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The Steiner ratio for the dual normed plane

1997
*
Discrete Mathematics
*

Du et al. (1993) conjectured that

doi:10.1016/s0012-365x(96)00080-5
fatcat:3t2mostjrbfljj6t5nagwyibba
*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*. ...##
###
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

1993
*
Discrete & Computational Geometry
*

In this note we investigate various properties of minimum

doi:10.1007/bf02189328
fatcat:xfbmfoyiljezzly6e3sbvq4bji
*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. ...##
###
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

1996
*
Discrete Mathematics
*

We intend to discuss these problems

doi:10.1016/0012-365x(94)00368-s
fatcat:il4t535xu5hy3asey4r3o64hpu
*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 }. ...##
###
Minimum Steiner Tree Construction*
[chapter]

2008
*
Handbook of Algorithms for Physical Design Automation
*

Figure 1 depicts an MST and an SMT

doi:10.1201/9781420013481.ch24
fatcat:r2ewkzgdebhypbvn2v3f7vrya4
*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 ...##
###
A Dual-Based Algorithm for Multi-Level Network Design

1994
*
Management science
*

This paper develops and tests a

doi:10.1287/mnsc.40.5.567
fatcat:cs6yv2lh3bawxkkort46wnpoty
*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*). ...##
###
On Isoperimetric Inequalities in Minkowski Spaces

2010
*
Journal of Inequalities and Applications
*

In 14 equivalences of some affine isoperimetric inequalities, such as "

doi:10.1155/2010/697954
fatcat:wmiz42bb7ndpbngz6ds23px2ua
*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. ...##
###
Optimal Networks
[article]

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 ...

##
###
Equiangular tight frames from hyperovals
[article]

2016
*
arXiv
*
pre-print

An equiangular tight frame (ETF) is a set of equal

arXiv:1602.05557v2
fatcat:ggfutxlahnbwtkyv6fctwjnnqe
*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. ...##
###
Coordinated cutting plane generation via multi-objective separation

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. ...

##
###
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]

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] . ...

##
###
Configurations of lines and models of Lie algebras

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. ...

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