# Algebraic and Uniqueness Properties of Parity Ordered Binary Decision Diagrams and Their Generalization

## Abstract

Ordered binary decision diagrams (OBDDs) and parity OBDDs are data structures representing Boolean functions. In addition, we study their generalization which we call parity AOBDDs, give their algebraic characterization and compare their minimal size to the size of parity OBDDs.

We prove that the constraint that no arcs test conditions of type *x* _{i = 0 does not affect the node-size of parity (A) OBDDs and we give an efficient algorithm for finding such parity (A) OBDDs. We obtain a canonical form for parity OBDDs and discuss similar results for parity AOBDDs}

Algorithms for minimization and transformation to the canonical form for parity OBDDs running in time *O*(*S* ^{3}) and space *O*(*S* ^{2}) or in time *O*(*S* ^{3}/*logS*) and space *O*(*S* ^{3}/ *logS*) and an algorithm for minimization of parity AOBDDs running in time *O*(*nS* ^{3}) and space *O*(*nS* ^{2}) are presented (*n* is the number of variables, *S* is the number of vertices).

All the results are extendable to case of shared parity (A) OBDDs — data structures for representation of Boolean function sequences.

## Keywords

Boolean Function Canonical Form Uniqueness Condition Binary Decision Diagram Shared Parity## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Arlazarov, L., Dinic, E. A., Kronrod, A., Faradzev, I. A.: On economical construction of the transitive closure of a directed graph. Dokl. Akad. Nauk USSR 1970, 194, pp. 487–488 (in Russian), Soviet. Math. Dokl. 11, pp. 1209-1210 (in English)MathSciNetGoogle Scholar
- 2.Bryant, R. E.: Graph-based algorithms for Boolean function manipulation. IEEE Trans. on Computers 1986, 35, pp. 677–691zbMATHCrossRefGoogle Scholar
- 3.Bryant, R. E.: Symbolic Boolean manipulation with ordered binary decision diagrams. ACM Comp. Surveys 1992, 24, pp. 293–318CrossRefGoogle Scholar
- 4.Coppersmith, D., Winograd, S.: Matrix multiplication via arithmetic progressions. J. Symbolic Computation 1990, 9, pp. 251–280zbMATHMathSciNetCrossRefGoogle Scholar
- 5.Gergov, J., Meinel, Ch.: Mod-2-OBDDs — a data structure that generalizes EXOR-Sum-of-Products and Ordered Binary Decision Diagrams. Formal Methods in System Design 1996, 8, pp. 273–282CrossRefGoogle Scholar
- 6.Král’, D.: Algebraic and Uniqueness Properties of Parity Ordered Binary Decision Diagrams and their Generalization. ECCC report TR00-013Google Scholar
- 7.Löbbing, M., Sieling, D., Wegener, I.: Parity OBDDs cannot be handled efficiently enough. Information Processing Letters 1998, 67, pp. 163–168CrossRefMathSciNetGoogle Scholar
- 8.Waack, St.: On the descriptive and algorithmic power of parity ordered binary decision diagrams. Proc. 14th STACS 1997, Lecture Notes in Computer Sci. 1200, Springer Verlag 1997, pp. 201–212Google Scholar
- 9.Wegener, I.: Branching Programs and Binary Decision Diagrams — Theory and Applications. To appear (2000) in the SIAM monograph series Trends in Discrete Mathematics and Applications ed. P. L. Hammer.Google Scholar
- 10.Wegener, I.: Efficient data structures for Boolean functions. Discrete Mathematics 1994, 136, pp. 347–372zbMATHCrossRefMathSciNetGoogle Scholar