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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 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 space in total.
3) Next we give a randomised -approximation algorithm for the
streaming k-mismatch problem which uses
space and runs in worst-case time per
arriving symbol.
4) Finally we combine our new results to derive a randomised
space algorithm for the streaming k-mismatch problem
which runs in 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)
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