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Order-Preserving Squares in Strings

Authors Paweł Gawrychowski , Samah Ghazawi, Gad M. Landau

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Author Details

Paweł Gawrychowski
  • Institute of Computer Science, University of Wrocław, Poland
Samah Ghazawi
  • Department of Computer Science, University of Haifa, Israel
Gad M. Landau
  • Department of Computer Science, University of Haifa, Israel
  • Department of Computer Science and Engineering, NYU Tandon School of Engineering, New York University, Brooklyn, NY, USA

Funding

  • Ghazawi, Samah: partially supported by the Israel Science Foundation grant 1475/18, and Grant No. 2018141 from the United States-Israel Binational Science Foundation (BSF).
  • Landau, Gad M.: partially supported by the Israel Science Foundation grant 1475/18, and Grant No. 2018141 from the United States-Israel Binational Science Foundation (BSF).

Cite As Get BibTex

Paweł Gawrychowski, Samah Ghazawi, and Gad M. Landau. Order-Preserving Squares in Strings. In 34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 259, pp. 13:1-13:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.CPM.2023.13

Abstract

An order-preserving square in a string is a fragment of the form uv where u ≠ v and u is order-isomorphic to v. We show that a string w of length n over an alphabet of size σ contains 𝒪(σn) order-preserving squares that are distinct as words. This improves the upper bound of 𝒪(σ²n) by Kociumaka, Radoszewski, Rytter, and Waleń [TCS 2016]. Further, for every σ and n we exhibit a string with Ω(σn) order-preserving squares that are distinct as words, thus establishing that our upper bound is asymptotically tight. Finally, we design an 𝒪(σn) time algorithm that outputs all order-preserving squares that occur in a given string and are distinct as words. By our lower bound, this is optimal in the worst case.

Subject Classification

ACM Subject Classification
  • Theory of computation → Pattern matching
Keywords
  • repetitions
  • distinct squares
  • order-isomorphism

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