Functions for LocalMat objects. A A.Mat A.Mat Mat Add(B, C, alpha) A.Mat Mat Mult(B, C, alpha, beta) A.Mat A.Mat Trans Vec Mult(b, c, alpha, beta) A.Mat Vec Solve(b, c) A.Mat Trans Vec Solve(b, c) 14 \Delta E. Chow and M. A. Heroux Table 4 lists the functions that we have determined to be required for implementing the block preconditioners listed in Table 1. The functions are invoked by a block object represented by A. B and C are blocks of the same type as A, b and c are components from a block vector object, and ff and fi are scalars. The default value for ff is 1 and for fi is 0. CreateEmpty() creates an empty block (0 by 0 dimensions) of the same class as that of A. This function is useful for constructing blocks in the preconditioner without knowing the types of blocks that are being used. SetToZero(dim1, dim2) sets A to zero, resetting its dimensions if necessary. This operation is not combined with CreateEmpty() because it is not always necessary to zero a block when creating it, and zeroing a block could be relatively expensive for some block types. copies its argument block to the invoking block. The original data held by the invoking block is released, and if the new block has a different size, the allocated space is resized. CreateInv(lprecon) provides a common interface for creating local preconditioners. lprecon is of a type that describes a local preconditioner with its arguments from Table 2. The exact or approximate inverse (explicit or implicit) of A is generated. The CreateEmpty and CreateInv functions create new objects (not just the real data space). These functions return pointers to the new objects to emphasize this point. Overloading of the arithmetic operators such as for blocks and local preconditioners has been sacrificed since chained operations such as would be inefficient if implemented as a sequence of elementary operations. In addition, these operators are difficult to implement without extra memory copying (for the first store the result into a temporary before the result is copied into A by the = operator). These are the functions that we have found to be useful for block preconditioners. For example, used in BTIF, used in BILUK, and other functions are useful, for example, in matrix-vector product and triangular solve operations. Note in particular that Mat Trans Mat Mult is not a useful function here, and has not been defined. Note that local preconditioner objects also inherit these functions, although they do not need them all. For objects that are implicit local preconditioners, no matrix is formed, and operations such as addition (Mat Mat Add) do not make sense. For blocks for which no local preconditioner has been created, solving a system with that block (Mat Vec Solve) is not allowed. Here, again, we had to give the parent classes all the specializations of their derived classes. Table 5 indicates when the functions are allowed. An error condition is raised at run-time if the functions are used incorrectly. Given these operations, a one-step block SOR code could be implemented as shown below. Ap is a pointer to a block matrix object which stores its block structure in CSR format (the ia array stores the block row pointers, and the ja array stores the block column indices). The pointers to the diagonal elements in idiag and the inverses of the diagonal elements diag were computed during the call to setup. V is a block vector object that allows blocks in a vector to be accessed as individual entries. The rest of the code is self-explanatory. 1. for (i=0; i!Ap-?numrow(); i++) Object-Oriented Block Preconditioning \Delta 15 Table 5. The types of objects that may be used with each function. Explicit Implicit Coarse local local Function blocks precon. precon. CreateEmpty * MatCopy * Mat Trans * Mat Mat Add * Mat Mat Mult * Mat Vec Mult * Mat Trans Vec Mult * Mat Vec Solve * Mat Trans Vec Solve * 2. - 3. for (j=ia[i]; j!idiag[i]; j++) 4. - 5. // 6. 7. Ap-?val(j).Mat-Vec-Mult(V(ja[j]), V(i), -omega, 1.0); 8. - 9. 10. diag[i]-?Mat-Vec-Solve(V(i), V(i)); 11. - A block matrix that mixes different block types must be used very carefully. First, the restrictions for the different block types (Section 2.2.2) must not be violated. Second, unless we define arithmetic operations between blocks of different types, the incomplete factorization preconditioners cannot be used. Our main design alternative was to create a block matrix class for each block type. The classes would be polymorphic and define a set of common operations that preconditioners may use to manipulate their blocks. A significant advantage of this design is that it is impossible to use local preconditioners of the wrong type (e.g., use incomplete factorization on a dense block). A disadvantage is that different block types (e.g., specialized types created for a particular application) cannot be used within the same block matrix. Another alternative was to implement meta-matrices, i.e., blocks are nested re- cursively. It would be complicated, however, for users to specify these types of matrices and the levels of local preconditioners that could be used. In addition, there is very little need for such complexity in actual applications, and the two-level design (coarse and fine blocks) described in Section 2.1 should be sufficient. 4. NUMERICAL TESTS The numerical tests were carried out on the matrices listed in Table 6. SHERMAN1 is a reservoir simulation matrix on a grid, with one unknown per grid point. This is a simple symmetric problem which we solve using partitioning by Chow and M. A. Heroux planes. WIGTO966 is from an Euler equation model and was supplied by Larry Wigton of Boeing. FIDAP019 models an axisymmetric 2-D developing pipe flow with the fully-coupled Navier-Stokes equations using the two-equation k-ffl model for turbulence. The BARTHT1A and BARTHT2A matrices were supplied by Tim Barth of NASA Ames and are from a 2-D, high Reynolds number aerofoil problem, with a 1-equation turbulence model. The BARTHT2A model is solved with a preconditioner based on the less accurate but sparser BARTHT1A model. Table 6. Test matrices, listed with their dimensions and numbers of nonzeros. Matrix n no. nonz Tables 7 and 9 show the results for SHERMAN1 with the block relaxation and incomplete factorization global preconditioners, using various local preconditioners. The arguments given for the global and local preconditioners in these tables correspond to those displayed in Tables 1 and 2 respectively. A block size of 100 was used. Since the matrix is block tridiagonal, BILUK and BTIF are equivalent. The tables show the number of steps of GMRES (FGMRES, if appropriate) that were required to reduce the residual norm by a factor of 10 \Gamma8 . A dagger (y) is used to indicate that this was not achieved in 600 steps. Right preconditioning, 20 Krylov basis vectors and a zero initial guess were used. The right-hand side was provided with the matrix. Since the local preconditioners have different costs, Tables 8 and 9 show the CPU timings (system and user times) for BSSOR(1.,3) and BTIF. The tests were run on one processor of a Sun Sparcstation 10. For this particular problem and choice of partitioning, the ILU local preconditioners required the least total CPU time with BSSOR(1.,3). With BTIF, an exact solve was most efficient (i.e., the preconditioner was an exact solve). Table 7. Number of GMRES steps for solving the SHERMAN1 problem with block relaxation global preconditioners and various local preconditioners. LP RILUK(0,0.) 93 71 53 402 48 Object-Oriented Block Preconditioning \Delta 17 Table 8. Number of GMRES steps and timings for solving the SHERMAN1 problem with and various local preconditioners. precon solve total LP ILUT(2,0.) 44 0.03 1.73 1.76 Table 9. Number of GMRES steps and timings for solving the SHERMAN1 problem with block incomplete factorization and various local preconditioners. precon solve total Tables show the number of GMRES steps for the BARTHT2A matrix. A random right-hand side was used, and the initial guess was zero. The GMRES tolerance was 10 \Gamma8 and 50 Krylov basis vectors were used. In Table 10, block incomplete factorization was used as the global preconditioner, and LU factorization was used as the local preconditioner. In Table 11, block SSOR with one iteration used as the global preconditioner, and level-3 ILU was used as the local preconditioner. Table 10. Number of GMRES steps for solving the BARTHT2A problem with BILUK-LP LU. block BILUK level Tables 12 and 13 show the results for WIGTO996 using block incomplete factor- ization. The right-hand side was the vector of all ones, and the GMRES tolerance . The other parameters were the same as those in the previous experiment. The failures in Table 12 are due to inaccuracy for low fill levels, and instability for high levels. In Table 13, LP SVD(0.1,0.) used as the local preconditioner gave the Chow and M. A. Heroux Table 11. Number of GMRES steps for solving the BARTHT2A problem with block GMRES size steps best results. LP SVD(0.1,0.) indicates that the singular values of the pivot blocks were thresholded at 0.1 times the largest singular value. Table 12. Number of GMRES steps for solving the WIGTO966 problem with BILUK-LP INVERSE. block BILUK level Table 13. Number of GMRES steps for solving the WIGTO966 problem with block BILUK level Now we show some results with block tridiagonal incomplete factorization preconditioners using general sparse approximate inverses. The matrix FIDAP019 was partitioned into a block tridiagonal system using a constant block size of 161 (the last block has size 91). Since the matrix arises from a finite element problem, a more careful selection of the partitioning could have yielded better results. The rows of the system were scaled by their 2-norms, and then their columns were scaled similarly, since the matrix contains different equations and variables. A Krylov subspace size of 50 for GMRES was used. The right-hand side was constructed so that the solution is the vector of all ones. We compare the result with the pair of global-local preconditioners BILUK(0)-LP SVD(0.5,0.), using a block size of 5 (LP SVD(0.5,0.) gave the best result after several trials). Table 14 shows the number of GMRES steps to convergence, timings for setting up the preconditioner and for the iterations, and the number of nonzeros in the preconditioner. The experiments were carried out on one processor of a Sun Sparcstation 10. The timings show that some combinations of the BTIF global preconditioner with the APINVS local preconditioner are comparable to BILUK(0)-LP SVD(0.5,0.), but use much less memory, since only the approximate inverses of the pivot blocks need to be stored. Although the actual number of nonzeros in the matrix is 259 879, there were 39 355 block nonzeros required for BILUK, and therefore almost a million Object-Oriented Block Preconditioning \Delta 19 Table 14. Test results for the FIDAP019 problem. GMRES CPU time steps precon solve total in precon entries which were needed to be stored. The APINVS method produced approximate inverses that were sparser than the original pivot blocks. See [Chow and Saad 1998] for more details. There is often heated debate over the use of C++ in scientific computing. Ideally, C++ and Fortran 77 programs that are coded similarly should perform similarly. However, by using object-oriented features in C++ to make a program more flexible and maintainable, researchers usually encounter a 10 to percent performance penalty [Jiang and Forsyth 1995]. If optimized kernels such as the BLAS are called, then the C++ performance penalty can be very small for large problems, as a larger fraction of the time is spent in the kernels. Since C++ and Fortran 77 programs will usually be coded differently, a practical comparison is made when a general code such as BPKIT is compared to a specialized Fortran 77 code. Here we compare BPKIT to an optimized block SSOR preconditioner with a GMRES accelerator. This code performs block relaxations of the form ii r i for a block row i, where A ii is the i-th diagonal block of A, A :;i is the i-th block column of A, x i is the i-th block of the current solution, and r is the current residual vector. Notice that the update of the residual vector is very fast if A is stored by sparse columns and not by blocks. Since BPKIT stores the matrix A by blocks for flexibility, it is interesting to see what the performance penalty would be for this case. Tables 15 and 16 show the timings for block SSOR on a Sun Sparcstation 10 and a Cray C90 supercomputer, for the WIGTO966 matrix. In this case, the right-hand side was constructed so that the solution is a vector of all ones; the other parameters were the same as before. All programs were optimized at the highest optimization level; clock was used to measure CPU time (user and system) for the C++ programs, and etime and timef were used to measure the times for the Fortran 77 programs on the Sun and Cray computers, respectively. One step of block SSOR with used in the tests. The local preconditioner was an exact LU factorization. Results are shown for a large range of block sizes, and in the case of BPKIT, for both DENSE and CSR storage schemes for the blocks. The last column of each table gives the average time to perform one iteration of GMRES. The results show that the specialized Fortran 77 code has better performance over a wide range of block sizes. This is expected because the update of the residual, which is the most major computation, is not affected by the blocking. Chow and M. A. Heroux If dense blocks are used, BPKIT can be competitive on the Cray by using large block sizes, such as 128. Blocks of this size contain many zero entries which are treated as general nonzero entries when a dense storage scheme is used. However, vectorization on the Cray makes operations with large dense blocks much more efficient. If sparse blocks are used, BPKIT can be competitive on the workstation with moderate block sizes of 8 or 16. Operations with smaller sparse blocks are inefficient, while larger blocks imply larger LU factorizations for the local preconditioner. This comparison using block SSOR is dramatic since two very different data structures are used. Comparisons of level-based block ILU in C++ and Fortran 77 show very small differences in performance, since the data structures used are similar [Jiang and Forsyth 1995]. In conclusion, the types and sizes of blocks must be chosen carefully in BPKIT to attain high performance on a particular machine. The types and sizes of blocks should also be chosen in conjunction with the requirements of the preconditioning algorithm and the block structure of the matrix. Based on the above experiments, Table 17 gives an idea of the approximate block sizes that should be used for BPKIT, given no other constraints. 5. CONCLUDING REMARKS This article has described an object-oriented framework for block preconditioning. Polymorphism was used to handle different block types and different local precon- ditioners. Block types and local preconditioners form a "kernel" on which the block preconditioners are built. Block preconditioners are written in a syntax comparable to that for non-block preconditioners, and they work for matrices containing any block type. BPKIT is easily extensible, as an object-oriented code would al- low. We have distinguished between explicit and implicit local preconditioners, and deduced the operations and semantics that are useful for polymorphically manipulating blocks. Timings against a specialized and optimized Fortran 77 code on both workstations and Cray supercomputers show that this framework can approach the efficiency of such a code, as long as suitable block sizes and block types are chosen. We believe we have found a suitable compromise between Fortran 77-like performance and C++ flexibility. A significant contribution of BPKIT is the collection of high-quality preconditioners under a common, concise interface. Block preconditioners can be more efficient and more robust than their non-block counterparts. The block size parameterizes between a local and global method, and is valuable for compromising between accuracy and cost, or combining the effect of two methods. The combination of local and global preconditioners leads to a variety of useful methods, all of which may be applicable in different circumstances. ACKNOWLEDGMENTS We wish to thank Yousef Saad, Kesheng Wu and Andrew Chapman for their codes and for helpful discussions. We also wish to thank Larry Wigton and Tim Barth for providing some of the test matrices, and Tim Peck for helping us with editing. This article has benefited substantially from the comments and suggestions of one of the anonymous referees, and we are grateful for his time and patience. Object-Oriented Block Preconditioning \Delta 21 Table 15. timings. Specialized Fortran 77 program block GMRES time size steps precon solve total average BPKIT, dense blocks block GMRES time size steps precon solve total average 128 212 3.66 559.05 562.71 2.6543 BPKIT, sparse blocks block GMRES time size steps precon solve total average 128 212 4.42 162.58 167. 22 \Delta E. Chow and M. A. Heroux Table 16. WIGTO966: BSSOR(0.5,1)-LP LU, Cray C90 timings. Specialized Fortran 77 program block GMRES time size steps precon solve total average BPKIT, dense blocks block GMRES time size steps precon solve total average BPKIT, sparse blocks block GMRES time size steps precon solve total average 128 212 5.39 132.92 138.31 0.6524 Table 17. Recommended block sizes. Block type Sun Cray CSR Object-Oriented Block Preconditioning \Delta 23 --R Iterative Solution Methods. On some versions of incomplete block-matrix factorization iterative methods Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. A revised proposal for a sparse BLAS toolkit. BPKIT Block preconditioning toolkit. Approximate inverse techniques for block-partitioned ma- trices Approximate inverse preconditioners via sparse-sparse iter- ations Block preconditioning for the conjugate gradient method. LAPACK: A portable linear algebra library for supercomputers. IEEE Control Systems Society Workshop on Computer-Aided Control System Design (December <Year>1989</Year>) A set of level 3 basic linear algebra subprograms. A sparse matrix library in C An object oriented design for high performance linear algebra on distributed memory architectures. Sparse matrix test problems. ParPre: A parallel preconditioners package. Efficient implementation of a class of preconditioned conjugate gradient methods. Performance issues for iterative solvers in device simulation. Robust linear and nonlinear strategies for solution of the transonic Euler equations. users manual: Scalable library software for the parallel solution of sparse linear systems. Block SSOR precon- ditionings for high-order 3D FE systems On a family of two-level preconditionings of the incomplete block factorization type Fortran 90: An entry to object-oriented programming for solution of partial differential equations SPARSKIT: A basic tool kit for sparse matrix computations. A flexible inner-outer preconditioned GMRES algorithm ILUT: A dual threshold incomplete ILU factorization. Iterative Methods for Sparse Linear Systems. PETSc 2.0 users' manual. An approximate factorization procedure based on the block Cholesky decomposition and its use with the conjugate gradient method. BKAT: An object-oriented block Krylov accelerator toolkit Presentation at Cray Research Private communication. --TR Sparse matrix test problems A set of level 3 basic linear algebra subprograms The C++ programming language (2nd ed.) A flexible inner-outer preconditioned GMRES algorithm Iterative solution methods Performance issues for iterative solvers in device simulation Fortran 90 Object-oriented design of preconditioned iterative methods in diffpack Approximate Inverse Techniques for Block-Partitioned Matrices Approximate Inverse Preconditioners via Sparse-Sparse Iterations Iterative Methods for Sparse Linear Systems --CTR Michael Gertz , Stephen J. Wright, Object-oriented software for quadratic programming, ACM Transactions on Mathematical Software (TOMS), v.29 n.1, p.58-81, March Iain S. Duff , Michael A. Heroux , Roldan Pozo, An overview of the sparse basic linear algebra subprograms: The new standard from the BLAS technical forum, ACM Transactions on Mathematical Software (TOMS), v.28 n.2, p.239-267, June 2002 Marzio Sala, An object-oriented framework for the development of scalable parallel multilevel preconditioners, ACM Transactions on Mathematical Software (TOMS), v.32 n.3, p.396-416, September 2006 Glen Hansen , Andrew Zardecki , Doran Greening , Randy Bos, A finite element method for unstructured grid smoothing, Journal of Computational Physics, v.194 n.2, p.611-631, March 2004 Mikel Lujn , T. L. Freeman , John R. Gurd, OoLALA: an object oriented analysis and design of numerical linear algebra, ACM SIGPLAN Notices, v.35 n.10, p.229-252, Oct. 2000 Glen Hansen , Andrew Zardecki , Doran Greening , Randy Bos, A finite element method for three-dimensional unstructured grid smoothing, Journal of Computational Physics, v.202 n.1, p.281-297, January 2005 Michele Benzi, Preconditioning techniques for large linear systems: a survey, Journal of Computational Physics, v.182 n.2, p.418-477, November 2002

preconditioners;block matrices

287640

A combined unifrontal/multifrontal method for unsymmetric sparse matrices.

We discuss the organization of frontal matrices in multifrontal methods for the solution of large sparse sets of unsymmetric linear equations. In the multifrontal method, work on a frontal matrix can be suspended, the frontal matrix can be stored for later reuse, and a new frontal matrix can be generated. There are thus several frontal matrices stored during the factorization, and one or more of these are assembled (summed) when creating a new frontal matrix. Although this means that arbitrary sparsity patterns can be handled efficiently, extra work is required to sum the frontal matrices together and can be costly because indirect addressing is requred. The (uni)frontal method avoids this extra work by factorizing the matrix with a single frontal matrix. Rows and columns are added to the frontal matrix, and pivot rows and columns are removed. Data movement is simpler, but higher fill-in can result if the matrix cannot be permuted into a variable-band form with small profile. We consider a combined unifrontal/multifrontal algorithm to enable general fill-in reduction orderings to be applied without the data movement of previous multifrontal approaches. We discuss this technique in the context of a code designed for the solution of sparse systems with unsymmetric pattern.

Introduction . We consider the direct solution of sets of linear equations the coefficient matrix A is sparse, unsymmetric, and has a general nonzero pattern. In a direct approach, a permutation of the matrix A is factorized into its LU factors, are permutation matrices chosen to preserve sparsity and maintain numerical accuracy. Many recent algorithms and software for direct solution of sparse systems are based on a multifrontal approach [17, 2, 10, 27]. In this paper, we will examine a frontal matrix strategy to be used within a multifrontal approach. Frontal and multifrontal methods compute the LU factors of A by using data structures that permit regular access of memory and the use of dense matrix kernels in their innermost loops. On supercomputers and high-performance workstations, this can lead to a significant increase in performance over methods that have irregular memory access and which do not use dense matrix kernels. We discuss frontal methods in Section 2 and multifrontal techniques in Section 3, before discussing the combination of the two methods in Section 4. We consider the influence of the principle parameter in our approach in Section 5, and the performance Computer and Information Science and Engineering Department, University of Florida, Gainesville, Florida, USA. (904) 392-1481, email: davis@cis.ufl.edu. Technical reports and matrices are available via the World Wide Web at http://www.cis.ufl.edu/~davis, or by anonymous ftp at ftp.cis.ufl.edu:cis/tech-reports. y Rutherford Appleton Laboratory, Chilton, Didcot, Oxon. 0X11 0QX England, and European Center for Research and Advanced Training in Scientific Computation (CERFACS), Toulouse, France. email: isd@letterbox.rl.ac.uk. Technical reports, information on HSL, and matrices are available via the World Wide Web at http://www.cis.rl.ac.uk/struct/ARCD/NUM.html, or by anonymous ftp at seamus.cc.rl.ac.uk/pub. T. A. DAVIS AND I. S. DUFF l l l l l l l Fig. 2.1. Frontal method example of our new approach in Section 6, before a few concluding remarks and information on the availability of our codes in Section 7. 2. Frontal methods. In a frontal scheme [15, 26, 31, 32], the factorization proceeds as a sequence of partial factorizations and eliminations on full submatrices, called frontal matrices. Although frontal schemes were originally designed for the solution of finite element problems [26], they can be used on assembled systems [15] and it is this version that we study in this paper. For general systems, the frontal matrices can be written as where all rows are fully summed (that is there is no further contributions to come to the rows in (2.1)) and the first two block columns are fully summed. This means that pivots can be chosen from anywhere in the fully summed block comprising the first two block columns and, within these columns, numerical pivoting with arbitrary row interchanges can be accommodated since all rows in the frontal matrix are fully- summed. We assume, without loss of generality, that the pivots that have been chosen are in the square matrix F 11 of (2.1). F 11 is factorized, multipliers are stored over F 21 and the Schur complement using full matrix kernels. The submatrix consisting of the rows and columns of the frontal matrix from which pivots have not yet been selected is called the contribution block. At the next stage, further entries from the original matrix are assembled with the Schur complement to form another frontal matrix. The overhead is low since each equation is only assembled once and there is never any assembly of two (or more) frontal matrices. An example is shown in Figure 2.1, where two pivot steps have already been performed on a 5-by-7 frontal matrix (computing the first two rows and columns of U and L, respectively). Nonzero entries in L and U are shown in lower case. Row 6 has just been assembled into the current 4-by-7 frontal matrix (shown as a solid box). Columns 3 and 4 are now fully-summed and can be eliminated. After this step, rows 7 and 8 must both be assembled before columns 5 and 6 can be eliminated (the dashed box, a 4-by-6 frontal matrix containing rows 5 through 8 and columns 5, 6, 7, 8, 9, and 12 of the active submatrix). The frontal matrices are, of course, stored with columns A COMBINED UNIFRONTAL/MULTIFRONTAL METHOD 3 packed together so no zero columns are held in the frontal matrix delineated by the dashed box. The dotted box shows the state of the frontal matrix when the next five pivots can be eliminated. To factorize the 12-by-12 sparse matrix in Figure 2.1, a working array of size 5-by-7 is required to hold the frontal matrix. Note that, in Figure 2.1, the columns are in pivotal order. One important advantage of the method is that only the frontal matrix need reside in memory. Entries in A can be read sequentially from disk into the frontal matrix, one row at a time. Entries in L and U can be written sequentially to disk in the order they are computed. The method works well for matrices with small profile, where the profile of a matrix is a measure of how close the nonzero entries are to the diagonal and is given by the expression: a ji 6=0 where it is assumed the diagonal is nonzero so all terms in the summation are non- negative. If numerical pivoting is not required, fill-in does not increase the profile U has the same profile as A). To reduce the profile, the frontal method is typically preceded by an ordering method such as reverse Cuthill-McKee (RCM) [5, 7, 29], which is typically faster than the sparsity-preserving orderings required by a more general technique like a multifrontal method (such as nested dissection [24] and minimum degree [1, 25]). A degree update phase, which is typically the most costly part of a minimum degree algorithm, is not required in the RCM algorithm. However, for matrices with large profile, the frontal matrix can be large, and an unacceptable level of fill-in can occur. 3. Multifrontal methods. In a multifrontal scheme for a symmetric matrix [2, 8, 9, 10, 17, 18, 28], it is normal to use an ordering such as minimum degree to reduce the fill-in. Such an ordering tends to reduce fill-in much more than profile reduction orderings. The ordering is combined with a symbolic analysis to generate an assembly or computational tree, where each node represents the assembly and elimination operations on a frontal matrix and each edge the transfer of data from child to parent node. When using the tree to drive the numerical factorization, assembly and eliminations at any node can proceed as soon as those at the child nodes have been completed, giving added flexibility for issues such as exploitation of parallelism. As in the frontal scheme, the complete frontal matrix (2.1) cannot normally be factorized but only a few steps of Gaussian elimination are possible, after which the remaining reduced matrix (the Schur complement needs to be summed (assembled) with other data at the parent node. In the unsymmetric-pattern multifrontal method, the tree is replaced by a directed acyclic graph (dag) [9], and a contribution block may be assembled into more than one subsequent frontal matrix. 4. Combining the two methods. Let us now consider an approach that combines some of the best features of the two methods. Assume we have chosen a pivot and determined a frontal matrix as in a normal multifrontal method. At this stage, a normal multifrontal method will select as many pivots as it can from the fully summed part of the frontal matrix, perform the eliminations corresponding to these pivots, store the contributions to the factors, and store the remaining frontal matrix for later assembly at the parent node of the assembly tree. The strategy we use in 4 T. A. DAVIS AND I. S. DUFF our combined method is, after forming the frontal matrix, to hold it as a submatrix of a larger working array and allow further pivots to be chosen from anywhere within the frontal matrix. If such a potential pivot lies in the non-fully summed parts of the frontal matrix then it is necessary to sum its row and column before it could be used as a pivot. This is possible so long as its fully summed row and column can be accommodated within the larger working array. In this way, we avoid some of the data movement and assemblies of the multifrontal method. Although the motivation is different, the idea of continuing with a frontal matrix for some steps before moving to another frontal matrix is similar to recent work in implementing frontal schemes within a domain decomposition environment, for example [23], where several fronts are used within a unifrontal context. However, in the case of [23], the ordering is done a priori and no attempt is made to use a minimum degree ordering. Another case where one continues with a frontal matrix for several assemblies is when relaxed supernode amalgamation is used [17]. However, this technique delays the selection of pivots and normally causes more fill-in and operations than our technique. Indeed, the use of our combined unifrontal/multifrontal approach does not preclude implementing relaxed supernode amalgamation as well. In comparison with the unifrontal method where the pivoting is determined entirely from the order of the assemblies, the combined method requires the selection of pivots to start each new frontal matrix. This implies either an a priori symbolic ordering or a pivot search during numerical factorization. The pivot search heuristic requires the degrees of rows and columns, and thus a degree update phase. The cost of this degree update is clearly a penalty we pay to avoid the poor fill-in properties of conventional unifrontal orderings. We now describe how this new frontal matrix strategy is applied in UMFPACK Version 1.1 [8, 10], in order to obtain a new combined method. However, the new frontal strategy can be applied to any multifrontal algorithm. UMFPACK does not use actual degrees but keeps track of upper bounds on the degree of each row and column. The symmetric analogue of the approximate degree update in UMFPACK has been incorporated into an approximate minimum degree algorithm (AMD) as discussed in [1], where the accuracy of our degree bounds is demonstrated. Our new algorithm consists of several major steps, each of which comprises several pivot selection and elimination operations. To start a major step, we choose a pivot, using the degree upper bounds and a numerical threshold test, and then define a working array which is used for pivoting and elimination operations so long as these can be performed within it. When this is no longer possible, the working array is stored and another major step is commenced. To start a major step, the new algorithm selects a few (by default columns from those of minimum upper bound degree and computes their patterns and true degrees. A pivot row is selected on the basis of the upper bound on the row degree from those rows with nonzero entries in the selected columns. Suppose the pivot row and column degrees are r and c, respectively. A k-by-l working array is allocated, typically has a value between 2 and 3, and is fixed for the entire factorization). The active frontal matrix is c-by-r but is stored in a k-by-l working array. The pivot row and column are fully assembled into the working array and define a submatrix of it as the active frontal matrix. The approximate degree update phase computes the bounds on the degrees of all the rows and columns in this active frontal matrix and assembles previous contribution blocks into the active frontal matrix. A row i in a previous contribution block is assembled into the active frontal matrix if A COMBINED UNIFRONTAL/MULTIFRONTAL METHOD 5 1. the row index i is in the nonzero pattern of the current pivot column, and 2. if the column indices of the remaining entries in the row are all present in the nonzero pattern of the current pivot row. Columns of previous contribution blocks are assembled in an analogous manner. The major step then continues with a sequence of minor steps at each of which another pivot is sought. These minor steps are repeated until the factorization can no longer continue within the current working array, at which point a new major step is started. When a pivot is chosen in a minor step, its rows and columns are fully assembled into the working array and redefine the active frontal matrix. The approximate degree update phase computes the bounds on all rows and columns in this active frontal matrix and assembles previous contribution blocks into the active frontal matrix. The corresponding row and column of U and L are computed, but the updates from this pivot are not necessarily performed immediately. For efficient use of the Level 3 BLAS it is better to accumulate a few updates (typically 16 or 32, if possible) and perform them at the same time. To find a pivot in this minor step, a single candidate column from the contribution block is first selected, choosing one with least value for the upper bound of the column degree, and any pending updates are applied to this column. The column is assembled into a separate work vector, and a pivot row is selected on the basis of upper bound on the row degrees and a numerical threshold test. Suppose the candidate pivot row and column have degrees are r 0 and c 0 , respectively. Three conditions apply: 1. If r 0 ? l or c 0 ? k, then factorization can no longer continue within the active frontal matrix. Any pending updates are applied. The LU factors are stored. The active contribution block is saved for later assembly into a subsequent frontal matrix. The major step is now complete. 2. If r then the candidate pivot can fit into the active frontal matrix without removing the p pivots already stored there. Factorization continues within the active frontal matrix by commencing another minor step. 3. Otherwise, if l then the candidate pivot can fit, but only if some of the previous p pivots are shifted out of the current frontal matrix. Any pending updates are applied. The LU factors corresponding to the pivot rows and columns are removed from the front and stored. The active contribution block is left in place. Set p / 1. Factorization continues within the active frontal matrix by commencing another minor step. We know of no previous multifrontal method that considers case 3 (in particular, UMFPACK Version 1.1 does not). Case 1 does not occur in frontal methods, which are given a working array large enough to hold the largest frontal matrix (unless numerical pivoting causes the frontal matrix to grow unexpectedly). Taking simultaneous advantage of both cases 1 and 3 can significantly reduce the memory requirements and number of assembly operations, while still allowing the use of orderings that reduce fill-in. Figure 4.1 illustrates how the working array is used in UMFPACK Version 1.1 and the combined method. The matrices L 1 , L 2 , U 1 , and U 2 in the figure are the columns and rows of the LU factors corresponding to the pivots eliminated within this frontal matrix. The matrix D is the contribution block. The matrix - U 2 is U 2 with rows in reverse order, and - columns in reverse order. The arrows denote how these matrices grow as new pivots are added. When pivots are removed from the working array in Figure 4.1(b), for case 4 above, the contribution block does 6 T. A. DAVIS AND I. S. DUFF (a) UMFPACK working array (b) Combined method working array empty empty U U LD U 121Fig. 4.1. Data structures for the active frontal matrix Table Test matrices. name n jAj sym. discipline comments gre 1107 1107 5664 0.000 discrete simul. computer system gemat11 4929 33185 0.001 electric power linear programming basis migration lns 3937 3937 25407 0.850 fluid flow linearized Navier-Stokes shyy161 76480 329762 0.726 fluid flow viscous fully-coupled Navier-Stokes hydr1 5308 23752 0.004 chemical eng. dynamic simulation rdist1 4134 94408 0.059 chemical eng. reactive distillation lhr04 4101 82682 0.015 chemical eng. light hydrocarbon recovery chemical eng. light hydrocarbon recovery not need to be repositioned. A similar data organization is employed by MA42 [22]. 5. Numerical experiments. We discuss some experiments on the selection of a value for G in this section and compare the performance of our code with other sparse matrix codes in the next section. We show in Table 5.1 the test matrices we will use in our experiments in this section and the next. The table lists the name, order, number of entries (jAj), symmetry, the discipline from which the matrix comes, and additional comments. The symmetry, or more correctly the structural symmetry, is the number of matched off-diagonal entries over the total number of off-diagonal entries. An entry, a ij (j 6= i), is matched if a ji is also an entry. We have performed our experiments on the effect of the value of G on a SUN SPARCstation 20 Model 41, using the Fortran compiler (f77 version 1.4, with -O4 and -libmil options), and the BLAS from [11] and show the results in Table 5.2. The table lists the matrix name, the growth factor G, the number of frontal matrices (major steps), the numerical factorization time in seconds, the total time in seconds, the number of nonzeros in the LU factors, the memory used, and the floating-point operation count. The number of frontal matrices is highest for decreases as G increases although, because the effect is local, this decrease may not be monotonic. Although the fill-in and operation count is typically the lowest when the minimum amount of memory is allocated for each frontal matrix 1), the factorization time is often high because of the additional data movement required to assemble the contribution blocks and the fact that the dense matrix kernels are more efficient for larger frontal matrices. These results show that our strategy of allocating more space than necessary for the frontal matrix and choosing pivots that are not from the initial fully summed A COMBINED UNIFRONTAL/MULTIFRONTAL METHOD 7 Table Effect of G on the performance of the combined method Matrix G fronts factor total jL memory op count gre 1107 1.0 926 0.51 0.89 0.048 0.160 2.6 2.0 406 0.43 0.74 0.079 0.196 6.7 2.5 323 0.70 1.01 0.100 0.224 9.0 7.0 122 0.75 0.85 0.121 0.235 9.5 2.0 988 0.37 0.70 0.067 0.287 0.8 2.5 873 0.39 0.69 0.077 0.299 1.0 3.0 776 0.33 0.66 0.087 0.312 1.3 5.0 586 0.41 0.70 0.119 0.353 2.0 7.0 538 0.41 0.77 0.148 0.381 2.8 2.0 2324 1.41 3.79 0.122 0.959 7.2 2.5 2326 1.37 3.96 0.122 0.883 7.2 3.0 2322 1.41 3.97 0.123 0.886 8.1 7.0 2280 2.82 5.87 0.153 0.852 16.0 lns 3937 1.0 3208 12.82 17.16 0.474 1.161 95.5 2.0 2011 5.68 9.02 0.494 1.170 84.1 2.5 3.0 1717 7.01 9.29 0.551 1.425 90.0 5.0 1446 6.26 9.30 0.591 1.750 96.4 7.0 1454 11.76 15.14 0.697 2.005 134.1 2.0 3947 0.52 1.57 0.112 0.384 2.7 2.5 3726 0.67 1.46 0.122 0.398 3.3 2.0 306 1.12 1.85 0.278 0.668 10.6 2.5 316 1.21 1.85 0.304 0.694 11.5 3.0 295 1.23 1.69 0.333 0.702 10.4 5.0 160 2.15 2.21 0.493 0.890 14.6 7.0 160 3.17 2.04 0.620 1.038 14.7 2.0 2223 1.90 4.03 0.313 0.824 16.4 2.5 2185 3.09 5.12 0.457 1.051 32.4 5.0 2153 2.51 4.10 0.444 0.995 26.4 7.0 2180 3.70 4.55 0.497 1.195 28.9 block can give substantial gains in execution time with sometimes a small penalty in storage, although occasionally the storage can be less as well. 6. Performance. In this section, we compare the performance of the combined unifrontal/multifrontal method (MA38) with the unsymmetric-pattern multifrontal 8 T. A. DAVIS AND I. S. DUFF method (UMFPACK Version 1.1 [8, 10]), a general sparse matrix factorization algorithm that is not based on frontal matrices (MA48, [19, 20]), the frontal method (MA42, [15, 22]), and the symmetric-pattern multifrontal method (MUPS [2]). All methods can factorize general unsymmetric matrices and all use dense matrix kernels to some extent [12]. We tested each method on a single processor of a CRAY C-98, although MUPS is a parallel code. Version 6.0.4.1 of the Fortran compiler (CF77) was used. Each method (except MA42, which we discuss later) was given 95 Mw of memory to factorize the matrices listed in Table 5.1. Each method has a set of input parameters that control its behavior. We used the recommended defaults for most of these, with a few exceptions that we now indicate. By default, three of the five methods (MA38, UMFPACK V1.1, and MA48) preorder a matrix to block triangular form (always preceded by finding a maximum transversal [14]), and then factorize each block on the diagonal [16]. This can reduce the work for unsymmetric matrices. We did not perform the preordering, since MA42 and MUPS do not provide these options. One matrix (lhr71) was so ill-conditioned that it required scaling prior to its factorization. The scale factors were computed by the Harwell Subroutine Library routine MC19A [6]. Each row was then subsequently divided by the maximum absolute value in the row (or column, depending on how the method implements threshold partial pivoting). No scaling was performed on the other matrices. By default, MUPS preorders each matrix to maximize the modulus of the smallest entry on the diagonal (using a maximum transversal algorithm). This is followed by a minimum degree ordering on the nonzero pattern of A +A T . For MA42, we first preordered the matrix to reduce its profile using a version of Sloan's algorithm [30, 21]. MA42 is able to operate both in-core and out-of- core, using direct access files. It has a finite-element entry option, for which it was primarily designed. We used the equation entry mode on these matrices, although some matrices are obtained from finite-element calculations. We tested MA42 both in-core and out-of-core. The CPU time for the out-of-core factorization was only slightly higher than the in-core factorization, but the memory usage was much less. We thus report only the out-of-core results. Note that the CPU time does not include any additional system or I/O device time required to read and write the direct access files. The only reliable way to measure this is on a dedicated system. On a CPU-bound multiuser system, the time spent waiting for I/O would have little effect on total system throughput. The symbolic analysis phase in MA42 determines a minimum front size (k-by-l, which assumes no increase due to numerical pivoting), and minimum sizes of other integer and real buffers (of total size b, say). We gave the numerical factorization phase a working array of size 2k-by-2l, and a buffer size of 2b, to allow for numerical pivoting. In our tables, we show how much space was actually used. The disk space taken by MA42 is not included in the memory usage. The results are shown in Table 6.1. For each matrix, the tables list the numerical factorization time, total factorization time, number of nonzeros in L+U (in millions), amount of memory used (in millions of words), and floating-point operation count (in millions of operations) for each method. The total time includes preordering, symbolic analysis and factorization, and numerical factorization. The time to compute the scale factors for the lhr71 matrix is not included, since we used the same scaling algorithm for all methods. For each matrix, the lowest time, memory usage, or operation count is shown in bold. We compared the solution vectors, x, for each method. We found A COMBINED UNIFRONTAL/MULTIFRONTAL METHOD 9 that all five methods compute the solutions with comparable accuracy, in terms of the norm of the residual. We do not give the residual in Table 6.1. All five codes have factorize-only options that are often much faster than the combined analysis+factorization phase(s), and indeed the design criterion for some codes (for example MA48) was to minimize factorize time even if this caused an increase in the initial analyse time. For MA42, however, the analysis is particularly simple so normally its overhead is much smaller than for the other codes. MUPS is shown as failing twice (on shyy161 and lhr71 which are both nearly singular). In both cases, numerical pivoting caused an increase in storage requirements above the 95 Mw allocated. MA42 failed to factorize the shyy161 matrix because of numerical problems. It ran out of disk space during the numerical factorization of the lhr71 matrix (about 5.8 Gbytes were required). For this matrix, we report the estimates of the number of entries in the LU factors and the memory usage, as reported by the symbolic analysis phase of MA42. Since the codes being compared all offer quite different capabilities and are designed for different environments, the results should not be interpreted as a direct comparison between them. However, what we would like to highlight is the improvements that our new technique brings to the UMFPACK code and that the new code is at least comparable in performance with other sparse matrix codes. The peak performance of MA38 is 607 Mflops for the numerical factorization of the psmigr 1 matrix (compared with the theoretical peak of about 1 Gflop for one C-90 processor). Our new code requires less storage than the original UMFPACK code and sometimes it requires less than half the storage. In execution time for both analyse/factor and only, it is generally faster (sometimes by nearly a factor of two) and when it is slower than the original UMFPACK code it is so by at most about 13% (for the hydr1 matrix). Over all the codes, MA38 has the fastest analyse+factorize time for five out of the ten matrices. Except for one matrix (lns 3937) it never takes more than twice the time of the fastest method. Its memory usage is comparable to the other in-core methods. MA42 can typically factorize these matrices with the least amount of core memory. MA42 was originally designed for finite-element entry and so is not optimized for equation entry. The experiments are run on only one computer which strongly favors direct addressing and may thus disadvantage MA48 which does not make such heavy use of the Level 3 BLAS. 7. Summary . We have demonstrated how the advantages of the frontal and multifrontal methods can be combined. The resulting algorithm performs well for matrices from a wide range of disciplines. We have shown that the combined unifrontal/multifrontal method gains over our earlier code because it avoids more indirect addressing and generally performs eliminations on larger frontal matrices. Other differences between UMFPACK Version 1.1 and the new code (MA38, or UMFPACK Version 2.0) include an option of overwriting the matrix A with its LU factors, printing of input and output parameters, a removal of the extra copy of the numerical values of A, iterative refinement with sparse backward error analysis [4], more use of Level 3 BLAS within the numerical factorization routine, and a simpler calling interface. These features improve the robustness of the code and result a modest decrease in memory usage. The combined unifrontal/multifrontal method is available as the Fortran 77 codes, UMFPACK Version 2.0 in Netlib [13], 1 and MA38 in Release 12 of the Harwell 1 UMFPACK Version 2.0, in Netlib, may only be used for research, education, or benchmarking T. A. DAVIS AND I. S. DUFF Subroutine Library [3]. 2 8. Acknowledgements . We would like to thank Nick Gould, John Reid, and Jennifer Scott from the Rutherford Appleton Laboratory for their helpful comments on a draft of this report. --R An approximate minimum degree ordering algorithm Vectorization of a multiprocessor multifrontal code Solving sparse linear systems with sparse backward error A linear time implementation of the reverse Cuthill-Mckee algorithm On the automatic scaling of matrices for Gaussian elimination Reducing the bandwidth of sparse symmetric matrices Users' guide to the unsymmetric-pattern multifrontal package (UMFPACK Distribution of mathematical software via electronic mail On algorithms for obtaining a maximum transversal An implementation of Tarjan's algorithm for the block triangularization of a matrix The design of MA48 The use of profile reduction algorithms with a frontal code The use of multiple fronts in Gaussian elimination Computer Solution of Large Sparse Positive Definite Systems A frontal solution program for finite element analysis The multifrontal method for sparse matrix solution: Theory and Practice The multifrontal method for sparse matrix solution: Theory and practice Comparative analysis of the Cuthill-Mckee and the reverse Cuthill-Mckee ordering algorithms for sparse matrices An algorithm for profile and wavefront reduction of sparse matrices frontal techniques for chemical process simulation on supercomputers Supercomputing strategies for the design and analysis of complex separation systems --TR Distribution of mathematical software via electronic mail Sparse matrix test problems The evolution of the minimum degree ordering algorithm A set of level 3 basic linear algebra subprograms The multifrontal method for sparse matrix solution Sparse matrix methods for chemical process separation calculations on supercomputers The design of a new frontal code for solving sparse, unsymmetric systems The design of MA48 Stable finite elements for problems with wild coefficients An Approximate Minimum Degree Ordering Algorithm An Unsymmetric-Pattern Multifrontal Method for Sparse LU Factorization The Multifrontal Solution of Indefinite Sparse Symmetric Linear Computer Solution of Large Sparse Positive Definite Reducing the bandwidth of sparse symmetric matrices A Supernodal Approach to Sparse Partial Pivoting --CTR Eero Vainikko , Ivan G. Graham, A parallel solver for PDE systems and application to the incompressible Navier-Stokes equations, Applied Numerical Mathematics, v.49 n.1, p.97-116, April 2004 J.-R. de Dreuzy , J. Erhel, Efficient algorithms for the determination of the connected fracture network and the solution to the steady-state flow equation in fracture networks, Computers & Geosciences, v.29 n.1, p.107-111, February Kai Shen, Parallel sparse LU factorization on different message passing platforms, Journal of Parallel and Distributed Computing, v.66 n.11, p.1387-1403, November 2006 Timothy A. Davis , John R. Gilbert , Stefan I. Larimore , Esmond G. Ng, A column approximate minimum degree ordering algorithm, ACM Transactions on Mathematical Software (TOMS), v.30 n.3, p.353-376, September 2004 Anshul Gupta, Recent advances in direct methods for solving unsymmetric sparse systems of linear equations, ACM Transactions on Mathematical Software (TOMS), v.28 n.3, p.301-324, September 2002 Xiaoye S. Li , James W. Demmel, SuperLU_DIST: A scalable distributed-memory sparse direct solver for unsymmetric linear systems, ACM Transactions on Mathematical Software (TOMS), v.29 n.2, p.110-140, June Timothy A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method, ACM Transactions on Mathematical Software (TOMS), v.30 n.2, p.165-195, June 2004 J. Candy , R. E. Waltz, An Eulerian gyrokinetic-Maxwell solver, Journal of Computational Physics, v.186 A. Guardone , L. Vigevano, Finite element/volume solution to axisymmetric conservation laws, Journal of Computational Physics, v.224 n.2, p.489-518, June, 2007

frontal methods;sparse unsymmetric matrices;linear equations;multifrontal methods

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On Two Interior-Point Mappings for Nonlinear Semidefinite Complementarity Problems.

Extending our previous work ( Monteiro and Pang 1996), this paper studies properties of two fundamental mappings associated with the family of interior-point methods for solving monotone nonlinear complementarity problems over the cone of symmetric positive semidefinite matrices. The first of these maps lead to a family of new continuous trajectories which include the central trajectory as a special case. These trajectories completely "fill up" the set of interior feasible points of the problem in the same way as the weighted central paths do the interior of the feasible region of a linear program. Unlike the approach based on the theory of maximal monotone maps taken by Shida and Shindoh (1996) and Shida, Shindoh, and Kojima (1995), our approach is based on the theory of local homeomorphic maps in nonlinear analysis.

Introduction In a series of recent papers (see Kojima, Shida and Shindoh 1995a, Kojima, Shida and Shindoh 1995b, Kojima, Shida and Shindoh 1996, Kojima, Shindoh and Hara 1997, Shida and Shindoh 1996, Shida, Shindoh and Kojima 1995, Shida, Shindoh and Kojima 1996), Hara, Kojima, Shida and Shindoh have introduced the monotone complementarity problem in symmetric matrices, studied its properties, and developed interior-point methods for its solution. A major source where this problem arises is a convex semidefinite program which has in the last couple years attracted a great deal of attention in the mathematical programming literature (see Alizadeh 1995, Alizadeh, Hae- berly and Overton 1994, Alizadeh, Haeberly and Overton 1995, Goldfarb and Scheinberg 1996, Luo, Sturm and Zhang 1996a, Luo, Sturm and Zhang 1996b, Monteiro 1995, Nesterov and Nemirovskii 1994, Nesterov and Todd 1995, Nesterov and Todd 1997, Potra and Sheng 1995, Ramana, Tun-cel and Wolkowicz 1997, Shapiro 1997, Vandenberghe and Boyd 1996, Zhang 1995). Our goal in this paper is to extend our previous work Monteiro and Pang (1996) to the monotone complementarity problem in symmetric matrices and to apply the results to a convex semidefinite program. Although the present analysis is significantly more involved (for one thing, a great deal of matrix-theoretic tools is employed), we obtain a large set of conclusions that extend those in Monteiro and Pang (1996). Similar to the motivation in this reference (which the reader is advised to consult), we undertook the present study in order to gain an in-depth understanding of the interior point methods, in particular the limiting behavior of certain solution trajectories, both new and known, for solving these important mathematical programs defined on the cone of positive semidefinite matrices. Let M n denote the vector space of n \Theta n matrices with real entries; let S n denote the subspace of M n consisting of the symmetric matrices; and let ! m denote the m-dimensional Euclidean space of real vectors. Let S n denote the subset of S n consisting of the positive semidefinite matrices. m be a given mapping which we assume to be continuous on S n (which is isomorphic to the Euclidean space ! n(n+1)=2 ). The complementarity problem which we shall study in this paper is to find a triple (X; Y; z) 2 S n \Theta S n \Theta ! m satisfying F (X; Y; (1) It is known (see the cited references) that there are several equivalent ways to represent the complementarity conditions in this problem; namely, (2) where "tr" denotes the trace of a matrix. Associated with these equivalent conditions, we can define mappings that will help us understand the limiting behavior of certain path-following interior point methods for solving the problem (1). In this paper, we focus on the first two conditions and the associated mappings. Specifically, a main objective of this paper is to examine properties of the mapping defined by Associated with this map, we define the set ++ \Theta S n We will give conditions on the mapping F which guarantee that the system has the following properties: (P1) it has a solution for every (A; B) 2 S n (P2) the solution, denoted (X(A; B); Y (A; B); z(A; B)), is unique when if a sequence ++ \Theta F (U \Theta ! m ) converges to a limit (A1 ++ \Theta F (U \Theta ! m ), then the sequence converges to the limit if a sequence ++ \Theta F (U \Theta ! m ) converges to a limit (A1 then the sequence implies the existence of a solution of (1) when We also consider the map ~ ~ and prove under suitable conditions that for every - 0 and B 2 F (S n ++ \Theta S n ~ has a solution, which is is unique when - ? 0. Clearly, this latter result implies that (1) has a solution under the weaker assumption that 0 2 F (S n ++ \Theta S n A major difference between the mappings H and ~ H lies in their ranges. Comparing the systems (5) and (7) with varying right-hand sides, we see that we are able to obtain results for a broader class of solution trajectories in the case of H than ~ H , with the right-side matrix A in (5) being an arbitrary symmetric matrix versus the restriction to a positive multiple of the identity matrix in (7). The main tool used to derive the above results is a known theory of local homeomorphic maps summarized in Monteiro and Pang (1996) that has been applied to a standard mixed complementarity problem defined on ! n As judged from the papers on semidefinite programming, the extension of the previous analysis Monteiro and Pang (1996) to the case of complementarity problems in symmetric positive semidefinite matrices is nontrivial; this thus necessitates our present investigation. The complementarity problem (1) arises as the set of first-order necessary optimality conditions for the following nonlinear semidefinite program (see Shapiro 1997): minimize '(x) subject to G(x) 2 \GammaS n are given smooth mappings. Indeed, it is well-known (see for example Shapiro 1997) that, under a suitable constraint qualification, if x is a local optimal solution of the semidefinite program, then there must exist j such that is the Lagrangian function defined by where denotes the scalar product of A; Letting F (U; V; x; r x L(x; U; we see that the first-order necessary optimality conditions for problem (8) are exactly in the form of the complementarity problem (1). For the mapping F in (11), our principal result is Theorem 4 which shows that under some fairly standard assumptions on the functions ', G, and h, the mapping H defined by (3) maps U \Theta ! m+p homeomorphically onto the convex set S n ++ \Theta F (U \Theta ! m+p ); moreover, H(S n Before proceeding further, we should relate the formulation (1) with the formulation in Kojima, Shida and Shindoh (1995a), Kojima, Shida and Shindoh (1995b), Kojima, Shida and Shindoh (1996), Kojima, Shindoh and Hara (1997), Shida and Shindoh (1996), Shida, Shindoh and Kojima (1995), Shida, Shindoh and Kojima (1996). Specifically by defining the set the problem (1) is of the form which is exactly the central problem studied in the cited references. So in principle the results in these references (especially Shida and Shindoh 1996, Shida, Shindoh and Kojima 1995 which deal with nonlinear problems) could be applied to the problem (1), provided that we can demonstrate that F is a "maximal monotone" subset of S n \Theta S n . By assuming that a certain monotonicity condition on F holds everywhere on S n \Theta S n \Theta ! m , the maximal monotonicity of F follows from Theorem 8 in Monteiro and Pang (1996). In this case, the existence and continuity of the solutions of system (7) follow from the theory presented in Shida, Shindoh and Kojima (1995). However, the requirement that the function F satisfy the monotonicity condition everywhere on S n \Theta S n \Theta ! m is quite restrictive; for example, the function F given by (11) satisfies the monotonicity condition only on the set S n Our analysis assumes that the monotonicity condition holds only on the latter set, and thus is valid for the special map F given by (11). Incidentally, under the weaker monotonicity condition imposed on F , we are able to establish that the set F defined by (12) is maximal monotone only in a restricted sense, namely with respect to the set U (see Section 6). Section 7 of this paper establishes the results of Shida, Shindoh and Kojima (1995) about the existence and continuity of the solutions to the system (7) under the weaker monotonicity condition on F . As far as the system (5) is concerned, properties (P1)-(P4) are shown to be valid here for the first time. The following notation is used throughout this paper. The symbols - and - denote, respectively, the positive semidefinite and positive definite ordering over the set of symmetric matrices; that is, for positive semidefinite, and X - Y (or Y OE X) means positive definite. We let M n ++ denote the set of matrices that respectively. Given U 2 M n and a differentiable function denote the m-vector whose i-th entry is (@G(x)=@x i is the partial derivative of G with respect to x i . Finally we let Preliminaries In this section, we introduce some further notation and describe some background concepts and results needed for the subsequent developments. The section is divided into two subsections. The first subsection summarizes a theory of local homeomorphic maps defined on metric spaces; the discussion is very brief. We refer the reader to Section 2 of Monteiro and Pang (1996), Chapter 5 of Ortega and Rheinboldt (1970), and Chapter 3 of Ambrosetti and Prodi (1993) for a thorough treatment of this theory. The second subsection introduces the key conditions on the mapping F in (1) and establishes some basic properties of the set U defined in (4). 2.1 Local homeomorphic maps If M and N are two metric spaces, we denote the set of continuous functions from M to N by C(M;N) and the set of homeomorphisms from M onto N by Hom(M;N ). For G 2 C(M;N ), Eg. Given such that G(D) ' E, the restricted mapping ~ defined by ~ denoted by Gj (D;E) ; if then we write this ~ G simply as GjD . We will also refer to Gj (D;E) as "G restricted to the pair (D; E)", and to GjD as "G restricted to D". The closure of a subset E of a metric space will be denoted by cl E. Any continuous function from a closed interval of the real line ! into a metric space will be called a path. We say that partition of the set V if space M is said to be connected if there exists no partition both O 1 and O 2 are non-empty and open. A metric space M is said to be path-connected if for any two points there exists a path M such that It is well-known that any path-connected metric space is connected; the converse however does not always hold. A metric space M is said to be simply-connected if it is path-connected and for any there exists a continuous mapping ff : [0; 1] \Theta [0; 1] !M such that ff(s; A subset C of a vector space is said be star-shaped if there exists c 0 2 C such that the line segment connecting c 0 to any other point in C is contained entirely in C. Clearly, every star-shaped set in a normed vector space is simply-connected. In the rest of this subsection, we will assume that M and N are two metric spaces and that 1 The mapping G 2 C(M;N) is said to be proper with respect to the set E ' N if is compact for every compact set K ' E. If G is proper with respect to N , we will simply say that G is proper. The proofs of the following three results can be found in Section 2 and the Appendix of Monteiro and Pang (1996). Proposition 1 Assume that G : M ! N is a local homeomorphism. If M is path-connected and N is simply-connected then G is proper if and only if G 2 Hom(M;N). Proposition be given sets satisfying the following conditions: GjM 0 is a local homeomorphism, G(M Assume that G is proper with respect to some set E such that N 0 ' E ' N . Then H restricted to the pair (M a proper local homeomorphism. If, in addition, N 0 is connected, then G(M 0 cl N 0 . Proposition 3 Let M be a path-connected metric space. Assume that is a local homeomorphism and that G \Gamma1 ([y is compact for any pair of points y and G(M) is convex. 2.2 Some key concepts We introduce the key conditions on the map F that will be assumed throughout the paper. mapping J(X; Y; z) defined on a subset dom (J) of M n \Theta M n \Theta ! m is said to be Y )-equilevel-monotone on a subset V ' dom (J) if for any (X; Y; z) 2 V and (X such that J(X; Y; dom (J), we will simply say that J is (X; Y )-equilevel-monotone. In the following two definitions, we assume that W , Z and N are three normed spaces and that OE(w; z) is a function defined on a subset of W \Theta Z with values in N . Definition 3 The function OE(w; z) is said to be z-bounded on a subset V ' dom (OE) if for every sequence f(w k ; z k )g ae V such that fw k g and fOE(w k ; z k )g are bounded, the sequence fz k g is also bounded. When dom (OE), we will simply say that OE is z-bounded. Definition 4 The function OE(w; z) is said to be z-injective on a subset V ' dom (OE) if the following implication holds: (w; dom (OE), we will simply say that OE is z-injective. In the next result, we collect a few technical facts which will be used later. Parts (a) and (b) are well-known consequences of the self-dual property of the cone S n . Their proofs are omitted. Parts (c), (d), and (e) pertain to properties of the set U . Lemma 1 The following statements hold: (a) U - 0 if and only if U ffl V - 0 for every V - 0; (b) if U - 0, (c) (d) if (X; Y then (e) the set U is star-shaped, thus simply-connected. Proof. We give a simple proof of (c). Let X; Y 2 S n be such that XY +Y X - 0. Then XY must be a P-matrix, and thus it has positive determinant. Hence X and Y are both nonsingular, and thus positive definite. By applying the proof of Theorem 3.1(iii) in Shida, Shindoh and Kojima (1996), we can establish part (d). The details are omitted. For part (e), observe that (I ; I) 2 U . Now let (X; Y ) be an arbitrary element in U . We will show that the line segment connecting (I ; I) to (X; Y ) is contained in U , from which part (e) follows. Indeed, any point on this segment is of the form (X It is easy to see ++ is a convex cone, it follows that (X t We have thus shown that U is star-shaped. The next lemma gives a consequence of the concepts introduced in the previous definitions. m be a continuous map and let H m be the map defined by (3). Assume that the map F is (X; Y )-equilevel-monotone and z-bounded. If the map H restricted to U \Theta ! m is a local homeomorphism, then H is proper with respect to S n \Theta F (U \Theta ! m ). Proof. Let K be a compact subset of S n \Theta F (U \Theta ! m ). We will show that H \Gamma1 (K) is compact, from which the result follows. The continuity of H implies that H \Gamma1 (K) is a closed set. Hence, it remains to show that H \Gamma1 (K) is bounded. Indeed, suppose for contradiction that there exists a sequence is compact and we may assume without loss of generality that there exists F 1 2 F (U \Theta ! m ) such that Clearly, we have F such that the set contains is an open set and every local homeomorphism maps open sets onto open sets, it follows from Lemma 4 that H(N1 ), and hence F (N1 ), is an open set. Thus, by (13), we conclude that for all k sufficiently large, say k hence that F (X monotone, we have (X This inequality together with fact that ( ~ imply for every k - k 0 . Using the fact that fH(X k ; Y k ; z k )g ae K and K is bounded, we conclude that g, is bounded. This fact together with the above inequality implies that the sequences fX k g and fY k g are bounded. Since lim k!1 k(X must have lim k!1 kz k is z-bounded, we conclude that lim k!1 1, thereby contradicting (13). 3 The Affine Case Beginning in this section, we develop our main theory for the complementarity problem (1) and the associated mapping H defined in (3). This section pertains to the case where F is affine; the treatment of the general case of a nonlinear map F is given in the next section. Apart from the fact that an affine map F considerably simplifies the analysis, through this case, we will be able to obtain a technical result (Lemma 5) that will play an important role in the analysis of a nonlinear F , which is the subject of the next section. The derivation of this technical lemma is definitely the main reason for us to consider the case of an affine map separately. We begin with a lemma that contains some elementary properties of affine maps. Lemma 3 Assume that F is an affine map and let F its linear part. Then the following statements hold: (a) F is (X; Y )-equilevel-monotone if and only if (b) F is z-injective if and only if (c) F is z-injective if and only if F is z-bounded. Proof. The proofs of (a) and (b) are straightforward. We next prove (c). Assume first that F is z-injective. To show that F is z-bounded assume for contradiction that f(X k ; Y k ; z k )g is a sequence in S n \Theta S n \Theta ! m such that f(X k are bounded and lim k!1 kz k By passing to a subsequence if necessary, we may assume that lim k!1 z k =kz k 1. Hence, we obtain z k Since \Deltaz 6= 0, this contradicts the fact that F is z-injective. Hence, the "only if " part of (c) follows. Assume now that F is not z-injective. Then there exists \Deltaz and \Deltaz 6= 0. The sequence f(X defined by X has the property that F 0 (X are bounded and lim k!1 kz k showing that F is not z-bounded. The 'if ' part of c) follows. The next lemma is an important step toward the main result in the affine case. Lemma 4 Assume that F is an affine map which is (X; Y )-equilevel- monotone and z-injective. Then the map H restricted to U \Theta ! m is a local homeomorphism. Proof. Since U \Theta ! m is an open set, it is sufficient to show that the derivative map H 0 (X; m is a isomorphism for every (X; Y; z) 2 U \Theta ! m . For this purpose, fix any (X; Y; z) 2 U \Theta ! m . Since H 0 (X; Y; z) is linear and is a map between identical spaces, it is enough to show that Indeed, assume that the left-hand side of the above implication holds. By the definition H , we have By (17) and Lemma 3(a), we have \DeltaX ffl \DeltaY - 0. In view of Lemma 1(d), this relation together with (18) imply that This conclusion together with (17) and Lemma 3(b) imply that We have thus shown that the implication (16) holds and the result follows. We are now ready to present the main result of this section. Theorem 1 Assume that F is an affine map which is (X; Y )-equilevel- monotone and z-injective. Then, the following statements hold: (a) H maps U \Theta ! m homeomorphically onto S n (b) H(S n Proof. The proof of the theorem follows from Proposition 2 as follows. Let M j S n show that these sets together with the map G j H j M satisfy the assumptions of Proposition 2. Indeed, first observe that GjM 0 is a local homeomorphism due to Lemma 4. The assumption that F is (X; Y )-equilevel-monotone and z-injective together with Lemma 3(c) and Lemma 4 imply that the assumptions of Lemma 2 are satisfied. Hence, the conclusion of this lemma implies that is proper with respect to E. Also the set H(M due to the fact that To show that H(MnM 0 assume for contradiction that there exists (X; Y; z) 2 MnM 0 such that H(X; Y; z) 2 N 0 . Then, by definition of H and the sets M , M 0 and N 0 , it follows that (XY +Y X)=2 2 S n ++ and (X; Y . But this contradicts Lemma 1(c), and hence we must have Lemma 1, the set U , and hence U star-shaped. Since the image of a star-shaped set under an affine map is star-shaped, it follows that F (U \Theta ! m ), and hence N 0 , is star-shaped. Since every star-shaped set is simply- connected, we conclude that N 0 is simply-connected. Hence, by Proposition 2 and the fact that restricted to the pair (M 0 is a proper local homeomorphism and that H(cl U \Theta ! m The theorem now follows from the two last conclusions, the fact that M 0 is path-connected, N 0 is simply-connected, and Proposition 1. We use the above result to prove the following very important technical lemma. be elements of U such that Proof. Let \DeltaX assume for contradiction that either \DeltaX 6= 0 or \DeltaY 6= 0. Without loss of generality, we may assume that \DeltaX 6= 0. We claim that there exists a linear \DeltaY and is monotone, that is W ffl M(W every W 2 S n . Indeed, if then the zero map satisfies the conditions of the claim. Assume then that \DeltaY 6= 0. We consider two cases depending on whether \DeltaX ffl \DeltaY ? 0 or \DeltaX ffl Consider first the case in which \DeltaX ffl \DeltaY ? 0. In this case, the subspace generated by \DeltaX and the subspace L orthogonal to \DeltaY span S n . Clearly, there exists a unique linear such that M (\DeltaX can be uniquely written as for every W 2 S n . Hence, M is a monotone map. Consider now the case in which \DeltaX ffl Let L 1 denote the subspace orthogonal to both \DeltaX and \DeltaY . Clearly, there exists a unique map \DeltaX; and M(V Any W 2 S n can be uniquely written as Then, we obtain \DeltaX for every W 2 S n . Hence, M is a monotone map. We have thus shown that the claim holds. Consider now the defined by F (X; Y . Using the fact that M is a monotone map and that M (\DeltaX see that F satisfies the assumptions of Theorem 1 (with By Theorem 1(a), it follows that the associated map H (with restricted to U is one-to-one. Moreover, (19) and the relation F (X imply that H(X last two conclusions together with the fact that (X imply that (X 4 The Nonlinear Case In this section we establish results for nonlinear maps F which are similar to Theorem 2. We also consider the case of a nonlinear map that arises from the mixed nonlinear complementarity problems in symmetric matrices. Theorem 2 Assume that F : S n m is a continuous map which is (X; Y )- equilevel-monotone, z-bounded (on S n z-injective on S n ++ \Theta S n Then, the following statements hold for the mapping H given by (3): (a) H is proper with respect to S n \Theta F (U \Theta ! m ); homeomorphically onto S n (c) H(S n Proof. The proof is close to the one given for Theorem 1. It consists of showing that the sets together with the map G j H j M satisfy the assumptions of Proposition 2. But instead of using Lemma 4 to show that GjM 0 is a local homeomorphism, we use Lemma 5 to prove that H j M 0 maps M 0 homeomorphically onto H(M 0 ). Since is a continuous map from an open subset of the vector space S n \Theta S n \Theta ! m into the same space, by the domain invariance theorem it suffices to show that H j M 0 is one-to-one. For this purpose, assume that z) for some ( - Then, by the definition of H , we have F ( - z) and - Y ~ X. Since F is (X; Y )-equilevel-monotone, we conclude that ( - Hence, by Lemma 5, we have Y ). This implies that F ( - and by the z-injectiveness of F on S n ++ \Theta S n ++ \Theta S n We have thus proved that H j M 0 maps M 0 homeomorphically onto H(M 0 ). To prove (b), it suffices to show that . As in Theorem 1, we can verify that H(M 0 Using the assumption that F is (X; Y )-equilevel-monotone and z-bounded and the fact that H j M 0 is a homeomorphism onto H(M 0 ), it follows from Lemma 2 that H is proper with respect to holds. Using the fact that U 0 is star-shaped and thus path- connected, we easily see that N 0 is path-connected, and hence connected. Hence, by Proposition 2 and the fact that H(M 0 and H(cl U 0 \Theta ! m Remark. The z-injectiveness of F is assumed only on S n ++ \Theta S n (and not on S n In the application to convex semidefinite programming to be discussed in the next section, we show that under appropriate convexity assumptions, the mapping F defined by (11) satisfies the former z-injectiveness property; thus Theorem 2 is applicable. Nevertheless, we do not know if this special map F is z-injective on the larger set. Theorem 2 establishes the claimed properties (P1)-(P4) of the map H stated in the Introduction. Indeed, (P1) follows from conclusion (c); (P2) and (P3) follow from (b); and (P4) follows from (a). In what follows, we give two important consequences of the above theorem, assuming that first one, Corollary 1, has to do with the central path for the semidefinite complementarity problem (1); the second one, Corollary 2, is a solution existence result for the same problem. Corollary 1 Assume that F : S n m is a continuous map which is (X; Y )- equilevel-monotone, z-bounded (on S n z-injective on S n ++ \Theta S n further that that ++ for every t 2 (0; 1]. Then there exist (unique) paths ++ and z : (0; Moreover, every accumulation point of (X(t); Y (t); z(t)) as t tends to 0 is a solution of the complementarity problem (1). Corollary 2 Assume that F : S n m is a continuous map which is (X; Y )- equilevel-monotone, z-bounded (on S n z-injective on S n ++ \Theta S n . If there exists m such that F (X , the system2 (XY has a solution, which is unique when A 2 S n ++ . Next, we introduce a strengthening of the equilevel-monotonicity concept and show that under this strengthened monotonicity property, the image F (U \Theta ! m ) is a convex set in S n \Theta ! m . Definition 5 The )-everywhere-monotone if there exist continuous functions OE that c(R; (R where Obviously, if F is (X; Y )-everywhere-monotone then F is (X; Y )-equilevel-monotone. Theorem 3 Suppose that F : S n m is a continuous map which is (X; Y )- everywhere monotone, z-bounded (on S n z-injective on S n ++ \Theta S n in addition to statements (a), (b) and (c) of Theorem 2, it holds that the set F (U \Theta ! m ) is convex. Proof. It suffices to establish the convexity of F (U \Theta ! m ). The proof is based on Proposition 3. Consider the set M j U \Theta ! m and the map G j H j U \Theta! m . With M and G defined this way, the convexity of F (U \Theta ! m ) follows from Proposition 3, provided that M and G satisfy the hypotheses of the corollary. The rest of proof is devoted to the verification of these hypotheses. First observe that G is a local homeomorphism due to Lemma 4. It remains to show that for any two triples ( ~ , the set G \Gamma1 (E) is compact, where E is the line segment connecting the points H( ~ z). It is enough to show that any sequence f(X has an accumulation point in G \Gamma1 (E). Indeed, let f(X every k - 0 we have Y ~ for some sequence f- k g ae [0; 1], where ~ z) and F k j F (X By the (X; Y )-everywhere monotonicity of F and (21), it follows that Hence, Similarly, we can show that Multiplying (22) by - k and (23) by and adding, we obtain By (20) and (21) and the fact that f- k g is bounded, we see that the sequences fX k ffl Y k g and are bounded. This observation, the fact that f- k g is bounded and the function c(\Delta; \Delta) is continuous imply that the right hand side of (24), and hence its left hand side, is bounded. A simple argument now shows that fX k g and fY k g are also bounded. This conclusion, the fact that is bounded and the map F is z-bounded imply that fz k g is bounded. Hence, the sequence is bounded and must have an accumulation point ( - z). It remains to show that closed since H is continuous and both the domain of H and the set E are closed. The closeness of H and the fact that f(X k imply that ( - Using (20) and the fact that ( ~ Y ) are in U , we easily see that ( - ++ . Morever, since we see that ( - . By Lemma 1(c), we conclude that ( - We have thus shown that ( - Consider now the special case in which the mapping F : S n has the following structure: F (X; Y; z) jB @ for some continuous mapping / . The following result gives conditions on / for the mapping F to satisfy all the assumptions of Theorem 3. This result will be used in the next section for the map F given by (11). Proposition 4 The following statements hold: (a) F is (X; Y )-everywhere-monotone on S n is a monotone mapping, that is \Theta for every (X; z); (X (b) F is z-injective on S n ++ \Theta S n is z-injective on S n (c) F is z-bounded on S n is z-bounded on S n Proof. We first prove (a). To prove that the mapping H is (X; Y )-everywhere-monotone, define OE(X; Y; z) j (X; \Gammaz); and c j 0: Using these two equalities and the fact that / is a monotone mapping, we obtain \Theta which implies (R The proofs of (b) and (c) are straightforward and therefore we omit the details. 5 Convex Semidefinite Programming In this section, we discuss the application of Theorem 3 to the mapping F given by (11). We wish to specify some conditions on the functions ', G, and h in order for the resulting function F to satisfy the assumptions of this theorem, and thus for the conclusions of the theorem to hold. The following correspondence of variables should be used to cast the mapping F defined by (11) into the form of the general mapping F considered in Section 4: (U; V ) Our first goal is to give a sufficient condition for the mapping F to be (X; Y )-monotone on As in Shapiro (1997), we say that the mapping G positive semidefinite convex (psd-convex) if The following technical result is useful for the analysis of this section. Lemma 6 Assume that G is an affine function. Then the following statements hold: (a) for every W 2 S n , the function x (b) the function x (c) if the set X j fx nonempty and bounded then, for every A 2 S n and fi 2 !, the set fx Proof. (a) Using (25) and Lemma 1(b), we obtain (b) Using (25), the fact that - max (\Delta) is a homogeneous convex function over the set of symmetric matrices and the implication U (c) Consider the function defined by v(x) j maxf- max (G(x)); kh(x)kg for x Using (b), we see that v is a convex function. Moreover, the level set v \Gamma1 (\Gamma1; 0] is nonempty and bounded since it coincides with the set X . By a well-known property of convex functions (Corollary 8.7.1 of Rockafellar 1970), it follows that every level set of v is bounded. Since fx In the next three lemmas we study properties of the mapping / defined by \GammaG(x) r x L(x; U; where the map L is defined in (10). Lemma 7 Suppose that continuously differentiable and convex, G continuously differentiable and psd-convex and is an affine function. Then, the map defined by (26) is monotone in the sense of Proposition 4(a). Proof. We have to show that for any (U; x; j); (U m+p , there holds 0: (27) The assumption of the proposition implies that the functions L(\Delta; U; are convex Hence, we have Adding these two inequalities, using the definition of L and simplifying, we obtain (27). Lemma 8 Suppose that continuous differentiable and convex, G continuously differentiable and psd-convex and is an affine function such that the gradient matrix rh(x) has full column rank. Then, / defined by (26) is (x; j)-injective on S n any one of the conditions below is satisfied: (a) ' is strictly convex; (b) for every U 2 S n ++ , the map x strictly convex; (c) the feasible set X j fx bounded and each G ij (x) is an analytic function. Proof. Let U 2 S n ++ and (x; j); m+p be such that /(U; x; We have to show that (x; since rh(x) has full column rank, we have Hence, it suffices to show that Note that by (28) and the definition of L, we have The assumption of the lemma implies that the functions L(\Delta; U; are convex on ! m . Hence, the two bracketed expressions in the right hand side of (29) are nonnegative and, since their sum is zero, both of them must be equal to zero. In particular, we have If (a) or (b) holds then the function L(\Delta; U; j) is strictly convex and, by (30), we must have Consider case (c). Since h is affine, we have h(x together with (30) and the definition of L imply 0: Since '(\Delta) and U ffl G(\Delta) are convex functions on that the two bracketed expressions in the above relation are nonnegative. Hence, we conclude that U ffl G(x It is now easy to see that this relation together with the convexity of U ffl G(x) and the fact that imply that U ffl using the fact that G is psd-convex and Thus, we have we must have is an analytic function, this implies that every y in the set L j f-x !g. Clearly, since and h is affine, we also have that Hence, L is contained in the set fy which is bounded in view of Lemma 6. This implies that Lemma 9 Suppose that the function ' is continuously differentiable and psd-convex and is an affine function such that the (constant) gradient matrix rh(x) has full column rank. Suppose also that the set X defined in Lemma 8(c) is nonempty and bounded. Then the map / defined by (26) is (x; j)-bounded on Proof. Assume that f(U k is a sequence in S n m+p such that fU k g and f/(U k are bounded. By the definition of /, it follows that fG(x k )g, fr x are bounded. Hence, there exists ff ? 0 such that or equivalently, fx k g ae fx ffg. By Lemma 6(c), we conclude that is bounded. This fact together with the fact that fU k g and fr x are bounded imply that frh(x k )j k g is bounded. Since rh(x) is constant and has full column rank, we conclude that is bounded. We have thus shown that f(x k ; j k )g is bounded, and hence that / is (x; j)- bounded on S n Combining the above lemmas with Theorem 3, we obtain the following theorem which is the main result of this section. Theorem 4 Suppose that the function ' continuously differentiable and convex, continuously differentiable and psd-convex, is an affine function such that the (constant) gradient matrix rh(x) has full column rank and the feasible set X defined in Lemma 8(c) is bounded. If any one of the following conditions holds: (a) ' is strictly convex; (b) for every U 2 S n ++ , the map x strictly convex; (c) each G ij is an analytic function, then the following statements hold for the maps F and H given by (11) and (3), respectively: (i) H is proper with respect to S n \Theta F (U \Theta ! m+p ); homeomorphically onto S n ++ \Theta F (U \Theta ! m+p ); (iii) the set F (U \Theta ! m+p ) is convex; (iv) H(S n Proof. In view of Lemmas 7, 8 and 9, the map / given by (26) is monotone on S n (x; j)-injective on S n j)-bounded on S n By Proposition 4, it follows that the map F given by (11) is (U; V )-everywhere-monotone on S n j)-injective on ++ \Theta S n j)-bounded on S n . The result now follows immediately from Theorem 3. By Theorem 4, if 0 2 F (U \Theta ! m+p ) then the system has a solution for every A 2 S n this solution is unique when A 2 S n ++ . Clearly, a solution of this system when yields a feasible solution of (8) satisfying the stationary condition (9). We henceforth give conditions on problem (8) for 0 to be an element of F (U \Theta ! m+p ). It turns out that the existence of a Slater point for (8) is one of the conditions required by the next result. Suppose that the function ' is continuously differentiable and psd-convex, is an affine function. Suppose also that the feasible set X defined in Lemma 8(c) is bounded and there exists a vector ~ x 2 X such that is given by (11). Proof. consider the problem subject to G(x) - \Gamma"I By Lemma 6(c) and the assumption that X is bounded, it follows that the set of feasible solutions of (32) is compact. Since its objective function is defined and continuous over the whole feasible region, we conclude that (32) has an optimal solution x . Observe that (32) satisfies the Slater constraint qualification since G(~x) OE \Gamma"I , is psd-convex and h is affine. Hence, there exist multipliers U 2 S n Letting it follows from (33) and (34) that or equivalently, F (U It remains to show that (U Clearly, we have ++ \Theta S n ++ due to the fact that G(x ) OE 0. Moreover, using (34) we obtain Clearly, this implies that (U 6 Maximal Monotonicity In this section, we show that a mapping F satisfying the assumptions of Theorem 2 defines a family of set-valued maps from S n into itself which are maximal monotone in a restricted sense. For this purpose, we introduce some definition and notation. If M and N are two metric spaces, we shall denote a set-valued map A from M into subsets of N by A the graph of A is the set is said to be a monotone set if for every (V; W there holds !W , with V and W being subsets of M n , is called a monotone map if Gr(A) is a monotone set. The monotone map A or its graph Gr(A), is said to be maximal monotone with respect to a subset T ' V \Theta W if there exists no monotone set \Gamma ' T that properly contains T " Gr(A). If simply say that the map A !W is maximal monotone. The following result establishes the maximal monotonicity with respect to the set U of certain set-valued maps associated with a mapping F : S n satisfying the assumptions of Theorem 2. Theorem 5 Suppose that F : S n m is a continuous map which is (X; Y )- equilevel-monotone, z-bounded (on S n z-injective on S n ++ \Theta S n defined by is monotone, and maximal monotone with respect to U . Proof. Using the fact that F is (X; Y )-equilevel-monotone, we easily see that AB is monotone. To show that AB is maximal monotone with respect to U , let \Gamma ' U be a monotone set containing We have to show that Theorem 2(b) implies the existence of a triple m such that are in \Gamma, we conclude that In view of Lemma 5, the last relation together with the first equality of (35) and the fact that both are in U imply that (X have thus shown that and the result follows. Observe that the set F given by (12) is equal to the set Gr(AB ) with hence F is a maximal monotone set with respect to set U . We end this section by giving the following close version of Theorem 5 for maps F defined over the whole space S n \Theta S n \Theta ! m . Theorem 6 Suppose that F m is a continuous map which is (X; Y )- monotone, z-bounded and z-injective. Then, for every B 2 F (S n \Theta S n \Theta ! m ), the set-valued map AB defined in Theorem 5 is maximal monotone. Proof. This follows immediately from Theorem 8 in Monteiro and Pang (1996) by using the fact that S n is isomorphic to ! n(n+1)=2 . 7 An Alternative Map Given a continuous map consider in this section the map defined by and present conditions on the map F which guarantee the existence of a (unique) solution of the system -I for every B 2 F (M n ++ \Theta S n first establish two useful lemmas. Lemma 11 Suppose that m is a continuous map which is (X; Y )- equilevel-monotone and z-injective on M n ++ \Theta S n Then, the map H defined by (36) maps ++ \Theta S n homeomorphically onto H(M n ++ \Theta S n Proof. Since M n ++ \Theta S n is an open subset of M n \Theta S n \Theta ! m , by the domain invariance theorem it suffices to show that that H j M n assume that (X; Y; z) are two triples in M n ++ \Theta S n m such that H(X; Y; have F (X; Y; Y \GammaY . Using (38) and the fact that F is (X; Y )-equilevel-monotone, we conclude that \DeltaX ffl \DeltaY - 0. Moreover, it is easy to see that (37) implies that (\DeltaX ) - or equivalently \DeltaX = \GammaX \DeltaY - Y \Gamma1 . Hence, we obtain where the penultimate equality follows from the fact that tr the matrix \DeltaY (X This observation, the fact that imply that \DeltaY (X Hence, we have We thus shown that Y . This conclusion together with (38) and the assumption that F is z-injective on M n ++ \Theta S n imply that z. Lemma 12 Suppose that m is a continuous map which is (X; Y )- equilevel-monotone, z-bounded on S n and z-injective on M n ++ \Theta S n . Then the defined by (36) satisfies the following two properties: (a) H is proper with respect to (!++ \Delta I) \Theta F (M n ++ \Theta S n (b) H restricted to S n proper with respect to (!+ \Delta I) \Theta F (M n ++ \Theta S n Proof. The proof of part (a) is very close to the one given for Lemma 2. Let K be a compact subset of (!++ \Delta I) \Theta F (M n ++ \Theta S n show that H \Gamma1 (K) is compact, from which the result follows. The continuity of H implies that H \Gamma1 (K) is a closed set. Hence, it remains to show that H \Gamma1 (K) is bounded. Indeed, suppose for contradiction that there exists a sequence each Y k is in S n and the product is a positive multiple of the identity matrix, it follows that both sequences fX k g and fY k g must belong to S n ++ . Since K is compact and fH(X we may assume without loss of generality that there exists F 1 2 F (M n ++ \Theta S n Clearly, we have F ++ \Theta S n such that the set ++ \Theta S n contains is an open set of M n ++ \Theta S n m and every homeomorphism maps open sets onto open sets, it follows from Lemma 11 that H(N1 ), and hence F (N1 ), is an open set. Thus, by (39), we conclude that for all k sufficiently large, say k Since F is (X; Y )-equilevel-monotone, we have (X This inequality together with fact that ( ~ imply for every k - k 0 . Using the fact that fH(X k ; Y k ; z k )g ae K and K is bounded, we conclude that g, and hence fX k ffl Y k g, is bounded. This fact together with the above inequality implies that the sequences fX k g and fY k g are bounded. Since lim k!1 k(X must have which implies that lim k!1 due to the fact that F is z-bounded. But this last conclusion contradicts (39). This establishes part (a). The proof of part (b) requires only a slight modification of a couple steps in the above proof. Indeed let K be a compact subset of (!+ \Delta I) \Theta F (M n ++ \Theta S n let the sequence f(X be such that (X We follow the above argument. Although we can not deduce that (X ++ \Theta S n ++ , but utilizing the fact that , we can still establish the boundedness of the sequence f(X )g. The details are not repeated. Thus (b) holds. Based on the above two lemmas, we can establish two additional properties of the mapping H . Theorem 7 Suppose that m is a continuous map which is (X; Y )- equilevel-monotone, z-bounded on S n and z-injective on M n ++ \Theta S n addition to the conclusions of Lemmas 11 and 12, we have: (a) H(S n ++ \Theta S n ++ \Theta S n (b) H(S n ++ \Theta S n Proof. Note that if we can show ++ \Theta S n ++ \Theta S n then part (a) follows. In turn the proof of the displayed inclusion consists of showing that the sets ++ \Theta S n ++ \Theta S n together with the map G j H satisfy the assumptions of Proposition 2. Clearly, is a local homeomorphism by Lemma 11. Also H(I ; I ; claim that H(MnM 0 assume that (X; Y; z) 2 M is such that H(X; Y; z) 2 N 0 . , we conclude that both X and Y are in S n ++ . Hence, (X; Y; z) 2 M 0 and the claim follows. By Lemma 12, we know that proper with respect to E. Using Lemma 1(e), it is easy to see that the set N 0 is path-connected, and thus connected. Hence, it follows from Proposition 2 that H(M 0 ) ' N 0 . This is precisely the inclusion (40). To prove (b), it suffices to show that (0; B) 2 H(S n ++ \Theta S n For each scalar - ? 0, there exists (X ++ \Theta S n must be symmetric positive definite. By part (b) of Lemma 12, we conclude that lim sup -!0 k(X Consequently, a simple limit argument completes the proof. As the final result of this paper, we present a corollary of the above theorem which summarizes the essential properties of the mapping ~ H defined in (6). This corollary is the analog of Theorem 2 for the alternative interior-point map ~ H . Corollary 3 Suppose that ~ m is a continuous map which is (X; Y )- equilevel-monotone, z-bounded on S n and z-injective on S n ++ \Theta S n Then, the defined by ~ ~ satisfies the following statements: (a) ~ H is proper with respect to (!+ \Delta I) \Theta ~ F (S n ++ \Theta S n (b) ~ H maps S n ++ \Theta S n homeomorphically onto ~ ++ \Theta S n (c) ~ F (S n ++ \Theta S n Proof. Consider the map defined by F (X; Y; . Using the assumptions about the map ~ F , it is easy to see that F is (X; Y )-equilevel-monotone, z-bounded on S n and z-injective on ++ \Theta S n Hence, it follows that the associated defined by (36) satisfies the conclusions of Lemma 11, Lemma 12 and Theorem 7, which easily imply the desired conclusions (a), (b), and (c). Corollary 4 Suppose that ~ m is a continuous map which is (X; Y )- equilevel-monotone, z-bounded on S n and z-injective on S n ++ \Theta S n ~ F (S n ++ \Theta S n Proof. Let F (S n ++ \Theta S n Corollary 3(c), we know that H is the map defined in Corollary 3. Hence, there exists (X; Y; z) 2 S n such that I and ~ the relation obviously implies that we conclude that B 2 ~ Acknowledgement During this research, Dr. Monteiro was supported by the National Science Foundation under grants INT-9600343 and CCR-970048 and the Office of Naval Research under grant N00014-94-1-0234; Dr. Pang was supported by the National Science Foundation under grants CCR-9213739 and by the Office of Naval Research under grant N00014-93-1-0228. --R Interior point methods in semidefinite programming with application to combinatorial optimization. Complementarity and nondegeneracy in semidefinite programming. A Primer of Nonlinear Analysis Interior point trajectories in semidefinite programming A predictor-corrector interior-point methods for the semidefinite linear complementarity problem using the Alizadeh-Haeberly-Overton search direction Properties of an interior-point mapping for mixed complementarity problems Interior Point Methods in Convex Programming: Theory and Application Iterative Solution of Nonlinear Equations in Several Variables A superlinearly convergent primal-dual infeasible-interior-point algorithm for semidefinite programming Strong duality for semidefinite programming Convex Analysis First and second order analysis of nonlinear semidefinite programs. Monotone semidefinite complementarity problems Centers of monotone generalized complementarity problems Existence of search directions in interior-point algorithms for the SDP and the monotone SDLCP programming. On extending some primal-dual interior-point algorithms from linear programming to semidefinite programming --TR --CTR Jong-Shi Pang , Defeng Sun , Jie Sun, Semismooth homeomorphisms and strong stability of semidefinite and Lorentz complementarity problems, Mathematics of Operations Research, v.28 n.1, p.39-63, February

interior point methods;maximal monotonicity;problems;mixed nonlinear complementarity problems;generalized complementarity;nonlinear semidefinite programming;weighted central path;monotone mappings;continuous trajectories

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Engineering Software Design Processes to Guide Process Execution.

AbstracttUsing systematic development processes is an important characteristic of any mature engineering discipline. In current software practice, Software Design Methodologies (SDMs) are intended to be used to help design software more systematically. This paper shows, however, that one well-known example of such an SDM, Booch Object-Oriented Design (BOOD), as described in the literature is too imprecise and incomplete to be considered as a fully systematic process for specific projects. To provide more effective and appropriate guidance and control in software design processes, we applied the process programming concept to the design process. Given two different sets of plausible design process requirements, we elaborated two more detailed and precise design processes that are responsive to these requirements. We have also implemented, experimented with, and evaluated a prototype (called Debus-Booch) that supports the execution of these detailed processes.

Introduction If software engineering is to make solid progress towards becoming a mature discipline, then it must move in the direction of establishing standardized, disciplined methods and processes that can be used systematically by practitioners in carrying out their routine software development tasks. We note that such standardized methods and processes should not be totally inflexible, but indeed must be tailorable and flexible to enable different practitioners to respond to what This research is supported by the Advanced Research Projects Agency, through ARPA Order #6100, Program Code 7E20, which was funded through grant #CCR-8705162 from the National Science Foundation. This work is also sponsored by the Advanced Research Projects Agency under Grant Number MDA972-91-J-1012. is known to be a very wide range of software development situations in correspondingly different ways. Regardless of this, however, the basis of a mature discipline of software engineering seems to us to entail being able to systematically execute a clearly defined process in carrying out these tasks. In this paper we refer to a process as being "systematic" if it provides precise and specific guidance for software practitioners to rationally carry out the routine parts of their work. As design is perhaps the most crucial task in software development, it seems particularly crucial that software design processes be clearly defined in such a way as to be more systematic. Humphrey [Hum93] says that "one of the great misconceptions about creative work is that it is all creative. Even the most advanced research and development involves a lot of routine. The role of a process is to make the routine aspects of a job truly routine." We agree with this, and believe that design as a creative activity still contains a lot of routine which can be systematized. For example, making each design decision is probably creative (e.g, deciding if an entity should be a class when using an object oriented design method). How- ever, the order of making each of these related design decisions can be relatively more systematic (e.g., identify each class first and then define its semantics and relations). We also anticipate that with progress in design communities, design methodologies will provide more routine which can be systematized. This will help adapt SDMs into practice more easily and thus improve productivity and software quality. This paper describes our work that is aimed at this goal-namely to make SDM processes more systematic and thus more effective in guiding designers. This work begins with the assumption that the large diversity of Software Design Methodologies (SDM's) provides at least a starting point in efforts to provide the software engineering community with such well-defined and systematic design processes. This paper concentrates on the Booch Object Oriented Design Methodology (BOOD) [Boo91] in order to provide specificity and focus. The paper shows, however, that BOOD, as described and defined in the litera- ture, is far too vague to provide specific guidance to designers, and is too imprecise and incomplete to be considered a very systematic process for the needs of specific projects. On the other hand, we did find that BOOD could be considered to be a methodological framework for a family of such processes. Our work builds upon the basic ideas of process programming [Ost87], which suggest that software processes should be thought of as software themselves, and that software processes should be designed, coded, and executed. That being the case, we found that BOOD, as described in the literature, is far closer to the architecture, or high-level design, of a design process than to the code of such a process. As such, BOOD is seen to be amenable to a variety of detailed designs and encodings, each representing an elaboration of the BOOD architecture, and each sufficiently detailed and specific that it can be systematic in a way that is consistent with superior engineering practice in older, more established engineering disciplines. In the remainder of this paper we indicate how and why we believe that BOOD should be considered a software design process architecture. We then suggest two significantly different detailed designs that can be elaborated from BOOD, each of which can be viewed as a more detailed elaboration upon the basic BOOD architecture. We show that these elaborations can be defined very precisely through the use of such accepted software design representations as OMT [RBP and through the use of process coding languages such as APPL/A. Indeed, this paper shows that the use of such formalisms is exactly what is needed in order to render these elaborations sufficiently complete and precise that they can be considered to be systematic. Thus, the paper indicates a path that needs to be traveled in order to take the work of software design methodologists and render it the adequate basis for a software engineering discipline. First (in Section 2), we define the process architecture provided by BOOD, and then describe two processes elaborated from the architecture. Second (in Section 3), we describe a prototype that supports designers in carrying out the execution of these pro- cesses, illustrating how these differently elaborated processes support different execution requirements. Third (in Section 4), we describe our experience of using the prototype and summarize some of the main issues that have arisen in our efforts to take the design process architectures that are described in the literature to the level of encoded, systematic design processes. 2 The BOOD Architecture and Two Elaborations 2.1 Overview of BOOD We decided to experiment with BOOD because BOOD is widely used, and provides a few application examples that are very useful in helping us to identify the key issues in elaborating BOOD to the level of ex- ecutable, systematic processes. A detailed description of BOOD can be found in [Boo91]. In this section, we present only a brief description of the architecture of the BOOD process. We believe that it can be summarized as consisting of the following steps: 1. Classes/Objects: Designers must first analyze the application requirements specification to identify the most basic classes and objects, which could be entities in the problem domain or mechanisms needed to support the re- quirement. This step produces a set of candidate classes and objects. 2. Determine Semantics of Classes/Objects: Designers must next determine which of the candidate classes should actually be defined in the design specification. If a class is to be defined, designers will determine its semantics, specifying its fields and operations. 3. Define Relations among Classes: This step is an extension of step 2. Designers must now define the relationships among classes, which include use, inheritance, instantiation and meta re- lationships. Steps 2 and 3 produce a set of class and object diagrams and templates, which might be grouped into class categories. 4. Implement Classes: Designers must finally select and then use certain mechanisms or programming constructs to implement the classes/objects. This step produces a set of modules, which might be grouped into subsystems. BOOD provides more hints and guidelines on how to carry out these steps. However, BOOD provides no further explicit elaboration on the details of these steps. Thus designers are left to fill in important details of how these complex, major activities are to be done. As a result, there is a considerable range of variation and success in carrying out BOOD. Further, the process carried out by those who are relatively more successful is not documented, defined or described in a way that helps them to repeat it effectively, or for them to pass on so that others can reuse it. We believe that this is the sense in which BOOD, as described in the literature, is a process architecture. It provides the broad features and outlines of how to produce a design. It supplies elements that can be thought of and used as building blocks for specific approaches to design creation. On the other hand, it provides no specific guidance, details or procedures. These are to be filled in by others who, we claim, then become design process designers (e.g., the authors of [HKK93]) and implementors when the method is applied to specific projects or organizations. 2.2 Process Definition Formalism Earlier experiences [KH88, CKO92] have shown that the State-Charts formalism [HLN + 90] is a powerful vehicle for modeling software processes. Thus, we use a variant of State-Charts [HLN + 90], the dynamic modeling notation of the Object Modeling Technique 91], to model the processes that we will elaborate from BOOD. As shown later, we believe these dynamic models of BOOD processes are sufficient to demonstrate our point Generally, our approach is to use the notion of a state (denoted as a labeled rounded box) to represent a step of a BOOD process, the notion of an activity (the text inside a rounded box and after "Do:") to represent a step which does not contain any other steps. According to OMT, the order for performing these activities can be sequential, parallel or some other forms. We use the order in which the activities is listed to recommend a plausible order for performing those activities. A transition (denoted as a solid arc) denotes moving from one design step to another. The text labels on a transition denote the events which cause the transi- tion. The text within brackets indicates guarding conditions for this transition. The text within parentheses denotes attributes passed along with the transition. A state could have sub-states, each of which denotes a sub-step of the step. Indeed, a modelling formalism is generally inadequate for characterizing certain details of processes. We found that sometimes it was necessary to specify these details in order to render the process we were attempting to specify sufficiently precisely that it could realistically be considered to be systematic. For ex- ample, OMT does not provide a capability for specifying the sequencing of two events which are sent by the same transition. Specification of this order might well be the basis for important guidance to a designer about which design issues ought to be considered before which others. Thus, we found it necessary to supplement OMT, by using a process coding language called APPL/A [SHO90b] to model such details. APPL/A is a superset of Ada that supports many features that we found to be useful. Some examples of APPL/A code are also provided in subsequent sections of this paper. Note that the goal of this work is to use these process models and codes to demonstrate the diversity and details of the processes that can be elaborated from an SDM. As shown later, these dynamic models of BOOD processes are sufficient to demonstrate this point. Thus, we did not develop OMT's object models and function models for BOOD processes. 2.3 Modeling the BOOD Architecture Fig. 1 represents an OMT model of the original BOOD process architecture, as described in [Boo91]. In this architecture, we merged step 2 and 3 of the original BOOD process because our experience shows that it is hard to separate those steps in practice (Booch himself also considers that step 3 is an extension of step 2 [Boo91]). We believe that this model is considerably more precise than the informal description originally provided. It is still quite vague and imprecise on many important issues, however. Booch [Boo91] claims that this vagueness is necessary in order to assure that users will be able to tailor and modify it as dictated by the specifics of particular design situations. For example, step 2 of Fig. 1 does not define the order for editing various BOOD diagrams and templates. It does not define clearly which of the diagrams or templates must be specified in order to move from step 2 to step 3. Booch claims that different designers might have important and legitimate needs to elaborate these details in different ways (Chapters 8-12 of [Boo91] provide a few examples). We found that there are indeed many ways in which these details might be elaborated precisely and that many of these different variants might offer better guidance. The differences might well arise from differences in application, differences in organization, differences in personnel expertise, and differences in the nature of specific project constraints. Once these differences have been understood and analyzed, however, the design process to be carried out should be defined with suitable precision. Such precise definitions are needed in order to support adequate improvements of the efforts of novices. In addition, we believe that there are expert designers who have internalized very 3. Implement Classes and Objects Do: Browse Class/Object Diagrams (Requirements) (The change list) User initiates the transition [No inconsistency is found] (Proposed modification) (Requirements exist) User initiates the transition Edit Object Diagram Edit State Transition Diagram Edit Module Diagram Edit Process Diagram Edit Moduel Template Edit Process Template [Candidate Classes/Objects are defined] User initiates the transition (The change list) Changes to candidates (List of candidates) Edit Class Template Edit Timing Diagram Do: Browse Candidate Classes/Objects 2. Identify the Semantics of Classes/Objects Edit Object Template Edit Class Diagram List candidate classes/objects Nouns 1. Do: Read requirements Edit Data Flow Diagram User initiates the transition (Class/object, and some other diagram/templates) [Class/Object diagram/template defined] Changes to the class/object diagram/templates Find inconsistency in class/object diagrams/templates (Proposed modification) User initiates the transition User initiates the transition (Proposed modification) Figure 1: A Process Architecture of BOOD specific and very effective elaborations of the BOOD architecture and that the more these are defined pre- cisely, the more important design expertise may be understood, reused, automated, and improved. In order to make the above remarks specific, we now discuss two possible elaborations of the BOOD architecture. In addition, using these examples we can show the need for, and power of, both design and code representations as vehicles for making design processes clear and thereby providing more effective guidance. 2.4 Two Examples of BOOD Process Refinement 2.4.1 Examples of Software Project Types First, we characterize two different types of projects for which we will elaborate variants of design pro- cesses, within the outlines of the BOOD architecture (see Project Properties columns in Table 1). The parameters of these characterizations are 1) implementation language, documentation requirements, project schedule, designer skill, 5) software operation domain, software domain, and 7) maturity of the software domain. Based upon our experiences, we identified these two project types as representatives of projects commonly encounted in software engineering practice (see Table 1). For example, an instance of project type 1 could be a defense-related or a medical systems project while an instance of project type 2 could be a civilian project. We expected that elaborating processes to fit the requirements of these two different types of projects would help us understand the range of processes that could be elaborated from BOOD. The seven characterization parameters were chosen because our earlier work indicated that these parameters are likely to have major and interesting effects upon design process elaboration. For example, when consulting with Siemens medical companies, we found that the U.S. Food and Drug Administration (FDA). has specific documentation requirements, and requires control and monitoring of corrective actions on the product design. [FDA89] says that "when corrective action is required, the action should be appropriately monitored. Schedule should be established for completing corrective action. Quick fixes should be pro- hibited." This certainly affects how an SDM should be applied to a specific project. The application examples described in [Boo91] also provide us with some details that seemed likely to be useful in employing these parameters to help us to derive these BOOD-based design processes. For exam- ple, one of Booch's examples indicates that if C++ is to be the eventual application coding language, then class/object diagrams would not need to be translated into module diagrams. In addition, Booch's problem report application example [Boo91] helps us to understand the process requirements for developing an information processing system. For instance, that example shows that the method must be tailored to support the design of database schemas. His traffic control example helps us to understand the process requirements for developing a large scale, device-embedded system. 2.4.2 The Processes Elaborated from BOOD In this section we present portions of the OMT diagrams used to define details of each of these two elaborations on the basic BOOD architecture. We then further refine parts of them down to the level of executable code. Each of these processes is clearly a "Booch Design Process", each represents what we consider to be a completely plausible design process, and each is quite completely and precisely defined-to the point of being systematic for the specific kinds of projects. These two processes demonstrate the point that there is a great deal of imprecision in the current definition of "Booch Object-Oriented Design." They also indicate how BOOD can be elaborated, and what the range of elaboration might be when it is applied to specific projects. We will refer to our first elaborated process as the Template Oriented Process (TOP). It emphasizes defining various BOOD templates (e.g., the class tem- plate) as it hypothesizes the importance of carrying 1. A Template Oriented Example 2. A Diagram Oriented Example Project Properties Process Requirements Project Properties Process Requirements Must be coded in Ada Specify Module Diagram Must be coded in C++ Guide designers not to specify Module Diagram since it is not needed in this case. Must incorporate very Requires specification of Only minimum documentation No need to enforce complete documentation all templates required specifying all templates Long-term Allow full documentation Short-term Encourage use of existing code Skilled design team Less process guidance Inexperienced design team More process guidance More process flexibility Less process flexibility Safety-critical More change control needed Non safety-critical Less change control needed (e.g., Medical Systems) to satisfy FDA's requirements Large scale, Use structured analysis Information processing system Single, familiar domain. device-embedded system Support partitioning domain Need to support schema design State of the art project Need to support prototyping Well-understood No support for prototyping needed. Need to support code reuse Table 1: Project Characteristics and Process Requirements out a design activity that delivers very complete doc- umentation. The TOP's emphasis on complete documentation can be seen by noting that we have refined steps 2 and 3 of Fig 1 into the more detailed model defined in Fig. 2. We further hypothesized in designing the TOP that the software to be developed is to be safety-critical, and that, therefore, the TOP should enforce more control over design change as this is often required by government agencies to ensure product quality. Accordingly note that the high level design of the TOP incorporates an approval cycle for all changes to previously defined artifacts. On the other hand, we hypothesized that the TOP is to be executed by skilled and experienced designers. Because of this, we did not refine the detailed design activities into lower level steps. Our expectation here is that such designers would insist upon freedom and flexibility that this would be given them. This also illustrates that it is possible to define a design process precisely, yet still provide considerable freedom and flexibility to practitioners. In addition we designed the TOP to allow for a certain degree of flexibility in making transitions from one step to another. We have also included the possibility of incorporating a prototyping subprocess into this process. We refer to our second elaborated process as the Diagram Oriented Process (DOP), as it emphasizes specifying BOOD diagrams. We derived this process from Booch's Home Heating System example [Boo91]. In the DOP we hypothesized that there are only weak requirements in the area of documentation, and we, therefore, do not design in the need for designers to specify BOOD's templates (see Figures 4). We also hypothesized that the product being designed will be coded in a language that provides direct support for programming classes and objects. For this reason, the DOP omits step 3 of the general model shown in Fig. 1 as part of its elaboration, leaving the model defined in Fig. 4. Note that this elaboration incorporates fewer top-level steps than the general BOOD model does. We also hypothesized that the DOP is aimed at supporting novice designers, and so the DOP provides detailed guidelines for identifying classes/objects (see Figures 5, 6, and 7). In addition, the DOP assumes that a great deal of importance is placed upon reuse. In response, the DOP incorporates steps that guide designers to reuse existing software components (see Fig. 7). The job of creating more specific and detailed elaborations of BOOD is not limited solely to modification of the processing steps of BOOD. It also entails specifying the flow of control between these steps and their substeps. A good example of the importance of these specifications can be seen by examining how change management is handled in these design processes. We use the term forward change management to denote a transition used to maintain consistency between a changed artifact and its dependent artifacts, that are normally specified at a later stage of the pro- cess. For example, a designer may add a class to a candidate class list (in step 1 of Fig. 2) . This results in forcing designers to redo step 2 to consider adding a corresponding class to the class diagram. There is virtually no guidance in BOOD about precisely how this is to be done, or how the critical and tricky issues of consistency management are to be addressed. Thus there is a clear need for more detailed guidance on automatic change control. One way this can be done is to refine this high-level transition further as shown in Fig. 8. In Fig. 8, a dotted line from a transition to a class represents an event sent by the transition. For example, the transition from Selected Class A to Rejected Class A, which is caused by updating candidate class A's field Needed to FALSE (i.e. class A is no longer needed), sends event Delete Class A to class Class. Clearly this refinement is simply one of a very large assortment of possible refinements. We do not claim that it is the only one or the "right" one. We do claim, however, that supplying details such as these provide specific guidance that is important for designers-especially for novice designers and for large design teams. Should it turn out that such a specifically designed process is shown to be particularly useful and desirable, then the detailed specification will also render it more amendable to computer support. We should also note that we did not stop at the level of design diagrams in refining the meaning of forward change management, but that we went further and defined it as actual executable process code. Our code was written in the APPL/A process coding language. Fig. 9 shows the APPL/A code for the process defined in Fig. 8. Note that this code provides even more de- tails. For example, note that this code specifies that changing a candidate class to a candidate object will cause an ordered sequence of events: 1) the insertion of an object template, 2) the removal of the class template and 3) the forwarding of that template to step 3 for editing of the object template. Again, we stress that these specific details are not to be considered the only feasible elaboration of BOOD-only one possible elaboration. We do believe, however, that in specifying the design process to this level of detail deeper understandings result, and the process becomes more systematic. In addition, by reducing the process to executable APPL/A code, it becomes possible to use the computer to provide a great deal of automated support (e.g., some types of automatic updating and consistency maintenance) to human designers. Another kind of control flow in BOOD is backward change management, which is aimed at maintaining consistency between a specified artifact and all the artifacts upon which the specified artifact should de- pend. These artifacts are normally defined at earlier stages of the process. For example, in step 2 of Fig. 2, designers may need to define a class in a class diagram and find that this class does not correspond to any candidate class because of an incomplete or faulty analysis of the application requirements. Thus, designers have to go back to earlier steps, reviewing the requirements and possibly redoing step 1 to add this class to the candidate class list. This transition can be refined and coded in a manner similar to what was described in the case of forward change management. Do: Browse Requirement Browse Candidate Classes/Objects Edit Class Diagram Edit State Transition Diagram Edit Object Diagram Do: Browse Class Diagram Browse Object Diagram Edit Class Templates Edit Class Utility Templates Edit Object Templates Do: Browse Object Diagram Browse Class Diagram Edit Module Diagram Edit Process Diagram Edit Device Template Edit Process Template Edit Module Template Do: Browse Module Diagram Determine Semantics of Class Step 3 Specify Class/Object Templates Develop Module Diagram Specify Module Template Step [No inconsistency is found] [Requirements exist] User initates the transition Class/Object Find inconsistencies in Candidate Class/Object Diagrams (List of Candidate Class/Objects) [Candidate Classes/Objects are defined and reviewed] User initiates the transition User initiates the transition (Class/Object diargam) User initiates the transition Step (Requirements) Changes to candidates (Changed List) [The changes are approved] Diagrams Changes on (Changed diagrams) [Class/Object diagrams are reviewed] [The changes are approved] (Changed List) Changes to candidates [The changes are approved] Find inconsistencies in class/object templates [Modification is approved] [Modification is approved] class/object diagrams Find inconsistencies in [Modification is approved] Find inconsistencies in Class Diagram Changes to class/object templates (Class/Object Templates) [The changes are approved] User initiates the transition (Class/Object Diagrams) (Class/Object Templates) [Class/Object Templates are reviewed] Find inconsistencies in module diagram [Modification is approved] User initiates the transition (Module diagrams) [Module diagrams are being defined] Changes to module diagram (Changed module diagram) Figure 2: Top-level Process Definition of the Template Oriented Process These process definitions, including both main flow and change management transitions, explicitly and clearly demonstrate how the published Booch Object Oriented Design description can be elaborated into a precisely defined process to provide more effective guidance for specific projects. Our research indicates that this observation is quite generally applicable to the range of SDM's that are currently being espoused widely in the community. There are a number of reasons for this imprecision. We have already noted that the imprecision is there intentionally to permit wide variation in design processes to match similarly wide design process contexts and requirements. While we neither doubt nor dispute this need, we believe that our work has shown that it can be met more effectively through tailoring SDMs for specific needs of projects. These processes resulting from the tailoring, and supported by the appropriate tools, provide more effective guidance and help implement various recommended practices (e.g, those recommended by FDA [FDA89]). In the next sections, we discuss how to support the execution of the elaborations of the BOOD architecture that we have just described. Do: Browse Requirement Do: Browse Requirements Edit Data Flow Diagram Do: Edit Object Diagram Edit Class Diagram User initiates the transition Inappropriate Problem Definition Step 1.1 Step 1.2 Problem Boundary Structured Analysis Step 1.4 Prototyping User initiates the transition Problem (Proposed new definition) (Problem User initiates the transition [Definition is approved] (Problem User initiates the transition Domain Analysis Step 1.3 Do: Browse Requirements Edit Candidate Class/Objects User initiates the transition(Data Flow Diagrams) (Problem [Definition is approved] (Candidate class/object) [Candidate Class/Objects exist] Step 1. Identify Candidate Class/Object Figure 3: Second-level Process Definition of Template Oriented Process: Refinement of Step 1 2. Identify the Semantics of Classes/Objects (Requirements) (The change list) (Requirements exist) User initiates the transition (Design specification) [No inconsistency is found] 1. Change to Candidates User initiates the transition (List of candidates) [Candidate Classes/Objects are defined] Do: Browse Candidate Classes/Objects Edit Class Diagram Edit Object Diagram Edit State Transition Diagram Edit Timing Diagram User initiates the transition Find inconsistency between candidates and class/object diagrams (Proposed modification) Figure 4: Top-level Process Definition of Diagram Oriented Process Do: Browse Requirement Problem Definition Inappropriate Problem Definition Step 1.1 Define Problem Boundary (Proposed definition) Domain Analysis Step 1.2 1.3 Reuse-based Design User initiates the transition and (Candidate Classes/Objects) User initiates the transition and [Problem is defined] (Candidate Abstract Class) User initiates the transition and Step 1. Identify Candidate Class/Object Figure 5: Second-level Process Definition of Diagram Oriented Process Search for Noun Search for Verb Search for Adjective Key Abstractions Do: Browse Requirement Candidate Classes/Objects Do: Identify Classes from Nouns Decide Operations from Verbs Objects from Nouns (Identified nouns, verbs, adjectives) User initiates the transition (Change List) Nouns/Verbs/Adjective Changes on Domain Analysis Step 1.2 Step 1.2.1 Step 1.2.2 Figure Third-level Process Definition of Diagram Oriented Process : Refinement of Domain Analysis Reusable Components Develop Object Diagram for the Edit the Concrete Classes Concrete Classes Instantiate the Abstract Class to User initiates the transition User initiates the transition User initiates the transition (Concrete Class) User initiates the transition Abstract Class Change on reusable components Change on semantics of the abstract class 1.3. Reuse-based Design Change on the object diagram (New components) (New object diagram) (Completed concrete classes) (Abstract classes) (Abstract classes) Class Abstract Classes, Reusable Components) (Reusable Components, Candidate Classes) User initiates the transition (Object Diagram, Candidate Classes) Find new sharable objects (Description of the objects) Figure 7: Third-level Process Definition of the Diagram Oriented Process : Refinement of Reuse-based Design, which is based on Booch's Home Heating System example 3 Support for Executing BOOD Pro- To experiment with our ideas and demonstrate how these processes should be supported appropriately, we have developed a research prototype, called Debus- Booch, to support the execution of design processes of the sort that have just been described. Execution of such processes is possible as a result of their encoding in APPL/A, a superset of Ada that can be translated into Ada, and then compiled into executable code. We note that BOOD addresses only issues concerned with supporting single users working on a single design project. As most designers must work in teams, and are often engaged in multiple projects simultane- ously, a practical system for support of such users must do more than simply execute straightforward encodings of BOOD elaborations. Our Debus-Booch prototype adapts an architecture used in a previous research prototype (Rebus [SHDH 91]). The architecture lets developers post (to be done) and submit (finished) tasks to a whiteboard to coordinate their task assign- ments. Since this work has been published and is not directly related to the topic of this paper, we will not describe it here. Selected Class A Insert Rejected Class A Update Rejected Class B Update Update Update [No existing class has name A] [No existing class has name A] Insert Selected Class B Terminated Update Delete Selected Object A Update Rejected Object A Class Object Insert the Class Update Delete Class A Delete Class A Delete Class A Insert Class A Insert Object A Figure 8: A Refinement of Forward Change Man- agement: illustrates more precisely how a change in the candidate list might affect the class dia- gram/templates. A candidate is recorded with three fields: name, needed (indicating if it is selected as an candidate), and kind (indicating that it is a class or In addition, there are a variety of difficult user interface issues to be faced in implementing a system such as this. Exhaustive treatment of all of these issues is well beyond the scope and limitations of this paper. An indication of our approaches to these and related problems can be seen from the following brief implementation discussion. 3.1 System Overview Debus-Booch provides four levels of process guidance and support to its end-users (see Fig. 11 for their user interface representations): 1. Process Selection (Accessed through a This enables users to select any of a range of elaborations of the BOOD ar- chitecture, or any non-atomic step of any such elaboration (as shown in Fig. 10). This is done by selecting a driver to perform a constrained sequence of steps at a certain level of the selected process step hierarchy. Debus-Booch helps users with this selection by furnishing users with access to information about the nature of these various processes and steps. with candidate-rel, class-template; trigger maintain-candidate; - maintain the product of step 1 trigger body maintain-candidate is begin loop trigger select upon candidate-rel.update needed new-name update-needed new-needed new-kind completion do change management is necessary only when - candidate is selected or being updated. case kind is when class =? - the candidate is no longer needed class-template.delete(name =? name); else - the candidate becomes needed class-template.insert (name =? name, .); query (pname, plen, sname, slen); define-class-proc(pname,plen,sname, slen); class-template.update (name =? name, update-name =? TRUE, new-name =? new-name); or (new-needed = TRUE and case new-kind is when object =? object-rel.insert (name =? name); class-template.delete (name); when operation =? . when abstract-class =? . case upon or select Figure 9: APPL/A code for defining Forward Change Management between candidate class list and class di- agram/template definitions Process 1 . Process N Step 1.1 Step 1.2 Step N.1 Step N.M Step 1.1.1 Step 1.1.1.1 Process 2 Console Driver 1.1 Driver 1.2 Driver N.1 Driver N.M Driver 1.1.1 Panel 1.1.1.1 Tool-Button 1 Tool-Button 2 Support Initiate Exclusive Unspecified Order Constrained Order Debus-Booch System Interface Architecture An SDM Model Criteria Guidline Display Display Figure 10: An SDM definition and support model 2. Process Step Execution (Accessed through a Panel): The user can obtain support for the sequencing and coordination of the driver activities to be performed in an elaborated design pro- cess. These activities can be divided into two cat- egories: required and optional activities. For ex- ample, in the step used to determine the semantics of classes, designers must use Class Diagram Editor, which therefore supports a required activ- ity; in the same step, designers may use a requirements browser, which therefore supports an optional activity. Designers can invoke all the tools that support the required activities by clicking on the Set Environment button. In using this access method, we help designers to set up a design environment more easily. Note that different processes may have different required activities. For example, in the template oriented process (TOP), editing the class template is a required activity. However, in contrast, using the diagram oriented process (DOP), the user cannot even access this editor. 3. Atomic activity/support (Ac- cessed through a Tool-Button): The user can obtain support for a specific activity in an atomic step. For example, the user can request access to a Class Diagram Editor in order to obtain support for defining a class diagram, which is an activity performed in determining the semantics of classes. 4. Documenation and Help Support (Ac- cessed through Displays): This support can be obtained in conjunction with the use of tools that support atomic activities. The displays that are made available convey a variety of informa- tion, such as the criteria, guidelines, examples, and measures [SO92] to be used to help designers understand how to carry out the activity. Debus-Booch provides the flexibility that is needed for experienced designers. Designers can use a console display to access all of the supports listed above. For example, a designer can click on the Console's Steps button to execute any step of any elaborated BOOD process (as long as the guarding condition for this step is satisfied, otherwise, the invocation will be rejected). Figure shows how these four types of support are made available to the designers who use Debus- Booch. In particular the figure indicates the degrees of interactions that are allowed among the supports for processes, steps, and activities. In particular, note that support for process execution will be provided on an exclusive basis only, as we believe it is reasonable to use only one process at a time to design any given sys- tem, or any major part of a system. Similarly, there are constraints on furnishing support for the simultaneous execution of process steps. This is because there are often data dependencies between steps. On the other hand, support for simultaneous execution of activities is unconstrained as many design process activities must often be highly cooperative in practice. Some sets of activities must indeed be carried out in constrained orders. In this case it is necessary to group them into composite steps. The decisions about allowable degrees of concurrency were made based on our observations of the nature and structure of the process models defined in Section 2.4.2. 3.2 Scenario for Use of Debus-Booch Here is a general scenario, which indicates how designers might use Debus-Booch (see Fig. 11): 1. Designers select a specific elaborated BOOD process from the menu popped up after pressing the Process button. They may select Process Selector to retrieve information about these processes. For each process, the Process Selector describes the most appropriate situations (e.g., the documentation requirements, project deadline) under which the process should be used. 2. Upon clicking on the menu item (i.e. a selected process), the corresponding driver will be initi- ated. Then, designers must enter the name of the subsystem to be designed. This subsystem can be assigned to them from a management process or a high-level system decomposition process (e.g., in our case, it is on the whiteboard [SHDH 3. When the subsystem name has been entered, the driver will check what design steps have been performed on this subsystem, and then automatically set the current sub-step in order to continue with the design of this subsystem. (This is tantamount to the process of restarting a suspended execution of the process from a previously stored checkpoint.) Then, the designer can click the Run button to invoke the corresponding sub-driver or atomic step support. 4. If a sub-driver is initiated, step 3 will be repeated except designers will not need to enter the sub-system name again. 5. If atomic step support is invoked, a panel appears and designers can click on its tool-buttons to invoke the tools to support the activities that should be carried out in this atomic step. 6. Having finished this step, designers can click on the next step using the Steps buttons of the driver to move the process forward. If the guarding condition (e.g., see Fig. 2) for the next step is true, the move will succeed, otherwise, the move will be rejected. After finishing the final step in the elaborated process, the designer may go back to the first step to start another iteration on the same subsystem, reviewing and revising the artifacts produced in the previous iteration. Thus, Debus-Booch also provides supports for process iteration. As this scenario illustrates, Debus-Booch provides different supports for users who are using different process elaborations. For example, using the template oriented process, the user will be guided by the driver, (with enforcement provided by the guarding condi- tion), to specify the module diagram as is useful when Ada is used as the implementation language. In con- trast, using the diagram oriented process, the user will be directed to not define the module diagram as it is not considered to be of value when an object-oriented language is used. 4 Experience and Evaluation In the past year, we have carried out two experiments and one evaluation with Debus-Booch. In the Figure 11: A Stack of Debus-Booch Windows Supporting the Booch Method first experiment, we used the prototype to develop a design example: an elevator control system. This is a real-time system that controls the moving of elevators in response to requests of users [RC92]. It was used as an example for demonstrating how the Arcadia consortium supports the whole software development life- cycle. The system requires full documentation, and is to be implemented in Ada. It is safety-critical and device-embedded. The design team was to include the lead author and students who had finished the software design course. Thus, this project has most of the characteristics described in the Template Oriented Example (see Table 1). Our experience with this experiment shows that the Template Oriented Process (TOP) supported our design development quite effectively. The process represented through the drivers and panels guided us to define the BOOD templates and the module di- agrams. For example, the designers were guided to define the problem boundary first and then identify candidate classes such as Controller, Button, Floor, and Door. In this experiment, we found that the Set- Environment button was most frequently used and was effective in guiding designers to define those required diagrams and templates. The flexibility offered by the process allowed the designers to modify some intermediate design specifications. For example, the designers often moved back to Step 1 from Step 2 (i.e., the Determine Semantics of Class step of Fig. 2) to modify the candidate classes. However, to ensure system safety, this process enforced stricter control over the other backward changes which directly affect the actual design documentation. For example, the transition from Step 3 to Step 2 of Fig. 2 was more strictly monitored. In using the prototype, we found the current implementation to be too restrictive. Thus, we think that Debus-Booch needs to provide a number of, rather than one, methods that can be selected for controlling the transition. Examples may include: 1) The modification triggers revision history recording, 2) The modication triggers change notification mech- anism, and 3) the modication triggers a change approval process. These example methods support different degrees of the control over the design process. In the second experiment we used Debus-Booch to develop a design for the problem reporting system as described in [Boo91]. This project fits five characteristics of the Diagram Oriented Example (See Table 1). The system is to be coded in C++, has minimum document requirements, and is not safety-critical. It is an information processing system and well-understood. The design team, including the lead author and a software engineer, however, is more experienced than that described in the Diagram Oriented Example. In this experiment our experience were similar to those in the first experiment. One additional, interesting experience is that for this well-understood domain (e.g., design of a relational database schema), the process (the Diagram Oriented Process (DOP)) could have been designed to be even more specific and therefore to provide more effective guidance. For ex- ample, Steps 1.2.2 and 2 should provide guidance to the normalization of the classes. This seems to indicate that for building a large system, an SDM might need to be tailored into a set of different processes, each of which is most effective for designing certain kinds of components of the system. For example, a large system might contain both an embedded system and a data processing system. That being the case, both DOP and TOP processes might need to be applied to developing this system. We have installed a version of Debus-Booch at Siemens Corporate Research (SCR). Some technologists there have used the prototype and evaluated it. These technologists are specialized and experienced in evaluating CASE tools and making recommendations to Siemens operating companies. During their evalu- ation, the technologists executed the tool and examined all its important features. Based upon their ex- perience, the technologists believe that Debus-Booch should be particularly useful for novice designers because the tool explicitly supports BOOD's concepts and processes. Their experience tells us that novice designers are much more interested in using a well de- fined, detailed process to guide their design. A tool, such as Debus-Booch, that explicitly supports an SDM process should help them to learn the SDM quickly. Some experiences coming out of these experiments and evaluation are: 1. Process execution hierarchy (the tree of drivers and panels in Fig. 10) cannot be too deep: There are two main reasons for having this sugges- tion: 1) A deep execution hierarchy needs too much effort in tracking the detailed process states. This problem is similar to the "getting lost in hyperspace" problem found in hypertext system [Con87]. 2) Need to minimize the time overhead from transiting between various tools that support various design steps. These suggestions clearly reinforce our observations about the problem of mental and resource overhead [SO93]. Novice designers are more willing to accept the overhead to trade for more guidance while skilled designers are not. However, the evaluation seems to indicate that even for the novice designers, the process execution tree cannot be too deep. The evaluation suggested that three levels seem to be maximal. 2. Designers had difficulty in selecting processes: Users need stronger support for selecting pro- cesses. The textual help message associated with each process seems to be not sufficient. A more readable and illustrative method must be developed to help users to understand the process requirements quickly, and thereby help users to select appropriate processes. 3. Support the coordination of designers working at different steps: Our model focuses on supporting designers to work in parallel in designing different software components, or supporting an individual designer to work in parallel on multiple software components. However, the current model is weak in coordinating two designers working on the same software component at different process steps. For example, we found that a finished class diagram might need to be passed to another designer for defining its module diagram. This often helps in utilizing the different skills of designers. 4. Need to have stronger support for tracking and coordinating processes: This suggestion is closely related to the first suggestion. The evaluation indicates that the process tracking mechanism is even more important when the process guides designers at the relatively low levels of the process. The process tracking must emphasize indicating the current state of the process and help designers understand the rationales and goal for performing the step. Summary Our work in developing elaborations of the BOOD architecture into more precise design process designs and code has brought a number of technical issues into sharpened focus. Generally, we have found that it is quite feasible and rewarding to develop design processes down to the level of executable code. Doing so raises a number of key issues that are all too easily swept under the rug by process architectures and process models. Many of these issues have tended to be resolved informally and in ad hoc ways in the past. This has stood in the way of putting into widespread practice superior software design processes. The following summarizes some of the more important and interesting findings of this work. 5.1 The Advantages of Detail in Process Definition Process modelers often struggle to choose between general process definitions and specific process defini- tions. Processes that are too general are often criticized for providing no useful guidelines. Processes that are too specific are often criticized as leaving no freedom to designers. We found that starting with a specific SDM such as BOOD, and then elaborating it and making it more specific to the needs of a particular situation represents a good blending of these two strategies. Doing this serves to make the resulting process sharper and more deterministic, and thus helps to make it more systematic and susceptible to computerized support. It seems worthwhile to note that taking this approach is tantamount to pursuing the process of developing a software design process as a piece of software, guided by a set of process requirements and an assumed architectural specification (in this case the BOOD architecture) We are therefore convinced of the importance of dealing with the details when elaborating design process architectures into designs and code. Here we summarize these process design issues, and describe how we addressed them in using our 1. Step selection: An SDM often describes many "you could do" activities in its process descrip- tion. In our work we turned many of them into "you should/must do" or "you should not do" activities in order to provide more effective guidance. For example, BOOD suggests specifying module diagrams. However, when using an implementation language that directly supports programming classes and objects, Debus-Booch guides designers to not specify these diagrams because they are useless in this specific application (see Fig. 4). With our process programming approach to the elaboration of specific processes we also found it straightforward to specify how to incorporate various other related processes (e.g., reuse, proto- typing) into the design process (see figures 3, 5 and 7 for example). 2. Refinement selection: An SDM generally provides its guidance as a set of high-level steps. Each high-level step has a set of guidelines. Designers are often left free to follow the guidelines closely or rely more upon their experience. Novice would tend to follow guidelines while skilled designers would rely more on their experience with some support from the guidelines. With our ap- proach, we provide both supports to novice and skilled designers. Novices can use the detailed process support to guide their design activities, while more skilled designers use only high-level process support. 3. Control condition selection: An SDM usually does not specify strictly how design changes should be managed. It usually does not specify precisely the conditions under which a step can be considered to be finished. With our approach of tailoring SDMs for specific projects, we can define the conditions quite precisely. For example, for a medical system which is often safety-critical and regulated by FDA, we decide to provide more strict control (see Fig. 2) to ensure system consistency and reliability. However, our experience in using Debus-Booch shows that such control mechanism should not be enabled until the specifications (e.g., class diagrams) are stable and have been used by other software components. 4. Control flow selection: An SDM usually does not specify all the possible transitions between steps, instead, it only specifies those that are likely to be done most frequently. Transitions that are the most crucial ones may also be the most difficult to explain, and thus not specified sufficiently precisely. Our approach makes it far easier to add precision to the specification of tran- sitions. For example, Fig. 8 shows the various transitions needed for modifying classes. 5. Concurrency specification: As noted earlier, most SDM's are intended only to specify how to support the efforts of a single designer working on one project at a time. It is clearly unrealistic to assume that this is the mode in which most designers work, and that, therefore, support for this mode of work is sufficient. In our work we adapted an architecture [SHDH + 91] that is capable of supporting group development. The activities which can be performed at each step allow individual designers to work on the same design in parallel. 5.2 Related Work We have not seen any work that is similar to our approach of developing design processes as software, then analyzing and contrasting the elaborated processes, and illustrating explicitly why currently existing SDM descriptions cannot be taken directly as a completely systematic process for specific projects. Our work is unique in that it indicates how one might use the process programming approach to modeling and coding an SDM into a family of more systematic processes used for a corresponding family of projects. It demonstrates how SDM processes can be defined more precisely. A more precisely defined SDM process is more likely to be effectively supported and thus provides more effective guidance. This experiment encourages us to be more confident in using the project-domain-specific process programming approach to solving many problems in sharpening and supporting software processes. Some work (e.g., studied mechanisms for supporting generic software processes. However, without studying specific generic and instantiated processes as we did in this work, the value of these mechanisms is hard to evaluate. This work is related to other projects aimed at developing a process-centered software environment, like those reported in [MS92, KF87, MR88, Phi89, ACM90, FO91, MGDS90, DG90]. The most significant difference between these efforts and our work is that our work, targeted at specific process require- ments, provides very specific strategies for supporting specific processes that emerge from the work of other acknowledged experts (in this case, these experts are in the domain of software design). For ex- ample, we provide very specific interface architecture and tool access methods for supporting SDMs and their various users. In contrast, most work in developing process-centered environments is aimed at developing general-purpose software development envi- ronments. For instance, [MR88] supports specifying any software development rules. Marvel [KF87] is a general purpose programming environment. It does not describe specifically how to provide effective guidance for using specific development method on specific kinds of projects. Another difference is that our work focuses on evaluating varied external behaviors of the system while other work focuses on the study of implementation mechanisms and process representation formalisms (e.g., [FO91]). The study of these mechanisms and formalisms is not the focus of our pa- per. Comparisons of our formalisms (e.g., APPL/A) to others can be found in [SHDH 6 Status and Future Work The current prototype version of Debus-Booch is implemented using C++, Guide (a user interface development tool), and APPL/A. It incorporates StP [AWM89] and Arcadia prototypes. The whole prototype consists of about 34 UNIX processes. Each of them supports a console, driver, panel, and other tools. It was also demonstrated at the tools fair of the Fifth International Conference on Software Development Environments 1 . At present, this prototype is being enhanced by the conversion of more of its code to APPL/A and by the incorporation of new features, new design process steps, and new design processes. We plan to carry out the following future work: 1. Focusing on more specific project domains, to elaborate still more specific process models and support environments. This should help deepen our understanding of the project domain's influences on process requirements and SDM elaborations 2. Collecting data about how these elaborated processes are used. Based on the analysis of these data, we would be able to adjust the processes more scientifically. 3. Developing a project-domain-specific process gen- erator. With the specification of project proper- ties, the corresponding process definitions and its support environment might eventually be automatically generated, at least in part. Acknowledgments We thank the members of the Arcadia software environment research consortium for their comments, particularly Stanley M. Sutton and Mark Maybee for their useful comments on the APPL/A code. We also thank those SCR researchers, particularly Wenpao Liao, who experimented with and evaluated our prototype. We are also very grateful to Tom Murphy and Dan Paulish for supporting us to continue this work at SCR. We thank Bill Sherman and Wen- pao Liao for reviewing the final version of this paper. --R Software process enactment in Oikos. Wasserman and R. Mechanisms for generic process support. The booch method: Process and pragmatics. Process modeling. An introduction and sur- vey Managing software processes in the environment melmac. Preproduction quality assurance planning: Recommendations for medical device manufacturers. Integration needs in process enacted environments. Formalizing specification modeling in ooa. Statemate: a working environment for the development of complex reactive sys- tems Using the personal software pro- cess A formalization of a design process. An architecture for intelligent assistence in software development. Software process modeling. Software development environment for law-governed systems Process integration in CASE environments. Software processes are software too. State change architecture: A prototype for executable process models. Object Oriented Modeling and Design. Rebus requirements on elevator control system. Object behavior analysis. Real time recursive design. Debus: a software design process program. Towards objective Challenges in executing design process. --TR --CTR Stanley Y. P. Chien, An object pattern for computer user interface systems, Information processing and technology, Nova Science Publishers, Inc., Commack, NY, 2001

software design process;design methodology;design methods

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Applications of non-Markovian stochastic Petri nets.

nets represent a powerful paradigm for modeling parallel and distributed systems. Parallelism and resource contention can easily be captured and time can be included for the analysis of system dynamic behavior. Most popular stochastic Petri nets assume that all firing times are exponentially distributed. This is found to be a severe limitation in many circumstances that require deterministic and generally distributed firing times. This has led to a considerable interest in studying non-Markovian models. In this paper we specifically focus on non-Markovian Petri nets. The analytical approach through the solution of the underlying Markov regenerative process is dealt with and numerical analysis techniques are discussed. Several examples are presented and solved to highlight the potentiality of the proposed approaches.

Introduction Over the past decade, stochastic and timed Petri nets of several kinds have been proposed to overcome limitations on the modeling capabilities of Petri nets (PNs). Although very powerful in capturing synchronization of events and contention for R.M. Fricks is with the SIMEPAR Laboratory and Ponticia Universidade Catolica do Parana, Curitiba/PR, Brazil. A. Puliato is with the Istituto di Informatica, Uni- versita di Catania, Catania, Italy. M. Telek is with the Department of Telecommunications, Technical University of Budapest, Budapest, Hungary. K.S. Trivedi is with the Department of Electrical and Computer Engineering, Duke University, Durham/NC, USA. E-mails: fricks@simepar.br, ap@iit.unict.it, telek@hit.bme.hu, and kst@ee.duke.edu. system resources, the original paradigm was not complete enough to capture other elements indispensable for dependability and performance modeling of systems. Thus, new extensions allowing for time and randomness abstractions became nec- essary. Despite the consensus on which elements to add, a certain uncertainty existed on where to aggregate the proposed extensions. From among several alternatives, a dominant one was soon established where the Petri nets could have transitions that once enabled would re according to exponential distributions with dierent rates (EXP transitions). This led to well known net types: Generalized Stochastic Petri Nets (GSPNs) [1] and Stochastic Reward Nets (SRNs) [2]. The resulting modeling framework allowed the denition and solution of stochastic problems enjoying the Markov property [3]: the probability of any particular future behavior of the process, when its current state is known exactly, is not altered by additional knowledge concerning its past behavior. These Markovian stochastic Petri nets (MSPNs) were very well accepted by the modeling community since a wide range of real dependability and performance models fall in the class of Markov models. Besides the ability to capture various types of system dependencies intrinsic to the underlying Markov models, other advantages of the Petri net framework also contributed to the popularity of the MSPNs. Among these reasons, we point out the power of concisely specifying very large Markov models, and the equal ease with which steady-state, transient, cummulative transient and sensitivity measures could be computed. One of the restrictions, however, is that only exponentially distributed ring times are captured. This led to the development of non-Markovian stochastic Petri nets. Non-Markovian stochastic Petri nets (NMSPNs) were then proposed to allow for the high level description of non-Markovian models. Likewise in the original evolutive chain, several alternative approaches to extend the Markovian Petri nets were proposed. Their distinctive feature was the underlying analytical technique used to solve the non-Markovian models. Candidate solution methods considered included the deployment of supplementary variables [4], the use of phase-type expansions approximations [5, 6], and the application of Markov renewal theory [7, 8]. Representative non-Markovian Petri nets proposed, listed according to the underlying solution techniques, are the Extended Stochastic Petri Nets (ESPNs) [9], the Deterministic and Stochastic Petri Nets (DSPNs) [10], the Stochastic Petri Nets with Phase-Type Distributed Transitions (ESPs) [11], and the Markov Regenerative Stochastic Petri Nets (MRSPNs) [12]. As a consequence of these evolutive steps, we observe that the restriction imposed on the distribution functions regulating the ring of timed transitions was progressively relaxed from exponential distributions to a combination of exponential and deterministic distributions, then to any distribution represented by phase type approximations, and nally to any general distribution function (GEN transitions). However, this exibility also brought a new requirement with it. If an enabled GEN transition is disabled before ring, a scheduling policy is needed to complete the model denition. Consider the generic client/server NMSPN model in Fig. 1 for instance. Requests from clients arrive according to a Poisson process (EXP transition t 1 ). Tokens in place clients already in the system. In a single server conguration only one of the queued requests will be serviced at a given time. The service requirement g of each request is sampled from a general distribution function G g (t) that coordinates the ring of the GEN transition t 2 . An age variables a g associated with a request keeps track of the amount of service actually received by the request. Service will be completed (i.e., transition will re) as soon as the age variable a g of the active request (the one receiving server's attention) reaches the value of its service requirement . After that, the request leaves the system and its associated age variable is destroyed. Furthermore, suppose that the server is failure- Figure 1: Fault-tolerant client/server model. prone with constant failure and repair rates. A token in place P 2 represents the active state of the server while a token in place P 3 indicates server being down (undergoing repair). Consequently, r- ing of the EXP transitions t 3 and t 4 correspond to the failure and end-of-repair events associated with the server. Whenever down, the server cannot service new clients or complete the service requirement of the current request, as shown by the inhibitor arc from place P 3 to transition t 2 . Clearly a scheduling policy is then necessary to precisely dene how the server must proceed when brought up again. In MSPNs with EXP transitions this was not a problem because of the memoryless property of the exponential distributions [3] 1 . The remaining processing time of an nterrupted request is also represented by the EXP transition t 2 . In the favorable case, the server is able to completely service the current request before a failure occurs (as shown in Fig. 2a). Otherwise the system behavior depends on the amount of remaining service at the time of the interruption, and whether the service already received by the request will be discarded. The service requirement may increase or decrease as an indirect consequence of system events responsible by the server interruption. For instance, the failure of the server in Fig. 1 may render certain activities of the client unnecessary, which would then reduce its service requirement to a lower value. Likewise, the age variable a g related to the active request may also be affected by the server interruption since the amount of service already provided to the request may be 1 If the scheduling policy is non-work-conserving and the service requirement of the client needs to be preserved then even the EXP transition has to be dealt with like a GEN transition. preserved or lost. We distinguish both situations calling the rst a work conserving scheme, and the second non-work-conserving. With these four conditions we constructed the table in Fig. 2b. Note that, although the service requirement is shown to be increasing after the interruption in the illustration in the bottom row of the table, the situation where g is also possible 2 . a) a g a g work conserving service preserved service modified a a a g task service started service completed prd pri prs non-working-conserving Figure 2: Dierent scheduling policies. Fig. 2b can be interpreted from two distinct per- spectives. From the clients' perspective, all curves correspond to the same client whose service is momentarily interrupted between times 2 and 3 . From the server's perpective, clients requests live only from interruption-to-interruption. There is a single age variable associated with the server, and what happens after interruptions is dened by the scheduling policy which may be preemptive or non- preemptive, depending on if the server swaps clients before nishing service or not. Preemptive policies are usually based on a hierarchical organization of requests (e.g., priority scheduling) or on an allocation of service based on time quotas (e.g., round-robin scheduling). In this case, system behavior is strongly aected by the preemptive policy and Naturally, ag at the time of the interruption needs to be always imposed. the overall performance will depend on the strategy adopted to deal with the preempted requests, as described in the following: The work done on the request prior to interruption is discarded so that the amount of work a g is lost. The server starts processing a new request which has a work requirement 0; i.e., a new sample is drawn from the service time distribution of the client. The server then starts serving this new request from the beginning (i.e., a shown in the bottom-right sketch in Fig. 2b. The server returns back to the preempted request with the original service requirement . No work is lost so that the age variable retains its value a g prior to the interruption. The request is resumed from the point of interruption as shown in the top-left sketch in Fig. 2b. The server also returns to the same request with the original service requirement . But the work done prior to the interruption is lost and the age variable a g is set to zero. The request processing starts from the beginning as shown in the top-right sketch in Fig. 2b. As in [13], the above policies are referred to as preemptive repeat dierent (prd), preemptive resume (prs) and preemptive repeat identical (pri), respectively 3 . The case shown in the bottom-left sketch in Fig. 2b is not considered in the literature as it is unrealistic. Note that in [15], the authors indicated the prd and prs type policies as enabling and age type. The pri policy of Petri net transitions was introduced for the rst time in [16]. The prd and prs (with phase-type distributed ring times) policies are the only ones considered in the available tools modeling NMSPNs [17, 11, 18, 19]. Note that when the scheduling is preemptive: (i) the prs and prd policies produce the same results with EXP transitions, but pri is dierent; (ii) The prd and pri policies have the same eect for transitions ring according to a deterministic random variable, but prs is dierent; and (iii) otherwise, all three policies will produce distinct results for otherwise same NMSPNs [14]. 3 The prd, prs and pri names were borrowed from queueing theory [14]. In this paper, we deal with the general class of non-Markovian Petri nets using examples of MR- SPNs, which can be analyzed by means of Markov regenerative processes. The remaining sections of the paper are organized as follows. The next section introduces Markov Regenerative Petri nets and describes how to deal with the underlying Markov Regenerative Process. Section 3 shows how to model a failure/repair process in a parallel machine through MRSPN. Section 4 further extends this model by adopting a dierent repair facility scheduling scheme. Preemption in a multi-tasking environment is analyzed in Section 5 through the WebSPN tool; the resulting model contains several concurrently enabled general transitions and dierent memory policies. Conclusions are nally presented in Section 6. Regenerative Petri Nets MRSPNs allow transitions with zero ring times (immediate transitions), exponentially distributed or generally distributed ring times. The dynamic behavior of an MRSPN is modeled by the execution of the underlying net, which is controlled by the position and movement of tokens. At any given time, the state of an MRSPN is dened by the number of tokens in each of its places, and is represented by a vector called its marking. The set of markings reachable from a given initial marking (i.e., the initial state of the system) by means of a sequence of transition rings denes the reachability set of the Petri net. This set together with arcs joining its markings and indicating the transition that cause the state transitions is called reachability graph. Two types of markings can be distinguished in the reachability graph. In a vanishing marking at least one immediate transition is enabled to re, while in a tangible marking no immediate transitions are enabled. Vanishing markings are eliminated before analysis of the MRSPN using elementary probability theory [12]. The resultant reduced reachability graph is a right-continuous, piecewise constant, continuous-time stochastic process represents the tangible marking of the MRSPN at time t. Choi, Kulkarni, and Trivedi [12] showed that this marking process is a Markov Regenerative Process (MRGP) (if the GEN transitions are of prd type and at most one GEN transition is enabled at a time), a member of a powerful paradigm generally grouped under the name Markov renewal theory [7, 8]. Mathematical denition and solution techniques for MRGP are summarized next. 2.1 Markov Renewal Sequence Assume a given system we are modeling is described by a stochastic process Z d taking values in a countable set . Suppose we are interested in a single event related with the system (e.g., when all system components fail). Additionally, assume the times between successive occurrences of this type of event are independent and identically distributed (i:i:d:) random variables. be the time instants of successive events to occur. The sequence of non-negative i:i:d: random variables, :::gg is a renewal process [20, 21]. Otherwise, if we do not start observing the system at the exact moment an event has occurred (i.e., S 0 6= 0) the stochastic process S is a delayed renewal process. However, suppose instead of a single event, we observe that certain transitions between identi- able system states Xn of a subset of , also resemble the behavior just described, when considered in isolation. Successive times Sn at which a xed state Xn is entered form a (possi- bly delayed) renewal process 4 . Additionally, when studying the system evolution we observe that at these particular times the stochastic process Z exhibits the Markov property, i.e., at any given moment Sn , n 2 N , we can forget the past history of the process. The future evolution of the process depends only on the current state at these embedded time points. In this scenario we are dealing with a countable collection of renewal processes progressing simultaneously such that successive states visited form an embedded discrete-time Markov chain (EMC) with state space The superposition of all the identied renewal processes gives the points known as Markov regeneration epochs (also called Markov renewal moments 5 ), and to- 4 We are assuming Xn is the system state at time Sn . 5 Note that these instants Sn are not renewal moments gether with the states of the EMC dene a Markov renewal sequence. In mathematical terms, the bivariate stochastic process (X; S) d is a Markov renewal sequence (MRS) provided that for all n 2 N , and t 0. We will always assume time-homogeneous MRS's; that is, the conditional transition probabilities are independent of n for any fore, we can always write The matrix of transition probabilities K(t) d is called the kernel of the MRS. 2.2 Markov Regenerative Processes A stochastic process fZ t ; t 0g is a Markov regenerative process i it exhibits an embedded MRS (X,S) with the additional property that all conditional nite distributions of fZSn+t ; t 0g given are the same as those of fZ As a special case, the denition implies that [8] This means that the MRGP fZ does not have the Markov property in gen- eral, but there is a sequence of embedded time points such that the states respectively of the process at these points satisfy the Markov property. It also implies that the future of the process Z from onwards depends on the past fZ only through The stochastic process between consecutive Markov regeneration epochs, usually refered to as described in renewal theory, since the distributions of the time interval between consecutive moments are not necessarily i.i.d. as subordinated process, can be any continuous-time discrete-state stochastic process over the same probability space. Recently published examples considered subordinated homogeneous CTMCs [12, 22], non-homogeneous CTMCs [23], semi-Markov processes (SMPs) [24], and MRGPs [25]. 2.3 Solution of Problems 0g be a stochastic process with discrete state space and embedded MRS K(t). For such a process we can dene a matrix of conditional transition probabilities as: In many problems involving Markov renewal pro- cesses, our primary concern is nding ways to effectively since several measures of interest (e.g., reliability and availability) are related to the conditional transition probabilities of the stochastic process. At any instant t, the conditional transition probabilities of Z can be written as [7, 8]: for all i 2 we construct a then the set of integral equations V ij (t) denes a Markov renewal equation, and can be expressed in matrix form as Z tdK(u)V(t u); (1) where the Lebesgue-Stieltjes integral 6 is taken term by term. To better distinguish the roles of matrices E(t) and K(t) in the description of the MRGP we callR tdK(u)V (t a density function dt . the matrix E(t) as the local kernel of the MRGP, since it describes the state probabilities of the subordinated process during the interval between successive Markov regeneration epochs. Since matrix K(t) describes the evolution of the process from the Markov regeneration epoch perspective, without describing what happens in between these moments we call it the global kernel of the MRGP. In the special case when the stochastic process Z does not experience state transitions between successive Markov regeneration epochs; i.e., Z is called a semi-Markov process and E(t) is a diagonal matrix with elements where is the sojourn time distribution in state i. Hence, the global kernel matrix alone (which in this case is usually denoted as Q(t)) completely describes the stochastic behavior of the SMP. The Markov renewal equation represents a set of coupled Volterra integral equations of the second kind [26] and can be solved in time-domain or in Laplace-Stieltjes domain. One possible time domain solution is based on a discretization approach to numerically evaluate the integrals presented in the Markov renewal equation. The integrals in Eqn. 1 are solved using some approximation rule such as trapezoidal rule, Simpson's rule or other higher order quadrature methods. Another time domain alternative is to construct a system of partial dierential equations (PDEs), using the method of supplementary variables [4]. This method has been considered for steady-state analysis of DSPNs in [22] and subsequently extended to the transient case in [27]. An alternative to the direct solution of the Markov renewal equation in time-domain is the use of transform methods. In particular, if we st dE(t) and V st dV(t), the Markov renewal equation become After solving the linear system for V (s), transform inversion is required 7 . In very simple cases, a closed-form inversion might be possible but in most cases of interest, numerical inversion will be necessary. The transform inversion however can encounter numerical di-culties especially if V has poles in the positive half of the complex plane. For a thorough discussion of Markov renewal equation solution techniques see [28, 29], and for generic Volterra integral equations numerical methods see [30, 31]. References for the application of Markov renewal theory in the solution of performance and reliability/availability models see [16, 32, 23, 28, 33, 34, 35, 36, 37]. Modeling Failure/Repair Activities in a Parallel Machine Conguration The use and analysis of MRSPNs is initially demonstrated using a computer system performability model. Two machines (a and b) are working in a parallel conguration sharing a single repair facility with a First-Come First-Served scheduling discipline. Due to the non-preemptive nature of this discipline, we do not need age variables in this case (once enabled all GEN transitions in the model will never be disabled until ring). We assume that both machines have exponential lifetime distributions with constant parameters a and b respectively. Whenever one of the machines fails it immediately requests repair. When the single repair facility is busy and a second failure occurs, the second machine to fail waits in a repair queue until the rst machine is put back into service. The repair-time of the machines is dened by the general distribution functions G a (t) and G b (t). The overall behavior of the system can be understood from the MRSPN illustrated in Fig. 3a. Machine a is working whenever there is a token in place P 1 . The EXP transition f a with rate a represents the failure of machine a. When machine a fails, a token is deposited in place P 6 and its repair is requested. If the repair facility is available (i.e., 7 This being the approach addopted in the solution of all examples presented in this paper. a f a f a f b a r f a f b f a f bc) a a r a b a f f r b a 2: 3: 4: 5: 7: a) b)r rb a Figure 3: Parallel system model: a) MRSPN; b) reachability graph; and c) state transition diagram. there is a token in place P 5 ), it is appropriated with the ring of immediate transition i a . The GEN transition r a , ring according to the distribution function G a (t), represents the random duration of repair. A token in place P 3 means that machine a is queued waiting for the availability of the single repair facility while machine b is undergoing repair (there is a token in place P 7 ). A symmetrical set of places and transitions describes the behavior of machine b. The system is down whenever there are no tokens in both the places P 1 and P 2 . The reachability graph corresponding to the Petri net is shown in Fig. 3b. Each marking in the graph is a 7-tuple keeping track of the number of tokens in places P 1 through P 7 . In the graph, solid arcs represent state changes due to the ring of immediate transitions or EXP tran- sitions, while dotted arcs denote the ring of GEN transitions. The vanishing markings (enclosed by dashed ellipses in the diagram) are eliminated when the reduced reachability graph is constructed (not shown), and based on the reduced version we constructed the state transition diagram of Fig. 3c. Dene the stochastic process to represent the system state at any instant, where both machines are working at t machine a is under repair while machine b is working at t 3 if machine b is under repair while machine a is working at t 4 if machine a is under repair while machine b is waiting for repair at t 5 if machine b is under repair while machine a is waiting for repair at t Note that possible values of Z t are the labels corresponding tangible markings in Fig. 3b. We are interested in computing performability measures associated with the system. To do so, we need to determine the conditional probabilities PrfZ 5g. Analysis of the resultant (reduced) reachability graph shows that Z is an MRGP with an EMC dened by the states 1, 2, and 3; i.e., 3g. We can observe that transitions to states 4 and 5 do not correspond to Markov renewal epochs because they occur while GEN transitions are enabled. An additional step adopted before starting the synthesis of the kernel matrices was the construction of a simplied state transition diagram. Fig. 3c shows a simplied version of the reduced reachability graph where the markings were replaced by the corresponding state indices. We preserved the convention for the arcs and extended the notation by representing states of the EMC by circles, and other states by squares. The construction of kernel matrices can proceed with the analysis of possible state transitions. The only non-zero elements in global kernel matrix K(t) correspond to the possible single-step transitions between states of the EMC. Consequently, we have the following structure of the matrix (identied directly from Fig. 3c): Let the random variables L a and L b be the respective time-to-failure of the two machines, we can determine K 1;2 (t) in the following way: is the first one to failg d Z te b a e a d Similarly, is the first one to failg Determination of the elements K 2;1 (t) and K 2;3 (t) is quite alike, so we only show how K 2;1 (t) is determined. The third row is completelly symmetrical to the second, so it can be easily undestood once K 2;1 (t) is understood. We need some auxiliary variables to help in the explanation of the constructive process of K 2;1 (t). Hence, we dene the random variables R a and R b to respectively represent times necessary to repair machines a and b. The distribution function of R a (R b ) is G a (G b ). Using this new variables we can compute K 2;1 (t): of a is finished by time t and b has not failed during the repair of ag Z tP rfL b > gdG a () dG a () Z te b dG a (): To summarize, the elements of the global kernel matrix are: Z te b dG a (); Z t1 e b dG a (); Z te a dG b (); and Z t1 e a Note that the global kernel will always be a square matrix. In this case with dimensions 3 3, since we have 3 states in the embedded Markov chain. However, the local kernel matrix is not necessarily a square matrix, since the cardinality of the state space of Z can be larger than the cardinality of the state space of the embedded Markov chain. This can be seen, for instance, in this system since the embedded Markov chain has only 3 states while the MRGP has 5 possible states. We construct the local kernel matrix E(t) following a similar inductive procedure. In this case we are looking for the probability that the MRGP will move to a given state before the next Markov renewal moment. Careful analysis of Fig. 3c reveals the structure of the local kernel matrix E(t):4 E 1;1 (t) Since in a single step the system can only go from state 1 to the other two states of the EMC then E 1;1 should be the complementary sojourn time distribution function in state 1, that is, The di-culty comes with the induction of E 2;2 (t) (complement of E 2;2 (t)). Once we solve for these, we have the solution for the remaining components of the matrix due to the symetry of the problem. Therefore, we explain the induction process that leads to E 2;2 (t): of a is not finished up to t and b has not failed until tg of a is not finished up to t P rfb has not failed until tg We can now express the remaining non-zero elements of the local kernel matrix as G c a (t) G c with G c a G c We can always verify our answers by summing the elements in each row of both kernel matrices. Corresponding row-sums of the two matrices must add to unity, condition that is easily veried to hold in the example. time (hours)0.9981.000 availability instantaneous interval time (hours)1.921.962.00 interval power time (hours)0.9981.000 availability instantaneous interval time (hours)1.921.962.00 interval power results: results: Figure 4: Numerical results for the parallel system with non-premptive repair. The kernel matrices determined can then be substituted in Equation (1) and the resultant system of coupled integral equations solved using one of the approaches described in [28, 29]. The resultant plots, labelled LST in Fig. 4, report system availability and performability computed when time to repair is deterministic; i.e., G a where U(t) is the unit step function; the failure rates (parameters a and b ) are identical b ) takes 5 hours. The interval availability is the expected proportion of time the system is operational during the period [0; t]: Z tE[X()]d; when the discrete random variable X represents the operational status of the system; i.e., the system is operatinal at time t, and 0 if it is not. The performability measure plotted in the gure corresponds to the interval processing capacity of the system, with the convention that a unit of computing capacity corresponds to that of one active machine. Following the approach used in [34], we also plotted corresponding Markovian system results, where each DET transition was replaced by an equivalent 25-stage Erlang subnet. The Markovian models were solved using the Stochastic Petri Net Package (SPNP) introduced in [38]. 4 Preemptive LCFS repair Fig. 5 shows the PN which describes the behavior of the system containing the same machines a and b of the previous example and applies the preemptive LCFS scheduling scheme. The repair of machine a (b), represented by a token at P 6 (P 7 ) is preempted as soon as machine b (a) fails, i.e., transition f b (f a ) res. In this case the repair facility is assigned to the machine which failed later (i 0 a or i 0 b res and a token is placed to P 8 or P 9 ). After the repair of the last failed machine (ring of r 0 a or r 0 b ) the repair facility returns to the completion of preempted repair action. Dierent memory policies can be considered depending on whether the repairman is able to \remember" the work already performed on the machine before preemption or not. In the case r r r 6 ?/ R f a i a r a a a Figure 5: Preemptive LCFS repair with non-identical machines that the prior work is lost due to the interruption and the repair must be repeated from scratch with an identical repair time requirement (pri policy) or with a repair time resampled from the original cumulative distribution function (prd policy). In the case that the prior work is not lost and the time to complete the preempted repair equals the residual repair time given the portion of work already completed before preemption (prs policy). The PN on Fig. 5 captures the dierent memory policies for repair by assigning transitions r a and r b the appropriate preemption policies. (The preemption policies of transitions r 0 a and r 0 b are not relevant since a and r 0 b cannot be preempted.) We analyze a simplied version of the two machine system with preemptive LCFS repair and with prs policy. We assume that the two machines are statistically identical, i.e., their failure and repair time distributions are the same. Fig. 6a shows a PN which describes the behavior of the system of two identical machines with LCFS scheduling. Tokens in place tokens in P 2 count failed machines (including the one under repair), and a token in place P 4 the availability of the single repair facility. In the initial marking is the only enabled transition. Firing of t 1 represents the failure of the rst machine and leads to state M In are competing. The GEN transition t 2 represents the repair of the failed machine and its ring returns the system to the initial state M 1 . The EXP transition t 3 represents the failure of the second machine and its ring disables q A A AU A A AU A A AU A A AU A A AU Figure Preemptive LCFS repair with identical machines removing one token from P 3 (the rst repair becomes dormant). In M is under repair and the other repair is dormant, and the only enabled transition is the repair of the last failed machine. Firing of the GEN transition t 4 leads the system again to M 2 , where the dormant repair is resumed. Assume that the failure times of both machines are exponentially distributed with parameter so that the EXP transitions t 1 and t 3 have ring rates 2 and , respectively. The preemptive policy of transition t 2 has to be assigned based on the system behavior to be eval- uated. (The preemptive policy of transition t 4 is irrelevant since t 4 can not be preempted.) Assigning a prd policy to t 2 means that each time t 2 is disabled by the failure of the second machine (t 3 res before t 2 ), the corresponding age variable a 2 is reset. As soon as t 2 becomes enabled again (the second repair completes and t 4 res) no memory is kept of the prior repair period, and the execution of the repair restarts from scratch. The prd service policies, like this one, are covered by the model definition in [39, 40]. The case when a pri policy is assigned to t 2 is very similar to the previous one except that as soon as t 2 becomes reenabled (the second repair completes and t 4 res), the same repair (same ring time sample) has to be completed from the begin- ning. This type of pri memory policy is covered by the model denition in [16], and can be analyzed by the transform domain method discussed there. Hereafter we assume that a prs policy is assigned to t 2 . When a prs policy is assigned to t 2 , each time t 2 is disabled without ring (t 3 res before t 2 ) the age variable a 2 is not reset. Hence, as the second repair completes (t 4 res), the system returns to keeping the value of a 2 , so that the time to complete the interrupted repair can be evaluated as the original repair requirement minus the current value of a 2 . The age variable a 2 counts the total time during which t 2 is enabled before r- ing, and is equal to the cumulative sojourn time in . The Markov renewal moments in the marking process correspond to the epochs of entrance to markings in which the age variables associated with all the transitions are equal to zero. By inspecting Fig. 6b, the Markov renewal moments are the epochs of entering M 1 and of entering M 2 from The subordinated process starting from marking 1 is a single step CTMC (since t 1 the only enabled EXP transition) and includes the only immediately reachable state M 2 (Markovian regeneration period). The subordinated process starting from marking includes all the states reachable from M 2 before ring of t is the only state in which t 2 is enabled, the age variable a 2 increases only in marking M 2 and maintains its value in M 3 . The ring of t 2 can only occur from leading to marking M 1 . Notice that the subordinated process starting from M 2 is semi-Markov since the ring time of t 4 is generally distributed. The age variable a 2 grows whenever the MRSPN is in marking M 2 , and the ring of t 2 occurs when a 2 reaches the actual value of the ring time (which is generally distributed with cumulative distribution function G(t)). If we condition that the ring time of t 2 to w, w acts an absorbing barrier for the accumulation functional represented by the age variable a 2 , the ring time of t 2 is determined by the rst passage time of a 2 across the absorbing barrier w. The closed form Laplace-Stieltjes transform expressions of the kernel matrices of the LCFS repair prs case are derived here in detail, applying the technique based on the Markov renewal theory. We build up the K row by row by considering separately all the states that can be regeneration states and can originate a subordinated process. M 3 can never be a regeneration state since t 2 is always active when entering to M 3 , g. The fact that M 3 is not a regeneration marking, means that the process can stay in M 3 only between two successive Markov renewal moments. The starting regeneration state is M 1 - (Markovian regeneration period) No general transition is enabled and the next regeneration state can only be state M 2 . The non-zero elements of the rst row of the kernel matrices are The starting regeneration state is M 2 - Transition t 2 is GEN so that the next regeneration time point is the epoch of ring of t 2 . The subordinated process starting from M 2 comprises states M 2 and M 3 and is an SMP (since t 4 is GEN) whose kernel is: G where G (s) is the LST of the distribution function of the ring time of t 4 . The transition t 2 res when the age variable a 2 reaches actual sample of the ring time 2 . In gen- eral, when a GEN transition is active the occurence of a Markov renewal epoch in the marking process of an NMSPN is due to one of the following two reasons: the GEN transition res, the GEN transition of prd type becomes disabled For the analysis of subordinated processes of this kind three matrix functions F i (t; w), D i (t; w) and denotes the time, w a xed r- ing time sample, and the superscript i refers to the initial (regeneration) state of the subordinated pro- cess) were introduced in [24]. F i (t; w) refers to the case when the next regeneration moment is because of the ring of the GEN transition with the (xed) ring time sample w. For the analysis of this case an additional matrix ( i referred to as branching probability matrix) is introduced, as well, to describe the state transition subsequent to the ring of the GEN transition. D i (t; w) captures the case when the next regeneration moment is caused by the disabling of the prd type GEN transition. And describes the state transition probabilities inside the regeneration period. Since transition t 2 is of prs type the matrix function does not play a role in the analysis of the subordinated process starting from marking . The remaining functions can be evaluated based on the kernel of the subordinated SMP u' (s; v) s u' (s; v) k' (t); s is the time variable and v is the barrier level variable in transform domain; r k is the indicator that the active GEN transition is enabled in state k; R i is the part of the state space reachable during the subordinated process; and the superscript () refers to Laplace-Stieltjes (Laplace) transform. Given that G g (t) is the distribution function of the ring time of the GEN transition, the elements of the i-th row of matrices K(t) and E(t) can be expressed as follows, as a function of the matrices dG g (w) To evaluate the 2nd row of the kernel matrices we are applying these results for the subordinated process starting from regeneration state M 2 . Doing so we obtain the following expressions for the non-zero matrix entries: F 22 (s; 22 (s; Unconditioning with respect to the ring time distribution of t 2 , and after inverting the Laplace transform (LT) with respect to v, the non-zero entries of the 2nd row of the LST matrix functions K e w(s+ G (s)) dG(w) 22 The LST of the state probabilities are obtained by solving the Markov renewal equation in transform domain. The time domain probabilities are calculated by numerically inverting the result by resorting to the Jagerman method [41]. To evaluate the performance of the dierent scheduling schemes, we compared the availability and processing power of the FCFS and the LCFS repair schemes with two dierent repair time dis- tributions. The FCFS scheme was evaluated by the time domain method introduced in the previous section and the LCFS scheme was evaluated by the above transform domain method. It is assumed that the system is available when at least one machine is working (marking M 1 and M 2 ) and that the system performance doubles when both machines are working. The failure times of both machines are exponentially distributed with rates 0:01. The repair times of both machines are assumed to be: deterministic G hyperexponentially distributed with G The mean repair time is 5 in both cases. Fig. 7a and 7b show the instantaneous and the interval measures of availability and processing power with deterministic repair time, respectively. The dotted line shows the instantaneous and the short dashed line shows the interval availability/power with LCFS repair, while the long dashed line shows the instantaneous and the solid line shows the interval availability/power with FCFS repair. It can be observed that the FCFS scheduling performs better in this case. The availability and processing power results for the hyperexponential repair time distribution are plotted on Fig. 7c and 7d, respectively. In these gures the dotted line shows the instantaneous availability/power with LCFS repair, while the dashed line shows the instantaneous availabil- ity/performability with FCFS repair. As can be seen from these gures, in contrast with the deterministic repair time the LCFS scheduling performs better with the hyperexponential repair time distribution Modeling preemption in a multi-tasking environment NMSPN require complex solution techniques mainly based on theory of Markov regenerative pro- cesses. Software packages are then required which can hide solution and implementation details. A big boost in this direction came from two well-known tools, DSPNexpress [42] and TimeNET [43, 44]. Recently, a new software package for non-Markovian Petri nets has been developed in a joint eort between the Universities of Catania and Bu- dapest. This tool, named WebSPN [45], provides a discrete time approximation of the stochastic behaviour of the marking process which results in the possibility to analyze a wider class of PN models with prd, prs and pri concurrently enabled generally distributed transitions. The approximation of the continuous time model at equispaced discrete time points involves the analysis of the system behavior over a time interval based on the system state at the beginning of the interval and the past history of the system. A Web- centered view has been adopted in its development in order to make it easily accessible from any node connected with the Internet as long as it possesses a Java-enabled Web browser. Sophisticated security mechanisms have also been implemented to regulate the access to the tool which are based on the use of public and private electronic keys. WebSPN is available at the following site: http://sun195.iit.unict.it/webspn/webspn2/ 5.1 Model description In this section we describe and solve a model of Petri net with several concurrently enabled GEN transitions and dierent memory policies. The system moves between an operative phase, where useful work is produced, and a phase of maintenance where the processing is temporarily interrupted. The Petri net shown in Fig. 8 represents the model of the system that consists of three functional blocks generically referred to as Block1, Block2 and Block3. Block1 models the alternation of the system between the operative phase and the maintenance phase. Block2 models the two sequential phases of processing of jobs. Finally, Block3 models the alternation of the system during the operative phase between the phase of pre-processing and the one of processing of jobs. Within Block1, the two states of operation where the system can be are represented by places user and system and by transitions U time and S time. A token in place user denotes the operative state, while a token in place system denotes the maintenance one. The duration of the operative phase is denoted by transition U time, while the maintenance one is denoted by transition S time. The inhibitor arcs outgoing from place system and leading to the timed and immediate transitions contained in Block2 and Block3 producer, cons1, busy prod, idle prod, busy2, idle2 are used for interrupting the activity of the system during the phase of maintenance Block2 models the processing of jobs. In partic- ular, the number of jobs to be processed is denoted by the number of tokens contained in place work, while the time of pre-processing of each job is represented by transition producer. Pre-processed jobs are queued in a buer (place bu1) waiting for the second phase of processing (transition cons1). In Block3, the alternation between the phases of pre-processing and processing of jobs is represented through places slot1 and slot2 and transitions busy brod, busy2, idle prod, idle2. A token in place slot1 denotes that the system is executing the pre-processing of a job, while a token in place slot2 denotes the execution of a phase of processing. An inhibitor arc between slot1 and cons1 deactivates the phase of processing when the pre-processing one is active. In the same way, the inhibitor arc between slot2 and producer deactivates the phase of pre-processing when the processing one is ac- tive. The time that the system alternately spends for these two activities is represented by transitions busy prod and busy2. The immediate transition idle prod (idle2) prevents the system to remain in phase 1 (2), even if no job is to be processed. The function of the inhibitor arcs from place work to transition idle prod and from place bu to transition idle2 is to enable such transitions when no job is to be processed in the corresponding phase of processing. Immediate transition end and place Stop are used for modeling the processing of all the jobs assigned to the system at the beginning. In fact, transition end is inhibited until at least one token is present in places work and bu. When all the jobs have been processed, transition end res, and immediately moves a token to place Stop. All the activities of the system are thus interrupted through the inhibitor arcs outgoing from place Stop. The measure that we evaluate from this model is the distribution of the time required for completing the set of jobs assigned to the system at the beginning. It can be obtained as the distribution of having a token in place Stop. With regard to the distributions of the ring times to be assigned to timed transitions, we assume that the ring times of transitions U time, S time, busy brod, busy2 are deterministic. We assume that the ring times of transitions producer and cons1 are respectively distributed uniformly and exponentially. The measures considered can therefore be evaluated by changing the memory policy associated with transitions producer and cons1. In the case of prd policy, the temporary interruption of the processing of a job (either because the whole system enters the phase of maintenance, or because, even if the system is in the production phase, it interrupts the pre-processing phase for changing to the processing one or vice versa) causes the interrupted job to be discarded. A new job is executed when the system is available again. The correspondence with a real system is perhaps hard to nd; however, we note that prd policy is the most commonly used one in the literature. Conversely, by adopting prs policy, we keep a memory of the work that we were executing. In this case, when transition producer is disabled, we keep a memory of the work that has already been executed on the job considered. When the system enters the operative state again, the pre-processing of the job continues from the point we had reached. In this case, the model can represent a system of manufacturing, where a machine used for production alternates cycles of production and cycles of maintenance, and production takes place in two sequential phases. We note that prd and prs policies are equivalent for transition cons1, since this one is and EXP transition. With pri policy, when transition producer is dis- abled, the work that had already been produced is lost, but we keep a memory of the job that we were processing. When the transition is enabled again we start from zero, but the amount of work to be produced on the job remains the same, because the job has not been changed. Such a behavior can be easily noted when accessing transactional databases, where each transaction is atomic (i.e., has to be processed with no interruption). If an interruption occurs, the transaction is entirely processed again. If we assume a memory policy like prs for transition cons1, the model could represent a client/server system where the accesses to database (transition producer) take place atomically, and the phase of processing of the query (transition cons1) requires a variable time, distributed exponentially. 5.2 Numerical Results For the solution of the model we assume that the ring time of transition producer is distributed uniformly between 0.5 and 1.5; the ring time of transitions time and S time are deterministic, with a ring time of 1; the ring time of transitions busy prod and busy2 are deterministic, with a r- ing time of 0.1; the ring time of transition cons1 is distributed exponentially, with a ring rate of transition end is immediate and has a priority of 2; transitions idle prod and idle2 are immediate and have a priority of 1; the total number of jobs to be processed is 3. In Fig. 9 we show the distribution of completion time for dierent memory policies assigned to transitions producer and cons1. The behavior of the system changes signicantly depending upon the memory policy adopted. The prs policy accrues the highest probability of completion within a given time. Both the prd and the prs policies accomplish the completion of jobs. In fact, curves eventually reach the value 1. Conversely, a dier- ent behavior can be observed if we assume a policy like pri. In fact, in that case, the resulting distribution is defective, since the unit value is never reached for t ! 1. This is closely connected with the choice of the parameters associated with transitions producer and U time. As we note in Fig. 10, when the ring time of transition U time is lower than 1.5, transition producer has a positive probability (50%)of not completing its work. Since in the case of pri policy the job is processed with the same work requirement, this causes a situation of impasse, which prevents the work assigned to the system to be completed. Fig. 11 shows how the overall system behavior changes if transition U time is assigned a ring time higher than 1.5 (for example 2.0). In such case, transition producer has a nite probability of ring before the system enters the phase of maintenance, and therefore the distribution of completion time with pri policy reaches the value 1. 6 Conclusion We discussed the need for more advanced techniques to capture generally distributed events which occur in everyday life. Among the dier- ent approaches proposed in the literature, non-Markovian nets represent a valid analytical alternative to numerical simulation. An approach based on the analysis of the underlying Markov Regenerative Process has been presented. Advanced preemption policies were introduced and several examples solved in detail. --R Modeling and Analysis of Stochastic Systems Renewal Theory An Introduction to Probability Theory and Its Applications The Theory of The Volterra Integral Equation of the Second Kind The Numerical Solution of Volterra Equations Analytical and Numerical Methods for Volterra Equations --TR --CTR Giacomo Bucci , Andrea Fedeli , Luigi Sassoli , Enrico Vicario, Timed State Space Analysis of Real-Time Preemptive Systems, IEEE Transactions on Software Engineering, v.30 n.2, p.97-111, February 2004

stochastic Petri nets;numerical analysis;markov regenerative processes;preemption policies