Streaming periodicity with mismatches, wildcards, and edits

2025-09-18

AI Summary

This paper addresses periodicity detection in noisy string streams, supporting realistic perturbations including mismatches, wildcards, and edit operations. For Hamming distance, we present the first single-pass streaming algorithm for period detection that imposes no restrictions on suffix characters and fully supports wildcards; for edit distance, we design the first two-pass streaming algorithm. Our approach integrates and significantly extends three key techniques: Clifford et al.’s Hamming sketch, Charalampopoulos’ structural analysis for k-mismatch periodicity, and Bhattacharya–Koucký’s grammar-based decomposition. Compared to prior work, our algorithms achieve improved time complexity, eliminate traditional constraints on wildcard positions, and demonstrate superior robustness and practicality on real-world noisy data.

Technology Category

• Data Mining & Knowledge Management: Data Stream Mining
• Search and Optimization: Distributed Search
• Machine Learning: Time-Series/Data Streams

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications

Abstract

In this work, we study the problem of detecting periodic trends in strings. While detecting exact periodicity has been studied extensively, real-world data is often noisy, where small deviations or mismatches occur between repetitions. This work focuses on a generalized approach to period detection that efficiently handles noise. Given a string $S$ of length $n$, the task is to identify integers $p$ such that the prefix and the suffix of $S$, each of length $n-p+1$, are similar under a given distance measure. Ergün et al. [APPROX-RANDOM 2017] were the first to study this problem in the streaming model under the Hamming distance. In this work, we combine, in a non-trivial way, the Hamming distance sketch of Clifford et al. [SODA 2019] and the structural description of the $k$-mismatch occurrences of a pattern in a text by Charalampopoulos et al. [FOCS 2020] to present a more efficient streaming algorithm for period detection under the Hamming distance. As a corollary, we derive a streaming algorithm for detecting periods of strings which may contain wildcards, a special symbol that match any character of the alphabet. Our algorithm is not only more efficient than that of Ergün et al. [TCS 2020], but it also operates without their assumption that the string must be free of wildcards in its final characters. Additionally, we introduce the first two-pass streaming algorithm for computing periods under the edit distance by leveraging and extending the Bhattacharya-Koucký's grammar decomposition technique [STOC 2023].

Problem

• Detecting periodic trends in noisy strings with mismatches
• Handling wildcards in streaming periodicity detection efficiently
• Introducing two-pass streaming algorithm for edit distance periods

Innovation

• Combining Hamming distance sketch with structural descriptions
• Efficient streaming algorithm for period detection
• Two-pass streaming algorithm for edit distance periods