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An improved algorithm for computing Steiner minimal trees in Euclidean d-space

Marcia Fampa, Kurt M. Anstreicher
2008 Discrete Optimization  
We describe improvements to Smith's branch-and-bound (B&B) algorithm for the Euclidean Steiner problem in R d .  ...  For a given topology we use a conic formulation for the problem of locating the Steiner points to obtain a rigorous lower bound on the minimal tree length.  ...  Introduction The Euclidean Steiner tree problem (ESTP) in R d is defined as follows: Given a set of points in R d , find a tree of minimal Euclidean length that spans these points but that can also utilize  ... 
doi:10.1016/j.disopt.2007.08.006 fatcat:i3h6kiosvbcf3djae2u5h53djy

An Efficient Rectilinear and Octilinear Steiner Minimal Tree Algorithm for Multidimensional Environments

Ming Che Lee, Gene Eu Jan, Chung Chin Luo
2020 IEEE Access  
This paper proposes an obstacle-avoiding 4/8/10/26-directional heuristic algorithm for this problem based on the Areibi's concept, Higher Geometry Maze Routing, and Sollin's minimal spanning tree algorithm  ...  INDEX TERMS Higher geometry maze routing, octilinear, rectilinear, Steiner tree.  ...  [24] proposed an obstacle-avoiding rectilinear Steiner minimal tree (OARSMT) construction algorithm that achieves an optimal solution for two-pin net.  ... 
doi:10.1109/access.2020.2977825 fatcat:il2zkodgznephfq65d2wxcbgqi

Page 6572 of Mathematical Reviews Vol. , Issue 2000i [page]

2000 Mathematical Reviews  
Our algorithm provides the first polynomial-time approxima- tion scheme for k-median instances in d-dimensional Euclidean space for any fixed d>2.  ...  “In this paper we provide a randomized approximation scheme for points in d-dimensional Euclidean space, with running time O(2'/*“nlognlogk), which is nearly linear for any fixed ¢ and d.  ... 

Deep-Steiner: Learning to Solve the Euclidean Steiner Tree Problem [article]

Siqi Wang, Yifan Wang, Guangmo Tong
2022 arXiv   pre-print
In this paper, we design discretization methods by leveraging the unique characteristics of the Steiner tree, and propose new training schemes for handling the dynamic Steiner points emerging during the  ...  In this paper, we present a study on solving the Euclidean Steiner tree problem using reinforcement learning enhanced by graph representation learning.  ...  Such problems can often be nicely reduced to the classic Euclidean Steiner tree (EST) problem, where we seek to connect a collection of points in the Euclidean space using Steiner minimal trees.  ... 
arXiv:2209.09983v1 fatcat:5jkj5e5axrenlcemdnyz3v523q

A novel approach to phylogenetic trees:d-Dimensional geometric Steiner trees

M. Brazil, D.A. Thomas, B.K. Nielsen, P. Winter, C. Wulff-Nilsen, M. Zachariasen
2009 Networks  
We suggest a novel distance-based method for the determination of phylogenetic trees. It is based on multidimensional scaling and Euclidean Steiner trees in high-dimensional spaces.  ...  Preliminary computational experience shows that the use of Euclidean Steiner trees for finding phylogenetic trees is a viable approach.  ...  This tree is an approximate solution to the d-dimensional Euclidean Steiner tree problem to determine the shortest tree interconnecting n given points in a d-dimensional space.  ... 
doi:10.1002/net.20279 fatcat:phj2lobpy5hdncoaicf7wgzmji

Page 5898 of Mathematical Reviews Vol. , Issue 97I [page]

1997 Mathematical Reviews  
For a given topology, we give a direct, geometric algorithm that computes an optimal rectilinear bottleneck Steiner tree in O(n’) time, which improves on the time complexity of previous algorithms.  ...  “In this paper, we first consider algorithms for computing op- timal bottleneck Steiner trees.  ... 

Terminal Steiner Tree with Bounded Edge Length

Zhiyong Lin
2007 Canadian Conference on Computational Geometry  
We present a constant approximation algorithm for this problem in this paper.  ...  In this paper, we study a Steiner tree related problem called "Terminal Steiner Tree with Bounded Edge Length": given a set of terminal points P in a plane, one is asked to find a Steiner tree T such that  ...  Then there is a polynomial time approximation scheme H d such that for every given natural number l ≥ 1, the algorithm H d l delivers a cover of n given points in a d-dimension Euclidean space by d-dimensional  ... 
dblp:conf/cccg/Lin07 fatcat:qcjwidlozzdrdme5egslh5ajhe

Page 3626 of Mathematical Reviews Vol. , Issue 95f [page]

1995 Mathematical Reviews  
Summary: “The geometric Steiner minimum tree problem (GSMT) is to connect at minimum cost n given points (called terminals) in d-dimensional Euclidean space.  ...  partitioning algorithm for the Steiner tree problem in R¢.  ... 

On Exact Solutions to the Euclidean Bottleneck Steiner Tree Problem [chapter]

Sang Won Bae, Chunseok Lee, Sunghee Choi
2009 Lecture Notes in Computer Science  
of the longest edge in the tree is minimized.  ...  We study the Euclidean bottleneck Steiner tree problem: given a set P of n points in the Euclidean plane and a positive integer k, find a Steiner tree with at most k Steiner points such that the length  ...  Acknowledgement The authors thank anonymous referees for valuable comments to improve our paper.  ... 
doi:10.1007/978-3-642-00202-1_10 fatcat:4iphryw6cnho3cphr2n7l3jerq

On exact solutions to the Euclidean bottleneck Steiner tree problem

Sang Won Bae, Chunseok Lee, Sunghee Choi
2010 Information Processing Letters  
of the longest edge in the tree is minimized.  ...  We study the Euclidean bottleneck Steiner tree problem: given a set P of n points in the Euclidean plane and a positive integer k, find a Steiner tree with at most k Steiner points such that the length  ...  Acknowledgement The authors thank anonymous referees for valuable comments to improve our paper.  ... 
doi:10.1016/j.ipl.2010.05.014 fatcat:qycwg2tnczdclkffy3kmgcwche

A Parallel Steiner Tree Construction on the Server-Client Model of Computation

Rashid Bin Muhammad
2008 Fifth International Conference on Information Technology: New Generations (itng 2008)  
In particular, this work analyzes the time delays among server and clients for the parallel Steiner tree algorithm and examines some consequences of time delays on the execution time by considering constant  ...  This paper presents the computational analysis of the Steiner tree problem on the Server-Client model of computation.  ...  Introduction A Steiner minimal tree (SMT) for a given set P of points in the Euclidean plane is the shortest interconnecting P .  ... 
doi:10.1109/itng.2008.167 dblp:conf/itng/Muhammad08a fatcat:5d57rcg76bcs3doflvcuahdffq

Page 3916 of Mathematical Reviews Vol. , Issue 2003e [page]

2003 Mathematical Reviews  
Summary (translated from the Russian): “For the Stoyan-Yakovlev algorithm of convex continuation of polynomials for a set of polypermutations in an arithmetic space, we give a modification for the case  ...  The bottleneck Steiner tree problem consists in connecting n given terminal nodes in the (rectilinear) plane by a tree of at most n + k nodes so that the length of the longest edge is minimized.  ... 

Page 398 of Mathematical Reviews Vol. , Issue 93a [page]

1993 Mathematical Reviews  
Werschulz (New York) 93a:68064 68Q35 05C05 68Q25 68RI10 Smith, Warren D. (1-BELL) How to find Steiner minimal trees in Euclidean d-space. The Steiner problem. Algorithmica 7 (1992), no. 2-3, 137-177.  ...  The paper provides several interesting results on the problem of finding a shortest Steiner tree for a collection of N points in a d- dimensional Euclidean space.  ... 

Steiner Tree Heuristic in the Euclidean d-Space Using Bottleneck Distances [chapter]

Stephan S. Lorenzen, Pawel Winter
2016 Lecture Notes in Computer Science  
Some of the most efficient heuristics for the Euclidean Steiner minimal trees in the d-dimensional space, d ≥ 2, use Delaunay tessellations and minimum spanning trees to determine small subsets of geometrically  ...  Computational experiments show a significant solution quality improvement. This is an extended abstract of a presentation given at EuroCG 2016.  ...  ., t n } in the Euclidean d-dimensional space R d , d ≥ 2, the Euclidean Steiner minimal tree (ESMT) problem asks for a shortest connected network T = (V, E), where N ⊆ V .  ... 
doi:10.1007/978-3-319-38851-9_15 fatcat:7x4p6bn5s5cjdipzmr34rlvdam

Image Segmentation using Various Approaches

Snehal B. Ranit, Dr. Nileshsingh V. Thakur
2013 International Journal of Image Processing and Vision Science  
In all these three approaches, finally the Steiner tree is created to identify the class of the region. For this purpose the Euclidean distance is used.  ...  Scope of improvement exists in the following areas: Image partitioning; color based feature; texture based feature, searching mechanism for similarity; cluster formation logic; pixel connectivity criterion  ...  Hougardy and Promel [9] , discussed an approximation algorithm for the Steiner Problem in graphs. Robins and Zelikovski [10] proposed an approximation algorithm for Steiner Tree in graphs.  ... 
doi:10.47893/ijipvs.2013.1074 fatcat:3674kqy5gjegvb7duqbri6yibm
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