Fundamental Structure of Optimal Cache Placement for Coded Caching with Nonuniform Demands

TL;DR

This paper characterizes the optimal cache placement structure for coded caching with nonuniform demands, revealing at most three file groups and providing a closed-form solution, which improves understanding and bounds in coded caching systems.

Contribution

It identifies the fundamental structure of optimal uncoded cache placement under nonuniform demands, introducing a closed-form solution and a new tighter information-theoretic bound.

Findings

Optimal cache placement involves at most three file groups.

Closed-form solutions are derived for each file group structure.

Maximum subpacketization scales as O(2^K/√K) in the worst case.

Abstract

This paper studies the caching system of multiple cache-enabled users with random demands. Under nonuniform file popularity, we thoroughly characterize the optimal uncoded cache placement structure for the coded caching scheme (CCS). Formulating the cache placement as an optimization problem to minimize the average delivery rate, we identify the file group structure in the optimal solution. We show that, regardless of the file popularity distribution, there are \emph{at most three file groups} in the optimal cache placement{, where files within a group have the same cache placement}. We further characterize the complete structure of the optimal cache placement and obtain the closed-form solution in each of the three file group structures. A simple algorithm is developed to obtain the final optimal cache placement by comparing a set of candidate closed-form solutions computed in parallel. We provide insight into the file groups formed by the optimal cache placement. The optimal placement solution also indicates that coding between file groups may be explored during delivery, in contrast to the existing suboptimal file grouping schemes. Using the file group structure in the optimal cache placement for the CCS, we propose a new information-theoretic converse bound for coded caching that is tighter than the existing best one. Moreover, we characterize the file subpacketization in the CCS with the optimal cache placement solution and show that the maximum subpacketization level in the worst case scales as $\mathcal{O}(2^K/\sqrt{K})$ for $K$ users.