Longest common substrings with k mismatches

TL;DR

This paper presents a practical linear-time, constant-space algorithm for finding the longest common substrings with up to k mismatches, and offers an improved theoretical solution for the case when k=1.

Contribution

It introduces a practical $O(nm)$ time, $O(1)$ space algorithm for the longest common substring with k mismatches, and a faster $O(n ext{log} m)$ solution for k=1.

Findings

Practical $O(nm)$ time, $O(1)$ space algorithm for general k.

Theoretical $O(n ext{log} m)$ time solution for k=1.

Improves over previous $O(nm)$ algorithms for k=1.

Abstract

The longest common substring with $k$-mismatches problem is to find, given two strings $S_1$ and $S_2$, a longest substring $A_1$ of $S_1$ and $A_2$ of $S_2$ such that the Hamming distance between $A_1$ and $A_2$ is $\le k$. We introduce a practical $O(nm)$ time and $O(1)$ space solution for this problem, where $n$ and $m$ are the lengths of $S_1$ and $S_2$, respectively. This algorithm can also be used to compute the matching statistics with $k$-mismatches of $S_1$ and $S_2$ in $O(nm)$ time and $O(m)$ space. Moreover, we also present a theoretical solution for the $k = 1$ case which runs in $O(n \log m)$ time, assuming $m\le n$, and uses $O(m)$ space, improving over the existing $O(nm)$ time and $O(m)$ space bound of Babenko and Starikovskaya.