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Dynamic half-space range reporting and its applications

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Abstract

We consider the half-space range-reporting problem: Given a setS ofn points in ℝd, preprocess it into a data structure, so that, given a query half-space γ, allk points ofS ∩ γ can be reported efficiently. We extend previously known static solutions to dynamic ones, supporting insertions and deletions of points ofS. For a given parameterm,n ≤m ≤n ⌊d/2⌋ and an arbitrarily small positive constant ɛ, we achieveO(m 1+ɛ) space and preprocessing time, O((n/m ⌊d/2⌋ logn+k) query time, and O(m1+ɛn) amortized update time (d ≳ 3). We present, among others, the following applications: an O(n1+ɛ)-time algorithm for computing convex layers in ℝ3, and an output sensitive algorithm for computing a level in an arrangements of planes in ℝ3, whose time complexity is O((b+n) nɛ, whereb is the size of the level.

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Authors and Affiliations

  1. Department of Computer Science, Duke University, Box 90129, 27708-0129, Durham, NC, USA

    P. K. Agarwal

  2. Department of Applied Mathematics, Charles University, Malostranskénam 25, 118 00, Praha 1, Czech Republic

    J. Matoušek

Authors
  1. P. K. Agarwal
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  2. J. Matoušek
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Additional information

Communicated by B. Chazelle.

Work by the first author has been supported by National Science Foundation Grant CCR-91-06514. A preliminary version of this paper appeared in Agarwalet al. [2], which also contains the results of [20] on dynamic bichromatic closest pair and minimum spanning trees.

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Agarwal, P.K., Matoušek, J. Dynamic half-space range reporting and its applications. Algorithmica 13, 325–345 (1995). https://doi.org/10.1007/BF01293483

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  • DOI: https://doi.org/10.1007/BF01293483

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