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Internal Pattern Matching in Small Space and Applications

Authors Gabriel Bathie , Panagiotis Charalampopoulos , Tatiana Starikovskaya

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Author Details

Gabriel Bathie
  • DIENS, École normale supérieure de Paris, PSL Research University, France
  • LaBRI, Université de Bordeaux, France
Panagiotis Charalampopoulos
  • Birkbeck, University of London, UK
Tatiana Starikovskaya
  • DIENS, École normale supérieure de Paris, PSL Research University, France

Funding

  • Bathie, Gabriel: Partially supported by the grant ANR-20-CE48-0001 from the French National Research Agency (ANR) and a Royal Society International Exchanges Award.
  • Charalampopoulos, Panagiotis: Research visits during which some of the presented ideas were conceived were funded by a Royal Society International Exchanges Award.
  • Starikovskaya, Tatiana: Partially supported by the grant ANR-20-CE48-0001 from the French National Research Agency (ANR) and a Royal Society International Exchanges Award.

Acknowledgements

We would like to dedicate this work to our dear friend and colleague Paweł Gawrychowski on the occasion of his 40th birthday. We thank Solon Pissis for helpful suggestions.

Cite AsGet BibTex

Gabriel Bathie, Panagiotis Charalampopoulos, and Tatiana Starikovskaya. Internal Pattern Matching in Small Space and Applications. In 35th Annual Symposium on Combinatorial Pattern Matching (CPM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 296, pp. 4:1-4:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.CPM.2024.4

Abstract

In this work, we consider pattern matching variants in small space, that is, in the read-only setting, where we want to bound the space usage on top of storing the strings. Our main contribution is 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 of length n that allows one to answer the following type of queries: Compute the occurrences of a fragment P of S inside another fragment T of S, provided that |T| < 2|P|. For any τ ∈ [1 . . n/log² n], we present a nearly-optimal Õ(n/τ)-size data structure that can be built in Õ(n) time using Õ(n/τ) extra space, and answers IPM queries in O(τ+log n log³ log n) time. IPM queries have been identified as a crucial primitive operation for the analysis of algorithms on strings. In particular, the complexities of several recent algorithms for approximate pattern matching are expressed with regards to the number of calls to a small set of primitive operations that include IPM queries; our data structure allows us to port these results to the small-space setting. We further showcase the applicability of our 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Pattern matching
Keywords
  • internal pattern matching
  • longest common substring
  • small-space algorithms

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