Dynamic Geometric Data Structures via Shallow Cuttings

Abstract

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)).