On Separating Points by Lines

TL;DR

This paper investigates the minimum number of lines needed to separate all pairs of points in a set, providing probabilistic bounds, an approximation algorithm, and exploring connections to partitions.

Contribution

It establishes the asymptotic bounds for the separability of random point sets and introduces a fast approximation algorithm for computing separability.

Findings

Minimum lines needed for random points is Θ(n^{2/3}) with polylogarithmic factors.

Provides a fast approximation algorithm for separability.

Highlights the connection between separability and partitions.

Abstract

Given a set $P$ of $n$ points in the plane, its separability is the minimum number of lines needed to separate all its pairs of points from each other. We show that the minimum number of lines needed to separate $n$ points, picked randomly (and uniformly) in the unit square, is $\Theta( n^{2/3})$, where $\Theta$ hides polylogarithmic factors. In addition, we provide a fast approximation algorithm for computing the separability of a given point set in the plane. Finally, we point out the connection between separability and partitions.