Abstract
We prove a theorem on partitioning point sets inE d (d fixed) and give an efficient construction of partition trees based on it. This yields a simplex range searching structure with linear space,O(n logn) deterministic preprocessing time, andO(n 1−1/d(logn)O(1)) query time. WithO(nlogn) preprocessing time, where δ is an arbitrary positive constant, a more complicated data structure yields query timeO(n1−1/d(log logn)O(1)). This attains the lower bounds due to Chazelle [C1] up to polylogarithmic factors, improving and simplifying previous results of Chazelleet al. [CSW].
The partition result implies that, forr d≤n 1−δ, a (1/r)-approximation of sizeO(r d) with respect to simplices for ann-point set inE d can be computed inO(n logr) deterministic time. A (1/r)-cutting of sizeO(r d) for a collection ofn hyperplanes inE d can be computed inO(n logr) deterministic time, provided thatr≤n 1/(2d−1).
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Part of this research was performed while the author was visiting at Freie Universität Berlin. The author was partially supported by the ESPRIT II Basic Research Actions Program of the EC under Contract No. 3075 (project Alcom). A preliminary version of this paper appeared in theProceedings of the Seventh ACM Symposium on Computational Geometry, pages 1–9, 1991.
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Matoušek, J. Efficient partition trees. Discrete Comput Geom 8, 315–334 (1992). https://doi.org/10.1007/BF02293051
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DOI: https://doi.org/10.1007/BF02293051