## Distributed Approaches to Triangulation and Embedding (2005)

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Venue: | In Proceedings 16th ACM-SIAM Symposium on Discrete Algorithms (SODA |

Citations: | 30 - 6 self |

### BibTeX

@INPROCEEDINGS{Slivkins05distributedapproaches,

author = {Aleksandrs Slivkins},

title = {Distributed Approaches to Triangulation and Embedding},

booktitle = {In Proceedings 16th ACM-SIAM Symposium on Discrete Algorithms (SODA},

year = {2005},

pages = {640--649}

}

### Years of Citing Articles

### OpenURL

### Abstract

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

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Citation Context ...e latencies in the Internet 1 (e.g. [9, 13, 8, 20, 37]) and, in particular, provide a number of quite successful distributed approaches that embed this distance into a low-dimensional Euclidean space =-=[30, 7, 32, 5, 25]-=-. 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. For instance, the Global Network P... |

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Citation Context ...on 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... |

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Citation Context ...te distance labeling scheme of Talwar [36]. 1 Introduction A number of recent papers in the networking community study the distance matrix defined by the node-tonode latencies in the Internet 1 (e.g. =-=[9, 13, 8, 20, 37]-=-) and, in particular, provide a number of quite successful distributed approaches that embed this distance into a low-dimensional Euclidean space [30, 7, 32, 5, 25]. In such algorithms it is feasible ... |

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Citation Context ...tudy here, and makes use of different sampling models and objective functions. Finally, our use of triangulation corresponds to the notion of triangle inequality bounds smoothing in distance geometry =-=[6]-=-, but our setting is different. Preliminaries Denote the uv-distance by duv. Let n be the number of nodes. Call a set of node pairs a ɛ-dense if for each node u it contains at least (1 − ɛ)n pairs uv.... |

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Citation Context ... be seen as an overlay network, our construction is similar in spirit to the overlay topologies constructed for locality-aware distributed hash tables and distributed nearest neighbor selection (e.g. =-=[34, 16, 15, 41]-=-), most notably to the topology constructed in [39]. However, our construction is quite different on the technical level since it is designed to yield provable guarantees on doubling metrics, and is t... |

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Citation Context ...e latencies in the Internet 1 (e.g. [9, 13, 8, 20, 37]) and, in particular, provide a number of quite successful distributed approaches that embed this distance into a low-dimensional Euclidean space =-=[30, 7, 32, 5, 25]-=-. 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. For instance, the Global Network P... |

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Citation Context ...g as the doubling dimension of the metric is small. Here the doubling dimension of a metric is defined as the smallest k such that every ball can be covered by 2 k other balls of half the radius (see =-=[2, 12]-=-); a metric with this property is also called 2 k -doubling or just doubling if k is a constant. Doubling metrics, which generalize the distance matrices of low-dimensional point sets in Lp, have been... |

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Citation Context ...te distance labeling scheme of Talwar [36]. 1 Introduction A number of recent papers in the networking community study the distance matrix defined by the node-tonode latencies in the Internet 1 (e.g. =-=[9, 13, 8, 20, 37]-=-) and, in particular, provide a number of quite successful distributed approaches that embed this distance into a low-dimensional Euclidean space [30, 7, 32, 5, 25]. In such algorithms it is feasible ... |

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Citation Context ...e latencies in the Internet 1 (e.g. [9, 13, 8, 20, 37]) and, in particular, provide a number of quite successful distributed approaches that embed this distance into a low-dimensional Euclidean space =-=[30, 7, 32, 5, 25]-=-. 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. For instance, the Global Network P... |

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Citation Context ...ubling metrics, which generalize the distance matrices of low-dimensional point sets in Lp, have been studied recently in the context of metric embeddings, nearest neighbour search and other problems =-=[12, 24, 36, 23, 22, 19, 28, 4]-=-. At the same time, several recent studies suggest the bounded growth rate of balls as a useful way to capture the structural properties of the Internet distance matrix (see e.g. [34, 8, 30, 31, 41, 1... |

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Citation Context ... δ = c/ log k and compute an (ɛ, δ)-triangulation using the algorithm from Section 2. Then the beacons measure distances to one another and broadcast them to the entire network using a uniform gossip =-=[33]-=-; in this phase each beacon broadcasts one message of size O(k), the total per-node load being at most O(k 2 log n). Upon receiving this information nodes update the bounds D + on their distances to b... |

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Citation Context ...ointed out by Kleinberg et al. [19], the above Internet-related setting gives rise to a new set of theoretical questions not covered by the existing rich theory of algorithmic metric embeddings (e.g. =-=[18]-=-) where a full access to the distance matrix is assumed and heavily used. In particular, in the embedding algorithm of Bourgain [3] and Linial et al. [26], the coordinates are formed by measuring the ... |

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Citation Context ...g as the doubling dimension of the metric is small. Here the doubling dimension of a metric is defined as the smallest k such that every ball can be covered by 2 k other balls of half the radius (see =-=[2, 12]-=-); a metric with this property is also called 2 k -doubling or just doubling if k is a constant. Doubling metrics, which generalize the distance matrices of low-dimensional point sets in Lp, have been... |

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Citation Context ... 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 Intro... |

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Citation Context ... communicate with (and, in particular, measure distance to) any other node given its ID. Secondly, every node can select a node ID independently and uniformly at random among all nodes in the network =-=[21, 38]-=-. Such operation induces load on multiple nodes; to account for it, let us assume that when each node selects one random node ID, this induces a per-node load of (log n). We call such networks uar-add... |

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