## A Sub-quadratic Sequence Alignment Algorithm for Unrestricted Cost Matrices (2002)

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### BibTeX

@MISC{Crochemore02asub-quadratic,

author = {Maxime Crochemore and Gad M. Landau and Michal Ziv-Ukelson},

title = {A Sub-quadratic Sequence Alignment Algorithm for Unrestricted Cost Matrices},

year = {2002}

}

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### 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 O(n 2 ) 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(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 1 is the entropy of the text. Institut Gaspard-Monge, Universite de Marne-la-Vallee, Cite Descartes, Champs-surMarne, 77454 Marne-la-Vallee Cedex 2, France, email: mac@univ-mlv.fr. y Department of Computer Science, Haifa University, Haifa 31905, Israel, phone: (972-4) 824-0103, FAX: (972-4) 824-9331; Department of Computer and Information Science, Polytechnic University, Six MetroTech Center, Brooklyn, NY 11201-3840; email: landau@poly.edu; partially supported by NSF grant CCR-0104307, by NATO Science Programme grant PST.CLG.977017, by the Israel Science Foundation (grants 173/98 and 282/01), by the FIRST Foundation of the Israel Academy of Science and Humanities, and by IBM Faculty Partnership Award. z Department of Computer Science, Haifa University, Haifa 31905, Israel; On Education Leave from the IBM T.J.W. Research Center; email: michal@cs.haifa.il; partially supported by by the Israel Science Foundation (grants 173/98 and 282/01), and by the FIRST Foundation of the Israel Academy of Science ...