Forbidden Configurations in Discrete Geometry

by David Eppstein (Author)

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.

Review
'David Eppstein has managed to unify a huge swath of research on planar point sets through monotone properties and forbidden configurations. For example, finding grid points that avoid the obstacle of a 3-point line is a century-old problem still not entirely resolved. The author's unification naturally uncovers research lacuna, several of which he fills, while others are formulated as sharp new open problems. This rare synthesis of previous work will reinvigorate and redirect the field.' Joseph O'Rourke, Smith College, Massachusetts

'David Eppstein takes us on an adventure tour to the study of point configurations in Discrete Geometry. It visits many different topics, connected by the original viewpoint of 'forbidden configurations'. This is interesting, instructive - and fun!' Günter M. Ziegler, Freie Universität Berlin

'This unique volume collects and unifies almost a century of work on point configurations on the plane, and their properties that depend on whether each subset of three points is oriented clockwise, oriented counterclockwise, or collinear. Beginning with the Happy Ending Theorem, the author takes us through entertaining problems and into computational geometry. A delight to read as well as a persuasive case for the method of forbidden configurations, the book will be a valuable addition to the library of any discrete or computational geometer.' Peter Winkler, Dartmouth College, New Hampshire

'This is a fun read on certain topics in discrete and computational geometry. It begins with 'A Happy Ending' and ends with 'Only the Beginning'. Eppstein's journey through various problems of pointset configurations offers a new view of the subject even to experts of the field. Recommended to everyone who likes geometry and computer science.' Jozsef Solymosi, University of British Columbia