%0 Journal Article
%T Longest common substrings with k mismatches
%A Tomas Flouri
%A Emanuele Giaquinta
%A Kassian Kobert
%A Esko Ukkonen
%J Computer Science
%D 2014
%I arXiv
%R 10.1016/j.ipl.2015.03.006
%X 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.
%U http://arxiv.org/abs/1409.1694v2