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
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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.