Range searching with efficient hierarchical cuttings

  title={Range searching with efficient hierarchical cuttings},
  author={Jiř{\'i} Matou{\vs}ek},
  journal={Discrete \& Computational Geometry},
  • 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… 

Tables from this paper

How hard is half-space range searching?

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

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

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

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

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

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

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

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

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

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.



Efficient partition trees

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

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.

Quasi-optimal upper bounds for simplex range searching and new zone theorems

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

Spanning trees with low crossing number

  • J. Matoušek
  • Mathematics, Computer Science
    RAIRO Theor. Informatics Appl.
  • 1991
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.

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.

Storing line segments in partition trees

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.

Approximations and optimal geometric divide-and-conquer

We give an efficient deterministic algorithm for computing ?-approximations and ?-nets for range spaces of bounded VC-dimension. We assume that an n-point range space ? = (X, R) of VC-dimension d is

Cutting hyperplanes for divide-and-conquer

  • B. Chazelle
  • Mathematics, Computer Science
    Discret. Comput. Geom.
  • 1993
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.

Cutting hyperplane arrangements

A deterministic algorithm for finding a (1/r)-cutting withO(rd) simplices with asymptotically optimal running time is given, which has numerous applications for derandomizing algorithms in computational geometry without affecting their running time significantly.

An Optimal Algorithm with Unknown Time Complexity for Convex Matrix Searching