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(n2 ) 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(n² / log n), algorithm for an input of constant alphabet size. For most texts, the time complexity is actually O(h n² / log n), where h ≤ 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.