K-Mismatch With Don'T Cares

We give the first non-trivial algorithms for the k-mismatch pattern matching problem with don't cares. Given a text t of length n and a pattern p of length m with don't care symbols and a bound k, our algorithms find all the places that the pattern matches the text with at most k mismatches. We first give an O(n(k + log n log log n) log M) time randomised solution which finds the correct answer with high probability. We then present a new deterministic O(nk(2) log(3) m) time solution that uses tools developed for group testing and finally an approach based on k-selectors that runs in O(nk polylog m) time but requires O(poly m) time preprocessing. In each case, the location of the mismatches at each alignment is also given at no extra cost.