On the minimum number of Steiner points of constrained 1-line-fixed Steiner tree in the Euclidean plane $${\mathbb {R}}^2$$

2020 ◽  
Author(s):  
Jianping Li ◽  
Yujie Zheng ◽  
Junran Lichen ◽  
Wencheng Wang
Keyword(s):  
Steiner Tree ◽  
Euclidean Plane ◽  
Steiner Points ◽  
Minimum Number

COMPUTING STEINER POINTS AND PROBABILITY STEINER POINTS IN ℓ1 AND ℓ2 METRIC SPACES

2009 ◽  
Vol 01 (04) ◽  
pp. 541-554
Author(s):  
J. F. WENG ◽  
I. MAREELS ◽  
D. A. THOMAS
Keyword(s):  
Steiner Tree ◽  
Euclidean Plane ◽  
Metric Spaces ◽  
Steiner Tree Problem ◽  
Smooth Functions ◽  
Steiner Points ◽  
New Type ◽  
Point Set ◽  
Steiner Minimum Tree ◽  
The Given

The Steiner tree problem is a well known network optimization problem which asks for a connected minimum network (called a Steiner minimum tree) spanning a given point set N. In the original Steiner tree problem the given points lie in the Euclidean plane or space, and the problem has many variants in different applications now. Recently a new type of Steiner minimum tree, probability Steiner minimum tree, is introduced by the authors in the study of phylogenies. A Steiner tree is a probability Steiner tree if all points in the tree are probability vectors in a vector space. The points in a Steiner minimum tree (or a probability Steiner tree) that are not in the given point set are called Steiner points (or probability Steiner points respectively). In this paper we investigate the properties of Steiner points and probability Steiner points, and derive the formulae for computing Steiner points and probability Steiner points in ℓ1- and ℓ2-metric spaces. Moreover, we show by an example that the length of a probability Steiner tree on 3 points and the probability Steiner point in the tree are smooth functions with respect to p in d-space.

scholarly journals On compatible triangulations with a minimum number of Steiner points

2020 ◽  
Vol 835 ◽  
pp. 97-107
Author(s):  
Anna Lubiw ◽  
Debajyoti Mondal
Keyword(s):  
Steiner Points ◽  
Minimum Number

GENERALIZED MELZAK'S CONSTRUCTION IN THE STEINER TREE PROBLEM

2002 ◽  
Vol 12 (06) ◽  
pp. 481-488 ◽  
Author(s):  
JIA F. WENG
Keyword(s):  
Steiner Tree ◽  
Euclidean Plane ◽  
Steiner Tree Problem ◽  
Minimal Tree ◽  
Steiner Minimal Tree ◽  
The Core ◽  
Steiner Minimal Trees ◽  
Minimum Networks ◽  
Set Of Points ◽  
The Given

For a given set of points in the Euclidean plane, a minimum network (a Steiner minimal tree) can be constructed using a geometric method, called Melzak's construction. The core of the Melzak construction is to replace a pair of terminals adjacent to the same Steiner point with a new terminal. In this paper we prove that the Melzak construction can be generalized to constructing Steiner minimal trees for circles so that either the given points (terminals) are constrained on the circles or the terminal edges are tangent to the circles. Then we show that the generalized Melzak construction can be used to find minimum networks separating and surrounding circular objects or to find minimum networks connecting convex and smoothly bounded objects and avoiding convex and smoothly bounded obstacles.

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

Networks ◽  
2014 ◽  
Vol 64 (1) ◽  
pp. 18-28 ◽  
Author(s):  
Marcus N. Brazil ◽  
Charl J. Ras ◽  
Doreen A. Thomas
Keyword(s):  
Steiner Tree ◽  
Euclidean Plane ◽  
Steiner Tree Problem

scholarly journals Fault-Tolerant Placement of Additional Mesh Nodes in Rural Wireless Mesh Networks: A Minimum Steiner Tree Based Centre of Mass With Bounded Edge Length

2021 ◽  
Vol 9 (4) ◽  
pp. 39-50
Author(s):  
Jean Louis Kedieng Ebongue Fendji ◽  
Patience Leopold Bagona
Keyword(s):  
Digital Divide ◽  
Wireless Mesh Networks ◽  
Steiner Tree ◽  
Mesh Networks ◽  
Fault Tolerant ◽  
Edge Length ◽  
Centre Of Mass ◽  
Steiner Points ◽  
Minimum Steiner Tree ◽  
Wireless Mesh

Wireless mesh networks are presented as an attractive solution to reduce the digital divide between rural and developed areas. In a multi-hop fashion, they can cover larger spaces. However, their planning is subject to many constraints including robustness. In fact, the failure of a node may result in the partitioning of the network. The robustness of the network is therefore achieved by carefully placing additional nodes. This work tackles the problem of additional nodes minimization when planning bi and tri-connectivity from a given network. We propose a vertex augmentation approach inspired by the placement of Steiner points. The idea is to incrementally determine cut vertices and bridges in the network and to carefully place additional nodes to ensure connectivity, bi and tri-connectivity. The approach relies on an algorithm using the centre of mass of the blocks derived after the partitioning of the network. The proposed approach has been compared to a modified version of a former approach based on the Minimum Steiner Tree. The different experiments carried out show the competitiveness of the proposed approach to connect, bi-connect, and tri-connect the wireless mesh networks.

scholarly journals Hybrid greedy genetic algorithm for the Euclidean Steiner tree problem

2019 ◽  
Author(s):  
Andrey Oliveira ◽  
Danilo Sanches ◽  
Bruna Osti
Keyword(s):  
Genetic Algorithm ◽  
Steiner Tree ◽  
Optimization Problem ◽  
Spanning Trees ◽  
Evolutionary Strategies ◽  
Steiner Tree Problem ◽  
Minimum Length ◽  
Genetic Operators ◽  
Steiner Points ◽  
Euclidean Steiner Tree Problem

This paper presents a genetic algorithm for the Euclidean Steiner tree problem. This is an optimization problem whose objective is to obtain a minimum length tree to interconnect a set of fixed points, and for this purpose to be achieved, new auxiliary points, called Steiner points, can be added. The proposed heuristic uses a genetic algorithm to manipulate spanning trees, which are then transformed into Steiner trees by inserting and repositioning the Steiner points. Greedy genetic operators and evolutionary strategies are tested. Results of numerical experiments for benchmark library problem (OR-Library) are presented and discussed.This paper presents a genetic algorithm for the Euclidean Steiner tree problem. This is an optimization problem whose objective is to obtain a minimum length tree to interconnect a set of fixed points, and for this purpose to be achieved, new auxiliary points, called Steiner points, can be added. The proposed heuristic uses a genetic algorithm to manipulate spanning trees, which are then transformed into Steiner trees by inserting and repositioning the Steiner points. Greedy genetic operators and evolutionary strategies are tested. Results of numerical experiments for benchmark library problem (OR-Library) are presented and discussed.

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