Table Of ContentLocating Lines and Hyperplanes
Applied Optimization
Volume25
Series Editors:
Panos M. Pardalos
University of Florida, U.SA.
Donald Hearn
University of Florida, U.SA.
The titles published in this series are listed at the end of this volume.
Locating Lines
and Hyperplanes
Theory and Algorithms
by
Anita Schobel
Universitiit Kaiserslautern,
Kaiserslautem, Germany
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Cata1ogue record for this book is available from the Library of Congress.
ISBN 978-1-4613-7428-2 ISBN 978-1-4615-5321-2 (eBook)
DOI 10.1007/978-1-4615-5321-2
Printed on acid-free paper
AII Rights Reserved
©1999 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers 1999
Softcover reprint of the hardcover 1s t edition 1999
No part of the materia] protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanica1,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner
For my children
Svenja and Malte
Contents
Preface IX
1. BASIC CONCEPTS 1
1.1 Introduction 1
1.2 Measuring Distances: Norms and Metrics 3
1.3 Problem Description and Notation 7
1.4 Related Problems in Computational Geometry 17
1.4.1 Transversal Theory 17
1.4.2 Set Width Problems 21
1.5 Piecewise Linear Programs with Restrictions 21
2. LINE LOCATION WITH VERTICAL DISTANCE 33
2.1 Results for the Vertical Distance 33
2.2 A Dual Interpretation 36
2.2.1 The Median Problem 37
2.2.2 The Center Problem 38
3. LOCATING LINES IN THE NORMED PLANE 47
3.1 Results for p-Norm Distances 47
3.1.1 The Rectangular Distance 47
3.1.2 The Chebyshev Distance 49
3.1.3 The Euclidean Distance 49
3.1.4 Other p-Norm Distances 50
3.2 Results for Arbitrary Norms 57
3.3 Algorithmic Approaches for Norms 67
3.4 A Fast Algorithm for Block Norm Distances 71
4. FINDING ALL OPTIMAL LINES 77
4.1 The Strong Incidence Property 77
4.2 The Strong Blockedness Property 83
4.3 Determining all Optimal Lines 87
5. LINE LOCATION WITH OTHER DISTANCES 95
5.1 Results for Gauges 95
vii
viii LOCATING LINES AND HYPERPLANES: THEORY AND ALGORITHMS
5.2 Results for Metrics 100
5.3 Results for Mixed Distance Functions 106
5.4 Algorithms for General Line Location Problems 110
5.5 Summary 112
6. RESTRICTED LINE LOCATION PROBLEMS 115
6.1 Restricted Problems with Vertical Distance 116
6.2 Generalization to Block Norms 122
6.3 Generalization to Arbitrary Norms 124
6.4 Locating a Line Segment with Vertical Distance 128
6.5 Other Types of Restrictions 135
7. LOCATING HYPERPLANES IN NORMED SPACES 139
7.1 Results for the Horizontal Distance in Rn 140
7.1.1 The Problem in the Primal Space 140
7.1.2 A Dual Interpretation in n dimensions 145
7.2 Results in Normed Spaces 148
rn.n
7.2.1 The Rectangular Distance in 148
rn.n
7.2.2 The Euclidean Distance in 148
7.2.3 Arbitrary Norms in rn.n 149
7.3 Algorithmic Approaches for Hyperplane Location 153
7.4 A Characterization of Smooth Norms 157
8. EXTENSIONS: LOCATING OTHER OBJECTS 161
8.1 Bicriteria Line Segment Problems 162
8.2 Location of a Circle 171
8.3 Planar Location of One-Dimensional Facilities 177
Appendices 181
A-Summary 181
B- List of Algorithms 183
C- List of Symbols 185
References 187
Index
197
A straight line is one which lies evenly with the points on itself.
-Euclid, The Elements
Straight lines have been a fascinating subject in mathematics right from the
beginning of geometry.
But even in modern mathematical disciplines affine subspaces like lines and
hyperplanes are still an exciting research object. In this text I am concerned
with the location of lines in the plane and of hyperplanes in normed spaces.
On a conference on transportation analysis I first heard of path location prob
lems in networks. Such problems deal with the location of a path which is as
close as possible to a given set of customer-nodes. In traffic planning these
problems arise when a new bus line is organized, or if a new railway line is
built. Since Professor Hamacher and my colleagues work on continuous loca
tion problems I decided to study path location problems, not in networks, but
in the plane. Unfortunately, the location of an arbitrary or of a polygonal path
is NP-hard (see Section 8.3) such that I turned my attention to the location of
straight lines.
Given a set of weighted points in the plane, the line location problem is to find
a straight line which minimizes the sum of distances or the maximum distance
to the given point set. Studying the literature about the location of lines, I
recognized that these problems play an important role not only in location
theory, but also in statistics when finding regression lines (orthogonal or
vertical L1-fit problem) and in computational geometry as a special case of
hyperplane approximation problems (£1- or L approximation problems).
00
In this text, I present my research on line and hyperplane location problems.
Figure 0.1 shows the relations between the chapters ofthis text. A brief outline
of all chapters is given in the following:
• In Chapter 1, I formally introduce hyperplane location problems with re
spect to different distance measures. I also discuss the relation of these
problems to transversal theory and to the set width problem in computa-
ix
x LOCATING LINES AND HYPERPLANES: THEORY AND ALGORJTHMS
Chapter 1
1.1' 1.2, 1.3
and Lemma 1. 9
Chapter 2
Chapter 3
Chapter 5 Chapter 7
Figure 0.1. Relations between the chapters of this text.
tiona! geometry (Section 1.4). In Section 1.5, I introduce the concept of
restricted piecewise linear convex programs which is needed in Chapter 6.
• Chapter 2 deals with line location problems with the vertical distance. Since
these problems are convex, they can easily be solved. In this chapter, I also
introduce a geometric duality which gives some new insight in line location
problems with vertical distance. Chapter 2 is an important building block
for all subsequent chapters.
• Then I turn to (non-convex) line location problems with arbitrary norms
(Chapter 3). After reviewing the known results about line location problems
with rectangular and Euclidean distance, I extend the results of Chapter 2
to all distances derived from norms. Algorithmic approaches for arbitrary
norms and for the case of block norm distances are given.
• In Chapter 4 I can strengthen the results of Chapter 3 in the case of smooth
norms. These results lead to algorithms that find the set of all optimal lines,
even in the case of arbitrary norms.
• Chapter 5 investigates distance functions that are not derived from norms.
In particular, I study the case of gauges, of metrics, and of mixed distance
functions.
