Geometric Separability using Orthogonal Objects

TL;DR

This paper develops polynomial-time algorithms for constructing optimal geometric separators of various orthogonal shapes to distinguish red and blue point sets.

Contribution

It introduces algorithms for finding optimal orthogonal geometric separators of different shapes, expanding the toolkit for geometric separability problems.

Findings

Polynomial algorithms for rectangular annulus separators

Polynomial algorithms for square annulus separators

Polynomial algorithms for orthogonal convex polygon separators

Abstract

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.