# Applications of random sampling in computational geometry, II

@inproceedings{Clarkson1988ApplicationsOR, title={Applications of random sampling in computational geometry, II}, author={Kenneth L. Clarkson}, booktitle={SCG '88}, year={1988} }

Random sampling is used for several new geometric algorithms. The algorithms are “Las Vegas,” and their expected bounds are with respect to the random behavior of the algorithms. One algorithm reports all the intersecting pairs of a set of line segments in the plane, and requires <italic>&Ogr;</italic>(<italic>A</italic> + <italic>n</italic> log <italic>n</italic>) expected time, where <italic>A</italic> is the size of the answer, the number of intersecting pairs reported. The algorithm…

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## References

SHOWING 1-10 OF 59 REFERENCES

Algorithms for diametral pairs and convex hulls that are optimal, randomized, and incremental

- MathematicsSCG '88
- 1988

An algorithm of this kind is given for computing the intersection of a set of halfspaces in three dimensions, resulting in a Las Vegas algorithm for the diameter requiring n expected time.

A deterministic algorithm for partitioning arrangements of lines and its application

- Computer ScienceSCG '89
- 1989

This paper presents a deterministic algorithm which is faster than Matousk's recent algorithm [Ma] for large values of r, and applies it to several problems involving lines or segments in the plane, and obtain deterministic algorithms which are faster than any previously known algorithms.

Halfspace range search: an algorithmic application of K-sets

- MathematicsSCG '85
- 1985

It is shown that the maximum number of&kgr;-sets realized by a set of n points in E3 is 3, which constitutes the only nontrivial upper bound, as a function of n and kgr, known to date.

Applications of random sampling in computational geometry, II

- Computer Science, MathematicsDiscret. Comput. Geom.
- 1989

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.

Epsilon-nets and simplex range queries

- Computer ScienceSCG '86
- 1986

A new technique for half-space and simplex range query using random sampling to build a partition-tree structure and introduces the concept of anε-net for an abstract set of ranges to describe the desired result of this random sampling.

On k-hulls and related problems

- MathematicsSTOC '84
- 1984

Efficient computation of the 'cut' guaranteed by the classical 'Ham Sandwich theorem', faster preprocessing time for polygon retrieval, and theoretical improvements to a problem of intersecting lines and points posed by Hopcroft.

Polling: a new randomized sampling technique for computational geometry

- Computer Science, MathematicsSTOC '89
- 1989

A new randomized sampling technique, called Polling, is introduced which has applications to deriving efficient parallel algorithms for fundamental problems like the convex hull in three dimensions, Voronoi diagram of point sites on a plane and Euclidean minimal spanning tree.

A probabilistic algorithm for the post office problem

- Computer ScienceSTOC '85
- 1985

The algorithm employs random sampling, so the expected time holds for any set of points, and approaches the preprocessing time required for any algorithm constructing the Voronoi diagram of the input points.

New applications of random sampling in computational geometry

- Computer ScienceDiscret. Comput. Geom.
- 1987

This paper gives several new demonstrations of the usefulness of random sampling techniques in computational geometry. One new algorithm creates a search structure for arrangements of hyperplanes by…

Partition trees for triangle counting and other range searching problems

- Computer ScienceSCG '88
- 1988

Bounds for space and query time are optimal up to polylog — factors and the preprocessing time for the data structures is polynomial according to recent results by Chazelle.