On the Combinatorial Version of the Slepian-Wolf Problem

TL;DR

This paper investigates the combinatorial Slepian-Wolf problem, establishing new lower bounds for linear encoding schemes and presenting an efficient randomized protocol that achieves optimal communication complexity.

Contribution

It introduces a new lower bound for syndrome coding schemes with linear encoding and provides the first polynomial-time randomized protocol matching the optimal communication complexity.

Findings

New lower bound for syndrome coding with linear encoding.

Polynomial-time randomized protocol achieving optimal communication.

Bridging the gap between theoretical bounds and practical protocols.

Abstract

We study the following combinatorial version of the Slepian-Wolf coding scheme. Two isolated Senders are given binary strings $X$ and $Y$ respectively; the length of each string is equal to $n$, and the Hamming distance between the strings is at most $\alpha n$. The Senders compress their strings and communicate the results to the Receiver. Then the Receiver must reconstruct both strings $X$ and $Y$. The aim is to minimise the lengths of the transmitted messages. For an asymmetric variant of this problem (where one of the Senders transmits the input string to the Receiver without compression) with deterministic encoding a nontrivial lower bound was found by A.Orlitsky and K.Viswanathany. In our paper we prove a new lower bound for the schemes with syndrome coding, where at least one of the Senders uses linear encoding of the input string. For the combinatorial Slepian-Wolf problem with randomized encoding the theoretical optimum of communication complexity was recently found by the first author, though effective protocols with optimal lengths of messages remained unknown. We close this gap and present a polynomial time randomized protocol that achieves the optimal communication complexity.