Pseudo-realtime Pattern Matching: Closing the Gap

Abstract

We consider the k-difference and k-mismatch problems in the pseudo-realtime model where the text arrives online and the time complexity measure is per arriving character and unamortised. The well-known k-difference/k-mismatch problems are those of finding all alignments of a pattern of length m with a text of length n where the edit/Hamming distance is at most k. Offline, the literature gives efficient solutions in O(nk) and $O(n \sqrt{k \log k})$ time, respectively. More recently, a pseudo-realtime solution was given for the former in O(k logm) time and the latter in $O(\sqrt{k \log k}\log m)$ time per arriving text character. Our work improves these complexities to O(k) time for the k-difference problem and $O(\sqrt{k}\log k + \log m)$ for the k-mismatch problem. In the process of developing the main results, we also give a simple solution with optimal time complexity for performing longest common extension queries in the same pseudo-realtime setting which may be of independent interest.