A subquadratic sequence alignment algorithm for unrestricted scoring matrices

Given two strings of size n over a constant alphabet, the classical algorithm for computing the similarity between two sequences [D. Sankoff and J. B. Kruskal, eds., Time Warps, String Edits, and Macromolecules; Addison - Wesley, Reading, MA, 1983; T. F. Smith and M. S. Waterman, J. Molec. Biol., 147 (1981), pp. 195 - 197] uses a dynamic programming matrix and compares the two strings in O(n(2)) time. We address the challenge of computing the similarity of two strings in subquadratic time for metrics which use a scoring matrix of unrestricted weights. Our algorithm applies to both local and global similarity 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(n(2)/log n), algorithm for an input of constant alphabet size. For most texts, the time complexity is actually O(hn(2)/log n), where h less than or equal to 1 is the entropy of the text. We also present an algorithm for comparing two run-length encoded strings of length m and n, compressed into m' and n' runs, respectively, in O(m' n + n' m) complexity. This result extends to all distance or similarity scoring schemes that use an additive gap penalty.