Computational aspects of disks enclosing many points

TL;DR

The paper studies the problem of finding a pair of points $p,q$ in a point set $S$ such that any disk containing $p$ and $q$ must contain a constant fraction of $S$, formalized via $C_S(p,q)$. It develops multiple algorithms in both Euclidean and geodesic settings, including a randomized $O(n\log n)$ approach with a tunable constant $c$, a quadratic-time method achieving the best known bound $C_S(p,q)\ge n/4.7$, and linear-time solutions for convex-position and diametral-disk variants, along with bichromatic and simple-polygon extensions. The methods rely on minimum-weight segments on bisectors, higher-order Voronoi diagrams, and geodesic Voronoi structures, adapting to polygonal domains with careful handling of centers, distances, and path structures. Collectively, the results bridge existential bounds and algorithmic constructions, offering efficient strategies across several problem variants and settings, with several open questions remaining for sub-quadratic general-position algorithms and optimization of pair-maximization.

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 $c = 1/2-\sqrt{(1+2α)/12}$, for any $0<α<1$. 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.

Figures (4)

• Figure 1: $x_i$ is to the left of $L(p,q)$, thus $\omega(s) = 5$ and $\omega(s') = 4$. The dashed disks are disks through $p$ and $q$ and centred at a point in $s$ and $s'$.
• Figure 2: Configuration of $n$ points where any disk through $p$ and $q$ contains at most $(n-2)/2$ other points inside it.
• Figure 3: Four points with the order-$1$ Voronoi diagram in red, the order-$2$ Voronoi diagram in green, and the order-$3$ Voronoi diagram in blue. Edges of the order-$k$ Voronoi diagram are the segments of weight $k-1$.
• Figure 4: A simple polygon and two points $u$ and $v$ whose bisector consists of a single segment of weight $0$. No geodesic disk with $u$ and $v$ on the boundary contains another point of the set.

Theorems & Definitions (42)

• Lemma 1
• proof
• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• Lemma 2