Empty Squares in Arbitrary Orientation Among Points

Abstract

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.