Compressed Communication Complexity of Hamming Distance

• Main Authors: Shiori Mitsuya, Yuto Nakashima, Shunsuke Inenaga, Hideo Bannai, Masayuki Takeda
• Format: Article
• Language: English
• Published: MDPI AG 2021-04-01
• Series: Algorithms
• Subjects: communication complexity; Hamming distance; Lempel-Ziv compression
• Online Access: https://www.mdpi.com/1999-4893/14/4/116

We consider the communication complexity of the <i>Hamming distance</i> 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. Although 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.