Compressed Communication Complexity of Hamming Distance

TL;DR

This paper introduces a randomized protocol for computing the Hamming distance between strings compressed with LZ77, utilizing advanced factorization techniques, and explores properties of LZ77 compression sizes.

Contribution

It presents a novel randomized protocol for Hamming distance in LZ77 compressed strings and analyzes the non-monotonicity of LZ77 compression sizes.

Findings

Protocol effectively computes Hamming distance from compressed data.

LZ77 compression size can increase when prefixes are removed.

Analysis uses Crochemore's C-factorization and Rytter's AVL-grammar.

Abstract

We consider the communication complexity of the Hamming distance of two strings. Bille et al. [SPIRE 2018] considered the communication complexity of the longest common prefix (LCP) problem in the setting where the two parties have their strings in a compressed form, i.e., represented by the Lempel-Ziv 77 factorization (LZ77) with/without self-references. We present a randomized public-coin protocol for a joint computation of the Hamming distance of two strings represented by LZ77 without self-references. While our scheme is heavily based on Bille et al.'s LCP protocol, our complexity analysis is original which uses Crochemore's C-factorization and Rytter's AVL-grammar. As a byproduct, we also show that LZ77 with/without self-references are not monotonic in the sense that their sizes can increase by a factor of 4/3 when a prefix of the string is removed.