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The Size of Reduced OBDD's and Optimal Read-Once Branching Programs for Almost all Boolean Functions
November 1994 (vol. 43 no. 11)
pp. 1262-1269

Boolean functions are often represented by ordered binary-decision diagrams (OBDDs) introduced by Bryant (1986). Liaw and Lin (1992) have proved upper and lower bounds on the minimal OBDD size of almost all Boolean functions. Now tight bounds are proved for the minimal OBDD size for arbitrary or optimal variable orderings and for the minimal read-once branching program size of almost all functions. Almost all Boolean functions have a sensitivity of almost 1, i.e., the minimal OBDD size for an optimal variable ordering differs from the minimal OBDD size for a worst variable ordering by a factor of at most 1+/spl epsi/(n) where /spl epsi/(n) converges exponentially fast to 0.

[1] S. B. Akers, "Binary decision diagrams,"IEEE Trans. Comput., vol. C-27, no. 6, pp. 509-516, June 1978.
[2] K. S. Brace, R. L. Rudell, and R. E. Bryant, "Efficient implementation of a BDD package," in27th ACM/IEEE Design Automat. Conf., 1990, pp. 40-45.
[3] R. E. Bryant, "Graph-based algorithms for Boolean function manipulation,"IEEE Trans. Comput., vol. C-35, no. 8, pp. 677-691, Aug. 1986.
[4] R. E. Bryant, "On the complexity of VLSI implementations and graph representations with application to integer multiplication,"IEEE Trans. Comput., vol. 40, no. 2, pp. 205-213, Feb. 1991.
[5] R. E. Bryant, "Symbolic boolean manipulation with ordered binary-decision diagrams,"ACM Computing Surveysvol. 24, no. 3, pp. 293-318, 1992.
[6] P. Erdös and J. Spencer,Probabilistic Methods in Combinatorics. New York: Academic Press, 1974.
[7] W. Feller,An Introduction to Probability Theory and its Applications. New York: Wiley, 1968.
[8] S. J. Friedman and K. J. Supowit, "Finding the optimal variable ordering for binary decision diagrams,"IEEE Trans. Comput., vol. 39, no. 5, pp. 710-713, May 1990.
[9] J. Gergov and C. Meinel, "Frontiers of feasible boolean manipulation with branching programs," in10th Symp. Theoretical Aspects of Comput. Sci., (Lecture Notes in Computer Science 665), 1993, pp. 576-585.
[10] V. F. Kolchin, B. A. Sevast'yanov, and V. P. Christyakov,Random Allocations. New York: Wiley, 1978.
[11] H.-T. Liaw and C.-S. Lin, "On the OBDD-representation of general boolean functions,"IEEE Trans. Comput., vol. 41, no. 6, pp. 661-664, June 1992.
[12] C. E. Shannon, "The synthesis of two-terminal switching circuits,"AT&T Bell Syst. Tech. J., vol. 28, pp. 59-98, 1949.
[13] D. Sieling and I. Wegener, "Graph driven BDD's--A new data structure for boolean functions," to appear inTheoretical Computer Science, 1995.
[14] I. Wegener,The Complexity of Boolean Functions. New York: Wiley, 1987.

Index Terms:
Boolean functions; directed graphs; computational complexity; programming theory; optimal read-once branching programs; Boolean functions; ordered binary-decision diagrams; minimal read-once branching program size; reduction rules; size complexity; variable ordering.
Citation:
I. Wegener, "The Size of Reduced OBDD's and Optimal Read-Once Branching Programs for Almost all Boolean Functions," IEEE Transactions on Computers, vol. 43, no. 11, pp. 1262-1269, Nov. 1994, doi:10.1109/12.324559
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