Matching with don't-cares and a small number of mismatches

In matching with don't-cares and k mismatches we are given a pattern of length m and a text of length n, both of which may contain don't-cares (a symbol that matches all symbols), and the goal is to find all locations in the text that match the pattern with at most k mismatches, where k is a parameter. We present new algorithms that solve this problem using a combination of convolutions and a dynamic programming procedure. We give randomized and deterministic solutions that run in time O(nk2logm) and O(nk3logm), respectively, and are faster than the most efficient extant methods for small values of k. Our deterministic algorithm is the first to obtain an O(polylog(k)⋅nlogm) running time.