The Communication Complexity of Pattern Matching with Edits Revisited

TL;DR

This paper improves the understanding of the communication complexity for pattern matching with edits, providing tighter bounds and matching lower bounds for the problem, especially over constant-sized alphabets.

Contribution

It refines the encoding size to match lower bounds and establishes a new tight lower bound for reporting edit sequences in pattern matching with edits.

Findings

Improved encoding size to O(n/m · k log(m|Σ|/k))

Established a new tight lower bound matching the encoding size

Matching the communication complexity for pattern matching with mismatches in streaming contexts

Abstract

In the decades-old Pattern Matching with Edits problem, given a length-$n$ string $T$ (the text), a length-$m$ string $P$ (the pattern), and a positive integer $k$ (the threshold), the task is to list the $k$-error occurrences of $P$ in $T$, that is, all fragments of $T$ whose edit distance to $P$ is at most $k$. The one-way communication complexity of Pattern Matching with Edits is the minimum number of bits that Alice, given an instance $(P, T, k)$ of the problem, must send to Bob so that Bob can reconstruct the answer solely from that message. For the natural parameter regime of $0 < k < m < n/2$, our recent work [STOC'24] yields that ${\Omega}(n/m \cdot k \log(m/k))$ bits are necessary and $O(n/m \cdot k \log^2 m)$ bits are sufficient for Pattern Matching with Edits. More generally, for strings over an alphabet ${\Sigma}$, our recent work [STOC'24] gives an $O(n/m \cdot k \log m \log(m|{\Sigma}|))$-bit encoding that allows one to recover a shortest sequence of edits for every $k$-error occurrence of $P$ in $T$. In this work, we revisit the original proof and improve the encoding size to $O(n/m \cdot k \log(m|{\Sigma}|/k))$, which matches the lower bound for constant-sized alphabets. We further establish a new tight lower bound of ${\Omega}(n/m \cdot k \log(m|{\Sigma}|/k))$ for the edit sequence reporting variant that we solve. Our encoding size also matches the communication complexity established for the simpler Pattern Matching with Mismatches problem in the context of streaming algorithms [Clifford, Kociumaka, Porat; SODA'19].