On the Efficiency of a Local Iterative Algorithm to Compute Delaunay Realizations

Greedy routing protocols for wireless sensor networks (WSNs) are fast and efficient but in general cannot guarantee message delivery. Hence researchers are interested in the problem of embedding WSNs in low dimensional space (e.g., R) in a way that guarantees message delivery with greedy routing. It is well known that Delaunay triangulations are such embeddings. We present the algorithm FindAngles, which is a fast, simple, local distributed algorithm that computes a Delaunay triangulation from any given combinatorial graph that is Delaunay realizable. Our algorithm is based on a characterization of Delaunay realizability due to Hiroshima et al. (IEICE 2000). When compared to the PowerDiagram algorithm of Chen et al. (SoCG 2007) that embeds triangulations in the plane so as to permit successful greedy routing, our algorithm requires on average 1/6 the number of iterations. FindAngles also scales linearly to larger networks and has a much faster distributed implementation than PowerDiagram, requiring just a single round of communication in each iteration. The PowerDiagram algorithm was proposed as an improvement on another algorithm due to Thurston (unpublished, 1988). Our experiments show that on average the PowerDiagram algorithm uses about 19% fewer iterations than the Thurston algorithm, whereas our algorithm uses about 89% fewer iterations. Experimentally, FindAngles exhibits well behaved convergence. Theoretically, we prove that with certain initial conditions the error term decreases monotonically. Taken together, these suggest our algorithm may have polynomial time convergence for certain classes of graphs. We note that our algorithm runs only on Delaunay realizable triangulations. This is not a significant concern because Hiroshima et al. (IEICE 2000) indicate that most combinatorial triangulations are indeed Delaunay realizable, which we have also observed experimentally: out of 5000 randomly generated combinatorial triangulations on 100 vertices, only one was not Delaunay realizable.