Algorithms in Combinatorial Geometry

1. Front Matter

   Pages I-XV

   PDF

2. Combinatorial Geometry

   1. Front Matter

      Pages 1-1

      PDF

   2. Fundamental Concepts in Combinatorial Geometry

      • Herbert Edelsbrunner

      Pages 3-28

   3. Permutation Tables

      • Herbert Edelsbrunner

      Pages 29-44

   4. Semispaces of Configurations

      • Herbert Edelsbrunner

      Pages 45-62

   5. Dissections of Point Sets

      • Herbert Edelsbrunner

      Pages 63-82

   6. Zones in Arrangements

      • Herbert Edelsbrunner

      Pages 83-96

   7. The Complexity of Families of Cells

      • Herbert Edelsbrunner

      Pages 97-118

3. Fundamental Geometric Algorithms

   1. Front Matter

      Pages 121-121

      PDF

   2. Constructing Arrangements

      • Herbert Edelsbrunner

      Pages 123-137

   3. Constructing Convex Hulls

      • Herbert Edelsbrunner

      Pages 139-176

   4. Skeletons in Arrangements

      • Herbert Edelsbrunner

      Pages 177-208

   5. Linear Programming

      • Herbert Edelsbrunner

      Pages 209-239

   6. Planar point Location Search

      • Herbert Edelsbrunner

      Pages 241-266

4. Geometric and Algorithmic Applications

   1. Front Matter

      Pages 269-269

      PDF

   2. Problems For Configurations and Arrangements

      • Herbert Edelsbrunner

      Pages 271-291

   3. Voronoi Diagrams

      • Herbert Edelsbrunner

      Pages 293-333

   4. Separation and Intersection in the Plane

      • Herbert Edelsbrunner

      Pages 335-357

   5. Paradigmatic Design of Algorithms

      • Herbert Edelsbrunner

      Pages 359-379

5. Back Matter

   Pages 381-423

   PDF

Computational geometry as an area of research in its own right emerged in the early seventies of this century. Right from the beginning, it was obvious that strong connections of various kinds exist to questions studied in the considerably older field of combinatorial geometry. For example, the combinatorial structure of a geometric problem usually decides which algorithmic method solves the problem most efficiently. Furthermore, the analysis of an algorithm often requires a great deal of combinatorial knowledge. As it turns out, however, the connection between the two research areas commonly referred to as computa­ tional geometry and combinatorial geometry is not as lop-sided as it appears. Indeed, the interest in computational issues in geometry gives a new and con­ structive direction to the combinatorial study of geometry. It is the intention of this book to demonstrate that computational and com­ binatorial investigations in geometry are doomed to profit from each other. To reach this goal, I designed this book to consist of three parts, acorn binatorial part, a computational part, and one that presents applications of the results of the first two parts. The choice of the topics covered in this book was guided by my attempt to describe the most fundamental algorithms in computational geometry that have an interesting combinatorial structure. In this early stage geometric transforms played an important role as they reveal connections between seemingly unrelated problems and thus help to structure the field.

• Notation
• Permutation
• algorithm
• algorithms
• combinatorial geometry
• complexity
• geometry
• programming
• algorithm analysis and problem complexity
• combinatorics

• Dept. of Computer Science, University of Illinois at Urbana-Champaign, Urbana, USA

  Herbert Edelsbrunner

Policies and ethics