Algorithms in Combinatorial GeometrySpringer Science & Business Media, 06.12.2012 - 423 Seiten 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. |
Inhalt
2 | |
1 | 26 |
6 | 40 |
2 | 47 |
8 | 53 |
Exercises and Research Problems | 60 |
Dissections of Point Sets | 63 |
6 | 78 |
Linear Programming | 209 |
Planar Point Location Search | 241 |
GEOMETRIC AND ALGORITHMIC APPLICATIONS 269 | 267 |
Voronoi Diagrams | 293 |
Separation and Intersection in the Plane 335 | 334 |
Paradigmatic Design of Algorithms | 359 |
380 | |
123 | 387 |
Asymptotically Tight Bounds in d Dimensions | 86 |
Exercises and Research Problems | 95 |
4 | 109 |
7 | 116 |
Constructing Convex Hulls 139 | 138 |
Skeletons in Arrangements | 177 |
APPENDIX A Definitions | 395 |
APPENDIX B Notational Conventions | 409 |
417 | |
420 | |
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Häufige Begriffe und Wortgruppen
arrangement A(H arrangements of hyperplanes belongs boundary called cell complex Chapter circular sequence collinear combinatorial compute contains convex hull convex polytope convP corresponding data structure defined Delaunay triangulation deletion denote determine directed edge DT(S dual Edelsbrunner endfor endif endpoint Exercise extremal query face ƒ facet Figure finite set following result half-plane half-spaces ham-sandwich cut ham-sandwich tree hyperplane h implies incidence graph infeasible integer numbers intersection k-faces layered DAG Lemma Let H line query line segments linear program lower bound minimum spanning tree monotone subdivision multiset n-sequence node non-empty non-vertical hyperplanes number of faces O(n² O(nlogn P₁ and P₂ pairs plane point location point sets pointer points in E² polygon Procedure Prove real number recursive search problem set of points set of sites skeleton solution solve stabbing region Step subface Theorem tion transversal triangulation unbounded upper bound vector vertex vertical line vertical projection Voronoi diagrams zonotope