Geometric Separability using Orthogonal Objects.

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.