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The Mono- and Bichromatic Empty Rectangle and Square Problems in All Dimensions

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6034))

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Abstract

The maximum empty rectangle problem is as follows: Given a set of red points in ℝd and an axis-aligned hyperrectangle B, find an axis-aligned hyperrectangle R of greatest volume that is contained in B and contains no red points. In addition to this problem, we also consider three natural variants: where we find a hypercube instead of a hyperrectangle, where we try to contain as many blue points as possible instead of maximising volume, and where we do both. Combining the results of this paper with previous results, we now know that all four of these problems (a) are NP-complete if d is part of the input, (b) have polynomial-time sweep-plane solutions for any fixed d ≥ 3, and (c) have near linear time solutions in two dimensions.

This research is supported by the Natural Sciences and Engineering Research Council (NSERC).

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Authors and Affiliations

  1. Department of Computer Science, University of Saskatchewan, Canada

    Jonathan Backer & J. Mark Keil

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  1. Jonathan Backer
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  2. J. Mark Keil
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Editors and Affiliations

  1. Faculty of Mathematics, Department of Computer Science, University of Waterloo, N2L 3G1, Waterloo, Ontario, Canada

    Alejandro López-Ortiz

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Backer, J., Keil, J.M. (2010). The Mono- and Bichromatic Empty Rectangle and Square Problems in All Dimensions. In: López-Ortiz, A. (eds) LATIN 2010: Theoretical Informatics. LATIN 2010. Lecture Notes in Computer Science, vol 6034. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12200-2_3

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  • DOI: https://doi.org/10.1007/978-3-642-12200-2_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-12199-9

  • Online ISBN: 978-3-642-12200-2

  • eBook Packages: Computer ScienceComputer Science (R0)

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