Fast and Efficient Restricted Delaunay Triangulation in Random Geometric Graphs

Let G = G(n, r) be a random geometric graph resulting from placing n nodes uniformly at random in the unit square (disk) and connecting every two nodes if and only if their Euclidean distance is at most r . Let rcon = q logn πn be the known critical radius guaranteeing connectivity when n → ∞. 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 networks the construction of RDG(G) requires O(n) messages, we show that when r = O(rcon) and G = G(n, r), with high probability, RDG(G) can be constructed locally in one round of communication with O( √ n logn) messages, and with only one hop neighborhood information. This proves that the existence of long Delaunay edges (an order larger than rcon) in the unit square (disk) does not significantly impact the efficiency with which good routing graphs can be maintained.