License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CPM.2023.13
URN: urn:nbn:de:0030-drops-179676
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Gawrychowski, Paweł ; Ghazawi, Samah ; Landau, Gad M.

Order-Preserving Squares in Strings

LIPIcs-CPM-2023-13.pdf (2 MB)


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 ?(σn) order-preserving squares that are distinct as words. This improves the upper bound of ?(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 ?(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.

BibTeX - Entry

  author =	{Gawrychowski, Pawe{\l} and Ghazawi, Samah and Landau, Gad M.},
  title =	{{Order-Preserving Squares in Strings}},
  booktitle =	{34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023)},
  pages =	{13:1--13:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-276-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{259},
  editor =	{Bulteau, Laurent and Lipt\'{a}k, Zsuzsanna},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-179676},
  doi =		{10.4230/LIPIcs.CPM.2023.13},
  annote =	{Keywords: repetitions, distinct squares, order-isomorphism}

Keywords: repetitions, distinct squares, order-isomorphism
Collection: 34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023)
Issue Date: 2023
Date of publication: 21.06.2023

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