Wiring knock-knee layouts: A global approach

  • Majid Sarrafzadeh
  • Dorothea Wagner
  • Frank Wagner
  • Karsten Weihe
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 650)


We present a global approach to solve the three-layer wirability problem for knock-knee layouts. In general, the problem is \(\mathcal{N}\mathcal{P}\)-complete. Only for very restricted classes of layouts polynomial three-layer wiring algorithms are known up to now. In this paper we show that for a large class of layouts a three-layer wiring can be constructed by solving a path problem in a special class of graphs or a two-satisfiability problem, and thus may be wired efficiently. Moreover, it is shown that a minimum stretching of the layout into a layout belonging to this class can be found by solving a clique cover problem in an interval graph. This problem is polynomially solvable as well. Altogether, the method also yields a good heuristic for the three-layer wirability problem for knock-knee layouts.


Interval Graph Disjoint Path Vertical Edge Path Graph Layout Area 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    M. L. Brady, D. J. Brown. VLSI routing: Four layers suffice. In: Advances in Computing Research 2 (VLSI Theory) (ed. F. P. Preparata) JAI Press (1984) 245–257Google Scholar
  2. 2.
    M. L. Brady, M. Sarrafzadeh. Stretching a knock-knee layout for multilayer wiring. IEEE Trans. Comput. 39 (1990) 148–152CrossRefGoogle Scholar
  3. 3.
    S. Even, A. Itai, A. Shamir. On the complexity of time table and multicommodity flow problems. SIAM J. Comput. 5 (1976) 691–703CrossRefGoogle Scholar
  4. 4.
    M. Formann, D. Wagner, F. Wagner. Routing through a dense channel with minimum total wire length. Proc. of the Second Ann. ACM-SIAM Symposium on Discrete Algorithms (1991) 475–482Google Scholar
  5. 5.
    A. Frank. Disjoint paths in a rectilinear grid. Combinatorica 2 (1982) 361–371Google Scholar
  6. 6.
    M. C. Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic Press (1980)Google Scholar
  7. 7.
    T. Gonzales, S. Zheng. Simple Three-layer channel routing algorithms. Proc. of Aegean Workshop on Computing LNCS 319 (ed. J. H. Reif) (1988) 237–246Google Scholar
  8. 8.
    T. Gonzales, S. Zheng. On ensuring three-layer wirability by stretching planar layouts. INTEGRATION: The VLSI Journal 8 (1989) 111–141CrossRefGoogle Scholar
  9. 9.
    M. Kaufmann, P. Molitor. Minimal stretching of a layout to ensure 2-layer wirability. to appear in INTEGRATION: The VLSI Journal Google Scholar
  10. 10.
    R. Kuchem, D. Wagner, F. Wagner. Area-optimal three-layer channel routing. Proc. of the 30th Ann. Symposium on Foundations of Computer Science (1989) 506–511Google Scholar
  11. 11.
    W. Lipski, Jr. On the structure of three-layer wirable layouts. In: Advances in Computing Research 2 (VLSI Theory) (ed. F.P. Preparata) JAI Press (1984) 231–243Google Scholar
  12. 12.
    W. Lipski, Jr., F. P. Preparata. A unified approach to layout wirability. Mathematical Systems Theory 19 (1987) 189–203CrossRefGoogle Scholar
  13. 13.
    K. Mehlhorn, F. P. Preparata. Routing through a rectangle. J. ACM 33 (1986) 60–85CrossRefGoogle Scholar
  14. 14.
    K. Mehlhorn, F. P. Preparata, M. Sarrafzadeh. Channel routing in knock-knee mode: Simplified algorithms and proofs. Algorithmica 1 (1986) 213–221Google Scholar
  15. 15.
    F. P. Preparata, W. Lipski, Jr. Optimal three-layer channel routing. IEEE Trans. on Computers 33 (1984) 427–437Google Scholar
  16. 16.
    I. G. Tollis. A new algorithm for wiring layouts. Proc. of the Aegean Workshop on Computing LNCS 319 (ed. J.H. Reif) (1988) 257–267Google Scholar
  17. 17.
    D. Wagner. A new approach to knock-knee channel routing. Proc. of the International Symposium on Algorithms LNCS 557 (eds. W. L. Hsu, R. C. T. Lee) (1991) 83–93Google Scholar
  18. 18.
    C. Wieners-Lummer. Three-layer channel routing in knock-knee mode. Preprint UniversitÄt PaderbornGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Majid Sarrafzadeh
    • 1
  • Dorothea Wagner
    • 2
  • Frank Wagner
    • 3
  • Karsten Weihe
    • 2
  1. 1.Department of Electrical Engineering and Computer ScienceNorthwestern UniversityEvanstonUSA
  2. 2.Fachbereich MathematikTechnische UniversitÄt BerlinBerlin 12Germany
  3. 3.Institut für Informatik, Fachbereich MathematikFreie UniversitÄt BerlinBerlin 33Germany

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