# 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} }

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.

## 363 Citations

An Algorithm for Constructing the Convex Hull of a Set of Spheres in Dimension D

- Computer ScienceComput. Geom.
- 1996

Derandomizing an Output-sensitive Convex Hull Algorithm in Three Dimensions

- MathematicsComput. Geom.
- 1995

Computation of Convex Hulls

- Computer Science, Mathematics
- 2013

When referring to “computation of convex hulls” we understand this as the task of computing the \(\mathcal {H}\)-representation of the convex hull of a given finite point set V⊆ℝ n . Depending on the…

Optimal Voronoi Diagram Construction with n Convex Sites in Three Dimensions

- Computer Science, MathematicsInt. J. Comput. Geom. Appl.
- 2007

This paper presents a worst-case optimal algorithm for constructing the Voronoi diagram for n disjoint convex and rounded semi-algebraic sites in 3 dimensions, based on a suboptimal 2-dimensional algorithm outlined by Lee and Drysdale and modified by Sharir25,30.

Efficient Computation of the Outer Hull of a Discrete Path

- Computer ScienceDGCI
- 2014

By combining the well-known wall follower algorithm for traversing mazes, the desired result with two passes is obtained resulting in a global linear time and space algorithm and the convex hull is obtained as a byproduct.

Densest Lattice Packings of 3 {

- Computer Science
- 2009

Based on Minkowski's work on critical lattices of 3-dimensional convex bodies we present an eecient algorithm for computing the density of a densest lattice packing of an arbitrary 3-polytope. As an…

Degenerate Convex Hulls On-Line in Any Fixed Dimension

- Computer ScienceDiscret. Comput. Geom.
- 1999

The main goal of this paper is to show how a simple modification of the formulation of an on-line algorithm by Boissonnat et al., based on that of Clarkson and Shor, can handle the case of degenerate sets of points for computing convex hulls on- line in Rd.

## References

SHOWING 1-10 OF 24 REFERENCES

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

- 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 Ultimate Planar Convex Hull Algorithm?

- Computer ScienceSIAM 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

- Computer ScienceDiscret. 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

- Mathematics, Computer ScienceDiscret. 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

- Computer Science, Mathematics
- 1981

It is shown that this algorithm is worst case optimal for even $d \geq 2$ and the main result is an O(n \log n + n^{\lfloor(d+1)/2\rfloor}) algorithm for the construction of the convex hull of n points in R^{d}.

Constructing higher-dimensional convex hulls at logarithmic cost per face

- Computer Science, MathematicsSTOC '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.

Probabilistic construction of deterministic algorithms: Approximating packing integer programs

- Computer Science, Mathematics27th Annual Symposium on Foundations of Computer Science (sfcs 1986)
- 1986

A deterministic view of random sampling and its use in geometry

- Computer Science, Mathematics[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

- Mathematics, Computer ScienceDiscret. 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 ScienceDiscret. 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.