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Geometric Hitting Set for Line-Constrained Disks

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Algorithms and Data Structures (WADS 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14079))

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Abstract

Given a set P of n weighted points and a set S of m disks in the plane, the hitting set problem is to compute a subset \(P'\) of points of P such that each disk contains at least one point of \(P'\) and the total weight of all points of \(P'\) is minimized. The problem is known to be NP-hard. In this paper, we consider a line-constrained version of the problem in which all disks are centered on a line \(\ell \). We present an \(O((m+n)\log (m+n)+\kappa \log m)\) time algorithm for the problem, where \(\kappa \) is the number of pairs of disks that intersect. For the unit-disk case where all disks have the same radius, the running time can be reduced to \(O((n + m)\log (m + n))\). In addition, we solve the problem in \(O((m + n)\log (m + n))\) time in the \(L_{\infty }\) and \(L_1\) metrics, in which a disk is a square and a diamond, respectively.

This research was supported in part by NSF under Grants CCF-2005323 and CCF-2300356. A full version of this paper is available at http://arxiv.org/abs/2305.09045.

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Authors and Affiliations

  1. University of Utah, Salt Lake City, UT, 84112, USA

    Gang Liu & Haitao Wang

Authors
  1. Gang Liu
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  2. Haitao Wang
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Correspondence to Gang Liu .

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Editors and Affiliations

  1. Carleton University, Ottawa, ON, Canada

    Pat Morin

  2. University of California, Santa Barbara, CA, USA

    Subhash Suri

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Liu, G., Wang, H. (2023). Geometric Hitting Set for Line-Constrained Disks. In: Morin, P., Suri, S. (eds) Algorithms and Data Structures. WADS 2023. Lecture Notes in Computer Science, vol 14079. Springer, Cham. https://doi.org/10.1007/978-3-031-38906-1_38

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  • DOI: https://doi.org/10.1007/978-3-031-38906-1_38

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-38905-4

  • Online ISBN: 978-3-031-38906-1

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