Voronoi diagrams—a survey of a fundamental geometric data structure
@article{Aurenhammer1991VoronoiDS,
title={Voronoi diagrams—a survey of a fundamental geometric data structure},
author={Franz Aurenhammer},
journal={ACM Comput. Surv.},
year={1991},
volume={23},
pages={345-405}
}Computational geometry is concerned with the design and analysis of algorithms for geometrical problems. In addition, other more practically oriented, areas of computer science— such as computer graphics, computer-aided design, robotics, pattern recognition, and operations research—give rise to problems that inherently are geometrical. This is one reason computational geometry has attracted enormous research interest in the past decade and is a well-established area today. (For standard sources…
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References
SHOWING 1-10 OF 261 REFERENCES
Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams
- Computer ScienceSTOC
- 1983
Two algorithms are given, one that constructs the Voronoi diagram of the given sites, and another that inserts a new site in O(n) time, based on the use of the Vor onoi dual, the Delaunay triangulation, and are simple enough to be of practical value.
Computational Geometry—A Survey
- Computer Science, MathematicsIEEE Transactions on Computers
- 1984
We survey the state of the art of computational geometry, a discipline that deals with the complexity of geometric problems within the framework of the analysis of algorithms. This newly emerged area…
Simplified Voronoi diagrams
- Computer ScienceSCG '87
- 1987
A Voronoi diagramV is defined based on a measure of distance which is not a true metric, which has lower algebraic complexity than the usual definition, which is a considerable advantage in motion-planning problems with many degrees of freedom.
Optimal Point Location in a Monotone Subdivision
- Computer ScienceSIAM J. Comput.
- 1986
A substantial refinement of the technique of Lee and Preparata for locating a point in $\mathcal{S}$ based on separating chains is exhibited, which can be implemented in a simple and practical way, and is extensible to subdivisions with edges more general than straight-line segments.
A simple on-line randomized incremental algorithm for computing higher order Voronoi diagrams
- Computer ScienceSCG '91
- 1991
The insertion or deletion of a site involves little more than the construction of a single convex hull in three-space, and the order-k Voronoi diagram for n sites can be computed in time and optimal space by an on-line randomized incremental algorithm.
Two-Dimensional Voronoi Diagrams in the Lp-Metric
- Computer ScienceJ. ACM
- 1980
Many proximity problems revolving a set of points, such as finding the nearest neighbor of a given point, finding the minimum spamung tree, findmg the smallest circle enclosing the point set, etc., can be solved very efficiently via the Voronoi diagram.
New applications of random sampling in computational geometry
- Computer ScienceDiscret. Comput. Geom.
- 1987
This paper gives several new demonstrations of the usefulness of random sampling techniques in computational geometry. One new algorithm creates a search structure for arrangements of hyperplanes by…
Further applications of random sampling to computational geometry
- Computer ScienceSTOC '86
- 1986
This paper gives several new demonstrations of the usefulness of random sampling techniques in computational geometry by creating a search structure for arrangements of hyperplanes by sampling the hyperplanes and using information from the resulting arrangement to divide and conquer.
Geometric transforms for fast geometric algorithms
- Computer Science
- 1979
This thesis describes several geometric problems whose solutions illustrate the use of geometric transforms, including fast algorithms for intersecting half-spaces, constructing Voronoi diagrams, and computing the Euclidean diameter of a set of points.