The Steiner Tree Problem
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The Steiner Tree Problem
Author: Hans Jürgen Prömel
language: en
Publisher: Springer Science & Business Media
Release Date: 2012-12-06
"A very simple but instructive problem was treated by Jacob Steiner, the famous representative of geometry at the University of Berlin in the early nineteenth century. Three villages A,B ,C are to be joined by a system of roads of minimum length. " Due to this remark of Courant and Robbins (1941), a problem received its name that actually reaches two hundred years further back and should more appropriately be attributed to the French mathematician Pierre Fermat. At the end of his famous treatise "Minima and Maxima" he raised the question to find for three given points in the plane a fourth one in such a way that the sum of its distances to the given points is minimized - that is, to solve the problem mentioned above in its mathematical abstraction. It is known that Evangelista Torricelli had found a geometrical solution for this problem already before 1640. During the last centuries this problem was rediscovered and generalized by many mathematicians, including Jacob Steiner. Nowadays the term "Steiner prob lem" refers to a problem where a set of given points PI, . . . ,Pn have to be connected in such a way that (i) any two of the given points are joined and (ii) the total length (measured with respect to some predefined cost function) is minimized.
The Steiner Tree Problem
The Steiner problem asks for a shortest network which spans a given set of points. Minimum spanning networks have been well-studied when all connections are required to be between the given points. The novelty of the Steiner tree problem is that new auxiliary points can be introduced between the original points so that a spanning network of all the points will be shorter than otherwise possible. These new points are called Steiner points - locating them has proved problematic and research has diverged along many different avenues.This volume is devoted to the assimilation of the rich field of intriguing analyses and the consolidation of the fragments. A section has been given to each of the three major areas of interest which have emerged. The first concerns the Euclidean Steiner Problem, historically the original Steiner tree problem proposed by Jarník and Kössler in 1934. The second deals with the Steiner Problem in Networks, which was propounded independently by Hakimi and Levin and has enjoyed the most prolific research amongst the three areas. The Rectilinear Steiner Problem, introduced by Hanan in 1965, is discussed in the third part. Additionally, a forth section has been included, with chapters discussing areas where the body of results is still emerging.The collaboration of three authors with different styles and outlooks affords individual insights within a cohesive whole.
Spanning Trees and Optimization Problems
The design of approximation algorithms for spanning tree problems has become an exciting and important area of theoretical computer science and also plays a significant role in emerging fields such as biological sequence alignments and evolutionary tree construction. While work in this field remains quite active, the time has come to collect under