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Linearly Representable Entropy Vectors and Their Relation to Network Coding Solutions

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Linearly Representable Entropy Vectors and Their Relation to Network Coding Solutions 2009 IEEE Information Theory Workshop Linearly Representable Entropy Vectors and their Relation to Network Coding Solutions Asaf Cohen, Michelle Effros, Salman Avestimehr and Ralf Koetter Abstract—In this work, we address the following question: importantly, may or may not be known to us in their entirety. “When can we guarantee the optimality of linear coding at all (For a comprehensive tutorial see [9].) For these reasons, internal nodes of a network?” While sufficient conditions for even in the absence of a complete characterization of the rate linear coding throughout the network are known, it is not clear whether relaxing the linearity constraints at the terminal nodes region, it is desirable to diminish the need for non-Shannon can result in simpler operations at the internal nodes. inequalities. We present a novel method to analyze the space of network so- Any network coding solution can be viewed as an optimiza- lutions using the constraints resulting from the network topology, tion over the entropic region subject to constraints imposed by and we identify sufficient conditions for an optimal linear solution the network topology and the receiver demands. Therefore, at all internal nodes. These conditions are also sufficient to characterize the rate region only in terms of Shannon information the focus has been on characterizing the entropic region. This inequalities. characterization turns out to be quite difficult, and remains an open problem for more than three variables. Our contribution I. INTRODUCTION is to give sufficient conditions under which the Shannon Optimal coding strategies for networks are well understood inequalities together with the network constraints result in only for some demands. For example, it is well known that an outer bound which is completely within the entropic linear coding suffices for multicast [1], [2]. For general de- region. In this case, there is no need to consider non-Shannon mands, mainly negative results are available, e.g. cases where inequalities. Moreover, properties which apply to the outer linear codes are suboptimal [3]. bound, immediately apply to any solution for the network. As In this work, we take advantage of the functional constraints a result, we are able to prove interesting properties for rate within the network in order to limit the space of possible regions on networks with non-multicast demands, a class of network solutions, and we derive sufficient conditions under networks which is generally unsolved. which the resulting space adheres a linear representation at The rest of the paper is organized as follows. Section II all internal nodes. Network codes which employ non-linear introduces definitions and previous results. Section III contains operations at the terminal nodes together with linear network the main result. Section IV includes two examples. coding at the internal nodes are useful, for example, in non- II. PRELIMINARIES multicast networks with dependent sources, such as networks with side information [4]. A. Networks and Codes While similar constraints where used in the LP bound [5] A network is defined as a directed graph (V, E), where V and the Ingleton-LP bound [6], [7], the focus in this work is is the set of vertices (nodes) and E ⊆ V × V is the set of not on deriving tight outer bounds, but rather on when can a edges (links). For each edge e = (a, b) ∈ E, we use o(e) = a linear representation be found for a given network and what and d(e) = b to denote the origin and destination vertices, are the applications such a representation has on the codes that respectively, of edge e. Associated with each edge e ∈ E is a can be used. capacity c(e) ≥ 0. We assume acyclic graphs and noise-free Furthermore, an important outcome of this representation links. ∞ is the ability to test whether Shannon-type inequalities suffice Let {(X1,i,...,XK,i)}i=1 be a sequence of independent in order to derive the rate region of a network. Non-Shannon and identically distributed K-tuples of discrete random vari- inequalities might not be linear (e.g. [8]), cannot be expressed ables with alphabet X1 × · · · × XK . We refer to Xj as in a canonical form to facilitate the analysis, and, most the j-th source in the network. The sources are statistically K independent with some known distribution Πi=1pi(xi). Let K 0Asaf Cohen was with the California Institute of Technology. He is now be the power set of {1,...,K} and denote by S : V 7→ K with the Department of Communication System Engineering, Ben-Gurion University of the Negev, Israel ([email protected]). the mapping defining the sources available at each node. Michelle Effros is with the Department of Electrical Engineering, California Analogously, denote by D : V 7→ K the mapping defining Institute of Technology, Pasadena, CA 91125 ([email protected]). the sources demanded at each node. We assume that nodes Salman Avestimehr is with the School of Electrical and Computer Engineer- ing, Cornell University, Ithaca, NY 14853 ([email protected]). for which S(v) 6= ∅, i.e. source nodes, have no incoming Ralf Koetter is with the Institute for Comm. Engineering, Technische Univer- edges and that nodes for which D(v) 6= ∅, i.e. terminal sitat¨ Munchen,¨ D-80290 Munchen,¨ Germany ([email protected]). nodes have no outgoing edges. A network with graph (V, E), This work is partially supported by DARPA Grant W911NF-07-I0029, Lee K Center for Advanced Networking at Caltech and the Center for Mathematics sources {Xi}i=1 and mappings S and D is denoted by K of Information at Caltech. (V, E, Πi=1pi(xi), S, D). 978-1-4244-4983-5/09/$25.00 © 2009 IEEE 549 Definition 1. For any vector of rates (Re)e∈E , a C. Polyhedral Cones nRe ((2 )e∈E , n) source code is the following set of mappings For an integer d > 0, a polyhedral cone C is the set of all vectors x ∈ Rd satisfying Ax ≥ 0 for some A ∈ Rm×d. That n n nRe ge :Πi∈S(o(e))Xi 7→ {1,..., 2 } e ∈ E,S(o(e)) 6= ∅ is n nRe0 nRe d ge :Πe0:d(e0)=o(e){1,..., 2 } 7→ {1,..., 2 } C = {x ∈ R : Ax ≥ 0}. e ∈ E,S(o(e)) = ∅ d Let u1,..., up be a set of vectors in R . Denote by n nRe n ht :Πe:d(e)=t{1,..., 2 } 7→ Πi∈D(t)Xi D(t) 6= ∅. NonNeg(u1,..., up) the non-negative, or conic, hull of u1,..., up, namely, ˆ n For each terminal node t ∈ V we use Xi,t to denote the n n NonNeg(u1,..., up) = {λ1u1 + ... + λpup : reproduction of Xi found by decoder ht . We are interested λ ∈ R, λ ≥ 0 i = 1, . , p}. (4) in the set of possible values (c(e))e∈E for which the sources i i can be reproduced at the terminals with a negligible error Theorem 1 (Minkowski, e.g. [11]). For any polyhedral cone probability. Precisely, we require that for any ² > 0 there d C in R there exist vectors u1,..., up such that nRe exists a sufficiently large n and a ((2 )e∈E , n) code with ˆ n n C = NonNeg(u ,..., u ). Re ≤ c(e) for all e ∈ E, such that Pr(Xi,t = Xi ) ≥ 1 − ² 1 p for all terminal nodes t and demands i ∈ D(t). We call the The current literature includes several algorithms (e.g. [11]) closure of this set of rate vectors the set of achievable rates, K to retrieve one representation of a polyhedral cone from the which we denote by R(V, E, Πi=1pi(xi), S, D). other. In [12], the authors used the representation of a poly- hedral cone given in Theorem 1 to show that the polymatroid B. Polymatroids, Entropic and Pseudo-Entropic Vectors axioms and Ingleton inequality characterize all inequalities for ranks of up to 4 linear subspaces. In this work, we use this Let N be an index set of size n and N be the power set of representation to analyze network coding solutions. N. A function g : N 7→ R defines a polymatroid (N, g) with a ground set N and rank function g if it satisfies the following D. Finite Alphabet Linear Representation three conditions [10]: Let L be a linear (vector) space over a finite field F . Let n n {Li}i=1 be subspaces of L. Let w({Li}) be the (2 − 1)- g(∅) = 0, (1) dimensional vector whose entries are the ranks of all possible g(I) ≤ g(J) for I ⊆ J ⊆ N, (2) unions of Li’s. That is g(I) + g(J) ≥ g(I ∪ J) + g(I ∩ J) for I,J ⊆ N. (3) w({Li}) = (rank(L1),..., rank(Ln), rank(L ∪ L ),..., rank(∪n L )) (5) For any polymatroid g with ground set N, we can represent g 1 2 i=1 i n by the vector (g(I): I ⊆ N) ∈ R2 −1 defined on the ordered, Theorem 2 ([12, Theorem 2]). For any finite linear space L non-empty subsets of N. We denote the set of all polymatroids over a finite field F or the real line R, and any set of subspaces with a ground set of size n by Γn. Thus w ∈ Γn if and only if {Li}, w({Li}) is entropic. w and w satisfy equations (1)–(3) for all I,J ⊆ N, where I J Proof: We include here a short description of how to w is the value of w at the entry corresponding to the subset I construct a random vector whose entropy vector is proportional I.
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