Incremental orthogonal graph drawing in three dimensions

  • Achilleas Papakostas
  • Ioannis G. Tollis
Drawings in the Air
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1353)

Abstract

We present two algorithms for orthogonal graph drawing in three dimensional space. For graphs of maximum degree six, the 3-D drawing is produced in linear time, has volume at most 4.66n3 and each edge has at most three bends. If the degree of the graph is arbitrary, the vertices are represented by solid 3-D boxes whose surface is proportional to their degree. The produced drawing has two bends per edge. Both algorithms guarantee no crossings and can be used under an interactive setting (i.e., vertices arrive and enter the drawing on-line), as well.

References

  1. 1.
    T. Biedl and G. Kant, A Better Heuristic for Orthogonal Graph Drawings, Proc. 2nd Ann. European Symposium on Algorithms (ESA '94), Lecture Notes in Computer Science, vol. 855, pp. 24–35, Springer-Verlag, 1994.Google Scholar
  2. 2.
    M. Brown and M. Najork, Algorithm animation using 3D interactive graphics, Proc. ACM Symp. on User Interface Software and Technology, 1993, pp. 93–100.Google Scholar
  3. 3.
    I. Bruss and A. Frick, Fast Interactive 3-D Visualization, Proc. of Workshop GD '95, Lecture Notes in Comp. Sci. 1027, Springer-Verlag, 1995, pp. 99–110.Google Scholar
  4. 4.
    R. Cohen, P. Eades, T. Lin, F. Ruskey, Three Dimensional Graph Drawing, Proc. of DIMACS Workshop GD '94, Lecture Notes in Comp. Sci. 894, Springer-Verlag, 1994, pp. 1–11.Google Scholar
  5. 5.
    I. Cruz and J. Twarog, 3D Graph Drawing with Simulated Annealing, Proc. of Workshop GD '95, Lecture Notes in Comp. Sci. 1027, Springer-Verlag, 1995, pp. 162–165.Google Scholar
  6. 6.
    G. Di Battista, P. Eades, R. Tamassia and I. Tollis, Algorithms for Drawing Graphs: An Annotated Bibliography, Computational Geometry: Theory and Applications, vol. 4, no 5, 1994, pp. 235–282. Also available via anonymous ftp from ftp.cs.brown.edu, gdbiblio.tex.Z and gdbiblio.ps.Z in /pub/papers/compgeo.MathSciNetGoogle Scholar
  7. 7.
    P. Eades, C. Stirk, S. Whitesides, The Techniques of Kolmogorov and Bardzin for Three Dimensional Orthogonal Graph Drawings, TR 95-07, Dept. of Computer Science, University of Newcastle, Australia, 1995. Also to appear in Information Processing Letters.Google Scholar
  8. 8.
    P. Eades, A. Symvonis, S. Whitesides, Two Algorithms for Three Dimensional Orthogonal Graph Drawing, Proc. of Workshop GD '96, Lecture Notes in Comp. Sci. 1190, Springer-Verlag, 1996, pp. 139–154.Google Scholar
  9. 9.
    A. Garg and R. Tamassia, GIOTT03D: A System for Visualizing Hierarchical Structures in 3D, Proc. of Workshop GD '96, Lecture Notes in Comp. Sci. 1190, Springer-Verlag, 1996, pp. 193–200.Google Scholar
  10. 10.
    Goos Kant, Drawing Planar Graphs Using the Canonical Ordering, Algorithmica, vol. 16, no. 1, 1996, pp. 4–32.Google Scholar
  11. 11.
    A. N. Kolmogorov and Y. M. Bardzin, About Realization of Sets in 3-dimensional Space, Problems in Cybernetics, 1967, pp. 261–268.Google Scholar
  12. 12.
    J. MacKinley, G. Robertson, S. Card, Cone Trees: Animated 3d visualizations of hierarchical information, In Proc. of SIGCHI Conf. on Human Factors in Computing, pp. 189–194, 1991.Google Scholar
  13. 13.
    A. Papakostas and I. G. Tollis, Algorithms for Area-Efficient Orthogonal Drawings, Technical Report UTDCS-06-95, The University of Texas at Dallas, 1995.Google Scholar
  14. 14.
    A. Papakostas and I. G. Tollis, Issues in Interactive Orthogonal Graph Drawing, Proc. of Workshop GD '95, Lecture Notes in Comp. Sci, 1027, Springer-Verlag, 1995, pp. 419–430.Google Scholar
  15. 15.
    A. Papakostas and I. G. Tollis, A Pairing Technique for Area-Efficient Orthogonal Drawings, Proc. of Workshop GD '96, Lecture Notes in Comp. Sci. 1190, Springer-Verlag, 1996, pp. 355–370.Google Scholar
  16. 16.
    A. Papakostas, J. Six and I. G. Tollis, Experimental and Theoretical Results in Interactive Graph Drawing, Proc. of Workshop GD '96, Lecture Notes in Comp. Sci. 1190, Springer-Verlag, 1996, pp. 371–386.Google Scholar
  17. 17.
    A. Papakostas and I. G. Tollis, Incremental Orthogonal Graph Drawing in Three Dimensions, Technical Report UTDCS-02-97, The University of Texas at Dallas, 1997. (available through www.utdallas.eduttollis)Google Scholar
  18. 18.
    S. Reiss, An engine for the 3D visualization of program information, J. Visual Languages and Computing, vol. 6, no. 3, 1995.Google Scholar
  19. 19.
    Markus Schäffter, Drawing Graphs on Rectangular Grids, Discr. Appl. Math. 63 (1995) pp. 75–89.CrossRefGoogle Scholar
  20. 20.
    R. Tamassia and I. Tollis, Planar Grid Embeddings in Linear Time, IEEE Trans. on Circuits and Systems CAS-36 (1989), pp. 1230–1234.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1997

Authors and Affiliations

  • Achilleas Papakostas
    • 1
  • Ioannis G. Tollis
    • 1
  1. 1.Dept. of Computer ScienceThe University of Texas at DallasRichardson

Personalised recommendations