Greedy convex embeddings for Ad-Hoc networks

Recent advances in networked systems and wireless communications have set the stage for applications with wide-ranging benefits. Perhaps the most natural problem in such systems is the "efficient" propagation of locally stored data. In order to address this problem, the notion of greedy embedding was defined by Papadimitriou and Ratajczak [1], where the authors conjectured that every 3-connected planar graph has a greedy embedding (possibly planar and convex) in the Euclidean plane. Recently, the greedy embedding conjecture was proved by Leighton and Moitra [2]. However, their algorithm does not result in a drawing that is planar and convex in the Euclidean plane for all 3-connected planar graphs. Here we consider the planar convex greedy embedding conjecture and give a probabilistic proof for the existence of such embeddings. In addition, we discuss a second proof which is almost immediate in the case of an embedding into the 3-dimensional sphere.