Greedy Routing in Tree-Decomposed Graphs

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 ${\mathcal D}$ such that greedy routing in the augmented graph $(G,{\mathcal D})$ performs in O(klog2n) expected number of steps. We also prove that if G has chordality ≤ k, then the tree-decomposition-based distribution ${\mathcal D}$ insures that greedy routing in $(G,{\mathcal D})$ 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 $(G,{\mathcal D})$ performs in O(log2n) expected number of steps.