New Algorithms for the Longest Common Subsequence Problem New Algorithms for the Longest Common Subsequence Problem New Algorithms for the Longest Common Subsequence Problem

Given two sequences A = a 1 a 2 : : :a m and B = b 1 b 2 : : :b n , m n, over some alphabet , a common subsequence C = c 1 c 2 : : :c l of A and B is a sequence that can be obtained from both A and B by deleting zero or more (not necessarily adjacent) symbols. Finding a common subsequence of maximallength is called the Longest CommonSubsequence (LCS) Problem. Two new algorithms based on the well-known paradigm of computing minimal matches are presented. One runs in time O(ns+minfds; pmg) and the other runs in time O(ns + minfp(n ? p); pmg) where s = jj is the alphabet size, p is the length of a longest common subsequence and d is the number of minimal matches. The ns term is charged by a standard preprocessing phase. When m n both algorithms are fast in situations when a LCS is expected to be short as well as in situations when a LCS is expected to be long. Further they show a much smaller degeneration in intermediate situations, especially the second algorithm. Thus, they are more exible than previous algorithms which, in general, specialized either on short or long longest common subsequences.