Dynamic Connectivity in Disk Graphs

TL;DR

This paper addresses the problem of maintaining the connectivity information of disk intersection graphs in the plane dynamically as sites are inserted or removed, which is crucial for applications in spatial networks.

Contribution

The paper introduces new data structures and algorithms for efficiently updating the connectivity of disk graphs under dynamic site insertions and deletions.

Findings

Achieved efficient update times for dynamic connectivity in disk graphs.

Provided theoretical bounds for the complexity of maintaining connectivity.

Extended previous static algorithms to dynamic scenarios.

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 $S$.