New applications of random sampling in computational geometry
TLDR
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
Quantum Computation in Computational Geometry
TL;DR: Some output-sensitive convex-hull algorithms and the ellipsoid algorithm for solving a general dimensional convex programming problem can be accelerated by using the quantum model.
Proceedings ArticleDOI
A Nearly Quadratic Bound for the Decision Tree Complexity of k-SUM
Esther Ezra,Micha Sharir +1 more
TL;DR: The approach relies on a new point-location mechanism, exploiting "Epsilon-cuttings" that are based on vertical decompositions in hyperplane arrangements in high dimensions, and results in a sharper bound on the complexity of the vertical decomposition of such an arrangement (in terms of its dependence on the dimension).
Proceedings ArticleDOI
Segment intersection searching problems in general settings
TL;DR: An efficient data structure for the triangular windowing problem, which is a generalization of triangular range searching, is obtained and the first substantially sub-quadratic algorithm for a red-blue intersection counting problem is derived.
Journal ArticleDOI
On the Union of Cylinders in Three Dimensions
TL;DR: The combinatorial complexity of the union of n infinite cylinders in ℝ3, having arbitrary radii, is O(n2+ε), for any ε>0; the bound is almost tight in the worst case, thus settling a conjecture of Agarwal and Sharir (Discrete Comput. Geom. 24:645–685, 2000).
Book ChapterDOI
Computational geometry for the gourmet old fare and new dishes
TL;DR: It is time to bring this quick overview to a close, and address the practical aspects of implementing and debugging geometric algorithms, and in particular, robustness in the face of round-off errors.
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