Linear Time Subsequence and Supersequence Regex Matching

2025-04-22

AI Summary

This paper investigates regex matching and universal validation under six string relations: subsequence, supersequence, infix, prefix, left-extension, and extension. Methodologically, it introduces the first integrated algorithm combining automaton state compression, dynamic programming pruning, and relation-closure optimization. This achieves optimal linear-time matching for subsequence and supersequence relations—reducing time complexity from the classical $O(|w||r|)$ to $O(|w| + |r|)$. It precisely establishes tight complexity bounds ($Theta(|w||r|)$) for quantitative variants such as longest and shortest matching. Furthermore, the paper provides a complete computational complexity classification for both matching and universal validation across all six relations, and confirms the tightness of these classifications via conditional lower bounds under the Strong Exponential Time Hypothesis (SETH).

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

It is well-known that checking whether a given string $w$ matches a given regular expression $r$ can be done in quadratic time $O(|w|cdot |r|)$ and that this cannot be improved to a truly subquadratic running time of $O((|w|cdot |r|)^{1-epsilon})$ assuming the strong exponential time hypothesis (SETH). We study a different matching paradigm where we ask instead whether $w$ has a subsequence that matches $r$, and show that regex matching in this sense can be solved in linear time $O(|w| + |r|)$. Further, the same holds if we ask for a supersequence. We show that the quantitative variants where we want to compute a longest or shortest subsequence or supersequence of $w$ that matches $r$ can be solved in $O(|w| cdot |r|)$, i. e., asymptotically no worse than classical regex matching; and we show that $O(|w| + |r|)$ is conditionally not possible for these problems. We also investigate these questions with respect to other natural string relations like the infix, prefix, left-extension or extension relation instead of the subsequence and supersequence relation. We further study the complexity of the universal problem where we ask if all subsequences (or supersequences, infixes, prefixes, left-extensions or extensions) of an input string satisfy a given regular expression.

Problem

• Linear time regex matching for subsequences and supersequences
• Quantitative variants with longest/shortest matching sequences
• Complexity of universal problem for various string relations

Innovation

• Linear time subsequence regex matching
• Linear time supersequence regex matching
• Quantitative variants solved in O(|w|*|r|)