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

  • Muhammad Jawaherul Alam
  • Therese Biedl
  • Stefan Felsner
  • Andreas Gerasch
  • Michael Kaufmann
  • Stephen G. Kobourov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7074)


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 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. Finally, we 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.


Planar Graph Simple Polygon Contact Representation Contact Graph Outer Vertex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alam, M.J., Biedl, T., Felsner, S., Gerasch, A., Kaufmann, M., Kobourov, S.G.: Linear-time algorithms for proportional contact graph representations. Technical Report CS-2011-19, University of Waterloo (2011)Google Scholar
  2. 2.
    Alam, M.J., Biedl, T., Felsner, S., Kaufmann, M., Kobourov, S., Ueckert, T.: Computing cartograms with optimal complexity (2011) (submitted)Google Scholar
  3. 3.
    Biedl, T., Ruiz Velázquez, L.E.: Drawing Planar 3-Trees with Given Face-Areas. In: Eppstein, D., Gansner, E.R. (eds.) GD 2009. LNCS, vol. 5849, pp. 316–322. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  4. 4.
    Biedl, T., Ruiz Velázquez, L.E.: Orthogonal Cartograms with Few Corners Per Face. In: Dehne, F., Iacono, J., Sack, J.-R. (eds.) WADS 2011. LNCS, vol. 6844, pp. 98–109. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  5. 5.
    Buchsbaum, A.L., Gansner, E.R., Procopiuc, C.M., Venkatasubramanian, S.: Rectangular layouts and contact graphs. ACM Transactions on Algorithms 4(1) (2008)Google Scholar
  6. 6.
    de Berg, M., Mumford, E., Speckmann, B.: On rectilinear duals for vertex-weighted plane graphs. Discrete Mathematics 309(7), 1794–1812 (2009)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Gansner, E.R., Hu, Y.F., Kaufmann, M., Kobourov, S.G.: Optimal Polygonal Representation of Planar Graphs. In: López-Ortiz, A. (ed.) LATIN 2010. LNCS, vol. 6034, pp. 417–432. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  8. 8.
    He, X.: On floor-plan of plane graphs. SIAM Journal of Computing 28(6), 2150–2167 (1999)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Heilmann, R., Keim, D.A., Panse, C., Sips, M.: Recmap: Rectangular map approximations. In: 10th IEEE Symp. on Information Visualization (InfoVis 2004), pp. 33–40 (2004)Google Scholar
  10. 10.
    Kawaguchi, A., Nagamochi, H.: Orthogonal Drawings for Plane Graphs with Specified Face Areas. In: Cai, J.-Y., Cooper, S.B., Zhu, H. (eds.) TAMC 2007. LNCS, vol. 4484, pp. 584–594. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  11. 11.
    Koźmiński, K., Kinnen, E.: Rectangular duals of planar graphs. Networks 15, 145–157 (1985)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Liao, C.-C., Lu, H.-I., Yen, H.-C.: Compact floor-planning via orderly spanning trees. Journal of Algorithms 48, 441–451 (2003)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Mondal, D., Nishat, R.I., Rahman, M.S., Alam, M.J.: Minimum-area drawings of plane 3-trees. In: CCCG, pp. 191–194 (2010)Google Scholar
  14. 14.
    Rahman, M.S., Miura, K., Nishizeki, T.: Octagonal drawings of plane graphs with prescribed face areas. Computational Geometry 42(3), 214–230 (2009)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Ringel, G.: Equiareal graphs. In: Bodendiek, R. (ed.) Contemporary Methods in Graph Theory, pp. 503–505. Wissenschaftsverlag (1990)Google Scholar
  16. 16.
    Rinsma, I.: Nonexistence of a certain rectangular floorplan with specified area and adjacency. Environment and Planning B: Planning and Design 14, 163–166 (1987)CrossRefGoogle Scholar
  17. 17.
    Schnyder, W.: Embedding planar graphs on the grid. In: SODA, pp. 138–148 (1990)Google Scholar
  18. 18.
    Sun, Y., Sarrafzadeh, M.: Floorplanning by graph dualization: L-shaped modules. Algorithmica 10(6), 429–456 (1993)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Thomassen, C.: Plane cubic graphs with prescribed face areas. Combinatorics, Probability & Computing 1, 371–381 (1992)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Ungar, P.: On diagrams representing graphs. J. London Math. Soc. 28, 336–342 (1953)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    van Kreveld, M.J., Speckmann, B.: On rectangular cartograms. Computational Geometry 37(3), 175–187 (2007)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Yeap, K.-H., Sarrafzadeh, M.: Floor-planning by graph dualization: 2-concave rectilinear modules. SIAM Journal on Computing 22, 500–526 (1993)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Muhammad Jawaherul Alam
    • 1
  • Therese Biedl
    • 2
  • Stefan Felsner
    • 3
  • Andreas Gerasch
    • 4
  • Michael Kaufmann
    • 4
  • Stephen G. Kobourov
    • 1
  1. 1.Department of Computer ScienceUniversity of ArizonaTucsonUSA
  2. 2.David R. Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada
  3. 3.Institut für MathematikTechnische Universität BerlinBerlinGermany
  4. 4.Wilhelm-Schickhard-Institut für InformatikTübingen UniversitätTübingenGermany

Personalised recommendations