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Drawing Unordered Trees on k-Grids

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

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.

Keywords

  • Binary Tree
  • Bottom Half
  • Outgoing Edge
  • Full Tree
  • Linear Time Algorithm

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.

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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)

    CrossRef  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)

    CrossRef  MATH  Google Scholar 

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

    CrossRef  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)

    CrossRef  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)

    CrossRef  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)

    CrossRef  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)

    CrossRef  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)

    CrossRef  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)

    CrossRef  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)

    CrossRef  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)

    CrossRef  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)

    CrossRef  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)

    CrossRef  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)

    CrossRef  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)

    CrossRef  MathSciNet  MATH  Google Scholar 

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

    CrossRef  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)

    CrossRef  MathSciNet  MATH  Google Scholar 

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

    CrossRef  MathSciNet  MATH  Google Scholar 

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

    CrossRef  MathSciNet  MATH  Google Scholar 

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

    CrossRef  Google Scholar 

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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

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  • 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

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