On Triangluar Separation of Bichromatic Point Sets in Polygonal Environment

TL;DR

This paper introduces algorithms for finding inscribed triangular separators in polygonal environments to distinguish between red and blue point sets, including cases with no perfect separator, with improved efficiency.

Contribution

It presents an output-sensitive algorithm for inscribed triangular separators and a constant-factor approximation for maximum separators in polygonal environments.

Findings

Efficient algorithm for inscribed triangular separators with complexity depending on separator count.

Algorithm for maximum triangular separators with sub-quadratic time complexity.

Analysis of conditions where no perfect separator exists and how to approximate in such cases.

Abstract

Let $\mathcal P$ be a simple polygonal environment with $k$ vertices in the plane. Assume that a set $B$ of $b$ blue points and a set $R$ of $r$ red points are distributed in $\mathcal P$. We study the problem of computing triangles that separate the sets $B$ and $R$, and fall in $\mathcal P$. We call these triangles \emph{inscribed triangular separators}. We propose an output-sensitive algorithm to solve this problem in $O(r \cdot (r+c_B+k)+h_\triangle)$ time, where $c_B$ is the size of convex hull of $B$, and $h_\triangle$ is the number of inscribed triangular separators. We also study the case where there does not exist any inscribed triangular separators. This may happen due to the tight distribution of red points around convex hull of $B$ while no red points are inside this hull. In this case we focus to compute a triangle that separates most of the blue points from the red points. We refer to these triangles as \emph{maximum triangular separators}. Assuming $n=r+b$, we design a constant-factor approximation algorithm to compute such a separator in $O(n^{4/3} \log^3 n)$ time. "Eligible for best student paper"