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.
From: Mark Joachim Krallmann [view email]
[v1]
Fri, 1 May 2026 15:47:02 UTC (606 KB)
[v2]
Tue, 5 May 2026 13:02:36 UTC (606 KB)
[v3]
Thu, 9 Jul 2026 12:40:35 UTC (687 KB)
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