# 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}
}
• F. Aurenhammer
• Published 1 September 1991
• Computer Science
• ACM Comput. Surv.
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…
4,281 Citations
This paper presents the projector algorithm: a new and simple algorithm which enables the (combinatorial) computation of 2D Voronoi diagrams and the computation of the induced Delaunay graph is obtained almost automatically.
The question is formalized precisely, and it is shown that the answer is positive in the case of Rd, or in (possibly infinite dimensional) uniformly convex normed spaces, assuming there is a common positive lower bound on the distance between the sites.
A sweep circle algorithm for construct Voronoi diagrams in parallel and the Saddle Vertex Graph to efficiently compute discrete geodesics on meshes are presented and one application of CVT and discrete geodeics, anisotropic shape distribution on meshes is proposed.
• Computer Science
Trans. Comput. Sci.
• 2010
A general framework for computing Voronoi diagrams of different classes of sites under various distance functions in R is presented, and it is proved that through randomization, the expected running time becomes near-optimal in the worst case.
• Computer Science
• 2002
The proposal outlines a challenging career development plan focusing on research in a broad cross-section of computational geometry, building on and significantly broadening the PI’s successful work in the field over the last several years.
This book is primarily a textbook introduction to various areas of discrete geometry, in which several key results and methods are explained, in an accessible and concrete manner, in each area.
• Daniel Reem
• Computer Science
2009 Sixth International Symposium on Voronoi Diagrams
• 2009
This work presents an efficient and simple algorithm for computing Voronoi diagrams in general normed spaces, possibly infinite dimensional and allows infinitely many generators of a general form.
A generalization of the sweep line method of Fortune was developed, and a sweep algorithm based on sweep circles was implemented; this algorithms runs with O(n · log n) time and O( n) space requirements, which is optimal.

## References

SHOWING 1-10 OF 261 REFERENCES

• Computer Science
STOC
• 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.
• Computer Science, Mathematics
IEEE 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
• Computer Science
SCG '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.
• Computer Science
SIAM 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.
• Computer Science
SCG '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.
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.
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
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.
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.