Empty Squares in Arbitrary Orientation Among Points

Authors Sang Won Bae , Sang Duk Yoon

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Sang Won Bae
  • Division of Computer Science and Engineering, Kyonggi University, Suwon, South Korea
Sang Duk Yoon
  • Department of Service and Design Engineering, Sungshin Women’s University, Seoul, South Korea

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Sang Won Bae and Sang Duk Yoon. Empty Squares in Arbitrary Orientation Among Points. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 13:1-13:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


This paper studies empty squares in arbitrary orientation among a set P of n points in the plane. We prove that the number of empty squares with four contact pairs is between Ω(n) and O(n²), and that these bounds are tight, provided P is in a certain general position. A contact pair of a square is a pair of a point p ∈ P and a side 𝓁 of the square with p ∈ 𝓁. The upper bound O(n²) also applies to the number of empty squares with four contact points, while we construct a point set among which there is no square of four contact points. We then present an algorithm that maintains a combinatorial structure of the L_∞ Voronoi diagram of P, while the axes of the plane continuously rotate by 90 degrees, and simultaneously reports all empty squares with four contact pairs among P in an output-sensitive way within O(slog n) time and O(n) space, where s denotes the number of reported squares. Several new algorithmic results are also obtained: a largest empty square among P and a square annulus of minimum width or minimum area that encloses P over all orientations can be computed in worst-case O(n² log n) time.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • empty square
  • arbitrary orientation
  • Erdős - Szekeres problem
  • L_∞ Voronoi diagram
  • largest empty square problem
  • square annulus


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