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Length of a Full Steiner Tree as a Function of Terminal Coordinates [article]

Alexei Yu. Uteshev, Elizaveta A. Semenova
2021 arXiv   pre-print
Given the coordinates of the terminals {(x_j,y_j)}_j=1^n of the full Euclidean Steiner tree, its length equals | ∑_j=1^n z_j U_j | , where {z_j:=x_j+ 𝐢 y_j}_j=1^n and {U_j}_j=1^n are suitably chosen 6th  ...  We also extend this result for the cost of the optimal Weber networks which are topologically equivalent to some full Steiner trees.  ...  The first aim of the present paper is express the length of a full Steiner tree as an explicit function of the terminal coordinates.  ... 
arXiv:2102.03303v1 fatcat:lqumxrgcefgkxbi4nxhklhn2ra

Some Analytics for Steiner Minimal Tree Problem for Four Terminals [article]

Alexei Yu. Uteshev
2015 arXiv   pre-print
Given the coordinates of four terminals in the Euclidean plane we present explicit formulas for Steiner point coordinates for Steiner minimal tree problem.  ...  We utilize the obtained formulas for evaluation of the influence of terminal coordinates on the loci of Steiner points.  ...  The author appreciate the courtesy of Elisabeth Semenova for designing Fig. 7 . This research was supported by the St.Petersburg State University research grant 9.38.674.2013.  ... 
arXiv:1505.03564v1 fatcat:ah2hc2dr6vbznjbiwzrbrwslni

Approximations and Lower Bounds for the Length of Minimal Euclidean Steiner Trees

J. H. Rubinstein, J. Weng, N. Wormald
2006 Journal of Global Optimization  
We give a new lower bound on the length of the minimal Steiner tree with a given topology joining given terminals in Euclidean space, in terms of toroidal images.  ...  We use the lower bound to prove bounds on the "error" e in the length of an approximate Steiner tree, in terms of the maximum deviation d of an interior angle of the tree from 120 • .  ...  tom gives a local maximum of the length function and a Steiner tree with length d(t n , m).  ... 
doi:10.1007/s10898-005-4207-8 fatcat:qsurxe7ldzcdvdfytn4grkuy6y

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  ...  Since the distance function in -geometry is a norm, it follows that ¡ ¡ , treated as a function of the positions of the Steiner points of , is a convex function.  ...  These nodes are referred to as Steiner points. A -tree is said to be full if all terminals have degree ¤ . Clearly any -tree can be decomposed into a union of full subtrees meeting only at terminals.  ... 
doi:10.1007/s00453-005-1178-6 fatcat:bh4sh33txrdm5h43gmnwfpnizy

Rectilinear Steiner Trees in Narrow Strips [article]

Henk Alkema, Mark de Berg
2021 arXiv   pre-print
The goal of Minimal Rectilinear Steiner Tree is to find a rectilinear Steiner tree with minimal total length.  ...  A rectilinear Steiner tree for a set P of points in ℝ^2 is a tree that connects the points in P using horizontal and vertical line segments.  ...  Introduction In the Minimum Steiner Tree problem in the plane, we are given as input a set P of points in the plane, called terminals, and the goal is to find a minimum-length tree that connects the terminals  ... 
arXiv:2103.08354v1 fatcat:zozhwmlf6ratpjopb3zgaktdby

On uniqueness in Steiner problem [article]

Mikhail Basok, Danila Cherkashin, Nikita Rastegaev, Yana Teplitskaya
2018 arXiv   pre-print
Moreover, we show that the Hausdorff dimension of n-points configurations on which some locally minimal trees have the same length is also at most 2n-1.  ...  We prove that the set of n-point configurations for which solution of the planar Steiner problem is not unique has Hausdorff dimension is at most 2n-1.  ...  Then while the tree is not full we turn one of the parts around the terminal of degree two as shown at the right two parts of Fig. 2 . Now let us show how to swap to points A and B in a full tree.  ... 
arXiv:1809.01463v2 fatcat:jxoyhgtbjfduxhahvnh5aq7c3y

Overlaid oriented Voronoi diagrams and the 1-Steiner tree problem [article]

Michael S. Payne, Charl Ras, Marcus Volz
2020 arXiv   pre-print
We then study the effect that the OOVD data has in reducing the complexity of 1-Steiner tree construction when compared to a naive approach.  ...  Overlaid oriented Voronoi diagrams (OOVDs) are known to provide useful data for the construction of optimal Euclidean 1-Steiner trees.  ...  Introduction The Euclidean Steiner tree problem is to construct the network with shortest total edge length that connects a set of input terminals in the plane.  ... 
arXiv:2002.06752v1 fatcat:cqagbi5rv5fizndoxequsgnk2m

Creating and exploiting flexibility in rectilinear steiner trees

E. Bozorgzadeh, R. Kastner, M. Sarrafzadeh
2003 IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems  
The main contribution of this paper is an algorithm which takes a stable Steiner tree as an input and maps it to a more flexible Steiner tree.  ...  We show that the flexibility of a Steiner tree is related to its routability.  ...  Stroobandt for his valuable feedback regarding various aspect of this work. They are also grateful to the reviewers for their detailed comments, technically and conceptually.  ... 
doi:10.1109/tcad.2003.810747 fatcat:kd3aa63cbvagrmynt52t4ogmp4

Approximating minimum Steiner point trees in Minkowski planes

M. Brazil, C. J. Ras, D. A. Thomas
2010 Networks  
Given a set of points, we define a minimum Steiner point tree to be a tree interconnecting these points and possibly some additional points such that the length of every edge is at most 1 and the number  ...  We also introduce a new canonical form for minimum Steiner point trees in the Euclidean plane; this demonstrates that minimum Steiner point trees are shortest total length trees with a certain discrete-edge-length  ...  The authors wish to thank Jamie Evans for partaking in many fruitful discussions during the development of this paper. We would also like to thank the referees for their insightful comments.  ... 
doi:10.1002/net.20376 fatcat:f3wpqkvkojhexh75kr3h6w5hbu

Approximation of Octilinear Steiner Trees Constrained by Hard and Soft Obstacles [chapter]

Matthias Müller-Hannemann, Anna Schulze
2006 Lecture Notes in Computer Science  
In this paper, we consider two versions of the shortest octilinear Steiner tree problem for a given point set K of terminals in the plane: (1) a version in the presence of hard octilinear obstacles, and  ...  But if the Steiner tree intersects some soft obstacle, then no connected component of the induced subtree may be longer than a given fixed length L.  ...  Acknowledgment The first author was partially supported by the DFG Focus Program 1126 "Algorithmic Aspects of Large and Complex Networks", grant Mu1482/2-2.  ... 
doi:10.1007/11785293_24 fatcat:gpyx2dx63fhonbukptbyg3vwdm

Approximating Minimum Steiner Point Trees in Minkowski Planes [article]

M. Brazil, C. J. Ras, D. A. Thomas
2013 arXiv   pre-print
Given a set of points, we define a minimum Steiner point tree to be a tree interconnecting these points and possibly some additional points such that the length of every edge is at most 1 and the number  ...  We also introduce a new canonical form for minimum Steiner point trees in the Euclidean plane; this demonstrates that minimum Steiner point trees are shortest total length trees with a certain discrete-edge-length  ...  The authors wish to thank Jamie Evans for partaking in many fruitful discussions during the development of this paper. We would also like to thank the referees for their insightful comments.  ... 
arXiv:1307.2987v1 fatcat:dbita6gtwbbhjh7shhnvcgg24e

Computing steiner minimum trees in Hamming metric

Ernst Althaus, Rouven Naujoks
2006 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm - SODA '06  
Computing Steiner minimum trees in Hamming metric is a well studied problem that has applications in several fields of science such as computational linguistics and computational biology.  ...  Among all methods for finding such trees, algorithms using variations of a branch and bound method developed by Penny and Hendy have been the fastest for more than 20 years.  ...  Given a set T ⊆ U of required points (terminals) in an universe U and a cost function c : U × U → R, a Steiner tree is a tree connecting T ∪ S for a subset S ⊆ U .  ... 
doi:10.1145/1109557.1109578 fatcat:6jyoht6ohrdrbbb6a34aogd5gi

Performance Analysis of the Algorithms forthe Construction of Multilayer Obstacle Avoiding Rectilinear Steiner Minimum Tree

Divyaprabha Divyaprabha, Dr. Prasad G.R
2014 IOSR Journal of Electrical and Electronics Engineering  
The main aim of routing in VLSI design is to interconnect the cells that have been assigned positions as a solution of the placement problem.  ...  This paper provides a survey of various multilayer obstacles avoiding rectilinear Steiner minimal tree algorithms proposed and thus there is a need for an algorithm to produce better solution quality (  ...  For a graph G (V, E) and a terminal set s cv , a minimum terminal spanning tree(MTST) connects all the vertices in s using a set of terminal paths among vertices in s such that the sum of lengths of those  ... 
doi:10.9790/1676-09610512 fatcat:sgwwxg3rmrgfhpw6j75ruvfsju

Additive Approximation for Near-Perfect Phylogeny Construction [chapter]

Pranjal Awasthi, Avrim Blum, Jamie Morgenstern, Or Sheffet
2012 Lecture Notes in Computer Science  
The problem is formulated as that of finding a minimum Steiner tree on n points over the Boolean hypercube of dimension d.  ...  We study the problem of constructing phylogenetic trees for a given set of species.  ...  In such cases, one would much prefer an algorithm whose excess could be written as a function of q only.  ... 
doi:10.1007/978-3-642-32512-0_3 fatcat:vzogjruyo5bhzkrrm3azvrf6xe

Additive Approximation for Near-Perfect Phylogeny Construction [article]

Pranjal Awasthi, Avrim Blum, Jamie Morgenstern, Or Sheffet
2012 arXiv   pre-print
The problem is formulated as that of finding a minimum Steiner tree on n points over the Boolean hypercube of dimension d.  ...  We study the problem of constructing phylogenetic trees for a given set of species.  ...  In such cases, one would much prefer an algorithm whose excess could be written as a function of q only.  ... 
arXiv:1206.3334v1 fatcat:7h43i6eqeneqfabmiv7uxdvvxi
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