Skip to main content
SpringerLink
Log in

Drawing Unordered Trees on k-Grids

  • Conference paper
WALCOM: Algorithms and Computation (WALCOM 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7157))

Included in the following conference series:

Abstract

We present almost linear area bounds for drawing complete trees on the octagonal grid. For 7-ary trees we establish an upper and lower bound of Θ(n 1.129) and for ternary trees the bounds of \(\O(n^{1.048})\) and Θ(n), where the latter needs edge bends. We explore the unit edge length and area complexity of drawing unordered trees on k-grids with k ∈ {4, 6, 8} and generalize the \(\mathcal{NP}\)-hardness results of the orthogonal and hexagonal grid to the octagonal grid.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 54.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 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)

    Chapter  Google Scholar 

  2. Bhatt, S.N., Cosmadakis, S.S.: The complexity of minimizing wire lengths in VLSI layouts. Inf. Process. Lett. 25(4), 263–267 (1987)

    Article  MATH  Google Scholar 

  3. Bloesch, A.: Aestetic layout of generalized trees. Softw. Pract. Exper. 23(8), 817–827 (1993)

    Article  Google Scholar 

  4. Brandenburg, F.J.: Nice Drawings of Graphs are Computationally Hard. In: Gorny, P., Tauber, M.J. (eds.) Informatics and Psychology 1988. LNCS, vol. 439, pp. 1–15. Springer, Heidelberg (1990)

    Chapter  Google Scholar 

  5. Brunner, W., Matzeder, M.: Drawing Ordered (k − 1)–Ary Trees on k–Grids. In: Brandes, U., Cornelsen, S. (eds.) GD 2010. LNCS, vol. 6502, pp. 105–116. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  6. Buchheim, C., Jünger, M., Leipert, S.: Improving Walker’s Algorithm to Run in Linear Time. In: Goodrich, M.T., Kobourov, S.G. (eds.) GD 2002. LNCS, vol. 2528, pp. 344–353. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  7. 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)

    Article  MathSciNet  MATH  Google Scholar 

  8. 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)

    Article  MathSciNet  MATH  Google Scholar 

  9. Eades, P.: Drawing free trees. Bull. Inst. Comb. Appl. 5, 10–36 (1992)

    MathSciNet  MATH  Google Scholar 

  10. Eades, P., Whitesides, S.: The logic engine and the realization problem for nearest neighbor graphs. Theor. Comput. Sci. 169(1), 23–37 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  12. Garey, M.R., Johnson, D.S.: Computers and Intractability; A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1990)

    MATH  Google Scholar 

  13. Garg, A., Goodrich, M.T., Tamassia, R.: Planar upward tree drawings with optimal area. Int. J. Comput. Geometry Appl. 6(3), 333–356 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  14. 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)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gregori, A.: Unit-length embedding of binary trees on a square grid. Inf. Process. Lett. 31(4), 167–173 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  16. 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 

  17. Kaufmann, M., Wagner, D.: Drawing Graphs. LNCS, vol. 2025. Springer, Heidelberg (2001)

    Book  MATH  Google Scholar 

  18. Kramer, M.R., Leeuwen, J.L.V.: The complexity of wire-routing and finding minimum area layouts for arbitrary VLSI circuits. Adv. Comput. Res. (1984)

    Google Scholar 

  19. Marriott, K., Stuckey, P.J.: NP-completeness of minimal width unordered tree layout. J. Graph Algorithms Appl. 8(2), 295–312 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Reingold, E.M., Tilford, J.S.: Tidier drawing of trees. IEEE Trans. Software Eng. 7(2), 223–228 (1981)

    Article  Google Scholar 

  21. 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)

    Article  MathSciNet  MATH  Google Scholar 

  22. Supowit, K.J., Reingold, E.M.: The complexity of drawing trees nicely. Acta Inf. 18, 377–392 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  23. Valiant, L.G.: Universality considerations in VLSI circuits. IEEE Trans. Computers 30(2), 135–140 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  24. Walker, J.Q.W.: A node-positioning algorithm for general trees. Softw. Pract. Exper. 20(7), 685–705 (1990)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Passau, Germany

    Christian Bachmaier & Marco Matzeder

Authors
  1. Christian Bachmaier
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marco Matzeder
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Graph Drawing and Information Visualization Laboratory, Department of Computer Science and Engineering, Bangladesh University of Engineering and Technology, Bangladesh

    Md. Saidur Rahman

  2. , Faculty of Engineering, Department of Computer Science, Gunma University, 1-5-1 Tenjin-Cho, 376-8515, Kiryu-Shi, Japan

    Shin-ichi Nakano

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Bachmaier, C., Matzeder, M. (2012). Drawing Unordered Trees on k-Grids. In: Rahman, M.S., Nakano, Si. (eds) WALCOM: Algorithms and Computation. WALCOM 2012. Lecture Notes in Computer Science, vol 7157. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28076-4_20

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-28076-4_20

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-28075-7

  • Online ISBN: 978-3-642-28076-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation