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Distance Estimation and Object Location via Rings of Neighbors
 In 24 th Annual ACM Symposium on Principles of Distributed Computing (PODC
, 2005
"... We consider four problems on distance estimation and object location which share the common flavor of capturing global information via informative node labels: lowstretch routing schemes [47], distance labeling [24], searchable small worlds [30], and triangulationbased distance estimation [33]. Fo ..."
Abstract

Cited by 66 (5 self)
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We consider four problems on distance estimation and object location which share the common flavor of capturing global information via informative node labels: lowstretch routing schemes [47], distance labeling [24], searchable small worlds [30], and triangulationbased 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
Distributed Approaches to Triangulation and Embedding
, 2006
"... A number of recent papers in the networking community study the distance matrix defined by the nodetonode latencies in the Internet and, in particular, provide a number of quite successful distributed approaches that embed this distance into a lowdimensional Euclidean space. In such algorithms it ..."
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Cited by 30 (7 self)
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A number of recent papers in the networking community study the distance matrix defined by the nodetonode latencies in the Internet and, in particular, provide a number of quite successful distributed approaches that embed this distance into a lowdimensional Euclidean space. In such algorithms it is feasible to measure distances among only a linear or nearlinear 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 ɛrelaxed embeddings and triangulations to infinite metrics and arbitrary measures, and to improve on the approximate distance labeling scheme of Talwar [36].
EMBEDDING, DISTANCE ESTIMATION AND OBJECT LOCATION IN NETWORKS
, 2006
"... Concurrent with numerous theoretical results on metric embeddings, a growing body of research in the networking community has studied the distance matrix defined by nodetonode latencies in the Internet, resulting in a number of recent approaches that approximately embed this distance matrix into l ..."
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Cited by 2 (0 self)
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Concurrent with numerous theoretical results on metric embeddings, a growing body of research in the networking community has studied the distance matrix defined by nodetonode latencies in the Internet, resulting in a number of recent approaches that approximately embed this distance matrix into lowdimensional Euclidean space. A fundamental distinction between the theoretical approaches to embeddings and this recent Internetrelated work is that the latter operates under the additional constraint that it is only feasible to measure a linear number of node pairs, and typically in a highly structured way. Indeed, the most common framework here is a beaconbased approach: one randomly chooses a small number of nodes (’beacons’) in the network, and each node measures its distance to these beacons only. Moreover, beaconbased algorithms are also designed for the more basic problem of triangulation, in which one uses the triangle inequality to infer the distances that have not been measured. We give algorithms with provable performance guarantees for triangulation and embedding. We show that in addition to multiplicative error in the distances, performance guarantees for beaconbased algorithms typically must include a notion of ”slack ” – a certain fraction of all distances may be arbitrarily distorted. For arbitrary metrics, we give a beaconbased embedding algorithm that achieves constant distortion on a (1 − ɛ)fraction of distances; this provides some theoretical justification for the success of the recent