关键词:
Dynamic data structures
Convex hulls
Nearest neighbor search
Closest pair
Shallow cuttings
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(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)).