Greedy Convex Embeddings for Ad-Hoc Networks

Abstract

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, 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. 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.