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Optimizing Squares Covering a Set of Points

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Combinatorial Optimization and Applications (COCOA 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8881))

Abstract

We investigate two kinds of optimization problems regarding points in the 2-dimensional plane that need to be enclosed by squares.

  1. (1)

    Given a set of \(n\) points, find a given number of squares that enclose all the points, minimizing the size of the largest square used.

  2. (2)

    Given a set of \(n\) points, enclose the maximum number of points, using a specified number of squares of a fixed size. We provide different techniques to solve the above problems in cases where squares are axis-parallel or of arbitrary orientation, disjoint or overlapping. All the algorithms we use run in time that is a low-order polynomial in \(n\).

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Notes

  1. 1.

    This polygon may be the convex hull of the points in \(\mathcal P\) computed in \(O(n \log n)\) time.

  2. 2.

    Since the points in \(\mathcal{P}'_l {\setminus }\{a, b, c, d\}\) cannot affect the size of the left square, they can be pruned.

  3. 3.

    To be exact, \(\mathcal{P}_1\) (resp. \(\mathcal{P}_2\), \(\mathcal{P}_3\)) covers the first \(\lceil n/3\rceil \) (resp. next \(\lfloor n/3\rfloor \), the remaining \(\lceil n/3\rceil \) or \(\lfloor n/3\rfloor \)) points. They can be determined in linear time [2]. So the difference is at most one.

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Author information

Authors and Affiliations

  1. School of Computing Science, Simon Fraser University, Burnaby, Canada

    Binay Bhattacharya & Tsunehiko Kameda

  2. Advanced Computing and Microelectronics Unit, Indian Statistical Institute, Kolkata, India

    Sandip Das

  3. Department of Computer Science and Engineering, University of Kalyani, Kalyani, India

    Priya Ranjan Sinha Mahapatra

  4. Department of Computer Science, University of Texas at Austin, Austin, USA

    Zhao Song

Authors
  1. Binay Bhattacharya
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  2. Sandip Das
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  3. Tsunehiko Kameda
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  4. Priya Ranjan Sinha Mahapatra
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  5. Zhao Song
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Corresponding author

Correspondence to Tsunehiko Kameda .

Editor information

Editors and Affiliations

  1. Zhejiang Normal University, Jinhua, Zhejiang, China

    Zhao Zhang

  2. University of Texas, Tyler, Texas, USA

    Lidong Wu

  3. University of Texas, Dallas, Texas, USA

    Wen Xu

  4. University of Texas, Dallas, USA

    Ding-Zhu Du

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Bhattacharya, B., Das, S., Kameda, T., Sinha Mahapatra, P.R., Song, Z. (2014). Optimizing Squares Covering a Set of Points. In: Zhang, Z., Wu, L., Xu, W., Du, DZ. (eds) Combinatorial Optimization and Applications. COCOA 2014. Lecture Notes in Computer Science(), vol 8881. Springer, Cham. https://doi.org/10.1007/978-3-319-12691-3_4

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  • DOI: https://doi.org/10.1007/978-3-319-12691-3_4

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-12690-6

  • Online ISBN: 978-3-319-12691-3

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