An algorithm with linear expected running time for string editing with substitutions and substring reversals

The edit distance between given two strings X and Y is the minimum number of edit operations that transform X into Y without performing multiple operations in the same position. Ordinarily, string editing is based on character insert, delete, and substitute operations. Motivated from the facts that substring reversals are observed in genomic sequences, and it is not always possible to transform a given sequence X into a given se-quence Y by reversals alone (e.g. X is all 0’s, and Y is all 1s), Muthukrishnan and Sahinalp [7, 8] considered a “simple ” well-defined edit distance model in which the edit operations are: replace a character, and reverse and replace a substring. A substring of X can only be reversed if the reversal results in a match in the same position in Y. The cost of each character replacement and substring reversal is 1. The distance in this case is defined only when |X | = |Y | = n. There is an algorithm for computing the distance in this model with worst-case time complexity O(n log2 n) [8]. We present a dynamic programming algorithm with worst-case time complexity O(n2) but its expected running-time is O(n). In our dynamic program-ming solution the weights of edit operations can vary for different substitutions, and the costs of reversals can be a function of the reversal-length