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。我们的目标是对 $S$ 进行预处理，以高效计算位于轴对齐查询矩形内的 $S$ 子集的最小包围圆。该问题的先前数据结构通过将点提升到三维空间、对偶化为多面体并搜索这些多面体的交点，实现了查询时间 $O(\log^6 n)$ 及预处理时间和空间 $O(n \log^2 n)$。我们提出了一种基于二维几何结构（特别是二维最远点沃罗诺伊图）的显著简化方法。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，在相同预处理边界下实现了期望 $O(\log^{5/2} n \log\log n)$ 的查询时间。