A pairing technique for area-efficient orthogonal drawings (extended abstract)

  • Achilleas Papakostas
  • Ioannis G. Tollis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1190)


An orthogonal drawing of a graph is a drawing such that vertices are placed on grid points and edges are drawn as sequences of vertical and horizontal segments. In this paper we present linear time algorithms that produce orthogonal drawings of graphs with n nodes. If the maximum degree is four, then the drawing produced by our first algorithm needs area no greater than 0.76n2, and introduces no more than 2n + 2 bends. Also, every edge of such a drawing has at most two bends. Our algorithm is based on forming and placing pairs of vertices of the graph. If the maximum degree is three, then the drawing produced by our second algorithm needs at most 1/4n2 area, and at most ILn/2 + 2l + 1⌋ bends (⌊n/2⌋ + 3 bends, if the graph is biconnected), where l is the number of biconnected components that are leaves in the block tree. For biconnected graphs, this algorithm produces optimal drawings with respect to the number of bends (within a constant of two), since there is a lower bound of n/2 + 1 in the number of bends for orthogonal drawings of degree 3 graphs.


  1. 1.
    Therese Biedl, “Embedding Nonplanar Graphs in the Rectangular Grid”, Rutcor Research Report 27–93, 1993.Google Scholar
  2. 2.
    Therese Biedl, “New Lower Bounds for Orthogonal Graph Drawings”, Proc. of GD '95, Lecture Notes in Comp, Sci., 1027, Springer-Verlag, 1995, pp. 28–39.Google Scholar
  3. 3.
    T. Biedl and G. Kant, “A Better Heuristic for Orthogonal Graph Drawings”, Technical Report, Utrecht Univ., Dept. of Comp. Sci., UU-CS-1995-04. Prelim. version appeared in Proc. 2nd Ann. European Symposium on Algorithms (ESA '94), Lecture Notes in Computer Science, vol. 855, pp. 24–35, Springer-Verlag, 1994.Google Scholar
  4. 4.
    G. DiBattista, 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, gdbiblio.tex.Z and in /pub/papers/compgeo.MathSciNetGoogle Scholar
  5. 5.
    G. DiBattista, A. Garg, G. Liotta, R. Tamassia, E. Tassinari and F. Vargiu, “An Experimental Comparison of Three Graph Drawing Algorithms”, Proc. of ACM Symp. on Computational Geometry, pp. 306–315, 1995. The version of the paper with the four algorithms can be obtained from http://www.cs.brown/people/rt.Google Scholar
  6. 6.
    G. DiBattista, G. Liotta and F. Vargiu, “Spirality of orthogonal representations and optimal drawings of series-parallel graphs and 3-planar graphs,” Proc. Workshop on Algorithms and Data Structures, Lecture Notes in Computer Science 709, Springer-Verlag, 1993, pp. 151–162.Google Scholar
  7. 7.
    S. Even and G. Granot, “Rectilinear Planar Drawings with Few Bends in Each Edge”, Tech. Report 797, Comp. Science Dept., Technion, Israel Inst. of Tech., 1994.Google Scholar
  8. 8.
    S. Even and R.E. Tarjan, “Computing an st-numbering”, Theor. Comp. Sci. 2 (1976), pp. 339–344.CrossRefGoogle Scholar
  9. 9.
    A. Garg and R. Tamassia, “On the Computational Complexity of Upward and Rectilinear Planarity Testing”, Proc. DIMACS Workshop GD '94, Lecture Notes in Comp. Sci. 894, Springer-Verlag, 1994, pp. 286–297.Google Scholar
  10. 10.
    Goos Kant, “Drawing planar graphs using the lmc-ordering”, Proc. 33th Ann. IEEE Symp. on Found. of Comp. Science, 1992, pp. 101–110.Google Scholar
  11. 11.
    F. T. Leighton, “New lower bound techniques for VLSI”, Proc. 22nd Ann. IEEE Symp. on Found. of Comp. Science, 1981, pp. 1–12.Google Scholar
  12. 12.
    Charles E. Leiserson, “Area-Efficient Graph Layouts (for VLSI)”, Proc. 21st Ann. IEEE Symp. on Found. of Comp. Science, 1980, pp. 270–281.Google Scholar
  13. 13.
    A. Papakostas and I. G. Tollis, “Algorithms for Area-Efficient Orthogonal Drawings”, Tech. Report UTDCS-06-95, The University of Texas at Dallas, 1995. Also available on the WWW at∼tollis.Google Scholar
  14. 14.
    A. Papakostas and I. G. Tollis, “Improved Algorithms and Bounds for Orthogonal Drawings”, Proc. DIMACS Workshop GD '94, Lecture Notes in Comp. Sci. 894, Springer-Verlag, 1994, pp. 40–51.Google Scholar
  15. 15.
    A. Papakostas and I. G. Tollis, “Issues in Interactive Orthogonal Graph Drawing”, Proc. of GD '95, Lecture Notes in Comp. Sci. 1027, Springer-Verlag, 1995, pp. 419–430.Google Scholar
  16. 16.
    Markus Schäffter, “Drawing Graphs on Rectangular Grids”, Discr. Appl. Math. 63 (1995), pp. 75–89.CrossRefGoogle Scholar
  17. 17.
    J. Storer, “On minimal node-cost planar embeddings”, Networks 14 (1984), pp. 181–212.Google Scholar
  18. 18.
    R. Tamassia, “On embedding a graph in the grid with the minimum number of bends”, SIAM J. Comput. 16 (1987), pp. 421–444.CrossRefGoogle Scholar
  19. 19.
    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
  20. 20.
    R. Tamassia, I. Tollis and J. Vitter, “Lower Bounds for Planar Orthogonal Drawings of Graphs”, Information Processing Letters 39 (1991), pp. 35–40.CrossRefGoogle Scholar
  21. 21.
    L. Valiant, “Universality Considerations in VLSI Circuits”, IEEE Trans. on Comp., vol. C-30, no 2, (1981), pp. 135–140.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

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

Personalised recommendations