# A sub-quadratic sequence alignment algorithm for unrestricted cost matrices

## Abstract

The classical algorithm for computing the similarity between two sequences [36, 39] uses a dynamic programming matrix, and compares two strings of size n in 0(n2) time. We address the challenge of computing the similarity of two strings in sub-quadratic time, for metrics which use a scoring matrix of unrestricted weights. Our algorithm applies to both local and global alignment computations. The speed-up is achieved by dividing the dynamic programming matrix into variable sized blocks, as induced by Lempel-Ziv parsing of both strings, and utilizing the inherent periodic nature of both strings. This leads to an O(n2/logn) algorithm for an input of constant alphabet size. For most texts, the time complexity is actually 0(hn21 logn) where h ≤ 1 is the entropy of the text.