We present new results on a number of fundamental problems about dynamic geometric data structures: (1) We describe the first fully dynamic data structures with sublinear amortized update time for maintaining (i) the number of vertices or the volume of the convex hull of a 3D point set, (ii) the largest empty circle for a 2D point set, (iii) the Hausdorff distance between two 2D point sets, (iv) the discrete 1-center of a 2D point set, (v) the number of maximal (i.e., skyline) points in a 3D point set. The update times are near $$n^{11/12}$$ for (i) and (ii), $$n^{5/6}$$ for (iii) and (iv), and $$n^{2/3}$$ for (v). Previously, sublinear bounds were known only for restricted “semi-online” settings (Chan in SIAM J. Comput. 32(3), 700–716 (2003)). (2) We slightly improve previous fully dynamic data structures for answering extreme point queries for the convex hull of a 3D point set and nearest neighbor search for a 2D point set. The query time is $$O(\log ^2\!n)$$ , and the amortized update time is $$O(\log ^4\!n)$$ instead of $$O(\log ^5\!n)$$ (Chan in J. ACM 57(3), # 16 (2010); Kaplan et al. in 28th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 2495–2504. SIAM, Philadelphia (2017)). (3) We also improve previous fully dynamic data structures for maintaining the bichromatic closest pair between two 2D point sets and the diameter of a 2D point set. The amortized update time is $$O(\log ^4\!n)$$ instead of $$O(\log ^7\!n)$$ (Eppstein in Discrete Comput. Geom. 13(1), 111–122 (1995); Chan in J. ACM 57(3), # 16 (2010); Kaplan et al. in 28th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 2495–2504. SIAM, Philadelphia (2017)).

中文翻译：

---

通过浅层切割的动态几何数据结构

我们提出了关于动态几何数据结构的许多基本问题的新结果：（1）我们描述了第一个具有次线性摊销更新时间的全动态数据结构，用于维护（i）顶点的数量或凸包的体积3D 点集，(ii) 2D 点集的最大空心圆，(iii) 两个 2D 点集之间的 Hausdorff 距离，(iv) 2D 点集的离散 1-中心，(v) 最大数量（即，天际线）3D 点集中的点。(i) 和 (ii) 的更新时间接近 $$n^{11/12}$$，(iii) 和 (iv) 的 $$n^{5/6}$$，以及 $$n^ {2/3}$$ 用于 (v)。以前，次线性边界仅适用于受限的“半在线”设置（Chan 在 SIAM J. Comput. 32(3), 700-716 (2003)）。(2) 我们稍微改进了以前的全动态数据结构，用于回答 3D 点集的凸包的极值点查询和 2D 点集的最近邻搜索。查询时间为 $$O(\log ^2\!n)$$ ，摊销更新时间为 $$O(\log ^4\!n)$$ 而不是 $$O(\log ^5\ !n)$$（Chan in J. ACM 57(3), # 16 (2010); Kaplan et al. in 28th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 2495–2504. SIAM, Philadelphia (2017)） . (3) 我们还改进了以前的全动态数据结构，以保持两个二维点集之间的双色最近对和二维点集的直径。摊销更新时间是 $$O(\log ^4\!n)$$ 而不是 $$O(\log ^7\!n)$$ (Eppstein in Discrete Comput. Geom. 13(1), 111– 122 (1995); Chan in J. ACM 57(3), # 16 (2010); Kaplan 等。在第 28 届年度 ACM-SIAM 离散算法研讨会上，第 2495-2504 页。暹罗，费城（2017 年））。