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

Bartal, Yair; Filtser, Arnold; Neiman, Ofer

doi:10.1137/1.9781611974331.ch62

Cited by 8 publications

(13 citation statements)

References 24 publications

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“…Hence the greedy spanner for general graphs has size and lightness at least as good as in Theorem 1. In [BFN16] it was proved that for any parameter 0 < δ < 1 and any stretch parameter t = t(n), if every n-vertex weighted graph admits a t-spanner with at most m(n, t) edges and lightness at most l(n, t), then for every such graph there also exists a t/δ-spanner with at most m(n, t) edges and lightness at most 1 + δ · l(n, t). 2 Applying Theorem 4 again, we derive the following result.…”

Section: The Basic Optimality Proofmentioning

confidence: 99%

The Greedy Spanner is Existentially Optimal

Filtser

Solomon

2016

Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing

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The greedy spanner is arguably the simplest and most well-studied spanner construction. Experimental results demonstrate that it is at least as good as any other spanner construction, in terms of both the size and weight parameters. However, a rigorous proof for this statement has remained elusive.In this work we fill in the theoretical gap via a surprisingly simple observation: The greedy spanner is existentially optimal (or existentially near-optimal) for several important graph families. Focusing on the weight parameter, the state-of-the-art spanner constructions for both general graphs (due to Chechik and Wulff-Nilsen [SODA'16]) and doubling metrics (due to Gottlieb [FOCS'15]) are complex. Plugging our observation on these results, we conclude that the greedy spanner achieves near-optimal weight guarantees for both general graphs and doubling metrics, thus resolving two longstanding conjectures in the area.Further, we observe that approximate-greedy algorithms are existentially near-optimal as well. Consequently, we provide an O(n log n)-time construction of (1 + )-spanners for doubling metrics with constant lightness and degree. Our construction improves Gottlieb's construction, whose runtime is O(n log 2 n) and whose number of edges and degree are unbounded, and remarkably, it matches the state-of-the-art Euclidean result (due to Gudmundsson et al. [SICOMP'02]) in all the involved parameters (up to dependencies on and the dimension).

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Section: The Basic Optimality Proofmentioning

confidence: 99%

The Greedy Spanner is Existentially Optimal

Filtser

Solomon

2016

Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing

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“…Following our work, [BFN16] discovered a connection between terminal distortion and coarse partial distortion. First recall the notion of coarse partial distortion, introduced in [KSW09, ABC + 05].…”

Section: Subsequent Workmentioning

confidence: 68%

Terminal embeddings

Elkin

Filtser

Neiman

2017

Theoretical Computer Science

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In this paper we study terminal embeddings, in which one is given a finite metric (X, d X ) (or a graph G = (V, E)) and a subset K ⊆ X of its points are designated as terminals. The objective is to embed the metric into a normed space, while approximately preserving all distances among pairs that contain a terminal. We devise such embeddings in various settings, and conclude that even though we have to preserve ≈ |K| · |X| pairs, the distortion depends only on |K|, rather than on |X|.We also strengthen this notion, and consider embeddings that approximately preserve the distances between all pairs, but provide improved distortion for pairs containing a terminal. Surprisingly, we show that such embeddings exist in many settings, and have optimal distortion bounds both with respect to X × X and with respect to K × X.Moreover, our embeddings have implications to the areas of Approximation and Online Algorithms. In particular, [ALN08] devised anÕ( √ log r)-approximation algorithm for sparsestcut instances with r demands. Building on their framework, we provide anÕ( log |K|)approximation for sparsest-cut instances in which each demand is incident on one of the vertices of K (aka, terminals). Since |K| ≤ r, our bound generalizes that of [ALN08]. * A preliminary version of this paper appeared in APPROX'15. †

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“…Moreover, builing ont the embedding of [5] this method implies a deterministic embedding in O(log n) dimension and distortion and constant q moments, for all fixed q < ∞. This further provides optimal prioritized embeddings [23,17].…”

Section: Introductionmentioning

confidence: 95%

Advances in Metric Ramsey Theory and its Applications

Bartal

2021

Preprint

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Metric Ramsey theory is concerned with finding large well-structured subsets of more complex metric spaces. For finite metric spaces this problem was first studies by Bourgain, Figiel and Milman [21], and studied further in depth by Bartal et. al [10]. In this paper we provide deterministic constructions for this problem via a novel notion of metric Ramsey decomposition. This method yields several more applications, reflecting on some basic results in metric embedding theory.The applications include various results in metric Ramsey theory including the first deterministic construction yielding Ramsey theorems with tight bounds, a well as stronger theorems and properties, implying appropriate distance oracle applications.In addition, this decomposition provides the first deterministic Bourgain-type embedding of finite metric spaces into Euclidean space, and an optimal multi-embedding into ultrametrics, thus improving its applications in approximation and online algorithms.The decomposition presented here, the techniques and its consequences have already been used in recent research in the field of metric embedding for various applications.* This is paper is still in stages of preparation, this version is not intended for distribution. A preliminary version of this article was written by the author in 2006, and was presented in the 2007 ICMS Workshop on Geometry and Algorithms [14]. The basic result on constructive metric Ramsey decomposition and metric Ramsey theorem has also appeared in the author's lectures notes, e.g. [15].

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

Cited by 8 publications

References 24 publications

The Greedy Spanner is Existentially Optimal

The Greedy Spanner is Existentially Optimal

Terminal embeddings

Advances in Metric Ramsey Theory and its Applications

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