The power of geometric duality
BIT - Ellis Horwood series in artificial intelligence
Computational geometry: an introduction
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New upper bounds for neighbor searching
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Halfspace range search: an algorithmic application of k-sets
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Reporting points in halfspaces
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On range reporting, ray shooting and k-level construction
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SFCS '93 Proceedings of the 1993 IEEE 34th Annual Foundations of Computer Science
Cache-oblivious range reporting with optimal queries requires superlinear space
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A general approach for cache-oblivious range reporting and approximate range counting
Proceedings of the twenty-fifth annual symposium on Computational geometry
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A dynamic data structure for 3-D convex hulls and 2-D nearest neighbor queries
Journal of the ACM (JACM)
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A general approach for cache-oblivious range reporting and approximate range counting
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Improved space bounds for cache-oblivious range reporting
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Processing a large number of continuous preference top-k queries
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Range searching on uncertain data
ACM Transactions on Algorithms (TALG)
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ESA'12 Proceedings of the 20th Annual European conference on Algorithms
Nearest neighbor searching under uncertainty II
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We give the first optimal solution to a standard problem in computational geometry: three-dimensional halfspace range reporting. We show that n points in 3-d can be stored in a linear-space data structure so that all k points inside a query halfspace can be reported in O(log n + k) time. The data structure can be built in O(n log n) expected time. The previous methods with optimal query time required superlinear (O(n log log n)) space. We also mention consequences, for example, to higher dimensions and to external-memory data structures. As an aside, we partially answer another open question concerning the crossing number in Matoušek's shallow partition theorem in the 3-d case (a tool used in many known halfspace range reporting methods).