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Linear-Time Algorithms for Hole-free Rectilinear Proportional Contact Graph Representations

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Abstract

In a proportional contact representation of a planar graph, each vertex is represented by a simple polygon with area proportional to a given weight, and edges are represented by adjacencies between the corresponding pairs of polygons. In this paper we first study proportional contact representations that use rectilinear polygons without wasted areas (white space). In this setting, the best known algorithm for proportional contact representation of a maximal planar graph uses 12-sided rectilinear polygons and takes O(nlogn) time. We describe a new algorithm that guarantees 10-sided rectilinear polygons and runs in O(n) time. We also describe a linear-time algorithm for proportional contact representation of planar 3-trees with 8-sided rectilinear polygons and show that this is optimal, as there exist planar 3-trees that require 8-sided polygons. We then show that a maximal outer-planar graph admits a proportional contact representation using rectilinear polygons with 6 sides when the outer-boundary is a rectangle and with 4 sides otherwise. Finally we study maximal series-parallel graphs. Here we show that O(1)-sided rectilinear polygons are not possible unless we allow holes, but 6-sided polygons can be achieved with arbitrarily small holes.

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Notes

  1. Any distribution of ε area among the empty rectangular regions is feasible, as long as they are all non-zero; we just give equal area ε′ to all these empty regions.

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Authors and Affiliations

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

    M. Jawaherul Alam & Stephen G. Kobourov

  2. David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, Canada

    Therese Biedl

  3. Institut für Mathematik, Technische Universität Berlin, Berlin, Germany

    Stefan Felsner

  4. Wilhelm-Schickhard-Institut für Informatik, Tübingen Universität, Tübingen, Germany

    Andreas Gerasch & Michael Kaufmann

Authors
  1. M. Jawaherul Alam
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  2. Therese Biedl
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  3. Stefan Felsner
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  4. Andreas Gerasch
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  5. Michael Kaufmann
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  6. Stephen G. Kobourov
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Corresponding author

Correspondence to M. Jawaherul Alam.

Additional information

This research was initiated at the Dagstuhl Seminar 10461 on Schematization. M.J. Alam’s and S.G. Kobourov’s research funded in part by NSF grants CCF-0545743 and CCF-1115971. T. Biedl’s research supported by NSERC. S. Felsner’s research partially supported by EUROGIGA project GraDR and DFG Fe 340/7-2.

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Alam, M.J., Biedl, T., Felsner, S. et al. Linear-Time Algorithms for Hole-free Rectilinear Proportional Contact Graph Representations. Algorithmica 67, 3–22 (2013). https://doi.org/10.1007/s00453-013-9764-5

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  • DOI: https://doi.org/10.1007/s00453-013-9764-5

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