Abstract
We investigate the size and structure of ordered binary decision diagrams (OBDDs) for random Boolean functions. Wegener [Weg94] proved that for “most” values of n, the expected OBDD-size of a random Boolean function with n variables equals the worst-case size up to terms of lower order. Our main result is that this phenomenon, also known as strong Shannon effect, shows a threshold behaviour: The strong Shannon effect does not hold within intervals of constant width around the values n = 2h + h, but it does hold outside these intervals. Also, the oscillation of the expected and the worst-case size is described. Methodical innovations of our approach are a functional equation to locate “critical levels” in OBDDs and the use of Azuma's martingale inequality and Chvátal's large deviation inequality for the hypergeometric distribution. This leads to significant improvements over Wegener's probability bounds.
Keywords
- Boolean Function
- Constant Width
- Hypergeometric Distribution
- Binary Decision Diagram
- Ordered 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.
Graduate school “Algorithmische Diskrete Mathematik”, supported by Deutsche Forschungsgemeinschaft, grant GRK 219/2-97.
supported by Deutsche Forschungsgemeinschaft, grant Pr 296/3-2.
This is a preview of subscription content, log in via an institution.
Preview
Unable to display preview. Download preview PDF.
References
Noga Alon, Joel H. Spencer: The Probabilistic Method; John Wiley & Sons, New York, 1991.
Randal E. Bryant: Graph-Based Algorithms for Boolean Function Manipulation; IEEE Transactions on Computers, vol. C-35, no. 8, 1986, 677–691.
Randal E. Bryant: Symbolic Boolean Manipulation with Ordered Binary-Decision Diagrams; ACM Computing Surveys, vol. 24, no. 3, 1992, 293–318.
Vašek Chvátai: The Tail of the Hypergeometric Distribution; Discrete Mathematics, vol. 25, 1979, 285–287.
Clemens Gröpl, Anand Srivastav, Hans Jürgen Prömel: Full version of this extended abstract, available via http://www.informatik.hu-berlin.de/Institut/struktur/ algorithmen/forschung/veri/shannon.ps.gz.
Mark A. Heap, Melvin Ray Mercer: Least Upper Bounds on OBDD Sizes; IEEE Transactions on Computers, vol. 43, no. 6, 1994, 764–767.
Valentin F. Kolchin, Boris A. Sevast'ianov, Vladimir P. Chistiakov: Random Allocations; John Wiley & Sons, 1978, Chapter 1.
Heh-Tyan Liaw, Chen-Shang Lin: On the OBDD-Representation of General Boolean Functions; NSC Rep., NSC79-0404-E002-35, 1990.
Heh-Tyan Liaw, Chen-Shang Lin: On the OBDD-Representation of General Boolean Functions; IEEE Transactions on Computers, vol. 41, no. 6, 1992, 661–664.
Martin Löbbing, Olaf Schröer, Ingo Wegener: The Theory of Zero-Suppressed BDDs and the Number of Knights Tours; Proceedings of the IFIP WG 10.5 Workshop on Applications of the Reed-Muller Expansion in Circuit Design (Reed-Muller'95), August 27.–29. 1995, Makuhari, Chiba, Japan, 1995, 38–45.
Shin-Ichi Minato. Zero-Suppressed BDDs for Set Manipulation in Combinatorial Problems; Proceedings of the 30th ACM/IEEE Design Automation Conference, 1993, 272–277.
Olaf Schröer, Ingo Wegener: The Theory of Zero-Suppressed BDDs and the Number of Knights Tours; Forschungsbericht Nr. 552/1994, Universität Dortmund, Fachbereich Informatik, 1994, 25 pages.
Ingo Wegener: The Size of Reduced OBDDs and Optimal Read-Once Branching Programs for Almost All Boolean Functions; IEEE Transactions on Computers, vol. 43, no. 11, 1994, 1262–1269.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1998 Springer-Verlag
About this paper
Cite this paper
Gröpl, C., Prömel, H.J., Srivastav, A. (1998). Size and structure of random ordered binary decision diagrams. In: Morvan, M., Meinel, C., Krob, D. (eds) STACS 98. STACS 1998. Lecture Notes in Computer Science, vol 1373. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0028565
Download citation
DOI: https://doi.org/10.1007/BFb0028565
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-64230-5
Online ISBN: 978-3-540-69705-3
eBook Packages: Springer Book Archive