Steiner Minimal Trees

Steiner Minimal Trees

Nonconvex Optimization and Its Applications Volume 23 Managing Editors: Panos Pardalos University of Florida, U.S.A. Reiner Horst University of Trier, Germany Advisory Board: Ding-Zhu Du University of Minnesota, U.S.A. C.A. Floudas Princeton University, U.S.A. G. Infanger Stanford University, U.S.A. J.Mockus Lithuanian Academy of Sciences, Lithuania P.D. Panagiotopoulos Aristotle University, Greece H.D. Sherali Virginia Polytechnic Institute and State University, U.S.A. The titles published in this series are listed at the end of this volume.

Steiner Minimal Trees by Dietmar Cieslik Ernst-Moritz-Arndt University, Greifswald, Germany... '' SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-1-4419-4790-1 ISBN 978-1-4757-6585-4 (ebook) DOI 10.1007/978-1-4757-6585-4 Printed on acid-free paper All Rights Reserved 1998 Springer Science+ Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1998 Softcover reprint of the hardcover 1st edition 1998 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner

CONTENTS PREFACE 1 INTRODUCTION 1.1 The Historical Genesis 1.2 Networks 1.3 Spaces 1.4 Steiner's Problem 1.5 How to Attack Steiner's Problem lx 2 8 15 42 46 2 3 SMT AND MST IN METRIC SPACES- A SURVEY 2.1 Basic Properties 2.2 The Algorithmic Complexity to find an MST 2.3 Examples 2.4 Approximations and Heuristics FERMAT'S PROBLEM IN BANACH MINKOWSKI SPACES 3.1 Introduction to Fermat's Problem 3.2 Hulls of a Finite Set and the Set of Torricelli Points 3.3 Fermat's Problem in Several Specific Planes 3.4 Geometric Constructions for Torricelli Points in Banach Minkowski Planes 3.5 General Solution Methods 3.6 Generalizations 49 50 54 71 83 91 92 95 99 110 112 122

VI STEINER MINIMAL TREES 4 THE DEGREES OF THE VERTICES IN SHORTEST TREES 125 4.1 Upper Bounds for the Degrees of the Vertices 125 4.2 The Degrees of Vertices in a k-smt 128 4.3 The Degrees of Vertices in an SMT 137 4.4 The First Special Case: Finite Dimensional.Cp-Spaces 143 4.5 The Second Special Case: Banach-Minkowski Planes 150 4.6 Degree Constraint Trees 154 4.7 Further Results and Remarks 160 5 I-STEINER-MINIMAL-TREES 163 5.1 A Polynomially Bounded Algorithm 164 5.2 Examples 168 5.3 The Generalization to k-smt 172 6 METHODS TO CONSTRUCT SHORTEST TREES 177 6.1 The Essentials of Steiner's Problem 178 6.2 The Geometric Structure of Shortest Trees 179 6.3 Reductions 189 6.4 Methods to Minimize the Function SB 195 6.5 A Discretization 204 6.6 Approximation of Shortest Trees by Replacing One Space by Another 213 6.7 A Graph Theoretical Approximation 218 6.8 Heuristic Approaches 224 6.9 Concluding Remarks 232 7 THE STEINER RATIO OF BANACH- MINKOWSKI SPACES 235 7.1 Basic Facts 236 7.2 An Upper Bound for the Steiner Ratio 239 7.3 Euclidean Spaces 242 7.4 A Lower Bound for the Steiner Ratio 246 7.5 The Steiner Ratio and several other Parameters in Discrete Geometry 251

Contents Vll 7.6 The Steiner ratio and the Embedding of Spaces 7.7 The Steiner Ratio for k-smt's 7.8 Concluding Remarks 256 264 267 8 GENERALIZATIONS 8.1 Reasons why Steiner's Problem Should be Generalized 271 8.2 Shortest Networks with a Given Combinatorial Structure 273 8.3 Spanners 276 8.4 Searching More than one Tree 278 8.5 Fermat's Problem with Weighted Nodes 278 8.6 Steiner's Problem which is Steiner Point Weighted 281 8.7 Steiner's Problem in the Plane with Obstacles 282 8.8 Steiner's Problem in Spaces with a Weaker Triangle Inequality 283 271 REFERENCES 287 INDEX 315

PREFACE The problem of "Shortest Connectivity", which is discussed here, has a long and convoluted history. Many scientists from many fields as well as laymen have stepped on its stage. Usually, the problem is known as Steiner's Problem and it can be described more precisely in the following way: Given a finite set of points in a metric space, search for a network that connects these points with the shortest possible length. This shortest network must be a tree and is called a Steiner Minimal Tree (SMT). It may contain vertices different from the points which are to be connected. Such points are called Steiner points. Steiner's Problem seems disarmingly simple, but it is rich with possibilities and difficulties, even in the simplest case, the Euclidean plane. This is one of the reasons that an enormous volume of literature has been published, starting in the seventeenth century and continuing until today. 1 The difficulty is that we look for the shortest network overall. Minimum spanning networks have been well-studied and solved eompletely in the case where only the given points must be connected. The novelty of Steiner's Problem is that new points, the Steiner points, may be introduced so that an interconnecting network of all these points will be shorter. This also shows that it is impossible to solve the problem with combinatorial and geometric methods alone. Steiner's Problem is one of the most intuitive subjects in mathematics: The problem can be easily understood, and it is easy to formulate heuristic solutions and make conjectures about this problem. Nevertheless, the solutions of Steiner's Problem, several derived subproblems, and of related problems have proven to be extremely difficult. Many mathematicians of the last centuries have found this problem to be a catalyst for many considerations and investigations in mathematics overall. It is often used as a problem to introduce Computational Geometry. Moreover, Steiner's Problem has a great influence on many questions in applied mathematics. In the last three decades, many facts have been published about Steiner's Problem. At the center of the investigations of most of the papers, Steiner's Problem 1 The history of Steiner's Problem started with P.Fennat early.in the 17th century and C.F.Gaufi.in 1836. Perhaps with the famous book What ts Mathematics by R.Courant and H.Robb.ins.in 1941, tltis problem became popularized under the name of Steiner.

X STEINER MINIMAL TREES is considered in the Euclidean plane, in rectilinear norm or in graphs. Sometimes we find investigations about more general spaces. This book is the result of 18 years of research into Steiner's Problem and its relatives in theory and application. Starting with investigations of shortest networks for VLSI-layout and, on the other hand, for certain facility location problems, the author of this book has found many common properties for Steiner's Problem in various spaces. The purpose of this book is to sum up and to generalize many of these results for arbitrary finite-dimensional Banach spaces. It will be shown that we can create a homogeneous and general theory when we consider two dimensions of such spaces; and that we can find many facts which are helpful in attacking Steiner's Problem in the higher-dimensional cases. The book will examine the underlying mathematical properties of this network design problem and demonstrate how this problem can be attacked by various methods of geometry, graph theory, calculus, optimization and theoretical computer science. Additionally, we investigate several relatives of Steiner's Problem. One of this relatives is Steiner's Problem with a restricted number k of Steiner points, where k is a predetermined given nonnegative integer. We call a solution of this problem a k-smt. In particular, the 0-SMT has no Steiner points, which means we only connect the given points. Hence the problem to find a 0-SMT is equivalent to the well-known Minimum Spanning Tree (MST) Problem. Since this problem has its own interest we will discuss it more extensively. Whereas an MST can be found easily, the determination of an SMT in general is still unknown or at least hard in the sense of computational complexity 2. We shall, however, describe methods to find an SMT in exponential time. On the other hand, the construction of a k-smt for a fixed number k is not too hard. A further relative of Steiner's Problem is the local version of Steiner Minimal Trees; that means that we search for a single point such that the sum of distances to the given points is minimal This problem is called Fermat's Problem and a point which creates this minimum is called a Torricelli point. There are methods to construct such a point. Since Fermat's Problem has an independent interest we will it explain more extensively. To satisfy this demand it is necessary to consider many problems. Sometimes a subproblem is impossible to solve. Then we shall give a discussed conjecture about the complexity of the problem. We give examples for important classes of metric spaces in which Steiner's Prob- 2but we will describe a small class of metric spaces in which Steiner's Problem is simple

Preface XI lem is taken into consideration. In particular, we consider Banach-Minkowski spaces as the main subject of this book. A complete description of Steiner's Problem in such spaces has not be given before. We hope that this book opens the door for forthcoming investigations and creating a powerful theory for locational analysis. Acknowledgements. I thank all persons who supported my research and gave me helpful advice on how to write this book. In particular,! thank (in alphabetic order): J.E. Beasley (London), D.Z.Du (Beijing/Minneapolis), R.L. Graham (Murray Hill), P. Gritzmann (Trier), U.Huckenbeck (Greifswald), R. Lang (Hamburg), H.Lenz (Munich/Berlin), J. Linhart (Salzburg), H. Martini (Chemnitz), F. Morgan (Williamstown), W.O.J. Moser (Montreal), H. Sachs (Ilmenau), J.J. Seidel (Eindhoven), P. Schreiber (Greifswald), R.H. Schulz (Berlin), L.Wenzel (Ludwigshafen), P. Widmayer (Zurich) and P. Winter (Copenhagen). Moreover, I thank my colleagues W. Girbardt, H. Kohler and H. Werner for helpful technical support.