Sparse Bounded Hop-Spanners for Geometric Intersection Graphs

2025-04-08

AI Summary

This paper investigates tight asymptotic bounds on the size of 2-hop and 3-hop sparse bounded-hop spanners in *d*-dimensional geometric intersection graphs—such as unit-ball graphs, fat axis-aligned boxes, and semi-algebraic objects. Leveraging a synthesis of combinatorial geometry, divide-and-conquer with grid-based partitioning, VC-dimension and ε-net theory, and semi-algebraic methods, it establishes near-optimal upper and lower bounds for classical geometric graph families in high dimensions. Key contributions include: (i) an upper bound of $O^*(n^{3/2 - 1/(2lfloor d/2 floor + 1)})$ on the size of 2-hop spanners for *d*-dimensional ball graphs; and (ii) a lower bound of $Omega(n^{4/3})$ for 3-hop spanners in 3D tetrahedron intersection graphs. Collectively, these results characterize the fundamental trade-offs between dimensionality, geometric complexity, and sparsity in bounded-hop spanner construction.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Security and Privacy: Large-scale security measurements

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 $O^*left(n^{frac{3}{2}-frac{1}{2(2lfloor d/2 floor +1)}} ight)$ 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 $O^*left(n^{frac{3}{2}-frac{1}{2(2D-1)}} ight)$, 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 $Omega(n^{frac{4}{3}})$. 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 $Omegaleft(n (frac{log n}{log log n})^{d-2} ight)$.

Problem

• Study 2- and 3-hop spanners for geometric intersection graphs
• Improve bounds for spanners in various geometric graphs
• Analyze size complexity for different object families

Innovation

• Improved 2-hop spanners for geometric graphs
• Enhanced 3-hop spanners for semi-algebraic objects
• Optimized bounds for axis-parallel box spanners