Sparse Bounded Hop-Spanners for Geometric Intersection Graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915233430241280 |
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| author | Bhore, Sujoy Chan, Timothy M. Huang, Zhengcheng Smorodinsky, Shakhar Toth, Csaba D. |
| author_facet | Bhore, Sujoy Chan, Timothy M. Huang, Zhengcheng Smorodinsky, Shakhar Toth, Csaba D. |
| contents | 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 $Ω\left(n (\frac{\log n}{\log \log n})^{d-2}\right)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_05861 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sparse Bounded Hop-Spanners for Geometric Intersection Graphs Bhore, Sujoy Chan, Timothy M. Huang, Zhengcheng Smorodinsky, Shakhar Toth, Csaba D. Computational Geometry Discrete Mathematics 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 $Ω\left(n (\frac{\log n}{\log \log n})^{d-2}\right)$. |
| title | Sparse Bounded Hop-Spanners for Geometric Intersection Graphs |
| topic | Computational Geometry Discrete Mathematics |
| url | https://arxiv.org/abs/2504.05861 |