A Simple Algorithm for the Constrained Sequence Problems

TL;DR

This paper introduces a simple, efficient algorithm for the constrained longest common subsequence problem, improving the time complexity from quadratic to linear in sequence lengths, and relates it to constrained multiple sequence alignment.

Contribution

The paper presents a new $O(n imes m imes r)$ algorithm for the constrained LCS problem, significantly improving over previous methods and establishing its equivalence to a special case of constrained multiple sequence alignment.

Findings

The new algorithm runs in $O(n imes m imes r)$ time.

The constrained LCS problem is equivalent to a specific constrained multiple sequence alignment case.

The approach simplifies solving constrained sequence problems efficiently.

Abstract

In this paper we address the constrained longest common subsequence problem. Given two sequences $X$, $Y$ and a constrained sequence $P$, a sequence $Z$ is a constrained longest common subsequence for $X$ and $Y$ with respect to $P$ if $Z$ is the longest subsequence of $X$ and $Y$ such that $P$ is a subsequence of $Z$. Recently, Tsai \cite{Tsai} proposed an $O(n^2 \cdot m^2 \cdot r)$ time algorithm to solve this problem using dynamic programming technique, where $n$, $m$ and $r$ are the lengths of $X$, $Y$ and $P$, respectively. In this paper, we present a simple algorithm to solve the constrained longest common subsequence problem in $O(n \cdot m \cdot r)$ time and show that the constrained longest common subsequence problem is equivalent to a special case of the constrained multiple sequence alignment problem which can also be solved.