Improved Sublinear-Time Edit Distance for Preprocessed Strings

TL;DR

This paper presents improved algorithms for approximating the edit distance between strings in sublinear time using preprocessing, achieving faster and simpler solutions than previous methods.

Contribution

It introduces new sublinear-time algorithms for gap edit distance that leverage preprocessing to improve efficiency and simplicity over prior approaches.

Findings

Preprocessing one string enables faster approximate edit distance computation.

Preprocessing both strings further reduces the query time.

The algorithms outperform previous methods in gap, query, and preprocessing times.

Abstract

We study the problem of approximating the edit distance of two strings in sublinear time, in a setting where one or both string(s) are preprocessed, as initiated by Goldenberg, Rubinstein, Saha (STOC '20). Specifically, in the $(k, K)$-gap edit distance problem, the goal is to distinguish whether the edit distance of two strings is at most $k$ or at least $K$. We obtain the following results: * After preprocessing one string in time $n^{1+o(1)}$, we can solve $(k, k \cdot n^{o(1)})$-gap edit distance in time $(n/k + k) \cdot n^{o(1)}$. * After preprocessing both strings separately in time $n^{1+o(1)}$, we can solve $(k, k \cdot n^{o(1)})$-gap edit distance in time $k \cdot n^{o(1)}$. Both results improve upon some previously best known result, with respect to either the gap or the query time or the preprocessing time. Our algorithms build on the framework by Andoni, Krauthgamer and Onak (FOCS '10) and the recent sublinear-time algorithm by Bringmann, Cassis, Fischer and Nakos (STOC '22). We replace many complicated parts in their algorithm by faster and simpler solutions which exploit the preprocessing.