A Dynamic Programming Solution to a Generalized LCS Problem

TL;DR

This paper introduces a new dynamic programming algorithm for the generalized longest common subsequence problem with a string-excluding constraint, correcting previous inaccuracies and achieving an efficient $O(nmr)$ solution.

Contribution

It presents a correct and efficient dynamic programming algorithm for the string-excluding constrained LCS problem, improving upon prior methods.

Findings

The new algorithm correctly solves the problem.

The time complexity is $O(nmr)$.

Previous algorithms were incorrect.

Abstract

In this paper, we consider a generalized longest common subsequence problem, the string-excluding constrained LCS problem. For the two input sequences $X$ and $Y$ of lengths $n$ and $m$, and a constraint string $P$ of length $r$, the problem is to find a common subsequence $Z$ of $X$ and $Y$ excluding $P$ as a substring and the length of $Z$ is maximized. The problem and its solution were first proposed by Chen and Chao\cite{1}, but we found that their algorithm can not solve the problem correctly. A new dynamic programming solution for the STR-EC-LCS problem is then presented in this paper. The correctness of the new algorithm is proved. The time complexity of the new algorithm is $O(nmr)$.