On the Index Coding Problem and its Relation to Network Coding and Matroid Theory

1 On the Index Coding Problem and its Relation to Network Coding and Matroid Theory Salim Y. El Rouayheb, Alex Sprintson, and Costas N. Georghiades Department of Electrical and Computer Engineering Texas A&M University College Station, TX, USA Email: {salim, spalex, c-georghiades}@ece.tamu.edu Abstract— The index coding problem has recently attracted a significant attention from the research community due to its theoretical significance and applications in wireless ad-hoc networks. An instance of the index coding problem includes a sender that holds a set of information messages X = {x1, . . . , xk} and a set of receivers R. Each receiver ρ = (x, H) ∈R needs to obtain a message x ∈X and has prior side information comprising a subset H of X. The sender uses a noiseless communication channel to broadcast encoding of messages in X to all clients. The objective is to find an encoding scheme that minimizes the number of transmissions required to satisfy the receivers’ demands with zero error. In this paper, we analyze the relation between the index coding problem, the more general network coding problem and the problem of finding a linear representation of a matroid. In particular, we show that any instance of the network coding and matroid representation problems can be efficiently reduced to an instance of the index coding problem. Our reduction implies that many important properties of the network coding and matroid representation problems carry over to the index coding problem. Specifically, we show that vector linear codes outperform scalar linear codes and that vector linear codes are insufficient for achieving the optimum number of transmissions. I. INTRODUCTION In recent years there has been a significant interest in utilizing the broadcast nature of wireless signals to improve the throughput and reliability of ad-hoc wireless networks. The wireless medium allows the sender node to deliver data to sev- eral neighboring nodes with a single transmission. Moreover, a wireless node can opportunistically listen to the wireless channel and store all the obtained packets, including those designated for different nodes. As a result, the wireless nodes can obtain side information which, in combination with proper encoding techniques, can lead to a substantial improvement in the performance of the wireless network. Several recent studies focused on wireless architectures that utilize the broadcast properties of the wireless channel by using coding techniques. In particular, [1], [2] proposed new architectures, referred to as COPE and MIXIT, in which routers mix packets from different information sources to increase the overall network throughput. Birk and Kol [3], [4] discussed applications of coding techniques in satellite networks with caching clients with a low-capacity reverse channel [3], [4]. The major challenge in the design of opportunistic wireless networks is to identify an optimal encoding scheme that ρ4(x4, {x1}) ρ3(x3, {x2, x4}) ρ1(x1, {x2, x3}) ρ2(x2, {x1, x3}) Fig. 1. An instance of the index coding problem with four messages and four clients. Each client is represented by a couple (x,H), where x ∈X is the packet demanded by the client, and H ⊆X represent its side information. minimizes the number of transmissions necessary to satisfy all client nodes. This can be formulated as the Index Coding problem that includes a single sender node s and a set of receiver nodes R. The sender has a set of information messages X = {x1, . . . , xk} that need to be delivered to the receiver nodes. Each receiver ρ = (x, H) ∈R needs to obtain a single message x in X and has prior side information comprising a subset H ⊆X. The sender can broadcast the encoding of messages in X to the receivers through a noiseless channel that has a capacity of one message per channel use. The objective is to find an optimal encoding scheme, referred to as an index code, that satisfies all receiver nodes with the minimum number of transmissions. With a linear encoding scheme, all messages in X are elements of a finite field and all encoding operations are linear over that field. Figure 1 depicts an instance of the index coding problem that includes a sender with four messages x1, . . . , x4 and four clients. We assume that each message is an element of GF(2n), represented by n bits. Note that the sender can satisfy the demands of all clients, in a straightforward way, by broadcasting all four messages over the wireless channel. The encoding operation achieves a reduction of the number of arXiv:0810.0068v1 [cs.IT] 1 Oct 2008 2 messages by a factor of two. Indeed, it is sufficient to send just two messages x1+x2+x3 and x1+x4 (all operations are over GF(2n)) to satisfy the requests of all clients. This example demonstrates that by using an efficient encoding scheme, the sender can significantly reduce the number of transmissions which, in turn, results in a reduction in delay and energy consumption. The above example utilizes a scalar linear encoding s

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