Near isometric terminal embeddings for doubling metrics

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

4 Scopus citations

Abstract

Given a metric space (X, d), a set of terminals K ⊆ X, and a parameter t ≥ 1, we consider metric structures (e.g., spanners, distance oracles, embedding into normed spaces) that preserve distances for all pairs in K × X up to a factor of t, and have small size (e.g. number of edges for spanners, dimension for embeddings). While such terminal (aka source-wise) metric structures are known to exist in several settings, no terminal spanner or embedding with distortion close to 1, i.e., t = 1 + ϵ for some small 0 < ϵ < 1, is currently known. Here we devise such terminal metric structures for doubling metrics, and show that essentially any metric structure with distortion 1 + ϵ and size s(|X|) has its terminal counterpart, with distortion 1 + O(ϵ) and size s(|K|) + 1. In particular, for any doubling metric on n points, a set of k = o(n) terminals, and constant 0 < ϵ < 1, there exists • A spanner with stretch 1 + ϵ for pairs in K × X, with n + o(n) edges. • A labeling scheme with stretch 1 + ϵ for pairs in K × X, with label size ≈ log k. • An embedding into ld with distortion 1 + ϵ for pairs in K × X, where d = O(log k). Moreover, surprisingly, the last two results apply if only K is a doubling metric, while X can be arbitrary.

Original languageEnglish
Title of host publication34th International Symposium on Computational Geometry, SoCG 2018
EditorsCsaba D. Toth, Bettina Speckmann
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Pages361-3615
Number of pages3255
ISBN (Electronic)9783959770668
DOIs
StatePublished - 1 Jun 2018
Event34th International Symposium on Computational Geometry, SoCG 2018 - Budapest, Hungary
Duration: 11 Jun 201814 Jun 2018

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume99
ISSN (Print)1868-8969

Conference

Conference34th International Symposium on Computational Geometry, SoCG 2018
Country/TerritoryHungary
CityBudapest
Period11/06/1814/06/18

Keywords

  • Doubling metrics
  • Metric embedding
  • Spanners

ASJC Scopus subject areas

  • Software

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