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Applications of Sheaf Cohomology and Exact Sequences on Network Codings

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Applications of Sheaf Cohomology and Exact Sequences on Network Codings Applications of Sheaf Cohomology and Exact Sequences on Network Codings Robert Ghrist Yasuaki Hiraoka Departments of Mathematics and Department of Mathematical Electrical & Systems Engineering and Life Sciences University of Pennsylvania Hiroshima University Philadelphia, PA, USA Higashi-Hiroshima, Hiroshima, Japan Email: [email protected] Email: [email protected] Abstract—Sheaf cohomology is a mathematical tool for cohomological tools are easily extendable on a higher collating local algebraic data into global structures. The dimensional base space, such as simplicial complexes purpose of this paper is to apply sheaf theory into network or product of them, which might be useful in situa- coding problems. After the definition of sheaves, we define so called network coding sheaves for a general multi source tions with spacial expanse like wireless settings or with network coding scenario, and consider various forms of sheaf time dependence (product with time axis R). As a first cohomologies. The main theorem states that 0-th network step to this direction, we study applications of sheaf coding sheaf cohomology is equivalent to information flows cohomology to some basic problems in network codings for the network coding. Then, this theorem is applied by means of some fundamental exact sequences in this to several practical problems in network codings such as maxflow bounds, global extendability, network robustness, paper. and data merging, by using some of the standard exact This paper is organized as follows. In Section II, we sequences of homological algebra. explain fundamental sheaf theory which will be required to understand this paper. In particular, sheaf cohomol- I. INTRODUCTION ogy and its basic operations will be discussed in detail. This paper introduces new tools for the analysis of It should be noted that our tools presented here are data flows over networks, especially focusing on net- computable by elementary module calculus (or by linear work codings. The problem of network coding is one of algebra for data in a field), although sheaf theory itself a host of problems in data analysis and management that is highly abstract subject in mathematics. An important require an understanding of local-to-global transitions. notion introduced in this section is a network coding sheaf A new set of tools we present in this paper is based on (NC sheaf for short), which gives a relationship between sheaf theory. sheaf theory and network coding problems. Especially, Sheaf theory was invented in the mid 1940s as a information theoretical meaning of NC sheaf cohomol- branch of algebraic topology to deal with the collation of ogy plays important roles for applications. In Section local data on topological spaces. Through the success in III, NC sheaf cohomology is applied into some practical the theory of functions of several complex variables and problems (maxflow bound, global extendability, network algebraic geometry, this theory is now indispensable in robustness, and data merging). All of the techniques modern mathematics. However, instead of its generality used in this section are standard long exact sequences dealing with local-to-global transitions, applications to in homological algebra. In Section IV, we explain some other areas in science or engineering have not been future directions of this work. Throughout the paper, we well established so far except for logic and semantics refer to [1][4][6] for general discussions on sheaf theory. in computer science with the notion of Topos (e.g., [2][8][9]). II. SHEAF FORMULATION OF NETWORK CODINGS On the other hand, there is a powerful mathematical A. Definition of Sheaves tool in sheaf theory, so called sheaf cohomology, which Let X be a topological space (e.g., network) and R be can treat local-to-global transition in algebraic data level. a commutative ring. One of the important messages in this paper is to show usefulness of sheaf cohomology for applications to anal- Definition 1. (Presheaf). A presheaf F on X consists of ysis of data on a network. This viewpoint provides us the following data: with a lot of analytical tools from homological algebra - an R-module F(U) for each open subset U ⊂ X. such as exact sequences for various forms of sheaf - an R-linear map ρVU : F(U) F(V) for each pair V ⊂ cohomology. Moreover, not only on a network, sheaf U ⊂ X. ! These data satisfy the following conditions: S(rj) corresponds to the decoding map. Let us denote the set of all local coding maps by Φ = fφ g. ρ = id ; ρ ◦ ρ = ρ for W ⊂ V ⊂ U; wv UU U WV VU WU In order to express decodable information flows on where idU is the identity map on F(U). a network as a network coding sheaf cohomology, we extend the graph G = (V; E) to X = (V; E˜ ), where E˜ is σ 2 F(U) F(U) An element is called a section on , and given by adding edges e = jrjsij in E from each receiver R ρ an -linear map VU is called a restriction map. We often rj to all of its requesting sources si 2 S(rj) with cap(e = write σjV instead of ρVU(σ), and call it the restriction of jrjsij) = nsi . For removing ambiguity, we denote the σ to V. set of incoming edges at v 2 V in E or E˜ by In(v; E) or Definition 2. (Sheaf). A presheaf F on X is called a sheaf In(v; E˜ ), respectively. Out(v; E) and Out(v; E˜ ) are similarly if it satisfies the following two conditions: defined. This extension enables one to compare decoded infor- 1) For any open set U ⊂ X, any open covering U = mation at each receiver rj with transmitted information [i2IUi, and any section σ 2 F(U), σjU = 0 for all i from s 2 S(r ) as the glueing condition of the network i 2 I implies σ = 0. i j coding sheaf on the added edge e = jr s j. 2) For any open set U ⊂ X, any open covering U = j i Let us equip X with the usual Euclidean topology. For [ U , any family σ 2 F(U ) satisfying σ j = i2I i i i i Ui\Uj the definition of a network coding sheaf F (NC sheaf for σ j for all pairs (i; j), there exists σ 2 F(U) j Ui\Uj short), we at first assign sections for some special open such that σj = σ for all i 2 I. Ui i sets. Remark 3. 1) Each R-module F(U) is regarded as local Definition 4. (Local Sections). data storage for applications. 1) For a connected open set U contained in an edge 2) From the conditions in Definition 2, a sheaf F allows e 2 E˜ , F(U) := kcap(e). one to glue a set of local data together into global 2) For a connected open set U which only contains data uniquely. nv lv one node v 2 V, F(U) := k ⊕ k , where lv = B. Network Coding Sheaves e2In(v;E) cap(e). Let us at first explain the problem setting of network DefinitionP 5. (Local Restriction Maps). codings (e.g.,[7][10]) on which we construct sheaves. Let 1) For connected open sets V ⊂ U ⊂ e for some edge k be an R-module, or simply a (finite) field. Let G = e, ρVU := id : F(U) F(V). (V; E) be a directed graph(not necessarily acyclic), where 2) For connected open sets V ⊂ U, where U contains V E ˜ and are finite sets of nodes and edges, respectively. only one node v and! V is located in e 2 In(v; E), A directed edge e 2 E from v 2 V to w 2 V is denoted ρVU : F(U) F(V) is given by the projection map by e = jvwj (Head(e) := w, Tail(e) := v). We assume induced by the product structure in Definition 4 2). S = fs ; ··· ; s g ⊂ V that there exists a subset 1 α of nodes 3) For connected! open sets W ⊂ U, where U contains nsi called sources which transmit elements in k ; nsi 2 N; only one node v and W is located in e 2 Out(v; E˜ ), as information for each si 2 S. We often identify a ρWU := φwv : F(U) F(W), where w = Head(e). graph G with a topological space by the geometrical From these local definitions of sections and restriction representation with the usual Euclidean topology. ! We also assume that there exists a subset R = maps, the network coding sheaf is defined by the sheafifi- cation. It is a process to construct F(U) for arbitrary open fr1; ··· ; rβg ⊂ V of nodes called receivers. Each receiver U ⊂ X ρ : F(U) F(V) requires information from some sources and this assign- set and the restriction map VU ment is determined by S : R 2S in the sense that by using the glueing condition in Definition 2. More precisely, it is explained in the following way. a receiver ri requires all transmitted information from ! S S(ri) 2 2 . ! Definition 6. (NC Sheaf). For an open set U ⊂ X, Let cap : E N be a capacity function which assigns F(U) is defined by the set of all equivalent classes for each edge e 2 E its edge capacity cap(e). The set σ = [(σi;Ui)i2I], where a representative (σi;Ui)i2I with of the incoming! (outgoing, resp.) edges in the sense of a covering U = [i2IUi is given by a family of sections v 2 V (v) edge directions at a node is denoted by In σi 2 F(Ui) satisfying σijUi\Uj = σjjUi\Uj , and the (Out(v), resp.).
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