Three Stories on a Two-sided Coin: Index Coding, Locally Recoverable Distributed Storage, and Guessing Games on Graphs

TL;DR

This paper explores the deep connections between index coding, distributed storage, and guessing games on graphs, revealing their complementary capacity and rate regions through graph-theoretic concepts and unifying their optimal solutions.

Contribution

It generalizes recent results to establish a fundamental complementarity between index coding and distributed storage, and links guessing games as special cases within this framework.

Findings

The capacity region of index coding is characterized by the fractional chromatic number of confusion graphs.

The optimal rate region of distributed storage is characterized by the independence number of confusion graphs.

Guessing game success probabilities are linked to index coding capacity and distributed storage rates.

Abstract

Three science and engineering problems of recent interests -index coding, locally recoverable distributed storage, and guessing games on graphs- are discussed and the connection between their optimal solutions is elucidated. By generalizing recent results by Shanmugam and Dimakis and by Mazumdar on the complementarity between the optimal broadcast rate of an index coding problem on a directed graph and the normalized rate of a locally recoverable distributed storage problem on the same graph, it is shown that the capacity region and the optimal rate region of these two problems are complementary. The main ingredients in establishing this result are the notion of confusion graph introduced by Alon et al. (2008), the vertex transitivity of a confusion graph, the characterization of the index coding capacity region via the fractional chromatic number of confusion graphs, and the characterization of the optimal rate region of the locally recoverable distributed storage via the independence number of confusion graphs. As the third and final facet of the complementarity, guessing games on graphs by Riis are discussed as special cases of the locally recoverable distributed storage problem, and it is shown that the winning probability of the optimal strategy for a guessing game and the ratio between the winning probabilities of the optimal strategy and a random guess can be characterized, respectively, by the capacity region for index coding and the optimal rate region for distributed storage.