Randomized incremental construction of Delaunay and Voronoi diagrams

@article{Guibas2005RandomizedIC,
  title={Randomized incremental construction of Delaunay and Voronoi diagrams},
  author={Leonidas J. Guibas and Donald Ervin Knuth and Micha Sharir},
  journal={Algorithmica},
  year={2005},
  volume={7},
  pages={381-413}
}
In this paper we give a new randomized incremental algorithm for the construction of planar Voronoi diagrams and Delaunay triangulations. The new algorithm is more “on-line” than earlier similar methods, takes expected timeO(nℝgn) and spaceO(n), and is eminently practical to implement. The analysis of the algorithm is also interesting in its own right and can serve as a model for many similar questions in both two and three dimensions. Finally we demonstrate how this approach for constructing… 
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372 Citations
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372 Citations

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    2012 Ninth International Symposium on Voronoi Diagrams in Science and Engineering
  • 2012
We show how to localize the Delaunay triangulation of a given planar point set, namely, bound the set of points which are possible Delaunay neighbors of a given point. We then exploit this…
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Theoretical analysis and experimental results show that the proposed algorithm based on Delaunay triangulation of randomly distributed points in the Euclidean plane is an efficient method of generating Voronoi diagram.
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A new approach to constructing the Delaunay Triangulation and the optimum algorithm for the case of multidimensional spaces (d ≥ 2) and offers for the solving this problem an effective algorithm that satisfies all the given requirements.
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A new incremental insertion algorithm for constructing a Delaunay triangulation that is fast and practically independent of the distribution of input points, it is not memory demanding, and it is numerically stable and easy to implement is presented.
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A heuristic point search method which can locate a random point within the current triangle efficiently and test the performance on a collection of point sample sets and demonstrate a 30% performance improvement compared to existing 3D randomized incremental algorithms.
ON THE APPLICATION OF VORONOI DIAGRAMS AND DELAUNAY TRIANGULATION TO 3D RECONSTRUCTION
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A short and fast algorithm to optimally compute the Delaunay triangulation is highlighted, used in the reconstruction of 3D geometric figures where the complexity of the problem is greater than the classical 2D plane case.
A faster circle-sweep Delaunay triangulation algorithm
A semidynamic construction of higher-order voronoi diagrams and its randomized analysis
TLDR
It is proved that a randomized construction of thek-Delaunay tree, and thus of all the order≤k Voronoi diagrams, can be done in O(n logn+k3n) expected time and O(k2n)expected storage in the plane, which is asymptotically optimal for fixedk.
An efficient sweep-line Delaunay triangulation algorithm
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A reduced-precision Voronoi diagram is defined that similarly supports proximity queries, and a randomized incremental construction using only three times the input precision is described.
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References

SHOWING 1-10 OF 25 REFERENCES
On the construction of abstract voronoi diagrams
TLDR
It is shown that the abstract Voronoi diagram of n sites in the plane can be constructed in timeO(n logn) by a randomized algorithm based on Clarkson and Shor's randomized incremental construction technique.
The hierarchical representation of objects: the Delaunay tree
TLDR
The Delaunay tree provides efficient solutions to several problems such as building theDelaunay triangulation of a finite set of n points in any dimension, locating a point in the triangulations, defining neighborhood relationships in the Triangulation and computing intersections.
Higher-dimensional Voronoi diagrams in linear expected time
A general method is presented for determining the mathematical expectation of the combinatorial complexity and other properties of the Voronoi diagram ofn independent and identically distributed…
Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams
TLDR
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.
A fast planar partition algorithm. I
  • K. Mulmuley
  • Computer Science
    [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science
  • 1988
TLDR
Though the algorithm itself is simple, the global evolution of the partition is complex, which makes the analysis of the algorithm theoretically interesting in its own right.
Optimal Expected-Time Algorithms for Closest Point Problems
TLDR
Algorithms for solving a number of closest-point problems in k- space, including nearest neighbor searching, finding all nearest neighbors, and computing planar minimum spanning trees can be implemented to solve practical problems very efficiently.
Applications of random sampling in computational geometry, II
TLDR
Asymptotically tight bounds for (≤k)-sets are given, which are certain halfspace partitions of point sets, and a simple proof of Lee's bounds for high-order Voronoi diagrams is given.
Optimal Search in Planar Subdivisions
TLDR
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.
Optimal Point Location in a Monotone Subdivision
TLDR
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.
Fully dynamic techniques for point location and transitive closure in planar structures
  • F. Preparata, R. Tamassia
  • Mathematics, Computer Science
    [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science
  • 1988
TLDR
Planar st-graphs provide the topological underpinning of a fully dynamic planar point location technique in monotone subdivisions, which is an interesting (unique) specialization of the chain method of Lee-Preparata (1977).
...
...