Mathematical Surveys and Monographs Volume 152 Combinatorial Geometry and Its Algorithmic Applications The Alcalá Lectures János Pach Micha Sharir American Mathematical Society Combinatorial Geometry and Its Algorithmic Applications The Alcalá Lectures Mathematical Surveys and Monographs Volume 152 Combinatorial Geometry and Its Algorithmic Applications The Alcalá Lectures János Pach Micha Sharir American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Jerry L. Bona Michael G. Eastwood Ralph L. Cohen J. T. Stafford, Chair Benjamin Sudakov 2000 Mathematics Subject Classification. Primary 05C35, 05C62, 52C10, 52C30, 52C35, 52C45, 68Q25, 68R05, 68W05, 68W20. For additional information and updates on this book, visit www.ams.org/bookpages/surv-152 Library of Congress Cataloging-in-Publication Data Pach, Janos. Combinatorial geometry and its algorithmic applications : The Alcala lectures / Janos Pach, Micha Sharir. p. cm. — (Mathematical surveys and monographs ; v. 152) Includes bibliographical references and index. ISBN 978-0-8218-4691-9 (alk. paper) 1. Combinatorial geometry. 2. Algorithms. I. Sharir, Micha. II. Title. QA167.p332 2009 516.13–dc22 2008038876 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. c 2009 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 141312111009 Contents Preface and Apology vii Chapter 1. Sylvester–Gallai Problem: The Beginnings of Combinatorial Geometry 1 1. James Joseph Sylvester and the Beginnings 1 2. Connecting Lines and Directions 3 3. Directions in Space vs. Points in the Plane 6 4. Proof of the Generalized Ungar Theorem 7 5. Colored Versions of the Sylvester–Gallai Theorem 10 Chapter 2. Arrangements of Surfaces: Evolution of the Basic Theory 13 1. Introduction 13 2. Preliminaries 16 3. Lower Envelopes 20 4. Single Cells 27 5. Zones 29 6. Levels 32 7. Many Cells and Related Problems 37 8. Generalized Voronoi Diagrams 40 9. Union of Geometric Objects 42 10. Decomposition of Arrangements 49 11. Representation of Arrangements 54 12. Computing Arrangements 56 13. Computing Substructures in Arrangements 58 14. Applications 63 15. Conclusions 70 Chapter 3. Davenport–Schinzel Sequences: The Inverse Ackermann Function in Geometry 73 1. Introduction 73 2. Davenport–Schinzel Sequences and Lower Envelopes 74 3. Simple Bounds and Variants 79 4. Sharp Upper Bounds on λs(n)81 5. Lower Bounds on λs(n)86 6. Davenport–Schinzel Sequences and Arrangements 89 Chapter 4. Incidences and Their Relatives: From Szemer´edi and Trotter to Cutting Lenses 99 1. Introduction 99 2. Lower Bounds 102 v vi CONTENTS 3. Upper Bounds for Incidences via the Partition Technique 104 4. Incidences via Crossing Numbers—Sz´ekely’s Method 106 5. Improvements by Cutting into Pseudo-segments 109 6. Incidences in Higher Dimensions 112 7. Applications 114 Chapter 5. Crossing Numbers of Graphs: Graph Drawing and its Applications 119 1. Crossings—the Brick Factory Problem 119 2. Thrackles—Conway’s Conjecture 120 3. Different Crossing Numbers? 122 4. Straight-Line Drawings 125 5. Angular Resolution and Slopes 126 6. An Application in Computer Graphics 127 7. An Unorthodox Proof of the Crossing Lemma 129 Chapter 6. Extremal Combinatorics: Repeated Patterns and Pattern Recognition 133 1. Models and Problems 133 2. A Simple Sample Problem: Equivalence under Translation 135 3. Equivalence under Congruence in the Plane 137 4. Equivalence under Congruence in Higher Dimensions 139 5. Equivalence under Similarity 141 6. Equivalence under Homothety or Affine Transformations 143 7. Other Equivalence Relations for Triangles in the Plane 144 Chapter 7. Lines in Space: From Ray Shooting to Geometric Transversals 147 1. Introduction 147 2. Geometric Preliminaries 149 3. The Orientation of a Line Relative to n Given Lines 152 4. Cycles and Depth Order 158 5. Ray Shooting and Other Visibility Problems 163 6. Transversal Theory 167 7. Open Problems 170 Chapter 8. Geometric Coloring Problems: Sphere Packings and Frequency Allocation 173 1. Multiple Packings and Coverings 173 2. Cover-Decomposable Families and Hypergraph Colorings 175 3. Frequency Allocation and Conflict-Free Coloring 178 4. Online Conflict-Free Coloring 181 Chapter 9. From Sam Loyd to L´aszl´o Fejes T´oth: The 15 Puzzle and Motion Planning 183 1. Sam Loyd and the Fifteen Puzzle 183 2. Unlabeled Coins in Graphs and Grids 185 3. L´aszl´o Fejes T´oth and Sliding Coins 187 4. Pushing Squares Around 194 Bibliography 197 Index 227 Preface and Apology These lecture notes are a compilation of surveys of the topics that were pre- sented in a series of talks at Alcal´a, Spain, August 31 – September 5, 2006, by J´anos Pach and Micha Sharir. To a large extent, these surveys are adapted from earlier papers written by the speakers and their collaborators. In their present form, the notes aptly describe both the history and the state of the art of these topics. The notes are arranged in an order that roughly parallels the order of the talks. Chapter 1 describes the beginnings of combinatorial geometry: Starting with Sylvester’s problem on the existence of “ordinary lines,” we introduce a number of exciting problems on incidences between points and lines in the plane and in space. This chapter uses the material in Pach, Pinchasi, and Sharir [588, 589]. In Chapter 2 we survey many aspects of the theory of arrangements of surfaces in higher dimensions. It is adapted from Agarwal and Sharir [51]. Readers that have some familiarity with the basic theory of arrangements can start their reading on this topic from this chapter, while those that are complete novices may find it useful to look first at Chapter 3, which studies arrangements of curves in the plane, with special emphasis on Davenport–Schinzel sequences and the major role they play in the theory of arrangements. This chapter is adapted from Agarwal and Sharir [52]. Chapter 4 covers the topic of incidences between points and curves and its many relatives, where a surge of activity has taken place in the past decade. It is adapted from two similar surveys by Pach and Sharir [600, 601]. The study of combinatorial and topological properties of planar arrangements of curves has become a separate discipline in discrete and computational geometry, under the name of Graph Drawing. Some basic aspects of this emerging discipline are dis- cussed in Chapter 5, which is based on the survey by Pach [583]. Some classical questions of Erd˝os on repeated interpoint distances in a finite point set can be re- formulated as problems on the maximum number of incidences between points and circles, spheres, etc. In fact, these questions motivated and strongly influenced the early development of the theory of incidences a quarter of a century ago and they led to the discovery of powerful new combinatorial and topological tools. Many of Erd˝os’s questions can be naturally generalized to problems on larger repeated subpatterns in finite point sets. Based on Brass and Pach [175], in Chapter 6 we outline some of the most challenging open problems of this kind, whose solution would have interesting consequences in pattern matching and recognition. Chapter 7 treats the special topic of lines in three-dimensional space, which is a nice application (or showpiece, if you will) of the general theory of arrange- ments on one hand, and shows up in a variety of only loosely related topics, ranging from ray shooting and hidden surface removal in computer graphics to geometric transversal theory. This chapter partially builds upon a somewhat old paper by Chazelle, Edelsbrunner, Guibas, Sharir, and Stolfi [220], but its second half is new, vii viii PREFACE AND APOLOGY and presents (some of) the recent developments. Some combinatorial properties of arrangements of spheres, boxes, etc., are discussed in Chapter 8. They raise difficult questions on the chromatic numbers and other similar parameters of certain geo- metrically defined graphs and hypergraphs, with possible applications to frequency allocation in cellular telephone networks. Here we borrowed some material from Pach, Tardos, and T´oth [608]. An old and rich area of applications of Davenport–Schinzel sequences and the theory of geometric arrangements is motion planning. Starting with Sam Loyd’s coin puzzles, in Chapter 9 we discuss a number of problems that can be regarded as discrete variants of the “piano movers’ problem” on graphs and grids. Some of the results have applications to the reconfiguration of metamorphic robotic systems. This chapter is based on recent joint papers with Bereg, C˘alinescu, and Dumitrescu [138, 190, 280]. While we have made our best attempts to make these notes comprehensive, they are not at the level of a polished monograph, nor do they provide full coverage of all the relevant recent results and developments. It is our hope that they be a useful source of reference to the rich and extensive theory presented at the Alcal´a series of talks. Apart from those friends and coauthors whose names were mentioned above, we freely borrowed from joint work with D.