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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. 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.
Highlights
Communication complexity, first introduced by Yao [1], is a well-studied sub-field of complexity theory which aims at quantifying the total amount of communication between the multiple parties who separately hold partial inputs of a function f
Bille et al [7] were the first to consider 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) [8] with/without self-references
We present a randomized public-coin protocol for a joint computation of the Hamming distance of two strings represented by non-self-referencing LZ77, with O(d log z) communication rounds and O(d logmax ) total bits of communication, where d is the Hamming distance between
Summary
Communication complexity, first introduced by Yao [1], is a well-studied sub-field of complexity theory which aims at quantifying the total amount of communication (bits) between the multiple parties who separately hold partial inputs of a function f. Bille et al [7] were the first to consider 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) [8] with/without self-references. Bille et al [7] proposed a randomized public-coin protocol for the LCP problem with O(log z ) communication rounds and O(log) total bits of communication, wheredenotes the length of the LCP of the two strings A and B and zdenotes the size of the non-self-referencing LZ77 factorization of the LCP A[1..`]. We present a randomized public-coin protocol for a joint computation of the Hamming distance of two strings represented by non-self-referencing LZ77, with O(d log z) communication rounds and O(d logmax ) total bits of communication, where d is the Hamming distance between. It was answered by Mantaci et al [16] that γ is non-monotonic
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