Shallow-low-light trees, and tight lower bounds for euclidean spanners

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Abstract

We show that for every n-point metric space M and positive integer k, there exists a spanning tree T with unweighted diameter O(k) and weight w(T) = O(k · n1/k) · w(MST(M)), and a spanning tree T′ with weight w(T′) = O(k) · w(MST(M)) and unweighted diameter O(k · n1/k). Moreover, there is a designated point rt such that for every other point v, both distT(rt, v) and distT(rt,v) are at most (1 + ε) · distM(rt,v), for an arbitrarily small constant ε > O. We prove that the above tradeoffs are tight up to constant factors in the entire range of parameters. Furthermore, our lower bounds apply to a basic one-dimensional Euclidean space. Finally, our lower bounds for the particular case of unweighted diameter O(log n) settle a long-standing open problem in Computational Geometry.

Original languageEnglish
Title of host publicationProceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008
Pages519-528
Number of pages10
DOIs
StatePublished - 31 Dec 2008
Event49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008 - Philadelphia, PA, United States
Duration: 25 Oct 200828 Oct 2008

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
ISSN (Print)0272-5428

Conference

Conference49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008
Country/TerritoryUnited States
CityPhiladelphia, PA
Period25/10/0828/10/08

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