Computational aspects of disks enclosing many points

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.

💡 Research Summary

The paper studies the computational problem of finding, in a set S of n points in the plane, a pair of points (p,q) such that every disk that contains both p and q must also contain a linear fraction cn of the points, where c>0 is a constant independent of n. While the existence of such pairs has been known for decades (starting with Neumann‑Lara and Urrutia, later improved by Edelsbrunner et al. to roughly n/4·7 points), no efficient algorithm for actually locating a suitable pair had been presented. This work fills that gap by giving several algorithms that work under different assumptions and that achieve various trade‑offs between running time and the guaranteed constant c.

Key definitions. For a pair (p,q) the authors define C_S(p,q) as the minimum number of points that any disk containing p and q must also contain, and ˜C_S(p,q) as the symmetric version that also accounts for points outside the disk (i.e., min{ω, n‑ω‑2} where ω is the number of interior points of a particular disk). Computing these values efficiently is the core technical challenge.

Computing C_S and ˜C_S in O(n log n). The set of all disks through p and q is parametrized by the centers lying on the perpendicular bisector b(p,q). Each third point x∈S{p,q} defines a unique center c_x on b(p,q); sorting these n‑2 centers yields n‑1 intervals on the bisector, each interval corresponding to a distinct set of interior points. The weight ω of an interval is simply the number of points on one side of the directed line L(p,q). Adjacent intervals differ by exactly one point, so after an O(n) initialization the weights of all intervals can be updated in constant time while scanning the sorted list. Consequently both C_S(p,q) and ˜C_S(p,q) are obtained in O(n log n) time (Algorithm 1).

Randomized O(n log n) algorithm. Using a combinatorial bound by Ramos and Viana (S_k ≤ 3(k+1)n – 3(k+1)(k+2)), the authors prove that for any 0<α<1 at least 6α·(n choose 2) pairs satisfy ˜C_S(p,q) ≥ (½ – √(1+2α)/12)·n. Therefore, repeatedly sampling a random pair and evaluating it with the O(n log n) subroutine yields a suitable pair in expected O(n log n) time. By repeating O(log n) times the success probability can be boosted to 1–1/n at the cost of O(n log² n) time.

Quadratic‑time exact algorithm. By fixing a point p and sorting the bisectors to all other points once, the algorithm can sweep all pairs (p,q) simultaneously, updating the interval weights for each q in O(1) time. This yields an O(n²) algorithm that guarantees a constant c = ½ – 1/√12 ≈ 0.224, which is very close to the best known existential lower bound n/4·7 ≈ 0.25 n. Moreover, by keeping track of the maximum C_S(p,q) during the sweep, the algorithm can also output the pair that maximizes the guaranteed number of interior points in O(n² log n) time.

Convex position and diametral disks. When the point set is in convex position, the authors exploit the cyclic order of the convex hull. By examining only adjacent hull vertices they prove that any such adjacent pair forces every containing disk to include at least n/3 points, and this pair can be found in linear time. The same technique applies to the “diametral disk” variant (the disk having pq as a diameter), yielding a linear‑time algorithm that guarantees n/3 interior points.

Bichromatic version. For a red‑blue balanced set (n/2 red, n/2 blue) the same random and quadratic algorithms can be restricted to red‑blue pairs. The combinatorial analysis still holds, so a constant‑fraction guarantee is retained without changing the asymptotic running times.

Geodesic disks inside a simple polygon. The authors extend all previous ideas to the setting where points lie inside a simple polygon P and disks are defined with respect to geodesic distance. The bisector of two points becomes a piecewise curve composed of line segments and hyperbolic arcs, but the same principle—sorting the intersection points of this curve with the set of circles defined by third points—remains valid. Additional care is required to handle polygon boundary intersections and to verify the existence of a geodesic disk through three points, yet the overall algorithms retain O(n log n) expected time (randomized) or O(n²) worst‑case time, and the convex‑position bound of n/3 still holds.

Overall contribution. The paper transforms a purely existential geometric result into concrete, implementable algorithms covering a spectrum of scenarios: general position, convex position, diametral disks, bichromatic inputs, and geodesic disks in polygons. It provides both fast randomized methods with adjustable constants and deterministic quadratic‑time methods that achieve near‑optimal constants. These techniques are likely to be useful in applications such as clustering, outlier detection, and geometric data summarization where one needs a small “representative” pair guaranteeing coverage of a large fraction of the data.