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Rectangular Tile Covers of 2D-Strings

Authors Jakub Radoszewski , Wojciech Rytter , Juliusz Straszyński , Tomasz Waleń , Wiktor Zuba

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Author Details

Jakub Radoszewski
  • University of Warsaw, Poland
Wojciech Rytter
  • University of Warsaw, Poland
Juliusz Straszyński
  • University of Warsaw, Poland
Tomasz Waleń
  • University of Warsaw, Poland
Wiktor Zuba
  • University of Warsaw, Poland
  • CWI, Amsterdam, The Netherlands


We thank Panagiotis Charalampopoulos for helpful discussions.

Cite AsGet BibTex

Jakub Radoszewski, Wojciech Rytter, Juliusz Straszyński, Tomasz Waleń, and Wiktor Zuba. Rectangular Tile Covers of 2D-Strings. In 33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 223, pp. 23:1-23:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


We consider tile covers of 2D-strings which are a generalization of periodicity of 1D-strings. We say that a 2D-string A is a tile cover of a 2D-string S if S can be decomposed into non-overlapping 2D-strings, each of them equal to A or to A^T, where A^T is the transpose of A. We show that all tile covers of a 2D-string of size N can be computed in 𝒪(N^{1+ε}) time for any ε > 0. We also show a linear-time algorithm for computing all 1D-strings being tile covers of a 2D-string.

Subject Classification

ACM Subject Classification
  • Theory of computation → Pattern matching
  • tile cover
  • periodicity
  • efficient algorithm


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