Towards the Exact Minimization of BDDs—An Elitism-Based Distributed Evolutionary Algorithm

Binary Decision Diagrams (BDDs) are the state-of-the-art data structure for representation and manipulation of Boolean functions. In general, exact BDD minimization is NP-complete. For BDD-based technology, a small improvement in the number of nodes often simplifies the follow-up problem tremendously. This paper proposes an elitism-based evolutionary algorithm (EBEA) for BDD minimization. It can efficiently find the optimal orderings of variables for all LGSynth91 benchmark circuits with a known minimum size. Moreover, we develop a distributed model of EBEA, DEBEA, which obtains the best-ever variable orders for almost all benchmarks in the LGSynth91. Experimental results show that DEBEA is able to achieve super-linear performance compared to EBEA for some hard benchmarks.

[1]  David E. Goldberg,et al.  AllelesLociand the Traveling Salesman Problem , 1985, ICGA.

[2]  Wolfgang Rosenstiel,et al.  Multilevel logic synthesis based on functional decision diagrams , 1992, [1992] Proceedings The European Conference on Design Automation.

[3]  F. Somenzi,et al.  Using lower bounds during dynamic BDD minimization , 2001 .

[4]  Fabio Somenzi,et al.  Who are the variables in your neighborhood , 1995, ICCAD.

[5]  Randal E. Bryant,et al.  Symbolic Boolean manipulation with ordered binary-decision diagrams , 1992, CSUR.

[6]  George Karypis,et al.  Introduction to Parallel Computing , 1994 .

[7]  Rolf Drechsler,et al.  History-based dynamic BDD minimization , 2001, Integr..

[8]  Hiroshi Sawada,et al.  Minimization of binary decision diagrams based on exchanges of variables , 1991, 1991 IEEE International Conference on Computer-Aided Design Digest of Technical Papers.

[9]  Beate Bollig,et al.  Improving the Variable Ordering of OBDDs Is NP-Complete , 1996, IEEE Trans. Computers.

[10]  Darrell Whitley,et al.  Scheduling problems and traveling salesman: the genetic edge recombination , 1989 .

[11]  Fabio Somenzi,et al.  CUDD: CU Decision Diagram Package Release 2.2.0 , 1998 .

[12]  R. Drechsler,et al.  Symmetry Based Variable Ordering for ROBDDs , 1995 .

[13]  D. J. Smith,et al.  A Study of Permutation Crossover Operators on the Traveling Salesman Problem , 1987, ICGA.

[14]  D. Fogel.  An evolutionary approach to the traveling salesman problem , 1988, Biological Cybernetics.

[15]  Rolf Drechsler,et al.  Fast exact minimization of BDDs , 1998, Proceedings 1998 Design and Automation Conference. 35th DAC. (Cat. No.98CH36175).

[16]  Rolf Drechsler,et al.  Minimization of BDDs by Evolutionary Algorithms , 1997 .

[17]  Kenneth J. Supowit,et al.  Finding the Optimal Variable Ordering for Binary Decision Diagrams , 1990, IEEE Trans. Computers.

[18]  F. Somenzi,et al.  Who are the variables in your neighbourhood , 1995, Proceedings of IEEE International Conference on Computer Aided Design (ICCAD).

[19]  Rolf Drechsler,et al.  Minimization of free BDDs , 2002, Integr..

[20]  David E. Goldberg,et al.  Alleles, loci and the traveling salesman problem , 1985 .

[21]  Rolf Drechsler,et al.  Fast exact minimization of BDD's , 2000, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst..

[22]  Lawrence Davis,et al.  Applying Adaptive Algorithms to Epistatic Domains , 1985, IJCAI.

[23]  R. Rudell.  Dynamic variable ordering for ordered binary decision diagrams , 1993, Proceedings of 1993 International Conference on Computer Aided Design (ICCAD).

[24]  Randal E. Bryant,et al.  Graph-Based Algorithms for Boolean Function Manipulation , 1986, IEEE Transactions on Computers.

[25]  Dirk Thierens,et al.  Elitist recombination: an integrated selection recombination GA , 1994, Proceedings of the First IEEE Conference on Evolutionary Computation. IEEE World Congress on Computational Intelligence.

[26]  Masahiro Fujita,et al.  On variable ordering of binary decision diagrams for the application of multi-level logic synthesis , 1991, Proceedings of the European Conference on Design Automation..

[27]  Joseph C. Culberson.  Crossover versus Mutation: Fueling the Debate: TGA versus GIGA , 1993, ICGA.

[28]  Rolf Drechsler,et al.  BDD minimization using symmetries , 1999, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst..

[29]  Beate Bollig,et al.  On the Effect of Local Changes in the Variable Ordering of Ordered Decision Diagrams , 1996, Inf. Process. Lett..

[30]  Larry J. Eshelman,et al.  The CHC Adaptive Search Algorithm: How to Have Safe Search When Engaging in Nontraditional Genetic Recombination , 1990, FOGA.

[31]  Xiaoyu Song,et al.  BDD minimization by scatter search , 2002, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst..

[32]  Jinn-Moon Yang,et al.  Integrating adaptive mutations and family competition into genetic algorithms as function optimizer , 2000, Soft Comput..

[33]  Tae Sun Kim,et al.  An Efficient Method for Optimal BDD Ordering Computation , 1993 .

[34]  D. Fogel.  Applying evolutionary programming to selected traveling salesman problems , 1993 .