Dynamic Connectivity in Disk Graphs

TL;DR

The paper advances dynamic connectivity for disk graphs by developing a proxy-graph framework built from hierarchical space partitions (quadtrees/quadforests), MBMs, and AWNNs, enabling connectivity queries without explicit edge enumeration. It delivers strong results for unit-disk graphs with $O(\log^2 n)$ updates and $O(\log n/\log\log n)$ queries, and extends to general disks with polynomial or logarithmic dependence on the radius ratio $\Psi$ via progressively sparser proxies and disk-revealing structures (RDS). It also introduces semi-dynamic (incremental/decremental) variants and a radius-agnostic (arbitrary $\Psi$) approach based on heavy-path decompositions, achieving query times of $O(\log n/\log\log n)$ and update bounds that are polylogarithmic in $n$ and suitably sublinear in $\Psi$. The disk-revealing data structure (RDS) is a key conceptual tool enabling efficient handling of deletions and may be of independent interest for geometric dynamic problems. Overall, the work significantly improves prior dynamic-connectivity results for disk graphs and provides a versatile framework for future geometric dynamic data-structure design.

Abstract

Let $S$ be a set of $n$ sites in the plane, so that every site $s \in S$ has an associated radius $r_s > 0$. Let $\mathcal{D}(S)$ be the disk intersection graph defined by $S$, i.e., the graph with vertex set $S$ and an edge between two distinct sites $s, t \in S$ if and only if the disks with centers $s$, $t$ and radii $r_s$, $r_t$ intersect.Our goal is to design data structures that maintain the connectivity structure of $\mathcal{D}(S)$ as sites are inserted and/or deleted in $S$.

Figures (23)

• Figure 1: Two levels of the hierarchical grid, a cell $\sigma$ at level $i$, its center $a(\sigma)$, and the $(5 \times 5)$-neighborhood of $\sigma$.
• Figure 2: The set $L_P$ induced by $P$. The unit disks are drawn dashed. If a site $b \in B$ lies above the lower envelope, the unit disks intersect.
• Figure 3: A solution for unit disks with $O(\log^{6} n)$ update time.
• Figure 4: The structure of our data structure for unit disks.
• Figure 5: The neighboring cells of the red cell in $\mathcal{G}_{i-1}$. The area of the neighboring cells in one level beneath is colored in a darker shade.

Theorems & Definitions (39)

• Theorem 1
• Theorem 2: Holm et al. holm_poly-logarithmic_2001
• Lemma 3: Kaplan et al. kaplan_dynamic_2020
• Lemma 4: Kaplan et al. kaplan_dynamic_2020, Liu Liu20
• Lemma 5
• Lemma 6: Agarwal et al. agarwal_dynamic_2019
• Lemma 7
• Lemma 8
• Theorem 9
• Lemma 10