Smallest Enclosing Disk Queries Using Farthest-Point Voronoi Diagrams

TL;DR

This paper introduces a simpler 2D geometric approach using farthest-point Voronoi diagrams to efficiently answer smallest enclosing disk queries within axis-aligned rectangles, improving query times over previous methods.

Contribution

A novel, simpler 2D geometric method based on farthest-point Voronoi diagrams for rectangle-restricted smallest enclosing disk queries, with improved query times.

Findings

Deterministic query time of O(log^4 n)

Expected randomized query time of O(log^{5/2} n log log n)

Preprocessing time and space remain O(n log^2 n)

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.