The k-mismatch problem revisited

We revisit the complexity of one of the most basic problems in pattern matching. In the k-mismatch problem we must compute the Hamming distance between a pattern of length m and every m-length substring of a text of length n, as long as that Hamming distance is at most k. Where the Hamming distance is greater than k at some alignment of the pattern and text, we simply output "No". We study this problem in both the standard offline setting and also as a streaming problem. In the streaming k-mismatch problem the text arrives one symbol at a time and we must give an output before processing any future symbols. Our main results are as follows: 1) Our first result is a deterministic $O(n k^2\log{k} / m+n \text{polylog} m)$ time offline algorithm for k-mismatch on a text of length n. This is a factor of k improvement over the fastest previous result of this form from SODA 2000 by Amihood Amir et al. 2) We then give a randomised and online algorithm which runs in the same time complexity but requires only $O(k^2\text{polylog} {m})$ space in total. 3) Next we give a randomised $(1+\epsilon)$-approximation algorithm for the streaming k-mismatch problem which uses $O(k^2\text{polylog} m / \epsilon^2)$ space and runs in $O(\text{polylog} m / \epsilon^2)$ worst-case time per arriving symbol. 4) Finally we combine our new results to derive a randomised $O(k^2\text{polylog} {m})$ space algorithm for the streaming k-mismatch problem which runs in $O(\sqrt{k}\log{k} + \text{polylog} {m})$ worst-case time per arriving symbol. This improves the best previous space complexity for streaming k-mismatch from FOCS 2009 by Benny Porat and Ely Porat by a factor of k. We also improve the time complexity of this previous result by an even greater factor to match the fastest known offline algorithm (up to logarithmic factors)

05291 Abstracts Collection -- Sublinear Algorithms

From 17.07.05 to 22.07.05, the Dagstuhl Seminar 05291 ``Sublinear Algorithms\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available