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(1/epsilon(2)n log(3) 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(n root m log m). In this paper, we discuss the problem of approximate edit distance with swap and mismatch. We show a randomized O(1/epsilon(3)n( log n log(3) m) time algorithm for the problem. The algorithm guarantees an approximation factor of (1 + epsilon) with probability of at least 1 - 1/n.