Proceedings ArticleDOI
How hard is halfspace range searching
Hervé Brönnimann,Bernard Chazelle +1 more
- pp 271-275
TLDR
The paper investigates the complexity of halfspace range searching and establishes a tradeoff between the storage and the worst-case query time in the Fredman/Yao arithmetic model of computation, establishing the first nontrivial lower bound for half space range searching.Abstract:
We investigate the complexity of halfspace range searching: Given n points in d-space, build a data structure that allows us to determine efficiently how many points lie in a query halfspace. We establish a tradeoff between the storage m and the worst-case query time t in the Fredman/Yao arithmetic model of computation. We show that t must be at least on the order of (n/log n)1-((d-1)/(d(d+1))m1/d.To our knowledge, this is the first nontrivial lower bound for halfspace range searching. Although the bound is unlikely to be optimal, it falls reasonably close to the recent O(n(log m/n)d+1/m1/d) upper bound established by Matouscek. We also show that it is possible to devise a sequence of n inserts and halfspace range queries that require a total time of n2-t(1/d). Our results imply nontrivial lower bounds for spherical range searching in any fixed dimension. For example they show that, with linear storage, circular range queries in the plane require O(n1/3) time (modulo a logarithmic factor).read more
Citations
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Geometric Range Searching and Its Relatives
Pankaj K. Agarwal,Je Erickson +1 more
TL;DR: This volume provides an excellent opportunity to recapitulate the current status of geometric range searching and to summarize the recent progress in this area.
Journal ArticleDOI
Reporting points in halfspaces
TL;DR: The halfspace itrange itreporting problem, given a finite set P of points in R d, can be solved substantially more efficiently that the more general simplex range searching problem.
Journal ArticleDOI
On Approximating the Depth and Related Problems
Boris Aronov,Sariel Har-Peled +1 more
TL;DR: This paper reduces the problem of finding a disk covering the largest number of red points, while avoiding all the blue points to a near-linear expected-time randomized approximation algorithm and proves that approximate range counting has roughly the same time and space complexity as answering emptiness range queries.
Proceedings ArticleDOI
Approximate range searching
Sunil Arya,David M. Mount +1 more
TL;DR: It is shown that if one is willing to allow approximate ranges, then it is possible to do much better than current state-of-the-art results, and empirical evidence is given showing that allowing small relative errors can significantly improve query execution times.
Journal ArticleDOI
Approximate range searching
Sunil Arya,David M. Mount +1 more
TL;DR: A lower bound for approximate range searching based on partition trees of Ω( log n+(1/e) d−1 ) , which implies optimality for convex ranges (assuming fixed dimensions), and empirical evidence showing that allowing small relative errors can significantly improve query execution times.
References
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Book
Integral geometry and geometric probability
Abstract: Part I. Integral Geometry in the Plane: 1. Convex sets in the plane 2. Sets of points and Poisson processes in the plane 3. Sets of lines in the plane 4. Pairs of points and pairs of lines 5. Sets of strips in the plane 6. The group of motions in the plane: kinematic density 7. Fundamental formulas of Poincare and Blaschke 8. Lattices of figures Part II. General Integral Geometry: 9. Differential forms and Lie groups 10. Density and measure in homogenous spaces 11. The affine groups 12. The group of motions in En Part III. Integral Geometry in En: 13. Convex sets in En 14. Linear subspaces, convex sets and compact manifolds 15. The kinematic density in En 16. Geometric and statistical applications: stereology Part IV. Integral Geometry in Spaces of Constant Curvature: 17. Noneuclidean integral geometry 18. Crofton's formulas and the kinematic fundamental formula in noneuclidean spaces 19. Integral geometry and foliated spaces: trends in integral geometry.
Journal ArticleDOI
ź-nets and simplex range queries
David Haussler,Emo Welzl +1 more
TL;DR: The concept of an ɛ-net of a set of points for an abstract set of ranges is introduced and sufficient conditions that a random sample is an Â-net with any desired probability are given.
Book
Ten lectures on the probabilistic method
TL;DR: The Janson Inequalities as discussed by the authors allow accurate approximation of extremely small probabilities, and have been shown to be useful for the probabilistic method in many problems. But they do not cover the complexity of the Janson inequalities.
Book
Data Structures and Algorithms 3 : Multi-dimensional Searching and Computational Geometry
TL;DR: It sounds good when knowing the data structures and algorithms 3 multi dimensional searching and computational geometry in this website, but it will not become a unity of the way for you to get amazing benefits at all.
Journal ArticleDOI
Reporting points in halfspaces
TL;DR: The halfspace itrange itreporting problem, given a finite set P of points in R d, can be solved substantially more efficiently that the more general simplex range searching problem.