Every Schnyder Drawing is a Greedy Embedding

Geographic routing is a routing paradigm, which uses geographic coordinates of network nodes to determine routes. Greedy routing, the simplest form of geographic routing forwards a packet to the closest neighbor towards the destination. A greedy embedding is a embedding of a graph on a geometric space such that greedy routing always guarantees delivery. A Schnyder drawing is a classical way to draw a planar graph. In this manuscript, we show that every Schnyder drawing is a greedy embedding, based on a generalized definition of greedy routing. 1 Greedy Routing on Planar Triangular Graphs In this manuscript, we establish some results on greedy routing on planar triangular graphs. Planar triangular graphs are a special class of graphs, where every face of the graph is a triangle, including the external face. Hence the outer-face consists of three nodes A1, A2 and A3. We consider the problem of greedy routing [1] on such graphs. In order to perform geometric routing, the graph has to be drawn on a certain geometric space. Such a drawing is called a greedy embedding [2], which is defined as follows. Definition 1. Greedy Embedding A greedy embedding is an embedding of a graph on a respective geometric space such that, greedy routing always succeeds. In other words, between every node pair u, v there is another node w adjacent to u, such that d(u, v) > d(w, v), where d(.) is the underlying metric on the geometric space. Dhandapani [3] showed that every planar triangulated graph can be drawn on the plane as a greedy embedding. They generalize the classical Schnyder drawing [4] leading to a family of planar drawings, then they show that there exists a greedy drawing in this set of drawings. In this work, we prove that every Schnyder drawing is a greedy embedding. We emphasize the use of a generalized definition of a greedy routing [5], on which our algorithm is based. 1 ar X iv :1 60 9. 04 17 3v 1 [ cs .C G ] 1 4 Se p 20 16