Document Open Access Logo

Single Family Algebra Operation on BDDs and ZDDs Leads to Exponential Blow-Up

Authors Kengo Nakamura , Masaaki Nishino , Shuhei Denzumi

PDF

File

Thumbnail PDF
PDF
LIPIcs.ISAAC.2024.52.pdf
  • Filesize: 1.19 MB
  • 17 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • Binary decision diagrams
  • family of sets
  • family algebra

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

Binary decision diagram (BDD) and zero-suppressed binary decision diagram (ZDD) are data structures to represent a family of (sub)sets compactly, and it can be used as succinct indexes for a family of sets. To build BDD/ZDD representing a desired family of sets, there are many transformation operations that take BDDs/ZDDs as inputs and output BDD/ZDD representing the resultant family after performing operations such as set union and intersection. However, except for some basic operations, the worst-time complexity of taking such transformation on BDDs/ZDDs has not been extensively studied, and some contradictory statements about it have arisen in the literature. In this paper, we show that many transformation operations on BDDs/ZDDs, including all operations for families of sets that appear in Knuth’s book, cannot be performed in worst-case polynomial time in the size of input BDDs/ZDDs. This refutes some of the folklore circulated in past literature and resolves an open problem raised by Knuth. Our results are stronger in that such blow-up of computational time occurs even when the ordering, which has a significant impact on the efficiency of treating BDDs/ZDDs, is chosen arbitrarily.

Cite As Get BibTex

Kengo Nakamura, Masaaki Nishino, and Shuhei Denzumi. Single Family Algebra Operation on BDDs and ZDDs Leads to Exponential Blow-Up. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 52:1-52:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ISAAC.2024.52

Author Details

Kengo Nakamura
  • NTT Communication Science Laboratories, Kyoto, Japan
Masaaki Nishino
  • NTT Communication Science Laboratories, Kyoto, Japan
Shuhei Denzumi
  • NTT Communication Science Laboratories, Kyoto, Japan

Funding

This work was generally supported by JSPS KAKENHI Grant Number JP20H05963 and JST CREST Grant Number JPMJCR22D3.
  • Denzumi, Shuhei: was supported by JSPS KAKENHI Grant Number JP23H04391.

Acknowledgements

We thank Hiromi Emoto and Shou Ooba for pointing out the issues regarding the complexity of performing family algebra operations on ZDDs. We also thank Shin-ichi Minato, Jun Kawahara, and Norihito Yasuda for valuable discussions on this topic. I am grateful to the reviewers of ISAAC for the comments to improve the manuscript.

References

  1. B. Bollig. A simpler counterexample to a long-standing conjecture on the complexity of Bryant’s apply algorithm. Inf. Process. Lett., 114(3):124-129, 2014. URL: https://doi.org/10.1016/j.ipl.2013.11.003.
  2. R. E. Bryant. Graph-based algorithms for Boolean function manipulation. IEEE Trans. Comput., C-35(8):677-691, 1986. URL: https://doi.org/10.1109/TC.1986.1676819.
  3. R. E. Bryant. On the complexity of VLSI implementations and graph representations of Boolean functions with application to integer multiplication. IEEE Trans. Comput., 40(2):205-213, 1991. URL: https://doi.org/10.1109/12.73590.
  4. O. Coudert. Solving graph optimization problems with ZBDDs. In Proc. of the European Design & Test Conference (ED&TC 97), pages 224-228, 1997. URL: https://doi.org/10.1109/EDTC.1997.582363.
  5. O. Coudert, J. C. Madre, and H. Fraisse. A new viewpoint on two-level logic minimization. In Proc. of the 30th ACM/IEEE Design Automation Conference (DAC 1993), pages 625-630, 1993. URL: https://doi.org/10.1145/157485.165071.
  6. A. Darwiche and P. Marquis. A knowledge compilation map. J. Artif. Intell. Res., 17(1):229-264, 2002. URL: https://doi.org/10.1613/jair.989.
  7. U. Gupta, P. Kalla, and V. Rao. Boolean Gröbner basis reductions on finite field datapath circuits using the unate cube set algebra. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst., 38(3):576-588, 2019. URL: https://doi.org/10.1109/TCAD.2018.2818726.
  8. T. Inoue, H. Iwashita, J. Kawahara, and S. Minato. Graphillion: software library for very large sets of labeled graphs. Int. J. Softw. Tools Technol. Trans., 18(1):57-66, 2016. URL: https://doi.org/10.1007/s10009-014-0352-z.
  9. T. Inoue, N. Yasuda, S. Kawano, Y. Takenobu, S. Minato, and Y. Hayashi. Distribution network verification for secure restoration by enumerating all critical failures. IEEE Trans. Smart Grid, 6(2):843-852, 2015. URL: https://doi.org/10.1109/TSG.2014.2359114.
  10. A. Ito, R. Ueno, and N. Homma. Efficient formal verification of Galois-Field arithmetic circuits using ZDD representation of Boolean polynomials. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst., 41(3):794-798, 2022. URL: https://doi.org/10.1109/TCAD.2021.3059924.
  11. J. Kawahara, T. Inoue, H. Iwashita, and S. Minato. Frontier-based search for enumerating all constrained subgraphs with compressed representation. IEICE Trans. Fundamentals, E100-A(9):1773-1784, 2017. URL: https://doi.org/10.1587/transfun.E100.A.1773.
  12. J. Kawahara, T. Saitoh, H. Suzuki, and R. Yoshinaka. Solving the longest oneway-ticket problem and enumerating letter graphs by augmenting the two representative approaches with ZDDs. In Proc. of the Computational Intelligence in Information System (CIIS 2016), pages 294-305, 2016. URL: https://doi.org/10.1007/978-3-319-48517-1_26.
  13. D. E. Knuth. The Art of Computer Programming: Vol. 4A. Combinatorial Algorithms, Part 1, volume 4A: Combinatorial Algorithms, Part I. Addison-Wesley Professional, 2011. Google Scholar
  14. S. Minato. Zero-suppressed BDDs for set manipulation in combinatorial problems. In Proc. of the 30th ACM/IEEE Design Automation Conference (DAC 1993), pages 272-277, 1993. URL: https://doi.org/10.1145/157485.164890.
  15. S. Minato. Calculation of unate cube set algebra using zero-suppressed BDDs. In Proc. of the 31st ACM/IEEE Design Automation Conference (DAC 1994), pages 420-424, 1994. URL: https://doi.org/10.1145/196244.196446.
  16. S. Minato, T. Uno, and H. Arimura. LCM over ZBDDs: Fast generation of very large-scale frequent itemsets using a compact graph-based representation. In Proc. of Advances in Knowledge Discovery and Data Mining (PAKDD 2008), pages 234-246, 2008. URL: https://doi.org/10.1007/978-3-540-68125-0_22.
  17. H. G. Okuno, S. Minato, and H. Isozaki. On the properties of combination set operations. Inf. Process. Lett., 66(4):195-199, 1998. URL: https://doi.org/10.1016/S0020-0190(98)00067-2.
  18. F. Somenzi. CUDD: CU decision diagram package, 1997. URL: https://github.com/ivmai/cudd.
  19. Y. Takenobu, N. Yasuda, S. Kawano, S. Minato, and Y. Hayashi. Evaluation of annual energy loss reduction based on reconfiguration scheduling. IEEE Trans. Smart Grid, 9(3):1986-1996, 2018. URL: https://doi.org/10.1109/TSG.2016.2604922.
  20. R. Yoshinaka, J. Kawahara, S. Denzumi, H. Arimura, and S. Minato. Counterexamples to the long-standing conjecture on the complexity of BDD binary operations. Inf. Process. Lett., 112(16):636-640, 2012. URL: https://doi.org/10.1016/j.ipl.2012.05.007.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact