The Optimal Memory-Rate Trade-off for the Non-uniform Centralized Caching Problem with Two Files under Uncoded Placement

TL;DR

This paper introduces an optimal caching scheme for two files with non-uniform demand, achieving the best possible memory-rate trade-off by combining equal sub-packetization, popularity-based cache allocation, and interference alignment.

Contribution

It presents a novel caching and delivery strategy that is optimal for two files with arbitrary popularity distributions under uncoded placement.

Findings

Achieves the optimal expected memory-rate trade-off for two files with non-uniform demands.

Develops a new delivery scheme based on interference alignment.

Extends converse bounds to non-uniform demand scenarios.

Abstract

A new scheme for the problem of centralized coded caching with non-uniform demands is proposed. The distinguishing feature of the proposed placement strategy is that it admits equal sub-packetization for all files while allowing the users to allocate more cache to the files which are more popular. This creates natural broadcasting opportunities in the delivery phase which are simultaneously helpful for the users who have requested files of different popularities. For the case of two files, we propose a new delivery strategy based on interference alignment which enables each user to decode his desired file following a two-layer peeling decoder. Furthermore, we extend the existing converse bounds for uniform demands under uncoded placement to the nonuniform case. To accomplish this, we construct $N!$ auxiliary users, corresponding to all permutations of the $N$ files, each caching carefully selected sub-packets of the files. Each auxiliary user provides a different converse bound. The overall converse bound is the maximum of all these $N!$ bounds. We prove that our achievable delivery rate for the case of two files meets this converse, thereby establishing the optimal expected memory-rate trade-off for the case of $K$ users and two files with arbitrary popularities under uncoded placement.