On the Combinatorial Version of the Slepian–Wolf Problem

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 the strings $X$ and $Y$ . The aim is to minimize 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 bound was found by Orlitsky and Viswanathany. In this 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 known earlier, even 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.