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

The recently introduced longest common substring with kmismatches (k-LCF) problem is to find, given two sequences S1 and S2 of length n each, a longest substring A1 of S1 and A2 of S2 such that the Hamming distance between A1 and A2 is at most k. So far, the only subquadratic time result for this problem was known for k = 1 [6]. We first present two output-dependent algorithms solving the k-LCF problem and show that for k = O(log n), where ε > 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 logmin(k+l0, σ)/ log n) time, where l0 is the length of the standard longest common substring.