On a convex operator for finite sets Abstract. Let S be a finite set with n elements in a real linear space. Let J S be a set of n intervals in R. We introduce a convex operator co(S, J S ) which generalizes the familiar concepts of the convex hull conv S and the affine hull aff S of S. We establish basic properties of this operator. It is proved that each homothet of conv S that is contained in aff S can be obtained using this operator. A variety of convex subsets of aff S can also be obtained. For example, this operator assigns a regular dodecagon to the 4-element set consisting of the vertices and the orthocenter of an equilateral triangle. For J S which consists of bounded intervals, we give the upper bound for the number of vertices of the polytope co(S, J S ). The set in (1.1) will not change if the conditions ξ j ≥ 0 are replaced by the conditions ξ j ∈ [0, 1], j = 1, . . . , m. This leads us to ask the following natural question: How will the set on the right-hand side of (1.1) change when the conditions ξ j ≥ 0 are replaced by the conditions ξ j ∈ I j , j = 1, . . . , m, where I j are arbitrary nonempty intervals in R? An immediate and obvious answer is that the resulting set will always be a subset of aff S. In this article we explore this question further. That is, we study the subsets of L introduced by the following definition.. It is clear that the convex interval hull co(S, J S ) coincides with conv S when all the intervals in J S are equal to [0, 1] and co(S, J S ) coincides with aff S when all the intervals in J S are equal to R. In this sense co(S, J S ) generalizes these two well-known concepts.. Our primary interest in this article is to explore the family of all convex interval hulls co(S, J S ) which are bounded. Examples in Section 2 show that a variety of convex sets appear in such families even if S is fixed. It is quite striking that when S is the set of only 4 points: the vertices and the orthocenter of an equilateral triangle, for example An inverse problem in this setting is as follows: For a given convex set K find a finite set S with minimal cardinality and a family J S of intervals such that K = co(S, J S ). Examples 2.7 and 2.10 suggest solutions of the inverse problem for K equal to a regular dodecagon and for K equal to a rhombic dodecahedron. This inverse problem and unbounded convex interval hulls will be considered elsewhere.. Let m be a positive integer. For a fixed family J of m nonempty intervals in R our operator S → co(S, J ) is a set-valued function defined on finite subsets of L with m elements. Recall that many set-valued functions f considered in convexity theory are described in the following way:. where F is a prescribed family of subsets of L. The convex hull itself and many well-known generalizations of it are obtained in this way, see for example [2] and [6] . An immediate consequence of definition (1.2) is the inclusion X ⊆ f (X). From examples in Section 2 and our results in Section 6 it is clear that the convex interval hull does not always satisfy the inclusion S ⊆ co(S, J S ). As a matter of fact, for every set S there are families of intervals J S for which S is not a subset of co(S, J S ). In this sense our operator differs from operators described by (1.2) . Definitions similar to Definition 1.1 appeared in [4] and [5] . We recall the following three definitions from [5, p. 363] . First, for nonempty sets Λ ⊂ R m and S ⊂ L denote by Λ · S ⊂ L the set of all m j=1 λ j s j , where (λ 1 , . . . , λ m ) ∈ Λ and s j ∈ S, j = 1 . . . , m. Second, a set S ⊂ L is called endo -Λ if Λ · S ⊆ S. Third, with F being the family of all endo-Λ sets, (1.2) defines the Λ-hull operator. A special case of Λ-hull operator with Λ = (ξ, 1 − ξ) : ξ ∈ ∆ ⊂ R 2 , where ∆ is any nonempty subset of [0, 1] containing at least one point interior to [0, 1], was considered in [4] . In [4] endo-Λ sets are called ∆-convex sets. (We notice that Motzkin in [5] does not refer to [4] .). In this paragraph we point out the differences between the definitions of Λ·S and co(S, J S ). To this end, let Λ be the intersection of I 1 × · · · × I m and the hyperplane m j=1 ξ j = 1, where I j are nonempty intervals in R, and let S = {x 1 , . . . , x m }, where x 1 , . . . , x m are distinct points in L. Then, in general, Λ · S contains more linear combinations than co(S, J S ). The first reason for this is that, with ξ 1 , . . . ξ m ∈ Λ, m j=1 ξ j s j ∈ Λ · S whenever s 1 , . . . , s m ∈ S, while for m j=1 ξ j x j ∈ co(S, J S ) it is essential that x 1 , . . . , x m are distinct points in S. For example, with s 1 = · · · = s m = s ∈ S, the condition m j=1 ξ j = 1 implies S ⊂ Λ · S, while S ⊂ co(S, J S ) is not true in general. The second reason is that in the definition of co(S, J S ) the point x j ∈ S, for fixed j ∈ {1, . . . , m}, is scaled only by scalars in I j , while there is no such restriction in the definition of Λ · S. We also remark that the geometry of the sets Λ · S and the properties of the operator S → Λ · S for a fixed Λ were not considered in [4] and [5] .. The article is organized as follows. In Section 2 we give several illustrative examples of convex interval hulls in R 2 and R 3 for sets S with three, four and five points. A justification for the adjective "convex" in the name of co(S, J S ) begins Section 3. Furthermore, in this section we characterize nonemptyness and boundedness of co(S, J S ). In Section 4 we prove that all bounded convex interval hulls are polytopes. We also give an upper bound for the number of vertices of such polytopes. As we have already noticed, different families of intervals can result in the same convex interval hulls. In Section 5 we study minimality conditions for a family of intervals, where minimality is understood in such a way that any further shrinking of intervals results in a smaller convex interval hull. In Section 6 we prove that a family of bounded convex interval hulls of a fixed finite set S is invariant under homotheties. As a special case of this result we obtain that for each homothet K of conv S there exists a family of intervals J S such that K = co(S, J S ). We use this result to give a detailed description of bounded convex interval hulls of finite affinely independent subsets of a linear space.. In this paragraph we introduce the notation. By R we denote the real numbers. The symbol L denotes a real linear space and · is a norm in this space. A specific linear space that we will encounter is R m , where m is a positive integer. The linear operations from L are extended to subsets of L in the following standard way. For subsets K and M of L and α, β ∈ R we put. For a mapping T : L → L, T (K) denotes the set of all T x, x ∈ K.

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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 O O O O O O O O O O O O O O O


Labelings and Encoders With the Uniform Bit Error Property With Applications to Serially Concatenated Trellis Codes Abstract-The well-known uniform error property for signal constellations and codes is extended to encompass information bits. We introduce a class of binary labelings for signal constellations, called bit geometrically uniform (BGU) labelings, for which the uniform bit error property holds, i.e., the bit error probability does not depend on the transmitted signal. Strong connections between the symmetries of constellations and binary Hamming spaces are involved. For block-coded modulation (BCM) and trellis-coded modulation (TCM) Euclidean-space codes, BGU encoders are introduced and studied. The properties of BGU encoders prove quite useful for the analysis and design of codes aimed at minimizing the bit, rather than symbol, error probability. Applications to the analysis and the design of serially concatenated trellis codes are presented, together with a case study which realizes a spectral efficiency of 2 b/s/Hz. A EUCLIDEAN-SPACE signal constellation has the uniform error property (UEP) if the symbol error probability does not depend on the transmitted signal. For geometrically uniform (GU) constellations [1] , the Voronoi (decision) regions of the signals are all congruent and the UEP holds. Most of the usual constellations (PSK, PAM, QAM) are either GU or approximately GU if we neglect boundary effects. The notion of geometrical uniformity can be extended to codes of finite or infinite length over GU constellations [1] . Also in this case, the Voronoi regions of GU code sequences are congruent and the symbol sequence error probability (often called error event probability) is independent from the transmitted signal sequence. UEP is of great utility, because only the all-zero Manuscript received December 24, 1999 ; revised April 12, 2001 . This work was supported in part by Qualcomm, Inc., under a Research Grant. The material in this paper was presented in part at the IEEE International Symposium on Information Theory, Cambridge, MA, August 1998. R. Garello is with the Dipartimento di Elettronica ed Automatica, Università di Ancona, 60131 Ancona, Italy (e-mail: roberto.garello@ieee.org).. G. Montorsi and S. Benedetto are with the Dipartimento di Elettronica, Politecnico di Torino, 10129 Torino, Italy (e-mail: montorsi@polito.it; benedetto @polito.it).. D. Divsalar and F. Pollara are with the Jet Propulsion Laboratory, Pasadena, CA 91109 USA (e-mail: dariush@shannon.jpl.nasa.gov; fabrizio@ shannon.jpl.nasa.gov).. Communicated by F. R. Kschischang, Associate Editor for Coding Theory.. Publisher Item Identifier S 0018-9448(02)00003-2.. sequence must be considered to evaluate any kind of code performance; analysis and design are thus dramatically simplified [2] , [3] . For most transmission systems, the bit error probability (or the frame error probability), involving the binary information sequences that must be transmitted to the receiver, is a more important performance measure than the symbol sequence error probability. Association between binary information sequences and signal sequences is provided by labelings for uncoded constellations, and by encoders for codes.. In classical coding theory, encoders are often not considered, or are somehow confused with the code. The main reason is that given a code, the performance of different encoders generating it are usually almost the same: at a fixed bit error rate of practical interest, the performance differences in terms of signal-to-noise ratios (SNRs) are limited to fractions of a decibel. As a consequence, the attention in design is focused on the code and, usually, on its (Hamming or Euclidean) distance properties.. In some applications, however, a clear separation between codes and encoders is essential. This is certainly the case of rotational invariance [4] , [5] . Another example is constituent codes for turbo codes [6] , [7] , where different encoders generating the same code lead to significantly different performance. In all these cases, the design must be focused on the encoder-code pair.. The goal of this paper is to extend the concept and the simplification provided by UEP to encompass binary information sequences. We look for conditions under which the bit error probability can be computed by considering any transmitted information sequence, for example, the all-zero sequence. While UEP applies to the code this property, which will be called uniform bit error property (UBEP) in the following, applies to the encoder-code pair. As UEP for codes, UBEP should prove very useful in all cases focused on the encoder-code pair, because design and analysis would be highly simplified.. In this following, we will introduce a class of binary labelings called bit geometrically uniform (BGU) labelings, and show that they possess the UBEP. Their properties will then be extended to binary encoders for block-coded modulation (BCM) and trelliscoded modulation (TCM) codes.. To prove the utility of UBEP, we will consider as a case study the class of serially concatenated codes with interleavers, which represent an alternative to turbo codes [6] : they have 0018-9448/02$17.00 © 2002 IEEE been proved to yield very good (in some cases superior to turbo codes) performance [8] , [9] . Serially concatenated codes with interleavers have been proposed as binary codes in [8] and as bandwidth-efficient trellis codes in [9] . For binary codes, all available performance evaluation tools, as well as design guidelines [8] , [10] , [11] , are based on the knowledge of the input-output weight-enumerating function (IOWEF), which enumerates the codewords weights according to the weights of the information sequences that generated them. Using Hamming weights instead of Hamming distances means that only the all-zero sequence can be considered to evaluate the code performance, which is certainly true for binary linear codes using binary modulation, that clearly posses the UBEP.. Things become more complex if we are interested in the analysis and design of serially concatenated TCMs realizing high spectral efficiencies, obtained by employing as inner code a TCM code over a generic constellation. Conditions for the UBEP to hold in this case are not so easy to satisfy. However, we will prove that, with BGU encoders, the "average" performance still depends only on the IOWEFs with respect to the all-zero sequence. A design approach consisting in maximizing the effective free Euclidean distance of the inner TCM code (defined as the minimum Euclidean weight of a code sequence generated by an information sequence with Hamming weight two) can then be adopted. Finally, the great simplification allowed by BGU encoders will be shown by the design of a code realizing a spectral efficiency of 2 b/s/Hz over an 8-PSK constellation.. In Section II, we briefly revisit the concept of Uniform Error Property and Geometrical Uniformity. In Section III we discuss the UBEP. BGU labelings are introduced in Section IV, where it is shown that they satisfy the UBEP. In Section IV-A, Hamming space symmetries are considered and it is shown that a -signal GU constellation admits a BGU labeling if and only if it has a generating group isomorphic to a generating group of the Hamming space . A potential application of these results, besides BGU labelings, could be the construction of nonlinear binary codes with the UEP obtained from group codes over a Hamming space generating group , a subject that received attention from several authors [12] - [14] in the recent past for the case . In Section V, BCM Euclidean-space codes with finite length are considered. In Section VI, TCM Euclidean-space codes with infinite length are considered and the distance rule is defined on the code trellis. Some necessary and sufficient conditions for BGU encoders are derived: they are simplified when Abelian generating groups are involved. It is shown that linearity can be a misleading framework when UBEP is required. In Sections VII and VIII, the serial concatenation of an outer (block or convolutional) binary code and an inner (BCM or TCM) Euclidean-space code are considered. It is shown that, thanks to the BGU properties, analysis and design can be focused on the properties of the inner code with respect to the all-zero sequence, which allows a great simplification. Finally, in Section VIII-A, the design of a code realizing a spectral efficiency of 2 b/s/Hz over an 8-PSK constellation is studied. The obtained results prove the significance of the BGU approach. The input-output encoder relationships between information and code sequences are crucial for some applications. For BGU labelings and encoders, introduced and characterized in this paper, the bit error probability does not depend on the transmitted sequence, a property that can be very useful for analysis and design. This theory has been applied, together with design techniques already developed for turbo code design, to the construction of new serially concatenated trellis codes.

O O O O O O O O O O O O O O B I O O B I I O O B O O O O O O O O O O O O O O O O O B 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 B 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 O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O B 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 B 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 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 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 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 O O O O O O O O O O O O O O O O O O O O O B 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 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 O O O O O O O O O O O O O B O O O O O O O B 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 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 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 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 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 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 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 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 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 B I 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 O O O B I O O O O O O O O O O O O O B I 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 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 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 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 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 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 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 O O O B I I 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 O O O O O O O O O O O O O B 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 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 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 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 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 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 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 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 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 B I


Communication in XYZ All-to-All Quantum Networks with a Missing Link We explicitate the relation between Hamiltonians for networks of interacting qubits in the XY Z model and graph Laplacians. We then study evolution in networks in which all sites can communicate with each other. These are modeled by the complete graph Kn and called all-to-all networks. It turns out that Kn does not exhibit perfect state transfer (PST). However, we prove that deleting an edge in Kn allows PST between the two non-adjacent sites, when n is a multiple of four. An application is routing a qubit over n different sites, by switching off the link between the sites that we wish to put in communication. Additionally, we observe that, in certain cases, the unitary inducing evolution in Kn is equivalent to the Grover operator. Networks of interacting qubits are a generalization of spin chains. These are of theoretical importance for the study of many-body quantum systems and could constitute a good test ground for technologies spanning from quantum key distribution in multi-user networks to various nano-scale devices.. Since the first works [2, 5, 10 ] (see also [3] , for a review), networks of interacting qubits are considered to be good candidates for engineering perfect quantum channels and allow information transfer between distant particles. In this perspective, such networks appear to be useful for the implementation of data buses in quantum mechanical devices, in particular because undergoing a free dynamics after an initial set-up. One of the problems arising in the scenario is given by natural dispersion effects, which determine a loss of information often proportional to the distance (i.e., the number of spins) between communicating sites. Ways to circumvent the issue are based on a local tuning of the couplings [5] or protocols for lifting the encoding of qubits into multiparticle states [4, 9] (see also [6] and the references therein).. Another approach is to study what network topologies guarantee a small decay of fidelity, when the couplings are homogeneous and constant during the entire evolution of the system. Results in this direction have provided examples of the so-called perfect state transfer (for short, PST ), in relation to combinatorial properties of the graphs modeling the networks (a list of references on this area is in [1] ). In the XY model, when considering a single excitation, it has been shown that PST depends essentially on the eigensystem of the adjacency matrix of the graph, because certain invariant eigenspaces of the total Hilbert space evolve independently. In the cases analyzed so far, even if PST occurs between two specific sites of a network, routing arbitrarily over the entire network still requires an external controller.. Here we will deal with PST in certain networks of spin-half particles with the XY Z interaction. Specifically, we will describe the structure of the Hamiltonian which governs the single excitation setting. This turns out to be proportional to the (combinatorial) Laplacian of the graph modeling the network. Thus, it may be worth to remark that Hamiltonian arising in the XY and theXYZ interactions are naturally associated to the adjacency matrix and the Laplacian respectively and that these objects are the two most common matrix representations of graphs.. We will study quantum evolution in a network in which all sites can communicate between each other (in both directions). Such a network is modeled by the complete graph on n vertices K n and it is called all-to-all network. The number of links in K n is n(n − 1)/2. It turns out that K n does not exhibit PST. However, we will show that deleting an edge in K n will allow PST between the two non-adjacent sites in the obtained network, when n is a multiple of 4 (apart the trivial case n = 2). This is modeled by a graph usually denoted by K − n , an all-toall network with a missing link. The phenomenon is due to interference effects and it is counterintuitive. On the side, we remark that in certain cases the unitary inducing evolution in K n is the Grover matrix up to an overall phase.. An application is routing a qubit on K − n (i.e., transferring the qubit between any two vertices of K − n , arbitrarily). The protocol involves an external agent, whose role is to switch OFF only the link between the sites that we wish to put in communication. When the process involves all n sites, it requires exactly 2n − 1 ON/OFF switching operations. This does not provide any direct advantage over considering the empty graph as a network (this is trivially the graph with no edges), with a single edge ij if communication between site i and site j is desired. On the other hand, dynamics on K − n is far more complex and the same edge configuration is potentially useful for other tasks beyond perfect communication of a qubit.. The structure of the paper is as follows. In Section 2 we give minimal background information on the XY Z model and the necessary mathematical definitions. We will clarify the exact relation between the XY Z Hamiltonian and the Laplacian matrix (this relation was mentioned in [7] , but not given explicitly). In Section 3 we study evolution on K n . Section 4 is devoted to K − n . Wi will also mention a generalization in which we delete an arbitrary number of vertex disjoint edges. A brief conclusion is drawn in Section 5. We have given an explicit relation between the Hamiltonian for networks of interacting qubits in XY Z model and the graph Laplacian. We have studied evolution in networks modeled by the complete graph K n and the complete graph minus one edge K − n . While K n does not allow PST, we have seen that K − n does allows PST between the two non-adjacent sites, when n is a multiple of four. This result can be used to arbitrarily route a qubit between any two vertices of K − n .

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 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 B I I 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 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 O O O O O O O O O O B I 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 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 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 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 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 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 O O O O O O O O B I I 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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


Adaptive Genetic Operators Based on Coevolution with Fuzzy Behaviors Abstract-This paper presents a technique for adapting control parameter settings associated with genetic operators. Its principal features are: 1) the adaptation takes place at the individual level by means of fuzzy logic controllers (FLCs) and 2) the fuzzy rule bases used by the FLCs come from a separate genetic algorithm (GA) that coevolves with the GA that applies the genetic operator to be controlled. The goal is to obtain fuzzy rule bases that produce suitable control parameter values for allowing the genetic operator to show an adequate performance on the particular problem to be solved. The empirical study of an instance of the technique has shown that it adapts the parameter settings according to the particularities of the search space allowing significant performance to be achieved for problems with different difficulties. The places in Table IV where these signs do not appear correspond with the performance values for T-FBs. • In general, the best and results are reached with (T-0.5). This agrees with [59] , where this value seemed a good choice for a large class of functions. So, with , FR introduces a suitable balance between the progress of the average fitness and the standard variance for most functions.

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On priority queues with priority jumps In this paper, we investigate a simplified head-of-the-line with priority jumps (HOL-PJ) scheduling discipline. Therefore, we consider a discrete-time single-server queueing system with two priority queues of infinite capacity and with a newly introduced HOL-PJ priority scheme. We derive expressions for the probability generating function of the system contents and the packet delay. Some performance measures (such as mean and variance) of these quantities are derived and are used to illustrate the impact and significance of the HOL-PJ priority scheduling discipline in an output queueing switch. We compare this dynamic priority scheduling discipline with a FIFO scheduling and a static priority scheduling (HOL) and we investigate the influence of the different parameters of the simplified HOL-PJ scheduling discipline. One of the main keys to a succesful network is the ability to support different applications. Different applications generate different types of traffic, which need different QoS (Quality of Service) standards. For real-time applications, it is important that mean delay and delay variance are not too large, while for non real-time applications, the packet loss ratio is the restrictive quantity. To support multiple classes of traffic in multimedia networks, priority mechanisms can be used. Multiple priority levels are provided, and different priority levels are given to different classes of traffic. There are two ways to use priorities; the first is to use a priority mechanism as a scheduling method (or a queueing discipline). Different delay requirements can be satisfied by scheduling delay-sensitive traffic first. The second way is to use a priority scheme to control congestion. In this case, when network congestion occurs, different packet loss requirements can be satisfied by selectively discarding low-priority packets. In this paper, we will only focus on a priority scheme as a scheduling method and we will thus not discuss local congestion control schemes.. The simplest priority scheme is a static priority scheme. In this discipline, priority is always given to the delay-sensitive class, and thus packets of this class are always scheduled for service before delay-insensitive packets. This scheme provides relatively low delays for the delay-sensitive class (see e.g. [1, 5, 9, 10, 13, 15] ). But if a large portion of the network traffic consists of highpriority traffic, the performance for the low-priority traffic can be severely degraded. Specifically, head-of-line (HOL) priority scheduling can cause excessive delays for the low-priority class, especially if the network is highly loaded. This problem is also known as the starvation problem. In order to find a solution for this problem, several dynamic priority schemes have been proposed in the literature. A class of dynamic priority schemes are the queue-length-treshold scheduling disciplines (QLT). In these disciplines, the priority queues are served depending on their queue length (analyses, see [4, 6, 7, 8, 11, 12] ). Another class of dynamic priority schemes are the head-of-line with priority jumps (HOL-PJ) disciplines. This discipline was introduced in [14] . In a queueing system with this type of discipline, high-priority packets arrive in a high-priority queue, while low-priority packets arrive in a low-priority queue (and the packets in the highpriority queue thus have service priority over packets in the low-priority queue). However, packets (at the head) of the low-priority queue can jump to (the end of) the high-priority queue. Many criteria can be used to decide when low-priority packets jump, e.g., a maximum queueing delay in the low-priority queue (see [14] ), a queue-length threshold of the low-priority queue, a random jumping probability per time unit,. . .. In [14] e.g., a limit is imposed on the maximum queueing delay for the packets in the lowpriority queue. The value of this limit is set to the difference between the delay requirement for the low-priority class and that for the high-priority class. Immediately after a packet's delay at the low-priority queue equals (or exceeds) the limit, the packet jumps to the tail of the high-priority queue. The jumped packet is treated in the high-priority queue as if it was a newly arriving packet to that queue. This jumping mechanism guarantees that at any time, the packets are queued according to the largeness of the queueing delays in excess of their respective delay requirements. Possible disadvantages of this scheme are the processing overhead required for monitoring packets for time-out and the additional implementation hardware, i.e., a local clock at the low-priority queue, a list of the arrival times of the low-priority packets, and circuitry for the timing mechanism to make the packet(s) at the head of the queue jump.. In this paper, we alter this HOL-PJ scheme, mainly in order to evade these disadvantages. We introduce a parameter β, which yields the probability that the total contents of the low-priority queue jumps to the end of the high-priority queue. The value of β can be chosen in such a way that the low-priority traffic stays within its delay requirements, i.e., the more stringent the delay requirement, the larger the value of β. Also, when β = 0 in our model, this is the well-known static priority queue (which has been fully analyzed in [15] ).. We have chosen to analyze this type of priority jumping for a number of reasons. Firstly, this scheme can be implemented relatively easy. Indeed, it is not necessary to keep track of the queueing delays of the low-priority packets, which makes a major difference with regard to the implementation (i.e., no clock, no list or no timing mechanism). Secondly, the analysis is kept analytically tractable. We use an analysis based on generating functions for assessing the performance of queues with the introduced HOL-PJ scheduling discipline. We obtain the joint generating function of the contents in the high-and low-priority queues and the generating functions of the delay of high-and low-priority packets. From the obtained generating functions, we can then easily calculate expressions for some interesting performance measures, such as mean value and variance of the queue contents and packet delay. These expressions require little computational effort and are well-suited for evaluating the impact of the various system parameters on the overall performance. We will also show that our results can be applied to the case of an output-queueing switch (see section 6) with a HOL-PJ priority scheduling. Thirdly, we will show that the delay of high-and low-priority packets in this scheme and the scheme of [14] are comparable. By means of simulation results, we will show that the performance of the output-queueing switch for the simplified scheme hardly differs from the performance for the original HOL-PJ scheme of [14] , with respect to the (mean) packet delay. This makes the newly introduced scheme quite attractive.. The contribution of this paper concerns the model that is considered, as well as the solution technique that is used. First, as far as the model is concerned, the main difference with most of the articles mentioned in this introduction is that, for the case of a multi-class output-queueing switch, the arrival processes of the different types of packets are not mutually independent, which complicates the analysis. Secondly, as discussed before, we 'summarize' the jumping process by a single parameter β. Finally, we want to show that a generating-function solution method is suitable for analyzing this type of buffers with a HOL-PJ priority scheduling.. The outline of the paper is as follows. In the following section, we describe the mathematical model. In sections 3 and 4, we analyze the steady-state system contents and packet delay of both classes. In section 5, we calculate the moments of the system contents and packet delay. We apply the obtained results to an output-queueing switch with Bernoulli arrivals, and discuss the impact of the HOL-PJ priority scheduling discipline in section 6. Some conclusions are formulated in section 7. In this paper, we have analyzed a queueing system with a newly introduced, simplified HOL-PJ priority scheduling discipline. A generating-function approach was adopted, which led to expressions of performance measures, such as mean and variance of the system contents in both queues and mean and variance of packet delay of both types of packets, that are easy to evaluate. The model includes possible correlation between the numbers of arrivals of the two traffic types during a slot. As a consequence, the results could be used to evaluate the performance of a dynamically prioritized output-queueing switch with Bernoulli arrivals. Finally, it is shown that simplifying the jumping mechanism hardly affects the original HOL-PJ scheme, while it is easier to implement and analyze.

O O O O O O O O O O O O O O O O O O O O O O O O B 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 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 O O O O B I I 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 O O O O O O O O O O O O O O O O O O O O


Multiple testing in statistical analysis of systems-based information retrieval experiments High-quality reusable test collections and formal statistical hypothesis testing together support a rigorous experimental environment for information retrieval research. But as Armstrong et al. [2009b] recently argued, global analysis of experiments suggests that there has actually been little real improvement in ad hoc retrieval effectiveness over time. We investigate this phenomenon in the context of simultaneous testing of many hypotheses using a fixed set of data. We argue that the most common approaches to significance testing ignore a great deal of information about the world. Taking into account even a fairly small amount of this information can lead to very different conclusions about systems than those that have appeared in published literature. We demonstrate how to model a set of IR experiments for analysis both mathematically and practically, and show that doing so can cause p-values from statistical hypothesis tests to increase by orders of magnitude. This has major consequences on the interpretation of experimental results using reusable test collections: it is very difficult to conclude that anything is significant once we have modeled many of the sources of randomness in experimental design and analysis. The past 20 years have seen a great improvement in the rigor of information retrieval experimentation, due primarily to two factors: high-quality, public, portable test collections such as those produced by TREC (the Text REtrieval Conference [Voorhees and Harman 2005]) , and the increased practice of statistical hypothesis testing to determine whether measured improvements can be ascribed to something other than random chance. Together these create a very useful standard for reviewers, program committees, and journal editors; work in information retrieval (IR) increasingly cannot be published unless it has been evaluated using a well-constructed test collection and shown to produce a statistically significant improvement over a good baseline.. But, as the saying goes, any tool sharp enough to be useful is also sharp enough to be dangerous. The outcomes of significance tests are themselves subject to random chance: their p-values depend partially on random factors in an experiment, such as the topic sample, violations of test assumptions, and the number of times a particular hypothesis has been tested before. If publication is partially determined by some selection criteria (say, p < 0.05) applied to a statistic with random variation, it is inevitable that ineffective methods will sometimes be published just because of randomness in the experiment. Moreover, repeated tests of the same hypothesis exponentially increase the probability that at least one test will incorrectly show a significant result. When many researchers have the same idea-or one researcher tests the same idea enough times with enough small algorithmic variations-it is practically guaranteed that significance will be found eventually. This is the Multiple Comparisons Problem. The problem is well known, but it has some far-reaching consequences on how we interpret reported significance improvements that are less well-known. The biostatistician John Ioannidis [2005a Ioannidis [ , 2005b argues that one major consequence is that most published work in the biomedical sciences should be taken with a large grain of salt, since significance is more likely to be due to randomness arising from multiple tests across labs than to any real effect.. Multiple comparisons are not the only sharp edge to worry about, though. Reusable test collections allow repeated application of "treatments" (retrieval algorithms) to "subjects" (topics), creating an experimental environment that is rather unique in science: we have a great deal of "global" information about topic effects and other effects separate from the system effects we are most interested in, but we ignore nearly all of it in favor of "local" statistical analysis of pairs of systems. Furthermore, as time goes on, we-both as individuals and as a community-learn about what types of methods work and don't work on a given test collection, but then proceed to ignore those learning effects in analyzing our experiments. Armstrong et al. [2009b] recently noted that there has been little real improvement in ad hoc retrieval effectiveness over time. That work largely explores the issue from a sociological standpoint, discussing researchers' tendencies to use below-average baselines and the community's failure to reimplement and comprehensively test against the best known methods. We explore the same issue from a theoretical statistical standpoint: what would happen if, in our evaluations, we were to model randomness due to multiple comparisons, global information about topic effects, and learning effects over time? As we will show, the result is that we actually expect to see that there is rarely enough evidence to conclude significance. In other words, even if researchers were honestly comparing against the strongest baselines and seeing increases in effectiveness, those increases would have to be very large to be found truly significant when all other sources of randomness are taken into consideration.. This work follows previous work on the analysis of significance tests in IR experimentation, including recent work in the TREC setting by Smucker et al. [2009 Smucker et al. [ , 2007 , Lynam [2007, 2006] , Sanderson and Zobel [2005] , and Zobel [1998] , and older work by Tague-Sutcliffe and Blustein [1994] , Hull [1993] , Wilbur [1994] , and Savoy [1997] . Our work approaches the problem from first principles, specifically treating hypothesis testing as the process of making simultaneous inferences within a model fit to experimental data and reasoning about the implications. To the best of our knowledge, ours is the first work that investigates testing from this perspective, and the first to reach the conclusions that we reach herein.. This article is organized as follows: We first set the stage for globally-informed evaluation with reusable test collections in Section 2, presenting the idea of statistical hypothesis testing as performing inference in a model fit to evaluation data. In Section 3 we describe the Multiple Comparisons Problem that arises when many hypotheses are tested simultaneously and present formally-motivated ways to address it. These two sections are really just a prelude that summarize widely known statistical methods, though we expect much of the material to be new to IR researchers. Sections 4 and 5 respectively demonstrate the consequences of these ideas on previous TREC experiments and argue about the implications for the whole evaluation paradigm based on reusable test collections. In Section 6 we conclude with some philosophical musings about statistical analysis in IR. We have presented two issues in statistical analysis that affect our conclusions about significance between pairs of runs: the first is the data used to fit a model in which inferences about runs will be made, and the second is the Multiple Comparisons Problem and the adjustment of p-values to account for it. Using runs submitted to TREC, we showed that these two issues combined can have a radical effect on the way we interpret experimental data in the Cranfield paradigm: depending on how we define a family of experiments for MCP adjustment, it is possible that most of the pairs of systems considered significantly different by a paired t-test would no longer be significant once our adjustments are applied. At a high level our results support the conclusions of Armstrong et al. [2009b] that there is little reason to believe that there has been much improvement in ad hoc retrieval effectiveness.. There are still many questions without objective answers: How should significance be considered in publication decisions? How should a family of experiments be defined for p-value adjustment? How can learning effects on a given test collection be accounted for? All of these questions factor into the construction of a model within which to perform inferences about systems, and therefore into our understanding about IR system effectiveness. Furthermore, we have not even considered all the other ways multiple comparisons can happen: by looking at multiple evaluation measures, by using multiple hypothesis tests, by evaluating over multiple collections, and so on. All of these could be incorporated into a model for inference, but of course the more factors we model, the less likely we are to find significant differences-or the more annotated data we will need in order to find them. In short, MCP is a deep problem that colors all of the evaluation we do, particularly with reusable test collections.. With all of these questions, it is hard to escape the feeling that statistical inference is ultimately highly subjective, only providing a thin veneer of objectivity that makes us feel a little more comfortable about experimental rigor. That does not mean we should throw out the entire edifice-on the contrary, though we believe our analysis conclusively shows that a p-value cannot have any objective meaning, we still believe p-values from paired t-tests provide a good heuristic that is useful for many of the purposes they are currently used for. We only argue that p-values and significance test results in general should be taken with a very large grain of salt, and in particular have a limited effect on publication decisions and community-wide decisions about "interesting" research directions.. In the long term, we believe that IR experimental analysis should transition to a fully Bayesian modeling approach [Gelman et al. 2004] . In the Bayesian framework, we specify a model first, then fit it to data and perform inference. The advantage of this is that we make all modeling assumptions explicit; even if the initial models are incorrect, over time they can be refined in a community effort to handle the specific challenges of information retrieval evaluation; we demonstrate this in another recent publication [Carterette 2011 ]. Gelman et al. [2009] have shown that a Bayesian approach can have a strong mitigating effect on MCP; this is one example of the advantage it provides. We are presenting this work partially as a research paper, partially as a set of recommendations, and partially as a position paper that we hope will inspire discussion on experimental design and analysis in IR. In this section we consider questions raised by our results above about the nature of statistical analysis and its effect on the actual research people decide to do.

O O O B I O O B I O O O B I O O O O O O O O O O O O B I 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 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 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 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 O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O B I B O O O O O O O O B I 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 O O O O O O O O O O O O O O O O O O O O B I 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 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 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 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 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 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 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 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 O O O O O O B I 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 O O O O O O O O O O O O O O O O O O O O O O B I 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 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 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 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 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 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 O O O O O O O B 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 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 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 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 B I 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 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 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 B I O O O O O O O O O O B I O O O O O O O O B I 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 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 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 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 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 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 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 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 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 O O O O B I 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 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 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 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 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 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 O O O O B I 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 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 O B I O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O


Encoding and Constructing 1-Nested Phylogenetic Networks with Trinets Phylogenetic networks are a generalization of phylogenetic trees that are used in biology to represent reticulate or non-treelike evolution. Recently, several algorithms have been developed which aim to construct phylogenetic networks from biological data using triplets, i.e. binary phylogenetic trees on 3-element subsets of a given set of species. However, a fundamental problem with this approach is that the triplets displayed by a phylogenetic network do not necessarily uniquely determine or encode the network. Here we propose an alternative approach to encoding and constructing phylogenetic networks, which uses phylogenetic networks on 3-element subsets of a set, or trinets, rather than triplets. More specifically, we show that for a special, well-studied type of phylogenetic network called a 1-nested network, the trinets displayed by a 1-nested network always encode the network. We also present an efficient algorithm for deciding whether a dense set of trinets (i.e. one that contains a trinet on every 3-element subset of a set) can be displayed by a 1-nested network or not and, if so, constructs that network. In addition, we discuss some potential new directions that this new approach opens up for constructing and comparing phylogenetic networks. In this paper, we have shown that we can recover a 1-nested network from 'perfect data', viz. the dense set of 1-nested trinets that is displayed by the network. In practice, we will not usually have access to such information for biological datasets. Even so, it should be quite straight-forward to at least compute a dense set of trinets for any given biological dataset using existing phylogenetic network methods. For example, given a multiple sequence alignment, one could compute the most parsimonious or most likely trinet for every sub-alignment of 3 sequences (using, e.g. methods described in [20, 21] ), which would be feasible as there are a bounded number of 1nested trinets. Note that this would have the advantage that no 'breakpoints' would need to be computed for the multiple alignment, which is a first (and sometimes quite difficult) step that is usually required when constructing phylogenetic networks from phylogenetic trees (cf. e.g. [14, Chap. 11] , [25, Sect. 2] ).. Given that computing dense sets of trinets is feasible for biological data, it could be reasonable to develop methods for finding 1-nested networks displaying as many trinets as possible from a dense set of trinets. Similar techniques have been developed for triplets e.g. [13, 15, 33] , although it is worth noting that it is NP-hard to find a tree displaying a maximum number of rooted triplets from an arbitrary set of triplets [2, 17, 35] (even if the set is dense [3] ). Alternatively, it might be of interest to investigate if there might be an 'Aho-type' algorithm [1, 11, 12, 29] to determine if an arbitrary subset of 1-nested trinets encodes a 1-nested network, and, if so, adapt this to give 'Min-Cut' type algorithms for building 1-nested networks from sets of trinets (cf. [26, 28] ). A first step in this direction could be to determine whether or not it is an NP-complete problem to decide if an arbitrary subset of 1-nested trinets encodes a 1-nested network (in particular, note that there are non-dense sets of 1-nested trinets that encode 1-nested networks-e.g. the 1-nested network N on {w, x, y, z} pictured in Fig. 11(a) is the only 1-nested network on {w, x, y, z} displaying the two trinets presented in Fig. 11(b) ).. In another direction, clearly we can ask for results along the lines of those presented above for level-k networks [9] , k ≥ 2, phylogenetic networks that have a bounded level of complexity depending on k (and also, of course, 'k-nested' networks). Note that there are non-recoverable level-2 networks (e.g. Fig. 4 ), and so this could be rather more technical. Moreover, it should be noted that, for k ≥ 3, there are level-k networks that are not of level-(k − 1) all of whose trinets have fixed level (see Fig. 12 ). Thus, the levels of the trinets displayed by a network do not necessarily determine the level of a network. For practical purposes, it might also be of interest to determine a way to enumerate the level-k trinets, k ≥ 2.. Another avenue worth exploring, could be to try generalizing the above results to 'r-nets', r ≥ 4, i.e. phylogenetic networks with r-leaves (note that in case r = 4 Fig. 11 The 1-nested network N on {w, x, y, z} depicted in (a) is uniquely determined by the two trinets pictured in (b). As before, directions are omitted for clarity when clear. Also only the vertices that are leaves are marked by a dot Fig. 12 A level-n phylogenetic network N on {x 1 , x 2 , . . . , x n }, n ≥ 4, for which every trinet in Tr(N ) is of level-3. For clarity arc directions are omitted when clear quartet trees are commonly used to build phylogenetic trees, e.g. [31] ). Note that it is straight-forward to extend Definition 1 to obtain a set of r-nets displayed by a phylogenetic network. This could be quite useful in practice since it might be possible to obtain more accurate estimates for r-nets than trinets (at least for r = 4) before we try to piece them together, although, technically speaking, this could be very challenging.. Finally, we conclude with what we consider to be a rather bold conjecture:. Conjecture If N is a recoverable phylogenetic network on X, then Tr(N ) encodes N , that is, if N a recoverable phylogenetic network on X such that Tr(N ) = Tr(N ) then N is isomorphic to N .. A first (and probably quite instructive!) 'exercise' could be to try and show that this conjecture at least holds for level-2 networks. Note that if this conjecture were true, then as in Corollary 4, we would immediately obtain a new proper metric on the set of recoverable phylogenetic networks on X.

O O O O O O O B 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 O O B O O O O O O O O O O O O O O O O O O O O O O O B O O O B I 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 B O O B O O O O O O O O O O O B I O O O O O B O O O O O O O O O O O O O O O O O O O O O O B 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 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 O O O O B 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 O O B O O O O O O O B I 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 O O O O O O O O O O O B 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 O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O B O O O O O O O B O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O B 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 O O O O O B O O O O O O O O O O O O O O O O O O O O O B 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 B O O O O O O O O O O O O O O B 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 B 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 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 O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B 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 B 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 O O O O O O O O O O O O O O O O O O O B 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 B I 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 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 B I O O O O O O O O O O O O O O O O O O O O B 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 O O O O O O O B I O O O O O O O O O O O O O O B I 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 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 O


Enhanced tracking and recognition of moving objects by reasoning about spatio-temporal continuity A framework for the logical and statistical analysis and annotation of dynamic scenes containing occlusion and other uncertainties is presented. This framework consists of three elements; an object tracker module, an object recognition/classification module and a logical consistency, ambiguity and error reasoning engine. The principle behind the object tracker and object recognition modules is to reduce error by increasing ambiguity (by merging objects in close proximity and presenting multiple hypotheses). The reasoning engine deals with error, ambiguity and occlusion in a unified framework to produce a hypothesis that satisfies fundamental constraints on the spatio-temporal continuity of objects. Our algorithm finds a globally consistent model of an extended video sequence that is maximally supported by a voting function based on the output of a statistical classifier. The system results in an annotation that is significantly more accurate than what would be obtained by frame-by-frame evaluation of the classifier output. The framework has been implemented and applied successfully to the analysis of team sports with a single camera. No computer vision algorithm for tracking or object recognition is perfect under real-world operating conditions. Object trackers have difficulty with complex occlusions (e.g. in crowded pedestrian scenes or on the sports field) and object recognition algorithms rarely give 100% accuracy, even on well posed data sets, let alone under unconstrained circumstances. This lack of reliability is one of the reasons for the slow commercial uptake of the complex visual surveillance systems presented at machine vision conferences. In this paper we propose a framework for taking the imperfect output of an object tracker and object classification system and refining it based on principles of logical consistency and the spatio-temporal continuity of physical objects, to produce a far more accurate scene annotation.. The lower level tracking and recognition systems explicitly model the possibility of ambiguity and error by assigning probabilities for the presence of objects within bounding boxes in each video frame. This output is passed to a reasoning engine which constructs a ranked set of possible models that are consitent with the requirements of object continuity. The the final output is then a globally consistent spatio-temporal description of the scene which is maximally supported by probabalistic information given by classifier.. A number of researchers have attempted to deal with object occlusion (and the resultant tracking problems) by attempting to track through occlusion. This involves reasoning about object ordering along the camera optical axis either using ground plane information [1, 2] or simply reasoning about relative spatial ordering [3] . Dynamic models such as the Kalman filter are often used to model the position of occluded objects [4, 5] , under the assumption of known dynamics (e.g. linear motion), when no visual information is available. Multiple cameras have also been used to bypass the occlusion problem [6] , however this is not always possible or practicable.. Our approach to occlusion handling differs from this body of work and has more similarity with the methods of McKenna et al. [7] and Sherrah and Gong [8] . These works do not attempt to disambiguate occluding objects, but instead reason about the occlusion taking place. McKenna et al. track 'blobs' that may be groups or individuals. In their work it is initially assumed all objects are separate (an assumption we do not make); and when blobs merge the resultant blob is recorded as a 'group' made up of the contributing individuals. A dynamically updated model of global object colour is used to disambiguate objects at the point at which blobs split. This model is also used to reason about object occlusion within a group that makes up a single blob. This is useful when a split group consists of more than two individuals, however it relies on an assumption that no object is completely occluded during the split. Sherrah and Gong [8] present work in a highly constrained scenario where the head and hands of a single individual are tracked as blobs. The hands may occlude each other or the face (to form a single blob). A handbuilt Bayesian network is used to perform frame-by-frame occlusion reasoning, based on available data (blob positions, velocities, number of blobs etc.). Perhaps the closest work to ours was presented recently by Yang et al. [9] . This system uses multiple cameras to provide a top view of the 'visual hull' of a crowd scene. Constraints on the number of pedestrians represented by each observed blob are determined according to the size of the blob's bounding box. These constraints are propagated from frame to frame to give an upper and lower limit on the number of objects present. All observed moving objects are assumed to be pedestrians, and no attempt is made to localise or identify individual pedestrians. Lipton et al. [10] present a system that uses simple object classification (pedestrian vs. car) to aid object tracking. Simple temporal consistency rules are used to prune transient objects resulting from noise. None of these systems performs more than frame-by-frame reasoning or allows for the possibility of error in the underlying low-level tracking and recognition algorithms. Our system performs long-term reasoning about object-blob associations over extended sequences of frames. By maintaining spatio-temporal consistency over sequences, many local imperfections and ambiguities in the low-level data are eliminated. To obtain the probability that an observation is of a particular identity, P (identity m |observation), the maximum value of P (model n |observation) over all exemplars relating to that identity is taken. Taking the maximum value is appropriate as the different exemplars relating to a partcular identity generally relate to different object configurations or viewpoints. As such, exemplars relating to object configurations or viewpoints other than the actual ones provide little information. These probabilities (along with the object locations and sizes from the object tracker) are passed on to the reasoning engine (described in section 5).

O O O O O O O O O O O B 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 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 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 B 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 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 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 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 O O O O O O O B I 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 O O O O O B 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 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 B 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 new family of proximity graphs: Class cover catch digraphs Motivated by issues in machine learning and statistical pattern classification, we investigate a class cover problem (CCP) with an associated family of directed graphs-class cover catch digraphs (CCCDs). CCCDs are a special case of catch digraphs. Solving the underlying CCP is equivalent to finding a smallest cardinality dominating set for the associated CCCD, which in turn provides regularization for statistical pattern classification. Some relevant properties of CCCDs are studied and a characterization of a family of CCCDs is given. © 2006 Elsevier B.V. All rights reserved.Keywords: Class cover problem; Pattern classification; Machine learning; DigraphConsider a set X with a dissimilarity measure d and two finite, non-empty sets X + , X − ⊆ X, with a distinction of target class given to one of the sets. Recall that a dissimilarity measure on X is a function d :1 ) 0. We will denote X + as the target class unless otherwise specified. In its most general form, the class cover problem (CCP) is to find a minimum cardinality set of open covering balls B i , with center c i and radius r i (B i = {x ∈ X : d(x, c i ) < r i }), whose union (the cover) contains all of the target class and does not contain any of the non-target class. The CCP is, therefore, a special case of the classic set cover problem [1] with constraints on the type of covering sets that are considered. We write the list ((X, d), X + , X − ) to denote an instance of the CCP.The above formulation allows several variations of the CCP. The CCP was introduced in [3] by Cannon and Cowen. They considered the constrained (all covering balls must be centered at target class points) homogeneous (all covering balls must have the same radius) CCP. The constrained inhomogeneous CCP (CICCP) was introduced in [17] and is the version we will focus on in this paper.The CCP was originally motivated by supervised pattern classification, and while the focus of this paper is not classification, we will review the topic and its connection to the CCP to demonstrate this motivation. We will describe a model of two-class supervised pattern classification. Consider a process in which we first draw, at random, a label from the set {−1, +1} and then (conditionally on the value of the chosen label) choose a random observation from the set X. Let Y be the random variable whose value is that of the chosen label and X be the random variable whose We have shown that for any CCCD D = (V , A), it is possible to find a set of |V | = n target class points and one nontarget class point in R n which induce (via the L 1 , L 2 , or L ∞ metric) a digraph isomorphic to D. A natural complement to this result is to prove the minimum dimension necessary to embed a given CCCD. Another characterization of interest is to give conditions on the dissimilarity measure such that "no simple cycles" is a necessary and sufficient condition to be a CCCD.. Throughout this paper, we have presented the CCP in a deterministic manner. In the spirit of the application of the CCP to the construction of classifiers, there is an interest in the study of the CCP and CCCDs when the random sets X + and X − drawn from some distributions F + and F − are considered directly, as opposed to the observed sets X + and X − studied herein. In this random case, CCCDs can be seen as vertex random graphs [12] . The theorems presented in this paper represent an effort to gain some understanding of the class of classifiers available based on CCDs and the regularization provided by the CCP. As mentioned in Section 1, a classifier built using the CCP has complexity directly related to the size of the solution to the CCP or, equivalently, the size of the minimum dominating set ( ) in the random CCCD. The exact distribution for in a family of one-dimensional CCCDs is calculated in [17] . In the general case, Corollary 3 shows an upper bound on . In order to better understand the complexity characteristics of the classifier derived from CCP solutions, additional information about the distribution of is being sought.

O O O O O O O O O O O O O O B I O O B I O O O B I I 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 O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O B I I B I B I 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 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 B I I 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 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 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 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 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 O O O O B I 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 B I 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 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 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 O O O O O O O B 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 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 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 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 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 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 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 O O O O O O O O O O O O O O O O O O O O O O


Scheduling garbage collection in real-time systems The key to successful deployment of garbage collection in real-time systems is to enable provably safe schedulability tests of the real-time tasks. At the same time one must be able to determine the total heap usage of the system. Schedulability tests typically require a uniformed model of timing assumptions (inter-arrival times, deadlines, etc.). Incorporating the cost of garbage collection in such tests typically requires both artificial timing assumptions of the garbage collector and restricted capabilities of the task scheduler. In this paper, we pursue a different approach. We show how the reactive object model of the programming language Timber enables us to decouple the cost of a concurrently running copying garbage collector from the schedulability of the real-time tasks. I.e., we enable any regular schedulability analysis without the need of incorporating the cost of an interfering garbage collector. We present the garbage collection demand analysis, which determines if the garbage collector can be feasibly scheduled in the system. As capabilities of embedded devices increase, the possibilities and demands of more complex systems arise. This leads to an ever increasing need of more sophisticated runtime system features. The ability to use shared dynamic 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. CODES+ISSS '10, October 24-29, 2010 , Scottsdale, Arizona, USA. Copyright 2010 ACM 978-1-60558-905-3/10/10 ...$10.00. data and parallel/concurrent execution of tasks are both examples of features that enables the programmer to more extensively utilize these new capabilities. However, the two features in conjunction makes automatic memory management (garbage collection) a necessity.. Garbage collection has widely been acknowledge for reducing development time as well as enhancing reliability of software systems. In the real-time system context, the work on garbage collection can roughly be divided into two categories. The first category concerns the development of garbage collection algorithms suitable for real-time systems. This line of work was initiated already in the late 70's by Baker [2] , and has since then received most of the attention. The second category, to which this paper belongs, concerns schedulability analyses of garbage collected real-time systems. This line of research was initiated by Henriksson [14] .. A real-time system is defined in terms of a set of tasks for which timing assumptions are uniformly defined (interarrival times and patterns, deadlines, etc.). Corresponding schedulability tests crucially depend on the uniformed task model [21] . Adapting such tests to incorporate the cost of garbage collection typically imposes unnecessary restrictions on both task model and scheduler. The main issue is that schedulability analysis requires execution time estimates of the tasks to be independent, whereas such estimate of a garbage collector depends on a combination of memory and timing behaviors of the real-time tasks.. In this paper, we pursue a different approach. We show how an incremental copying garbage collector [18] deployed as a concurrent process in the reactive environment of the programming language Timber [23] enables us to decouple the cost of garbage collection from the feasibility of the realtime tasks. I.e., we enable any schedulability analysis for a real-time system, without the need of incorporating the cost of garbage collection into that analysis. Instead, we offer another analysis, decoupled from the regular schedulability analysis. Given a schedulable set of tasks, our analysis provides a sufficient test to determine if the garbage collector can be scheduled in the system. We call this analysis garbage collection demand analysis. Our approach relies on the following two key properties:. 1. Each atomic increment of the garbage collector can be executed within a small and constant time (including root-set scanning and synchronization operations).. 2. In order to preserve schedulability of the task set, the garbage collector runs as the lowest priority process.. The running time of a concurrent copying garbage collector depends on the amount of live heap memory and the synchronization work due to interrupting mutators. Besides timing assumptions for the task set, our analysis requires upper-bounds on global live heap space and heap allocation of each task as input parameters. Such analyses does not fit within the scope of this paper. However, a forthcoming paper presenting live heap space analysis for real-time systems is in preparation [19] .. The general motivation behind this paper is pervaded by the ambition to achieve provable schedulability properties of a real-time garbage collector, as opposed to improve performance. Although improving schedulability is important, it is not within the scope of this paper to embark that field. Instead, our objective is to enable schedulability tests for garbage collection independently of which scheduling policy that is employed for the real-time tasks. We have shown how the reactive object model of Timber enables us to decouple the schedulability analysis of the real-time tasks from the cost of garbage collection. We have, based on the incremental copying garbage collector presented in [18] and the real-time programming language Timber [23] , developed a feasibility test for the collector. We call this test garbage collection demand analysis, which contains only clearly identified parameters of the real-time system and their corresponding effect on garbage collection time. Apart from formal verification of correctness, we have confirmed the validity of the model through an experimental study run on the LPC2468 Developer's Kit from Embedded Artists.. The key motivation behind our approach to schedule garbage collection is to enable any scheduling policy (with corresponding feasibility test) for the real-time tasks. This is achieved by identifying and exploiting key properties of the run-time behavior of real-time tasks defined in Timber, which allow us to decouple the cost of garbage collection from the processor demand of the real-time tasks. The key issue in scheduling concurrent garbage collection in real-time systems is undoubtedly how the garbage collector should be able to compete with the real-time tasks. Typically, this amounts to finding appropriate timing assumptions for the collector task. Although the garbage collector does not really have such timing properties, more or less artificial ones are necessary in order to determine schedulability of the whole system. Due to the quite complex dependency between properties of the real-time tasks and the required execution time for garbage collection, much of the focus in this field have unfortunately diverged from pure schedulability analysis towards improving measured performance. In [12] , Fu and Hauser presents a framework for describing a broad class of real-time garbage collectors and their corresponding scheduling premises. They also accurately identify the key to enable provable schedulability guarantees of a garbage collected system, to wit comprehensive knowledge about the memory behavior of the real-time tasks as well as the inter-dependencies between them and the garbage collector.. In [13] , Henriksson presents a feasibility test for garbage collection based on response time analysis [16] . In contrast to our work, he assumes a per task parameter for worstcase garbage collection time required after one invocation (G i). This parameter slightly corresponds to our execution time parameter due to new allocations (T A i ). However, he does not present the connection between Gi and the actual garbage collection algorithm used. This work was later extended by Gestegård-Robertz and Henriksson to compute an upper-bound on the cycle time of the garbage collector in order to meet total heap memory limits [26] , which corresponds to a rearrangement of our memory needs formula (Equation 4).. In [20] , Kim et al. presents upper-bound estimates on the execution time of a garbage collector based on Brook's [6] evacuation strategy. They present a schedulability test for the whole system based on a worst-case response time analysis of a sporadic server. In addition, they also present a live memory analysis to determine the worst-case local live heap memory of each task. In contrast to our work, they do not present a detailed connection between the parameters used in the execution time estimate and the actual garbage collection algorithm used. They are also limited to use rate monotonic priorities to enable the sporadic server schedulability test.. In [8] , Chang presents a hybrid approach based on a lazy freeing reference counting collector and a backup mark-sweep collector. External fragmentation is avoided by using a fixed block size. He also presents a schedulability test based on a dual-priority scheduling scheme including the worst-case cost for both the reference counting collector and the marksweep collector. This is achieved by integrating the two garbage collectors into the real-time scheduling framework as tasks (i.e., derive appropriate timing assumptions for them). The main difference between his approach and ours is that he integrates the cost of garbage collection in the regular schedulability analysis. This makes it more difficult to extend the regular schedulability test with more features (e.g., shared resources) without affecting the schedulability test of the garbage collector.. Recently, Kalibera et al. developed schedulability tests for both time-based and slack-based scheduling of time-triggered garbage collection [17] . They show that none of them are superior to the other (in terms of schedulability) and they draw the conclusion that the choice of scheduling policy is a key part of designing garbage collected real-time systems.

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