New efficient algorithms for the merged LCS problem with and without block constraints using sparse dynamic programming

The longest common subsequence problem has been widely studied and used to find out the relationship between sequences. In this paper, we study the interleaving relationship between sequences. Given a target sequence T and two merging sequences A and B, we need to find out the LCS between M(A, B) and T, where M(A, B) denotes the merging sequence of A and B. We first present a O((Rr + Pm)log log r) time algorithm where |T| = n, |A| = m, |B| = r, R is the total number of ordered pairs of positions at which the two strings A and T match and P denotes the total number of ordered pairs of positions at which the two strings B and T match. We also propose an algorithm to solve a variation of the problem where block constraint arises. The running time of the blocked version is O(max{Rβ log log r, Pαlog log r}), where α denotes the number of blocks in A and β denotes the number of blocks in B.