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Empty Squares in Arbitrary Orientation Among Points

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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 \(\Omega (n)\) and \(O(n^2)\), and that these bounds are tight, provided P is in general position. A contact pair of a square is a pair of a point \(p\in P\) and a side \(\ell \) of the square with \(p\in \ell \). The upper bound \(O(n^2)\) also applies to the number of empty squares with four contact points. Meanwhile, the lower bound becomes 0 as we can construct a point set among which there is no square of four contact points. These combinatorial results are based on new observations on the \(L_\infty \) Voronoi diagram with the axes rotated and its close connection to empty squares in arbitrary orientation. We then present an algorithm that maintains a combinatorial structure of the \(L_\infty \) 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 \(O(s\log n)\) 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 \(O(n^2 \log n)\) time.

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Authors and Affiliations

  1. Division of Computer Science and Engineering, Kyonggi University, Suwon, Korea

    Sang Won Bae

  2. Department of Service and Design Engineering, Sungshin Women’s University, Seoul, Korea

    Sang Duk Yoon

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  1. Sang Won Bae
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A preliminary version of this work was presented at the 36th International Symposium on Computational Geometry (SoCG 2020) [10]. S.W.Bae was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2018R1D1A1B07042755). S.D.Yoon was supported by the Sungshin Women’s University Research Grant of H20190003.

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Bae, S.W., Yoon, S.D. Empty Squares in Arbitrary Orientation Among Points. Algorithmica 85, 29–74 (2023). https://doi.org/10.1007/s00453-022-01002-1

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