Combinatorial Geometry and Its Algorithmic Applications: The Alcalá Lectures
About this Title
János Pach, Courant Institute of Mathematical Sciences, New York, NY and Micha Sharir, Tel Aviv University, Tel Aviv, Israel
Publication: Mathematical Surveys and Monographs
Publication Year 2009: Volume 152
ISBNs: 978-0-8218-4691-9 (print); 978-1-4704-1379-8 (online)
MathSciNet review: MR2469102
MSC: Primary 05-02; Secondary 05C35, 05Cxx, 52-02, 52C35, 68R05, 68U05
Based on a lecture series given by the authors at a satellite meeting of the 2006 International Congress of Mathematicians and on many articles written by them and their collaborators, this volume provides a comprehensive up-to-date survey of several core areas of combinatorial geometry. It describes the beginnings of the subject, going back to the nineteenth century (if not to Euclid), and explains why counting incidences and estimating the combinatorial complexity of various arrangements of geometric objects became the theoretical backbone of computational geometry in the 1980s and 1990s. The combinatorial techniques outlined in this book have found applications in many areas of computer science from graph drawing through hidden surface removal and motion planning to frequency allocation in cellular networks.
Combinatorial Geometry and Its Algorithmic Applications is intended as a source book for professional mathematicians and computer scientists as well as for graduate students interested in combinatorics and geometry. Most chapters start with an attractive, simply formulated, but often difficult and only partially answered mathematical question, and describes the most efficient techniques developed for its solution. The text includes many challenging open problems, figures, and an extensive bibliography.
Graduate students and research mathematicians interested in combinatorial geometry and algorithmic applications.
Table of Contents
- 1. Sylvester-Gallai problem: The beginnings of combinatorial geometry
- 2. Arrangements of surfaces: Evolution of the basic theory
- 3. Davenport-Schinzel sequences: The inverse Ackermann function in geometry
- 4. Incidences and their relatives: From Szemerédi and Trotter to cutting lenses
- 5. Crossing numbers of graphs: Graph drawing and its applications
- 6. Extremal combinatorics: Repeated patterns and pattern recognition
- 7. Lines in space: From ray shooting to geometric transversals
- 8. Geometric coloring problems: Sphere packings and frequency allocation
- 9. From Sam Loyd and László Fejes Tóth: The 15 puzzle and motion planning