Given a bichromatic point set $P=\textbf{R}$ $ \cup$ $ \textbf{B}$ of red and blue points, a separator is an object of a certain type that separates $\textbf{R}$ and $\textbf{B}$. We study the geometric separability problem when the separator is a) rectangular annulus of fixed orientation b) rectangular annulus of arbitrary orientation c) square annulus of fixed orientation d) orthogonal convex polygon. In this paper, we give polynomial time algorithms to construct separators of each of the above type that also optimizes a given parameter. Specifically, we give an $O(n^3 \log n)$ algorithm that computes (non-uniform width) separating rectangular annulus in arbitrary orientation, of minimum possible width. Further, when the orientation is fixed, we give an $O(n\log n)$ algorithm that constructs a uniform width separating rectangular annulus of minimum possible width and an $O(n\log^2 n)$ algorithm that constructs a minimum width separating concentric square annulus. We also give an optimal algorithm that computes a separating orthogonal convex polygon with minimum number of edges, that runs in $O(n\log n)$ time.

中文翻译：

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使用正交对象的几何可分性

给定红点和蓝点的双色点集 $P=\textbf{R}$ $ \cup$ $ \textbf{B}$，分隔符是将 $\textbf{R}$ 和$\textbf{B}$。我们研究了当分隔符是 a) 固定方向的矩形环 b) 任意方向的矩形环 c) 固定方向的方形环 d) 正交凸多边形时的几何可分性问题。在本文中，我们给出多项式时间算法来构造上述每种类型的分隔符，同时优化给定参数。具体来说，我们给出了一个 $O(n^3 \log n)$ 算法，该算法计算（非均匀宽度）在任意方向上分离矩形环的最小可能宽度。此外，当方向固定时，我们给出了一个 $O(n\log n)$ 算法，该算法构造了一个最小可能宽度的均匀宽度分隔矩形环，以及一个 $O(n\log^2 n)$ 算法，该算法构造了一个最小宽度分隔同心方形环。我们还给出了一个最佳算法，该算法计算具有最少边数的分离正交凸多边形，该算法在 $O(n\log n)$ 时间内运行。