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.

中文翻译：

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汉明距离的压缩通信复杂度

我们考虑了两个字符串的汉明距离的通信复杂性。Bille等。[SPIRE 2018]考虑了在最长的公共前缀（LCP）问题的通信复杂性的情况下，当事双方以压缩形式表示其字符串时，即由带/不带自噪声的Lempel-Ziv 77因式分解（LZ77）表示参考。我们提出了一种随机公共硬币协议，用于联合计算以LZ77表示的两个字符串的汉明距离，而无需自我参照。虽然我们的方案主要基于Bille等人的LCP协议，但我们的复杂度分析却是原始的，它使用了Crochemore的C分解和Rytter的AVL语法。作为副产品，