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Orthogonal Layout with Optimal Face Complexity

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Book cover SOFSEM 2016: Theory and Practice of Computer Science (SOFSEM 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9587))

Abstract

We study a problem motivated by rectilinear schematization of geographic maps. Given a biconnected plane graph G and an integer \(k\ge 0\), does G have a strict-orthogonal drawing with at most k reflex angles per face? For \(k=0\) the problem is equivalent to realizing each face as a rectangle. The problem can be reduced to a max-flow problem in some linear-size nonplanar network, but the best solutions require \(\varOmega (n^{1.5} \log n\log k)\) time. We describe a graph matching approach that can decide strict-orthogonal drawability for arbitrary reflex complexity k in \(O((nk)^{1.5})\) time, which is faster for constant values of k. In contrast, if the embedding is not fixed, we prove that it is NP-complete to decide whether a planar graph admits a strict-orthogonal drawing with reflex face complexity 4.

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Acknowledgement

We thank the anonymous reviewers from our previous submission for pointing out how the network-flow formulations from earlier work can be modified to compute orthogonal drawings with bounded reflex face complexities, and for the suggestions on improving the NP-hardness result.

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Authors and Affiliations

  1. Department of Computer Science, University of Arizona, Tucson, USA

    M. Jawaherul Alam & Stephen G. Kobourov

  2. Department of Computer Science, University of Manitoba, Winnipeg, Canada

    Debajyoti Mondal

Authors
  1. M. Jawaherul Alam
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  2. Stephen G. Kobourov
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  3. Debajyoti Mondal
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Correspondence to Debajyoti Mondal .

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Editors and Affiliations

  1. University of Latvia, Riga, Latvia

    Rūsiņš Mārtiņš Freivalds

  2. University of Paderborn, Paderborn, Germany

    Gregor Engels

  3. University of Genoa, Genoa, Italy

    Barbara Catania

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Jawaherul Alam, M., Kobourov, S.G., Mondal, D. (2016). Orthogonal Layout with Optimal Face Complexity. In: Freivalds, R., Engels, G., Catania, B. (eds) SOFSEM 2016: Theory and Practice of Computer Science. SOFSEM 2016. Lecture Notes in Computer Science(), vol 9587. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-49192-8_10

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  • DOI: https://doi.org/10.1007/978-3-662-49192-8_10

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  • Print ISBN: 978-3-662-49191-1

  • Online ISBN: 978-3-662-49192-8

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