Greedy routing in tree-decomposed graphs

Abstract

We propose a new perspective on the small world phenomenon by considering arbitrary graphs augmented according to probabilistic distributions guided by tree-decompositions of the graphs. We show that, for any n-node graph G of treewidth ≤ k, there exists a tree-decomposition-based distribution D such that greedy routing in the augmented graph (G, D) performs in O(k log 2 n) expected number of steps. We also prove that if G has chordality ≤ k, then the tree-decomposition-based distribution D insures that greedy routing in (G, D) performs in O((k + log n) log n) expected number of steps. In particular, for any n-node graph G of chordality O(log n) (e.g., chordal graphs), greedy routing in the augmented graph (G, D) performs in O(log 2 n) expected number of steps. © Springer-Verlag Berlin Heidelberg 2005.