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1. Semialgebraic Range Stabbing, Ray Shooting, and Intersection Counting in the Plane

   https://doi.org/10.4230/LIPIcs.SoCG.2024.33  

   Chan, Timothy M; Cheng, Pingan; Zheng, Da Wei (June 2024, Proc. 40th Sympos. Computational Geometry (SoCG))

   Mulzer, Wolfgang; Phillips, Jeff M (Ed.)

   Polynomial partitioning techniques have recently led to improved geometric data structures for a variety of fundamental problems related to semialgebraic range searching and intersection searching in 3D and higher dimensions (e.g., see [Agarwal, Aronov, Ezra, and Zahl, SoCG 2019; Ezra and Sharir, SoCG 2021; Agarwal, Aronov, Ezra, Katz, and Sharir, SoCG 2022]). They have also led to improved algorithms for offline versions of semialgebraic range searching in 2D, via lens-cutting [Sharir and Zahl (2017)]. In this paper, we show that these techniques can yield new data structures for a number of other 2D problems even for online queries: 1) Semialgebraic range stabbing. We present a data structure for n semialgebraic ranges in 2D of constant description complexity with O(n^{3/2+ε}) preprocessing time and space, so that we can count the number of ranges containing a query point in O(n^{1/4+ε}) time, for an arbitrarily small constant ε > 0. (The query time bound is likely close to tight for this space bound.) 2) Ray shooting amid algebraic arcs. We present a data structure for n algebraic arcs in 2D of constant description complexity with O(n^{3/2+ε}) preprocessing time and space, so that we can find the first arc hit by a query (straight-line) ray in O(n^{1/4+ε}) time. (The query bound is again likely close to tight for this space bound, and they improve a result by Ezra and Sharir with near n^{3/2} space and near √n query time.) 3) Intersection counting amid algebraic arcs. We present a data structure for n algebraic arcs in 2D of constant description complexity with O(n^{3/2+ε}) preprocessing time and space, so that we can count the number of intersection points with a query algebraic arc of constant description complexity in O(n^{1/2+ε}) time. In particular, this implies an O(n^{3/2+ε})-time algorithm for counting intersections between two sets of n algebraic arcs in 2D. (This generalizes a classical O(n^{3/2+ε})-time algorithm for circular arcs by Agarwal and Sharir from SoCG 1991.) 

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2. 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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3. Unit-disk range searching and applications

   https://doi.org/10.20382/jocg.v14i1a13  

   Wang, Haitao (December 2023, Journal of computational geometry)

   Given a set $$P$$ of $$n$$ points in the plane, we consider the problem of computing the number of points of $$P$$ in a query unit disk (i.e., all query disks have the same radius). We show that the main techniques for simplex range searching in the plane can be adapted to this problem. For example, by adapting Matoušek's results, we can build a data structure of $O(n)$ space in $$O(n^{1+\delta})$$ time (for any $$\delta>0$$) so that each query can be answered in $$O(\sqrt{n})$$ time; alternatively, we can build a data structure of $$O(n^2/\log^2 n)$$ space with $$O(n^{1+\delta})$$ preprocessing time (for any $$\delta>0$$) and $$O(\log n)$$ query time. Our techniques lead to improvements for several other classical problems in computational geometry. 1. Given a set of $$n$$ unit disks and a set of $$n$$ points in the plane, the batched unit-disk range counting problem is to compute for each disk the number of points in it. Previous work [Katz and Sharir, 1997] solved the problem in $$O(n^{4/3}\log n)$$ time. We give a new algorithm of $$O(n^{4/3})$$ time, which is optimal as it matches an $$\Omega(n^{4/3})$$-time lower bound. For small $$\chi$$, where $$\chi$$ is the number of pairs of unit disks that intersect, we further improve the algorithm to $$O(n^{2/3}\chi^{1/3}+n^{1+\delta})$$ time, for any $$\delta>0$$. 2. The above result immediately leads to an $$O(n^{4/3})$$ time optimal algorithm for counting the intersecting pairs of circles for a set of $$n$$ unit circles in the plane. The previous best algorithms solve the problem in $$O(n^{4/3}\log n)$$ deterministic time [Katz and Sharir, 1997] or in $$O(n^{4/3}\log^{2/3} n)$$ expected time by a randomized algorithm [Agarwal, Pellegrini, and Sharir, 1993]. 3. Given a set $$P$$ of $$n$$ points in the plane and an integer $$k$$, the distance selection problem is to find the $$k$$-th smallest distance among all pairwise distances of $$P$$. The problem can be solved in $$O(n^{4/3}\log^2 n)$$ deterministic time [Katz and Sharir, 1997] or in $$O(n\log n+n^{2/3}k^{1/3}\log^{5/3}n)$$ expected time by a randomized algorithm [Chan, 2001]. Our new randomized algorithm runs in $$O(n\log n +n^{2/3}k^{1/3}\log n)$$ expected time. 4. Given a set $$P$$ of $$n$$ points in the plane, the discrete $$2$$-center problem is to compute two smallest congruent disks whose centers are in $$P$$ and whose union covers $$P$$. An $$O(n^{4/3}\log^5 n)$$-time algorithm was known [Agarwal, Sharir, and Welzl, 1998]. Our techniques yield a deterministic algorithm of $$O(n^{4/3}\log^{10/3} n\cdot (\log\log n)^{O(1)})$$ time and a randomized algorithm of $$O(n^{4/3}\log^3 n\cdot (\log\log n)^{1/3})$$ expected time. 

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4. Constructing many faces in arrangements of lines and segments

   https://doi.org/10.20382/jocg.v14i1a11  

   Wang, Haitao (December 2023, Journal of computational geometry)

   We present new algorithms for computing many faces in arrangements of lines and segments. Given a set $$S$$ of $$n$$ lines (resp., segments) and a set $$P$$ of $$m$$ points in the plane, the problem is to compute the faces of the arrangements of $$S$$ that contain at least one point of $$P$$. For the line case, we give a deterministic algorithm of $$O(m^{2/3}n^{2/3}\log^{2/3} (n/\sqrt{m})+(m+n)\log n)$$ time. This improves the previously best deterministic algorithm [Agarwal, 1990] by a factor of $$\log^{2.22}n$$ and improves the previously best randomized algorithm [Agarwal, Matoušek, and Schwarzkopf, 1998] by a factor of $$\log^{1/3}n$$ in certain cases (e.g., when $$m=\Theta(n)$$). For the segment case, we present a deterministic algorithm of $$O(n^{2/3}m^{2/3}\log n+\tau(n\alpha^2(n)+n\log m+m)\log n)$$ time, where $$\tau=\min\{\log m,\log (n/\sqrt{m})\}$$ and $$\alpha(n)$$ is the inverse Ackermann function. This improves the previously best deterministic algorithm [Agarwal, 1990] by a factor of $$\log^{2.11}n$$ and improves the previously best randomized algorithm [Agarwal, Matoušek, and Schwarzkopf, 1998] by a factor of $$\log n$$ in certain cases (e.g., when $$m=\Theta(n)$$). We also give a randomized algorithm of $$O(m^{2/3}K^{1/3}\log n+\tau(n\alpha(n)+n\log m+m)\log n\log K)$$ expected time, where $$K$$ is the number of intersections of all segments of $$S$$. In addition, we consider the query version of the problem, that is, preprocess $$S$$ to compute the face of the arrangement of $$S$$ that contains any given query point. We present new results that improve the previous work for both the line and the segment cases. In particulary, for the line case, we build a data structure of $$O(n\log n)$$ space in $$O(n\log n)$$ randomized time, so that the face containing the query point can be obtained in $$O(\sqrt{n\log n})$$ time with high probability (more specifically, the query returns a binary search tree representing the face so that standard binary-search-based queries on the face can be handled in $$O(\log n)$$ time each and the face itself can be output explicitly in time linear in its size). 

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5. Lower Envelopes of Surface Patches in 3-Space

   https://doi.org/10.4230/LIPICS.ESA.2024.6  

   Agarwal, Pankaj K; Ezra, Esther; Sharir, Micha (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Chan, Timothy; Fischer, Johannes; Iacono, John; Herman, Grzegorz (Ed.)

   Let Σ be a collection of n surface patches, each being the graph of a partially defined semi-algebraic function of constant description complexity, and assume that any triple of them intersect in at most s = 2 points. We show that the complexity of the lower envelope of the surfaces in Σ is O(n² log^{6+ε} n), for any ε > 0. This almost settles a long-standing open problem posed by Halperin and Sharir, thirty years ago, who showed the nearly-optimal albeit weaker bound of O(n²⋅ 2^{c√{log n}}) on the complexity of the lower envelope, where c > 0 is some constant. Our approach is fairly simple and is based on hierarchical cuttings and gradations, as well as a simple charging scheme. We extend our analysis to the case s > 2, under a "favorable cross section" assumption, in which case we show that the bound on the complexity of the lower envelope is O(n² log^{11+ε} n), for any ε > 0. Incorporating these bounds with the randomized incremental construction algorithms of Boissonnat and Dobrindt, we obtain efficient constructions of lower envelopes of surface patches with the above properties, whose overall expected running time is O(n² polylog), as well as efficient data structures that support point location queries in their minimization diagrams in O(log²n) expected time. 

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