Dynamic Pattern Matching with Wildcards

2026-01-14

AI Summary

This work addresses the fully dynamic pattern matching problem with wildcards, where both the text and the wildcard-containing pattern can be updated dynamically. For patterns with a small number $k$ of wildcards, the paper presents the first truly sublinear dynamic algorithm, achieving $O(kn^{k/(k+1)} + k^2 \log n)$ update and query time, and establishes a conditional lower bound under the Strong Exponential Time Hypothesis (SETH). When the number $w$ of non-wildcard characters is small, the authors design an efficient solution with $O(w + \log n)$ update time; notably, for the case of only two non-wildcards, they obtain a deterministic sublinear update algorithm running in $O(n^{0.8} \log n)$ time. The approach combines block decomposition, fast Fourier transform (FFT), and carefully engineered data structures.

Technology Category

• Search and Optimization: Distributed Search
• Data Mining & Knowledge Management: Data Compression
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

We study the fully dynamic pattern matching problem where the pattern may contain up to kwildcard symbols, each matching any symbol of the alphabet. Both the text and the pattern are subject to updates (insert, delete, change). We design an algorithm with O(nlog^2 n) preprocessing and update/query time O(knk/k+1 + k2 log n). The bound is truly sublinear for a constant k, and sublinear when k= o(log n). We further complement our results with a conditional lower bound: assuming subquadratic preprocessing time, achieving truly sublinear update time for the case k = {\Omega}(log n) would contradict the Strong Exponential Time Hypothesis (SETH). Finally, we develop sublinear algorithms for two special cases: - If the pattern contains w non-wildcard symbols, we give an algorithm with preprocessing time O(nw) and update time O(w + log n), which is truly sublinear whenever wis truly sublinear. - Using FFT technique combined with block decomposition, we design a deterministic truly sublinear algorithm with preprocessing time O(n^1.8) and update time O(n^0.8 log n) for the case that there are at most two non-wildcards.

Problem

• dynamic pattern matching
• wildcards
• text updates
• pattern updates
• sublinear algorithms

Innovation

• dynamic pattern matching
• wildcards
• sublinear algorithms
• FFT
• conditional lower bounds