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.
Cordasco al @cite_19 extend the result of Xu al @cite_16 by showing that routing with distance function @math in a uniform graph over @math nodes satisfies the inequality @math , where @math denotes the out-degree of each node, @math is the length of the longest path, and @math denotes the @math Fibonacci number. It is well-known that @math , where @math is the Golden ratio and @math denotes the integer closest to real number @math . It follows that @math . Cordasco al show that the inequality is strict if @math . For @math , they construct uniform graphs based upon Fibonacci numbers which achieve an optimal tradeoff between @math and @math .
@cite_19 @cite_16
1599788664 2031684765
We propose a family of novel schemes based on Chord retaining all positive aspects that made Chord a popular topology for routing in P2P networks. The schemes, based on the Fibonacci number system, allow to improve on the maximum average number of hops for lookups and the routing table size per node. We study a fundamental tradeoff issue in designing a distributed hash table (DHT) in peer-to-peer (P2P) networks: the size of the routing table versus the network diameter. Observing that existing DHT schemes have either 1) a routing table size and network diameter both of O(log sub 2 n), or 2) a routing table of size d and network diameter of O(n sup 1 d ), S. (2001) asked whether this represents the best asymptotic "state-efficiency" tradeoffs. We show that some straightforward routing algorithms achieve better asymptotic tradeoffs. However, such algorithms all cause severe congestion on certain network nodes, which is undesirable in a P2P network. We rigorously define the notion of "congestion" and conjecture that the above tradeoffs are asymptotically optimal for a congestion-free network. The answer to this conjecture is negative in the strict sense. However, it becomes positive if the routing algorithm is required to eliminate congestion in a "natural" way by being uniform. We also prove that the tradeoffs are asymptotically optimal for uniform algorithms. Furthermore, for uniform algorithms, we find that the routing table size of O(log sub 2 n) is a magic threshold point that separates two different "state-efficiency" regions. Our third result is to study the exact (instead of asymptotic) optimal tradeoffs for uniform algorithms. We propose a new routing algorithm that reduces the routing table size and the network diameter of Chord both by 21.4 without introducing any other protocol overhead, based on a novel number-theory technique. Our final result is to present Ulysses, a congestion-free nonuniform algorithm that achieves a better asymptotic "state-efficiency" tradeoff than existing schemes in the probabilistic sense, even under dynamic node joins leaves.