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
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Proceedings ArticleDOI
An upper bound on the number of planar k-sets
TL;DR: It is proved that O(n square root k/log/sub */k) is an upper bound for the number of k-sets in the plane, thus improving the previous bound of P. Erdos et al. (A Survey of Combinatorial Theory, North-Holland, 1983, p.139-49).
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A subexponential algorithm for abstract optimization problems
TL;DR: The author presents a randomized algorithm that solves any AOP with an expected number of O(e/sup O( square root mod H mod )/) oracle calls, which gives the first subexponential bound in d for this problem.
Proceedings ArticleDOI
Polling: a new randomized sampling technique for computational geometry
John H. Reif,Sandeep Sen +1 more
TL;DR: 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.
Posted Content
The discrete yet ubiquitous theorems of Carath\'eodory, Helly, Sperner, Tucker, and Tverberg
TL;DR: In this paper, the lemmas of Sperner and Tucker from combinatorial topology and the theorems of Carath-eodory, Helly, and Tverberg are discussed.
Proceedings ArticleDOI
Locally lifting the curse of dimensionality for nearest neighbor search (extended abstract)
TL;DR: The idea of aggressive pruning is introduced and a family of practical algorithms, an idealized analysis, and experiments are described that may contribute to improved general purpose algorithms for high dimensions.
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