This paper introduces a lightweight, scalable and accurate framework, called Meridian, for performing node selection based on network location. The framework consists of an overlay network structured around multi-resolution rings, query routing with direct measurements, and gossip protocols for dissemination. We show how this framework can be used to address three commonly encountered problems, namely, closest node discovery, central leader election, and locating nodes that satisfy target latency constraints in large-scale distributed systems without having to compute absolute coordinates. We show analytically that the framework is scalable with logarithmic convergence when Internet latencies are modeled as a growthconstrained metric, a low-dimensional Euclidean metric, or a metric of low doubling dimension. Large scale simulations, based on latency measurements from 6.25 million node-pairs as well as an implementation deployed on PlanetLab show that the framework is accurate and effective.

We present a near linear time algorithm for constructing hierarchical nets in finite metric spaces with constant doubling dimension. This data-structure is then applied to obtain improved algorithms for the following problems: approximate nearest neighbor search, well-separated pair decomposition, spanner construction, compact representation scheme, doubling measure, and computation of the (approximate) Lipschitz constant of a function. In all cases, the running (preprocessing) time is near linear and the space being used is linear. 1

This paper studies compact routing schemes for networks with low doubling dimension. Two variants are explored, name-independent routing and labeled routing. The key results obtained for this model are the following. First, we provide the first name-independent solution. Specifically, we achieve constant stretch and polylogarithmic storage. Second, we obtain the first truly scale-free solutions, namely, the network’s aspect ratio is not a factor in the stretch. Scale-free schemes are given for three problem models: name-independent routing on graphs, labeled routing on metric spaces, and labeled routing on graphs. Third, we prove a lower bound requiring linear storage for stretch < 3 schemes. This has the important ramification of separating for the first time the name-independent problem model from the labeled model for these networks, since compact stretch-1+ε labeled schemes are known to be possible.

The study of complex networks has emerged over the past several years as a theme spanning many disciplines, ranging from mathematics and computer science to the social and biological sciences. A significant amount of recent work in this area has focused on the development of random graph models that capture some of the qualitative properties observed in large-scale network data; such models have the potential to help us reason, at a general level, about the ways in which real-world networks are organized. We survey one particular line of network research, concerned with small-world phenomena and decentralized search algorithms, that illustrates this style of analysis. We begin by describing a well-known experiment that provided the first empirical basis for the "six degrees of separation" phenomenon in social networks; we then discuss some probabilistic network models motivated by this work, illustrating how these models lead to novel algorithmic and graph-theoretic questions, and how they are supported by recent empirical studies of large social networks.

We study a novel separator property called k-path separable. Roughly speaking, a k-path separable graph can be recursively separated into smaller components by sequentially removing k shortest paths. Our main result is that every minor free weighted graph is k-path separable. We then show that k-path separable graphs can be used to solve several object location problems: (1) a small-worldization with an average poly-logarithmic number of hops; (2) an (1 + ε)approximate distance labeling scheme with O(log n) space labels; (3) a stretch-(1 + ε) compact routing scheme with tables of poly-logarithmic space; (4) an (1+ε)-approximate distance oracle with O(n log n) space and O(log n) query time. Our results generalizes to much wider classes of weighted graphs, namely to bounded-dimension isometric sparable graphs.

This paper concerns compact routing schemes with arbitrary node names. We present a compact name-independent routing scheme for unweighted networks with n nodes excluding a fixed minor. For any fixed minor, the scheme, constructible in polynomial time, has constant stretch factor and requires routing tables with poly-logarithmic number of bits at each node. For shortest-path labeled routing scheme in planar graphs, we prove an Ω(n ɛ) space lower bound for some constant ɛ>0. This lower bound holds even for bounded degree triangulations, and is optimal for polynomially weighted planar graphs (ɛ =1/2).

We investigate the problem of optimizing the routing performances of a virtual network by adding extra random links. Our asynchronous and distributed algorithm ensures, by adding a single extra link per node, that the resulting network is a navigable small world, i.e., in which greedy routing, using the distance in the original network, computes paths of polylogarithmic length between any pair of nodes with probability 1 − O(1/n). Previously known small world augmentation processes require the global knowledge of the network and centralized computations, which is unrealistic for large decentralized networks. Our algorithm, based on a careful multi-layer sampling of the nodes and the construction of a light overlay network, bypasses these limitations. For bounded growth graphs, i.e., graphs where, for any node u and any radius r the number of nodes within distance 2r from u is at most a constant times the number of nodes within distance r, our augmentation process proceeds with high probability in O(log n log D) communication rounds, with O(log n log D) messages of size O(log n) bits sent per node and requiring only O(log n log D) bit space in each node, where n is the number of nodes, and D the diameter. In particular, with the only knowledge of original distances, greedy routing computes,

We consider the problem of name-independent routing in doubling metrics. A doubling metric is a metric space whose doubling dimension is a constant, where the doubling dimension of a metric space is the least value α such that any ball of radius r can be covered by at most 2 α balls of radius r/2. Given any δ> 0 and a weighted undirected network G whose shortest path metric d is a doubling metric with doubling dimension α, we present a name-independent routing scheme for G with (9+δ)-stretch, (2+ 1 δ)O(α) (log ∆) 2 (log n)bit routing information at each node, and packet headers of size O(log n), where ∆ is the ratio of the largest to the smallest shortest path distance in G. In addition, we prove that for any ǫ ∈ (0, 8), there is a doubling metric network G with n nodes, doubling dimension α ≤ 6 − log ǫ, and ∆ = O(2 1/ǫ n) such that any name-independent routing scheme on G with routing information at each node of size o(n (ǫ/60)2)-bits has stretch larger than 9 − ǫ. Therefore assuming that ∆ is bounded by a polynomial on n, our algorithm basically achieves optimal stretch for name-independent routing in doubling metrics with packet header size and routing information at each node both bounded by a polylogarithmic function of n.

One of the fundamental trade-offs in compact routing schemes is between the space used to store the routing table on each node and the stretch factor of the routing scheme – the maximum ratio over all pairs between the cost of the route induced by the scheme and the cost of a minimum cost path between the same pair. All previous routing schemes required storage that is dependent on the diameter of the network. We present a new scale-free routing scheme, whose storage and header sizes are independent of the aspect ratio of the network. Our scheme is based on a decomposition into sparse and dense neighborhoods. Given an undirected network with arbitrary weights and n arbitrary node names, for any integer k ≥ 1 we present the first scale-free routing scheme with asymptotically optimal space-stretch tradeoff that does not require edge weights to be polynomially bounded. The scheme uses � O(n 1/k) space routing table at each node, and routes along paths of asymptotically optimal linear stretch O(k).

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.