An Umbrella Converse for Data Exchange: Applied to Caching, Computing, and Shuffling

TL;DR

This paper introduces a universal converse bound for data exchange problems in multi-user communication systems, encompassing various specific problems like caching and shuffling, and demonstrates its effectiveness in recovering and extending known results.

Contribution

It presents a general, parameter-independent converse bound for data exchange, unifying and extending existing bounds in related problems such as caching and computing.

Findings

Successfully recovers known converses in caching and shuffling.

Derives a new general converse for heterogeneous cache sizes.

Relates the bound to the generalized independence number in index coding.

Abstract

The problem of data exchange between multiple nodes with storage and communication capabilities models several current multi-user communication problems like Coded Caching, Data Shuffling, Coded Computing, etc. The goal in such problems is to design communication schemes which accomplish the desired data exchange between the nodes with the optimal (minimum) amount of communication load. In this work, we present a converse to such a general data exchange problem. The expression of the converse depends only on the number of bits to be moved between different subsets of nodes, and does not assume anything further specific about the parameters in the problem. Specific problem formulations, such as those in Coded Caching, Coded Data Shuffling, Coded Distributed Computing, can be seen as instances of this generic data exchange problem. Applying our generic converse, we are able to efficiently recover known important converses in these formulations. Further, for a generic coded caching problem with heterogeneous cache sizes at the clients with or without a central server, we obtain a new general converse, which subsumes some existing results. Finally we relate a `centralized' version of our bound to the known generalized independence number bound in index coding, and discuss our bound's tightness in this context.