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The Steiner ratio for the dual normed plane

Peng-Jun Wan, Ding-Zhu Du, Ronald L. Graham
1997 Discrete Mathematics  
Du et al. (1993) conjectured that the Steiner ratio on a normed plane is equal to the Steiner ratio on its dual plane. In this paper we show that this conjecture is true for Ixl ~<5.  ...  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.  ...  If for any two points x and y, IIx -yH = 2Ix* -y*l* for some constant 2, then it follows immediately that the Steiner ratio for the normed plane is equal to the Steiner ratio for its dual normed plane.  ... 
doi:10.1016/s0012-365x(96)00080-5 fatcat:3t2mostjrbfljj6t5nagwyibba

Page 4769 of Mathematical Reviews Vol. , Issue 98H [page]

1998 Mathematical Reviews  
Graham, Ronald L. (1-BELL; Murray Hill, NJ) The Steiner ratio for the dual normed plane.  ...  The authors, B. Gao and Z. C. Liu [Discrete Comput. Geom. 9 (1993), no. 4, 351-370; MR 94b:05053] conjectured that the Steiner ratio on a normed plane is equal to the Steiner ratio on its dual plane.  ... 

Minimum steiner trees in normed planes

Ding-Zhu Du, Biao Gao, Ronald L. Graham, Zi-Cheng Liu, Peng-Jun Wan
1993 Discrete & Computational Geometry  
In this note we investigate various properties of minimum Steiner trees for general normed planes M(D). In particular we study the Steiner ratio p(D) for M(D), defined by  ...  We also investigate the Steiner ratio p(D) for D, and show that, for any D, 0.623 < p(D) < 0.8686. I.  ...  Acknowledgment The authors wish to express their appreciation to Professor Frank Morgan for his insightful comments on an earlier draft of this paper.  ... 
doi:10.1007/bf02189328 fatcat:xfbmfoyiljezzly6e3sbvq4bji

Page 594 of Mathematical Reviews Vol. , Issue 93b [page]

1993 Mathematical Reviews  
The Steiner ratio is the minimum ratio r(E) between the lengths of a Steiner minimum tree and a minimum spanning tree for the same set of points.  ...  Ding Zhu Du (1-MN-C) 93b:05043 05C05 51M05 Cieslik, Dietmar (D-EMAU) The Steiner-ratio in Banach-Minkowski planes.  ... 

Steiner minimal trees in Lp2

Dietmar Cieslik, Johann Linhart
1996 Discrete Mathematics  
We intend to discuss these problems for all planes with p-norm, i.e. the affine plane with norm II (tl, t2)II p = (I tx I ~ + I t2 IP) 1/p for 1 ~< p < oo and Ij(tl,t2)lt~ = max{Itll, It21}.  ...  For a finite set of points in a metric space a Steiner Minimal Tree (SMT) is a shortest tree which interconnects these points.  ...  We consider shortest trees for finite sets of points in L 2, where L 2 is the affine plane with norm [l(tl,t2)l[ p = (Itll p + It2lP) lip for 1 ~< p < ~ and II(tl,t2)lb~ --Max { I t 1 I, I t21 }.  ... 
doi:10.1016/0012-365x(94)00368-s fatcat:il4t535xu5hy3asey4r3o64hpu

Minimum Steiner Tree Construction* [chapter]

Gabriel Robins, Alexander Zelikovsky
2008 Handbook of Algorithms for Physical Design Automation  
Figure 1 depicts an MST and an SMT for the same pointset in the Manhattan plane.  ...  |x 1 − x 2 | + |y 1 − y 2 | We will focus on the rectilinear Steiner minimal tree problem, where every edge is embedded in the plane using a path of one or more alternating horizontal and vertical segments  ...  Thus, the 3 2 bound is tight for a wide range of MST-based strategies in the rectilinear plane [56] , which resolved the performance ratios for a number of heuristics in the literature with previously  ... 
doi:10.1201/9781420013481.ch24 fatcat:r2ewkzgdebhypbvn2v3f7vrya4

A Dual-Based Algorithm for Multi-Level Network Design

Anantaram Balakrishnan, Thomas L. Magnanti, Prakash Mirchandani
1994 Management science  
This paper develops and tests a dual-based algorithm for the Multi-level Network Design (MLND) problem.  ...  Given an undirected network with L possible facility types for each edge, and a partition of the nodes into L levels, the Multi-level Network Design (MLND) problem seeks a fixed cost minimizing design  ...  , and Loo norms).  ... 
doi:10.1287/mnsc.40.5.567 fatcat:cs6yv2lh3bawxkkort46wnpoty

On Isoperimetric Inequalities in Minkowski Spaces

Horst Martini, Zokhrab Mustafaev
2010 Journal of Inequalities and Applications  
In 14 equivalences of some affine isoperimetric inequalities, such as "duals" of L p versions of Petty's projection inequality and "duals" of L p versions of the Busemann-Petty inequality, are established  ...  Here we also mention the paper 16 , where the method of Steiner symmetrization is discussed and many references are given.  ...  Acknowledgments The authors wish to thank the referees for their valuable comments and suggestions.  ... 
doi:10.1155/2010/697954 fatcat:wmiz42bb7ndpbngz6ds23px2ua

Optimal Networks [article]

A.O. Ivanov, A.A. Tuzhilin
2012 arXiv   pre-print
The aim of this mini-course is to give an introduction in Optimal Networks theory.  ...  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?  ...  Steiner ratio was firstly defined for the Euclidean plane in [11] , and during the following years the problem of Steiner ratio calculation is one of the most attractive, interesting and difficult problems  ... 
arXiv:1210.6228v1 fatcat:levrir2vpzhkfenrh24tbfk25y

Equiangular tight frames from hyperovals [article]

Matthew Fickus, Dustin G. Mixon, John Jasper
2016 arXiv   pre-print
An equiangular tight frame (ETF) is a set of equal norm vectors in a Euclidean space whose coherence is as small as possible, equaling the Welch bound.  ...  In this paper, we present a new infinite family of complex ETFs that arises from hyperovals in finite projective planes.  ...  That is, mathematically speaking, flat vectors provide the largest possible ratio of 2-norm to ∞-norm. In light of this, it is natural to investigate ETFs that have flat synthesis operators.  ... 
arXiv:1602.05557v2 fatcat:ggfutxlahnbwtkyv6fctwjnnqe

Coordinated cutting plane generation via multi-objective separation

Edoardo Amaldi, Stefano Coniglio, Stefano Gualandi
2012 Mathematical programming  
The impact of our coordinated cutting plane generation scheme is assessed in a pure cutting plane setting when separating Stable Set and Cut Set inequalities for, respectively, the Max Clique and Min Steiner  ...  As cut diversity measure, we consider an aggregate of the 1-norm distances w.r.t. the normal vectors of the previous cuts.  ...  Acknowledgements We would like to thank two anonymous referees for useful comments that helped to improve the quality of the paper.  ... 
doi:10.1007/s10107-012-0596-x fatcat:gohrjxgbgrdnrlawq5kyanef4y

Page 6799 of Mathematical Reviews Vol. , Issue 2001I [page]

2001 Mathematical Reviews  
Let Le be the plane equipped with the p-norm.  ...  ratio of L,-planes.  ... 

Page 3929 of Mathematical Reviews Vol. , Issue 96g [page]

1996 Mathematical Reviews  
the Steiner ratio in Minkowski planes.  ...  The Steiner ratio for M(D) is defined by p(D) = inf{Ls(X)/Lu(X): X c E*}. If P is a (critical) point-set, let p,(D) = min{Ls(P)/Ly(P): |P| =k}.  ... 

A New Approach to Output-Sensitive Voronoi Diagrams and Delaunay Triangulations [article]

Gary L. Miller, Donald R. Sheehy
2013 arXiv   pre-print
The running time of our algorithm is O(f n Δ) where f is the output complexity of the Voronoi diagram and Δ is the spread of the input, the ratio of largest to smallest pairwise distances.  ...  Despite the simplicity of the algorithm and its analysis, it improves on the state of the art for all inputs with polynomial spread and near-linear output size.  ...  In this respect, the problem more resembles the Delaunay triangulation splitting problem for which Chazelle et al. gave a linear time algorithm for the plane [8] .  ... 
arXiv:1212.5098v2 fatcat:edncvjas3reznc7p4jpfdobiqa

Configurations of lines and models of Lie algebras

L. Manivel
2006 Journal of Algebra  
For e 7 and e 8 we are lead to beautiful models graded over the octonions, which display these algebras as plane projective geometries of subalgebras.  ...  The automorphism groups of the 27 lines on the smooth cubic surface or the 28 bitangents to the general quartic plane curve are well-known to be closely related to the Weyl groups of E 6 and E 7 .  ...  Chaput for their useful comments.  ... 
doi:10.1016/j.jalgebra.2006.04.029 fatcat:a6hxnwcs3redlezee576il6nv4
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