Geometric range searching

  title={Geometric range searching},
  author={Jir{\'i} Matousek},
  journal={ACM Comput. Surv.},
  • J. Matousek
  • Published 21 September 1994
  • Computer Science, Mathematics
  • ACM Comput. Surv.
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… 

Figures from this paper

Non-orthogonal Range Searching: A Review

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

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

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

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

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

  • B. Zhu
  • Computer Science
  • 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

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

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.

Arrangements and Their Applications




Applications of parametric searching in geometric optimization

The goal is to demonstrate the versatility of the parametric searching technique and obtain efficient solutions to several problems of Megiddo's parametric search technique.

Dynamic partition trees

An output-sensitive method for hidden surface removal in a set ofn triangles that runs in timeO(nlogn+n·kγ) whereγ=log2((1+√5)/2) ≈ 0.695 andk is the size of the visibility map obtained.

Lower bounds on the complexity of polytope range searching

It is proved that the worst case query time is Q(n/l/Hi) in the Euclidean plane, and more generally, Q((n/ log n)/m'l/d) in d-space, for d > 3, where n is the number of points and m is the amount of storage available.

Geometric retrieval problems

  • R. ColeC. Yap
  • Computer Science, Mathematics
    24th Annual Symposium on Foundations of Computer Science (sfcs 1983)
  • 1983

Quasi-optimal range searching in spaces of finite VC-dimension

It is proved that any set ofn points inEd admits a spanning tree which cannot be cut by any hyperplane (or hypersphere) through more than roughlyn1−1/d edges, and this result yields quasi-optimal solutions to simplex range searching in the arithmetic model of computation.

The power of geometric duality

A new formulation of the notion of duality that allows the unified treatment of a number of geometric problems is used, to solve two long-standing problems of computational geometry and to obtain a quadratic algorithm for computing the minimum-area triangle with vertices chosen amongn points in the plane.

Diameter, width, closest line pair, and parametric searching

This work applies Megiddo's parametric searching technique to several geometric optimization problems and derive significantly improve solutions for them, including an algorithm for computing the diameter of a point set in 3-space, and a very simple solution which bypasses parametric search altogether.

Partitioning Space for Range Queries

It is shown that, given a set S of n points in R3, one can always find three planes that form an eight-partition of S, that is, a partition where at most n/8 points of S lie in each of the eight open

Efficient ray shooting and hidden surface removal

The ray-shooting structure for curtains is used to obtain an algorithm for computing the view of a set of nonintersecting prolyhedra, the first output-sensitive algorithm for this problem that does not need a depth order on the faces of the polyhedra.

New applications of random sampling in computational geometry

This paper gives several new demonstrations of the usefulness of random sampling techniques in computational geometry. One new algorithm creates a search structure for arrangements of hyperplanes by