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 ≤ 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≤ n, and uses O(m) space, improving over the existing O(nm) time and O(m) space bound of Babenko and Starikovskaya.