Skip graphs are a novel distributed data structure, based on skip lists, that provide the full functionality of a balanced tree in a distributed system where resources are stored in separate nodes that may fail at any time. They are designed for use in searching peer-to-peer systems, and by providing the ability to perform queries based on key ordering, they improve on existing search tools that provide only hash table functionality. Unlike skip lists or other tree data structures, skip graphs are highly resilient, tolerating a large fraction of failed nodes without losing connectivity. In addition, simple and straightforward algorithms can be used to construct a skip graph, insert new nodes into it, search it, and detect and repair errors in a skip graph introduced due to node failures.

This paper examines graph-theoretic properties of existing peer-to-peer architectures and proposes a new infrastructure based on optimal-diameter de Bruijn graphs. Since generalized de Bruijn graphs possess very short average routing distances and high resilience to node failure, they are well suited for structured peer-to-peer networks. Using the example of Chord, CAN, and de Bruijn, we first study routing performance, graph expansion, and clustering properties of each graph. We then examine bisection width, path overlap, and several other properties that affect routing and resilience of peer-to-peer networks. Having confirmed that de Bruijn graphs offer the best diameter and highest connectivity among the existing peer-to-peer structures, we offer a very simple incremental building process that preserves optimal properties of de Bruijn graphs under uniform user joins/departures. We call the combined peer-to-peer architecture

Several peer-to-peer networks are based upon randomized graph topologies that permit e#cient 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 #(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.

We analyze the properties of Small-World networks, where links are much more likely to connect “neighbor nodes ” than distant nodes. In particular, our analysis provides new results for Kleinberg’s Small-World model and its extensions. Kleinberg adds a number of directed long-range random links to an n × n lattice network (vertices as nodes of a grid, undirected edges between any two adjacent nodes). Links have a non-uniform distribution that favors arcs to close nodes over more distant ones. He shows that the following phenomenon occurs: between any two nodes a path with expected length O(log 2 n) can be found using a simple greedy algorithm which has no global knowledge of long-range links. We show that Kleinberg’s analysis is tight: his algorithm achieves θ(log 2 n) delivery time. Moreover, we show that the expected diameter of the graph is θ(log n), a log n factor

Routing topologies for distributed hashing in peer-to-peer networks are classified into two categories: deterministic and randomized. A general technique for constructing determin-istic routing topologies is presented. Using this technique, classical parallel interconnection networks can be adapted to handle the dynamic nature of participants in peer-to-peer networks. A unified picture of randomized routing topolo-gies is also presented. Two new protocols are described which improve average latency as a function of out-degree. One of the protocols can be shown to be optimal with high probability. Finally, routing networks for distributed hash-ing are revisited from a systems perspective and several open design problems are listed.

We consider the problem of designing a compact communication network that supports e#cient routing in an Euclidean plane. Our network design and routing scheme achieves 1+# stretch, logarithmic diameter, and constant out degree. This improves upon the best known result so far that requires a logarithmic out-degree. Furthermore, our scheme is asymptotically optimal in Euclidean metrics whose diameter is polynomial.

We investigate an approach for routing in p2p networks called neighbor-of-neighbor greedy. We show that this approach may reduce significantly the number of hops used, when routing in skip graphs and small worlds. Furthermore we show that a simple variation of Chord is degree optimal. Our algorithm is implemented on top of the conventional greedy algorithms, thus it maintains the good properties of greedy routing. Implementing it may only improve the performance of the system.

We propose a self-stabilizing and modeless peer-topeer(P2P) network construction and maintenance protocol, called the Ring Network(RN) protocol. The RN protocol, when started on a network of peers that are in an arbitrary state, will cause the network to converge to a structured P2P system with a directed ring topology, where peers are ordered according to their identifiers. Furthermore, the RN protocol maintains this structure in the face of peer joins and departures. The RN protocol is a distributed and asynchronous message-passing protocol, which fits well the autonomous behavior of peers in a P2P system. The RN protocol requires only the existence of a bootstrapping system which is weakly connected. Peers do not need to be informed of any global network state, nor do they need to assist in repairing the network topology when they leave. We provide a proof of the self-stabilizing nature of the protocol, and experimentally measure the average cost (in time and number of messages) to achieve convergence. 1.

Abstract. In his seminal work, Kleinberg showed how to augment meshes using random edges, so that they become navigable; that is, greedy routing computes paths of polylogarithmic expected length between any pairs of nodes. This yields the crucial question of determining wether such an augmentation is possible for all graphs. In this paper, we answer negatively to this question by exhibiting a threshold on the doubling dimension, above which an infinite family of graphs cannot be augmented to become navigable whatever the distribution of random edges is. Precisely, it was known that graphs of doubling dimension at most O(log log n) are navigable. We show that for doubling dimension ≫ log log n, an infinite family of graphs cannot be augmented to become navigable. Finally, we complete our result by studying the special case of square meshes, that we prove to always be augmentable to become navigable.

We consider small world graphs as de ned by Kleinberg (2000), i.e., graphs obtained from a d-dimensional mesh by adding links chosen at random according to the d-harmonic distribution, d 1. This model aims at giving formal support to the \six degrees of separation" between individuals experienced by Milgram (1967), and veri ed recently by Dodds, Muhamad, and Watts (2003). In particular, Kleinberg shows that greedy routing performs in (log number of steps in d-dimensional augmented meshes, with O(log n) bits of topological awareness per node, for any d 1. We show that giving O(log n) bits of topological awareness per node decreases the expected number of steps of greedy routing to O(log n) in d-dimensional augmented meshes. We also show that, independently of the amount of topological awareness given to the nodes, greedy routing performs in 31 n) expected number of steps. In particular, augmenting the topological awareness above this optimum of O(log n) bits would drastically decrease the performances of greedy routing. Moreover, our model demonstrates that the eciency of greedy routing is sensible to the \world's dimension", in the sense that high dimensional worlds enjoy faster greedy routing than low dimensional ones. This could not be observed in Kleinberg's model.