Computing Maximal Repeating Subsequences in a String

2026-01-18

AI Summary

This work extends the maximal repeated subsequence problem from the multi-string setting to the single-string case and investigates efficient algorithms for computing square and k-repetitive subsequences. Addressing the constraint of containing a specified subsequence, the authors propose a novel framework that integrates combinatorial optimization with string algorithms. By introducing a computable function \( f(k) \) to govern parameter dependence, the approach achieves near-linear time complexity: \( O(n \log n) \) for square subsequences and \( O(f(k) \, n \log n) \) for k-repetitive subsequences. These results substantially improve upon prior algorithms with complexities of \( O(n^2) \) and \( O(n^{2k-1}) \), respectively, thereby overcoming the longstanding efficiency bottleneck for this problem in the single-string context.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Security and Privacy: Large-scale security measurements

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.

Problem

• maximal repeating subsequence
• string algorithms
• square subsequence
• k-repeating subsequence
• single string

Innovation

• maximal repeating subsequence
• string algorithms
• time complexity improvement
• square subsequence
• parameterized complexity