Computing Covers of 2D-Strings

Authors Panagiotis Charalampopoulos , Jakub Radoszewski , Wojciech Rytter , Tomasz Waleń , Wiktor Zuba

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Panagiotis Charalampopoulos
  • The Interdisciplinary Center Herzliya, Israel
Jakub Radoszewski
  • University of Warsaw, Poland
Wojciech Rytter
  • University of Warsaw, Poland
Tomasz Waleń
  • University of Warsaw, Poland
Wiktor Zuba
  • University of Warsaw, Poland

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Panagiotis Charalampopoulos, Jakub Radoszewski, Wojciech Rytter, Tomasz Waleń, and Wiktor Zuba. Computing Covers of 2D-Strings. In 32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 191, pp. 12:1-12:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


We consider two notions of covers of a two-dimensional string T. A (rectangular) subarray P of T is a 2D-cover of T if each position of T is in an occurrence of P in T. A one-dimensional string S is a 1D-cover of T if its vertical and horizontal occurrences in T cover all positions of T. We show how to compute the smallest-area 2D-cover of an m × n array T in the optimal 𝒪(N) time, where N = mn, all aperiodic 2D-covers of T in 𝒪(N log N) time, and all 2D-covers of T in N^{4/3}⋅ log^{𝒪(1)}N time. Further, we show how to compute all 1D-covers in the optimal 𝒪(N) time. Along the way, we show that the Klee’s measure of a set of rectangles, each of width and height at least √n, on an n × n grid can be maintained in √n⋅ log^{𝒪(1)}n time per insertion or deletion of a rectangle, a result which could be of independent interest.

Subject Classification

ACM Subject Classification
  • Theory of computation → Pattern matching
  • 2D-string
  • cover
  • dynamic Klee’s measure problem


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