one way communication complexity and the neciporuk lower bound on formula size in this paper the neciporuk method for proving lower bounds on the size of boolean formulas is reformulated in terms of one way communication complexity we investigate the settings of probabilistic formulas nondeterministic formulas and quantum formulas in all cases we can use results about one way communication complexity to prove lower bounds on formula size the main results regarding formula size are as follows we show a polynomial size gap between probabilistic quantum and deterministic formulas a near quadratic gap between the sizes of nondeterministic formulas with limited access to nondeterministic bits and nondeterministic formulas with access to slightly more such bits and a near quadratic lower bound on quantum formula size furthermore we give a polynomial separation between the sizes of quantum formulas with and without multiple read random inputs the lower bound methods for quantum and probabilistic formulas employ a variant of the neciporuk bound in terms of the vapnik chervonenkis dimension to establish our lower bounds we show optimal separations between one way and two way protocols for limited nondeterministic and quantum communication complexity and we show that zero error quantum one way communication complexity asymptotically equals deterministic one way communication complexity for total functions

span programs are equivalent to quantum query algorithms span programs form a linear algebraic model of computation that is studied to prove classical lower bounds quantum query complexity is a coherent generalization for quantum algorithms of decision tree complexity it is characterized by a semidefinite program known as the general adversary bound we connect these classical and quantum models by proving that for any boolean function the optimal witness size of a span program equals the general adversary bound therefore span program witness size and quantum query complexity are equivalent measures in particular quantum algorithms can be designed based on span programs

new limits to classical and quantum instance compression given an instance of a hard decision problem a limited goal is to compress that instance into a smaller equivalent instance of a second problem as one example consider the problem where given

a robust criterion for the modified gram schmidt algorithm with selective reorthogonalization a new criterion for selective reorthogonalization in the modified gram schmidt algorithm is proposed we study its behavior in the presence of rounding errors we give some counterexample matrices which prove that the standard criteria might fail through numerical experiments we illustrate that our new criterion seems to be suitable also for the classical gram schmidt algorithm with selective reorthogonalization

a modified gram schmidt algorithm with iterative orthogonalization and column pivoting abstract iterative orthogonalization is aimed to ensure small deviation from orthogonality in the gram schmidt process former applications of this technique are restricted to classical gram schmidt cgs and column oriented modified gram schmidt mgs the major aim of this paper is to explain how iterative orthogonalization is incorporated into row oriented mgs the interest that we have in a row oriented iterative mgs comes from the observation that this method is capable of performing column pivoting the use of column pivoting delays the deteriorating effects of rounding errors and helps to handle rank deficient least squares problems a second modification proposed in this paper considers the use of gram schmidt qr factorization for solving linear least squares problems the standard solution method is based on one orthogonalization of the r h s vector b against the columns of q the outcome of this process is the residual vector r and the solution vector x the modified scheme is a natural extension of the standard solution method that allows it to apply iterative orthogonalization this feature ensures accurate computation of small residuals and helps in cases when q has some deviation from orthogonality

the space complexity of recognizing well parenthesized expressions in the streaming model the index function revisited we show an p n t lower bound for the space required by any unidirectional constanterror randomized t pass streaming algorithm that recognizes whether an expression over two types of parenthesis is well parenthesized this proves a conjecture due to magniez mathieu and nayak 2009 and rigorously establishes the peculiar power of bi directional streams over unidirectional ones observed in the algorithms they present the lower bound is obtained by analysing the information that is necessarily revealed by the players about their respective inputs in a two party communication protocol for a variant of the index function

on the parity complexity measures of boolean functions the parity decision tree model extends the decision tree model by allowing the computation of a parity function in one step we prove that the deterministic parity decision tree complexity of any boolean function is polynomially related to the non deterministic complexity of the function or its complement we also show that they are polynomially related to an analogue of the block sensitivity we further study parity decision trees in their relations with an intermediate variant of the decision trees as well as with communication complexity

an improved lower bound on the sensitivity complexity of graph properties turan 1984 11 initiated the study of the sensitivity complexity of graph properties he conjectured that for any non trivial graph properties on n vertices the sensitivity complexity is at least n 1 he proved an email protected lower bound for sensitivity in his paper turan 1984 11 wegener 1985 12 proved this conjecture for all monotone graph properties in this paper we improve turan s lower bound to 617n 0 35n we hope that this will shed some light on the proof of turan s conjecture

on the competitive ratio of evaluating priced functions let f be a function on a set of variables v for each x i v let c x be the cost of reading the value of x an algorithm for evaluating f is a strategy for adaptively identifying and reading a set of variables u s v whose values uniquely determine the value of f we are interested in finding algorithms which minimize the cost incurred to evaluate f in the above sense competitive analysis is employed to measure the performance of the algorithms we address two variants of the above problem we consider the basic model in which the evaluation algorithm knows the cost c x for each x i v we also study a novel model where the costs of the variables are not known in advance and some preemption is allowed in the reading operations this model has applications for example when reading a variable coincides with obtaining the output of a job on a cpu and the cost is the cpu time for the model where the costs of the variables are known we present a polynomial time algorithm with the best possible competitive ratio cf for each function f that is representable by a threshold tree and for each fixed cost function c remarkably the best known result for the same class of functions is a pseudo polynomial algorithm with competitiveness 2 cf still in the same model we introduce the linear programming approach lpa a framework that allows the design of efficient algorithms for evaluating functions we show that different implementations of this approach lead in general to the best algorithms known so far and in many cases to optimal algorithms for different classes of functions considered before in the literature via the lpa we are able to determine exactly the optimal extremal competitiveness of monotone boolean functions remarkably the upper bound which leads to this result holds for a much broader class of functions which also includes the whole set of boolean functions we also show how to extend the lpa together with these results to the model where the costs of the variables are not known beforehand in particular we show how to employ the extended lpa to design a polynomial time optimal with respect to competitiveness algorithm for the class of monotone boolean functions representable by threshold trees

numerical aspects of gram schmidt orthogonalization of vectors abstract several variants of gram schmidt orthogonalization are reviewed from a numerical point of view it is shown that the classical and modified variants correspond to the gauss jacobi and gauss seidel iterations for linear systems further it is shown that orthogonalization with respect to elliptic norms and biorthogonalization can be formulated as orthogonalization by oblique projections

sensitivity block sensitivity and l block sensitivity of boolean functions sensitivity is one of the simplest and block sensitivity one of the most useful invariants of a boolean function nisan siam j comput 20 6 1991 999 and nisan and szegedy comput complexity 4 4 1994 301 have shown that block sensitivity is polynomially related to a number of measures of boolean function complexity the main open question is whether or not a polynomial relationship exists between sensitivity and block sensitivity we define the intermediate notion of l block sensitivity and show that for any fixed l this new quantity is polynomially related to sensitivity we then achieve an improved though still exponential upper bound on block sensitivity in terms of sensitivity as a corollary we also prove that sensitivity and block sensitivity are polynomially related when the block sensitivity is n

a deim induced cur factorization we derive a cur approximate matrix factorization based on the discrete empirical interpolation method deim for a given matrix bf a such a factorization provides a low rank approximate decomposition of the form bf a approx bf c bf u bf r where bf c and bf r are subsets of the columns and rows of bf a and bf u is constructed to make bf c bf u bf r a good approximation given a low rank singular value decomposition bf a approx bf v bf s bf w t the deim procedure uses bf v and bf w to select the columns and rows of bf a that form bf c and bf r through an error analysis applicable to a general class of cur factorizations we show that the accuracy tracks the optimal approximation error within a factor that depends on the conditioning of submatrices of bf v and bf w for very large problems bf v and bf w can be approximated well using an incremental qr algorithm that makes only one pass through bf a numer

maximal sensitivity of boolean nested canalizing functions abstract boolean nested canalizing functions ncf have important applications in molecular regulatory networks engineering and computer science in the literature there are two sensitivities to measure the complexity of a boolean function one is average sensitivity the other one is maximal sensitivity we follow the tradition and omit the word maximal in other words in this paper sensitivity is always maximal sensitivity using the past work of the authors and their coauthors on a characterization of ncf we obtain the formula of the sensitivity of any ncf we find that the sensitivity of any ncf is between n 2 2 and n both lower and upper bounds are tight we prove that the block sensitivity hence the l block sensitivity is the same as the sensitivity for ncf it is well known that monotone boolean functions mbf also have this property we characterize all functions which are both monotone and nested canalizing mncf the closed formula of the cardinality of the set of mncfs is also provided

the learnability of quantum states traditional quantum state tomography requires a number of measurements that grows exponentially with the number of qubits n but using ideas from computational learning theory we show that one can do exponentially better in a statistical setting in particular to predict the outcomes of most measurements drawn from an arbitrary probability distribution one needs only a number of sample measurements that grows linearly with n this theorem has the conceptual implication that quantum states despite being exponentially long vectors are nevertheless reasonable in a learning theory sense the theorem also has two applications to quantum computing first a new simulation of quantum one way communication protocols and second the use of trusted classical advice to verify untrusted quantum advice

crew prams and decision trees this paper gives a full characterization of the time needed to compute a boolean function on a crew pram with an unlimited number of processors the characterization is given in terms of a new compl

error bounds on complex floating point multiplication given floating point arithmetic with t digit base beta significands in which all arithmetic operations are performed as if calculated to infinite precision and rounded to a nearest representable value we prove that the product of complex values z 0 and z 1 can be computed with maximum absolute error abs z 0 abs z 1 frac 1 2 beta 1 t sqrt 5 in particular this provides relative error bounds of 2 24 sqrt 5 and 2 53 sqrt 5 for ieee 754 single and double precision arithmetic respectively provided that overflow underflow and denormals do not occur we also provide the numerical worst cases for ieee 754 single and double precision arithmetic

polynomial approximation and quadrature on geographic rectangles abstract using some recent results on subperiodic trigonometric interpolation and quadrature and the theory of admissible meshes for multivariate polynomial approximation we study product gaussian quadrature hyperinterpolation and interpolation on some regions of s d d 2 such regions include caps zones slices and more generally spherical rectangles defined on s 2 by longitude and co latitude geographic rectangles we provide the corresponding matlab codes and discuss several numerical examples on s 2

quantum query as a state decomposition abstract the quantum query model is a framework that allows us to express most known quantum algorithms algorithms represented by this model consist of a set of unitary operators acting over a finite hilbert space and a final measurement step consisting of a set of projectors in this work we prove that the application of these unitary operators before the measurement step is equivalent to decomposing a unit vector into a sum of vectors and then inverting some of their relative phases we also prove that the vectors of that sum must fulfill a list of properties and we call such vectors a block set if we define the measurement step for the block set formulation similarly to the quantum query model then we prove that both formulations give the same gram matrix of output states although the block set formulation allows a much more explicit form therefore the block set reformulation of the quantum query model gives us an alternative interpretation of how quantum algorithms work finally we apply our approach to the analysis and complexity of quantum exact algorithms

complexity measures and decision tree complexity a survey we discuss several complexity measures for boolean functions certificate complexity sensitivity block sensitivity and the degree of a representing or approximating polynomial we survey the relations and biggest gaps known between these measures and show how they give bounds for the decision tree complexity of boolean functions on deterministic randomized and quantum computers

reorthogonalized block classical gram schmidt a reorthogonalized block classical gram schmidt algorithm is proposed that factors a full column rank matrix a into a qr where q is left orthogonal has orthonormal columns and r is upper triangular and nonsingular this block gram schmidt algorithm can be implemented using matrix matrix operations making it more efficient on modern architectures than orthogonal factorization algorithms based upon matrix vector operations and purely vector operations gram schmidt orthogonal factorizations are important in the stable implementation of krylov space methods such as gmres and in approaches to modifying orthogonal factorizations when columns and rows are added or deleted from a matrix with appropriate assumptions about the diagonal blocks of r the algorithm when implemented in floating point arithmetic with machine unit varepsilon m produces q and r such that vert i q t q vert o varepsilon m and vert a qr vert o varepsilon m vert a vert the first of these bounds has not been shown for a block gram schmidt procedure before as consequence of these results we provide a different analysis with a slightly different assumption that re establishes a bound of giraud et al num math 101 1 87 100 2005 for the cgs2 algorithm

fast solving of weighted pairing least squares systems this paper presents a generalization of the weighted least squares wls named weighted pairing least squares wpls which uses a rectangular weight matrix and is suitable for data alignment problems two fast solving methods suitable for solving full rank systems as well as rank deficient systems are studied computational experiments clearly show that the best method in terms of speed accuracy and numerical stability is based on a special 1 2 3 inverse whose computation reduces to a very simple generalization of the usual cholesky factorization backward substitution method for solving linear systems

communication optimal parallel and sequential qr and lu factorizations we present parallel and sequential dense qr factorization algorithms that are both optimal up to polylogarithmic factors in the amount of communication they perform and just as stable as householder qr we prove optimality by deriving new lower bounds for the number of multiplications done by non strassen like qr and using these in known communication lower bounds that are proportional to the number of multiplications we not only show that our qr algorithms attain these lower bounds up to polylogarithmic factors but that existing lapack and scalapack algorithms perform asymptotically more communication we derive analogous communication lower bounds for lu factorization and point out recent lu algorithms in the literature that attain at least some of these lower bounds the sequential and parallel qr algorithms for tall and skinny matrices lead to significant speedups in practice over some of the existing algorithms including lapack and scalapack for example up to 6 7 times over scalapack a performance model for the parallel algorithm for general rectangular matrices predicts significant speedups over scalapack

letter to the editor experiments on orthogonalization by biorthogonal representations of orthogonal projectors a number of experiments are performed with the aim of enhancing a particular feature arising when biorthogonal sequences are used for the purpose of orthogonalization it is shown that an orthogonalization process executed by biorthogonal sequences and followed by a re orthogonalization step admits four numerically different realizations the four possibilities are originated by the fact that although an orthogonal projector is by definition a self adjoint operator due to numerical errors in finite precision arithmetic the biorthogonal representation does not fulfil such a property in the experiments presented here one of the realizations is shown clearly numerically superior to the remaining three

quantum conditional mutual information and approximate markov chains a state on a tripartite quantum system a otimes b otimes c forms a markov chain if it can be reconstructed from its marginal on a otimes b by a quantum operation from b to b otimes c we show that the quantum conditional mutual information i a c b of an arbitrary state is an upper bound on its distance to the closest reconstructed state it thus quantifies how well the markov chain property is approximated

algorithms for boolean function query properties we investigate efficient algorithms for computing boolean function properties relevant to query complexity such properties include for example deterministic randomized and quantum query complexities block sensitivity certificate complexity and degree as a real polynomial the algorithms compute the properties given an n variable function s truth table of size n 2n as input our main results are the following o n log2 3 log n algorithms for many common properties an o n log2 5 log n algorithm for block sensitivity an o n algorithm for testing quasi symmetry a notion of a tree decomposition of a boolean function proof that the decomposition is unique and an o nlog2 3 log n algorithm for finding it a subexponential time approximation algorithm for space bounded quantum query complexity to develop this algorithm we give a new way to search systematically through unitary matrices using finite precision arithmetic the algorithms discussed have been implemented in a linkable library

block gram schmidt orthogonalization the classical gram schmidt algorithm for computing the qr factorization of a matrix x requires at least one pass over the current orthogonalized matrix q as each column of x is added to the factorization when q becomes so large that it must be maintained on a backing store each pass involves the costly transfer of data from the backing store to main memory however if one orthogonalizes the columns of x in blocks of m columns the number of passes is reduced by a factor of 1 m moreover matrix vector products are converted into matrix matrix products allowing level 3 blas cache performance in this paper we derive such a block algorithm and give some experimental results that suggest it can be quite effective for large scale problems even when the matrix x is rank degenerate

nondeterministic quantum query and communication complexities we study nondeterministic quantum algorithms for boolean functions f such algorithms have positive acceptance probability on input x iff f x 1 in the setting of query complexity we show that the nondeterministic quantum complexity of a boolean function is equal to its nondeterministic polynomial degree we also prove a quantum vs classical gap of 1 vs n for nondeterministic query complexity for a total function in the setting of communication complexity we show that the nondeterministic quantum complexity of a two party function is equal to the logarithm of the rank of a nondeterministic version of the communication matrix this implies that the quantum communication complexities of the equality and disjointness functions are n 1 if we do not allow any error probability we also exhibit a total function in which the nondeterministic quantum communication complexity is exponentially smaller than its classical counterpart

a note on the error analysis of classical gram schmidt an error analysis result is given for classical gram schmidt factorization of a full rank matrix a into a qr where q is left orthogonal has orthonormal columns and r is upper triangular the work presented here shows that the computed r satisfies rt r at a e where e is an appropriately small backward error but only if the diagonals of r are computed in a manner similar to cholesky factorization of the normal equations matrix at the end of the article implications for classical gram schmidt with reorthogonalization are noted a similar result is stated in giraud et al numer math 101 1 87 100 2005 however for that result to hold the diagonals of r must be computed in the manner recommended in this work

backward error analysis of the shift and invert arnoldi algorithm we perform a backward error analysis of the inexact shift and invert arnoldi algorithm we consider inexactness in the solution of the arising linear systems as well as in the orthonormalization steps and take the non orthonormality of the computed krylov basis into account we show that the computed basis and hessenberg matrix satisfy an exact shift and invert krylov relation for a perturbed matrix and we give bounds for the perturbation we show that the shift and invert arnoldi algorithm is backward stable if the condition number of the small hessenberg matrix is not too large this condition is then relaxed using implicit restarts moreover we give notes on the hermitian case considering hermitian backward errors and finally we use our analysis to derive a sensible breakdown condition

on the decisional complexity of problems over the reals we consider the role of randomness for the decisional complexity in algebraic decision or computation trees i e the number of comparisons ignoring all other computation recently ting and yao showed that the problem of finding the maximum of n elements has decisional complexity o log2n 1994 inform process lett 49 39 43 in contrast rabin showed in 1972 an n bound for the deterministic case 1972 j comput system sci 6 639 650 we point out that their technique is applicable to several problems for which corresponding n lower bounds hold we show that in general the randomized decisional complexity is logarithmic in the size of the decision tree we then turn to the question of the number of random bits needed to obtain the ting and yao result we provide a deterministic o k log n algorithm for finding the elements which are larger than a given element given a bound k on the number of these elements we use this algorithm to obtain an o log2n random bits and o log2n queries algorithm for finding the maximum

on the sensitivity complexity of bipartite graph properties sensitivity complexity introduced by cook dwork and reischuk 1982 1986 in 2 3 is an important complexity measure of boolean functions turan 1984 7 initiated the study of sensitivity complexity for graph properties he conjectured that for any non trivial graph property on n vertices the sensitivity complexity is at least n 1 he proved that it is greater than n 4 in his paper wegener 1985 8 verified this conjecture for all monotone graph properties recently sun 2011 6 improved the lower bound to 617n for general graph properties we follow their steps and investigate the sensitivity complexity of bipartite graph properties in this paper we propose the following conjecture about the sensitivity of bipartite graph properties which can be considered as the bipartite analogue of turan s conjecture for any non trivial nxm bipartite graph property f s f max email protected email protected we prove this conjecture for all nx2 bipartite graph properties for general nxm bipartite graph properties we show a max n email protected m email protected lower bound we also prove this conjecture when the bipartite graph property can be written as a composite function

a parallel and streaming dynamic mode decomposition algorithm with finite precision error analysis for large data abstract a novel technique based on the full orthogonalization arnoldi foa is proposed to perform dynamic mode decomposition dmd for a sequence of snapshots a modification to foa is presented for situations where the matrix a is unknown but the set of vectors a i 1 v 1 i 1 n are known the modified foa is the kernel for the proposed projected dmd algorithm termed foa based dmd the proposed algorithm to compute dmd modes and eigenvalues i does not require singular value decomposition svd for snapshot matrices x with 2 x 1 m where 2 x is the 2 norm condition number of the snapshot matrix and m is the relative round off error or machine epsilon ii has an optional rank truncation step motivated by round off error analysis for snapshot matrices x with 2 x 1 m iii requires only one snapshot at a time thus making it a streaming method even with the optional rank truncation step iv consumes less memory and requires less floating point operations to obtain the projected matrix than existing projected dmd methods and v lends itself to easy parallelism as the main computational kernel involves only vector additions dot products and matrix vector products the new technique is therefore well suited for dmd of large datasets on parallel computing platforms we show both theoretically and using numerical examples that for foa based dmd without rank truncation the finite precision error in the computed projection of the linear mapping is o m 2 x the proposed method is also compared to existing projected dmd methods for computational cost memory consumption and relative round off error error indicators are presented that are useful to decide when to stop acquiring new snapshots the proposed method is applied to several examples of numerical simulations of fluid flow

trade offs between depth and width in parallel computation a new technique for proving lower bounds for parallel computation is introduced this technique enables us to obtain for the first time nontrivial tight lower bounds for shared memory models of parallel computation that allow several processors to have simultaneous access to the same memory location specifically we use a concurrent read concurrent write model of parallel computation it has p processors each has access to a common memory of size m also called communication width or width in short the input to the problem is located in an additional read only portion of the common memory for a wide variety of problems including parity majority and summation we show that the time complexity t depth and the communication width m are related by the trade off curve mt 2 omega n where n is the size of the input regardless of the number of processors moreover for every point on this curve with m o n log 2 n we give a matching upper bound with the optimal number of processors we e

on read once threshold formulae and their randomized decision tree complexity abstract tc 0 is the class functions computable by polynomial size constant depth formulae with threshold gates read once tc 0 ro tc 0 is the subclass of tc 0 which restricts every variable to occur exactly once in the formula our main result is a tight linear lower bound on the randomized decision tree complexity of any function in ro tc 0 this relationship between threshold circuits and decision trees bears significance on both models of computation regarding decision trees this is the first class of functions for which such a strong bound is known regarding threshold circuits it may be considered as a possible first step towards proving tc 0 nc 1 generalizing our lower bound to all functions in tc 0 would establish this separation another structural result we obtain is that a read once threshold formula uniquely represents the function it computes

quantum lower bounds by polynomials we examine the number of queries to input variables that a quantum algorithm requires to compute boolean functions on 0 1 n in the black box model we show that the exponential quantum speed up obtained for partial functions i e problems involving a promise on the input by deutsch and jozsa simon and shor cannot be obtained for any total function if a quantum algorithm computes some total boolean function f with small error probability using t black box queries then there is a classical deterministic algorithm that computes f exactly with o ts6 queries we also give asymptotically tight characterizations of t for all symmetric f in the exact zero error and bounded error settings finally we give new precise bounds for and or and parity our results are a quantum extension of the so called polynomial method which has been successfully applied in classical complexity theory and also a quantum extension of results by nisan about a polynomial relationship between randomized and deterministic decision tree complexity

upper bounds on fourier entropy given a function f 0 1 n rightarrow mathbb r its fourier entropy is defined to be sum s widehat f 2 s log widehat f 2 s where hat f denotes the fourier transform of f this quantity arises in a number of applications especially in the study of boolean functions an outstanding open question is a conjecture of friedgut and kalai 1996 called the fourier entropy influence fei conjecture asserting that the fourier entropy of any boolean function f is bounded above up to a constant factor by the total influence average sensitivity of f

on the power of ambainis lower bounds the polynomial method and the ambainis lower bound or alb for short method are two main quantum lower bound techniques while recently ambainis showed that the polynomial method is not tight the present paper aims at studying the power and limitation of alb s we first use known alb s to derive n1 5 lower bounds for bipartiteness bipartiteness matching and graph matching in which the lower bound for bipartiteness improves the previous n one we then show that all the three known ambainis lower bounds have a limitation n min c0 f c1 f where c0 f and c1 f are the 0 and 1 certificate complexities respectively this implies that for many problems such as triangle k clique bipartiteness and bipartite graph matching which draw wide interest and whose quantum query complexities are still open the best known lower bounds cannot be further improved by using ambainis techniques another consequence is that all the ambainis lower bounds are not tight for total functions this upper bound for alb s can be further improved to min c0 f c1 f n ci f where ci f is the size of max intersection of a 0 and a 1 certificate set again this implies that alb s cannot improve the best known lower bound for some specific problems such as and or tree whose precise quantum query complexity is still open finally we generalize the three known alb s and give a new alb style lower bound method which may be easier to use for some problems

computing approximate fekete points by qr factorizations of vandermonde matrices we propose a numerical method implemented in matlab for computing approximate fekete points on compact multivariate domains it relies on the search of maximum volume submatrices of vandermonde matrices computed on suitable discretization meshes and uses a simple greedy algorithm based on qr factorization with column pivoting the method gives also automatically an algebraic cubature formula provided that the moments of the underlying polynomial basis are known numerical tests are presented for the interval and the square which show that approximate fekete points are well suited for polynomial interpolation and cubature

shifted cholesky qr for computing the qr factorization of ill conditioned matrices the cholesky qr algorithm is an efficient communication minimizing algorithm for computing the qr factorization of a tall skinny matrix x in mathbb r m times n where m gg n unfortunately i

sensitive functions and approximate problems properties of functions that are good measures of the crcw pram complexity of computing them are investigated while theblock sensitivityis known to be a good measure of the crew pram complexity no such measure is known for crcw prams it is shown that the complexity of computing a function is related to itseverywhere sensitivity introduced by vishkin and wigderson specifically the time required to compute a functionf dn rof everywhere sensitivityes f withpprocessors and unbounded memory is logloges f log d 4p es f this improves results of azar and of vishkin and wigderson this lower bound is used to derive new lower bounds for someapproximate problems these problems can often be solved faster than their exact counterparts and for many applications it is sufficient to solve the approximate problem it is shown thatapproximate selection approximate counting approximate compaction andpadded sortingall require time loglogn with a linear number of processors if the level of accuracy desired is moderately high for these levels of accuracy no lower bounds were known for these problems on the pram model the lower bounds for some of the problems are tight

rounding error analysis of the classical gram schmidt orthogonalization process this paper provides two results on the numerical behavior of the classical gram schmidt algorithm the first result states that provided the normal equations associated with the initial vectors are numerically nonsingular the loss of orthogonality of the vectors computed by the classical gram schmidt algorithm depends quadratically on the condition number of the initial vectors the second result states that provided the initial set of vectors has numerical full rank two iterations of the classical gram schmidt algorithm are enough for ensuring the orthogonality of the computed vectors to be close to the unit roundoff level

streaming universal distortion free entanglement concentration this paper presents a streaming sequential protocol for universal entanglement concentration at the shannon bound alice and bob begin with n identical but unknown two qubit pure states each containing e ebits of entanglement they each run a reversible algorithm on their qubits and end up with y perfect epr pairs where y ne o n our protocol is streaming so the n input systems are fed in one at a time and perfect epr pairs start popping out almost immediately it matches the optimal block protocol exactly at each stage so the average yield after n inputs is y ne o logn so somewhat surprisingly there is no tradeoff between yield and lag our protocol optimizes both in contrast the optimal n qubit block protocol achieves the same yield but since no epr pairs are produced until the entire input block is read its lag is o n finally our algorithm runs in o log n space so a lot of entanglement can be efficiently concentrated using a very small e g current or near future technology quantum processor along the way we find an optimal streaming protocol for extracting randomness from classical i i d sources and a more space efficient implementation of the schur transform

any monotone property of 3 uniform hypergraphs is weakly evasive for a boolean function f let d f denote its deterministic decision tree complexity i e minimum number of adaptive queries required in worst case in order to determine f in a classic paper rivest and vuillemin 11 show that any non constant monotone property p 0 1 n 2 0 1 of n vertex graphs has d p n 2 we extend their result to 3 uniform hypergraphs in particular we show that any non constant monotone property p 0 1 n 3 0 1 of n vertex 3 uniform hypergraphs has d p n 3 our proof combines the combinatorial approach of rivest and vuillemin with the topological approach of kahn saks and sturtevant 6 interestingly our proof makes use of vinogradov s theorem weak goldbach conjecture inspired by its recent use by babai et al 1 in the context of the topological approach our work leaves the generalization to k uniform hypergraphs as an intriguing open question

quantum walk algorithm for element distinctness we use quantum walks to construct a new quantum algorithm for element distinctness and its generalization for element distinctness the problem of finding two equal items among n given items we get an o n 2 3 query quantum algorithm this improves the previous o n 3 4 quantum algorithm of buhrman et al siam j comput 34 2005 pp 1324 1330 and matches the lower bound of aaronson and shi j acm 51 2004 pp 595 605 we also give an o n k k 1 query quantum algorithm for the generalization of element distinctness in which we have to find k equal items among n items

polynomial interpolation and cubature over polygons we have implemented a matlab code to compute discrete extremal sets of fekete and leja type on convex or concave polygons together with the corresponding interpolatory cubature formulas the method works by qr and lu factorizations of rectangular vandermonde matrices on weakly admissible meshes wams of polygons constructed by polygon quadrangulation

separations in query complexity based on pointer functions in 1986 saks and wigderson conjectured that the largest separation between deterministic and zero error randomized query complexity for a total boolean function is given by the function f on n 2k bits defined by a complete binary tree of nand gates of depth k which achieves r0 f o d f 0 7537 we show that this is false by giving an example of a total boolean function f on n bits whose deterministic query complexity is n while its zero error randomized query complexity is o n we further show that the quantum query complexity of the same function is o n1 4 giving the first example of a total function with a super quadratic gap between its quantum and deterministic query complexities we also construct a total boolean function g on n variables that has zero error randomized query complexity n log n and bounded error randomized query complexity r g o n this is the first super linear separation between these two complexity measures the exact quantum query complexity of the same function is qe g o n these functions show that the relations d f o r1 f 2 and r0 f o r f 2 are optimal up to polylogarithmic factors further variations of these functions give additional separations between other query complexity measures a cubic separation between q and r0 a 3 2 power separation between qe and r and a 4th power separation between approximate degree and bounded error randomized query complexity all of these examples are variants of a function recently introduced by goos pitassi and watson which they used to separate the unambiguous 1 certificate complexity from deterministic query complexity and to resolve the famous clique versus independent set problem in communication complexity

polynomial fitting and interpolation on circular sections we construct weakly admissible polynomial meshes wams on circular sections such as symmetric and asymmetric circular sectors circular segments zones lenses and lunes the construction resorts to recent results on subperiodic trigonometric interpolation the paper is accompanied by a software package to perform polynomial fitting and interpolation at discrete extremal sets on such regions

dense quantum coding and quantum finite automata we consider the possibility of encoding m classical bits into many fewer n quantum bits qubits so that an arbitrary bit from the original m bits can be recovered with good probability we show that nontrivial quantum codes exist that have no classical counterparts on the other hand we show that quantum encoding cannot save more than a logarithmic additive factor over the best classical encoding the proof is based on an entropy coalescence principle that is obtained by viewing holevo s theorem from a new perspective in the existing implementations of quantum computing qubits are a very expensive resource moreover it is difficult to reinitialize existing bits during the computation in particular reinitialization is impossible in nmr quantum computing which is perhaps the most advanced implementation of quantum computing at the moment this motivates the study of quantum computation with restricted memory and no reinitialization that is of quantum finite automata it was known that there are languages that are recognized by quantum finite automata with sizes exponentially smaller than those of corresponding classical automata here we apply our technique to show the surprising result that there are languages for which quantum finite automata take exponentially more states than those of corresponding classical automata

a property of quantum relative entropy with an application to privacy in quantum communication we prove the following information theoretic property about quantum states substate theorem let and be quantum states in the same hilbert space with relative entropy s c tr log log c then for all 0 there is a state such that the trace distance tr tr 2 and 2o c 2 it states that if the relative entropy of and is small then there is a state close to i e with small trace distance tr that when scaled down by a factor 2o c sits inside or becomes a substate of this result has several applications in quantum communication complexity and cryptography using the substate theorem we derive a privacy trade off for the set membership problem in the two party quantum communication model here alice is given a subset a n bob an input i i n and they need to determine if i i a privacy trade off for set membership in any two party quantum communication protocol for the set membership problem if bob reveals only k bits of information about his input then alice must reveal at least n 2o k bits of information about her input we also discuss relationships between various information theoretic quantities that arise naturally in the context of the substate theorem

lower bounds on probabilistic linear decision trees abstract the power of probabilistic linear decision trees is examined it is shown that the standard arguments used to prove lower bounds on deterministic linear decision trees apply to probabilistic linear decision trees as well examples are next given of problems which randomization helps solving

the critical complexity of all monotone boolean functions and monotone graph properties crew pram s are a powerful model of parallel computers lower bounds for this model are rather general cook dwork and reischuk upper and lower time bounds for parallel random access machines without simultaneous writes siam j comput in press proved that the crew pram complexity of boolean functions is bounded by log b c f where b 4 79 and c f is the critical complexity of f this lower bound is often even tight for a class of functions f the critical complexity c f the minimum of all c f where f f is the best general lower bound on the critical complexity of all f f we determine the critical complexity of the set of all nondegenerate boolean functions and all monotone nondegenerate boolean functions up to a small additive term and we compute exactly the critical complexity of the class of all monotone graph properties proving partially a conjecture of turan 1984

on the isomorphism problem for decision trees and decision lists we study the complexity of isomorphism testing for boolean functions that are represented by decision trees or decision lists our results are the following isomorphism testing of rank 1 decision trees is complete for logspace for any constant r 2 isomorphism testing for rank r decision trees is polynomial time equivalent to graph isomorphism as a consequence of our reduction we obtain our main result for decision trees a 2 n log s o 1 time algorithm for isomorphism testing of decision trees of size s over n variables the isomorphism problem for decision lists admits a schaefer type trichotomy depending on the class of base functions the isomorphism problem is either in l or polynomial time equivalent to graph isomorphism or conp hard

computing boolean functions from multiple faulty copies of input bits suppose we want to compute a boolean function f but instead of receiving the input we only get l e faulty copies of each input bit a typical solution in this case is to take the majority value of the faulty bits for each individual input bit and apply f on the majority values we call this the trivial construction we show that if f 0 1 n 0 1 and e are known the best function construction f is often not the trivial one in particular in many cases the best f cannot be written as a composition of f with some functions and in addition it is better to use a randomized f than a deterministic one we also prove that the trivial construction is optimal in some rough sense if we denote by l f the number of 1 10 biased copies we need from each input to reliably compute f using the best randomized recovery function f and we denote by ltriv f the analogous number for the trivial construction then ltriv f l f moreover both quantities are in log s f where s f is the sensitivity of f a quantity related to l f is dstat erand f min i 1n li where li is the number of 1 10 biased copies of xi such that the above number of readings is sufficient to recover f with high probability this quantity was first introduced by reischuk and schmeltz 14 in order to provide lower bounds for the noisy circuit size of f in this article we give a complete characterization of dstat erand f through a combinatorial lemma that can be interesting on its own right

the loss of orthogonality in the gram schmidt orthogonalization process in this paper we study numerical behavior of several computational variants of the gram schmidt orthogonalization process we focus on the orthogonality of computed vectors which may be significantly lost in the classical or modified gram schmidt algorithm while the gram schmidt algorithm with reorthogonalization has been shown to compute vectors which are orthogonal to machine precision level the implications for practical implementation and its impact on the efficiency in the parallel computer environment are considered

on the modulo degree complexity of boolean functions abstract for each integer m 2 every boolean function f can be expressed as a unique multilinear polynomial modulo m and the degree of this multilinear polynomial is called its modulo m degree in this paper we investigate the modulo degree complexity of total boolean functions initiated by parikshit gopalan et al 9 in which they asked the following question whether the degree complexity of a boolean function is polynomially related with its modulo m degree for m be a power of primes it is already known that the module m degree can be arbitrarily smaller compare to the degree complexity see section 2 for details when m has at least two distinct prime factors the question remains open towards this question our results include 1 we obtain some nontrivial equivalent forms of this question 2 we affirm this question for some special classes of functions 3 we prove a no go theorem explaining why this problem is difficult to attack from the computational complexity point of view 4 we show a super linear separation between the degree complexity and the modulo m degree

a tighter relation between sensitivity complexity and certificate complexity abstract the sensitivity conjecture proposed by nisan and szegedy in 1994 which asserts that for any boolean function its sensitivity complexity is polynomially related to the block sensitivity complexity is one of the most important and challenging problems in the study of decision tree complexity despite a lot of efforts the best known upper bound of block sensitivity as well as the certificate complexity is still exponential in terms of sensitivity in this paper we give a better upper bound for certificate complexity and block sensitivity b s f c f 8 9 o 1 s f 2 s f 1 where b s f c f and s f are the block sensitivity certificate complexity and sensitivity respectively the proof is based on a deep investigation on the structure of the sensitivity graph we also provide a tighter relationship between the 0 certificate complexity c 0 f and 0 sensitivity s 0 f for functions with small 1 sensitivity s 1 f

block sensitivity of minterm transitive functions boolean functions with a high degree of symmetry are interesting from a complexity theory perspective extensive research has shown that these functions if nonconstant must have high complexity according to various measures in a recent work of this type sun 2007 9 gave lower bounds on the block sensitivity of nonconstant boolean functions invariant under a transitive permutation group sun showed that all such functions satisfy bs f email protected n 1 3 he also showed that there exists such a function for which bs f o n 3 7lnn his example belongs to a subclass of transitively invariant functions called minterm transitive functions defined by chakraborty 2005 3 we extend these results in two ways first we show that nonconstant minterm transitive functions satisfy bs f email protected n 3 7 thus sun s example has nearly minimal block sensitivity for this subclass second we improve sun s example we exhibit a minterm transitive function for which bs f o n 3 7ln 1 7n

the influence of orthogonality on the arnoldi method abstract many algorithms for solving eigenproblems need to compute an orthonormal basis the computation is commonly performed using a qr factorization computed using the classical or the modified gram schmidt algorithm the householder algorithm the givens algorithm or the gram schmidt algorithm with iterative reorthogonalization for the eigenproblem although textbooks warn users about the possible instability of eigensolvers due to loss of orthonormality few theoretical results exist in this paper we prove that the loss of orthonormality of the computed basis can affect the reliability of the computed eigenpair when we use the arnoldi method we also show that the stopping criterion based on the backward error and the value computed using the arnoldi method can differ because of the loss of orthonormality of the computed basis of the krylov subspace we also give a bound which quantifies this difference in terms of the loss of orthonormality

accurate and efficient evaluation of failure probability for partial different equations with random input data several computational challenges arise when evaluating the failure probability of a given system in the context of risk prediction or reliability analysis when the dimension of the uncertainties becomes high well established direct numerical methods can not be employed because of the curse of dimensionality many surrogate models have been proposed with the aim of reducing computational effort however most of them fail in computing an accurate failure probability when the limit state surface defined by the failure event in the probability space lacks smoothness in addition for a stochastic system modeled by partial differential equations pdes with random input only a limited number of the underlying pdes order of a few tens are affordable to solve in practice due to the considerable computational effort therefore preventing the application of many numerical methods especially for high dimensional random inputs in this work we develop hybrid and goal oriented adaptive reduced basis methods to tackle these challenges by accurately and efficiently computing the failure probability of a stochastic pde the curse of dimensionality is significantly alleviated by reduced basis approximation whose bases are constructed by goal oriented adaptation moreover an accurate evaluation of the failure probability for pde system with solution of low regularity in probability space is guaranteed by the certified a posteriori error bound for the output approximation error at the end of this paper we suitably extend our proposed method to deal with more general pde models finally we perform several numerical experiments to illustrate its computational accuracy and efficiency c 2013 elsevier b v all rights reserved

pseudorandom generators hard for k dnf resolution and polynomial calculus resolution a pseudorandom generator gn f0 1g n f0 1g m is hard for a propositional proof system p if roughly speaking p cannot eciently

the pattern matrix method we develop a novel technique for communication lower bounds the pattern matrix method specifically fix an arbitrary function f 0 1 n to 0 1 and let a f be the matrix whose columns are each an application of f to some subset of the variables x 1 x 2 ldots x 4n we prove that a f has bounded error communication complexity omega d where d is the approximate degree of f this result remains valid in the quantum model regardless of prior entanglement in particular it gives a new and simple proof of razborov s breakthrough quantum lower bounds for disjointness and other symmetric predicates we further characterize the discrepancy approximate rank and approximate trace norm of a f in terms of well studied analytic properties of f broadly generalizing several recent results on small bias communication and agnostic learning the method of this paper has also enabled important progress in multiparty communication complexity

revisiting deutsch jozsa algorithm abstract the deutsch jozsa algorithm is essentially faster than any possible deterministic classical algorithm for solving a promise problem that is in fact a symmetric partial boolean function named as the deutsch jozsa problem the deutsch jozsa problem can be equivalently described as a partial function d j n 0 0 1 n 0 1 defined as d j n 0 x 1 for x n 2 d j n 0 x 0 for x 0 n and it is undefined for the remaining cases where n is even and x is the hamming weight of x the deutsch jozsa algorithm needs only one query to compute d j n 0 but the classical deterministic algorithm requires n 2 1 queries to compute it in the worse case we present all symmetric partial boolean functions with degree 1 and 2 we prove the exact quantum query complexity of all symmetric partial boolean functions with degree 1 and 2 we prove deutsch jozsa algorithm can compute any symmetric partial boolean function f with exact quantum 1 query complexity

least squares polynomial approximation on weakly admissible meshes disk and triangle we construct symmetric polar wams weakly admissible meshes with low cardinality for least squares polynomial approximation on the disk these are then mapped to an arbitrary triangle numerical tests show that the growth of the least squares projection uniform norm is much slower than the theoretical bound and even slower than that of the lebesgue constant of the best known interpolation points for the triangle as opposed to good interpolation points such meshes are straightforward to compute for any degree the construction can be extended to polygons by triangulation

interaction in quantum communication in some scenarios there are ways of conveying information with many fewer even exponentially fewer qubits than possible classically moreover some of these methods have a very simple structure they involve only few message exchanges between the communicating parties it is therefore natural to ask whether every classical protocol may be transformed to a simpler quantum protocol one that has similar efficiency but uses fewer message exchanges we show that for any constant k there is a problem such that its k 1 message classical communication complexity is exponentially smaller than its k message quantum communication complexity this in particular proves a round hierarchy theorem for quantum communication complexity and implies via a simple reduction an omega n1k lower bound for k message quantum protocols for set disjointness for constant k enroute we prove information theoretic lemmas and define a related measure of correlation the informational distance that we believe may be of significance in other contexts as well

does ignorance of the whole imply ignorance of the parts large violations of noncontextuality in quantum theory a central question in our understanding of the physical world is how our knowledge of the whole relates to our knowledge of the individual parts one aspect of this question is the following to what extent does ignorance about a whole preclude knowledge of at least one of its parts relying purely on classical intuition one would certainly be inclined to conjecture that a strong ignorance of the whole cannot come without significant ignorance of at least one of its parts indeed we show that this reasoning holds in any noncontextual nc hidden variable model hv curiously however such a conjecture is false in quantum theory we provide an explicit example where a large ignorance about the whole can coexist with an almost perfect knowledge of each of its parts more specifically we provide a simple information theoretic inequality satisfied in any nc hv but which can be arbitrarily violated by quantum mechanics

computing multivariate fekete and leja points by numerical linear algebra we discuss and compare two greedy algorithms that compute discrete versions of fekete like points for multivariate compact sets by basic tools of numerical linear algebra the first gives the so called approximate fekete points by qr factorization with column pivoting of vandermonde like matrices the second computes discrete leja points by lu factorization with row pivoting moreover we study the asymptotic distribution of such points when they are extracted from weakly admissible meshes

average case quantum query complexity we compare classical and quantum query complexities of total boolean functions it is known that for worst case complexity the gap between quantum and classical can be at most polynomial we show that for average case complexity under the uniform distribution quantum algorithms can be exponentially faster than classical algorithms under non uniform distributions the gap can even be super exponential we also prove some general bounds for average case complexity and show that the average case quantum complexity of majority under the uniform distribution is nearly quadratically better than the classical complexity a preliminary version of this paper appeared in the proceedings of the 17th annual symposium on theoretical aspects of computer science stacs 2000 lecture notes in computer science vol 1770 berlin springer