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.

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 Fixed Points of g-Interpolative Ćirić–Reich–Rus-Type Contractions in b-Metric Spaces

Axioms ◽  
2020 ◽  
Vol 9 (4) ◽  
pp. 132
Author(s):  
Youssef Errai ◽  
El Miloudi Marhrani ◽  
Mohamed Aamri
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Common Fixed Point ◽  
Metric Space ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
New Type ◽  
The Given ◽  
Type Contraction

We use interpolation to obtain a common fixed point result for a new type of Ćirić–Reich–Rus-type contraction mappings in metric space. We also introduce a new concept of g-interpolative Ćirić–Reich–Rus-type contractions in b-metric spaces, and we prove some fixed point results for such mappings. Our results extend and improve some results on the fixed point theory in the literature. We also give some examples to illustrate the given results.

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 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.

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.

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