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Symmetry in Network Coding Symmetry in Network Coding Jayant Apte, John MacLaren Walsh Drexel University, Dept. of ECE, Philadelphia, PA 19104, USA [email protected], [email protected] Abstract—We establish connections between graph theoretic will associate a subgroup of Sn (group of all permutations symmetry, symmetries of network codes, and symmetries of rate of n symbols) called the network symmetry group (NSG) that regions for k-unicast network coding and multi-source network contains information about the symmetries of that instance. §II coding. We identify a group we call the network symmetry group as the common thread between these notions of symmetry is devoted to the preliminaries and the definition of NSG. The and characterize it as a subgroup of the automorphism group connections between NSGs and symmetries of network codes of a directed cyclic graph appropriately constructed from the and rate regions are discussed in §III. Secondly, we provide underlying network’s directed acyclic graph. Such a charac- means to compute this group given an instance of a multi- terization allows one to obtain the network symmetry group source network coding problem. This is achieved through a using algorithms for computing automorphism groups of graphs. We discuss connections to generalizations of Chen and Yeung’s graph theoretic characterization of the NSG, specifically, as a partition symmetrical entropy functions and how knowledge of subgroup of automorphism group of a directed cyclic graph the network symmetry group can be utilized to reduce the called the dual circulation graph that we construct from the complexity of computing the LP outer bounds on network coding directed acyclic graph underlying the MSNC instance. The k- capacity as well as the complexity of polyhedral projection for unicast network coding problem (k-UNC), which is a special computing rate regions. Index Terms—network coding, symmetry, graph automorphism case of MSNC problem, has a simpler characterization of NSGs, which is covered in §IV, followed by the more general I. INTRODUCTION characterization of NSG for MSNC in §V. We then provide Many important pragmatic problems, including the efficient pointers to algorithms for computing automorphism groups transfer of information over networks, the design of efficient of graphs that can be readily used for computing NSGs. distributed information storage systems, and the design of Finally, in §VII we discuss connection to Chen and Yeung’s streaming media systems, have been shown to involve deter- partition symmetrical entropy functions [6] and use of NSGs mining the capacity region of an abstracted network under in polyhedral projection to compute polyhedral bounds on rate multi-source multi-sink network coding (MSNC). Yan et al. regions and other applications. [1] provided an implicit characterization of the rate region of MSNC over directed acyclic graphs in terms of the entropy II. NETWORK SYMMETRY GROUP ∗ function region Γn, however, the problems of characterizing We begin by defining the problems of interest for this work. Γ∗ and its closure Γ∗ remain open to date. Nonetheless, [1] n n Definition 1. An instance of multisource network coding provides a method, in principle, for at least bounding the problem (MSNC) is described by the tuple (G; S; T ; β) where MSNC capacity region of a network by substituting in known G = (V; E) is a directed acyclic graph, S; T ⊆ V are the polyhedral inner and outer bounds for Γ∗ [2]–[5]. When the n sets of source and sink nodes respectively, S\T = ;, and resulting inner and outer bounds match, the capacity region β : T! 2S n ; is a map giving the demands for each of the has been determined. sink nodes. The notion of network symmetry can be helpful for deter- mining these bounds and capacity regions in at least two ways. The directed acyclic graphs considered in this work are First of all, certain inner and outer bounds can be shown to assumed to be simple (i.e. there exists at most one directed match, yielding exact calculation of the rate region, when the edge between any u; v 2 V ). network has certain symmetries. In this vein, Chen and Yeung Definition 2. An instance of MSNC problem is an instance of [6] have shown that certain symmetrical parts of Γ¯∗ fixed N k-unicast problem if jβ(t)j = 1; 8t 2 T , jSj = jT j = k and under the action of symmetric groups defined by certain types β is a bijection between T and S. of partitions are equivalent to the same symmetrical parts of the Shannon outer bound ΓN . A second important way that With each source node s 2 S and edge e 2 E we network symmetry can be helpful in determining rate regions associate discrete random variables Xs and Xe respectively. for networks under MSNC is via the reduction of complexity Altogether, between the edge and source random variables, of computing their polyhedral bounds. we have a set of n = jEj + jSj random variables, collected Bearing this in mind, the first goal of this work is to for- into the set Xn. For each edge e = (u; v) 2 E, define malize the notion of structural symmetry in MSNC problems. head(e) = v and tail(e) = u. For each node v 2 V we A natural way to obtain such a formalization is via groups, define sets In(v); Out(v) ⊆ Xn. For v 2 V n S, In(v) is the and with each instance of multi-source network coding we collection of random variables associated with edges e 2 E g s.t. head(e) = v. For s 2 S, In(s) is the set fXsg. For to stabilize X setwise if X = X. The collection of all group each v 2 V n T , Out(v) is the collection of random variables elements g 2 G that setwise stabilize a subset X ⊆ S forms associated with edges e 2 E s.t. tail(e) = v whereas for t 2 T , a group called stabilizer subgroup, denoted as GX . 2n−1 0 2n−1+n Out(t) is same as β(t). Let Hn , R and Hn , R . The basic group action that we consider in this work is The network coding constraints for MSNC problem [1] can that of subgroup of Sn on Xn, or more properly, its indices be classified into 3 sets. The first set L1 contains the source S[E: such a group acts on Xn by permuting subscripts of independence constraint random variables in Xn via the map φ :(g; Xi) 7! Xig . This X fundamental group action induces an action on several other hs = hS : (1) sets that are defined using Xn. One such induced action is the s2S Xn action on the power set 2 of all subsets of Xn via the map L Xn The second set 2 contains node constraints :(g; A) 7! fφ(g; Xi) j Xi 2 Ag, for any A 2 2 . Let L^ be the set of all linear constraints of the form specified hIn(i) = hIn(i)[Out(i)); i 2 V: (2) in equations (1), (2) and (3) amongst entropies of subsets Let the subsets of L2 associated with s 2 S,v 2 V n (S[T ) of random variables in Xn and !; r that arise from MSNC 1 2 3 and t 2 T be denoted as L2, L2 and L2 respectively. The third instances. G ≤ Sn acts on each constraint C 2 L^ via the map type of constraints is the rate constraints on information rates π :(g; C) 7! C0 where C0 is is obtained from C by replacing Xn of edge random variables. For each edge e 2 E and source joint entropies hA of sets A ⊆ 2 appearing in C by joint s 2 S we have rate constraints entropy (A) and by replacing rate variables Re and !s by rate variable Reg and !sg , respectively. For example, let g = he ≤ Re; hs ≥ !s: (3) (1; 2)(3; 4)(7; 8) 2 G ≤ S8, S = [jSj], E = [jSj + jEj] n [jSj] Let the set of all rate constraints be denoted as L3. Define and C be the constraint hf1;3g = hf1;3;4;8g. Then π(g; C) g L123 as, L123 = L1 [L2 [L3. When considering rate regions, or C is the constraint hf2;4g = hf2;4;3;7g and if C is the g we purposefully remain agnostic w.r.t. the application (i.e. constraint h8 ≤ R8 then C is the constraint h7 ≤ R7. Note information flow vs. information storage) by allowing both that L123 ⊂ L^. source entropies and edge rates to be variables, which leads We define the network symmetry group (NSG) as follows. to an extended formulation of the rate region. This allows us I to develop a unified framework for symmetry in MSNC. Let Definition 4. The network symmetry group G of a MSNC \ instance I = (G = (V; E); S; T ; β) is the subgroup of Sn, L123 denote the intersection of halfspaces and hyperplanes in 0 n = jSj + jEj, that stabilizes L123 setwise under its induced Hn corresponding to constraints in the set L123. Similarly, 0 00 1 2 action on L^. let L and L denote such intersection for L1 [L2 [L2 and 3 I for L3 [L2 respectively. Under this extended formulation, the When defined this way, G also stabilizes sets L1, L2 and L3 exact rate region in [1] can be restated as setwise. Furthermore, since L1 is stablized setwise, subsets 1 2 3 ∗ 0 00 L ; L and L of constraints in L2 are also stabilized setwise. R∗ = proj (con(Γ \L ) \L ) (4) 2 2 2 !;r n A natural question now arises: given an instance I of MSNC I where proj!;r(·) corresponds to linear projection onto ! = problem, how does one compute the NSG G ? We answer [!sjs 2 S] and r = [Reje 2 E] and each set it viewed as n this question in sections IV and V, using a graph theoretic a subset of R2 −1+n with variables not appearing in its characterization of GI involving two more concepts below.
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