Efficient Algorithms for the Longest Common Subsequence Problem with Sequential Substring Constraints

In this paper, we generalize the inclusion constrained longest common subsequence (CLCS) problem to the hybrid CLCS problem which is the combination of the sequence inclusion CLCS and the string inclusion CLCS, called the sequential sub string constrained longest common subsequence (SSCLCS) problem. In the SSCLCS problem, we are given two strings A and B of lengths m and n, respectively, formed by alphabet Σ and a constraint sequence C formed by ordered strings (C¹, C², C³, · · · C^l) with total length τ. We are to find the longest common subsequence D of A and B containing C¹,C²,C³, · · · , C^l as substrings and the order of C´s are retained. This problem have two variants that the strings in C may or may not overlap. We proposed algorithms with O(mnl + (m+ n)(|Σ| + r)) and O(mnr + (m+ n)|Σ|) time for the two variants of the problem. For the special case with one or two constraints, our algorithms runs in O(mn+(m+n)(|Σ|+r)) and O(mnr +(m+n)|Σ|) time, which are an order faster than the algorithm proposed by Chen and Chao [1].