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Generalised k-Steiner Tree Problems in Normed Planes

2013
*
Algorithmica
*

We also extend their approach further to encompass other

doi:10.1007/s00453-013-9780-5
fatcat:hsoebwee55hopncq53g7ony6h4
*normed**planes*, and to solve a much wider class of*problems*, including the*k*-bottleneck*Steiner**tree**problem*and other*generalised**k*-*Steiner**tree*...*In*this paper we*generalise*their approach*in*order to solve the*k*-*Steiner**tree**problem*,*in*which the*Steiner*minimum*tree*may contain up to*k**Steiner*points for a given constant*k*. ... Our algorithm for solving the*generalised**k*-*Steiner**tree**problem**in**normed**planes*has three primary phases. The first phase constructs a set of feasible internal topologies. ...##
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Designing Optimal Flow Networks
[article]

2009
*
arXiv
*
pre-print

We characterise the local topological structure of

arXiv:0903.2124v1
fatcat:kv7n6s4objcvni4yymqha5yadm
*Steiner*points*in*MGAs for linear cost functions. This*problem*has applications to the design of drains, gas pipelines and underground mine access. ... The network may contain other unprescribed nodes, known as*Steiner*points. ... Given a set N of terminals, the*Steiner**problem*(or*Steiner*Minimum*Tree**problem*) asks for an SMT spanning N . ...##
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Survivable minimum bottleneck networks

2015
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Computational geometry
*

We show that the survivable bottleneck

doi:10.1016/j.comgeo.2015.06.002
fatcat:f7sasho7bje4tka6b7bral4qv4
*Steiner**tree**problem**in**normed**planes*can be solved*in*polynomial time when the number of*Steiner*points is constant. ...*In*particular, under the Euclidean and rectilinear*norms*our algorithm constructs an optimal solution*in*O(n 2k+3 log n) steps, where n is the number of terminals and*k*is the number of*Steiner*points. ... Conclusion We present the first polynomial time algorithm for the survivable bottleneck*Steiner**tree**problem**in**normed**planes*. ...##
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Steiner trees for fixed orientation metrics

2008
*
Journal of Global Optimization
*

We consider the

doi:10.1007/s10898-008-9305-y
fatcat:qxgpo43vkfhqvn6u5u3z4lvfaq
*problem*of constructing*Steiner*minimum*trees*for a metric defined by a polygonal unit circle (corresponding to σ ≥ 2 weighted legal orientations*in*the*plane*). ... We provide a simple proof that the angle configuration for a*Steiner*point extends to all*Steiner*points*in*a full*Steiner*minimum*tree*, such that at most six orientations suffice for edges*in*a full*Steiner*...*In*Section 2 we define the*Steiner**tree**problem**in**normed**planes*formally, present some known results and fix our notation. ...##
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The Gilbert arborescence problem

2012
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Networks
*

We investigate the

doi:10.1002/net.21475
fatcat:tgg3e35f2bexjcxuc6cdvimq2i
*problem*of designing a minimum cost flow network interconnecting n sources and a single sink, each with known locations*in*a*normed*space and with associated flow demands. ... We characterise the local topological structure of*Steiner*points*in*MGAs, showing,*in*particular, that for a wide range of metrics, and for some typical real-world cost-functions, the degree of each*Steiner*... A*Steiner**tree*(ST) is a*tree*whose length cannot be shortened by a small perturbation of its Gilbert flows Gilbert [5] proposed the following*generalisation*of the*Steiner**problem**in*Euclidean space ...##
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Quantitative illumination of convex bodies and vertex degrees of geometric Steiner minimal trees
[article]

2004
*
arXiv
*
pre-print

Let v(d) be the maximum degree of a vertex, and s(d) of a

arXiv:math/0410144v1
fatcat:tdgm6rmawzcxvhwbhrkgldufla
*Steiner*point,*in*a*Steiner*minimal*tree**in*a d-dimensional*normed*space, where both maxima are over all*norms*. F. ... The second involves*Steiner*minimal*trees*. ...*Steiner*minimal*trees*have been studied mostly*in*the Euclidean*plane*and the rectilinear*plane*(*K*a parallelogram) [HRW92] . ...##
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The Local Steiner Problem in Finite-Dimensional Normed Spaces

2007
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Discrete & Computational Geometry
*

We develop a general method of proving that certain star configurations

doi:10.1007/s00454-006-1298-z
fatcat:hd3ipc77tfbqvay6ob2oqljkgq
*in*finit e-dimensional*normed*spaces are*Steiner*minimal*trees*. ... We determine the maximum degree of a given point*in*a*Steiner*minimal*tree**in*this*norm*. The proof makes essential use of extremal finite set theory. ... Cockayne [9] considered Minkowski*planes*(two-dimensional*normed*spaces). Cieslik initiated the study of*Steiner*minimal*trees**in*general finite-dimensional*normed*spaces [5, 6] . ...##
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Canonical Forms and Algorithms for Steiner Trees in Uniform Orientation Metrics

2005
*
Algorithmica
*

We present some fundamental structural properties for minimum length networks (known as

doi:10.1007/s00453-005-1178-6
fatcat:bh4sh33txrdm5h43gmnwfpnizy
*Steiner*minimum*trees*) interconnecting a given set of points*in*an environment*in*which edge segments are restricted ... Acknowledgments The authors would like to thank Benny*K*. Nielsen and Pawel Winter for many fruitful discussions. Also we thank the anonymous referees for their comments and suggestions. ... A Minkowski*plane*(or*normed**plane*) is a two-dimensional real*normed*space £ ¥ § with unit ball ¤ . (For more background on Minkowski*planes*, see [14] .) ...##
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The bottleneck 2-connected k-Steiner network problem for k≤2

2012
*
Discrete Applied Mathematics
*

The geometric bottleneck

doi:10.1016/j.dam.2012.01.006
fatcat:yjbf5h6gyzgsdpx2q6drsd2rrm
*Steiner*network*problem*on a set of vertices X embedded*in*a*normed**plane*requires one to construct a graph G spanning X and a variable set of*k*≥ 0 additional points, such that ...*In*this paper, we consider the Euclidean bottleneck*Steiner*network*problem*for*k*≤ 2, where G is constrained to be 2-connected. ... This paper presents algorithms for solving the bottleneck*Steiner**problem**in*the Euclidean*plane*when the solution graph is required to be 2-connected and contains exactly*k*= 1 or*k*= 2*Steiner*points ...##
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The bottleneck 2-connected k-Steiner network problem for k≤ 2
[article]

2011
*
arXiv
*
pre-print

The geometric bottleneck

arXiv:1108.3655v1
fatcat:xixblw44yne23aoyiofu5c73l4
*Steiner*network*problem*on a set of vertices X embedded*in*a*normed**plane*requires one to construct a graph G spanning X and a variable set of*k*≥ 0 additional points, such that ...*In*this paper we consider the Euclidean bottleneck*Steiner*network*problem*for*k*≤ 2, where G is constrained to be 2-connected. ... This paper presents algorithms for solving the bottleneck*Steiner**problem**in*the Euclidean*plane*when the solution graph is required to be 2-connected and contains exactly*k*= 1 or*k*= 2*Steiner*points ...##
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A flow-dependent quadratic steiner tree problem in the Euclidean plane

2014
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Networks
*

We introduce a flow-dependent version of the quadratic

doi:10.1002/net.21553
fatcat:s3so3k6xdvehdiwbbkcavp3hei
*Steiner**tree**problem**in*the*plane*. ... The output is a set of*Steiner*points (satisfying the given bound)*in*the*plane*and a*tree*T interconnecting all nodes such that the sum of P(u, v) over all u, v is minimised. ... Introduction Given a set of points Z*in*a*normed**plane*⟨R 2 , || · ||⟩ and a real number p > 0, the geometric power-p*Steiner**tree**problem*(or geometric p-STP) seeks a finite set of points S ⊂ R 2 (the ...##
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A Flow-dependent Quadratic Steiner Tree Problem in the Euclidean Plane
[article]

2011
*
arXiv
*
pre-print

We introduce a flow-dependent version of the quadratic

arXiv:1111.2109v1
fatcat:t32ojbkc75fozfo3lwpn5kcvve
*Steiner**tree**problem**in*the*plane*. ... An instance of the*problem*on a set of embedded sources and a sink asks for a directed*tree*T spanning these nodes and a bounded number of*Steiner*points, such that ∑_e ∈ E(T)f(e)|e|^2 is a minimum, where ... Introduction Given a set of points Z*in*a*normed**plane*R 2 , || · || and a real number p > 0, the geometric power-p*Steiner**tree**problem*(or geometric p-STP) seeks a finite set of points S ⊂ R 2 (the*Steiner*...##
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Algorithms for the power-p Steiner tree problem in the Euclidean plane

2018
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Revista de Informática Teórica e Aplicada
*

We study the

doi:10.22456/2175-2745.80525
fatcat:nqkz73ciofdi5dwibz744fetn4
*problem*of constructing minimum power-$p$ Euclidean $*k*$-*Steiner**trees**in*the*plane*. ...*in*the*plane*. ... Therefore the power-p Euclidean*k*-*Steiner**tree**problem**generalises*the classical Euclidean*Steiner**tree**problem*(set p = 1 and*k*= n − 2). ...##
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Minimal curvature-constrained networks
[article]

2016
*
arXiv
*
pre-print

The Dubins network

arXiv:1606.02026v1
fatcat:yt2fqdlfknhlrawcgasq2yxify
*problem*is similar to the*Steiner**tree**problem*, except that the terminals are directed and there is a curvature constraint. ... We propose the minimum curvature-constrained*Steiner*point algorithm for determining the optimal location of the*Steiner*point*in*a 3-terminal network. ... A*Steiner**tree*with n − 2*Steiner*points is called a full*Steiner**tree*. Now consider the Dubins*problem**in*the*plane*. It is convenient to introduce some notation. ...##
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The total Steiner $k$-distance for $b$-ary recursive trees and linear recursive trees

2010
*
Discrete Mathematics & Theoretical Computer Science
*

The total

doi:10.46298/dmtcs.2779
fatcat:rks7vokpvvevfbxubcciov5up4
*Steiner*$*k*$-distance is the sum of all*Steiner*$*k*$-distances*in*a*tree*and it*generalises*the Wiener index. ...*In*a second step we prove a transformation of the total*Steiner*$*k*$-distance of $b$-ary*trees*with weighted edges to arbitrary recursive*trees*. ... Acknowledgements The author thanks Ralph Neininger for posing the*problem*and leading towards an approach to the*Steiner**k*-distance. ...
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