Two algorithms for three dimensional orthogonal graph drawing

  • Peter Eades
  • Antonios Symvonis
  • Sue Whitesides
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1190)


We use basic results from graph theory to design two algorithms for constructing 3-dimensional, intersection-free orthogonal grid drawings of n vertex graphs of maximum degree 6. Our first algorithm gives drawings bounded by an O(√n× O(√n) × O(√n) box; each edge route contains at most 7 bends. The best previous result generated edge routes containing up to 16 bends per route. Our second algorithm gives drawings having at most 3 bends per edge route. The drawings lie in an O(n)×O(n) × O(n) bounding box. Together, the two algorithms initiate the study of bend/bounding box trade-off issues for 3-dimensional grid drawings.


  1. 1.
    C. Berge, Graphs and Hypergraphs, North Holland, Amsterdam, 1973.Google Scholar
  2. 2.
    S. Bhatt, S. Cosmadakis, “The Complexity of Minimizing Wire Lengths in VLSI Layouts”, Information Processing Letters, Vol. 25, 1987, pp. 263–267.CrossRefGoogle Scholar
  3. 3.
    T. Biedl, Embedding Nonplanar Graphs in the Rectangular Grid, Rutcor Research Report 27-93, 1993.Google Scholar
  4. 4.
    T. Biedl and G. Kant, “A Better Heuristic for Orthogonal Graph Drawings”, Proc. 2 nd European Symposium on Algorithms (ESA '94), Lecture Notes in Computer Science, Vol. 855, Springer Verlag, 1994, pp. 24–35.Google Scholar
  5. 5.
    T. Biedl, “New Lower Bounds for Orthogonal Graph Drawings”, Graph Drawing, Lecture Notes in Computer Science, Vol. 1027, Springer Verlag, 1995, pp. 28–39.Google Scholar
  6. 6.
    J. A. Bondy, U. S. R. Murty, Graph Theory with Applications, North Holland, Amsterdam, 1976.Google Scholar
  7. 7.
    G. DiBattista, P. Eades, R. Tamassia, I. Tollis, “Algorithms for Drawing Graphs: An Annotated Bibliography”, Computational Geometry: Theory and Applications, Vol. 4, 1994, pp. 235–282.MathSciNetGoogle Scholar
  8. 8.
    D. Dolev, F. T. Leighton, H. Trickey, “Planar Embeddings of Planar Graphs”, Advances in Computing Research 2 (ed. F. P. Preparata), JAI Press Inc., Greenwich CT, USA, 1984, pp. 147–161.Google Scholar
  9. 9.
    P. Eades, C. Stirk, S. Whitesides, The Techniques of Komolgorov and Bardzin for Three Dimensional Orthogonal Graph Drawings, TR 95-07, Dept. of Computer Science, University of Newcastle NSW, Australia, October 1995.Google Scholar
  10. 10.
    Shimon Even, Graph Algorithms, Computer Science Press, 1979.Google Scholar
  11. 11.
    S. Even and G. Granot, “Rectilinear Planar Drawings with Few Bends in Each Edge”, Technical Report 797, Computer Science Department, Technion, 1994.Google Scholar
  12. 12.
    A. Garg, R. Tamassia, On the Computational Complexity of Upward and Rectilinear Planarity Testing, TR CS-94-10, Dept. of Computer Science, Brown University, 1994.Google Scholar
  13. 13.
    P. Hall, “On Representation of Subsets”, J. London Mathematical Society, Vol. 10, 1935, pp. 26–30.Google Scholar
  14. 14.
    J. E. Hopcroft, R. M. Karp, “An n 5/2 Algorithm for Maximum Matchings in Bipartite Graphs”, SIAM J. Comput., Vol. 2 (4), 1973, pp. 225–231.CrossRefGoogle Scholar
  15. 15.
    Goos Kant, “Drawing Planar Graphs Using the lmc-Ordering”, Proc. 33rd IEEE Symp. on Foundations of Computer Science, 1992, pp. 101–110.Google Scholar
  16. 16.
    A. N. Komolgorov, Ya. M. Bardzin, “About Realisation of Sets in in 3-Dimensional Space”, Problems in Cybernetics, March 1967, pp. 261–268.Google Scholar
  17. 17.
    A. Papakostas and I. Tollis, “A Pairing Technique for Area-Efficient Orthogonal Drawings”, these proceedings.Google Scholar
  18. 18.
    F. P. Preparata, “Optimal Three-Dimensional VLSI Layouts”, Mathematical Systems Theory, Vol. 16, 1983, pp.1–8.CrossRefGoogle Scholar
  19. 19.
    A. L. Rosenberg, “Three-Dimensional Integrated Circuitry”, Advanced Research in VLSI (eds. Kung, Sproule, Steele), 1981, pp. 69–80.Google Scholar
  20. 20.
    A. L. Rosenberg, “Three-Dimensional VLSI: A Case Study”, Journal of the ACM, Vol. 30 (3), 1983, pp. 397–416.CrossRefGoogle Scholar
  21. 21.
    Markus Schäffter, “Drawing Graphs on Rectangular Grids”, Discrete Applied Math. Vol. 63, 1995, pp. 75–89.CrossRefGoogle Scholar
  22. 22.
    R. Tamassia, “On Embedding a Graph in the Grid with a Minimum Number of Bends”, SIAM J. Comput., Vol. 16 (3), 1987, pp. 421–443.CrossRefGoogle Scholar
  23. 23.
    R. Tamassia and I. Tollis, “A Unified Approach to Visibility Representations of Planar Graphs”, Discrete and Computational Geometry, Vol. 1, 1986, pp. 321–341.Google Scholar
  24. 24.
    R. Tamassia and I. Tollis, “Efficient Embeddings of Planar Graphs in Linear Time”, IEEE Symposium on Circuits and Systems, 1987, pp. 495–498.Google Scholar
  25. 25.
    R. Tamassia and I. Tollis, “Planar Grid Embedding in Linear Time”, IEEE Transactions on Circuits and Systems, Vol. 36 (9), 1989, pp. 1230–1234.CrossRefGoogle Scholar
  26. 26.
    David Wood, “On Higher-Dimensional Orthogonal Graph Drawing”, manuscript, 1996, Dept. of Computer Science, Monash U.Google Scholar

Copyright information

© Springer-Verlag 1997

Authors and Affiliations

  • Peter Eades
    • 1
  • Antonios Symvonis
    • 2
  • Sue Whitesides
    • 3
  1. 1.Dept. of Computer ScienceUniversity of NewcastleAustralia
  2. 2.Basser Dept. of Computer ScienceUniversity of SydneyAustralia
  3. 3.School of Computer ScienceMcGill UniversityMontrealCanada

Personalised recommendations