k-Mismatch with Don’t Cares

Abstract

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 + lognloglogn)logm) time randomised solution which finds the correct answer with high probability. We then present a new deterministic O(nk ²log³ 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 $O(\mbox{poly } m)$ time preprocessing. In each case, the location of the mismatches at each alignment is also given at no extra cost.