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.

IDENTIFYING STEINER MINIMAL TREES ON FOUR POINTS IN SPACE

2009 ◽  
Vol 01 (03) ◽  
pp. 401-411 ◽  
Author(s):  
J. F. WENG ◽  
D. A. THOMAS ◽  
I. MAREELS
Keyword(s):  
Euclidean Space ◽  
Euclidean Plane ◽  
Sufficient Conditions ◽  
Steiner Tree Problem ◽  
Minimum Length ◽  
Minimal Tree ◽  
Steiner Minimal Tree ◽  
Steiner Minimal Trees ◽  
Necessary And Sufficient ◽  
Set Of Points

A Steiner minimal tree is a network with minimum length spanning a given set of points in space. There are several criteria for identifying the Steiner minimal tree on four points in the Euclidean plane. However, it has been proved that the length of the Steiner minimal tree on four points cannot be computed using radicals if the four points lie in Euclidean space. This unsolvability implies that it is unlikely that similar necessary and sufficient conditions exist in the spatial case. Hence, a problem arises: Is it possible to generalize the known planar criteria to space in the sense that they are sufficient to identify Steiner minimal trees on four points in space? This problem is investigated and some sufficient conditions are proved in this paper. These sufficient conditions can help us to solve the general Steiner tree problem on n(> 4) points in Euclidean space.

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 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 Optimization of Cable Layout Design in a Wind Farm: A Hybrid Approach

2015 ◽  
Vol 11 (2) ◽  
Keyword(s):  
Convex Hull ◽  
Steiner Tree ◽  
Wind Farm ◽  
Multicast Routing ◽  
Hybrid Approach ◽  
Steiner Tree Problem ◽  
Steiner Minimal Tree ◽  
Minimal Graph ◽  
Short Time ◽  
Optimal Graph

In this paper, a hybrid algorithm based on modified Ants Colony Optimization (ACO) and Artificial Immune Algorithm (AIA) for solving the Steiner Minimal Tree Problem (SMTP) is introduced. Since the Steiner Tree Problem is NP-hard, we design an algorithm to construct high quality Steiner trees in a short time which is suitable for real time multicast routing in networks. After the breadth - first traversal of the minimal graph obtained by ACO, the terminal points are divided into different convex hull sets, and the full Steiner tree is structured from the convex hull sets partition. The Steiner points can be vaccinated by AIA to get an optimal graph. The average optimization effect of AIA is shorter than the minimal graph obtained using ACO, and the performance of the algorithm is shown. We give an example of application in wind farm network design.

Convex relaxation and variational approximation of the Steiner problem: theory and numerics

2018 ◽  
Vol 3 (1) ◽  
pp. 19-27 ◽  
Author(s):  
M. Bonafini
Keyword(s):  
Steiner Tree ◽  
Convex Relaxation ◽  
Steiner Tree Problem ◽  
Variational Approximation ◽  
Steiner Problem ◽  
Convex Relaxations ◽  
Numerical Schemes ◽  
Steiner Minimal Trees ◽  
Euclidean Steiner Tree Problem

Abstract We survey some recent results on convex relaxations and a variational approximation for the classical Euclidean Steiner tree problem and we see how these new perspectives can lead to effective numerical schemes for the identification of Steiner minimal trees.

scholarly journals Robust Reoptimization of Steiner Trees

Algorithmica ◽  
2020 ◽  
Vol 82 (7) ◽  
pp. 1966-1988
Author(s):  
Keshav Goyal ◽  
Tobias Mömke
Keyword(s):  
Steiner Tree ◽  
Optimal Solution ◽  
Approximate Solutions ◽  
Problem Instance ◽  
Steiner Tree Problem ◽  
Performance Guarantees ◽  
Improved Performance ◽  
Algorithmic Technique ◽  
The Cost ◽  
The Given

Abstract In reoptimization, one is given an optimal solution to a problem instance and a (locally) modified instance. The goal is to obtain a solution for the modified instance. We aim to use information obtained from the given solution in order to obtain a better solution for the new instance than we are able to compute from scratch. In this paper, we consider Steiner tree reoptimization and address the optimality requirement of the provided solution. Instead of assuming that we are provided an optimal solution, we relax the assumption to the more realistic scenario where we are given an approximate solution with an upper bound on its performance guarantee. We show that for Steiner tree reoptimization there is a clear separation between local modifications where optimality is crucial for obtaining improved approximations and those instances where approximate solutions are acceptable starting points. For some of the local modifications that have been considered in previous research, we show that for every fixed $$\varepsilon > 0$$ ε > 0 , approximating the reoptimization problem with respect to a given $$(1+\varepsilon )$$ ( 1 + ε ) -approximation is as hard as approximating the Steiner tree problem itself. In contrast, with a given optimal solution to the original problem it is known that one can obtain considerably improved results. Furthermore, we provide a new algorithmic technique that, with some further insights, allows us to obtain improved performance guarantees for Steiner tree reoptimization with respect to all remaining local modifications that have been considered in the literature: a required node of degree more than one becomes a Steiner node; a Steiner node becomes a required node; the cost of one edge is increased.

scholarly journals The Euclidean Steiner Tree Problem: Simulated Annealing and Other Heuristics

2021 ◽  
Author(s):  
◽  
Geoffrey Ross Grimwood
Keyword(s):  
Simulated Annealing ◽  
Steiner Tree ◽  
Heuristic Method ◽  
Distribution Networks ◽  
Network Design Problem ◽  
Steiner Tree Problem ◽  
Viable Solution ◽  
Euclidean Steiner Tree Problem ◽  
Set Of Points

<p>In this thesis the Euclidean Steiner tree problem and the optimisation technique called simulated annealing are studied. In particular, there is an investigation of whether simulated annealing is a viable solution method for the problem. The Euclidean Steiner tree problem is a topological network design problem and is relevant to the design of communication, transportation and distribution networks. The problem is to find the shortest connection of a set of points in the Euclidean plane. Simulated annealing is a generally applicable method of finding solutions of combinatorial optimisation problems. The results of the investigation are very satisfactory. The quality of simulated annealing solutions compare favourably with those of the best known tailored heuristic method for the Euclidean Steiner tree problem</p>

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