# Geometric range searching

@article{Matousek1994GeometricRS, title={Geometric range searching}, author={Jir{\'i} Matousek}, journal={ACM Comput. Surv.}, year={1994}, volume={26}, pages={421-461} }

In geometric range searching, algorithmic problems of the following type are considered. Given an n-point set P in the plane, build a data structure so that, given a query triangle R, the number of points of P lying in R can be determined quickly. Similar questions can be asked for point sets in higher dimensions, with triangles replaced by simplices or by more complicated shapes. Algorithms of this type are of crucial importance in computational geometry, as they can be used as subroutines in…

## 96 Citations

### Non-orthogonal Range Searching: A Review

- Computer Science
- 2008

The long history of nonorthogonal range searching is reviewed from its beginning to the present and the latest advances in this area are shown, including heuristic algorithms, small improvements for higher dimensions, and new problems on range searching with a variety of applications.

### Simplex Range Searching and Its Variants: A Review

- Mathematics
- 2017

A central problem in computational geometry, range searching arises in many applications, and numerous geometric problems can be formulated in terms of range searching. A typical range-searching…

### On the relative complexities of some geometric problems

- Mathematics, Computer ScienceCCCG
- 1995

This paper considers the relative complexities of a large number of computational geometry problems whose complexities are believed to be roughly (n4=3), and surveys known reductions among problems involving lines in three-space, and among higher dimensional closestpair problems.

### The Effect of Corners on the Complexity of Approximate Range Searching

- Mathematics, Computer ScienceDiscret. Comput. Geom.
- 2009

Lower bounds on the worst-case complexity of approximate range searching in the semigroup arithmetic model for ranges consisting of d-dimensional unit hypercubes under rigid motions are established and the improvements offered by idempotence do apply to smooth convex ranges.

### Geometric Multicut

- Computer Science, MathematicsICALP
- 2019

The problem is referred to as geometrick-cut, as it is a geometric analog to the well-studied multicut problem on graphs and a randomised approximation algorithm for polygons and any number of colours is given.

### Fast Range Searching with Delaunay Triangulations

- Computer ScienceGeoInformatica
- 1997

A new algorithm for generating random simple polygons within a given domain and the empirical results show that the constant coefficient of the algorithm is small, at least for the special cases when the query polygon is either a triangle or an axis-parallel box.

### Enumerating Collinear Points in Higher Dimensions

- Mathematics, Computer ScienceSci. Ann. Comput. Sci.
- 2019

A practical algorithm can be used to detect if any three of the points are collinear or find the line that intersects the most points in S, and one with space complexity O(n) and time complexity O (dn log n) is presented.

### Dynamic Data Structures for Geometric Search and Retrieval

- Computer Science
- 2013

This dissertation introduces two dynamic quadtree-based data structures for storing a set of points in space, called the quadtreap and the splay quadtree, and presents the first output sensitive algorithm for maintaining a well-separated pair decomposition (WSPD) for a dynamic point set.

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