TY - JOUR

T1 - Low-light trees, and tight lower bounds for Euclidean spanners

AU - Dinitz, Yefim

AU - Elkin, Michael

AU - Solomon, Shay

N1 - Funding Information:
Research of M. Elkin and S. Solomon has been supported by the Israeli Academy of Science, grant 483/06.
Funding Information:
Partially supported by the Lynn and William Frankel Center for Computer Sciences.

PY - 2010/6/1

Y1 - 2010/6/1

N2 - 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{dot operator}n1/k){dot operator}w(MST(M)), and a spanning tree T′ with weight w(T′)=O(k){dot operator}w(MST(M)) and unweighted diameter O(k{dot operator}n1/k). These trees also achieve an optimal maximum degree. Furthermore, we demonstrate that these trees can be constructed efficiently. We prove that these tradeoffs between unweighted diameter and weight are tight up to constant factors in the entire range of parameters. Moreover, our lower bounds apply to a basic one-dimensional Euclidean space. Our lower bounds for the particular case of unweighted diameter O(log n) are of independent interest, settling a long-standing open problem in Computational Geometry. In STOC'95 Arya et al. devised a construction of Euclidean Spanners with unweighted diameter O(log n) and weight O(log n){dot operator}w(MST(M)). In SODA'05 Agarwal et al. showed that this result is tight up to a factor of O(log log n). We close this gap and show that the result of Arya et al. is tight up to constant factors. Finally, our upper bounds imply improved approximation algorithms for the minimumh-hop spanning tree and bounded diameter minimum spanning tree problems for metric spaces.

AB - 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{dot operator}n1/k){dot operator}w(MST(M)), and a spanning tree T′ with weight w(T′)=O(k){dot operator}w(MST(M)) and unweighted diameter O(k{dot operator}n1/k). These trees also achieve an optimal maximum degree. Furthermore, we demonstrate that these trees can be constructed efficiently. We prove that these tradeoffs between unweighted diameter and weight are tight up to constant factors in the entire range of parameters. Moreover, our lower bounds apply to a basic one-dimensional Euclidean space. Our lower bounds for the particular case of unweighted diameter O(log n) are of independent interest, settling a long-standing open problem in Computational Geometry. In STOC'95 Arya et al. devised a construction of Euclidean Spanners with unweighted diameter O(log n) and weight O(log n){dot operator}w(MST(M)). In SODA'05 Agarwal et al. showed that this result is tight up to a factor of O(log log n). We close this gap and show that the result of Arya et al. is tight up to constant factors. Finally, our upper bounds imply improved approximation algorithms for the minimumh-hop spanning tree and bounded diameter minimum spanning tree problems for metric spaces.

KW - Approximation algorithms

KW - Computational geometry

KW - Euclidean spanners

KW - Hop-diameter

KW - Low distortion embeddings

UR - http://www.scopus.com/inward/record.url?scp=77952008812&partnerID=8YFLogxK

U2 - 10.1007/s00454-009-9230-y

DO - 10.1007/s00454-009-9230-y

M3 - Article

AN - SCOPUS:77952008812

VL - 43

SP - 736

EP - 783

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 4

ER -