In a series of papers, Avraham, Filtser, Kaplan, Katz, and Sharir (SoCG'14), Kaplan, Katz, Saban, and Sharir (ESA'23), and Katz, Saban, and Sharir (ESA'24) studied a class of geometric optimization problems -- including reverse shortest path in unweighted and weighted unit-disk graphs, discrete Fréchet distance with one-sided shortcuts, and reverse shortest path in visibility graphs on 1.5-dimensional terrains -- for which standard parametric search does not work well due to a lack of efficient parallel algorithms for the corresponding decision problems. The best currently known algorithms for all the above problems run in $O^*(n^{6/5})=O^*(n^{1.2})$ time (ignoring subpolynomial factors), and they were obtained using a technique called \emph{shrink-and-bifurcate}. We improve the running time to $\tilde{O}(n^{8/7}) \approx O(n^{1.143})$ for these problems. Furthermore, specifically for reverse shortest path in unweighted unit-disk graphs, we improve the running time further to $\tilde{O}(n^{9/8})=\tilde{O}(n^{1.125})$.

We present the first fully dynamic connectivity data structures for geometric intersection graphs achieving constant query time and sublinear amortized update time for most types 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 $n^{7/8}$).

In SoCG 2022, Conroy and Tóth presented several constructions of sparse, low-hop spanners in geometric intersection graphs, including an $O(n\log n)$-size 3-hop spanner for $n$ disks (or fat convex objects) in the plane, and an $O(n\log^2 n)$-size 3-hop spanner for $n$ axis-aligned rectangles in the plane. Their work left open two major questions: (i) can the size be made closer to linear by allowing larger constant stretch? and (ii) can near-linear size be achieved for more general classes of intersection graphs? We address both questions simultaneously, by presenting new constructions of constant-hop spanners that have almost linear size and that hold for a much larger class of intersection graphs. More precisely, we prove the existence of an $O(1)$-hop spanner for arbitrary string graphs with $O(nα_k(n))$ size for any constant $k$, where $α_k(n)$ denotes the $k$-th function in the inverse Ackermann hierarchy. We similarly prove the existence of an $O(1)$-hop spanner for intersection graphs of $d$-dimensional fat objects with $O(nα_k(n))$ size for any constant $k$ and $d$. We also improve on some of Conroy and Tóth's specific previous results, in either the number of hops or the size: we describe an $O(n\log n)$-size 2-hop spanner for disks (or more generally objects with linear union complexity) in the plane, and an $O(n\log n)$-size 3-hop spanner for axis-aligned rectangles in the plane. Our proofs are all simple, using separator theorems, recursion, shifted quadtrees, and shallow cuttings.

In the colored orthogonal range reporting problem, we want a data structure for storing n colored points so that given a query axis-aligned rectangle, we can report the distinct colors among the points inside the rectangle. This natural problem has been studied in a series of papers, but most prior work focused on the static case. In this paper, we give a dynamic data structure in the 2D case which can answer queries in O(log^{1+o(1)} n + klog^{1/2+o(1)}n) time, where k denotes the output size (the number of distinct colors in the query range), and which can support insertions and deletions in O(log^{2+o(1)}n) time (amortized) in the standard RAM model. This is the first fully dynamic structure with polylogarithmic update time whose query cost per color reported is sublogarithmic (near √{log n}). We also give an alternative data structure with O(log^{1+o(1)} n + klog^{3/4+o(1)}n) query time and O(log^{3/2+o(1)}n) update time (amortized). We also mention extensions to higher constant dimensions.