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Characterizing the Complexity of Boolean Functions represented by Well-Structured Graph-Driven Parity-FBDDs

Published online by Cambridge University Press:  15 December 2002

Henrik Brosenne ,
Matthias Homeister  and
Stephan Waack
Henrik Brosenne
Affiliation:
Institut für Numerische und Angewandte Mathematik, Georg-August-Universität Göttingen, Lotzestr. 16-18, 37083 Göttingen, Germany; [email protected]. [email protected].
Matthias Homeister
Affiliation:
Institut für Numerische und Angewandte Mathematik, Georg-August-Universität Göttingen, Lotzestr. 16-18, 37083 Göttingen, Germany; [email protected]. [email protected].
Stephan Waack
Affiliation:
Institut für Numerische und Angewandte Mathematik, Georg-August-Universität Göttingen, Lotzestr. 16-18, 37083 Göttingen, Germany; [email protected]. [email protected].
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Abstract

We investigate well-structured graph-driven parity-FBDDs, which strictly generalize the two well-known models parity OBDDs and well-structured graph-driven FBDDs. The first main result is a characterization of the complexity of Boolean functions represented by well-structured graph-driven parity-FBDDs in terms of invariants of the function represented and the graph-ordering used. As a consequence, we derive a lower bound criterion and prove an exponential lower bound for certain linear code functions. The second main result of this paper is a polynomial time algorithm that minimizes the number of nodes in a graph-driven parity-FBDD.

Keywords

Well-structured graph-driven parity-FBDDslower boundsminimization algorithmcomplexity theorydata structures for Boolean functions.
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 36 , Issue 3 , July 2002 , pp. 229 - 247
Copyright
© EDP Sciences, 2002

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References

B. Bollig, St. Waack and P. Woelfel, Parity graph-driven read-once branching programs and an exponential lower bound for integer multiplication, in Proc. 2nd IFIP International Conference on Theoretical Computer Science (2002).
Breitbart, Y., Hunt, H.B. and Rosenkrantz, D., The size of binary decision diagrams representing Boolean functions. Theoret. Comput. Sci. 145 (1995) 45-69. CrossRef
Brosenne, H., Homeister, M. and Waack, St., Graph-driven free parity BDDs: Algorithms and lower bounds, in Proc. 26th MFCS. Springer Verlag, Lecture Notes in Comput. Sci. 2136 (2001) 212-223. CrossRef
Bryant, R., On the complexity of VLSI implementations of Boolean functions with applications to integer multiplication. IEEE Trans. Comput. 40 (1991) 205-213. CrossRef
R.E. Bryant, Symbolic manipulation of Boolean functions using a graphical representation, in Proc. 22nd DAC. Piscataway, NJ (1985) 688-694.
Bryant, R.E., Graph-based algorithms for Boolean function manipulation. IEEE Trans. Comput. 35 (1986) 677-691. CrossRef
Coppersmith, D. and Winograd, S., Matrix multiplication via arithmetic progressions. J. Symb. Comput. 9 (1990) 251-280. CrossRef
Gergov, J. and Meinel, Ch., Frontiers of feasible and probabilistic feasible Boolean manipulation with branching programs, in Proc. 10th STACS. Springer Verlag, Lecture Notes in Comput. Sci. 665 (1993) 576-585. CrossRef
Gergov, J. and Meinel, Ch., Mod-2-OBDDs - A data structure that generalizes exor-sum-of-products and ordered binary decision diagrams. Formal Methods in System Design 8 (1996) 273-282. CrossRef
Jukna, S., Entropy of contact circuits and lower bounds on their complexit. Theoret. Comput. Sci. 57 (1988) 113-129. CrossRef
Jukna, S., Linear codes are hard for oblivious read-once parity branching programs. Inform. Process. Lett. 69 (1999) 267-269. CrossRef
E.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes. Elsevier (1977).
Sieling, D., Lower bounds for linear transformed OBDDs and FBDDs, in Proc. 19th FSTTCS. Springer Verlag, Lecture Notes in Comput. Sci. 1738 (1999) 356-368. CrossRef
Sieling, D. and Wegener, I., Graph driven BDDs - A new data structure for Boolean functions. Theoret. Comput. Sci. 141 (1995) 238-310. CrossRef
St. Waack, On the descriptive and algorithmic power of parity ordered binary decision diagrams. Inform. Comput. 166 (2001) 61-70. CrossRef
I. Wegener, Branching Programs and Binary Decision Diagrams - Theory and Applications. SIAM, Philadelphia, SIAM Monogr. Discrete Math. Appl. (2000).