A General Coded Caching Scheme for Scalar Linear Function Retrieval

TL;DR

This paper characterizes when linear coded caching schemes for scalar linear function retrieval are optimal, revealing that optimal decoding coefficients depend on demand and encoding, and can be found via graph spanning tree solutions.

Contribution

It provides conditions for optimality of linear schemes in scalar linear function retrieval and links scheme design to graph spanning tree problems.

Findings

Optimal decoding coefficients are products of demand and encoding terms.

Relationships among encoding coefficients are represented by graph cycles.

Designing schemes reduces to solving a spanning tree problem.

Abstract

Coded caching aims to minimize the network's peak-time communication load by leveraging the information pre-stored in the local caches at the users. The original single file retrieval setting by Maddah-Ali and Niesen has been recently extended to general Scalar Linear Function Retrieval (SLFR) by Wan et al., who proposed a linear scheme that surprisingly achieves the same optimal load (under the constraint of uncoded cache placement) as in single file retrieval. This paper's goal is to characterize the conditions under which a general SLFR linear scheme is optimal and gain practical insights into why the specific choices made by Wan et al. work. This paper shows that the optimal decoding coefficients are necessarily the product of two terms, one only involving the encoding coefficients and the other only the demands. In addition, the relationships among the encoding coefficients are shown to be captured by the cycles of certain graphs. Thus, a general linear scheme for SLFR can be found by solving a spanning tree problem.