Drawing with Fat Edges

  • Christian A. Duncan
  • Alon Efrat
  • Stephen G. Kobourov
  • Carola Wenk
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2265)


In this paper, we introduce the problem of drawing with “fat” edges. Traditionally, graph drawing algorithms represent vertices as circles and edges as closed curves connecting the vertices. In this paper we consider the problem of drawing graphs with edges of variable thickness. The thickness of an edge is often used as a visualization cue, to indicate importance, or to convey some additional information. We present a model for drawing with fat edges and a corresponding polynomial time algorithm that uses the model. We focus on a restricted class of graphs that occur in VLSI wire routing and show how to extend the algorithm to general planar graphs. We show how to take an arbitrary wire routing and convert it into a homotopic equivalent routing such that the distance between any two wires is maximized. Moreover, the routing uses the minimum length wires. Maximizing the distance between wires is equivalent to finding the drawing in which the edges are drawn as thick as possible. To the best of our knowledge this is the first algorithm that finds the maximal distance between any two wires and allows for wires of variable thickness. The previous best known result for the corresponding decision problem with unit wire thickness is the algorithm of Gao et al., which runs in O(kn 2 log(kn)) time and uses O(kn 2) space, where n is the number of wires and k is the maximum of the input and output complexities. The running time of our algorithm is O(kn + n 3) and the space required is O(k+n). The algorithm generalizes naturally to general planar graphs as well.


  1. 1.
    B. Chazelle. A theorem on polygon cutting with applications. In 23th Annual Symposium on Foundations of Computer Science, pages 339–349, Los Alamitos, Ca., USA, Nov. 1982. IEEE Computer Society Press.Google Scholar
  2. 2.
    R. Cole and A. Siegel. River routing every which way, but loose. In 25th Annual Symposium on Foundations of Computer Science, pages 65–73, Los Angeles, Ca., USA, Oct. 1984. IEEE Computer Society Press.Google Scholar
  3. 3.
    D. Dolev, K. Karplus, A. Siegel, A. Strong, and J. D. Ullman. Optimal wiring between rectangles. In Conference Proceedings of the Thirteenth Annual ACM Symposium on Theory of Computation, pages 312–317, Milwaukee, Wisconsin, 11–13 May 1981.Google Scholar
  4. 4.
    S. Gao, M. Jerrum, M. Kaufmann, K. Mehlhorn, W. Rülling, and C. Storb. On continuous homotopic one layer routing. In Proceedings of the Fourth Annual Symposium on Computational Geometry (Urbana-Champaign, IL, June 6-8, 1988), pages 392–402, New York, 1988. ACM, ACM Press.Google Scholar
  5. 7.
    M. Kaufmann and K. Mehlhorn. Routing through a generalized switchbox. Journal of Algorithms, 7(4):510–531, Dec. 1986.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 8.
    D. T. Lee and F. P. Preparata. Euclidean Shortest Paths in the Presence of Rectilinear Barriers. Networks, 14(3):393–410, 1984.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 9.
    C. E. Leiserson and F. M. Maley. Algorithms for routing and testing routability of planar VLSI layouts. In Proceedings of the Seventeenth Annual ACM Symposium on Theory of Computing, pages 69–78, Providence, Rhode Island, 6–8 May 1985.Google Scholar
  8. 10.
    C. E. Leiserson and R. Y. Pinter. Optimal placement for river routing. SIAM Journal on Computing, 12(3):447–462, Aug. 1983.zbMATHCrossRefGoogle Scholar
  9. 11.
    F. M. Maley. Single-Layer Wire Routing. PhD thesis, Massachusetts Institute of Technology, 1987.Google Scholar
  10. 12.
    A. Mirzaian. River routing in VLSI. Journal of Computer and System Sciences, 34(1):43–54, Feb. 1987.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 13.
    J. Pach and R. Wenger. Embedding planar graphs at fixed vertex locations. In Proc. 6th Int. Symp. Graph Drawing (GD’ 98), pages 263–274, 1998.Google Scholar
  12. 14.
    R. Pinter. River-routing: Methodology and analysis, 1983.Google Scholar
  13. 15.
    D. Richards. Complexity of single layer routing. IEEE Transactions on Computers, 33:286–288, 1984.zbMATHCrossRefGoogle Scholar
  14. 17.
    A. Schrijver. Edge-disjoint homotopic paths in straight-line planar graphs. SIAM Journal on Discrete Mathematics, 4(1):130–138, Feb. 1991.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Christian A. Duncan
    • 1
  • Alon Efrat
    • 2
  • Stephen G. Kobourov
    • 2
  • Carola Wenk
    • 3
  1. 1.Department of Computer ScienceUniversity of MiamiCoral Gables
  2. 2.Department of Computer ScienceUniversity of ArizonaTucson
  3. 3.Institut für InformatikFreie Universität BerlinBerlinGermany

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