A Simple Sublinear Algorithm for Gap Edit Distance

TL;DR

This paper presents a simple, efficient sublinear algorithm for the gap edit distance problem, distinguishing strings with small or large edit distances in near-linear time, advancing the understanding of approximate string similarity computations.

Contribution

The paper introduces a straightforward sublinear time algorithm for the gap edit distance problem, resolving an open question for the entire relevant range of parameters.

Findings

Runs in time $ ilde O(n/\sqrt{k})$ for the gap problem

Provides a $k$-vs-$k^2$ algorithm with $ ilde O(n)$ preprocessing and $ ilde O(n/k)$ query time

Improves query time over previous algorithms for the same problem

Abstract

We study the problem of estimating the edit distance between two $n$-character strings. While exact computation in the worst case is believed to require near-quadratic time, previous work showed that in certain regimes it is possible to solve the following {\em gap edit distance} problem in sub-linear time: distinguish between inputs of distance $\le k$ and $>k^2$. Our main result is a very simple algorithm for this benchmark that runs in time $\tilde O(n/\sqrt{k})$, and in particular settles the open problem of obtaining a truly sublinear time for the entire range of relevant $k$. Building on the same framework, we also obtain a $k$-vs-$k^2$ algorithm for the one-sided preprocessing model with $\tilde O(n)$ preprocessing time and $\tilde O(n/k)$ query time (improving over a recent $\tilde O(n/k+k^2)$-query time algorithm for the same problem [GRS'20].