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  "path": "/abs/2605.00743v1",
  "publishedAt": "2026-05-04T00:00:00.000Z",
  "site": "https://arxiv.org",
  "tags": [
    "Kevin Buchin",
    "Mark Joachim Krallmann",
    "Frank Staals"
  ],
  "textContent": "**Authors:** Kevin Buchin, Mark Joachim Krallmann, Frank Staals\n\nLet $S$ 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.",
  "title": "Smallest Enclosing Disk Queries Using Farthest-Point Voronoi Diagrams"
}