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

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&#39;24), replacing the linear dependency on $Ψ$ by a polylogarithmic dependency. Finally, we generalise our results to $d$-dimensional hypercubes.