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Near-optimal distortion bounds for embedding doubling spaces into L₁

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Anastasios SidiropoulosAuthors Info & Claims

STOC '11: Proceedings of the forty-third annual ACM symposium on Theory of computing

Pages 765 - 772

https://doi.org/10.1145/1993636.1993737

Published: 06 June 2011 Publication History

Abstract

We exhibit an infinite doubling metric space (X,d) such that for any non-expansive f : X -> L₁, there exists a pair x,y ∈ X with d(x,y) arbitrarily large, and such that |f(x)-f(y)\|₁/d(x,y) ≲ √log log d(x,y)}/(log d(x,y)).

As a consequence, we show that there are n-point doubling metrics which require distortion Ω(√{log n/(log log n)}) into L₁, matching the upper bound of [Gupta-Krauthgamer-Lee, FOCS'03] up to a factor of O(√log log n). The best previous lower bound for doubling spaces, due to [Cheeger-Kleiner-Naor, FOCS'09] was of the form (log n)^δ for some small, unspecified value of δ > 0.

Furthermore, this gives a nearly optimal integrality gap for a weak version of the SDP for the general Sparsest Cut Problem. The weak SDP suffices for all known rounding algorithms, and the best previous gap was of the order (log n)^1/4/(log log n) [Lee-Moharrami, STOC'10].

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Cited By

Naor AYoung R(2018)Vertical perimeter versus horizontal perimeterAnnals of Mathematics10.4007/annals.2018.188.1.4188:1(171-279)Online publication date: 1-Jul-2018
https://doi.org/10.4007/annals.2018.188.1.4
Naor AYoung RHatami HMcKenzie PKing V(2017)The integrality gap of the Goemans-Linial SDP relaxation for sparsest cut is at least a constant multiple of √log nProceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing10.1145/3055399.3055413(564-575)Online publication date: 19-Jun-2017
https://dl.acm.org/doi/10.1145/3055399.3055413
Bartal YGottlieb LNeiman OCheng SDevillers O(2014)On the Impossibility of Dimension Reduction for Doubling Subsets of ℓpProceedings of the thirtieth annual symposium on Computational geometry10.1145/2582112.2582170(60-66)Online publication date: 8-Jun-2014
https://dl.acm.org/doi/10.1145/2582112.2582170
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Index Terms

Near-optimal distortion bounds for embedding doubling spaces into L₁
1. Theory of computation
  1. Design and analysis of algorithms

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cover image ACM Conferences

STOC '11: Proceedings of the forty-third annual ACM symposium on Theory of computing

June 2011

840 pages

ISBN:9781450306911

DOI:10.1145/1993636

General Chair:
Lance Fortnow
Northwestern University
,
Program Chair:
Salil Vadhan
Harvard University

Copyright © 2011 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 06 June 2011

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STOC'11: Symposium on Theory of Computing

June 6 - 8, 2011

California, San Jose, USA

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STOC '11 Paper Acceptance Rate 84 of 304 submissions, 28%;

Overall Acceptance Rate 1,469 of 4,586 submissions, 32%

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Cited By

Naor AYoung R(2018)Vertical perimeter versus horizontal perimeterAnnals of Mathematics10.4007/annals.2018.188.1.4188:1(171-279)Online publication date: 1-Jul-2018
https://doi.org/10.4007/annals.2018.188.1.4
Naor AYoung RHatami HMcKenzie PKing V(2017)The integrality gap of the Goemans-Linial SDP relaxation for sparsest cut is at least a constant multiple of √log nProceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing10.1145/3055399.3055413(564-575)Online publication date: 19-Jun-2017
https://dl.acm.org/doi/10.1145/3055399.3055413
Bartal YGottlieb LNeiman OCheng SDevillers O(2014)On the Impossibility of Dimension Reduction for Doubling Subsets of ℓpProceedings of the thirtieth annual symposium on Computational geometry10.1145/2582112.2582170(60-66)Online publication date: 8-Jun-2014
https://dl.acm.org/doi/10.1145/2582112.2582170
Neiman O(2014)Low Dimensional Embeddings of Doubling MetricsTheory of Computing Systems10.1007/s00224-014-9567-358:1(133-152)Online publication date: 5-Sep-2014
https://doi.org/10.1007/s00224-014-9567-3
Neiman O(2014)Low Dimensional Embeddings of Doubling MetricsApproximation and Online Algorithms10.1007/978-3-319-08001-7_2(12-23)Online publication date: 2014
https://doi.org/10.1007/978-3-319-08001-7_2
Gupta ATalwar KWitmer DBoneh DRoughgarden TFeigenbaum J(2013)Sparsest cut on bounded treewidth graphsProceedings of the forty-fifth annual ACM symposium on Theory of Computing10.1145/2488608.2488644(281-290)Online publication date: 1-Jun-2013
https://dl.acm.org/doi/10.1145/2488608.2488644
Arora SGe RSinop A(2013)Towards a Better Approximation for Sparsest Cut?Proceedings of the 2013 IEEE 54th Annual Symposium on Foundations of Computer Science10.1109/FOCS.2013.37(270-279)Online publication date: 26-Oct-2013
https://dl.acm.org/doi/10.1109/FOCS.2013.37

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