On Succinctness of Geometric Greedy Routing in Euclidean Plane

Geometric greedy routing uses (virtual) geographic point locations as addresses for the purposes of routing, while routing decisions are made by each node individually based only on local information. It requires that the network topology is greedily embedded in a metric space through assigning virtual coordinates to the nodes in the network. Recent research pointed out that such scheme is significantly more efficient than other traditional routing schemes only if the virtual coordinates are succinctly represented (i.e., using O(poly(log n)) bits per coordinate) in the greedy embedding, an issue that has been overlooked for a long time. In this paper we prove that such succinctness is not achievable in Euclidean plane, assuming that Cartesian or polar coordinate system is used, the network topology is a planar graph, and the planar embedding is preserved during the process of greedy embedding. Indeed, we provide a counter example to show that in this situation, some coordinates must take ¿(n) bits to represent.