An optimal convex hull algorithm in any fixed dimension

@article{Chazelle1993AnOC,
title={An optimal convex hull algorithm in any fixed dimension},
author={Bernard Chazelle},
journal={Discrete \& Computational Geometry},
year={1993},
volume={10},
pages={377-409}
}
• B. Chazelle
• Published 1 December 1993
• Mathematics, Computer Science
• Discrete & Computational Geometry
We present a deterministic algorithm for computing the convex hull ofn points inEd in optimalO(n logn+n⌞d/2⌟) time. Optimal solutions were previously known only in even dimension and in dimension 3. A by-product of our result is an algorithm for computing the Voronoi diagram ofn points ind-space in optimalO(n logn+n⌜d/2⌝) time.
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References

SHOWING 1-10 OF 24 REFERENCES
Convex hulls of finite sets of points in two and three dimensions
• Mathematics, Computer Science
CACM
• 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 Ultimate Planar Convex Hull Algorithm?
• Mathematics, Computer Science
SIAM J. Comput.
• 1986
We present a new planar convex hull algorithm with worst case time complexity $O(n \log H)$ where $n$ is the size of the input set and $H$ is the size of the output set, i.e. the number of vertices
Small-dimensional linear programming and convex hulls made easy
• R. Seidel
• Mathematics, Computer Science
Discret. Comput. Geom.
• 1991
Two randomized algorithms solve linear programs involvingm constraints ind variables in expected timeO(m) and constructs convex hulls ofn points in ℝd,d>3, in expectedTimeO(n[d/2]).
Partitioning arrangements of lines I: An efficient deterministic algorithm
• P. Agarwal
• Mathematics, Computer Science
Discret. Comput. Geom.
• 1990
A deterministic algorithm for partitioning the plane into O(r2) triangles so that no triangle meets more thanO(n/r) lines of ℒ is presented.
A Convex Hull Algorithm Optimal for Point Sets in Even Dimensions
Finding the convex hull of a finite set of points is important not only for practical applications but also for theoretical reasons: a number of geometrical problems, such as constructing Voronoi
Constructing higher-dimensional convex hulls at logarithmic cost per face
• R. Seidel
• Mathematics, Computer Science
STOC '86
• 1986
The main tool in this new approach is the notion of a straight line shelling of a polytope in convex hull problems, which has best case time complexity O(m 2 -tFlogm), which is an improvement over the best previously achieved bounds for a large range of values of F.
A deterministic view of random sampling and its use in geometry
• Mathematics, Computer Science
[Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science
• 1988
It is shown how to compute, in polynomial time, a simplicial packing of size O(r/sup d/) that covers d-space, each of whose simplices intersects O(n/r) hyperplanes.
Cutting hyperplane arrangements
• J. Matousek
• Mathematics, Computer Science
Discret. Comput. Geom.
• 1991
A deterministic algorithm for finding a (1/r)-cutting withO(rd) simplices with asymptotically optimal running time is given, which has numerous applications for derandomizing algorithms in computational geometry without affecting their running time significantly.
Voronoi diagrams and arrangements
• Mathematics, Computer Science
Discret. Comput. Geom.
• 1986
It turns out that the standard Euclidean Voronoi diagram of point sets inRd along with its order-k generalizations are intimately related to certain arrangements of hyperplanes, and this fact can be used to obtain new Vor onoi diagram algorithms.
Voronoi Diagrams from Convex Hulls
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.