Longest common substrings with k mismatches

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 ¿k. We introduce a practical O ( n m ) 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 ( n m ) 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 ( n m ) time and O ( m ) space bound of Babenko and Starikovskaya 1]. Two new algorithms for the longest common substring with k mismatches problem.A practical solution for arbitrary k which uses constant space.A theoretical solution for one mismatch which runs in quasilinear time.