arXiv subjects

Zhengcheng Huang

Publications and source records attributed to Zhengcheng Huang.

6 recordsLinked to original sources

arXiv2025

Sparse Bounded Hop-Spanners for Geometric Intersection Graphs

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 $\mathbb{R}^d$. For example, we show that the intersection graph of $n$ balls in $\mathbb{R}^d$ admits a $2$-hop spanner of size $O^*\left(n^{\frac{3}{2}-\frac{1}{2(2\lfloor d/2\rfloor +1)}}\right)$ and the intersection graph of $n$ fat axis-parallel boxes in $\mathbb{R}^d$ 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 $\mathbb{R}^d$ admits a $3$-hop spanner of size $O^*\left(n^{\frac{3}{2}-\frac{1}{2(2D-1)}}\right)$, where $D$ is a parameter associated with the description complexity of the objects. For such families (or more specifically, for tetrahedra in $\mathbb{R}^3$), we provide a lower bound of $Ω(n^{\frac{4}{3}})$. For $3$-hop and axis-parallel boxes in $\mathbb{R}^d$, we provide the upper bound $O(n \log ^{d-1}n)$ and lower bound $Ω\left(n (\frac{\log n}{\log \log n})^{d-2}\right)$.

cs.CG↗

arXiv2025

Faster Algorithms for Reverse Shortest Path in Unit-Disk Graphs and Related Geometric Optimization Problems: Improving the Shrink-and-Bifurcate Technique

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 $\tilde{O}(n^{9/8})=\tilde{O}(n^{1.125})$.

cs.DS↗

arXiv2024

Shortest Path Separators in Unit Disk Graphs

We introduce a new balanced separator theorem for unit-disk graphs involving two shortest paths combined with the 1-hop neighbours of those paths and two other vertices. This answers an open problem of Yan, Xiang and Dragan [CGTA '12] and improves their result that requires removing the 3-hop neighborhood of two shortest paths. Our proof uses very different ideas, including Delaunay triangulations and a generalization of the celebrated balanced separator theorem of Lipton and Tarjan [J. Appl. Math. '79] to systems of non-intersecting paths.

cs.CG↗

arXiv2024

Dynamic Geometric Connectivity in the Plane with Constant Query Time

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 $n^{7/8}$).

cs.CG↗

arXiv2023

Constant-Hop Spanners for More Geometric Intersection Graphs, with Even Smaller Size

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.

cs.CG↗

arXiv2020

Improved Upper and Lower Bounds for LR Drawings of Binary Trees

In SODA'99, Chan introduced a simple type of planar straight-line upward order-preserving drawings of binary trees, known as LR drawings: such a drawing is obtained by picking a root-to-leaf path, drawing the path as a straight line, and recursively drawing the subtrees along the paths. Chan proved that any binary tree with $n$ nodes admits an LR drawing with $O(n^{0.48})$ width. In SODA'17, Frati, Patrignani, and Roselli proved that there exist families of $n$-node binary trees for which any LR drawing has $Ω(n^{0.418})$ width. In this note, we improve Chan's upper bound to $O(n^{0.437})$ and Frati et al.'s lower bound to $Ω(n^{0.429})$.

cs.CG↗