Partial delaunay triangulation-based asynchronous planarization of quasi unit disk graphs

We present a distributed and fully asynchronous algorithm for construction of the partial Delaunay triangulation over quasi unit disk graphs. Provided that the ratio of the maximum to the minimum communication range of nodes is bounded from above by square root of two, our algorithm outputs a connected and planar overlay graph of the input graph, which enables the use of localized geographic routing algorithms that guarantee message delivery. Moreover, under the assumption that the input graph is civilized (i.e., any two network nodes have non-zero minimum Euclidean distance), we show that our algorithm is localized. We show by means of simulation that our approach yields output graphs whose Euclidean spanning ratio is on average significantly smaller compared to those constructed by all other asynchronous approaches.