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.
Kleinberg's construction has found applications in the design of overlay routing networks for Distributed Hash Tables. Symphony @cite_13 is an adaptation of Kleinberg's construction in a single dimension. The idea is to place @math nodes in a virtual circle and to equip each node with @math out-going links. In the resulting network, the average path length of routes with distance function @math is @math hops. Note that unlike Kleinberg's network, the space here is virtual and so are the distances and the sense of routing. The same complexity was achieved with a slightly different Kleinberg-style construction by Aspnes al @cite_18 . In the same paper, it was also shown that any symmetric, randomized degree- @math network has @math routing complexity.
@cite_18 @cite_13
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We consider the problem of designing an overlay network and routing mechanism that permits finding resources efficiently in a peer-to-peer system. We argue that many existing approaches to this problem can be modeled as the construction of a random graph embedded in a metric space whose points represent resource identifiers, where the probability of a connection between two nodes depends only on the distance between them in the metric space. We study the performance of a peer-to-peer system where nodes are embedded at grid points in a simple metric space: a one-dimensional real line. We prove upper and lower bounds on the message complexity of locating particular resources in such a system, under a variety of assumptions about failures of either nodes or the connections between them. Our lower bounds in particular show that the use of inverse power-law distributions in routing, as suggested by Kleinberg [5], is close to optimal. We also give heuristics to efficiently maintain a network supporting efficient routing as nodes enter and leave the system. Finally, we give some experimental results that suggest promising directions for future work. Consider @math nodes connected by wires to make an n-dimensional binary cube. Suppose that initially the nodes contain one packet each addressed to distinct nodes of the cube. We show that there is a distributed randomized algorithm that can route every packet to its destination without two packets passing down the same wire at any one time, and finishes within time @math with overwhelming probability for all such routing requests. Each packet carries with it @math bits of bookkeeping information. No other communication among the nodes takes place.The algorithm offers the only scheme known for realizing arbitrary permutations in a sparse N node network in @math time and has evident applications in the design of general purpose parallel computers. We study approximate distributed solutions to the weighted all-pairs-shortest-paths (APSP) problem in the CONGEST model. We obtain the following results. @math A deterministic @math -approximation to APSP in @math rounds. This improves over the best previously known algorithm, by both derandomizing it and by reducing the running time by a @math factor. In many cases, routing schemes involve relabeling, i.e., assigning new names to nodes and require that these names are used in distance and routing queries. It is known that relabeling is necessary to achieve running times of @math . In the relabeling model, we obtain the following results. @math A randomized @math -approximation to APSP, for any integer @math , running in @math rounds, where @math is the hop diameter of the network. This algorithm simplifies the best previously known result and reduces its approximation ratio from @math to @math . Also, the new algorithm uses uses labels of asymptotically optimal size, namely @math bits. @math A randomized @math -approximation to APSP, for any integer @math , running in time @math and producing compact routing tables of size @math . The node lables consist of @math bits. This improves on the approximation ratio of @math for tables of that size achieved by the best previously known algorithm, which terminates faster, in @math rounds. We analyze decentralized routing in small-world networks that combine a wide variation in node degrees with a notion of spatial embedding. Specifically, we consider a variation of Kleinberg's augmented-lattice model (STOC 2000), where the number of long-range contacts for each node is drawn from a power-law distribution. This model is motivated by the experimental observation that many "real-world" networks have power-law degrees. In such networks, the exponent α of the power law is typically between 2 and 3. We prove that, in our model, for this range of values, 2