New applications of random sampling in computational geometry
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This paper gives several new demonstrations of the usefulness of random sampling techniques in computational geometry by creating a search structure for arrangements of hyperplanes by sampling the hyperplanes and using information from the resulting arrangement to divide and conquer.Abstract:
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 sampling the hyperplanes and using information from the resulting arrangement to divide and conquer. This algorithm requiresO(sd+?) expected preprocessing time to build a search structure for an arrangement ofs hyperplanes ind dimensions. The expectation, as with all expected times reported here, is with respect to the random behavior of the algorithm, and holds for any input. Given the data structure, and a query pointp, the cell of the arrangement containingp can be found inO(logs) worst-case time. (The bound holds for any fixed ?>0, with the constant factors dependent ond and ?.) Using point-plane duality, the algorithm may be used for answering halfspace range queries. Another algorithm finds random samples of simplices to determine the separation distance of two polytopes. The algorithm uses expectedO(n[d/2]) time, wheren is the total number of vertices of the two polytopes. This matches previous results [10] for the cased = 3 and extends them. Another algorithm samples points in the plane to determine their orderk Voronoi diagram, and requires expectedO(s1+?k) time fors points. (It is assumed that no four of the points are cocircular.) This sharpens the boundO(sk2 logs) for Lee's algorithm [21], andO(s2 logs+k(s?k) log2s) for Chazelle and Edelsbrunner's algorithm [4]. Finally, random sampling is used to show that any set ofs points inE3 hasO(sk2 log8s/(log logs)6) distinctj-sets withj≤k. (ForS ?Ed, a setS? ?S with |S?| =j is aj-set ofS if there is a half-spaceh+ withS? =S ?h+.) This sharpens with respect tok the previous boundO(sk5) [5]. The proof of the bound given here is an instance of a "probabilistic method" [15].read more
Citations
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Book ChapterDOI
Computing the Hausdorff Distance of Geometric Patterns and Shapes
TL;DR: Algorithms for computing the Hausdorff distance in a very general case in which geometric objects are represented by finite collections of k-dimensional simplices in d-dimensional space are developed.
Journal ArticleDOI
Multi-Pass Geometric Algorithms
Timothy M. Chan,Eric Y. Chen +1 more
TL;DR: This paper proposes the study of exact geometric algorithms that require limited storage and make only a small number of passes over the input.
Proceedings ArticleDOI
A dynamic data structure for 3-d convex hulls and 2-d nearest neighbor queries
TL;DR: This is the first method that guarantees polylogarithmic update and query cost for arbitrary sequences of insertions and deletions, and improves the previous O(n)(ε))-time method by Agarwal and Matoušek a decade ago.
Journal ArticleDOI
An upper bound on the number of planar K -sets
TL;DR: It is proved that O(n√k/log*k) is an upper bound for the number ofk-sets in the plane, thus improving the previous bound of Erdös, Lovász, Simmons, and Strauss by a factor of log*k.
Book ChapterDOI
Approximation Algorithms for k-Line Center
TL;DR: In this article, the running time of the algorithm is O(n log n), with the constant of proportionality depending on k, d, and?, where k is the number of points in the set P of n points in Rd and d is an integer k? 1, where w* denotes the minimum value so that P can be covered by k cylinders of radius at most w *.
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