Communication-Efficient Construction of the Plane Localized Delaunay Graph

Let V be a finite set of points in the plane. We present a 2-local algorithm that constructs a plane $\frac{4 \pi \sqrt{3}}{9}$-spanner of the unit-disk graph $\mathord{\it UDG}(V)$. Each node can only communicate with nodes that are within unit-distance from it. This algorithm only makes one round of communication and each point of V broadcasts at most 5 messages. This improves on all previously known message-bounds for this problem.