An Efficient Dynamic Programming Algorithm for STR-IC-SEQ-EC-LCS Problem

TL;DR

This paper introduces a new dynamic programming algorithm to solve a generalized longest common subsequence problem involving inclusion and exclusion constraints, with improved time complexity.

Contribution

It presents an efficient $O(nmst)$ dynamic programming algorithm for a complex LCS variant with inclusion and exclusion constraints, and proves its correctness.

Findings

Algorithm runs in $O(nmst)$ time.

Time complexity can be further reduced to cubic time.

Correctness of the algorithm is formally proved.

Abstract

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 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.