Longest Common Subsequence in at Least $k$ Length Order-Isomorphic Substrings

TL;DR

This paper introduces an efficient $O(mn)$ time algorithm for the longest common subsequence problem with at least $k$ length order-isomorphic substrings, improving upon existing methods and providing practical solutions.

Contribution

The paper presents the first $O(mn)$ time algorithms for LCS with at least $k$ length order-isomorphic substrings, enhancing computational efficiency.

Findings

Achieved $O(mn)$ worst-case time complexity for the problem.

Provided an easy-to-implement algorithm for the order-isomorphic substrings variant.

Improved upon previous algorithms with higher worst-case running times.

Abstract

We consider the longest common subsequence (LCS) problem with the restriction that the common subsequence is required to consist of at least $k$ length substrings. First, we show an $O(mn)$ time algorithm for the problem which gives a better worst-case running time than existing algorithms, where $m$ and $n$ are lengths of the input strings. Furthermore, we mainly consider the LCS in at least $k$ length order-isomorphic substrings problem. We show that the problem can also be solved in $O(mn)$ worst-case time by an easy-to-implement algorithm.