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1. Dynamic Geometric Data Structures via Shallow Cuttings

   https://doi.org/10.4230/LIPIcs.SoCG.2019.24  

   Chan, Timothy M. ( January 2019 , Leibniz international proceedings in informatics)

   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]. 

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2. Sparse Approximation via Generating Point Sets

   https://doi.org/10.1137/1.9781611974331.ch40  

   Blum, Avrim ; Har-Peled, Sariel ; Raichel, Benjamin ( December 2015 , Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms (SODA2016))

   For a set P of n points in the unit ball b ⊆ R d , consider the problem of finding a small subset T ⊆ P such that its convex-hull ε-approximates the convex-hull of the original set. Specifically, the Hausdorff distance between the convex hull of T and the convex hull of P should be at most ε. We present an efficient algorithm to compute such an ε ′ -approximation of size kalg, where ε ′ is a function of ε, and kalg is a function of the minimum size kopt of such an ε-approximation. Surprisingly, there is no dependence on the dimension d in either of the bounds. Furthermore, every point of P can be ε- approximated by a convex-combination of points of T that is O(1/ε2 )-sparse. Our result can be viewed as a method for sparse, convex autoencoding: approximately representing the data in a compact way using sparse combinations of a small subset T of the original data. The new algorithm can be kernelized, and it preserves sparsity in the original input. 

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3. Fast and Exact Convex Hull Simplification

   https://doi.org/10.4230/LIPIcs.FSTTCS.2021.26  

   Klimenko, Georgiy ; Raichel, Benjamin ( October 2021 , Foundations of Software Technology and Theoretical Computer Science (FSTTCS))

   Bojanczyk, Mikolaj ; Chekuri, Chandra (Ed.)

   Given a point set P in the plane, we seek a subset Q ⊆ P, whose convex hull gives a smaller and thus simpler representation of the convex hull of P. Specifically, let cost(Q,P) denote the Hausdorff distance between the convex hulls CH(Q) and CH(P). Then given a value ε > 0 we seek the smallest subset Q ⊆ P such that cost(Q,P) ≤ ε. We also consider the dual version, where given an integer k, we seek the subset Q ⊆ P which minimizes cost(Q,P), such that |Q| ≤ k. For these problems, when P is in convex position, we respectively give an O(n log²n) time algorithm and an O(n log³n) time algorithm, where the latter running time holds with high probability. When there is no restriction on P, we show the problem can be reduced to APSP in an unweighted directed graph, yielding an O(n^2.5302) time algorithm when minimizing k and an O(min{n^2.5302, kn^2.376}) time algorithm when minimizing ε, using prior results for APSP. Finally, we show our near linear algorithms for convex position give 2-approximations for the general case. 

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4. Spherical convex hull of random points on a wedge

   https://doi.org/10.1007/s00208-023-02704-9  

   Besau, Florian ; Gusakova, Anna ; Reitzner, Matthias ; Schütt, Carsten ; Thäle, Christoph ; Werner, Elisabeth M. ( August 2023 , Mathematische Annalen)

   Consider two half-spaces$$H_1^+$$$H_1^{+}$and$$H_2^+$$$H_2^{+}$in$${\mathbb {R}}^{d+1}$$$R^{d+1}$whose bounding hyperplanes$$H_1$$$H_1$and$$H_2$$$H_2$are orthogonal and pass through the origin. The intersection$${\mathbb {S}}_{2,+}^d:={\mathbb {S}}^d\cap H_1^+\cap H_2^+$$$S_{2,+}^d:=S^d\cap H_1^{+}\cap H_2^{+}$is a spherical convex subset of thed-dimensional unit sphere$${\mathbb {S}}^d$$$S^d$, which contains a great subsphere of dimension$$d-2$$$d-2$and is called a spherical wedge. Choosenindependent random points uniformly at random on$${\mathbb {S}}_{2,+}^d$$$S_{2,+}^d$and consider the expected facet number of the spherical convex hull of these points. It is shown that, up to terms of lower order, this expectation grows like a constant multiple of$$\log n$$$logn$. A similar behaviour is obtained for the expected facet number of a homogeneous Poisson point process on$${\mathbb {S}}_{2,+}^d$$$S_{2,+}^d$. The result is compared to the corresponding behaviour of classical Euclidean random polytopes and of spherical random polytopes on a half-sphere.

    

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5. Active Learning a Convex Body in Low Dimensions

   Har-Peled, Sariel ; Jones, Mitchell ; Saladi, Rahul ( July 2020 , Leibniz international proceedings in informatics)

   Czumaj, Artur ; Dawar, Anuj ; Merelli, Emanuela (Ed.)

   Consider a set P ⊆ ℝ^d of n points, and a convex body C provided via a separation oracle. The task at hand is to decide for each point of P if it is in C using the fewest number of oracle queries. We show that one can solve this problem in two and three dimensions using O(⬡_P log n) queries, where ⬡_P is the largest subset of points of P in convex position. In 2D, we provide an algorithm which efficiently generates these adaptive queries. Furthermore, we show that in two dimensions one can solve this problem using O(⊚(P,C) log² n) oracle queries, where ⊚(P,C) is a lower bound on the minimum number of queries that any algorithm for this specific instance requires. Finally, we consider other variations on the problem, such as using the fewest number of queries to decide if C contains all points of P. As an application of the above, we show that the discrete geometric median of a point set P in ℝ² can be computed in O(n log² n (log n log log n + ⬡(P))) expected time. 

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