Sparse Bounded Hop-Spanners for Geometric Intersection Graphs
Sujoy Bhore, Timothy M. Chan, Zhengcheng Huang, Shakhar Smorodinsky, Csaba D. Toth
TL;DR
This work studies sparse $2$- and $3$-hop spanners in geometric intersection graphs across diverse object families. It develops subquadratic constructions for $2$-hop spanners of fat objects (notably balls) and sub-$n^{3/2}$ $3$-hop spanners for general semi-algebraic objects, employing polynomial partitioning and divide-and-conquer; it also proves $3$-hop upper bounds for axis-aligned boxes and derives dimension- and complexity-dependent refinements for various families. Complementing these upper bounds are strong lower bounds, including $\Omega(n^{4/3})$ for tetrahedra in $\mathbb{R}^3$ and $\Omega(n(\log n/\log\log n)^{d-2})$ type bounds for axis-aligned boxes in higher dimensions, illustrating the role of geometry and description complexity. Collectively, the results advance understanding of how geometry, dimension, and object description complexity govern the sparsity of hop-spanners, with implications for efficient routing and shortest-path computations in geometric networks.
Abstract
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 $Ω\left(n (\frac{\log n}{\log \log n})^{d-2}\right)$.