Filters








225 Hits in 4.4 sec

Generalised k-Steiner Tree Problems in Normed Planes

Marcus Brazil, Charl J. Ras, Konrad J. Swanepoel, Doreen A. Thomas
2013 Algorithmica  
We also extend their approach further to encompass other 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.  ... 
doi:10.1007/s00453-013-9780-5 fatcat:hsoebwee55hopncq53g7ony6h4

Designing Optimal Flow Networks [article]

M. G. Volz, M. Brazil, K. J. Swanepoel, D. A. Thomas
2009 arXiv   pre-print
We characterise the local topological structure of 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 .  ... 
arXiv:0903.2124v1 fatcat:kv7n6s4objcvni4yymqha5yadm

Survivable minimum bottleneck networks

C.J. Ras
2015 Computational geometry  
We show that the survivable bottleneck 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.  ... 
doi:10.1016/j.comgeo.2015.06.002 fatcat:f7sasho7bje4tka6b7bral4qv4

Steiner trees for fixed orientation metrics

M. Brazil, M. Zachariasen
2008 Journal of Global Optimization  
We consider the 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.  ... 
doi:10.1007/s10898-008-9305-y fatcat:qxgpo43vkfhqvn6u5u3z4lvfaq

The Gilbert arborescence problem

M. G. Volz, M. Brazil, C. J. Ras, K. J. Swanepoel, D. A. Thomas
2012 Networks  
We investigate the 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  ... 
doi:10.1002/net.21475 fatcat:tgg3e35f2bexjcxuc6cdvimq2i

Quantitative illumination of convex bodies and vertex degrees of geometric Steiner minimal trees [article]

Konrad J Swanepoel
2004 arXiv   pre-print
Let v(d) be the maximum degree of a vertex, and s(d) of a 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] .  ... 
arXiv:math/0410144v1 fatcat:tdgm6rmawzcxvhwbhrkgldufla

The Local Steiner Problem in Finite-Dimensional Normed Spaces

Konrad J. Swanepoel
2007 Discrete & Computational Geometry  
We develop a general method of proving that certain star configurations 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] .  ... 
doi:10.1007/s00454-006-1298-z fatcat:hd3ipc77tfbqvay6ob2oqljkgq

Canonical Forms and Algorithms for Steiner Trees in Uniform Orientation Metrics

M. Brazil, D. A. Thomas, J. F. Weng, M. Zachariasen
2005 Algorithmica  
We present some fundamental structural properties for minimum length networks (known as 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] .)  ... 
doi:10.1007/s00453-005-1178-6 fatcat:bh4sh33txrdm5h43gmnwfpnizy

The bottleneck 2-connected k-Steiner network problem for k≤2

M. Brazil, C.J. Ras, D.A. Thomas
2012 Discrete Applied Mathematics  
The geometric bottleneck 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  ... 
doi:10.1016/j.dam.2012.01.006 fatcat:yjbf5h6gyzgsdpx2q6drsd2rrm

The bottleneck 2-connected k-Steiner network problem for k≤ 2 [article]

M. Brazil, C.J. Ras, D.A. Thomas
2011 arXiv   pre-print
The geometric bottleneck 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  ... 
arXiv:1108.3655v1 fatcat:xixblw44yne23aoyiofu5c73l4

A flow-dependent quadratic steiner tree problem in the Euclidean plane

Marcus N. Brazil, Charl J. Ras, Doreen A. Thomas
2014 Networks  
We introduce a flow-dependent version of the quadratic 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  ... 
doi:10.1002/net.21553 fatcat:s3so3k6xdvehdiwbbkcavp3hei

A Flow-dependent Quadratic Steiner Tree Problem in the Euclidean Plane [article]

Marcus Brazil, Charl Ras, Doreen Thomas
2011 arXiv   pre-print
We introduce a flow-dependent version of the quadratic 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  ... 
arXiv:1111.2109v1 fatcat:t32ojbkc75fozfo3lwpn5kcvve

Algorithms for the power-p Steiner tree problem in the Euclidean plane

Christina Burt, Alysson Costa, Charl Ras
2018 Revista de Informática Teórica e Aplicada  
We study the 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).  ... 
doi:10.22456/2175-2745.80525 fatcat:nqkz73ciofdi5dwibz744fetn4

Minimal curvature-constrained networks [article]

David Kirszenblat, Kashyapa Sirinanda, Marcus Brazil, Peter Grossman, J. Hyam Rubinstein, Doreen Thomas
2016 arXiv   pre-print
The Dubins network 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.  ... 
arXiv:1606.02026v1 fatcat:yt2fqdlfknhlrawcgasq2yxify

The total Steiner $k$-distance for $b$-ary recursive trees and linear recursive trees

Götz Olaf Munsonius
2010 Discrete Mathematics & Theoretical Computer Science  
The total 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.  ... 
doi:10.46298/dmtcs.2779 fatcat:rks7vokpvvevfbxubcciov5up4
« Previous Showing results 1 — 15 out of 225 results