Embedding metrics into ultrametrics and graphs into spanning trees with constant average distortion

Ittai Abraham, Yair Bartal, Ofer Neiman

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

28 Scopus citations

Abstract

This paper addresses the basic question of how well can a tree approximate distances of a metric space or a graph. Given a graph, the problem of constructing a spanning tree in a graph which strongly preserves distances in the graph is a fundamental problem in network design. We present scaling distortion embeddings where the distortion scales as a function of ∈, with the guarantee that for each ∈ the distortion of a fraction 1 - ∈ of all pairs is bounded accordingly. Such a bound implies, in particular, that the average distortion and ℓq-distortions are small. Specifically, our embeddings have constant average distortion and O(√log n)ℓ2-distortion. This follows from the following results: we prove that any metric space embeds into an ultrametric with scaling distortion O(√1/∈). For the graph setting we prove that any weighted graph contains a spanning tree with scaling distortion O(√1/∈). These bounds are tight even for embedding in arbitrary trees. For probabilistic embedding into spanning trees we prove a scaling distortion of Õ (log2(1/∈)), which implies constant ℓq-distortion for every fixed q < ∞.

Original languageAmerican English
Title of host publicationProceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007
PublisherAssociation for Computing Machinery
Pages502-511
Number of pages10
ISBN (Electronic)9780898716245
StatePublished - 2007
Event18th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007 - New Orleans, United States
Duration: 7 Jan 20079 Jan 2007

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
Volume07-09-January-2007

Conference

Conference18th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007
Country/TerritoryUnited States
CityNew Orleans
Period7/01/079/01/07

Bibliographical note

Publisher Copyright:
Copyright © 2007 by the Association for Computing Machinery, Inc. and the Society for Industrial and Applied Mathematics.

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