Faster Space-Efficient STR-IC-LCS Computation

TL;DR

This paper introduces three space-efficient algorithms for the constrained longest common subsequence problem, specifically the STR-IC-LCS, optimizing for time and space based on string lengths and constraints.

Contribution

It presents novel algorithms that improve time and space efficiency for the STR-IC-LCS problem, including solutions for special cases and alternative approaches.

Findings

Achieves $O(n^2)$ time with space depending on LCS length

Provides a faster algorithm with $O(nr/\log r + n(n- ext{LCS})$ time

Offers an alternative algorithm with optimized time and space based on STR-IC-LCS length

Abstract

One of the most fundamental method for comparing two given strings $A$ and $B$ is the longest common subsequence (LCS), where the task is to find (the length) of an LCS of $A$ and $B$. In this paper, we deal with the STR-IC-LCS problem which is one of the constrained LCS problems proposed by Chen and Chao [J. Comb. Optim, 2011]. A string $Z$ is said to be an STR-IC-LCS of three given strings $A$, $B$, and $P$, if $Z$ is a longest string satisfying that (1) $Z$ includes $P$ as a substring and (2) $Z$ is a common subsequence of $A$ and $B$. We present three efficient algorithms for this problem: First, we begin with a space-efficient solution which computes the length of an STR-IC-LCS in $O(n^2)$ time and $O((\ell+1)(n-\ell+1))$ space, where $\ell$ is the length of an LCS of $A$ and $B$ of length $n$. When $\ell = O(1)$ or $n-\ell = O(1)$, then this algorithm uses only linear $O(n)$ space. Second, we present a faster algorithm that works in $O(nr/\log{r}+n(n-\ell+1))$ time, where $r$ is the length of $P$, while retaining the $O((\ell+1)(n-\ell+1))$ space efficiency. Third, we give an alternative algorithm that runs in $O(nr/\log{r}+n(n-\ell'+1))$ time with $O((\ell'+1)(n-\ell'+1))$ space, where $\ell'$ denotes the STR-IC-LCS length for input strings $A$, $B$, and $P$.