# Range searching with efficient hierarchical cuttings

@article{Matouek1992RangeSW,
title={Range searching with efficient hierarchical cuttings},
author={Jiř{\'i} Matou{\vs}ek},
journal={Discrete \& Computational Geometry},
year={1992},
volume={10},
pages={157-182}
}
• J. Matoušek
• Published 1 July 1992
• Computer Science
• Discrete & Computational Geometry
AbstractWe present an improved space/query-time tradeoff for the general simplex range searching problem, matching known lower bounds up to small polylogarithmic factors. In particular, we construct a linear-space simplex range searching data structure withO(n1−1/d) query time, which is optimal ford=2 and probably also ford>2. Further, we show that multilevel range searching data structures can be built with only a polylogarithmic overhead in space and query time per level (the previous…
260 Citations

## Tables from this paper

• Computer Science
Discret. Comput. Geom.
• 1993
The results imply the first nontrivial lower bounds for spherical range searching in any fixed dimension and establish a tradeoff between the storagem and the worst-case query timet in the Fredman/Yao arithmetic model of computation.
A new method is given that achieves simultaneously O(nlogn) preprocessing time, O(n) space, and O( n1−1/d) query time with high probability, which leads to more efficient multilevel partition trees, which are needed in many data structuring applications.
We prove a theorem on partitioning point sets inEd (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
• Mathematics, Computer Science
SODA
• 2023
A collection of new results are presented which improve previous bounds by multiple logarithmic factors that were caused by the use of multi-level data structures in simplex range searching and related problems in computational geometry.
Several new algorithms for constructing box-trees with small worst-case query complexity with respect to queries with axis-parallel boxes and with points are described and lower bounds on the worst- case query complexity for box-Trees are proved.
This thesis obtains the first optimal data structure for approximate halfspace range counting in 3D and provides two nontrivial methods to approximate the simplicial depth of a given point in higher dimension.
This work shows that the main techniques for simplex range searching in the plane can be adapted to the problem of computing the number of points in a query unit disk (i.e., all query disks have the same radius), and builds a data structure of $O(n)$ space so that each query can be answered in O(\sqrt{n}) time.
• Computer Science, Mathematics
SODA '07
• 2007
A reduction from matrix multiplication to the offline version of problem shows that in R2 the authors' time-space tradeoffs are close to optimal in the sense that improving them substantially would improve the best exponent of matrix multiplication.
It is shown how idempotence can be used to improve not only approximate, but also exact halfspace range searching, because its data structures are much simpler than both their exact and relative model counterparts, and so are amenable to efficient implementation.
This work considers variations of range searching in which, given a query range, the goal is to compute some function based on frequencies of points that lie in the range to exhibit the hardness of these problems by reducing Boolean matrix multiplication to the construction and use of a range mode or least frequent element data structure.

## References

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This paper presents quasi-optimal upper bounds for simplex range searching. The problem is to preprocess a setP ofn points in ℜd so that, given any query simplexq, the points inP ∩q can be counted or
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A A (H) Monte Carlo algorithm for this problem is obtained, improving a resuit of Edelsbrunner é tal and has numerous conséquences for the construction offurther randomized algorithms, using the above problems as a subroutine.
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Two variants of partition trees are designed that can be used for storing arbitrarily oriented line segments in the plane in an efficient way and it is shown how to use these structures for solving line segment intersection queries, triangle stabbing queries and ray shooting queries in reasonably efficient ways.
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A deterministic algorithm for computing a (1/r)-cutting ofO(rd) size inO(nrd−1) time is presented, based on a hierarchical construction of cuttings, which also provides a simple optimal data structure for locating a point in an arrangement of hyperplanes.
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