# Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams

@article{Guibas1983PrimitivesFT, title={Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams}, author={Leonidas J. Guibas and Jorge Stolfi}, journal={Proceedings of the fifteenth annual ACM symposium on Theory of computing}, year={1983} }

We discuss the following problem: given n points in the plane (the “sites”), and an arbitrary query point q, find the site that is closest to q. This problem can be solved by constructing the Voronoi diagram of the given sites, and then locating the query point in one of its regions. We give two algorithms, one that constructs the Voronoi diagram in O(n lg n) time, and another that inserts a new site in O(n) time. Both are based on the use of the Voronoi dual, the Delaunay triangulation, and…

## Figures and Topics from this paper

## 947 Citations

A practical algorithm for computing the Delaunay triangulation for convex distance functions

- Computer ScienceSODA '90
- 1990

A number of generalizations of the Voronoi diagram have been presented, including an algorithm for general distance measures that Minkowski called convex distance functions, where the “unit circle” can be defined to be any convex shape.

Lecture 7: Voronoi diagrams

- 2010

We introduce a critical idea of EGC, the technique of root bounds. This will be addressed through the study of Fortune's algorithm for computing the Voronoi diagram of a set of points. The algorithms…

Constructing Two-Dimensional Voronoi Diagrams via Divide-and-Conquer of Envelopes in Space

- Computer Science, Mathematics2009 Sixth International Symposium on Voronoi Diagrams
- 2009

A general framework for computing two-dimensional Voronoi diagrams of different classes of sites under various distance functions is presented and it is proved that through randomization, the expected running time becomes near-optimal in the worst case.

Constructing Two-Dimensional Voronoi Diagrams via Divide-and-Conquer of Envelopes in Space

- Mathematics, Computer ScienceISVD
- 2009

A general framework for computing two-dimensional Voronoi diagrams of different classes of sites under various distance functions is presented and it is proved that through randomization, the expected running time becomes near-optimal in the worst case.

Storing the Subdivision o f a Polyhedral Surface

- 2005

A common structure arising in computational geometry is the subdivision of a plane defined by the faces of a straight-line planar graph. We consider a natural generalization of this structure on a…

Intrinsic computation of voronoi diagrams on surfaces and its application

- Computer Science
- 2015

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.

The power of geometric duality

- Computer Science, Mathematics24th Annual Symposium on Foundations of Computer Science (sfcs 1983)
- 1983

A new formulation of the notion of duality that allows the unified treatment of a number of geometric problems is used, to solve two long-standing problems of computational geometry and to obtain a quadratic algorithm for computing the minimum-area triangle with vertices chosen among n points in the plane.

A Parallel Algorithm for Finding the Constrained Voronoi Diagram of Line Segments in the Plane

- Mathematics, Computer ScienceWADS
- 1999

An O(1/αlog n) time parallel algorithm for constructing the constrained Voronoi diagram of a set L of n non-crossing line segments in E2, using O(n1+α) processors on a CREW PRAM model, which constructs the constrained Delaunay triangulation of L in the same time and processor bound by the duality.

The power of geometric duality

- Mathematics
- 1985

A new formulation of the notion of duality that allows the unified treatment of a number of geometric problems is used, to solve two long-standing problems of computational geometry and to obtain a quadratic algorithm for computing the minimum-area triangle with vertices chosen amongn points in the plane.

Updates on Voronoi Diagrams

- Mathematics, Computer Science2011 Eighth International Symposium on Voronoi Diagrams in Science and Engineering
- 2011

Two sweep algorithms for updating Voronoi diagrams, one for deleting and another for inserting a site, which are applicable to points on the sphere surface or on the plane and are better than or similar to those of the CGAL library, which work on Delaunay triangulations.

## References

SHOWING 1-10 OF 36 REFERENCES

Proximity and reachability in the plane.

- Mathematics
- 1978

Abstract : The Voronoi diagram for a set of N points in the Cartesian plane in which the L sub p-metric is the distance measure (p, 1 < or = p < or = infinity, is a real number) is defined and an…

Voronoi Diagrams from Convex Hulls

- Mathematics, Computer ScienceInf. Process. Lett.
- 1979

The fundamental result is that a K-dimensional Euclidean Voronoi diagram of N points can be constructed by transforming the points to K + I-space, which extends straightforwardly to higher dimensions.

Closest-point problems

- Mathematics, Computer Science16th Annual Symposium on Foundations of Computer Science (sfcs 1975)
- 1975

The purpose of this paper is to introduce a single geometric structure, called the Voronoi diagram, which can be constructed rapidly and contains all of the relevant proximity information in only linear space, and is used to obtain O(N log N) algorithms for most of the problems considered.

Finding the Intersection of two Convex Polyhedra

- Computer Science, MathematicsTheor. Comput. Sci.
- 1978

An algorithm to test whether their intersection is empty, and if so to find a separating plane, and to construct their intersection polyhedron is developed, which runs in timeO (n log n), where n is the sum of the numbers of vertices of the two polyhedra.

Optimal Search in Planar Subdivisions

- Computer Science, MathematicsSIAM J. Comput.
- 1983

This work presents a practical algorithm for subdivision search that achieves the same (optimal) worst case complexity bounds as the significantly more complex algorithm of Lipton and Tarjan, namely $O(\log n)$ search time with $O(n)$ storage.

Efficient computation of continuous skeletons

- Computer Science20th Annual Symposium on Foundations of Computer Science (sfcs 1979)
- 1979

An O(n lgn) algorithm is presented for the construction of skeletons of arbitrary n-line polygonal figures that employs a linear time algorithm for the merging of two arbitrary (standard) Voronoi diagrams.

Convex hulls of finite sets of points in two and three dimensions

- Mathematics, Computer ScienceCACM
- 1977

The presented algorithms use the “divide and conquer” technique and recursively apply a merge procedure for two nonintersecting convex hulls to ensure optimal time complexity within a multiplicative constant.

The Art of Computer Programming, Volume I: Fundamental Algorithms, 2nd Edition

- Computer Science
- 1968

A container closure assembly for maintaining a sterile sealed container is provided having a ferrule having a top annular portion and a depending skirt portion for securing a resilient stopper for…

A polyhedron representation for computer vision

- Computer ScienceAFIPS '75
- 1975

My approach to computer vision is best characterized as inverse computer graphics, where perceived television pictures are analyzed to compute detailed geometric models to generate synthetic television images.

The Art of Computer Programming

- Computer Science
- 1968

The arrangement of this invention provides a strong vibration free hold-down mechanism while avoiding a large pressure drop to the flow of coolant fluid.