基于浅切割的动态几何数据结构
Dynamic Geometric Data Structures via Shallow Cuttings
摘要：
我们提出了一些新的结果动态几何数据结构的基本问题：1。我们描述了具有次线性摊销更新时间的第一完全动态数据结构，用于维护（i）3D点集的凸包的顶点数或体积，（ii）2D点集的最大空圆，（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（即，天际线）点。对于（i）和（ii），更新时间接近$n^{11/12}$，对于（iii）和（iv），更新时间接近$n^{7/8}$，对于（v），更新时间接近$n^{2/3}$。以前，次线性界限只知道有限的“半在线”设置[Chan，SODA 2002]。2.我们稍微改进了以前的全动态数据结构回答极端点查询的凸包的三维点集和最近邻搜索的二维点集。查询时间是$O(\log^2n)$，分期更新时间是$O(\log^4n)$而不是$O(\log^5n)$ [Chan，SODA 2006; Kaplan等人，SODA 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一个二维点集的直径。分期更新时间是$O(\log^4n)$而不是$O(\log^7n)$ [Eppstein 1995; Chan，SODA 2006; Kaplan等人，SODA 2017]。
英文摘要：
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), $n^{7/8}$ for (iii) and (iv), and $n^{2/3}$ for (v). Previously, sublinear bounds were known only for restricted `semi-online' settings [Chan, SODA 2002]. 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 $O(\log^2n)$, and the amortized update time is $O(\log^4n)$ instead of $O(\log^5n)$ [Chan, SODA 2006; Kaplan et al., SODA 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 $O(\log^4n)$ instead of $O(\log^7n)$ [Eppstein 1995; Chan, SODA 2006; Kaplan et al., SODA 2017].
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