Efficient Internal Pattern Matching in Small Space

Core Concepts

The authors present a space-time trade-off for the Internal Pattern Matching (IPM) problem, where the goal is to construct a data structure over a string S that allows efficiently answering queries about occurrences of a fragment P of S inside another fragment T of S, provided that |T| < 2|P|.

Abstract

The authors consider the Internal Pattern Matching (IPM) problem in the read-only setting, where the goal is to bound the space usage on top of storing the input strings. Their main contribution is a space-time trade-off for IPM queries.
For any integer τ = O(n/ log^2 n), the authors present a data structure that can be built using O(n log(n/τ) + (n/τ) log^4 n log log n) time and O((n/τ) log n (log log n)^3) extra space, and can answer IPM queries in O(τ + log n log^3 log n) time. This data structure is nearly optimal in the sense that the product of the query time and space is optimal up to polylogarithmic factors.
The authors achieve this result by utilizing the concept of τ-partitioning sets as anchor points for identifying pattern occurrences, and employing sparse suffix trees and a three-dimensional range searching structure. For patterns that do not avoid a specific periodic structure, they leverage the periodic structure to construct the necessary anchor points.
The authors further showcase the applicability of their IPM data structure by using it to obtain space-time trade-offs for the longest common substring and circular pattern matching problems in the asymmetric streaming setting.

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Key Insights Distilled From

Internal Pattern Matching in Small Space and Applications

by Gabriel Bath... at arxiv.org 04-29-2024

https://arxiv.org/pdf/2404.17502.pdf

Deeper Inquiries

How can the techniques developed for the IPM problem be extended to handle more general types of internal string queries, such as period queries or cyclic equivalence queries

The techniques developed for the Internal Pattern Matching (IPM) problem can be extended to handle more general types of internal string queries by leveraging the concept of anchor points and partitioning sets. For example, for period queries, we can identify specific patterns within the string that repeat at regular intervals. By using anchor points to mark the starting positions of these periodic patterns, we can efficiently determine the periods of the substrings. This approach allows us to answer period queries in a similar fashion to IPM queries, by utilizing the anchor points to locate and analyze the repeating patterns.
Similarly, for cyclic equivalence queries, where we need to find rotations of a pattern within a text, we can apply the concept of anchor points to identify potential rotation points. By creating anchor points at strategic locations based on the properties of cyclic equivalence, we can efficiently determine the rotations of the pattern within the text. This approach enables us to handle cyclic equivalence queries by utilizing the anchor points to locate and verify the rotations of the pattern.
In essence, the techniques developed for the IPM problem can be extended to handle more general types of internal string queries by adapting the concept of anchor points and partitioning sets to suit the specific requirements of each query type. By strategically identifying anchor points and utilizing them effectively, we can efficiently address a variety of internal string queries beyond the scope of traditional pattern matching.

Are there other fundamental string processing problems that can benefit from the efficient small-space IPM data structure presented in this work

The efficient small-space IPM data structure presented in this work can benefit several other fundamental string processing problems by providing a space-efficient solution for handling internal queries on strings. Some of the fundamental string processing problems that can benefit from this data structure include:

Shortest Unique Substring: By utilizing the space-efficient IPM data structure, we can efficiently identify the shortest unique substrings within a given text. The ability to handle internal queries in small space allows for a more optimized solution to this problem.

Dictionary Matching: The small-space IPM data structure can be applied to efficiently match multiple patterns from a dictionary within a text. By leveraging the internal query capabilities, we can streamline the process of dictionary matching while conserving space.

String Covers: The concept of internal queries can be extended to address string cover problems, where the goal is to find substrings that cover a given text. The small-space IPM data structure can enhance the efficiency of identifying string covers within a text.

Dynamic Longest Common Substring: Dynamic variations of the longest common substring problem, where the input text is subject to changes, can benefit from the space-efficient IPM data structure. The ability to handle internal queries in small space enables dynamic updates and efficient computation of the longest common substring.

Overall, the small-space IPM data structure can be a valuable tool for various fundamental string processing problems, providing optimized solutions and efficient processing of internal queries on strings.

What are the potential applications of the space-efficient algorithms for longest common substring and circular pattern matching in the asymmetric streaming setting, and how can these techniques be further developed or applied in other domains

The space-efficient algorithms for the longest common substring and circular pattern matching in the asymmetric streaming setting offer several potential applications across different domains. These techniques can be further developed and applied in the following ways:

Bioinformatics: In bioinformatics, the algorithms for the longest common substring and circular pattern matching can be utilized for sequence analysis, alignment, and comparison. The efficient space-time trade-offs in the asymmetric streaming setting can enhance the processing of biological sequences and patterns.

Data Compression: The algorithms can be applied in data compression techniques where identifying common substrings or patterns is crucial for reducing redundancy and optimizing storage space. The space-efficient algorithms can improve the compression process by efficiently handling substring matching and circular patterns.

Network Security: In cybersecurity applications, the techniques for the longest common substring and circular pattern matching can be used for intrusion detection, malware analysis, and pattern recognition in network traffic. The space-efficient algorithms can enhance the detection and analysis of patterns in network data streams.

Text Mining: The algorithms can be applied in text mining and information retrieval tasks where identifying common substrings or circular patterns is essential for analyzing and extracting meaningful information from text data. The space-efficient techniques can improve the efficiency of text processing and pattern recognition tasks.

By exploring these potential applications and further developing the space-efficient algorithms for the longest common substring and circular pattern matching, we can enhance various domains such as bioinformatics, data compression, network security, and text mining with optimized string processing capabilities.