License: 
Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2018.36
URN: urn:nbn:de:0030-drops-87498
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/8749/
Elkin, Michael ;
Neiman, Ofer
Near Isometric Terminal Embeddings for Doubling Metrics
Abstract
Given a metric space (X,d), a set of terminals K subseteq 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 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+epsilon for some small 0<epsilon<1, is currently known.
Here we devise such terminal metric structures for doubling metrics, and show that essentially any metric structure with distortion 1+epsilon and size s(|X|) has its terminal counterpart, with distortion 1+O(epsilon) and size s(|K|)+1. In particular, for any doubling metric on n points, a set of k=o(n) terminals, and constant 0<epsilon<1, there exists
- A spanner with stretch 1+epsilon for pairs in K x X, with n+o(n) edges.
- A labeling scheme with stretch 1+epsilon for pairs in K x X, with label size ~~ log k.
- An embedding into l_infty^d with distortion 1+epsilon for pairs in K x 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.
BibTeX - Entry
@InProceedings{elkin_et_al:LIPIcs:2018:8749,
author = {Michael Elkin and Ofer Neiman},
title = {{Near Isometric Terminal Embeddings for Doubling Metrics}},
booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)},
pages = {36:1--36:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-066-8},
ISSN = {1868-8969},
year = {2018},
volume = {99},
editor = {Bettina Speckmann and Csaba D. T{\'o}th},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/8749},
URN = {urn:nbn:de:0030-drops-87498},
doi = {10.4230/LIPIcs.SoCG.2018.36},
annote = {Keywords: metric embedding, spanners, doubling metrics}
}
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Keywords: |
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metric embedding, spanners, doubling metrics |
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Collection: |
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34th International Symposium on Computational Geometry (SoCG 2018) |
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Issue Date: |
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2018 |
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Date of publication: |
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08.06.2018 |
