IT ] 12 A pr 2 01 6 Distributed Index Coding

In this paper, we study the capacity region of the general distributed index coding. In contrast to the traditional centralized index coding where a single server contains all n messages requested by the receivers, in the distributed ind ex coding there are 2 − 1 servers, each containing a unique non-empty subset J of the messages and each is connected to all receivers via a noiseless independent broadcast link with an arbitrary capacity CJ ≥ 0. First, we generalize the existing polymatroidal outer bound on the capacity region of the centralized problem to the distributed case. Next, building upon the existing centralized composite coding scheme, we propo se three distributed composite coding schemes and derive the corresponding inner bounds on the capacity region. We prese nt a number of interesting numerical examples, which highlight the subtleties and challenges of dealing with the distributed i ndex coding, even for very small problem sizes of n = 3 and n = 4. I. I NTRODUCTION The index coding problem is a canonical problem in network information theory and has been studied over the past two decades using tools from various disciplines such a s combinatorics, algebra, and information theory [1], [2]. I n the traditional index coding problem, it is assumed that one ser ver has all messages requested by the receivers. However, in man y practical circumstances this assumption might not be true a nd the messages might be distributed over multiple servers. We refer to this more general version as the distributed index coding problem compared to the traditional centralized index coding problem. The distributed index coding problem was first studied by Ong, Ho, and Lim [3], where lower and upper bounds on the optimal codelength were derived in the special case in which each message is only known to one receiver and each receiver only knows one message a priori and it is shown that the bounds match if no two servers have any messages in common. In this paper, we study the distributed index coding problem in its general form, which to the best of our knowledge has not been investigated before. First, we generalize the polymatroidal outer bound on the capacity region of the centralized problem [4] to the distributed case. Next, buil ding upon the existing centralized composite coding scheme [4], we propose three distributed composite coding schemes, derive the corresponding inner bounds on the capacity regio n and show their use via examples. Although the polymatroidal outer bound is tight for all centralized index coding proble ms with up to n = 5 messages [4], there exist instances of the distributed problem withn = 3 messages for which the polymatroidal outer bound is not tight. Nevertheless, using customized Shannon-type inequalities, we show that t he proposed distributed composite coding scheme achieves the capacity region for all instances of the problem with n = 3 messages. In this paper,[n] denotes the set {1, 2, . , n} and the set of all nonempty subsets of [n] is N . = {J ⊆ [n] : J 6= ∅}. II. SYSTEM MODEL AND PROBLEM SETUP Consider the following index coding problem. There are n messages in the system, x1, x2, · · · , xn, wherexj ∈ {0, 1}j for j ∈ [n] and sometj . There aren receivers, where receiver j wants to obtain message xj and knows a subset of the messages a priori, denoted by x(Aj) for someAj ⊆ [n]\{j}. For simplicity of notation throughout the paper and where the context is clear, we will refer to j as the wanted message and toAj as the side information of receiver j, respectively. Any instance of this problem can be specified by a side information graphG with n nodes, in which a directed edge i → j represents that receiver j has messagei as side information (i ∈ Aj). For instance, Fig. 1 shows the directed graph representing the index coding problem with A1 = ∅, A2 = {3}, andA3 = {2}. The main difference in the system model compared to traditional (centralized) index coding problem is in the se rver setup. Instead of a single server which contains all message , there are2−1 servers. For each J ∈ N , there is a server that contains all messages j ∈ J and the capacity of the broadcast link connecting serverJ to all receivers is denoted by CJ . Hence, we assume that there are 2− 1 ideal bit pipes to the receivers with arbitrary link capacities. This is a fairly g eneral model that allows for all possible message availabilities o n different servers. IfCJ = 1 only for J = [n] and is zero otherwise, we recover the centralized index coding problem . A special normalized symmetric case is where CJ = 1 for all J ∈ N . ServerJ sends sequence yJ ∈ {0, 1}J for some uJ to all receivers which is a function of the messages at that server. Based on the side information Aj and the received bits yJ ∈ {0, 1} uJ from all servers, receiver j finds the estimate x̂j of the messagexj (see Fig. 2). We say that rate-capacity PSfrag replacements 1