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تسجيل مستخدم جديدEmpty 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 a certain general position. A contact pair of a square is a pair of a point $pin P$ and a side $ell$ of the square with $pin ell$. The upper bound $O(n^2)$ 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. 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 rotates 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 $O(n^2 log n)$ time.
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