The approximate swap and mismatch edit distance

Abstract

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 ( n m ) , 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 have been analyzed independently. Karloff [10] showed an algorithm that approximates the edit distance problem with only the mismatch operation in time O ( 1 ϵ 2 n log 3 m ) . Amir et al. [4] 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 m log m ) . In this paper, we discuss the problem of approximate edit distance with swap and mismatch. We show a randomized O ( 1 ϵ 3 n log n log 3 m ) time algorithm for the problem. The algorithm guarantees an approximation factor of ( 1 + ϵ ) with probability of at least 1 − 1 n .