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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Journal ArticleDOI
Discrepancy and approximations for bounded VC-dimension
TL;DR: It is shown that if for anym-point subset $$Y \subseteq X$$ the number of distinct subsets induced by ℛ onY is bounded byO(md) for a fixed integerd, then there are improved upper bounds on the size of ε-approximations for (X,ℛ).
Proceedings ArticleDOI
A subexponential bound for linear programming
TL;DR: A simple randomized algorithm which solves linear programs with n constraints and d variables in expected O ( nde ( d ln( n +1) 1/4 ) time in the unit cost model and the subexponential complexity bound is presented.
Proceedings ArticleDOI
Constructing levels in arrangements and higher order Voronoi diagrams
TL;DR: A simple lazy randomized incremental algorithm is given to compute ≤k-levels in arrangements of x-monotone Jordan curves in the plane, and in arrangement of planes in three-dimensional space.
Posted Content
Improved Approximation Algorithms for Geometric Set Cover
TL;DR: In this article, it was shown that for a random subset R of S and function f, there is a decomposition of the portion of U not covered by R into an expected f(|R|) regions, each region of a particular simple form.
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Partitioning Arrangements of Lines: II. Applications
TL;DR: An algorithm that preprocesses a set ofn points in the plane, into a data structure of sizeO(m) forn logn≤m≤n2, so that the number of points ofS lying inside a query triangle can be computed inO((n/√m) log3/2n) time.
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