Information and Control
Algorithms in combinatorial geometry
Algorithms in combinatorial geometry
A randomized algorithm for closest-point queries
SIAM Journal on Computing
Applications of random sampling in computational geometry, II
Discrete & Computational Geometry - Selected papers from the fourth ACM symposium on computational geometry, Univ. of Illinois, Urbana-Champaign, June 6 8, 1988
Euclidean minimum spanning trees and bichromatic closest pairs
SCG '90 Proceedings of the sixth annual symposium on Computational geometry
SCG '91 Proceedings of the seventh annual symposium on Computational geometry
Randomized multidimensional search trees (extended abstract): dynamic sampling
SCG '91 Proceedings of the seventh annual symposium on Computational geometry
Computational Geometry: Theory and Applications
Small-dimensional linear programming and convex hulls made easy
Discrete & Computational Geometry
Randomized multidimensional search trees: lazy balancing and dynamic shuffling (extended abstract)
SFCS '91 Proceedings of the 32nd annual symposium on Foundations of computer science
Dynamic maintenance of geometric structures made easy
SFCS '91 Proceedings of the 32nd annual symposium on Foundations of computer science
Reporting points in halfspaces
SFCS '91 Proceedings of the 32nd annual symposium on Foundations of computer science
Randomized multidimensional search trees: further results in dynamic sampling (extended abstract)
SFCS '91 Proceedings of the 32nd annual symposium on Foundations of computer science
Ray shooting and parametric search
STOC '92 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing
Applications of random sampling to on-line algorithms in computational geometry
Discrete & Computational Geometry
How hard is halfspace range searching?
SCG '92 Proceedings of the eighth annual symposium on Computational geometry
Fully dynamic Delaunay triangulation in logarithmic expected time per operation
Computational Geometry: Theory and Applications
Point location among hyperplanes and unidirectional ray-shooting
Computational Geometry: Theory and Applications
A Singly-Expenential Stratification Scheme for Real Semi-Algebraic Varieties and Its Applications
ICALP '89 Proceedings of the 16th International Colloquium on Automata, Languages and Programming
Four Results on Randomized Incremental Constructions
STACS '92 Proceedings of the 9th Annual Symposium on Theoretical Aspects of Computer Science
Dynamic Half-Space Range Reporting and Its Application
Dynamic Half-Space Range Reporting and Its Application
SCG '92 Proceedings of the eighth annual symposium on Computational geometry
On point location and motion planning among simplices
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
ACM Computing Surveys (CSUR)
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Let H be a set of n halfspaces in Ed (where the dimension d ≥ 4 is fixed), and P(H) the convex polytope defined as their intersection. We show that P(H) can be preprocessed in time and space O(n[d/2]/(log n)[d/2]-&egr;) (for any fixed &egr; 0) so that ray shooting queries with rays starting in P(H) can be answered in time O(log n). This improves previous bounds by obtaining optimal query time and by improving the product Q(n)S(n)1/[d/2] (Q(n) and S(n) denoting query time and storage of a data structure for this problem) to O(n(log n)&egr;) for an arbitrarily small &egr; 0. By a well known lifting transformation, the results imply the same bounds for nearest (or furthest) neighbor queries in space of one dimension lower, which is an improvement in itself. We furthermore show that a structure for the ray shooting problem can be dynamically maintained under a sequence of random insertions, using the history of the maintenance process for the polytope as a point location data structure. The expected update time is O(m[d/2]-1(log m)O(1)), the query time for ray shooting queries is O(log2 m) with high probability, where m is the current number of points.