# 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} }

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

### How hard is half-space range searching?

- Computer ScienceDiscret. 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.

### Optimal Partition Trees

- Computer ScienceSCG
- 2010

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.

### Efficient partition trees

- Computer ScienceSCG '91
- 1991

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…

### Simplex Range Searching Revisited: How to Shave Logs in Multi-Level Data Structures

- Mathematics, Computer ScienceSODA
- 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.

### Results on geometric networks and data structures

- Computer Science, Mathematics
- 2004

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.

### On Geometric Range Searching, Approximate Counting and Depth Problems

- Computer Science
- 2008

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.

### Unit-Disk Range Searching and Applications

- Computer ScienceSWAT
- 2022

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.

### Counting colors in boxes

- Computer Science, MathematicsSODA '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.

### Approximate Range Searching: The Absolute Model

- MathematicsWADS
- 2007

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.

### Adaptive Range Counting and Other Frequency-Based Range Query Problems

- Computer Science
- 2012

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.

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