On Triangluar Separation of Bichromatic Point Sets in Polygonal Environment

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 $O(r\cdot (r+c_B+k)+h_{△})$ time, where $c_B$ is the size of convex hull of $B$, and $h_{△}$ 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"