Narrow-shallow-low-light trees with and without steiner points

Michael Elkin, Shay Solomon

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

Abstract

We show that for every set of n points in the plane and a designated point , there exists a tree T that has small maximum degree, depth and weight. Moreover, for every point , the distance between rt and v in T is within a factor of (1+ε) close to their Euclidean distance ||rt,v||. We call these trees narrow-shallow-low-light (NSLLTs). We demonstrate that our construction achieves optimal (up to constant factors) tradeoffs between all parameters of NSLLTs. Our construction extends to point sets in , for an arbitrarily large constant d. The running time of our construction is O(n •logn). We also study this problem in general metric spaces, and show that NSLLTs with small maximum degree, depth and weight can always be constructed if one is willing to compromise the root-distortion. On the other hand, we show that the increased root-distortion is inevitable, even if the point set resides in a Euclidean space of dimension Θ(logn). On the bright side, we show that if one is allowed to use Steiner points then it is possible to achieve root-distortion (1+ε) together with small maximum degree, depth and weight for general metric spaces. Finally, we establish some lower bounds on the power of Steiner points in the context of Euclidean spanning trees and spanners.

Original languageEnglish
Title of host publicationAlgorithms - ESA 2009 - 17th Annual European Symposium, Proceedings
Pages215-226
Number of pages12
DOIs
StatePublished - 2009
Externally publishedYes
Event17th Annual European Symposium on Algorithms, ESA 2009 - Copenhagen, Denmark
Duration: 7 Sep 20099 Sep 2009

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume5757 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference17th Annual European Symposium on Algorithms, ESA 2009
Country/TerritoryDenmark
CityCopenhagen
Period7/09/099/09/09

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