Faster STR-EC-LCS Computation

Abstract

The longest common subsequence (LCS) problem is a central problem in stringology that finds the longest common subsequence of given two strings A and B. More recently, a set of four constrained LCS problems (called generalized constrained LCS problem) were proposed by Chen and Chao [J. Comb. Optim, 2011]. In this paper, we consider the substring-excluding constrained LCS (STR-EC-LCS) problem. A string Z is said to be an STR-EC-LCS of two given strings A and B excluding P if, Z is one of the longest common subsequences of A and B that does not contain P as a substring. Wang et al. proposed a dynamic programming solution which computes an STR-EC-LCS in O(mnr) time and space where \(m = |A|, n = |B|, r = |P|\) [Inf. Process. Lett., 2013]. In this paper, we show a new solution for the STR-EC-LCS problem. Our algorithm computes an STR-EC-LCS in \(O(n|\varSigma | + (L+1)(m-L+1)r)\) time where \(|\varSigma | \le \min \{m, n\}\) denotes the set of distinct characters occurring in both A and B, and L is the length of the STR-EC-LCS. This algorithm is faster than the O(mnr)-time algorithm for short/long STR-EC-LCS (namely, \(L \in O(1)\) or \(m-L \in O(1)\)), and is at least as efficient as the O(mnr)-time algorithm for all cases.