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.
Distributed hash table (DHT) systems are an important class of peer-to-peer routing infrastructures. They enable scalable wide-area storage and retrieval of information, and will support the rapid development of a wide variety of Internet-scale applications ranging from naming systems and file systems to application-layer multicast. DHT systems essentially build an overlay network, but a path on the overlay between any two nodes can be significantly different from the unicast path between those two nodes on the underlying network. As such, the lookup latency in these systems can be quite high and can adversely impact the performance of applications built on top of such systems.In this paper, we discuss a random sampling technique that incrementally improves lookup latency in DHT systems. Our sampling can be implemented using information gleaned from lookups traversing the overlay network. For this reason, we call our approach lookup-parasitic random sampling (LPRS). LPRS is fast, incurs little network overhead, and requires relatively few modifications to existing DHT systems.For idealized versions of DHT systems like Chord, Tapestry and Pastry, we analytically prove that LPRS can result in lookup latencies proportional to the average unicast latency of the network, provided the underlying physical topology has a power-law latency expansion. We then validate this analysis by implementing LPRS in the Chord simulator. Our simulations reveal that LPRS-Chord exhibits a qualitatively better latency scaling behavior relative to unmodified Chord.Finally, we provide evidence which suggests that the Internet router-level topology resembles power-law latency expansion. This finding implies that LPRS has significant practical applicability as a general latency reduction technique for many DHT systems. This finding is also of independent interest since it might inform the design of latency-sensitive topology models for the Internet. Scalable overlay networks such as Chord, CAN, Pastry, and Tapestry have recently emerged as flexible infrastructure for building large peer-to-peer systems. In practice, such systems have two disadvantages: They provide no control over where data is stored and no guarantee that routing paths remain within an administrative domain whenever possible. SkipNet is a scalable overlay network that provides controlled data placement and guaranteed routing locality by organizing data primarily by string names. SkipNet allows for both fine-grained and coarse-grained control over data placement: Content can be placed either on a pre-determined node or distributed uniformly across the nodes of a hierarchical naming sub-tree. An additional useful consequence of SkipNet's locality properties is that partition failures, in which an entire organization disconnects from the rest of the system, can result in two disjoint, but well-connected overlay networks. Several peer-to-peer networks are based upon randomized graph topologies that permit efficient greedy routing, e. g., randomized hypercubes, randomized Chord, skip-graphs and constructions based upon small-world percolation networks. In each of these networks, a node has out-degree Θ(log n), where n denotes the total number of nodes, and greedy routing is known to take O(log n) hops on average. We establish lower-bounds for greedy routing for these networks, and analyze Neighbor-of-Neighbor (NoN)- greedy routing. The idea behind NoN, as the name suggests, is to take a neighbor's neighbors into account for making better routing decisions.The following picture emerges: Deterministic routing networks like hypercubes and Chord have diameter Θ(log n) and greedy routing is optimal. Randomized routing networks like randomized hypercubes, randomized Chord, and constructions based on small-world percolation networks, have diameter Θ(log n log log n) with high probability. The expected diameter of Skip graphs is also Θ(log n log log n). In all of these networks, greedy routing fails to find short routes, requiring Ω(log n) hops with high probability. Surprisingly, the NoN- greedy routing algorithm is able to diminish route-lengths to Θ(log n log log n) hops, which is asymptotically optimal. The various proposed DHT routing algorithms embody several different underlying routing geometries. These geometries include hypercubes, rings, tree-like structures, and butterfly networks. In this paper we focus on how these basic geometric approaches affect the resilience and proximity properties of DHTs. One factor that distinguishes these geometries is the degree of flexibility they provide in the selection of neighbors and routes. Flexibility is an important factor in achieving good static resilience and effective proximity neighbor and route selection. Our basic finding is that, despite our initial preference for more complex geometries, the ring geometry allows the greatest flexibility, and hence achieves the best resilience and proximity performance. We present Symphony, a novel protocol for maintaining distributed hash tables in a wide area network. The key idea is to arrange all participants along a ring and equip them with long distance contacts drawn from a family of harmonic distributions. Through simulation, we demonstrate that our construction is scalable, flexible, stable in the presence of frequent updates and offers small average latency with only a handful of long distance links per node. The cost of updates when hosts join and leave is small. 