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International Workshop on Combinatorial Algorithms

IWOCA 2014: Combinatorial Algorithms pp 286–297Cite as

Minimum r-Star Cover of Class-3 Orthogonal Polygons

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8986))

Abstract

We are interested in the problem of covering simple orthogonal polygons with the minimum number of r-stars; an orthogonal polygon is an r-star if it is star-shaped. The problem has been considered by Worman and Keil [13] who described an algorithm running in \(O(n^{17} \hbox {poly-log}\, n)\) time where n is the size of the input polygon.

In this paper, we consider the above problem on simple class-3 orthogonal polygons, i.e., orthogonal polygons that have dents along at most 3 different orientations. By taking advantage of geometric properties of these polygons, we give an output-sensitive \(O(n + k \log k)\)-time algorithm where k is the size of a minimum r-star cover; this is the first purely geometric algorithm for this problem. Ideas in this algorithm may be generalized to yield faster algorithms for the problem on general simple orthogonal polygons.

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References

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Acknowledgments

This research has been co-financed by the European Union (European Social Fund - ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) - Research Funding Program: THALIS UOA (MIS 375891) - Investing in knowledge society through the European Social Fund.

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Authors and Affiliations

  1. Department of Computer Science and Engineering, University of Ioannina, 45110, Ioannina, Greece

    Leonidas Palios & Petros Tzimas

Authors
  1. Leonidas Palios
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  2. Petros Tzimas
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Corresponding author

Correspondence to Leonidas Palios .

Editor information

Editors and Affiliations

  1. Faculty of Mathematics and Physics, Charles University, Praha, Czech Republic

    Kratochvíl Jan

  2. University of Newcastle, Newcastle, New South Wales, Australia

    Mirka Miller

  3. University of Minnesota, Duluth Department of Mathematics, Duluth, Minnesota, USA

    Dalibor Froncek

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Palios, L., Tzimas, P. (2015). Minimum r-Star Cover of Class-3 Orthogonal Polygons. In: Jan, K., Miller, M., Froncek, D. (eds) Combinatorial Algorithms. IWOCA 2014. Lecture Notes in Computer Science(), vol 8986. Springer, Cham. https://doi.org/10.1007/978-3-319-19315-1_25

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  • DOI: https://doi.org/10.1007/978-3-319-19315-1_25

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-19314-4

  • Online ISBN: 978-3-319-19315-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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