Pseudo-realtime Pattern Matching: Closing the Gap

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 wellknown 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 √ 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( √ k log k logm) time per arriving text character. Our work improves these complexities to O(k) time for the k-difference problem and O( √ k log k+logm) 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.