# 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…

## 4,281 Citations

### The projector algorithm: a simple parallel algorithm for computing Voronoi diagrams and Delaunay graphs

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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 Geometric Stability of Voronoi Diagrams with Respect to Small Changes of the Sites

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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.

### Intrinsic computation of voronoi diagrams on surfaces and its application

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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.

### Transactions on Computational Science IX - Special Issue on Voronoi Diagrams in Science and Engineering

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- 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.

### NSF CAREER Proposal : Approximation Algorithms for Geometric Computing 1 Overview

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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.

### Lectures on discrete geometry

- MathematicsGraduate texts in mathematics
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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.

### An Algorithm for Computing Voronoi Diagrams of General Generators in General Normed Spaces

- Computer Science2009 Sixth International Symposium on Voronoi Diagrams
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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.

### Calculating Voronoi Diagrams using Convex Sweep Curves

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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

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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…

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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.

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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.

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