Greedy Routing via Embedding Graphs onto Semi-metric Spaces

In this paper, we generalize the greedy routing concept to use semimetric spaces. We prove that any connected n-vertex graph G admits a greedy embedding onto an appropriate semi-metric space such that (1) each vertex v of the graph is represented by up to k virtual coordinates (where the numbers are from 1 to 2n - 1 and k <= deg(v)); and (2) using an appropriate distance definition, there is always a distance decreasing path between any two vertices in G. In particular, 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; and (2) each vertex can be represented by at most 3 virtual coordinates from 1 to 2n - 1. 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.