Swap and mismatch edit distance

There is no known algorithm that solves the general case of approximate string matching problem with the extended edit distance, 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 where analysed independently. It turns out that the approximate matching problem with only the mismatch operation can be solved in time O(n√mlogm). If the only edit operation allowed is the swap, then the problem can be solved in time O (n log m log σ), where σ = min(m, |Σ|). In this paper we show that the approximate string matching problem with the swap and mismatch as the edit operations, can be computed in time O(n√m log m).