A number of recent papers in the networking community study the distance matrix defined by the node-to-node latencies in the Internet and, in particular, provide a number of quite successful distributed approaches that embed this distance into a low-dimensional Euclidean space. In such algorithms it is feasible to measure distances among only a linear or near-linear number of node pairs; the rest of the distances are simply not available. Moreover, for applications it is desirable to spread the load evenly among the participating nodes. Indeed, several recent studies use this ’fully distributed ’ approach and achieve, empirically, a low distortion for all but a small fraction of node pairs. This is concurrent with the large body of theoretical work on metric embeddings, but there is a fundamental distinction: in the theoretical approaches to metric embeddings, full and centralized access to the distance matrix is assumed and heavily used. In this paper we present the first fully distributed embedding algorithm with provable distortion guarantees for doubling metrics (which have been proposed as a reasonable abstraction of Internet latencies), thus providing some insight into the empirical success of the recent Vivaldi algorithm [7]. The main ingredient of our embedding algorithm is an improved fully distributed algorithm for a more basic problem of triangulation, where the triangle inequality is used to infer the distances that have not been measured; this problem received a considerable attention in the networking community, and has also been studied theoretically in [19]. We use our techniques to extend ɛ-relaxed embeddings and triangulations to infinite metrics and arbitrary measures, and to improve on the approximate distance labeling scheme of Talwar [36]. 1

Random peer selection is used by numerous P2P applications; examples include application-level multicast, unstructured file sharing, and network location mapping. In most of these applications, support for a heterogeneous capacity distribution among nodes is desirable: in other words, nodes with higher capacity should be selected proportionally more often. Random peer selection can be performed over both structured and unstructured graphs. This paper compares these two basic approaches using a candidate example from each approach. For unstructured heterogeneous random peer selection, we use Swaplinks, from our previous work. For the structured approach, we use the Bamboo DHT adapted to heterogeneous selection using our extensions to the item-balancing technique by Karger and Ruhl. Testing the two approaches over graphs of 1000 nodes and a range of network churn levels and heterogeneity distributions, we show that Swaplinks is the superior random selection approach: (i) Swaplinks enables more accurate random selection than does the structured approach in the presence of churn, and (ii) The structured approach is sensitive to a number of hard-to-set tuning knobs that affect performance, whereas Swaplinks is essentially free of such knobs. 1