We study greedy routing over @math nodes placed in a ring, with the between two nodes defined to be the clockwise or the absolute distance between them along the ring. Such graphs arise in the context of modeling social networks and in routing networks for peer-to-peer systems. We construct the first network over @math nodes in which greedy routing takes @math hops in the worst-case, with @math out-going links per node. Our result has the first asymptotically optimal greedy routing complexity. Previous constructions required @math hops.
Papillon outperforms all of the above randomized constructions, using degree @math and achieving @math routing. It should be possible to randomize Papillon along similar principles to the Viceroy @cite_14 randomized construction of the butterfly network, though we do not pursue this direction here.
@cite_14
1970564778
We propose a family of constant-degree routing networks of logarithmic diameter, with the additional property that the addition or removal of a node to the network requires no global coordination, only a constant number of linkage changes in expectation, and a logarithmic number with high probability. Our randomized construction improves upon existing solutions, such as balanced search trees, by ensuring that the congestion of the network is always within a logarithmic factor of the optimum with high probability. Our construction derives from recent advances in the study of peer-to-peer lookup networks, where rapid changes require efficient and distributed maintenance, and where the lookup efficiency is impacted both by the lengths of paths to requested data and the presence or elimination of bottlenecks in the network.