Drawing Ordered (k − 1)–Ary Trees on k–Grids

  • Wolfgang Brunner
  • Marco Matzeder
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6502)


We explore the complexity of drawing ordered (k − 1)–ary trees on grids with k directions for \(k \in \left\{4,6,8\right\}\) and within a given area. This includes, e.g., ternary trees drawn on the orthogonal grid. For aesthetically pleasing tree drawings on these grids, we additionally present various restrictions similar to the common hierarchical case. First, we generalize the \({\mathcal{NP}}\)–hardness of minimal width in hierarchical drawings of ordered trees to (k − 1)–ary trees on k–grids and then we generalize the Reingold and Tilford algorithm to k–grids.


Binary Tree Outgoing Edge Minimal Width Boolean Expression Incoming Edge 
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  1. 1.
    Akkerman, T., Buchheim, C., Jünger, M., Teske, D.: On the complexity of drawing trees nicely: Corrigendum. Acta Inf. 40(8), 603–607 (2004)CrossRefMATHGoogle Scholar
  2. 2.
    Aziza, S., Biedl, T.C.: Hexagonal grid drawings: Algorithms and lower bounds. In: Graph Drawing, pp. 18–24 (2004)Google Scholar
  3. 3.
    Bachmaier, C., Brandenburg, F.J., Brunner, W., Hofmeier, A., Matzeder, M., Unfried, T.: Tree drawings on the hexagonal grid. In: Tollis, I.G., Patrignani, M. (eds.) GD 2008. LNCS, vol. 5417, pp. 372–383. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  4. 4.
    Bachmaier, C., Brandenburg, F.J., Forster, M., Holleis, P., Raitner, M.: Gravisto: Graph visualization toolkit. In: Graph Drawing, pp. 502–503 (2004)Google Scholar
  5. 5.
    Bhatt, S.N., Cosmadakis, S.S.: The complexity of minimizing wire lengths in VLSI layouts. Inf. Process. Lett. 25(4), 263–267 (1987)CrossRefMATHGoogle Scholar
  6. 6.
    Chan, T.M., Goodrich, M.T., Kosaraju, S.R., Tamassia, R.: Optimizing area and aspect ratio in straight-line orthogonal tree drawings. Comput. Geom. Theory Appl. 23(2), 153–162 (2002)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Crescenzi, P., Di Battista, G., Piperno, A.: A note on optimal area algorithms for upward drawings of binary trees. Comput. Geom. Theory Appl. 2, 187–200 (1992)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Eades, P.: Drawing free trees. Bulletin of the Institute of Combinatorics and its Applications 5, 10–36 (1992)MathSciNetMATHGoogle Scholar
  9. 9.
    Eades, P., Lin, T., Lin, X.: Minimum size h-v drawings. In: Advanced Visual Interfaces, pp. 386–394 (1992)Google Scholar
  10. 10.
    Eades, P., Whitesides, S.: The logic engine and the realization problem for nearest neighbor graphs. Theor. Comput. Sci. 169(1), 23–37 (1996)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Frati, F.: Straight-line orthogonal drawings of binary and ternary trees. In: Hong, S.-H., Nishizeki, T., Quan, W. (eds.) GD 2007. LNCS, vol. 4875, pp. 76–87. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  12. 12.
    Garg, A., Rusu, A.: Straight-line drawings of binary trees with linear area and arbitrary aspect ratio. J. Graph Algo. App. 8(2), 135–160 (2004)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kant, G.: Hexagonal grid drawings. In: Mulkers, A. (ed.) Live Data Structures in Logic Programs. LNCS, vol. 675, pp. 263–276. Springer, Heidelberg (1993)Google Scholar
  14. 14.
    Kaufmann, M., Wagner, D. (eds.): Drawing Graphs. LNCS, vol. 2025. Springer, Heidelberg (2001)MATHGoogle Scholar
  15. 15.
    Marriott, K., Stuckey, P.J.: NP-completeness of minimal width unordered tree layout. J. Graph Algorithms Appl. 8(2), 295–312 (2004)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Reingold, E.M., Tilford, J.S.: Tidier drawing of trees. IEEE Trans. Software Eng. 7(2), 223–228 (1981)CrossRefGoogle Scholar
  17. 17.
    Shin, C.S., Kim, S.K., Chwa, K.Y.: Area-efficient algorithms for straight-line tree drawings. Comput. Geom. Theory Appl. 15(4), 175–202 (2000)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Supowit, K.J., Reingold, E.M.: The complexity of drawing trees nicely. Acta Inf. 18, 377–392 (1982)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Tamassia, R.: On embedding a graph in the grid with the minimum number of bends. SIAM J. Comput. 16(3), 421–444 (1987)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Walker, J.Q.W.: A node-positioning algorithm for general trees. Softw. Pract. Exper. 20(7), 685–705 (1990)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Wolfgang Brunner
    • 1
  • Marco Matzeder
    • 1
  1. 1.University of PassauPassauGermany

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