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.2021.12
URN: urn:nbn:de:0030-drops-139635
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Charalampopoulos, Panagiotis ; Radoszewski, Jakub ; Rytter, Wojciech ; Waleń, Tomasz ; Zuba, Wiktor

Computing Covers of 2D-Strings

LIPIcs-CPM-2021-12.pdf (0.9 MB)


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 ?(N) time, where N = mn, all aperiodic 2D-covers of T in ?(N log N) time, and all 2D-covers of T in N^{4/3}⋅ log^{?(1)}N time. Further, we show how to compute all 1D-covers in the optimal ?(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^{?(1)}n time per insertion or deletion of a rectangle, a result which could be of independent interest.

BibTeX - Entry

  author =	{Charalampopoulos, Panagiotis and Radoszewski, Jakub and Rytter, Wojciech and Wale\'{n}, Tomasz and Zuba, Wiktor},
  title =	{{Computing Covers of 2D-Strings}},
  booktitle =	{32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021)},
  pages =	{12:1--12:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-186-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{191},
  editor =	{Gawrychowski, Pawe{\l} and Starikovskaya, Tatiana},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-139635},
  doi =		{10.4230/LIPIcs.CPM.2021.12},
  annote =	{Keywords: 2D-string, cover, dynamic Klee’s measure problem}

Keywords: 2D-string, cover, dynamic Klee’s measure problem
Collection: 32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021)
Issue Date: 2021
Date of publication: 30.06.2021

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