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.

翻译：我们针对动态盘集合的盘交集图维护一个$(1+\varepsilon)$-稀疏骨架。假设所有盘的直径均位于固定且已知的区间$[4,Ψ]$内。所得到的$(1+\varepsilon)$-稀疏骨架规模为$O(n \varepsilon^{-2} \log Ψ\log (\varepsilon^{-1}))$，其中$n$为当前盘的数量。我们提出了一种持久化数据结构的创新应用，用于动态维护该$(1+\varepsilon)$-稀疏骨架。该方法所需空间为$O(\varepsilon^{-2} n \log^4 n \log Ψ)$，期望平摊更新时间为$O( \left( \fracΨ{\varepsilon} \right)^2 \log^4 n \log^2 Ψ\log^2 (\varepsilon^{-1}) )$。对于常数$\varepsilon$和$Ψ$，该稀疏骨架具有近线性规模、近线性空间复杂度及多对数级更新时间。此外，我们观察到对于任意$\varepsilon < 1$，该稀疏骨架亦可作为连通性数据结构。通过对技术方案进行微调，该结果在动态支持盘交集图连通性查询方面获得了更优的界。具体而言，与（Baumann等，DCG'24）的动态数据结构相比，我们改进了空间使用率，将原本对$Ψ$的线性依赖替换为多对数依赖。最后，我们将结论推广至$d$维超立方体情形。