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
More filters
A Theory-Based Decision Heuristic for Disjunctive Linear Arithmetic
TL;DR: This work studies the decision problem of Disjunctive Linear Arithmetic over the Reals from the perspective of computational geometry and shows a branching heuristic that is based on approximating T -implications, based on a geometric analysis.
Book ChapterDOI
An efficient algorithm for the sign condition problem in the semi-algebraic context
TL;DR: The dialytic method solves (non-uniformly) the sign condition problem for a family of s polynomials in R[X1,...,Xn] given by an arithmetic circuit $\Gamma_{\mathcal F}$ of non-scalar complexity L performing L performing arithmetic operations.
Posted Content
Hopcroft's Problem, Log-Star Shaving, 2D Fractional Cascading, and Decision Trees.
Timothy M. Chan,Da Wei Zheng +1 more
TL;DR: In this paper, the problem of counting the number of point-line incidence pairs or point-above-line pairs in the plane was solved in O(n 4/3 ) time.
Book ChapterDOI
Dynamic Construction of Additively Weighted Network Voronoi Diagram
Xin Liu,Yi-li Tan,Hongmei Yang +2 more
TL;DR: This paper uses discrete construction of network Voronoi diagrams and shows that the algorithm can get over all kinds of shortcomings, and it is more useful and effective than the traditional ones.
Posted Content
Low-Rank Methods in Event Detection and Subsampled Point-to-Subspace Proximity Tests
TL;DR: In this article, the authors use a variant of low-rank factorization, which considers interval uncertainty sets around "known entries", on a suitable flattening of the input data to obtain a lowrank model, and perform event detection by running point-to-subspace distance query for incoming data.
References
More filters
Book
The Art of Computer Programming
TL;DR: The arrangement of this invention provides a strong vibration free hold-down mechanism while avoiding a large pressure drop to the flow of coolant fluid.
Computational geometry. an introduction
TL;DR: This book offers a coherent treatment, at the graduate textbook level, of the field that has come to be known in the last decade or so as computational geometry.
Book ChapterDOI
On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities
TL;DR: This chapter reproduces the English translation by B. Seckler of the paper by Vapnik and Chervonenkis in which they gave proofs for the innovative results they had obtained in a draft form in July 1966 and announced in 1968 in their note in Soviet Mathematics Doklady.
Book
Computational Geometry: An Introduction
TL;DR: In this article, the authors present a coherent treatment of computational geometry in the plane, at the graduate textbook level, and point out the way to the solution of the more challenging problems in dimensions higher than two.