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

2008
*
Discrete Optimization
*

We describe

doi:10.1016/j.disopt.2007.08.006
fatcat:i3h6kiosvbcf3djae2u5h53djy
*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 ...##
###
An Efficient Rectilinear and Octilinear Steiner Minimal Tree Algorithm for Multidimensional Environments

2020
*
IEEE Access
*

This paper proposes

doi:10.1109/access.2020.2977825
fatcat:il2zkodgznephfq65d2wxcbgqi
*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. ...##
###
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]

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*. ...

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

2009
*
Networks
*

We suggest a novel distance-based method

doi:10.1002/net.20279
fatcat:phj2lobpy5hdncoaicf7wgzmji
*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*. ...##
###
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

2007
*
Canadian Conference on Computational Geometry
*

We present a constant approximation

dblp:conf/cccg/Lin07
fatcat:qcjwidlozzdrdme5egslh5ajhe
*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 ...##
###
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]

2009
*
Lecture Notes in Computer Science
*

of the longest edge

doi:10.1007/978-3-642-00202-1_10
fatcat:4iphryw6cnho3cphr2n7l3jerq
*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. ...##
###
On exact solutions to the Euclidean bottleneck Steiner tree problem

2010
*
Information Processing Letters
*

of the longest edge

doi:10.1016/j.ipl.2010.05.014
fatcat:qycwg2tnczdclkffy3kmgcwche
*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. ...##
###
A Parallel Steiner Tree Construction on the Server-Client Model of Computation

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 . ...

##
###
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]

2016
*
Lecture Notes in Computer Science
*

Some of the most efficient heuristics

doi:10.1007/978-3-319-38851-9_15
fatcat:7x4p6bn5s5cjdipzmr34rlvdam
*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 . ...##
###
Image Segmentation using Various Approaches

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. ...

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