Efficient Computation for the Longest Common Subsequence with Substring Inclusion and Subsequence Exclusion Constraints

A generalized longest common subsequence problem is studied in this paper. In the generalized longest common subsequence problem, a constraining sequence of length s must be included as a substring and the other constraining sequence of length t must be excluded 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.