New Trends in Discrete and Computational Geometry

Discrete and computational geometry are two fields which in recent years have benefitted from the interaction between mathematics and computer science. The results are applicable in areas such as motion planning, robotics, scene analysis, and computer aided design. The book consists of twelve chapters summarizing the most recent results and methods in discrete and computational geometry. All authors are well-known experts in these fields. They give concise and self-contained surveys of the most efficient combinatorical, probabilistic and topological methods that can be used to design effective geometric algorithms for the applications mentioned above. Most of the methods and results discussed in the book have not appeared in any previously published monograph. In particular, this book contains the first systematic treatment of epsilon-nets, geometric tranversal theory, partitions of Euclidean spaces and a general method for the analysis of randomized geometric algorithms. Apart from mathematicians working in discrete and computational geometry this book will also be of great use to computer scientists and engineers, who would like to learn about the most recent results.

Contenu

I. Combinatorics and Algorithms of Arrangements.- 1. Introduction.- 2. Arrangements of Curves in the Plane.- 3. Lower Envelopes and Davenport-Schinzel Sequences.- 3.1 A Few Initial Applications.- 3.2 Applications of Lower Envelopes and Davenport-Schinzel Sequences.- 4. Faces in Arrangements.- 4.1 The Case of a Single Face.- 4.2 Many Faces in Arrangements and Related Problems.- 4.3 Partitioning Arrangements.- 5. Arrangements in Higher Dimensions.- 5.1 Lower Envelopes of Bivariate Functions and Their Applications.- 5.2 The Complexity of a Single Cell and Related Problems.- 6. Summary.- References.- II. Backwards Analysis of Randomized Geometric Algorithms.- 1. Introduction.- 2. Delaunay Triangulations of Convex Polygons.- 3. Intersecting Line Segments.- 4. Constructing Planar Convex Hulls.- 5. Backwards Analysis of QUICKSORT.- 6. A Bad Example.- 7. Linear Programming for Small Dimension.- 8. Welzl's Minidisk Algorithm.- 9. Clarkson's Backwards Analysis of the Conflict Graph Based on the Convex Hull Algorithm.- 10. Odds and Ends.- References.- III. Epsilon-Nets and Computational Geometry.- 1. Range Spaces and ?-Nets.- 2. Geometric Range Spaces.- 3. A Sample of Applications.- 4. Removing Logarithms.- 5. Removing the Randomization.- References.- IV. Complexity of Polytope Volume Computation.- 1. Jumps of the Derivatives.- 2. Exact Volume Computation is Hard.- 3. Volume Approximation.- References.- V. Allowable Sequences and Order Types in Discrete and Computational Geometry.- 1. Introduction.- 2. Combinatorial Types of Configurations in the Plane and Allowable Sequences.- 3. Arrangements of Lines and Pseudolines.- 4. Applications of Allowable Sequences.- 4.1 Stretchability of Pseudoline Arrangements.- 4.2 The Directions Problem.- 4.3 k-Sets.- 5. Order Types of Points in Rd and "Geometric Sorting".- 6. The Number of Order Types in Rd.- 7. Isotopy and Realizability Questions.- 8. Lattice Realization of Order Types and the Problem of Robustness in Computational Geometry.- References.- VI. Hyperplane Approximation and Related Topics.- 1. Introduction.- 2. MINSUM Problem: Orthogonal L1-Fit.- 2.1 Basic Facts.- 2.2 General Case: O(nd log n) Time and Linear Space Algorithm.- 2.3 General Case: O(nd) Time.- 2.4 Restrictions and Extensions.- 3. MINSUM Problem: Vertical L1-Fit.- 4. MINMAX Problem: Orthogonal L?-Fit.- 4.1 Unweighted Case.- 4.2 Weighted Case.- 4.3 MINMAX Problem: k-Flats.- 5. MINMAX Problem: Vertical L?-Fit.- 6. Related Issues.- 6.1 Stabbing Lines/Hyperplanes.- 6.2 k-Hyperplane Problems.- References.- VII. Geometric Transversal Theory.- 1. Introduction.- 2. Hadwiger-Type Theorems.- 3. The Combinatorial Complexity of the Space of Transversals.- 4. Translates of a Convex Set.- 5. Transversal Algorithms.- 6. Other Directions.- References.- VIII. Hadwiger-Levi's Covering Problem Revisited.- 0. Introduction.- 1. On I0(K) and I?(K).- 2. On Il(K) and k-fold Illumination.- 3. Some Simple Remarks on H(B).- 4. On Convex Bodies with Finitely Many Corner Points.- 5. Solution of Hadwiger-Levi's Covering Problem for Convex Polyhedra with Affine Symmetry.- References.- IX. Geometric and Combinatorial Applications of Borsuk's Theorem.- 1. Introduction.- 2. Van Kampen-Flores Type Results.- 3. The Ham-Sandwich Theorem.- 4. Centrally Symmetric Polytopes.- 5. Kneser's Conjecture.- 6. Sphere Coverings.- References.- X. Recent Results in the Theory of Packing and Covering.- 1. Introduction.- 2. Preliminaries and Basic Concepts.- 3. A Review of Some Classical Results in the Plane.- 4. Economical Packing in and Covering of the Plane.- 5. Multiple Packing and Covering.- 6. Some Computational Aspects of Packing and Covering.- 7. Restrictions on the Number of Neighbors in a Packing.- 8. Selected Topics in 3 Dimensions.- References.- XI. Recent Developments in Combinatorial Geometry.- 1. The Distribution of Distances.- 2. Graph Dimensions.- 3. Geometric Graphs.- 4. Arrangements of Lines in Space.- References.- XII. Set Theoretic Constructions in Euclidean Spaces.- 0. Introduction.- 1. Simple Transfinite Constructions.- 2. Closed Sets or Better Well-Orderings.- 3. Extending the Coloring More Carefully.- 4. The Use of the Continuum Hypothesis.- 5. The Infinite Dimensional Case.- 6. Large Paradoxical Sets in Another Sense.- References.- Author Index.