Optimal Degree Distributions for Uniform Small World Rings

Motivated by Kleinberg's (2000) and subsequent work, we consider the performance of greedy routing on a directed ring of $n$ nodes augmented with long-range contacts. In this model, each node $u$ is given an additional $D_u$ edges, a degree chosen from a specified probability distribution. Each such edge from $u$ is linked to a random node at distance $r$ ahead in the ring with probability proportional to $1/r$, a "harmonic" distance distribution of contacts. Aspnes et al. (2002) have shown an $O(\log^2 n / \ell)$ bound on the expected length of greedy routes in the case when each node is assigned exactly $\ell$ contacts and, as a consequence of recent work by Dietzfelbinger and Woelfel (2009), this bound is known to be tight. In this paper, we generalize Aspnes' upper bound to show that any degree distribution with mean $\ell$ and maximum value $O(\log n)$ has greedy routes of expected length $O(\log^2n / \ell)$, implying that any harmonic ring in this family is asymptotically optimal. Furthermore, for a more general family of rings, we show that a fixed degree distribution is optimal. More precisely, if each random contact is chosen at distance $r$ with a probability that decreases with $r$, then among degree distributions with mean $\ell$, greedy routing time is smallest when every node is assigned $\floor{\ell}$ or $\ceiling{\ell}$ contacts.