Linear programming in O(n × 3d2) time
Information Processing Letters
An Algorithm for Convex Polytopes
Journal of the ACM (JACM)
Linear Programming in Linear Time When the Dimension Is Fixed
Journal of the ACM (JACM)
Optimal Expected-Time Algorithms for Closest Point Problems
ACM Transactions on Mathematical Software (TOMS)
Computational geometry.
Geometric transforms for fast geometric algorithms
Geometric transforms for fast geometric algorithms
Application of computational geometry to pattern recognition problems
Application of computational geometry to pattern recognition problems
Applications of a planar separator theorem
SFCS '77 Proceedings of the 18th Annual Symposium on Foundations of Computer Science
Epsilon-nets and simplex range queries
SCG '86 Proceedings of the second annual symposium on Computational geometry
Further applications of random sampling to computational geometry
STOC '86 Proceedings of the eighteenth annual ACM symposium on Theory of computing
Lines in space-combinators, algorithms and applications
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
Polling: a new randomized sampling technique for computational geometry
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
Randomized algorithms for binary search and load balancing with geometric applications
SPAA '90 Proceedings of the second annual ACM symposium on Parallel algorithms and architectures
Optimal sample cost residues for differential database batch query problems
Journal of the ACM (JACM)
SCG '91 Proceedings of the seventh annual symposium on Computational geometry
A lower bound on the complexity of approximate nearest-neighbor searching on the Hamming cube
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Dense point sets have sparse Delaunay triangulations: or "…but not too nasty"
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
A note on point location in arrangements of hyperplanes
Information Processing Letters
Adaptive stratified reservoir sampling over heterogeneous data streams
Information Systems
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The post office problem is the following: points in d-dimensional space, so that given an arbitrary point p, the closest points in S to p can be found quickly.We consider the case of this problem where the Euclidean norm is the measure of distance. The previous best algorithm for this problem for d2 requires &Ogr;(n2d+1) preprocessing time to build a data structure allowing an &Ogr;(log n query time. We will show that a data structure can be built in expected &Ogr;(n(d-1)(1+k)) time, for any fixed k;&Ogr;, so that closest-point queries can be answered in &Ogr;(log n) worstcase time. (The constant factors depend on d and k.) The algorithm employs random sampling, so the expected time holds for any set of points. A variant of this algorithm (for the variant problem where only one closest point of S to the query point is desired) requires &Ogr;(n⌈d/2⌉) &ogr;(n⌈d/2⌉) preprocessing time for &ogr;(nt) worst-case query time, for any fixed &egr;0. These results approach the &OHgr;(n⌈d/2⌉) preprocessing time required for any algorithm constructing the Voronoi diagram of the input points. Implementation of these algorithms requires not too much more than a random sampling procedure and a procedure for constructing the Voronoi diagram of that random sample.