Index coding: An interference alignment perspective
2012 IEEE International Symposium on Information Theory Proceedings
The index coding problem is studied from an interference alignment perspective, providing new results as well as new insights into, and generalizations of, previously known results. An equivalence is established between multiple unicast index coding where each message is desired by exactly one receiver, and multiple groupcast index coding where a message can be desired by multiple receivers, which settles the heretofore open question of insufficiency of linear codes for the multiple unicast
... x coding problem by equivalence with multiple groupcast settings where this question has previously been answered. Necessary and sufficient conditions for the achievability of rate half per message are shown to be a natural consequence of interference alignment constraints, and generalizations to feasibility of rate 1 L+1 per message when each destination desires at least L messages, are similarly obtained. Finally, capacity optimal solutions are presented to a series of symmetric index coding problems inspired by the local connectivity and local interference characteristics of wireless networks. The solutions are based on vector linear coding. * Presented in part at ISIT 2012. Hamed Maleki and Syed Jafar (email: firstname.lastname@example.org, email@example.com) are with the Center for Pervasive Communications and Computing (CPCC) at Much progress in network information theory can be attributed to the pursuit of the capacity of simpleto-describe canonical network communication models. Simplicity in the network communication models often affords a clear formulation of techniques involved in the communication system. The focus of this paper is the index coding problem which is arguably the simplest multiuser capacity problem because it is a communication network that has only one link with finite capacity. Yet, this turns out to be the proverbial case where appearances can be quite deceiving. More than a decade after it was introduced by Birk and Kol in [1, 2] , the index coding problem not only remains open, but also has been shown to include as special cases a number of difficult problems in both wired and wireless settings -such as the general multiple unicast problem with linear network coding , multi-way relay networks  , and the blind cellular interference alignment problem in wireless networks  , to name a few. Remarkably, the index coding problem is also the origin of the fundamental idea of interference alignment , which was re-discovered, extensively studied and developed in a variety of forms in wireless networks [6, 7, 8] and has recently found applications in network coding problems such as the distributed data storage exact repair problem [9, 10] and the 3 unicast problem [11, 12] . In this paper, we attempt to bring this idea "home", by applying the understanding of the principles of interference alignment, into the original setting -the index coding problem. The essence of the index coding problem lies in its focus on a single bottleneck network. Having only one link with finite capacity concentrates the challenge of network coding in one place, highlighting some of the most fundamental, challenging, and surprising aspects of the network coding problem. Understanding the role of a single bottleneck edge in a network when the rest of the network is composed only of trivial links (of infinite capacity), is a natural stepping stone toward a broader understanding of communication networks 1 . We start with a discussion of similarly motivated single-bottleneck settings for both wired and wireless networks. Single Bottleneck Wireless Networks -Wireless Index Coding Consider a wireless network shown in Figure 2 (a) comprised of the source nodes shown on the left, which communicate with destination nodes shown on the right, through an intermediate network of relay nodes. Depending on propagation path loss different pairs of nodes may be connected or disconnected. Because this is a wireless setting, signals emerging from the same transmitter are broadcast, and signals arriving at the same receiver interfere. All transmitters are subject to power constraint P , and generally the receivers experience additive white Gaussian noise (AWGN) in addition to the superposition of fading signals from connected transmitters. As an analogue to the index coding problem defined by a single bottleneck link, let us assume only one of the receivers in the intermediate network experiences AWGN, e.g., of unit variance, while all other receivers experience no noise, i.e., have infinite resolution of the complex valued signals, essentially providing them infinite capacity links to their respective connected transmitters. Eliminating messages that have infinite capacity paths between their sources and all their desired destinations, what remains is the wireless index coding problem, introduced in . While, depending on the wireless network topology, the resulting wireless index coding problem can in general be quite involved, e.g., if the wireless network contains cycles that provide feedback from the output of the bottleneck receiver to the distributed or partially cooperating nodes transmitting to the bottleneck receiver, Fig. 2(b) shows a relatively simple example of the wireless index coding problem that corresponds to the index coding problem of Fig. 1(b) , in the sense that the capacity of the index coding problem maps directly to the degrees of freedom (DoF) of the wireless index coding problem. The capacity per message of the index coding problem in Figure 1(b) is 2/5, as is the DoF value per message for the wireless index coding problem in Fig. 2(b) , and in both cases the "unit" for measurement is the capacity/DoF of the bottleneck link/receiver. The index coding problem normalizes the bottleneck link capacity to unity, so that all rates are measured as multiples of the bottleneck link capacity, and the wireless index coding problem normalizes the number of signal dimensions (DoF) Figure 2: (a) Wireless network: If only one receiver (shown with incoming signals in black) in the intermediate network has non-zero (unit) AWGN variance, and all the other receivers have zero noise (infinite capacity), then the remaining problem is the wireless index coding problem. (b) Example of a wireless index coding setting. available to the bottleneck receiver to unity, and all DoF are measured as multiples of the bottleneck DoF. As explained in  , the relationship between the index coding problem and the wireless index coding problem goes much further, and much more can be said about their similarities and differences. For instance, if full cooperation is allowed between all sources directly transmitting to the bottleneck receiver in a wireless index coding problem, the DoF of the resulting network is the same as the capacity of the corresponding index coding problem (in their respective units). The DoF of the wireless index coding problem are, in general, bounded above by the capacity of the index coding problem. It also highlights the main difference between the index coding problem and the wireless index coding problem -all sources are necessarily allowed to fully cooperate in the former because the bottleneck transmitter has full knowledge of all messages, but not necessarily in the latter (depending on the topology of the original network in Figure 2(a) ). However, if the index coding problem has a capacity optimal vector linear coding solution that can be translated to the complex field, then the same solution may be applied in the wireless index coding problem as well. This is because vector linear solutions are comprised of a superposition of separately encoded messages, and a superposition over complex field is naturally provided by the wireless medium  . Somewhat surprisingly, this is a very common situation, e.g., all the instances of the index coding problems studied in this paper have capacity optimal vector linear coding solutions that translate to the complex field, thereby simultaneously providing the DoF characterization for the corresponding wireless index coding problem. In the wireless index coding problem discussed above, the bottleneck is concentrated at one receiver, lending the bottleneck a multiple access character. Another formulation of the wireless index coding problem is also conceivable where the bottleneck may be concentrated at one transmitter, e.g., all receivers experience additive noise and there is only one transmitter with finite power (all other transmitters have infinite power), which would lend the bottleneck a broadcast character, and which could be a similarly interesting and promising research avenue.