Side Information Graph Research Articles

Strategies for Optimal Transmission and Delay Reduction in Dynamic Index Coding Problem

Dynamic index coding is a practical generalization of conventional index coding that deals with real dynamic traffic streams. We identify the code-constrained capacity region of a dynamic index coding problem with a complete bi-directional side information graph and introduce the performance metric of dynamic index coding gain to measure how dynamic index coding reduces the required data transmissions. A greedy dynamic index coding scheme is proposed that achieves the maximum coding gain almost everywhere in the identified capacity region. Although the greedy scheme attains the maximum coding gain, its selfish nature may unacceptably increase transmission delay. To address this issue, a time-shared friendly dynamic index coding scheme is introduced that achieves the maximum coding gain over the entire capacity region and offers a lower delay than its greedy counterpart. To obtain the minimum delay, a constrained optimization problem is formulated to tune time-sharing weights in the friendly scheme. The closed-form solution of the optimization is derived for the special two-flow case. Furthermore, the results and analysis are extended to dynamic index coding problems with arbitrary side information graphs. We also use analytical and simulation results to provide graphical intuition for the obtained results.

Locally Decodable Index Codes

An index code for broadcast channel with receiver side information is locally decodable if each receiver can decode its demand by observing only a subset of the transmitted codeword symbols instead of the entire codeword. Local decodability in index coding is known to reduce receiver complexity, improve user privacy and decrease decoding error probability in wireless fading channels. Conventional index coding solutions assume that the receivers observe the entire codeword, and as a result, for these codes the number of codeword symbols queried by a user per decoded message symbol, which we refer to as locality , could be large. In this paper, we pose the index coding problem as that of minimizing the broadcast rate for a given value of locality (or vice versa) and designing codes that achieve the optimal trade-off between locality and rate. We identify the optimal broadcast rate corresponding to the minimum possible value of locality for all single unicast problems. We present new structural properties of index codes which allow us to characterize the optimal trade-off achieved by: vector linear codes when the side information graph is a directed cycle; and scalar linear codes when the minrank of the side information graph is one less than the order of the problem. We also identify the optimal trade-off among all codes, including non-linear codes, when the side information graph is a directed 3-cycle. Finally, we present techniques to design locally decodable index codes for arbitrary single unicast problems and arbitrary values of locality.

Generalized Lexicographic Products and the Index Coding Capacity

The index coding problem studies the fundamental limit on broadcasting multiple messages to their respective receivers with different sets of side information that are represented by a directed graph. The generalized lexicographic product structure in the side information graph is introduced as a natural condition under which the corresponding index coding problem can be decomposed into multiple interacting subproblems, each consisting of vertices with the same adjacency pattern with respect to other subproblems. For side information graphs with this structure, the capacity region is characterized in terms of the subproblem capacity regions combined in the same product structure. The proof is based on dual uses of random coding-one for a new multiletter characterization of the capacity region of a general index coding problem via joint typicality decoding and the other for a construction of a new multiletter code of matching rates from a single-letter code via joint typicality encoding. Several special cases are discussed that recover and strengthen known structural properties of the index coding capacity region.

Linear Programming Approximations for Index Coding

Index coding, a source coding problem over broadcast channels, has been a subject of both theoretical and practical interests since its introduction (by Birk and Kol, 1998). In short, the problem can be defined as follows: there is an input $P \triangleq (p_{1}, {\dots }, p_{n})$ , a set of $n$ clients who each desire a single entry $p_{i}$ of the input, and a broadcaster whose goal is to send as few messages as possible to all clients so that each one can recover its desired entry. Additionally, each client has some predetermined “side information,” corresponding to the certain entries of the input $P$ , which we represent as the “side information graph” $ \mathcal {G}$ . The graph $ \mathcal {G}$ has a vertex $v_{i}$ for client $i$ and a directed edge $(v_{i}, v_{j})$ , indicating that client $i$ knows the $j$ th entry of the input. Given a fixed side information graph $ \mathcal {G}$ , we are interested in determining or approximating the “broadcast rate” of index coding on the graph, i.e., the least number of messages the broadcaster can transmit so that every client recovers its desired information. The complexity of determining this broadcast rate in the most general case is open, and the best-known approximations are barely better than the trivial $O(n)$ -approximation corresponding to sending each client their information directly without performing any coding. Using index coding schemes based on linear programs (LPs), we take a two-pronged approach to approximating the broadcast rate. First, extending earlier work on planar graphs, we focus on approximating the broadcast rate for special graph families, such as graphs with the small chromatic number and disk graphs. In certain cases, we are able to show that simple LP-based schemes give constant-factor approximations of the broadcast rate, which seem extremely difficult to obtain in the general case. Second, we provide several LP-based schemes for the general case, which are not constant-factor approximations, but which strictly improve on the best-known schemes. These can be viewed as both a strengthening of the constant-factor approximations proven for special graph families (as these schemes strictly improve on those which we prove are good approximations), as well as another tool that can be used either in practice or in future theoretical analyses.

Linear Index Coding With Multiple Senders and Extension to a Cellular Network

In this paper, linear index codes with multiple senders are studied, where every receiver receives encoded messages from all senders. A new fitting matrix for the multiple senders is proposed and it is proved that the minimum rank of the proposed fitting matrices is the optimal codelength of linear index codes for the multiple senders. In addition, a new type of a side information graph related with the optimal codelength is proposed and whether given side information is critical or not is studied. Furthermore, linear index codes for the cellular network scenario are studied, where each receiver can receive a subset of sub-codewords. Since some receivers cannot receive the entire codeword in the cellular network scenario, the encoding method based on the fitting matrix has to be modified. In the cellular network scenario, we propose another fitting matrix and prove that an optimal generator matrix can be found based on these fitting matrices. In addition, some properties on the optimal codelength of linear index codes for the cellular network case are studied.

On the Security of Index Coding With Side Information

Security aspects of the index coding with side information (ICSI) problem are investigated. Building on the results of Bar-Yossef (2006), the properties of linear index codes are further explored. The notion of weak security, considered by Bhattad and Narayanan (2005) in the context of network coding, is generalized to block security. It is shown that the linear index code based on a matrix <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> , whose column space code <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> ) has length <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> , minimum distance <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> , and dual distance <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">⊥</sup> , is ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> -1- <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i> ) -block secure (and hence also weakly secure) if the adversary knows in advance <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i> ≤ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> -2 messages, and is completely insecure if the adversary knows in advance more than <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> - <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">⊥</sup> messages. Strong security is examined under the conditions that the adversary: 1) possesses <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i> messages in advance; 2) eavesdrops at most μ transmissions; 3) corrupts at most δ transmissions. We prove that for sufficiently large <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> , an optimal linear index code which is strongly secure against such an adversary has length κ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> +μ+2δ . Here, κ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> is a generalization of the min-rank over F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> of the side information graph for the ICSI problem in its original formulation in the work of Bar-Yossef et al.

Index Coding With Side Information

Motivated by a problem of transmitting supplemental data over broadcast channels (Birk and Kol, INFOCOM 1998), we study the following coding problem: a sender communicates with n receivers R1,..., Rn. He holds an input x ∈ {0,01l}n and wishes to broadcast a single message so that each receiver Ri can recover the bit xi. Each Ri has prior side information about x, induced by a directed graph Grain nodes; Ri knows the bits of a; in the positions {j | (i,j) is an edge of G}.G is known to the sender and to the receivers. We call encoding schemes that achieve this goal INDEXcodes for {0,1}n with side information graph G. In this paper we identify a measure on graphs, the minrank, which exactly characterizes the minimum length of linear and certain types of nonlinear INDEX codes. We show that for natural classes of side information graphs, including directed acyclic graphs, perfect graphs, odd holes, and odd anti-holes, minrank is the optimal length of arbitrary INDEX codes. For arbitrary INDEX codes and arbitrary graphs, we obtain a lower bound in terms of the size of the maximum acyclic induced subgraph. This bound holds even for randomized codes, but has been shown not to be tight.