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Title: Dynamic Geometric Data Structures via Shallow Cuttings
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].  more » « less
Award ID(s):
1814026
PAR ID:
10141567
Author(s) / Creator(s):
Date Published:
Journal Name:
Leibniz international proceedings in informatics
Volume:
129
ISSN:
1868-8969
Page Range / eLocation ID:
24:1-24:13
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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