Greedy Routing on Virtual Raw Anchor Coordinate (VRAC) System

Geographic routing is an appealing routing strategy that uses the location information of the nodes to route the data. This technique uses only local information of the communication graph topology and does not require computational effort to build routing table or equivalent data structures. A particularly efficient implementation of this paradigm is greedy routing, where along the data path the nodes forward the data to a neighboring node that is closer to the destination. The decreasing distance to the destination implies the success of the routing scheme. A related problem is to consider an abstract graph and decide whether there exists an embedding of the graph in a metric space, called a greedy embedding, such that greedy routing guarantees the delivery of the data. A common approach to assign geographic coordinates is to measure distances (for instance the distances between neighboring nodes) and compute(virtual) coordinates. The rationale of the Virtual raw Anchor Coordinate System(VRAC) is to use the (raw) measured distances as coordinates in order to avoid further computations. More precisely, each node needs to measure three distances. In this paper, we investigate the existence of greedy routing in the VRAC coordinate system using a metric free characterization of greedy paths that is more general than in previous works. We show that if the graph is saturated (see definition in the text) then the greedy algorithm guarantees delivery. Interestingly, the approach of greediness here applies to Schnyder drawings of planar triangulations. Indeed, by choosing the measured distances appropriately Schnyder drawings of planar triangulations are always saturated and hence our greedy routing algorithm succeeds. The VRAC coordinates have conditions to satisfy to make greedy routing successful. These conditions can be inferred from geometric considerations. However, we formulate these conditions in an abstract way in order to avoid geometric considerations and in order to make possible further derivation of virtual VRAC coordinate systems, i.e. using only the abstract graph description. In particular using only local information would lead to distributed algorithm.