Dynamic Geometric Data Structures via Shallow Cuttings

We present new results on a number of fundamental problems about dynamic geometric data structures:

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].
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, SODA 2006; Kaplan et al., SODA 2017].
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 1995; Chan, SODA 2006; Kaplan et al., SODA 2017].

We present new results on a number of fundamental problems about dynamic geometric data structures:

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].
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, SODA 2006; Kaplan et al., SODA 2017].
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 1995; Chan, SODA 2006; Kaplan et al., SODA 2017].

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