Two algorithms for the longest common subsequence of three (or more) strings

Abstract

Various algorithms have been proposed, over the years, for the longest common subsequence problem on 2 strings (2-LCS), many of these improving, at least for some cases, on the classical dynamic programming approach. However, relatively little attention has been paid in the literature to the k-LCS problem for k > 2, a problem that has interesting applications in areas such as the multiple alignment of sequences in molecular biology. In this paper, we describe and analyse two algorithms with particular reference to the 3-LCS problem, though each algorithm can be extended to solve the k-LCS problem for general k. The first algorithm, which can be viewed as a “lazy” version of dynamic programming, has time and space complexity that is O(n(n−1)2) for 3 strings, and O(kn(n−1)k-1) for k strings, where n is the common length of the strings and l is the length of an LCS. The second algorithm, which involves evaluating entries in a “threshold” table in diagonal order, has time and space complexity that is O(l(n−1)2+sn) for 3 strings, and O(kl(n−1)k−1+ksn) for k strings, where s is the alphabet size. For simplicity, the algorithms are presented for equal-length strings, though extension to unequal-length strings is straightforward. Empirical evidence is presented to show that both algorithms show significant improvement on the basic dynamic programming approach, and on an earlier algorithm proposed by Hsu and Du, particularly, as would be expected, in the case where l is relatively large, with the balance of evidence being heavily in favour of the threshold approach.