We consider four problems on distance estimation and object location which share the common flavor of capturing global information via informative node labels: low-stretch routing schemes [47], distance labeling [24], searchable small worlds [30], and triangulation-based distance estimation [33]. Focusing on metrics of low doubling dimension, we approach these problems with a common technique called rings of neighbors, which refers to a sparse distributed data structure that underlies all our constructions. Apart from improving the previously known bounds for these problems, our contributions include extending Kleinberg’s small world model to doubling metrics, and a short proof of the main result in Chan et al. [14]. Doubling dimension is a notion of dimensionality for general metrics that has recently become a useful algorithmic concept in the theoretical computer science literature. 1

Abstract—A large amount of algorithms has recently been designed for the Internet under the assumption that the distance defined by the round-trip delay (RTT) is a metric. Moreover, many of these algorithms (e.g., overlay network construction, routing scheme design, sparse spanner construction) rely on the assumption that the metric has bounded ball growth or bounded doubling dimension. This paper analyzes the validity of these assumptions and proposes a tractable model matching experimental observations. On the one hand, based on Skitter data collected by CAIDA and King matrices of Meridian and P2PSim projects, we verify that the ball growth of the Internet, as well as its doubling dimension, can actually be quite large. Nevertheless, we observed that the doubling dimension is much smaller when restricting the measures to balls of large enough radius. Moreover, by computing the number of balls of radius r required to cover balls of radius R> r, we observed that this number grows with R much slower than what is predicted by a large doubling dimension. On the other hand, based on data collected on the PlanetLab platform by the All-Sites-Pings project, we confirm that the triangle inequality does not hold for a significant fraction of the nodes. Nevertheless, we demonstrate that RTT measures satisfy a weak version of the triangle inequality: there exists a small constant ρ such that for any triple u, v, w, we have RTT(u,v) ≤ ρ ·max{RTT(u,w), RTT(w,v)}. (Smaller bounds on ρ can even be obtained when the triple u, v, w is skewed). We call inframetric a distance function satisfying this latter inequality. Inframetrics subsume standard metrics and ultrametrics. Based on inframetrics and on our observations concerning the doubling dimension, we propose an analytical model for Internet RTT latencies. This model is tuned by a small set of parameters concerning the violation of the triangle inequality and the geometrical dimension of the network. We demonstrate the tractability of our model by designing a simple and efficient compact routing scheme with low stretch. Precisely, the scheme has constant multiplicative stretch and logarithmic additive stretch. I.

Small-world properties, such as small-diameter and clustering, and the power-law property are widely recognized as common features of large-scale real-world networks. Recent studies also notice two important geographical factors which play a significant role, particularly in Internet related setting. These two are the distance-bias tendency (links tend to connect to closer nodes) and the property of bounded growth in localities. However, existing formal models for real-world complex networks usually don’t fully consider these geographical factors. We describe a flexible approach using a standard augmented graph model (e.g. Watt and Strogatz’s [33], and Kleinberg’s [20] models) and present important initial results on a refined model where we focus on the small-diameter characteristic and the above two geographical factors. We start with a general model where an arbitrary initial node-weighted graph H is augmented with additional random links specified by a generic ‘distribution rule ’ τ and the weights of nodes in H. We consider a refined setting where the initial graph H is associated with a growth-bounded metric, and τ has a distance-bias characteristic, specified by parameters as follows. The base graph H has neighborhood growth bounded from both below and above, specified by parameters β1, β2> 0. (These parameters can be thought of as the dimension of the graph, e.g. β1 = 2 and β2 = 3 for a graph modeling a setting with nodes in both 2D and 3D settings.) That is 2β1 Nu(2r) ≤ Nu(r) ≤ 2β2 where Nu(r) is the number of nodes v within metric distance r from u: d(u, v) ≤ r. When we add random links using distribution τ, this distribution is specified by parameter α> 0 such that the probability that 1 a link from u goes to v � = u is ∝ dα (u,v). We show which parameters produce a small-diameter graph and how the diameter changes depending on the relationship between the distance-bias parameter α and the two bounded growth parameters β1, β2> 0. In particular, for most connected base graphs, the diameter of our aug-

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Small-world properties are common in many large-scale real-world networks such as social networks, the Internet, or biological networks. In 2000, Kleinberg produced a new model for a striking aspect of acquaintance networks: that short chains can be found using limited local information only (e.g. a search based on a first-name basis). This model added random links to a 2D grid, such that the random links were more likely to connect closer nodes. We expand Kleinberg’s work to a more general study of graphs formed by adding a generic distribution (often non-uniform) of random links to a class of local-contact graphs. We study general rules and characteristics which produce small-world (and related) properties. We also use our observations to design practical networks. We focus on the abstract properties of the random link distributions which introduce short paths (typically, with length as a poly-log of network size) between the sites of the local-contact graph. By finding such general properties, we give a thorough analysis of Kleinberg’s small-world models. We also develop new techniques for analyzing

We present a new lookup algorithm for Viceroy, a peer-to-peer system, which outperforms the existing algorithm for the system, yet it is much simpler and easier to implement. We also present our fully-functional graphical simulation of Viceroy, and propose several other improved lookup (routing) algorithms for this network. We show experimental results to support our claims, and discuss the implications of a simplified algorithm for the system.

We design network topologies and routing strategies which optimize several measures simultaneously: low cost, small routing diameter, bounded degree and low congestion. This set of design issues is broader than traditional network design and hence, our work is useful and relevant to a set of traditional and emerging design problems. Surprisingly, a simple idea from the research on small-world models, inspires a fruitful approach and useful techniques here. Starting with a simple model we consider adding long links to an n ×n grid graph. Ideally, for a given budget to buy additional long links, we consider mechanisms for choosing links such that the routing diameter is small enough (poly-log of n) while the congestion ratio (between the most used link and the average one) is minimized, assuming uniform traffic between any two of the n 2 nodes. We show that by adding O(1) long links to each node we achieve an almost logarithmic routing diameter and maintain a near optimal trade-off between congestion ratio and average weight (of long links): W eight × CongestionRatio = O(n). Our results are comparable to the best similar network structures when the trade-off space we consider is reduced to those in the compared designs (with fewer trade-off factors). We also consider extensions of our results to more general settings. We propose two construction schemes: 1) a static (fixed link) design and 2) a dynamic (random link) design. While the former provides our best trade-off results, the later is more scalable, better suited for dynamic and fault-tolerance issues, and can be useful for wireless ad-hoc networks. 1

Abstract — We design network topologies and routing strategies which optimize several measures simultaneously: low cost, small routing diameter, bounded degree and low congestion. This set of design issues is broader than traditional network design and hence, our work is useful and relevant to a set of traditional and emerging design problems. Surprisingly, a simple idea from the research on small-world models, inspires a fruitful approach and useful techniques here. Starting with a simple model we consider adding long links to an n×n grid graph. Ideally, for a given budget to buy additional long links, we consider mechanisms for choosing links such that the routing diameter is small enough (poly-log of n) while the congestion ratio (between the most used link and the average one) is minimized, assuming uniform traffic between any two of the n 2 nodes. We show that by adding O(1) long links to each node we achieve an almost logarithmic routing diameter and maintain a near optimal trade-off between congestion ratio and average weight (of long links): Weight×CongestionRatio = O(n). Our results are comparable to the best similar network structures when the trade-off space we consider is reduced to those in the compared designs (with fewer trade-off factors). We also consider extensions of our results to more general settings. We propose two construction schemes: 1) a static (fixed link) design and 2) a dynamic (random link) design. While the former provides our best trade-off results, the later is more scalable, better suited for dynamic and fault-tolerance issues, and can be useful for wireless ad-hoc networks. I.