Fast algorithm for computing longest common subsequences of small alphabet size

Abstract

Given two strings of lengths m and n≥m on an alphabet of size s, the longest common subsequence (LCS) problem is to determine the longest subsequence that can be obtained by deleting zero or more symbols from either string. The first O(mn) algorithm was given by Hirschberg in 1975. The algorithm was later revised to O(ln), where l is the length of an LCS between the two strings. Another strategy given by Hunt and Szymanski takes O(rlogn) time, where r≤mn is the total number of matches between the two strings. Apostolico and Guerra combined the two approaches and derived an O(mlogn + dlog(mn/d)) algorithm, where d≤r is the number of dominant matches (minimal candidates) between the two strings. Efficient algorithms for two similar strings were devised by Nakatsu et al. [7] and Myers [6] with time complexities of O(n(m-l)) and O(n(n-l)), respectively. This paper presents a new algorithm for this problem, which requires preprocessing that is nearly standard for the LCS problem and has time and space complexity of O(ns + min {ds, lm}) and O(ns + d), respectively. This algorithm is particularly efficient when s (the alphabet size) is small. Different data structures are used to obtain variations of the basic algorithm that require different time and space complexities.