Drawing clustered graphs on an orthogonal grid

  • Peter Eades
  • Qing-Wen Feng
Clustering and Labelling
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1353)

Abstract

Clustered graphs are graphs with recursive clustering structures over the vertices. For graphical representation, the clustering structure is represented by a simple region that contains the drawing of all the vertices which belong to that cluster. In this paper, we present an algorithm which produces planar drawings of clustered graphs in a convention known as orthogonal grid rectangular cluster drawings. We present an algorithm which produces such drawings with On2 area and with at most 3 bends in each edge. This result is as good as existing results for classical planar graphs. Further, we show that our algorithm is optimal in terms of the number of bends in each edge.

References

  1. 1.
    C. Batini, L. Furlani, and E. Nardelli.What is a good diagram? a pragmatic approach. In Proc. 4th Int. Conf. on the Entity Relationship Approach, 1985.Google Scholar
  2. 2.
    C. Batini, M. Talamo, and R. Tamassia. Computer aided layout of entity-relationship diagrams. Journal of Systems and Software, 4:163–173, 1984.CrossRefGoogle Scholar
  3. 3.
    G. di Battista and R. Tamassia. Algorithms for plane representations of acyclic digraphs. Theoretical Computer Science, 61:175–198, 1988.CrossRefGoogle Scholar
  4. 4.
    G. di Battista, R. Tamassia, and I.G. Tollis. Constrained visibility representations of graphs. Information Processing Letters, 41:1–7, 1992.CrossRefGoogle Scholar
  5. 5.
    T. Biedl. New lower bounds for orthogonal graph drawings. In Franz J. Brandenburg, editor, GD'95, volume 1027 of Lecture Notes in Computer Science, pages 28–39. Springer-Verlag, 1995.Google Scholar
  6. 6.
    T. Biedl and G. Kant. A better heuristic for orthogonal graph drawings. In ESA'94 volume 855 of Lecture Notes in Computer Science, pages 24–35. Springer-Verlag, 1994.Google Scholar
  7. 7.
    P. Bose, A. Dean, J. Hutchinson., and T. Shermer. On rectangle visibility graphs. In Stephen C. North, editor, GD'96, volume 1190 of Lecture Notes in Computer Science, pages 25–44. Springer-Verlag, 1997.Google Scholar
  8. 8.
    P. Eades, Q. Feng, and X. Lin. Straight-line drawing algorithms for hierarchical graphs and clustered graphs. In Stephen C. North, editor, GD'96, volume 1190 of Lecture Notes in Computer Science, pages 113–128. Springer-Verlag, 1997.Google Scholar
  9. 9.
    S. Even and G. Granot. Rectilinear planar drawings with few bends in each edge. Technical Report 797, Computer Science Department, Technion, Israel Institute of Technology, 1994.Google Scholar
  10. 10.
    S. Even and G. Granot. Grid layout of block diagrams — bounding the number of bends in each connection. In R. Tamassia and I. G. Tollis, editors, GD'94, volume 894 of Lecture Notes in Computer Science, pages 64–75. Springer-Verlag, 1995.Google Scholar
  11. 11.
    Q. Feng, R. Cohen, and P. Eades. How to draw a planar clustered graph. In COCOON'95, volume 959 of Lecture Notes in Computer Science, pages 21–31. Springer-Verlag, 1995.Google Scholar
  12. 12.
    Q. Feng, R. Cohen, and P. Eades. Planarity for clustered graphs. In ESA'95, volume 979 of Lecture Notes in Computer Science, pages 213–226. Springer-Verlag, 1995.Google Scholar
  13. 13.
    U. Fößmeier, G. Kant, and M. Kaufmann. 2-visibility drawings of planar graphs. In Stephen C. North, editor, GD'96, volume 1190 of Lecture Notes in Computer Science, pages 155–168. Springer-Verlag, 1997.Google Scholar
  14. 14.
    A. Garg and R. amassia. On the computational complexity of upward and rectilinear planarity testing. In R. Tamassia and I. G. Tollis, editors, GD'94 volume 894 of Lecture Notes in Computer Science, pages 286–297. Springer-Verlag, 1995.Google Scholar
  15. 15.
    D. Harel. On visual formalisms. Communications of the ACM, 31(5):514–530, 1988.CrossRefGoogle Scholar
  16. 16.
    G. Kant. Drawing planar graphs using the lmc-ordering. In Proc. 33th IEEE Symp. on Foundations of Computer Science, pages 101–110, 1992.Google Scholar
  17. 17.
    G. Kant. Drawing planar graphs using the canonical ordering. Algorithmica, 16:432, 1996.Google Scholar
  18. 18.
    G. Kar, B.P. Madden, and R.S. Gilbert. Heuristic layout algorithms for network management presentation services. IEEE Network, pages 29–36, November 1988.Google Scholar
  19. 19.
    J. Kawakita. The KJ method — a scientific approach to problem solving. Technical report, Kawakita Research Institute, Tokyo, 1975.Google Scholar
  20. 20.
    M.R. Kramer and J. van Leeuwen. The complexity of wire-routing and finding minimum area layouts for arbitrary VLSI circuits. In F.P. Preparata, editor, Advances in Computing Research, volume 2, pages 129–146. JAI Press, Greenwich, Conn., 1985.Google Scholar
  21. 21.
    C. E. Leiserson. Area-efficient graph layouts (for VLSI). In Proceedings of the IEEE Symposium on the Foundations of Computer Science, pages 270–281, 1980.Google Scholar
  22. 22.
    K. Misue and K. Sugiyama.An overview of diagram based idea organizer: D-abductor. Technical Report IIAS-RR-93-3E, ISIS, Fujitsu Laboratories, 1993.Google Scholar
  23. 23.
    S. North. Drawing ranked digraphs with recursive clusters. In Proc. ALCOM Workshop on Graph Drawing '93, September 1993.Google Scholar
  24. 24.
    J. Nummenmaa and J. Tuomi. Constructing layouts for er-diagrams from visibility representations. In Proc. 9th Int. Conf. on Entity-Relationship Approach, pages 303–317, 1990.Google Scholar
  25. 25.
    A. Papakostas and I. G. Tollis. Improved algorithms and bounds for orthogonal drawings. In R. Tamassia and I. G. Tollis, editors, GD'94 volume 894 of Lecture Notes in Computer Science, pages 40–51. Springer-Verlag, 1994.Google Scholar
  26. 26.
    A. Papakostas and I. G. Tollis. A pairing technique for area-efficient orthogonal drawings. In Stephen C. North, editor, GD'96, volume 1190 of Lecture Notes in Computer Science, pages 355–370. Springer-Verlag, 1997.Google Scholar
  27. 27.
    D. Reiner, G. Brown, M. Friedell, J. Lehman, R. McKee, P. Rheingans, and A. Rosenthal. A database designer's workbench. In S. Spaccapietra, editor, Entity-Relationship Approach: Proc. 5th Int. Conf. on Entity-Relationship Approach (Dijon France 1987), pages 347–360, New York, N.Y., 1987. North-Holland.Google Scholar
  28. 28.
    P. Rosenstiehl and R.E. Tarjan. Rectilinear planar layouts and bipolar orientations of planar graphs. Discrete and Computational Geometry, 1(4):343–353, 1986.Google Scholar
  29. 29.
    J.A. Storer. On minimal node-cost planar embeddings. Networks, 14:181–212, 1984.Google Scholar
  30. 30.
    K. Sugiyama and K. Misue. Visualization of structural information: Automatic drawing of compound digraphs. IEEE Transactions on Software Engineering, 21(4):876–892, 1991.Google Scholar
  31. 31.
    R. Tamassia. New layout techniques for entity-relationship diagrams. In Proc. 4th Int. Conf. on Entity-Relationship Approach, pages 304–311, 1985.Google Scholar
  32. 32.
    R. Tamassia. On embedding a graph in the grid with the minimum number of bends. SIAM J. Computing, 16(3):421–444, 1987.CrossRefGoogle Scholar
  33. 33.
    R. Tamassia and I.G. Tollis. A unified approach to visibility representations of planar graphs. Discrete and Computational Geometry, 1(4):321–341, 1986.Google Scholar
  34. 34.
    R. Tamassia and I.G. Tollis. Efficient embedding of planar graphs in linear time. In Proc. IEEE Int. Symp. on Circuits and Systems, pages 495–498, 1987.Google Scholar
  35. 35.
    R. Tamassia and I.G. Tollis. Planar grid embedding in linear time. IEEE Trans. on Circuits and Systems, CAS-36(9):1230–1234, 1989.CrossRefGoogle Scholar
  36. 36.
    R. Tamassia, I.G. Tollis, and J.S. Vitter. Lower bounds for planar orthogonal drawings of graphs. Information Processing Letters, 39:35–40, 1991.CrossRefGoogle Scholar
  37. 37.
    J.D. Ullman. Computational Aspects of VLSI. Principles of Computer Science. Computer Science Press, Rockville, Md., 1984.Google Scholar
  38. 38.
    L. Valiant. Universality considerations in VLSI circuits. IEEE Transactions on Computers, C-30(2):135–140, 1981.Google Scholar
  39. 39.
    C. Williams, J. Rasure, and C. Hansen. The state of the art of visual languages for visualization. In Visualization 92, pages 202–209, 1992.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1997

Authors and Affiliations

  • Peter Eades
    • 1
  • Qing-Wen Feng
    • 2
  1. 1.Department of Computer Science and Software EngineeringUniversity of NewcastleAustralia
  2. 2.Tom Sawyer SoftwareBerkeleyUSA

Personalised recommendations