Greedy Routing on Virtual Raw Anchor Coordinate (VRAC) System

Abstract

Abstract-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.

Every Schnyder drawing is a greedy embedding

“…In [7], we introduced the concept of saturated graphs and show that there exists a local greedy routing algorithm with guaranteed delivery on such graphs. Definition of a saturated graph is purely combinatorial, but to construct those properties we use an underlying virtual coordinate system namely, virtual raw anchor coordinate (VRAC) system [7]. In the following, we describe the construction of a saturated graph, given a Schnyder drawing of a planar triangular graph.…”

“…A saturated graph is defined without a reference to an embedding. Our greedy routing algorithm in [7], uses the saturated graph property to prove the delivery guarantees. Following lemma constitutes a straight forward relationship between a Schnyder drawing and a saturated graph.…”

“…In [7], we devised a greedy routing algorithm which guarantees delivery, when the graph is saturated. Note that in [7], we do not use a metric to define the greedy path, instead use a generalized definition of a greedy path. Following lemma concludes the resulting connection between greedy embeddings and a Schnyder drawing of a planar triangular graph.…”

“…A planar graph is saturated if there exists exactly one edge in each sector s u 2i−1 , i = 1, 2, 3 for each node u. We present the following result from [7] on greedy routing on saturated graphs. Theorem 2.…”

“…In [7], we devised a greedy routing algorithm which guarantees delivery, when the graph is saturated. Note that in [7], we do not use a metric to define the greedy path, instead use a generalized definition of a greedy path.…”