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The Steiner ratio of several discrete metric spaces

2003
*
Discrete Mathematics
*

We look for estimates and exact values for

doi:10.1016/s0012-365x(02)00762-8
fatcat:2tmfstv3pvewlonbpysvuznshy
*the**Steiner**ratio*in*several**discrete**metric**spaces*. ... Particularly, we determine*the**Steiner**ratio*for*spaces**of*words, and we estimate*the**Steiner**ratio*for speciÿc graphs. ... For*the**Steiner**ratio**of*every*metric**space*1¿m(X; )¿ 1 2 holds. ...##
###
Page 8936 of Mathematical Reviews Vol. , Issue 2003m
[page]

2003
*
Mathematical Reviews
*

We look for estimates and exact values for

*the**Steiner**ratio*in*several**discrete**metric**spaces*. ... See also 12012, 51009, 51014, 51015, 52013, 52024, 62211, 70020, 94082, 94087, 94095. 05C Graph theory 2003m:05050 05C05 68R10 Cieslik, Dietmar (D-EMAU-IMI,; Greifswald)*The**Steiner**ratio**of**several**discrete*...##
###
Branched Coverings and Steiner Ratio
[article]

2014
*
arXiv
*
pre-print

For a branched locally isometric covering

arXiv:1412.5433v1
fatcat:5ljsccvz4bcm3cjvcjkseut7g4
*of**metric**spaces*with intrinsic*metrics*, it is proved that*the**Steiner**ratio**of**the*base is not less than*the**Steiner**ratio**of**the*total*space**of**the*covering. ... As applications, it is shown that*the**Steiner**ratio**of**the*surface*of*an isosceles tetrahedron is equal to*the**Steiner**ratio**of**the*Euclidean plane, and that*the**Steiner**ratio**of*a flat cone with angle ...*The*work is partly supported by*the*Grant*of*President*of*RF for supporting*of*leading scientific schools*of*Russia, Project NSh-581.2014.1, and by RFBR, Project 13-01-00664a. A. ...##
###
The Steiner ratio of high-dimensional Banach–Minkowski spaces

2004
*
Discrete Applied Mathematics
*

*The*

*Steiner*

*ratio*is

*the*greatest lower bound

*of*

*the*

*ratios*

*of*

*the*

*Steiner*Minimal Tree-by

*the*Minimum Spanning Tree-lengths running over all ÿnite subsets

*of*a

*metric*

*space*. ... Particularly, let

*the*quantity C d deÿned as

*the*upper bound

*of*

*the*

*Steiner*

*ratio*

*of*all d-dimensional Banach

*spaces*, then lim d→∞ C d = lim d→∞ m d (B(2)); where m d (B(2)) denotes

*the*

*Steiner*

*ratio*

*of*... Chung and Gilbert [3] gave an upper and Du [7] found a lower bound for Euclidean

*spaces*

*of*high dimension: 0:61582 : : : 6 lim d→∞ m d (B(2)) 6 0:66984 : : : : For

*the*

*Steiner*

*ratio*

*of*

*several*other ...

##
###
On Robotic Optimal Path Planning in Polygonal Regions With Pseudo-Euclidean Metrics

2007
*
IEEE Transactions on Systems Man and Cybernetics Part B (Cybernetics)
*

Here, > 0 is

doi:10.1109/tsmcb.2007.896021
pmid:17702290
fatcat:mvy3o3ovizb5xdsz6sihfi645q
*the*user-defined error tolerance*ratio*. ... We also introduce an empirical method called*the*adaptive*discretization*method that improves*the*performance*of**the*approximation algorithms by placing*discretization*points densely only in areas that ... ACKNOWLEDGMENT Part*of**the*research was done when Z. Sun was at Duke University and Hong Kong Baptist University. ...##
###
Page 5237 of Mathematical Reviews Vol. , Issue 93i
[page]

1993
*
Mathematical Reviews
*

Finding a shortest network interconnecting a given set

*of*points in a*metric**space*is called*the**Steiner*minimum tree problem. ...*The**Steiner**ratio*p is*the*largest lower bound for*the**ratio**of**the*length*of*a*Steiner*minimum tree divided by*the*length*of*a minimum spanning tree for*the*same set*of*points. ...##
###
The Steiner ratio of L2kd

1999
*
Discrete Applied Mathematics
*

*The*

*Steiner*

*ratio*m(d; 2k)

*of*L d 2k is a measure

*of*how good an MST approximates an SMT. We estimate this quantity. ? 0166-218X/99/$ -see front matter ? 1999 Elsevier Science B.V. ... Let L d 2k be

*the*d-dimensional

*space*with 2k-norm. Given a ÿnite set N

*of*points in this

*space*. Find a connected graph G = (V; E) such that N ⊆ V and

*the*total length

*of*G is minimal. ... Our goal is to estimate this quantity with help

*of*

*the*knowledge

*of*good bounds for

*the*

*Steiner*

*ratio*

*of*Euclidean

*spaces*and

*the*embeddings

*of*Euclidean in

*the*

*spaces*L d 2k . 1 E d denotes

*the*d-dimensional ...

##
###
On-line steiner trees in the Euclidean plane

1993
*
Discrete & Computational Geometry
*

There are known on-line algorithms whose competitive

doi:10.1007/bf02573969
fatcat:jrg4rzqzsbaetfjwthfzjiare4
*ratio*is O(log n) even for all*metric**spaces*, but*the*only lower bound known is*of*[IW] for some contrived*discrete**metric**space*. ... total length*of**the*best*Steiner*tree that connects all*the*points. ... For*the**Steiner*tree problem (in any*metric**space*)*the*competitive*ratio**of*an on-line algorithm is*the*supremum, over all possible sets S,*of**the**ratio*between*the*weight*of**the*connected graph constructed ...##
###
Page 2221 of Mathematical Reviews Vol. , Issue 99c
[page]

1991
*
Mathematical Reviews
*

*The*problem

*of*convexity in

*discrete*

*spaces*related to

*metric*convexity and graphs has been

*of*great interest to re- searchers. ... Summary: “

*The*

*Steiner*tree problem asks for

*the*shortest tree connecting a given set

*of*terminal points in a

*metric*

*space*. ...

##
###
Contents

2003
*
Discrete Mathematics
*

Cieslik

doi:10.1016/s0012-365x(02)00779-3
fatcat:c6ruhvugjzazdi7o6mlx4drqrq
*The**Steiner**ratio**of**several**discrete**metric**spaces*189 M. Fischermann, D. Rautenbach and L. Volkmann Maximum graphs with a unique minimum dominating set 197 P. Frankl and N. ... CostaLabelings*of*Lee and Hamming*spaces*119T.A. McKeeChordal bipartite, strongly chordal, and strongly chordal bipartite graphs 231 Partitioning Boolean lattices into antichains 45 M. ...##
###
Author index to volume 260

2003
*
Discrete Mathematics
*

.,

doi:10.1016/s0012-365x(02)00785-9
fatcat:xjvlplv3ezalvfsvh7xhhrcl2y
*The**Steiner**ratio**of**several**discrete**metric**spaces*(Note) (1-3) 189-196 Cockayne, E.J., M.A. Henning and C.M. ... Tokushige,*The*game*of*n-times nim (Note) , T.W., M.A. Henning and P.J. ...##
###
Page 5575 of Mathematical Reviews Vol. , Issue 94j
[page]

1994
*
Mathematical Reviews
*

*The*problem is to find a point which minimizes

*the*sum

*of*distances from

*the*point to n given points in a

*metric*

*space*. ...

*Several*heuristics are analysed in this paper. It is found that

*the*best performance

*ratio*among these heuristics is |P|,

*the*number

*of*vertices in P. ...

##
###
Optimal Networks
[article]

2012
*
arXiv
*
pre-print

Optimal networks appear as solutions

arXiv:1210.6228v1
fatcat:levrir2vpzhkfenrh24tbfk25y
*of**the*following natural problem: How to connect a finite set*of*points in a*metric**space*in an optimal way? ... This mini-course was given in*the*First Yaroslavl Summer School on*Discrete*and Computational Geometry in August 2012, organized by International Delaunay Laboratory "*Discrete*and Computational Geometry ... Ovsyannikov)*The**Steiner**ratio*and*the**Steiner*-Gromov*ratio**of**the**metric**space**of*all compact subsets*of*Euclidean plane endowed with Hausdorff*metric*are equal to 1/2. ...##
###
On-line Steiner trees in the Euclidean plane

1992
*
Proceedings of the eighth annual symposium on Computational geometry - SCG '92
*

There are known on-line algorithms whose competitive

doi:10.1145/142675.142744
dblp:conf/compgeom/AlonA92
fatcat:ml42mh2tundhffscqmttvspkoi
*ratio*is O(log n) even for all*metric**spaces*, but*the*only lower bound known is*of*[IW] for some contrived*discrete**metric**space*. ... total length*of**the*best*Steiner*tree that connects all*the*points. ... For*the**Steiner*tree problem (in any*metric**space*)*the*competitive*ratio**of*an on-line algorithm is*the*supremum, over all possible sets S,*of**the**ratio*between*the*weight*of**the*connected graph constructed ...##
###
A tight lower bound for the Steiner ratio in Minkowski planes

1995
*
Discrete Mathematics
*

*The*

*Steiner*

*ratio*for a

*metric*

*space*is

*the*largest lower bound for

*the*

*ratio*

*of*lengths between a minimum

*Steiner*tree and a minimum spanning tree on

*the*same set

*of*points in

*the*

*metric*

*space*. ... In this note, we show that for any Minkowski plane,

*the*

*Steiner*

*ratio*is at least 2/3. This settles a conjecture

*of*Cieslik (1990) and also Du et al. (1991) . ... Graham and Hwang [12] conjectured that m-dimensional rectilinear

*space*has

*the*

*Steiner*

*ratio*m/(2m--1). ...

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