Advertisement

Complexity Theoretical Aspects of OFDDs

  • Beate Bollig
  • Martin Löbbing
  • Martin Sauerhoff
  • Ingo Wegener

Abstract

Experimental results have shown that OFDDs (ordered functional decision diagrams) are a representation of Boolean functions which are sometimes superior to OBDDs (ordered binary decision diagrams). Most of the complexity theoretical problems have been solved for OBDDs. Here some results for OFDDs are proved. It is NP-complete to decide whether a function represented by some OFDD can be represented by an OFDD of size s using another variable ordering. Given an OFDD representation for an incompletely specified function, it is NP-hard to compute an optimal OFDD cover for this function respecting the same variable ordering. The replacement of variables by constants may cause an exponential blow-up of the OFDD size. Finally, it is investigated how a local change of the variable ordering may change the OFDD size. This leads to simulated annealing algorithms to improve variable orderings.

Keywords

Boolean Function Variable Ordering Simulated Annealing Algorithm Reachable State Binary Decision Diagram 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    B. Becker and R. Drechsler, “On the computational power of functional decision diagrams.” Technical report, Universität Frankfurt, 1993.Google Scholar
  2. [2]
    B. Becker, R. Drechsler, and R. Werchner, “On the relation between BDDs and FDDs,” Latin American Theoretical Informatics, LNCS 911, pp. 72–83, 1995.Google Scholar
  3. [3]
    M. Bellare, O. Goldreich, and M. Sudan. “Free bits, PCP and non-approximability — towards tight results,” IEEE Symp. on Foundations of Computer Science, pp. 422–431, 1995.Google Scholar
  4. [4]
    B. Bollig, M. Löbbing, and I. Wegener, “Simulated annealing to improve variable orderings for OBDDs,” International Workshop on Logic Synthesis, 1995.Google Scholar
  5. [5]
    B. Bollig, M. Löbbing, M. Sauerhoff and I. Wegener, “Complexity theoretical aspects of OFDDs,” Proc. of the Workshop on Applications of the Reed-Muller Expansion in Circuit Design, IF IP WG 10.5, pp. 198–205, 1995Google Scholar
  6. [6]
    B. Bollig and I. Wegener, “Improving the variable ordering of OBDDs is NP-complete,” submitted to IEEE Trans. Computers, 1994.Google Scholar
  7. [7]
    R.E. Bryant, “Graph-based algorithms for Boolean function manipulation,” IEEE Trans. Computers, Vol. C-35, No. 8, pp. 677–691, 1986.CrossRefGoogle Scholar
  8. [8]
    S.-C. Chang, D.I. Cheng, and M. Marek-Sadowska, “Minimizing ROBDD size of incompletely specified multiple output functions,” Proc. of the European Design Automation Conference, pp. 620–624, 1994.Google Scholar
  9. [9]
    O. Coudert, C. Berthet, and J.C. Madre, “Verification of sequential machines using Boolean functional vectors,” IMEC/IFIP International Workshop on Applied Formal Methods for Correct VLSI Design, pp. 111–128, 1989.Google Scholar
  10. [10]
    S. Fortune, J. Hopcroft, and E.M. Schmidt, “The complexity of equivalence and containment for free single variable program schemes,” Proc. ICALP, LNCS 62, pp. 227–240, 1978.Google Scholar
  11. [11]
    M.R. Garey and D.S. Johnson, “Computers and Intractability — A Guide to the Theory of NP-Completeness,” Freeman and Company, New York, 1979.MATHGoogle Scholar
  12. [12]
    A.J. Hu, G. York, and D.L. Dill, “New techniques for efficient verification with implicitly conjoined BDDs,” Proc. of the 81st ACM/IEEE Design Automation Conference, pp. 276–282, 1994.Google Scholar
  13. [13]
    U. Kebschull, E. Schubert, and W. Rosenstiel, “Multilevel logic synthesis based on functional decision diagrams,” Proc. of the European Design Automation Conference, pp. 43–47, 1992.Google Scholar
  14. [14]
    R.L. Rudell, “Dynamic variable ordering for ordered binary decision diagrams,” Proc. of the ACM/IEEE Int. Conf, on Computer Aided Design, pp. 42–47, 1993.Google Scholar
  15. [15]
    M. Sauerhoff and I. Wegener, “On the complexity of minimizing the OBDD size for incompletely specified functions,” submitted to IEEE Trans, on CAD of Int. Circuits and Systems, 1994.Google Scholar
  16. [16]
    T.R. Shiple, R. Hojati, A.L. Sangiovanni-Vincentelli, and R.K. Brayton, “Heuristic minimization of BDDs using don’t cares,” Proc. of the 31st ACM/IEEE Design Automation Conference, pp. 225–231, 1994.Google Scholar
  17. [17]
    S. Tani, K. Hamaguchi, and S. Yajima, “The complexity of the optimal variable ordering problems of shared binary decision diagrams,” Proc. of the 4th Int. Symp. on Algorithms and Computation, LNCS 162, pp. 389–398, 1993.Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Beate Bollig
    • 1
  • Martin Löbbing
    • 1
  • Martin Sauerhoff
    • 1
  • Ingo Wegener
    • 1
  1. 1.Department of Computer ScienceUniversity of DortmundDortmundGermany

Personalised recommendations