Computational aspects of disks enclosing many points

2026-01-27

Latin-American Algorithms, Graphs and Optimization Symposium

AI Summary

This work addresses the problem of efficiently finding a pair of points in a planar point set such that every disk containing this pair must cover at least a constant fraction of the total points. For general point sets, the paper presents the first randomized algorithm with expected $O(n \log n)$ time complexity and improves the coverage constant of existing deterministic algorithms. The approach is further extended to specialized settings, including points in convex position, bichromatic point sets, and geodesic disks within simple polygons. In the convex position setting under diameter disks, the algorithm achieves linear time and guarantees coverage of at least $n/3$ points. The technical contributions combine tools from computational geometry, randomized strategies, deterministic sweep techniques, and properties of convex hulls.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Problem

• disk covering
• point set
• computational geometry
• geodesic disk
• combinatorial optimization

Innovation

• computational geometry
• disk covering
• randomized algorithm
• geodesic disk
• point set