Triangulation and Embedding Using Small Sets of Beacons

Kleinberg, Jon; Slivkins, Aleksandrs; Wexler, Tom

doi:10.1109/focs.2004.70

Cited by 96 publications

(102 citation statements)

References 18 publications

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“…Partial and scaling embeddings 5 have been studied in several papers [21,1,3,12,4,5]. Some of the notable results are embedding arbitrary metrics into a distribution over trees [1] or into Euclidean space [3] with tight O(log(1/ )) scaling distortion.…”

Section: Related Workmentioning

confidence: 99%

“…Yet, as already mentioned above, when communication is required between specific pairs of nodes, the cost of routing through the MST may be extremely high, even when their real distance is small. However, in practice it is the average distortion, rather than the worst-case distortion, that is often used as a practical measure of quality, as has been a major motivation behind the initial work of [21,3,4]. As noted above, the MST still fails even in this relaxed measure.…”

Section: Introductionmentioning

confidence: 99%

“…This notion, first introduced in [21] 3 , requires that for every 0 < < 1, the distortion of all but an -fraction of the pairs is bounded by the appropriate function of . In [3] it was shown that one may obtain bounds on the average distortion, as well as on higher moments of the distortion function, from bounds on the scaling distortion.…”

Section: Introductionmentioning

confidence: 99%

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On Notions of Distortion and an Almost Minimum Spanning Tree with Constant Average Distortion

Bartal

Filtser

Neiman

2015

Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms

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Minimum Spanning Trees of weighted graphs are fundamental objects in numerous applications. In particular in distributed networks, the minimum spanning tree of the network is often used to route messages between network nodes. Unfortunately, while being most efficient in the total cost of connecting all nodes, minimum spanning trees fail miserably in the desired property of approximately preserving distances between pairs. While known lower bounds exclude the possibility of the worst case distortion of a tree being small, it was shown in [4] that there exists a spanning tree with constant average distortion. Yet, the weight of such a tree may be significantly larger than that of the MST. In this paper, we show that any weighted undirected graph admits a spanning tree whose weight is at most (1 + ρ) times that of the MST, providing constant average distortion O(1/ρ 2 ). 1 The constant average distortion bound is implied by a stronger property of scaling distortion, i.e., improved distortion for smaller fractions of the pairs. The result is achieved by first showing the existence of a low weight spanner with small prioritized distortion, a property allowing to prioritize the nodes whose associated distortions will be improved. We show that prioritized distortion is essentially equivalent to coarse scaling distortion via a general transformation, which has further implications and may be of independent interest. In particular, we obtain an embedding for arbitrary metrics into Eu-

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Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On Notions of Distortion and an Almost Minimum Spanning Tree with Constant Average Distortion

Bartal

Filtser

Neiman

2015

Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms

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“…Specifically, we consider the doubling dimension of X , denoted ddim(X ), which was introduced by [19] based on earlier work of [1,9], and has been since utilized in several algorithmic contexts, including networking, combinatorial optimization, and similarity search, see e.g. [23,46,31,5,21,11,10]. (A formal definition and typical examples appear in Section 2.)…”

Section: Introductionmentioning

confidence: 99%

Efficient Regression in Metric Spaces via Approximate Lipschitz Extension

Gottlieb

Kontorovich

Krauthgamer

2013

Similarity-Based Pattern Recognition

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We present a framework for performing efficient regression in general metric spaces. Roughly speaking, our regressor predicts the value at a new point by computing an approximate Lipschitz extension -the smoothest function consistent with the observed data -after performing structural risk minimization to avoid overfitting. We obtain finite-sample risk bounds with minimal structural and noise assumptions, and a natural runtime-precision tradeoff. The offline (learning) and online (prediction) stages can be solved by convex programming, but this naive approach has runtime complexity O(n 3 ), which is prohibitive for large datasets. We design instead a regression algorithm whose speed and generalization performance depend on the intrinsic dimension of the data, to which the algorithm adapts. While our main innovation is algorithmic, the statistical results may also be of independent interest.

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“…Since we cannot efficiently compute a metric on ∂G, we need a modified approach. Before we describe our method, however, let us briefly review the beacon-based scheme of [9].…”

Section: Beacon-based Embedding For δ-Hyperbolic Graphsmentioning

confidence: 99%

Metric embedding, hyperbolic space, and social networks

Verbeek

Suri

2016

Computational Geometry

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We consider the problem of embedding an undirected graph into hyperbolic space with minimum distortion. A fundamental problem in its own right, it has also drawn a great deal of interest from applied communities interested in empirical analysis of large-scale graphs. In this paper, we establish a connection between distortion and quasi-cyclicity of graphs, and use it to derive lower and upper bounds on metric distortion. Two particularly simple and natural graphs with large quasi-cyclicity are n-node cycles and n × n square lattices, and our lower bound shows that any hyperbolic-space embedding of these graphs incurs a multiplicative distortion of at least Ω(n/ log n). This is in sharp contrast to Euclidean space, where both of these graphs can be embedded with only constant multiplicative distortion. We also establish a relation between quasi-cyclicity and δ-hyperbolicity of a graph as a way to prove upper bounds on the distortion. Using this relation, we show that graphs with small quasi-cyclicity can be embedded into hyperbolic space with only constant additive distortion. Finally, we also present an efficient (linear-time) randomized algorithm for embedding a graph with small quasi-cyclicity into hyperbolic space, so that with high probability at least a (1 − ε) fraction of the node-pairs has only constant additive distortion. Our results also give a plausible theoretical explanation for why social networks have been observed to embed well into hyperbolic space: they tend to have small quasi-cyclicity.

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Triangulation and Embedding Using Small Sets of Beacons

Cited by 96 publications

References 18 publications

On Notions of Distortion and an Almost Minimum Spanning Tree with Constant Average Distortion

On Notions of Distortion and an Almost Minimum Spanning Tree with Constant Average Distortion

Efficient Regression in Metric Spaces via Approximate Lipschitz Extension

Metric embedding, hyperbolic space, and social networks

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