Close to optimal decentralized routing in long-range contact networks

In order to explain the ability of individuals to find short paths to route messages to an unknown destination, based only on their own local view of a social network (the small world phenomenon), Kleinberg [The small-world phenomenon: an algorithmic perspective, Proc. 32nd ACM Symp. on Theory of Computing, 2000, pp. 163-170] proposed a network model based on a d-dimensional lattice of size n augmented with k long-range directed links per node. Individuals behavior is modeled by a greedy algorithm that, given a source and destination, forwards a message to the neighbor of the current holder, which is the closest to the destination. This algorithm computes paths of expected length Θ(log2 n/k) between any pair of nodes. Other topologies have been proposed later on to improve greedy algorithm performance. But, Aspnes et al. [Fault-tolerant routing in peer-to-peer systems, in: Proc. of ACM 3st Symp. on Princ. of Distr. Comp. (PODC 2002), Vol. 31, 2002, pp. 223-232] shows that for a wide class of long-range link distributions, the expected length of the path computed by this algorithm is always Ω(log2 n/(k2 log log n)).We design and analyze a new decentralized routing algorithm, in which nodes consult their neighbors near by, before deciding to whom forward the message. Our algorithm uses similar amount of computational resources as Kleinberg's greedy algorithm: it is easy to implement, visits O(log2 n/log2 (1 +k)) nodes on expectation and requires only Θ(log2 n/log(1 +k)) bits of memory--note that [G.S. Manku, M. Naor, U. Wieder, Know thy neighbor's neighbor: the power of lookahead in randomized P2P networks, in: Proc. of 36th ACM STOC 2004, 2004, to appear], shows that any decentralized algorithm visits at least Ω(log2 n/k) on expectation. Our algorithm computes however a path of expected length O(log n (log log n)2/log2 (1 + k)) between any pair of nodes. Our algorithm might fit better some human social behaviors (such as web browsing) and may also have successful applications to peer-to-peer networks where the length of the path along which the files are downloaded, is a critical parameter of the network performance.

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