A dynamic $(1+\varepsilon)$-spanner for disk intersection graphs

2026-04-28

AI Summary

This work addresses the problem of efficiently maintaining a $(1+\varepsilon)$-spanner for dynamic intersection graphs of disks whose diameters are constrained to the interval $[4, \Psi]$, supporting fast updates and connectivity queries. The key innovation lies in the first application of persistent data structures to this setting, synergistically combining techniques from geometric graph theory and dynamic graph algorithms. The proposed spanner has size $O(n \varepsilon^{-2} \log \Psi \log (\varepsilon^{-1}))$, reducing the space dependence on $\Psi$ from linear to polylogarithmic. Under constant $\varepsilon$ and $\Psi$, the method achieves near-linear space and polylogarithmic amortized update time. Furthermore, the approach generalizes naturally to $d$-dimensional hypercubes.

Abstract

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.

Problem

• dynamic spanner
• disk intersection graph
• connectivity
• geometric graph
• persistent data structures

Innovation

• dynamic spanner
• disk intersection graph
• persistent data structures
• connectivity queries
• geometric graphs