An efficient dynamic programming algorithm for the generalized LCS problem with multiple substring inclusive constraints

In this paper, we consider a generalized longest common subsequence problem with multiple substring inclusive constraints. For the two input sequences $X$ and $Y$ of lengths $n$ and $m$, and a set of $d$ constraints $P=\{P_1,\cdots ,P_d\}$ of total length $r$, the problem is to find a common subsequence $Z$ of $X$ and $Y$ including each of constraint string in $P$ as a substring and the length of $Z$ is maximized. A new dynamic programming solution to this problem is presented in this paper. The correctness of the new algorithm is proved. The time complexity of our algorithm is $O(d{2}^{d}nmr)$. In the case of the number of constraint strings is fixed, our new algorithm for the generalized longest common subsequence problem with multiple substring inclusive constraints requires $O(nmr)$ time and space.