Constructing arrangements of lines and hyperplanes with applications
SIAM Journal on Computing
A general approach to d-dimensional geometric queries
STOC '85 Proceedings of the seventeenth annual ACM symposium on Theory of computing
Algorithms in combinatorial geometry
Algorithms in combinatorial geometry
A randomized algorithm for closest-point queries
SIAM Journal on Computing
Lines in space-combinators, algorithms and applications
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
Applications of random sampling in computational geometry, II
SCG '88 Proceedings of the fourth annual symposium on Computational geometry
SCG '89 Proceedings of the fifth annual symposium on Computational geometry
A deterministic algorithm for partitioning arrangements of lines and its application
SCG '89 Proceedings of the fifth annual symposium on Computational geometry
Ray shooting and other applications of spanning trees with low stabbing number
SCG '89 Proceedings of the fifth annual symposium on Computational geometry
Implicitly representing arrangements of lines or segments
Discrete & Computational Geometry - Selected papers from the fourth ACM symposium on computational geometry, Univ. of Illinois, Urbana-Champaign, June 6 8, 1988
The Design and Analysis of Computer Algorithms
The Design and Analysis of Computer Algorithms
Selecting distances in the plane
SCG '90 Proceedings of the sixth annual symposium on Computational geometry
Approximations and optimal geometric divide-and-conquer
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
SCG '91 Proceedings of the seventh annual symposium on Computational geometry
Counting circular arc intersections
SCG '91 Proceedings of the seventh annual symposium on Computational geometry
On lines missing polyhedral sets in 3-space
SCG '93 Proceedings of the ninth annual symposium on Computational geometry
Efficient NC algorithms for set cover with applications to learning and geometry
Proceedings of the 30th IEEE symposium on Foundations of computer science
Reduction rules deliver efficient FPT-algorithms for covering points with lines
Journal of Experimental Algorithmics (JEA)
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We will consider an arrangement H of n hyperplanes in Ed (where the dimension d is fixed). An &egr;-cutting for H will be a collection of (possibly unbounded) d-dimensional simplices with disjoint interiors, which cover all Ed and such that the interior of any simplex is intersected by at most &egr;n hyperplanes of H. We give a deterministic algorithm, finding a (1/r)-cutting with &Ogr;(rd(log r)C) simplices in time &Ogr;(n(log n)Ard-1 (log r)B) (A,B,C are constants dependent on dimension). In a similar time bound (with an additional &Ogr;(r&Ogr;(1)) overhead) we can also find a (1/r)-net for the range space (X, H(X)), where X is a n-point set in Ed and H(X) denotes the set of all subsets of X which can be cut by a halfspace. This (1/r)-net has size &Ogr;(r log r), which matches the best known existence result; in fact, the method gives a constructive existence proof. In the plane, we can obtain a (1/r)-cutting of optimal size &Ogr;(r2) in time &Ogr;(nr) (which is optimal if we want to compute also the collection of lines intersecting each simplex of the cutting). This improves the result of Agarwal, and our algorithm is conceptually simpler.