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.
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References
A. V. Aho, J. E. Hopcroft, and J. D. Ullman,The Design and Analysis of Computer Algorithms, Addison-Wesley, Reading, MA, 1974.
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.
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.
J. A. Bondy and U. S. R. Murty,Graph Theory with Applications, Macmillan, New York, 1976.
P. Eades and D. Kelly, Heuristics for Reducing Crossings in 2-Layered Networks,Ars Combinatoria 21A (1986), 89–98.
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.
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.
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.
P. Eades and K. Sugiyama, How to Draw a Directed Graph,Journal of Information Processing 13(4) (1990), 424–437.
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).
P. Eades and N. Wormald, The Median Heuristic for Drawing 2-Layered Networks, Technical Report 69, Department of Computer Science, University of Queensland, 1986.
M. R. Garey and D. S. Johnson,Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman, San Francisco, CA, 1979.
M. R. Garey and D. S. Johnson, Crossing Number is NP-Complete,SIAM Journal on Algebraic and Discrete Methods 4(3) (1983), 312–316.
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.
D. Kelly, A View to Graph Layout Problems, Masters Thesis, Department of Computer Science, University of Queensland, 1987.
D. Kelly, Fundamentals of Planar Ordered Sets,Discrete Mathematics 63 (1987), 197–216.
E. Makinen, Experiments of Drawing 2-Level Hierarchical Graphs, Technical Report A-1988-1. Department of Computer Science, University of Tampere, Finland, 1988.
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.
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.
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.
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.
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.
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.
R. Tamassia, Drawing Algorithms for Planar st-Graphs,Australian Journal of Combinatorics 2 (1990), 217–236.
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.
J. N. Warfield, Crossing Theory and Hierarchy Mapping,IEEE Transactions on Systems, Man and Cybernetics 7 (1977), 502–523.
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Communicated by F. Thomson Leighton.
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Eades, P., Wormald, N.C. Edge crossings in drawings of bipartite graphs. Algorithmica 11, 379–403 (1994). https://doi.org/10.1007/BF01187020
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DOI: https://doi.org/10.1007/BF01187020