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Near-Linear Algorithms for Geometric Hitting Sets and Set Covers

Abstract

Given a finite range space \(\Sigma =(\mathsf {X},\mathcal {R})\), with \(N= |\mathsf {X}| + |\mathcal {R}|\), we present two simple algorithms, based on the multiplicative-weight method, for computing a small-size hitting set or set cover of \(\Sigma \). The first algorithm is a simpler variant of the Brönnimann–Goodrich algorithm but more efficient to implement, and the second algorithm can be viewed as solving a two-player zero-sum game. These algorithms, in conjunction with some standard geometric data structures, lead to near-linear algorithms for computing a small-size hitting set or set cover for a number of geometric range spaces. For example, they lead to \(O(N\mathrm {polylog}(N))\) expected-time randomized O(1)-approximation algorithms for both hitting set and set cover if \(\mathsf {X}\) is a set of points and \(\mathcal {R}\) a set of disks in \(\mathbb {R}^2\).

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Notes

  1. 1.

    The VC-dimension of a range space \(\Sigma = (\mathsf {X}, \mathcal {R})\) is the size of the largest subset \(A \subseteq X\) such that \(|\{A\cap R \mid R \in \mathcal {R}\}| = 2^{|A|}\).

  2. 2.

    Throughout this paper, we use \(\log x\) to denote \(\log _2 x\).

  3. 3.

    Using a sweep-plane technique, the running time of the algorithm for (P5) can be improved by a logarithmic factor. However, since this step is not the bottleneck, we use the above approach, which is simpler.

  4. 4.

    Recall that k is an integer satisfying \(k/2 < \kappa \le k\).

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Acknowledgements

The authors thank Kamesh Munagala for useful discussions and the two reviewers for their helpful comments.

Author information

Correspondence to Jiangwei Pan.

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Work on this paper is supported by NSF under Grants CCF-11-61359, IIS-14-08846, CCF-15-13816, and ISS-14-47554, by an ARO Grant W911NF-15-1-0408, and by Grant 2012/229 from the U.S.-Israel Binational Science Foundation. A preliminary version of the paper appeared in Proc. 30th Annual Symp. Comput. Geom. (SoCG’14).

Editor in Charge: Kenneth Clarkson

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Agarwal, P.K., Pan, J. Near-Linear Algorithms for Geometric Hitting Sets and Set Covers. Discrete Comput Geom (2019) doi:10.1007/s00454-019-00099-6

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Keywords

  • Geometric set cover
  • Near-linear algorithms
  • Multiplicative weight method
  • Disks
  • Rectangles

Mathematics Subject Classification

  • 68U05
  • 52C17