On steiner ratio conjectures

Ding-Zhu Du

doi:10.1007/bf02071981

The steiner ratio conjecture is true for five points

Journal of Combinatorial Theory Series A ◽

10.1016/0097-3165(85)90073-1 ◽

1985 ◽

Vol 38 (2) ◽

pp. 230-240 ◽

Cited By ~ 22

Author(s):

D.Z Du ◽

F.K Hwang ◽

E.Y Yao

Keyword(s):

Steiner Ratio

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Branched coverings and Steiner ratio

International Transactions in Operational Research ◽

10.1111/itor.12182 ◽

2015 ◽

Vol 23 (5) ◽

pp. 875-882

Author(s):

A. O. Ivanov ◽

A. A. Tuzhilin

Keyword(s):

Steiner Ratio ◽

Branched Coverings

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ON THE STEINER RATIO IN $\mathcal{R}_{n}$

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830911001358 ◽

2011 ◽

Vol 03 (04) ◽

pp. 473-489

Author(s):

HAI DU ◽

WEILI WU ◽

ZAIXIN LU ◽

YINFENG XU

Keyword(s):

Euclidean Space ◽

Spanning Tree ◽

Minimum Spanning Tree ◽

Dimensional Euclidean Space ◽

Minimum Length ◽

Steiner Ratio ◽

Line Segments ◽

Finite Set ◽

Steiner Minimum Tree ◽

Set Of Points

The Steiner minimum tree and the minimum spanning tree are two important problems in combinatorial optimization. Let P denote a finite set of points, called terminals, in the Euclidean space. A Steiner minimum tree of P, denoted by SMT(P), is a network with minimum length to interconnect all terminals, and a minimum spanning tree of P, denoted by MST(P), is also a minimum network interconnecting all the points in P, however, subject to the constraint that all the line segments in it have to terminate at terminals. Therefore, SMT(P) may contain points not in P, but MST(P) cannot contain such kind of points. Let [Formula: see text] denote the n-dimensional Euclidean space. The Steiner ratio in [Formula: see text] is defined to be [Formula: see text], where Ls(P) and Lm(P), respectively, denote lengths of a Steiner minimum tree and a minimum spanning tree of P. The best previously known lower bound for [Formula: see text] in the literature is 0.615. In this paper, we show that [Formula: see text] for any n ≥ 2.

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