Bandwidth and low dimensional embedding

Yair Bartal; Douglas E. Carroll; Adam Meyerson; Ofer Neiman

doi:10.1007/978-3-642-22935-0_5

Bandwidth and low dimensional embedding

Yair Bartal, Douglas E. Carroll, Adam Meyerson, Ofer Neiman

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

We design an algorithm to embed graph metrics into ℓ_p with dimension and distortion both dependent only upon the bandwidth of the graph. In particular we show that any graph of bandwidth k embeds with distortion polynomial in k into O(log k) dimensional ℓ_p, 1 ≤ p ≤ ∞. Prior to our result the only known embedding with distortion independent of n was into high dimensional ℓ₁ and had distortion exponential in k. Our low dimensional embedding is based on a general method for reducing dimension in an ℓ_p embedding, satisfying certain conditions, to the intrinsic dimension of the point set, without substantially increasing the distortion. As we observe that the family of graphs with bounded bandwidth are doubling, our result can be viewed as a positive answer to a conjecture of Assouad [2], limited to this family. We also study an extension to graphs of bounded tree-bandwidth.

Original language	English
Title of host publication	Approximation, Randomization, and Combinatorial Optimization
Subtitle of host publication	Algorithms and Techniques - 14th International Workshop, APPROX 2011 and 15th International Workshop, RANDOM 2011, Proceedings
Pages	50-61
Number of pages	12
DOIs	https://doi.org/10.1007/978-3-642-22935-0_5
State	Published - 8 Sep 2011
Externally published	Yes
Event	14th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2011 and the 15th International Workshop on Randomization and Computation, RANDOM 2011 - Princeton, NJ, United States Duration: 17 Aug 2011 → 19 Aug 2011

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	6845 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	14th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2011 and the 15th International Workshop on Randomization and Computation, RANDOM 2011
Country/Territory	United States
City	Princeton, NJ
Period	17/08/11 → 19/08/11

ASJC Scopus subject areas

Theoretical Computer Science
General Computer Science

Access to Document

10.1007/978-3-642-22935-0_5

Cite this

Bartal, Y., Carroll, D. E., Meyerson, A., & Neiman, O. (2011). Bandwidth and low dimensional embedding. In Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 14th International Workshop, APPROX 2011 and 15th International Workshop, RANDOM 2011, Proceedings (pp. 50-61). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6845 LNCS). https://doi.org/10.1007/978-3-642-22935-0_5

Bartal, Yair ; Carroll, Douglas E. ; Meyerson, Adam et al. / Bandwidth and low dimensional embedding. Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 14th International Workshop, APPROX 2011 and 15th International Workshop, RANDOM 2011, Proceedings. 2011. pp. 50-61 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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title = "Bandwidth and low dimensional embedding",

abstract = "We design an algorithm to embed graph metrics into ℓp with dimension and distortion both dependent only upon the bandwidth of the graph. In particular we show that any graph of bandwidth k embeds with distortion polynomial in k into O(log k) dimensional ℓp, 1 ≤ p ≤ ∞. Prior to our result the only known embedding with distortion independent of n was into high dimensional ℓ1 and had distortion exponential in k. Our low dimensional embedding is based on a general method for reducing dimension in an ℓp embedding, satisfying certain conditions, to the intrinsic dimension of the point set, without substantially increasing the distortion. As we observe that the family of graphs with bounded bandwidth are doubling, our result can be viewed as a positive answer to a conjecture of Assouad [2], limited to this family. We also study an extension to graphs of bounded tree-bandwidth.",

author = "Yair Bartal and Carroll, {Douglas E.} and Adam Meyerson and Ofer Neiman",

note = "Funding Information: E-mail addresses: [email protected] (Y. Bartal), [email protected] (D.E. Carroll), [email protected] (A. Meyerson), [email protected] (O. Neiman). 1 The work was done in part while the author was at the Center for the Mathematics of Information, Caltech, and the Institute for Pure and Applied Mathematics, UCLA. Supported in part by a grant from the Israeli Science Foundation (195/02) and in part by a grant from the National Science Foundation (NSF CCF-065253). 2 Research done while a student at UCLA. 3 Research supported by the National Science Foundation under Grant No. CCF-106540. 4 Supported by ISF grant No. 523/12 and by the European Union{\textquoteright}s Seventh Framework Programme (FP7/2007-2013) under grant agreement No. 303809.; 14th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2011 and the 15th International Workshop on Randomization and Computation, RANDOM 2011 ; Conference date: 17-08-2011 Through 19-08-2011",

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language = "English",

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Bartal, Y, Carroll, DE, Meyerson, A & Neiman, O 2011, Bandwidth and low dimensional embedding. in Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 14th International Workshop, APPROX 2011 and 15th International Workshop, RANDOM 2011, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 6845 LNCS, pp. 50-61, 14th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2011 and the 15th International Workshop on Randomization and Computation, RANDOM 2011, Princeton, NJ, United States, 17/08/11. https://doi.org/10.1007/978-3-642-22935-0_5

Bandwidth and low dimensional embedding. / Bartal, Yair; Carroll, Douglas E.; Meyerson, Adam et al.
Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 14th International Workshop, APPROX 2011 and 15th International Workshop, RANDOM 2011, Proceedings. 2011. p. 50-61 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6845 LNCS).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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N1 - Funding Information: E-mail addresses: [email protected] (Y. Bartal), [email protected] (D.E. Carroll), [email protected] (A. Meyerson), [email protected] (O. Neiman). 1 The work was done in part while the author was at the Center for the Mathematics of Information, Caltech, and the Institute for Pure and Applied Mathematics, UCLA. Supported in part by a grant from the Israeli Science Foundation (195/02) and in part by a grant from the National Science Foundation (NSF CCF-065253). 2 Research done while a student at UCLA. 3 Research supported by the National Science Foundation under Grant No. CCF-106540. 4 Supported by ISF grant No. 523/12 and by the European Union’s Seventh Framework Programme (FP7/2007-2013) under grant agreement No. 303809.

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N2 - We design an algorithm to embed graph metrics into ℓp with dimension and distortion both dependent only upon the bandwidth of the graph. In particular we show that any graph of bandwidth k embeds with distortion polynomial in k into O(log k) dimensional ℓp, 1 ≤ p ≤ ∞. Prior to our result the only known embedding with distortion independent of n was into high dimensional ℓ1 and had distortion exponential in k. Our low dimensional embedding is based on a general method for reducing dimension in an ℓp embedding, satisfying certain conditions, to the intrinsic dimension of the point set, without substantially increasing the distortion. As we observe that the family of graphs with bounded bandwidth are doubling, our result can be viewed as a positive answer to a conjecture of Assouad [2], limited to this family. We also study an extension to graphs of bounded tree-bandwidth.

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DO - 10.1007/978-3-642-22935-0_5

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SN - 9783642229343

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

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BT - Approximation, Randomization, and Combinatorial Optimization

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Bartal Y, Carroll DE, Meyerson A, Neiman O. Bandwidth and low dimensional embedding. In Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 14th International Workshop, APPROX 2011 and 15th International Workshop, RANDOM 2011, Proceedings. 2011. p. 50-61. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/978-3-642-22935-0_5

Bandwidth and low dimensional embedding

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