Metric Embeddings with Relaxed Guarantees

Abraham, Ittai; Bartal, Yair; Kleinberg, Jon; Chan, T.-H. Hubert; Neiman, Ofer; Dhamdhere, Kedar; Slivkins, Aleksandrs; Gupta, Anupam

doi:10.1109/sfcs.2005.51

Cited by 48 publications

(81 citation statements)

References 35 publications

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“…Applying a result of Balcan et al [6] we show that the resulting clustering obtained is indeed the optimal one. Albeit being very different, our approach resembles, in spirit, the work of Bartal [7], Abraham et al [1] and Räcke [22] in the sense that we reduce the problem of retrieving an optimal solution from a general instance to a tree-like instance (where it is poly-time solvable).…”

Section: Resultsmentioning

confidence: 99%

Center-based clustering under perturbation stability

Awasthi

Blum

Sheffet

2012

Information Processing Letters

157

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Clustering under most popular objective functions is NP-hard, even to approximate well, and so unlikely to be efficiently solvable in the worst case. Recently, Bilu and Linial [11] suggested an approach aimed at bypassing this computational barrier by using properties of instances one might hope to hold in practice. In particular, they argue that instances in practice should be stable to small perturbations in the metric space and give an efficient algorithm for clustering instances of the Max-Cut problem that are stable to perturbations of size O(n 1/2 ). In addition, they conjecture that instances stable to as little as O(1) perturbations should be solvable in polynomial time. In this paper we prove that this conjecture is true for any center-based clustering objective (such as k-median, k-means, and k-center). Specifically, we show we can efficiently find the optimal clustering assuming only stability to factor-3 perturbations of the underlying metric in spaces without Steiner points, and stability to factor 2 + √ 3 perturbations for general metrics. In particular, we show for such instances that the popular Single-Linkage algorithm combined with dynamic programming will find the optimal clustering. We also present NP-hardness results under a weaker but related condition.

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Section: Resultsmentioning

confidence: 99%

Center-based clustering under perturbation stability

Awasthi

Blum

Sheffet

2012

Information Processing Letters

157

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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%

“…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. These results imply constant average distortion and O(q) bound on the q -distortion.…”

Section: Related Workmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

“…A natural relaxation of this demand is that the spanning tree approximates all pairwise distances on average. Formally, the distortion of the pair u, v ∈ V in T is defined as d T (u,v) d G (u,v) , and the average distortion is 1 ( n 2 ) {u,v}∈( V 2 )…”

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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Testing Properties of Sets of Points in Metric Spaces

Onak

2008

Automata, Languages and Programming

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Abstract. Given query access to a set of points in a metric space, we wish to quickly check if it has a specific property. More precisely, we wish to distinguish sets of points that have the property from those that need to have at least an ε fraction of points modified to achieve it. We show one-sided error testers that immediately follow from known characterizations of metric spaces. Among other things, we give testers for tree metrics and ultrametrics which are optimal among one-sided error testers. Our tester for embeddability into the line is optimal even among two-sided error testers, and runs in sublinear time. We complement our algorithms with several lower bounds. For instance, we present lower bounds for testing dimensionality reduction in the ℓ1 and ℓ∞ metrics, which improve upon lower bounds given by Krauthgamer and Sasson (SODA 2003). All our lower bounds are constructed by using a generic approach. We also look at the problem from a streaming perspective, and give a method for converting each of our property testers into a streaming tester.

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Metric Embeddings with Relaxed Guarantees

Abstract: We consider the problem of embedding finite metrics with slack: we seek to produce embeddings with small dimension and distortion while allowing a (small)

Cited by 48 publications

References 35 publications

Center-based clustering under perturbation stability

Center-based clustering under perturbation stability

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

Testing Properties of Sets of Points in Metric Spaces

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