Compressed Communication Complexity of Longest Common Prefixes

TL;DR

This paper explores how the compressibility of the longest common prefix (LCP) can be exploited to reduce communication rounds and bits in distributed LCP problems, especially when the prefix is highly compressible.

Contribution

It introduces communication protocols that leverage LCP compressibility to optimize round complexity and communication cost in two-party and multi-party settings.

Findings

Reducing rounds to O(log z) when LCP is compressible

Achieving O(log z * log k + log l) communication in multi-string case

Protocols extendable from public-coin to private-coin models

Abstract

We consider the communication complexity of fundamental longest common prefix (Lcp) problems. In the simplest version, two parties, Alice and Bob, each hold a string, $A$ and $B$, and we want to determine the length of their longest common prefix $l=\text{Lcp}(A,B)$ using as few rounds and bits of communication as possible. We show that if the longest common prefix of $A$ and $B$ is compressible, then we can significantly reduce the number of rounds compared to the optimal uncompressed protocol, while achieving the same (or fewer) bits of communication. Namely, if the longest common prefix has an LZ77 parse of $z$ phrases, only $O(\lg z)$ rounds and $O(\lg \ell)$ total communication is necessary. We extend the result to the natural case when Bob holds a set of strings $B_1, \ldots, B_k$, and the goal is to find the length of the maximal longest prefix shared by $A$ and any of $B_1, \ldots, B_k$. Here, we give a protocol with $O(\log z)$ rounds and $O(\lg z \lg k + \lg \ell)$ total communication. We present our result in the public-coin model of computation but by a standard technique our results generalize to the private-coin model. Furthermore, if we view the input strings as integers the problems are the greater-than problem and the predecessor problem.