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.

翻译：设 $S$ 为 $\mathbb{R}^2$ 中 $n$ 个点的集合。我们的目标是预处理 $S$，以高效计算落在轴对齐查询矩形内部 $S$ 中各点所对应的最小包围圆。针对该问题的先前数据结构通过将点提升至三维空间、对偶化为多面体并搜索其交集，实现了 $O(\log^6 n)$ 的查询时间以及 $O(n \log^2 n)$ 的预处理时间和空间。我们提出了一种显著简化的方法，完全基于二维几何结构，特别是二维最远点沃罗诺伊图。该方法实现了确定性的 $O(\log^4 n)$ 查询时间，并通过随机化方法，在相同预处理界下实现了期望的 $O(\log^{5/2} n \log\log n)$ 查询时间。