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How hard is halfspace range searching?

Authors:

Hervé Brönnimann,

Bernard ChazelleAuthors Info & Claims

SCG '92: Proceedings of the eighth annual symposium on Computational geometry

Pages 271 - 275

https://doi.org/10.1145/142675.142730

Published: 01 July 1992 Publication History

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))}m^1/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/m^1/d) upper bound established by Matousˇek. We also show that it is possible to devise a sequence of n inserts and halfspace range queries that require a total time of n^2-θ(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 Ω(n^1/3) time (modulo a logarithmic factor).

References

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Cited By

Matoušek JSchwarzkopf O(1993)On ray shooting in convex polytopesDiscrete & Computational Geometry10.1007/BF0257397510:2(215-232)Online publication date: 1-Aug-1993
https://doi.org/10.1007/BF02573975
Schwarzkopf OAvis D(1992)Ray shooting in convex polytopesProceedings of the eighth annual symposium on Computational geometry10.1145/142675.142734(286-295)Online publication date: 1-Jul-1992
https://dl.acm.org/doi/10.1145/142675.142734

Index Terms

How hard is halfspace range searching?
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
      1. Graph algorithms
2. Theory of computation
  1. Randomness, geometry and discrete structures
    1. Computational geometry

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cover image ACM Conferences

SCG '92: Proceedings of the eighth annual symposium on Computational geometry

July 1992

384 pages

ISBN:0897915178

DOI:10.1145/142675

Chairman:
David Avis

Copyright © 1992 ACM.

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Published: 01 July 1992

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SOCG92: 8th Annual Symposium on Computational Geometry 1992

June 10 - 12, 1992

Berlin, Germany

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Cited By

Matoušek JSchwarzkopf O(1993)On ray shooting in convex polytopesDiscrete & Computational Geometry10.1007/BF0257397510:2(215-232)Online publication date: 1-Aug-1993
https://doi.org/10.1007/BF02573975
Schwarzkopf OAvis D(1992)Ray shooting in convex polytopesProceedings of the eighth annual symposium on Computational geometry10.1145/142675.142734(286-295)Online publication date: 1-Jul-1992
https://dl.acm.org/doi/10.1145/142675.142734

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