Computing Maximal Repeating Subsequences in a String

TL;DR

This paper introduces efficient algorithms for finding maximal repeating subsequences in a string, significantly improving previous bounds for specific cases like square and k-repeating subsequences.

Contribution

It presents new algorithms with improved time complexities for computing maximal square and k-repeating subsequences in a string, extending the study to single strings.

Findings

Maximal square subsequence can be computed in O(n log n) time.

Bound for k-repeating subsequences is improved to O(f(k)n log n).

Results also apply to constrained cases with subsequence requirements.

Abstract

In this paper we initiate the study of computing a maximal (not necessarily maximum) repeating pattern in a single input string, where the corresponding problems have been studied (e.g., a maximal common subsequence) only in two or more input strings by Hirota and Sakai starting 2019. Given an input string $S$ of length $n$, we can compute a maximal square subsequence of $S$ in $O(n\log n)$ time, greatly improving the $O(n^2)$ bound for computing the longest square subsequence of $S$. For a maximal $k$-repeating subsequence, our bound is $O(f(k)n\log n)$, where \(f(k)\) is a computable function such that $f(k) < k\cdot 4^k$. This greatly improves the $O(n^{2k-1})$ bound for computing a longest $k$-repeating subsequence of $S$, for $k\geq 3$. Both results hold for the constrained case, i.e., when the solution must contain a subsequence $X$ of $S$, though with higher running times.