A dynamic $(1+\varepsilon)$-spanner for disk intersection graphs
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866918502661619712 |
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| author | de Berg, Sarita van der Hoog, Ivor Rotenberg, Eva Vistisen, Johanne M. Wong, Sampson |
| author_facet | de Berg, Sarita van der Hoog, Ivor Rotenberg, Eva Vistisen, Johanne M. Wong, Sampson |
| contents | We maintain a $(1+\varepsilon)$-spanner over the disk intersection graph of a dynamic set of disks. We restrict all disks to have their diameter in $[4,Ψ]$ for some fixed and known $Ψ$. The resulting $(1+\varepsilon)$-spanner has size $O(n \varepsilon^{-2} \log Ψ\log (\varepsilon^{-1}))$, where $n$ is the present number of disks.
We develop a novel use of persistent data structures to dynamically maintain our $(1+\varepsilon)$-spanner. Our approach requires $O(\varepsilon^{-2} n \log^4 n \log Ψ)$ space and has an $O( \left( \fracΨ{\varepsilon} \right)^2 \log^4 n \log^2 Ψ\log^2 (\varepsilon^{-1}))$ expected amortised update time. For constant $\varepsilon$ and $Ψ$, this spanner has near-linear size, uses near-linear space and has polylogarithmic update time. Furthermore, we observe that for any $\varepsilon < 1$, our spanner also serves as a connectivity data structure. With a slight adaptation of our techniques, this leads to better bounds for dynamically supporting connectivity queries in a disk intersection graph. In particular, we improve the space usage when compared to the dynamic data structure of (Baumann et al., DCG'24), replacing the linear dependency on $Ψ$ by a polylogarithmic dependency. Finally, we generalise our results to $d$-dimensional hypercubes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_25397 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A dynamic $(1+\varepsilon)$-spanner for disk intersection graphs de Berg, Sarita van der Hoog, Ivor Rotenberg, Eva Vistisen, Johanne M. Wong, Sampson Computational Geometry Data Structures and Algorithms We maintain a $(1+\varepsilon)$-spanner over the disk intersection graph of a dynamic set of disks. We restrict all disks to have their diameter in $[4,Ψ]$ for some fixed and known $Ψ$. The resulting $(1+\varepsilon)$-spanner has size $O(n \varepsilon^{-2} \log Ψ\log (\varepsilon^{-1}))$, where $n$ is the present number of disks. We develop a novel use of persistent data structures to dynamically maintain our $(1+\varepsilon)$-spanner. Our approach requires $O(\varepsilon^{-2} n \log^4 n \log Ψ)$ space and has an $O( \left( \fracΨ{\varepsilon} \right)^2 \log^4 n \log^2 Ψ\log^2 (\varepsilon^{-1}))$ expected amortised update time. For constant $\varepsilon$ and $Ψ$, this spanner has near-linear size, uses near-linear space and has polylogarithmic update time. Furthermore, we observe that for any $\varepsilon < 1$, our spanner also serves as a connectivity data structure. With a slight adaptation of our techniques, this leads to better bounds for dynamically supporting connectivity queries in a disk intersection graph. In particular, we improve the space usage when compared to the dynamic data structure of (Baumann et al., DCG'24), replacing the linear dependency on $Ψ$ by a polylogarithmic dependency. Finally, we generalise our results to $d$-dimensional hypercubes. |
| title | A dynamic $(1+\varepsilon)$-spanner for disk intersection graphs |
| topic | Computational Geometry Data Structures and Algorithms |
| url | https://arxiv.org/abs/2604.25397 |