Smallest Enclosing Disk Queries Using Farthest-Point Voronoi Diagrams

2026-05-01

AI Summary

This work addresses the problem of efficiently preprocessing a planar point set to support fast queries for the minimum enclosing circle of points lying within an axis-aligned rectangular range. We propose a novel approach based on the two-dimensional farthest-point Voronoi diagram, which circumvents the intricate three-dimensional lifting and duality transformations commonly used in prior methods, thereby substantially simplifying the algorithmic framework. Relying solely on two-dimensional geometric constructions and incorporating randomized data structures, our method achieves O(n log²n) preprocessing time and space, while guaranteeing deterministic query time of O(log⁴n) and expected query time of O(log^{5/2}n log log n), significantly improving query efficiency.

Abstract

Let $S$ be a set of $n$ points in $\mathbb{R}^2$. Our goal is to preprocess $S$ to efficiently compute the smallest enclosing disk of the points in $S$ that lie inside an axis-aligned query rectangle. Previous data structures for this problem achieve a query time of $O(\log^6 n)$ with $O(n \log^2 n)$ preprocessing time and space by lifting the points to 3D, dualizing them into polyhedra, and searching through their intersections. We present a significantly simpler approach, solely based on 2D geometric structures, specifically 2D farthest-point Voronoi diagrams. Our approach achieves a deterministic query time of $O(\log^4 n)$ and, via randomization, an expected query time of $O(\log^{5/2} n \log\log n)$ with the same preprocessing bounds.

Problem

• smallest enclosing disk
• farthest-point Voronoi diagram
• range query
• computational geometry
• preprocessing

Innovation

• smallest enclosing disk
• farthest-point Voronoi diagram
• range query
• computational geometry
• randomized data structure