Dynamic Geometric Data Structures via Shallow Cuttings
Dynamic Geometric Data Structures via Shallow Cuttings
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DOI：
10.1007/s00454-020-00229-5
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发表时间：
2020-07-24
影响因子：
0.8
通讯作者：
Chan, Timothy M.
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文献类型：
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作者：
Chan, Timothy M.
我们为动态几何数据结构的许多基本问题提供了新的结果：（1）我们描述了第一个完全动态的数据结构，其中具有sublinear amortized更新时间以维持（i）（i）（i）一个顶点的数量或凸壳的体积3D点集，（ii）2D点集的最大空圆，（iii）两个2D点组之间的Hausdorff距离，（iv）2D点集的离散1中心，（v）最大数量（即天际线）在3D点集中点。对于（i）和（ii），（iii）和（ii）和（iv）的n（i）和（ii），n（5/6）和（v）的n（2/3），更新时间接近n（i）和（ii），n（5/6）。以前，仅因受限制的“半对线”设置而闻名（Siam J. Comput.32（3），700-716（2003））。 （2）我们稍微改善了以前的完全动态数据结构，用于回答3D点集的凸壳的极端查询，并最近邻居搜索2D点集。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. acm57（3），＃16（ 2010年; （3）我们还改进了以前的完全动态数据结构，以维持两个2D点集和2D点集的直径之间的最接近的两对。摊销的更新时间为o（log（4）n），而不是o（log（7）n）（离散计算中的eppstein。 ）＃16（2010年）;
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. ACM57(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. ACM57(3), # 16 (2010); Kaplan et al. in 28th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 2495-2504. SIAM, Philadelphia (2017)).