An efficient dynamic programming algorithm for a new generalized LCS problem

In this paper, we consider a generalized longest common subsequence problem, in which a constraining sequence of length s must be included as a substring and the other constraining sequence of length t must be included as a subsequence of two main sequences and the length of the result must be maximal. For the two input sequences X and Y of lengths n and m, and the given two constraining sequences of length s and t, we present an O(nmst) time dynamic programming algorithm for solving the new generalized longest common subsequence problem. The time complexity can be reduced further to cubic time in a more detailed analysis. The correctness of the new algorithm is proved.