A note on the longest common substring with $k$-mismatches problem

TL;DR

This paper introduces new algorithms for the longest common substring with k-mismatches problem, achieving subquadratic time for certain k values and providing practical solutions for sequence comparison.

Contribution

The paper presents two output-dependent algorithms and a tabulation-based method, extending subquadratic solutions to a broader range of k values in the k-LCF problem.

Findings

At least one algorithm runs in subquadratic time for k = O(log^{1-ε} n).

Algorithms operate in O(n) space and can be chosen after linear preprocessing.

Tabulation-based algorithm achieves near-quadratic time depending on parameters.

Abstract

The recently introduced longest common substring with $k$-mismatches ($k$-LCF) problem is to find, given two sequences $S_1$ and $S_2$ of length $n$ each, 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 at most $k$. So far, the only subquadratic time result for this problem was known for $k = 1$~\cite{FGKU2014}. We first present two output-dependent algorithms solving the $k$-LCF problem and show that for $k = O(\log^{1-\varepsilon} n)$, where $\varepsilon > 0$, at least one of them works in subquadratic time, using $O(n)$ words of space. The choice of one of these two algorithms to be applied for a given input can be done after linear time and space preprocessing. Finally we present a tabulation-based algorithm working, in its range of applicability, in $O(n^2\log\min(k+\ell_0, \sigma)/\log n)$ time, where $\ell_0$ is the length of the standard longest common substring.