Longest common substrings with k mismatches

The longest common substring with $k$-mismatches problem is to find, given two strings $S1$ and $S2$, a longest substring $A1$ of $S1$ and $A2$ of $S2$ such that the Hamming distance between $A1$ and $A2$ 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 $S1$ and $S2$, respectively. This algorithm can also be used to compute the matching statistics with $k$-mismatches of $S1$ and $S2$ 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.