Metric embedding via shortest path decompositions

Ittai Abraham; Anupam Gupta; Arnold Filtser; Ofer Neiman

doi:10.1145/3188745.3188808

Metric embedding via shortest path decompositions

Ittai Abraham
, Anupam Gupta
, Arnold Filtser
, Ofer Neiman

Dell
Carnegie Mellon University
Ben-Gurion University of the Negev

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

11 Scopus citations

Abstract

We study the problem of embedding weighted graphs of pathwidth k into ℓ_p spaces. Our main result is an O(k^min {1/p^{, 1}/2 ^})-distortion embedding. For p = 1, this is a super-exponential improvement over the best previous bound of Lee and Sidiropoulos. Our distortion bound is asymptotically tight for any fixed p > 1. Our result is obtained via a novel embedding technique that is based on low depth decompositions of a graph via shortest paths. The core new idea is that given a geodesic shortest path P, we can probabilistically embed all points into 2 dimensions with respect to P. For p > 2 our embedding also implies improved distortion on bounded treewidth graphs (O((k log n)¹/^p)). For asymptotically large p, our results also implies improved distortion on graphs excluding a minor.

Original language	English
Title of host publication	STOC 2018 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing
Editors	Monika Henzinger, David Kempe, Ilias Diakonikolas
Publisher	Association for Computing Machinery
Pages	912-919
Number of pages	8
ISBN (Electronic)	9781450355599
DOIs	https://doi.org/10.1145/3188745.3188808
State	Published - 20 Jun 2018
Externally published	Yes
Event	50th Annual ACM Symposium on Theory of Computing, STOC 2018 - Los Angeles, United States Duration: 25 Jun 2018 → 29 Jun 2018

Publication series

Name	Proceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)	0737-8017

Conference

Conference	50th Annual ACM Symposium on Theory of Computing, STOC 2018
Country/Territory	United States
City	Los Angeles
Period	25/06/18 → 29/06/18

Bibliographical note

Publisher Copyright:
© 2018 Association for Computing Machinery.

Funding

Arnold Filtser is Partially supported by the Lynn and William Frankel Center for Computer Sciences, ISF grant 1817/17, and by BSF Grant 2015813. Anupam Gupta is supported in part by NSF awards CCF-1536002, CCF-1540541, and CCF-1617790. Ofer Neiman is supported in part by ISF grant 1817/17, and by BSF Grant 2015813.

Funders	Funder number
Barth Syndrome Foundation	2015813
National Stroke Foundation	CCF-1536002, CCF-1617790, CCF-1540541
Israel Science Foundation	1817/17

Keywords

Metric embeddings
Normed spaces
Pathwidth
Shortest path decomposition
Treewidth

Access to Document

10.1145/3188745.3188808

Cite this

Abraham, I., Gupta, A., Filtser, A., & Neiman, O. (2018). Metric embedding via shortest path decompositions. In M. Henzinger, D. Kempe, & I. Diakonikolas (Eds.), STOC 2018 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing (pp. 912-919). (Proceedings of the Annual ACM Symposium on Theory of Computing). Association for Computing Machinery. https://doi.org/10.1145/3188745.3188808

Abraham, Ittai ; Gupta, Anupam ; Filtser, Arnold et al. / Metric embedding via shortest path decompositions. STOC 2018 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing. editor / Monika Henzinger ; David Kempe ; Ilias Diakonikolas. Association for Computing Machinery, 2018. pp. 912-919 (Proceedings of the Annual ACM Symposium on Theory of Computing).

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title = "Metric embedding via shortest path decompositions",

abstract = "We study the problem of embedding weighted graphs of pathwidth k into ℓp spaces. Our main result is an O(kmin \{1/p, 1/2 \})-distortion embedding. For p = 1, this is a super-exponential improvement over the best previous bound of Lee and Sidiropoulos. Our distortion bound is asymptotically tight for any fixed p > 1. Our result is obtained via a novel embedding technique that is based on low depth decompositions of a graph via shortest paths. The core new idea is that given a geodesic shortest path P, we can probabilistically embed all points into 2 dimensions with respect to P. For p > 2 our embedding also implies improved distortion on bounded treewidth graphs (O((k log n)1/p)). For asymptotically large p, our results also implies improved distortion on graphs excluding a minor.",

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Abraham, I, Gupta, A, Filtser, A & Neiman, O 2018, Metric embedding via shortest path decompositions. in M Henzinger, D Kempe & I Diakonikolas (eds), STOC 2018 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing. Proceedings of the Annual ACM Symposium on Theory of Computing, Association for Computing Machinery, pp. 912-919, 50th Annual ACM Symposium on Theory of Computing, STOC 2018, Los Angeles, United States, 25/06/18. https://doi.org/10.1145/3188745.3188808

Metric embedding via shortest path decompositions. / Abraham, Ittai; Gupta, Anupam; Filtser, Arnold et al.
STOC 2018 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing. ed. / Monika Henzinger; David Kempe; Ilias Diakonikolas. Association for Computing Machinery, 2018. p. 912-919 (Proceedings of the Annual ACM Symposium on Theory of Computing).

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

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Abraham I, Gupta A, Filtser A, Neiman O. Metric embedding via shortest path decompositions. In Henzinger M, Kempe D, Diakonikolas I, editors, STOC 2018 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing. Association for Computing Machinery. 2018. p. 912-919. (Proceedings of the Annual ACM Symposium on Theory of Computing). doi: 10.1145/3188745.3188808

Metric embedding via shortest path decompositions

Abstract

Publication series

Conference

Bibliographical note

Funding

Keywords

Access to Document

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Cite this