Algorithmica

, Volume 11, Issue 4, pp 379–403 | Cite as

Edge crossings in drawings of bipartite graphs

  • Peter Eades
  • Nicholas C. Wormald
Article

Abstract

Systems engineers have recently shown interest in algorithms for drawing directed graphs so that they are easy to understand and remember. Each of the commonly used methods has a step which aims to adjust the drawing to decrease the number of arc crossings. We show that the most popular strategy involves an NP-complete problem regarding the minimization of the number of arcs in crossings in a bipartite graph. The performance of the commonly employed “barycenter” heuristic for this problem is analyzed. An alternative method, the “median” heuristic, is proposed and analyzed. The new method is shown to compare favorably with the old in terms of performance guarantees. As a bonus, we show that the median heuristic performs well with regard to the total length of the arcs in the drawing.

Key words

Graph Bipartite graph Directed graph Edge crossing Median 

References

  1. [1]
    A. V. Aho, J. E. Hopcroft, and J. D. Ullman,The Design and Analysis of Computer Algorithms, Addison-Wesley, Reading, MA, 1974.Google Scholar
  2. [2]
    G. Di Battista and E. Nardelli,An Algorithm for Testing Planarity of Hierarchical Graphs, Lecture Notes in Computer Science, Vol. 246, Springer-Verlag, Berlin, 1987, pp. 277–289.Google Scholar
  3. [3]
    B. Berger and P. Shor, Approximation Algorithms for the Maximum Acyclic Subgraph Problem,Proceedings of the First Annual ACM-SIAM Symposium on Discrete Algorithms, San Francisco, 1990, pp. 236–243.Google Scholar
  4. [4]
    J. A. Bondy and U. S. R. Murty,Graph Theory with Applications, Macmillan, New York, 1976.Google Scholar
  5. [5]
    P. Eades and D. Kelly, Heuristics for Reducing Crossings in 2-Layered Networks,Ars Combinatoria 21A (1986), 89–98.Google Scholar
  6. [6]
    P. Eades and X. Lin, Notes on the Layer Assignment Problem for Drawing Directed Graphs,Australian Computer Science Communications 13 (1) (1991), 26-1–26-10.Google Scholar
  7. [7]
    P. Eades, B. D. McKay, and N. Wormald, On an Edge Crossing Problem,Proceedings of the Ninth Australian Computer Science Conference, Australian National University, 1986, pp. 327–334.Google Scholar
  8. [8]
    P. Eades, W. Smyth, and X. Lin, Heuristics for the Feedback Arc Set Problem, Technical Report 1-1989, School of Computing Science, Curtin University of Technology, 1989.Google Scholar
  9. [9]
    P. Eades and K. Sugiyama, How to Draw a Directed Graph,Journal of Information Processing 13(4) (1990), 424–437.Google Scholar
  10. [10]
    P. Eades, R. Tamassia, G. Di Battista, and I. Tollis, Algorithms for Drawing Graphs: an Annotated Bibliography, available as /pub/gdbiblio.tex.z from wilma.cs.brown.edu (to appear inComputational Geometry: Theory and Applications).Google Scholar
  11. [11]
    P. Eades and N. Wormald, The Median Heuristic for Drawing 2-Layered Networks, Technical Report 69, Department of Computer Science, University of Queensland, 1986.Google Scholar
  12. [12]
    M. R. Garey and D. S. Johnson,Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman, San Francisco, CA, 1979.Google Scholar
  13. [13]
    M. R. Garey and D. S. Johnson, Crossing Number is NP-Complete,SIAM Journal on Algebraic and Discrete Methods 4(3) (1983), 312–316.Google Scholar
  14. [14]
    E. R. Gasner, S. C. North, and K. P. Vo, DAG-A Program that Draws Directed Graphs,Software Practice and Experience 18(11) (1988), 1047–1062.Google Scholar
  15. [15]
    D. Kelly, A View to Graph Layout Problems, Masters Thesis, Department of Computer Science, University of Queensland, 1987.Google Scholar
  16. [16]
    D. Kelly, Fundamentals of Planar Ordered Sets,Discrete Mathematics 63 (1987), 197–216.Google Scholar
  17. [17]
    E. Makinen, Experiments of Drawing 2-Level Hierarchical Graphs, Technical Report A-1988-1. Department of Computer Science, University of Tampere, Finland, 1988.Google Scholar
  18. [18]
    D. J. Rosencrantz, R. E. Stearns, and P. M. Lewis, An Analysis of Several Heuristics for the Traveling Salesman Problem,SIAM Journal of Computing 6 (1977), 563–581.Google Scholar
  19. [19]
    L. A. Rowe, M. Davis, E. Messinger, C. Meyer, C. Spirakis, and A. Tuan, A Browser for Directed Graphs,Software Practice and Experience 17(1) (1987), 61–76.Google Scholar
  20. [20]
    K. Sugiyama, A Readability Requirement in Drawing Digraphs: Level Assignment and Edge Removal for Reducing the Total Length of Lines, Research Report 45, International Institute for Advanced Study of Social Information Science, Numazu, Japan, 1984.Google Scholar
  21. [21]
    K. Sugiyama, Drawing and Understanding Systems Structures: An Introduction to the SKETCH System, Working Paper WP-82-97, International Institute for Systems Analysis, Laxenburg, Austria, p. 52, 1982.Google Scholar
  22. [22]
    K. Sugiyama, S. Tagawa, and M. Toda, Methods for Visual Understanding of Hierarchical System Structures,IEEE Transactions on Systems, Man and Cybernetics 11(2) (1981), 109–125.Google Scholar
  23. [23]
    R. Tamassia, C. Batini, and G. Di Battista, Automatic Graph Drawing and Readability of Diagrams,IEEE Transactions on Systems, Man and Cybernetics 18 (1988), 61–79.Google Scholar
  24. [24]
    R. Tamassia, Drawing Algorithms for Planar st-Graphs,Australian Journal of Combinatorics 2 (1990), 217–236.Google Scholar
  25. [25]
    H. Trickey, DRAG: A Graph Drawing System, inDocument Manipulation and Typography (Proceedings of the International Conference on Electronic Publishing, Nice, 1988) (ed. J. C. van Vliet), Cambridge University Press, Cambridge, 1988, pp. 171–182.Google Scholar
  26. [26]
    J. N. Warfield, Crossing Theory and Hierarchy Mapping,IEEE Transactions on Systems, Man and Cybernetics 7 (1977), 502–523.Google Scholar

Copyright information

© Springer-Verlag New York Inc 1994

Authors and Affiliations

  • Peter Eades
    • 1
  • Nicholas C. Wormald
    • 2
  1. 1.Department of Computer ScienceUniversity of QueenslandQueenslandAustralia
  2. 2.Department of MathematicsUniversity of MelbourneParkvilleAustralia

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