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We look for estimates and exact values for 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. ...doi:10.1016/s0012-365x(02)00762-8 fatcat:2tmfstv3pvewlonbpysvuznshy
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 ...
For a branched locally isometric covering 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. ...arXiv:1412.5433v1 fatcat:5ljsccvz4bcm3cjvcjkseut7g4
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  gave an upper and Du  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 ...doi:10.1016/s0166-218x(03)00267-1 fatcat:xskwmmrwn5aibfwrr4j4t3l2na
Here, > 0 is 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. ...doi:10.1109/tsmcb.2007.896021 pmid:17702290 fatcat:mvy3o3ovizb5xdsz6sihfi645q
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 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 ...doi:10.1016/s0166-218x(99)00076-1 fatcat:hlnsagtofbds3pbu4olwcv5424
There are known on-line algorithms whose competitive 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 ...doi:10.1007/bf02573969 fatcat:jrg4rzqzsbaetfjwthfzjiare4
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. ...
Cieslik 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. ...doi:10.1016/s0012-365x(02)00779-3 fatcat:c6ruhvugjzazdi7o6mlx4drqrq
., 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. ...doi:10.1016/s0012-365x(02)00785-9 fatcat:xjvlplv3ezalvfsvh7xhhrcl2y
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 appear as solutions 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. ...arXiv:1210.6228v1 fatcat:levrir2vpzhkfenrh24tbfk25y
There are known on-line algorithms whose competitive 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 ...doi:10.1145/142675.142744 dblp:conf/compgeom/AlonA92 fatcat:ml42mh2tundhffscqmttvspkoi
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  conjectured that m-dimensional rectilinear space has the Steiner ratio m/(2m--1). ...doi:10.1016/0012-365x(95)00005-h fatcat:xctcjb56hnatjouuotr6eb2kwq
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