Forbidden Configurations in Discrete Geometry

Unifies discrete and computational geometry by using forbidden patterns of points to characterize many of its problems.

David Eppstein (Author)

9781108439138, Cambridge University Press

Paperback / softback, published 17 May 2018

238 pages
22.8 x 15.2 x 1 cm, 0.46 kg

'This book is distinguished by a number of attractive features. Perhaps most prominent is its strong unity of approach. The first 7 chapters establish a coherent foundation and language for expressing and investigating the subjects studied in the remaining 10 … Another is its clarity of presentation and reader-friendliness. In most chapters the author adopts the strategy of introducing the topic in terms of an easily-understood problem that is accessible to virtually any reader … If you have any interest in learning about this field, I highly recommend this book.' Frederic Green, SIGACT News

This book surveys the mathematical and computational properties of finite sets of points in the plane, covering recent breakthroughs on important problems in discrete geometry, and listing many open problems. It unifies these mathematical and computational views using forbidden configurations, which are patterns that cannot appear in sets with a given property, and explores the implications of this unified view. Written with minimal prerequisites and featuring plenty of figures, this engaging book will be of interest to undergraduate students and researchers in mathematics and computer science. Most topics are introduced with a related puzzle or brain-teaser. The topics range from abstract issues of collinearity, convexity, and general position to more applied areas including robust statistical estimation and network visualization, with connections to related areas of mathematics including number theory, graph theory, and the theory of permutation patterns. Pseudocode is included for many algorithms that compute properties of point sets.

1. A happy ending
2. Overview
3. Configurations
4. Subconfigurations
5. Properties, parameters, and obstacles
6. Computing with configurations
7. Complexity theory
8. Collinearity
9. General position
10. General-position partitions
11. Convexity
12. More on convexity
13. Integer realizations
14. Stretched permutations
15. Configurations from graphs
16. Universality
17. Stabbing
18. The big picture.

Subject Areas: Mathematical theory of computation [UYA], Combinatorics & graph theory [PBV], Geometry [PBM], Discrete mathematics [PBD]