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We present new results on 2- and 3-hop spanners for geometric intersection graphs. These include improved upper and lower bounds for 2- and 3-hop spanners for many geometric intersection graphs in ℝ^d. For example, we show that the intersection graph of n balls in ℝ^d admits a 2-hop spanner of size O^*(n^{3/2 - 1/(2(2⌊d/2⌋ + 1))}) and the intersection graph of n fat axis-parallel boxes in ℝ^d admits a 2-hop spanner of size O(n log^{d+1}n). Furthermore, we show that the intersection graph of general semi-algebraic objects in ℝ^d admits a 3-hop spanner of size O^*(n^{3/2 - 1/(2(2D-1))}), where D is a parameter associated with the description complexity of the objects. For such families (or more specifically, for tetrahedra in ℝ³), we provide a lower bound of Ω(n^{4/3}). For 3-hop and axis-parallel boxes in ℝ^d, we provide the upper bound O(n log ^{d-1}n) and lower bound Ω(n ({log n}/{log log n})^{d-2}). 

We present the first fully dynamic connectivity data structures for geometric intersection graphs achieving constant query time and sublinear amortized update time for many classes of geometric objects in 2D . Our data structures can answer connectivity queries between two objects, as well as "global" connectivity queries (e.g., deciding whether the entire graph is connected). Previously, the data structure by Afshani and Chan (ESA'06) achieved such bounds only in the special case of axis-aligned line segments or rectangles but did not work for arbitrary line segments or disks, whereas the data structures by Chan, Pătraşcu, and Roditty (FOCS'08) worked for more general classes of geometric objects but required n^{Ω(1)} query time and could not handle global connectivity queries. Specifically, we obtain new data structures with O(1) query time and amortized update time near n^{4/5}, n^{7/8}, and n^{20/21} for axis-aligned line segments, disks, and arbitrary line segments respectively. Besides greatly reducing the query time, our data structures also improve the previous update times for axis-aligned line segments by Afshani and Chan (from near n^{10/11} to n^{4/5}) and for disks by Chan, Pătraşcu, and Roditty (from near n^{20/21} to n^{7/8}). 

We revisit a standard polygon containment problem: given a convex k-gon P and a convex n-gon Q in the plane, find a placement of P inside Q under translation and rotation (if it exists), or more generally, find the largest copy of P inside Q under translation, rotation, and scaling. Previous algorithms by Chazelle (1983), Sharir and Toledo (1994), and Agarwal, Amenta, and Sharir (1998) all required Ω(n²) time, even in the simplest k = 3 case. We present a significantly faster new algorithm for k = 3 achieving O(n polylog n) running time. Moreover, we extend the result for general k, achieving O(k^O(1/ε) n^{1+ε}) running time for any ε > 0. Along the way, we also prove a new O(k^O(1) n polylog n) bound on the number of similar copies of P inside Q that have 4 vertices of P in contact with the boundary of Q (assuming general position input), disproving a conjecture by Agarwal, Amenta, and Sharir (1998). 

Let X be a set of points in ℝ² and 𝒪 be a set of geometric objects in ℝ², where |X| + |𝒪| = n. We study the problem of computing a minimum subset 𝒪^* ⊆ 𝒪 that encloses all points in X. Here a point x ∈ X is enclosed by 𝒪^* if it lies in a bounded connected component of ℝ²∖(⋃_{O ∈ 𝒪^*} O). We propose two algorithmic frameworks to design polynomial-time approximation algorithms for the problem. The first framework is based on sparsification and min-cut, which results in O(1)-approximation algorithms for unit disks, unit squares, etc. The second framework is based on LP rounding, which results in an O(α(n)log n)-approximation algorithm for segments, where α(n) is the inverse Ackermann function, and an O(log n)-approximation algorithm for disks. 

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.)