Empty Squares in Arbitrary Orientation Among Points

Sang Won Bae, Sang Duk Yoon, Danny Z. Chen, Sergio Cabello
2020 International Symposium on Computational Geometry  
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
more » ... re 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.
doi:10.4230/lipics.socg.2020.13 dblp:conf/compgeom/BaeY20 fatcat:dqffcalnovbmfffwvif2kqkk7i