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

TL;DR

This paper presents a new dynamic $(1+ ext{epsilon})$-spanner for disk intersection graphs that is efficient in size, space, and update time, with applications to connectivity queries and generalizations to higher dimensions.

Contribution

It introduces a novel persistent data structure-based approach for maintaining a dynamic spanner with improved space and update time bounds, extending to hypercubes.

Findings

Achieves near-linear size and space for constant epsilon and Psi.

Provides polylogarithmic expected amortised update time.

Improves existing bounds for dynamic connectivity in 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 $[4,\Psi]$ for some fixed and known $\Psi$. The resulting $(1+\varepsilon)$-spanner has size $O(n \varepsilon^{-2} \log \Psi \log (\varepsilon^{-1}))$, 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 $O(\varepsilon^{-2} n \log^4 n \log \Psi)$ space and has an $O( \left( \frac{\Psi}{\varepsilon} \right)^2 \log^4 n \log^2 \Psi \log^2 (\varepsilon^{-1}))$ expected amortised update time. For constant $\varepsilon$ and $\Psi$, 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 $\Psi$ by a polylogarithmic dependency. Finally, we generalise our results to $d$-dimensional hypercubes.