Restricted Transposition Invariant Approximate String Matching Under Edit Distance

Let A and B be strings with lengths m and n, respectively, over a finite integer alphabet. Two classic string mathing problems are computing the edit distance between A and B, and searching for approximate occurrences of A inside B. We consider the classic Levenshtein distance, but the discussion is applicable also to indel distance. A relatively new variant [8] of string matching, motivated initially by the nature of string matching in music, is to allow transposition invariance for A. This means allowing A to be “shifted” by adding some fixed integer t to the values of all its characters: the underlying string matching task must then consider all possible values of t. Mäkinen et al. [12, 13] have recently proposed O(mn log log m) and O(dn log log m) algorithms for transposition invariant edit distance computation, where d is the transposition invariant distance between A and B, and an O(mn log log m) algorithm for transposition invariant approximate string matching. In this paper we first propose a scheme to construct transposition invariant algorithms that depend on d or k. Then we proceed to give an O(n + d) algorithm for transposition invariant edit distance, and an O(kn) algorithm for transposition invariant approximate string matching.