Optimal Degree Distributions for Uniform Small World Rings
msra(2010)
摘要
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.
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关键词
probability distribution,upper bound,degree distribution,cluster computing
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