Greedy routing via embedding graphs onto semi-metric spaces

In this paper, we generalize the greedy routing concept to use semi-metric spaces. We prove that any connected n-vertex tree T admits a greedy embedding, onto an appropriate semi-metric space such that (1) each vertex v of the tree is represented by k=deg(v) virtual coordinates (where the numbers are from 1 to 2n−2); and (2) using an appropriate distance definition, the unique simple path between any two vertices in T is always a distance decreasing path between the two vertices. Applying the result to an arbitrary connected graph G, such a greedy embedding of any of its spanning tree T onto a semi-metric space becomes a greedy embedding of G onto the same semi-metric space. In particular, for 3-connected plane graphs, we prove that, for a 3-connected plane graph G, there is a greedy embedding of G such that: (1) the greedy embedding can be obtained in linear time; (2) each vertex can be represented by at most 3 virtual coordinates from 1 to 2n−2; and (3) the distance between two vertices can be computed in constant time. To our best knowledge, this is the first greedy routing algorithm for 3-connected plane graphs, albeit with non-standard notions of greedy embedding and greedy routing, such that: (1) it runs in linear time to compute the virtual coordinates for the vertices; and (2) the virtual coordinates are represented succinctly in O(logn) bits.