The Power of Lookahead in Small-World Routing Networks

Topologies and routing protocols for peer-to-peer networks have witnessed a flurry of research activity recently. Almost all designs employ greedy routing in one form or the other. Randomized topologies that are currently under investigation fall into two categories: (a) small-world network topologies based upon Kleinberg’s construction [Kle00], and (b) randomized variants of deterministic topologies like hypercubes and Chord [SMK01]. Our results pertain to the performance of greedy routing with/without 1-lookahead in these network topologies. greedy routing permits a node to consider only its own sets of connections for making routing decisions. The idea of 1-lookahead is for a node to consider its neighbor’s neighbors as well to make better routing decisions. In practice, such information is available (almost) for free since pairs of nodes are connected by TCP connections which anyways exchange keep-alive messages. The picture that emerges from our investigations is quite interesting: Deterministic topologies like hypercubes and Chord have Θ(logn) links per node where n denotes the number of nodes; the diameter is Θ(logn); greedy routing is optimal as it routes along shortest paths; 1-lookahead offers no improvement. Randomized variants of both the networks, defined quite naturally, shrink the diameter to Θ(logn/ log logn) in expectation. greedy routing fails to find short routes, requiring Θ(logn) on average. Surprisingly, augmented with just 1-lookahead, routes are Θ(logn/ log logn) on average, which is asymptotically optimal. The same behavior is true for adaptations of Kleinberg’s construction, which we analyze for smaller values of out-degree per node as well. In the process, we bring out the connections between Kleinberg’s construction, randomized hypercubes and Chord. We also present a deterministic variant of hypercubes that can route in worst case Θ(logn/ log logn) hops with only Θ(logn/ log logn) links per node.