Fast and Efficient Restricted Delaunay Triangulation in Random Geometric Graphs

Abstract

Let $G = \mathcal{G}(n,r)$ be a random geometric graph resulting from placing $n$ nodes uniformly at random in the unit square (or the unit disk) and connecting every two nodes if and only if their Euclidean distance is at most $r$. Let $r_\mathrm{con} = \sqrt{\frac{\log n}{\pi n}}(1+o(1))$ be the known critical radius for connectivity when $n \to \infty$. The restricted Delaunay graph RDG$(G)$ is a subgraph of $G$ with the following properties: it is a planar graph and a spanner of $G$, and in particular it contains all the short edges of the Delaunay triangulation of $G$. While in general graphs, the construction of RDG$(G)$ requires $\Theta(n)$ messages, we show that when $r = O(r_\mathrm{con})$ and $G=\mathcal{G}(n,r)$, then with high probability, RDG$(G)$ can be constructed locally in one round of communication with $O(\sqrt{n \log n})$ messages, and with only one-hop neighborhood information. This size of $r$ proves that the existence of long Delaunay edges (an order larger than $r_\mathrm{con}$) in the unit square (disk) does not significantly affect the efficiency with which good routing graphs can be maintained.