Approximate swap and mismatch edit distance

There is no known algorithm that solves the general case of the Approximate Edit Distance problem, where the edit operations are: insertion, deletion, mismatch, and swap, in time o(nm), where n is the length of the text and m is the length of the pattern. In the effort to study this problem, the edit operations were analyzed independently. Karloff [10] showed an algorithm that approximates the edit distance problem with only the mismatch operation in time O(1ϵ2nlog3m)O(1ϵ2nlog3⁡m). Amir et. al. [3] showed that if the only edit operations allowed are swap and mismatch, then the exact edit distance problem can be solved in time O(nm−−√logm)O(nmlog⁡m). In this paper, we discuss the problem of approximate edit distance with swap and mismatch. We show a randomized O(1ϵ3nlognlog3m)O(1ϵ3nlog⁡nlog3⁡m) time algorithm for the problem. The algorithm guarantees an approximation factor of (1 + ε) with probability of at least 1−1n1−1n.