Computational aspects of disks enclosing many points

TL;DR

This paper introduces several algorithms for finding point pairs in a set such that any disk containing them also contains a significant number of other points, with applications to convex and polygonal point sets.

Contribution

It presents new randomized and deterministic algorithms with improved constants and runtime complexities for identifying such point pairs in various geometric configurations.

Findings

Randomized algorithm finds pairs with at least 1/2 - sqrt((1+2α)/12) fraction of points in expected O(n log n) time.

Deterministic quadratic-time algorithm improves the constant to approximately 1/4.7.

Linear-time algorithms for convex position and diametral disk variants.

Abstract

Let $S$ be a set of $n$ points in the plane. We present several different algorithms for finding a pair of points in $S$ such that any disk that contains that pair must contain at least $cn$ points of $S$, for some constant $c>0$. The first is a randomized algorithm that finds a pair in $O(n\log n)$ expected time for points in general position, and $c = 1/2-\sqrt{(1+2\alpha)/12}$, for any $0<\alpha<1$. The second algorithm, also for points in general position, takes quadratic time, but the constant $c$ is improved to $1/2-1/{\sqrt{12}} \approx 1/4.7$. The second algorithm can also be used as a subroutine to find the pair that maximizes the number of points inside any disk that contains the pair, in $O(n^2\log n)$ time. We also consider variants of the problem. When the set $S$ is in convex position, we present an algorithm that finds in linear time a pair of points such that any disk through them contains at least $n/3$ points of $ S $. For the variant where we are only interested in finding a pair such that the diametral disk of that pair contains many points, we also have a linear-time algorithm that finds a disk with at least $n/3$ points of $S$. Finally, we present a generalization of the first two algorithms to the case where the set $S$ of points is coloured using two colours. We also consider adapting these algorithms to solve the same problems when $S$ is a set of points inside of a simple polygon $P$, with the notion of a disk replaced by that of a geodesic disk.