Skip to main content
SpringerLink
Log in

Predicting Memory Demands of BDD Operations Using Maximum Graph Cuts

  • Conference paper
  • First Online:
Automated Technology for Verification and Analysis (ATVA 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14216))

Abstract

The BDD package Adiar manipulates Binary Decision Diagrams (BDDs) in external memory. This enables handling big BDDs, but the performance suffers when dealing with moderate-sized BDDs. This is mostly due to initializing expensive external memory data structures, even if their contents can fit entirely inside internal memory.

The contents of these auxiliary data structures always correspond to a graph cut in an input or output BDD. Specifically, these cuts respect the levels of the BDD. We formalise the shape of these cuts and prove sound upper bounds on their maximum size for each BDD operation.

We have implemented these upper bounds within Adiar. With these bounds, it can predict whether a faster internal memory variant of the auxiliary data structures can be used. In practice, this improves Adiar’s running time across the board. Specifically for the moderate-sized BDDs, this results in an average reduction of the computation time by \(86.1\%\) (median of \(89.7\%\)). In some cases, the difference is even \(99.9\%\). When checking equivalence of hardware circuits from the EPFL Benchmark Suite, for one of the instances the time was decreased by 52 h.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 59.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 74.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Aggarwal, A., Vitter, J.S.: The input/output complexity of sorting and related problems. Commun. ACM 31(9), 1116–1127 (1988). https://doi.org/10.1145/48529.48535

    Article  MathSciNet  Google Scholar 

  2. Amarú, L., Gaillardon, P.E., De Micheli, G.: The EPFL combinational benchmark suite. In: 24th International Workshop on Logic & Synthesis (2015)

    Google Scholar 

  3. Amparore, E.G., Donatelli, S., Gallà, F.: starMC: an automata based CTL* model checker. PeerJ Comput. Sci. 8, e823 (2022)

    Article  Google Scholar 

  4. Arge, L.: The I/O-complexity of ordered binary-decision diagram manipulation. In: Staples, J., Eades, P., Katoh, N., Moffat, A. (eds.) ISAAC 1995. LNCS, vol. 1004, pp. 82–91. Springer, Heidelberg (1995). https://doi.org/10.1007/BFb0015411

    Chapter  Google Scholar 

  5. Arge, L.: The I/O-complexity of ordered binary-decision diagram. In: BRICS RS preprint series, vol. 29. Department of Computer Science, University of Aarhus (1996). https://doi.org/10.7146/brics.v3i29.20010

  6. Ashar, P., Cheong, M.: Efficient breadth-first manipulation of binary decision diagrams. In: IEEE/ACM International Conference on Computer-Aided Design (ICCAD), pp. 622–627. IEEE Computer Society Press (1994). https://doi.org/10.1109/ICCAD.1994.629886

  7. Brace, K.S., Rudell, R.L., Bryant, R.E.: Efficient implementation of a BDD package. In: 27th Design Automation Conference (DAC), pp. 40–45. Association for Computing Machinery (1990). https://doi.org/10.1109/DAC.1990.114826

  8. Bryant, R.E.: Graph-based algorithms for Boolean function manipulation. IEEE Trans. Comput. C-35(8), 677–691 (1986). https://doi.org/10.1109/TC.1986.1676819

  9. Bryant, R.E., Biere, A., Heule, M.J.H.: Clausal proofs for pseudo-Boolean reasoning. In: TACAS 2022. LNCS, vol. 13243, pp. 443–461. Springer, Cham (2022). https://doi.org/10.1007/978-3-030-99524-9_25

    Chapter  Google Scholar 

  10. Bryant, R.E., Heule, M.J.H.: Dual proof generation for quantified Boolean formulas with a BDD-based solver. In: Platzer, A., Sutcliffe, G. (eds.) CADE 2021. LNCS (LNAI), vol. 12699, pp. 433–449. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-79876-5_25

    Chapter  Google Scholar 

  11. Bryant, R.E., Heule, M.J.H.: Generating extended resolution proofs with a BDD-based SAT solver. In: TACAS 2021. LNCS, vol. 12651, pp. 76–93. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-72016-2_5

    Chapter  Google Scholar 

  12. Chen, J., Revels, J.: Robust benchmarking in noisy environments. arXiv (2016). https://arxiv.org/abs/1608.04295

  13. Chiang, Y.J., Goodrich, M.T., Grove, E.F., Tamassia, R., Vengroff, D.E., Vitter, J.S.: External-memory graph algorithms. In: Proceedings of the Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 139–149. SODA 1995, Society for Industrial and Applied Mathematics (1995)

    Google Scholar 

  14. Ciardo, G., Miner, A.S., Wan, M.: Advanced features in SMART: the stochastic model checking analyzer for reliability and timing. SIGMETRICS Perform. Evaluation Rev. 36(4), 58–63 (2009)

    Article  Google Scholar 

  15. Cimatti, A., Clarke, E., Giunchiglia, F., Roveri, M.: NuSMV: a new symbolic model checker. Int. J. Softw. Tools Technol. Transfer 2, 410–425 (2000). https://doi.org/10.1007/s100090050046

    Article  MATH  Google Scholar 

  16. Van Dijk, T., Van de Pol, J.: Sylvan: multi-core framework for decision diagrams. Int. J. Softw. Tools Technol. Transfer 19, 675–696 (2016). https://doi.org/10.1007/s10009-016-0433-2

    Article  Google Scholar 

  17. Gammie, P., van der Meyden, R.: MCK: model checking the logic of knowledge. In: Alur, R., Peled, D.A. (eds.) CAV 2004. LNCS, vol. 3114, pp. 479–483. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-27813-9_41

    Chapter  Google Scholar 

  18. He, L., Liu, G.: Petri net based symbolic model checking for computation tree logic of knowledge. arXiv (2020). https://arxiv.org/abs/2012.10126

  19. Hellings, J., Fletcher, G.H., Haverkort, H.: Efficient external-memory bisimulation on DAGs. In: Proceedings of the 2012 ACM SIGMOD International Conference on Management of Data, pp. 553–564. SIGMOD 2012, Association for Computing Machinery (2012). https://doi.org/10.1145/2213836.2213899, https://doi.org/10.1145/2213836.2213899

  20. Kant, G., Laarman, A., Meijer, J., van de Pol, J., Blom, S., van Dijk, T.: LTSmin: high-performance language-independent model checking. In: Baier, C., Tinelli, C. (eds.) TACAS 2015. LNCS, vol. 9035, pp. 692–707. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46681-0_61

    Chapter  Google Scholar 

  21. Kunkle, D., Slavici, V., Cooperman, G.: Parallel disk-based computation for large, monolithic binary decision diagrams. In: 4th International Workshop on Parallel Symbolic Computation (PASCO), pp. 63–72 (2010). https://doi.org/10.1145/1837210.1837222

  22. Lampis, M., Kaouri, G., Mitsou, V.: On the algorithmic effectiveness of digraph decompositions and complexity measures. Discrete Optim. 8(1), 129–138 (2011). https://doi.org/10.1016/j.disopt.2010.03.010. parameterized Complexity of Discrete Optimization

  23. Lomuscio, A., Qu, H., Raimondi, F.: MCMAS: an open-source model checker for the verification of multi-agent systems. Int. J. Softw. Tools Technol. Transfer 19(1), 9–30 (2015). https://doi.org/10.1007/s10009-015-0378-x

    Article  Google Scholar 

  24. Meyer, U., Sanders, P., Sibeyn, J.: Algorithms for Memory Hierarchies: Advanced Lectures. Springer, Heidelberg (2003). https://doi.org/10.1007/3-540-36574-5

  25. Milvang-Jensen, K., Hu, A.J.: BDDNOW: a parallel BDD package. In: Gopalakrishnan, G., Windley, P. (eds.) FMCAD 1998. LNCS, vol. 1522, pp. 501–507. Springer, Heidelberg (1998). https://doi.org/10.1007/3-540-49519-3_32

    Chapter  Google Scholar 

  26. Minato, S.I.: Zero-suppressed BDDs for set manipulation in combinatorial problems. In: 30th Design Automation Conference (DAC), pp. 272–277. Association for Computing Machinery (1993). https://doi.org/10.1145/157485.164890

  27. Minato, S.I., Ishiura, N., Yajima, S.: Shared binary decision diagram with attributed edges for efficient Boolean function manipulation. In: 27th Design Automation Conference (DAC), pp. 52–57. Association for Computing Machinery (1990). https://doi.org/10.1145/123186.123225

  28. Mølhave, T.: Using TPIE for processing massive data sets in C++. Duke University, Durham, NC, Technical report (2012)

    Google Scholar 

  29. Ochi, H., Yasuoka, K., Yajima, S.: Breadth-first manipulation of very large binary-decision diagrams. In: International Conference on Computer Aided Design (ICCAD), pp. 48–55. IEEE Computer Society Press (1993). https://doi.org/10.1109/ICCAD.1993.580030

  30. Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation, and complexity classes. J. Comput. Syst. Sci. 43(3), 425–440 (1991). https://doi.org/10.1016/0022-0000(91)90023-X

    Article  MathSciNet  MATH  Google Scholar 

  31. Sanghavi, J.V., Ranjan, R.K., Brayton, R.K., Sangiovanni-Vincentelli, A.: High performance BDD package by exploiting memory hierarchy. In: 33rd Design Automation Conference (DAC), pp. 635–640. Association for Computing Machinery (1996). https://doi.org/10.1145/240518.240638

  32. Sølvsten, S.C.: BDD Benchmark. Zenodo (2023). https://doi.org/10.5281/zenodo.7040263

  33. Sølvsten, S.C., van de Pol, J.: Adiar 1.0.1 : Experiment data (11 2021). https://doi.org/10.5281/zenodo.5638551

  34. Sølvsten, S.C., van de Pol, J.: Adiar 1.1: zero-suppressed decision diagrams in external memory. In: Rozier, K.Y., Chaudhuri, S. (eds.) NFM 2023. LNCS, vol. 13903, pp. 464–471. Springer, Heidelberg (2023). https://doi.org/10.1007/978-3-031-33170-1_28

    Chapter  Google Scholar 

  35. Sølvsten, S.C., van de Pol, J.: Adiar 1.1.0 : experiment data (2023). https://doi.org/10.5281/zenodo.7709134

  36. Sølvsten, S.C., van de Pol, J.: Adiar 1.2.0 : experiment data (2023). https://doi.org/10.5281/zenodo.8124120

  37. Sølvsten, S.C., van de Pol, J.: Predicting memory demands of BDD operations using maximum graph cuts (extended paper) (2023)

    Google Scholar 

  38. Sølvsten, S.C., de Pol, J., Jakobsen, A.B., Thomasen, M.W.B.: Adiar binary decision diagrams in external memory. In: TACAS 2022. LNCS, vol. 13244, pp. 295–313. Springer, Cham (2022). https://doi.org/10.1007/978-3-030-99527-0_16

    Chapter  Google Scholar 

  39. Somenzi, F.: CUDD: CU decision diagram package, 3.0. Technical report, University of Colorado at Boulder (2015)

    Google Scholar 

  40. Stornetta, T., Brewer, F.: Implementation of an efficient parallel BDD package. In: Design Automation Conference Proceedings, vol. 33, pp. 641–644 (1996). https://doi.org/10.1109/DAC.1996.545653

  41. Vengroff, D.E.: A transparent parallel I/O environment. In: In Proceedings of 1994 DAGS Symposium on Parallel Computation, pp. 117–134 (1994)

    Google Scholar 

  42. Yang, B., O’Hallaron, D.R.: Parallel breadth-first BDD construction. SIGPLAN Not. 32(7), 145–156 (1997). https://doi.org/10.1145/263767.263784

    Article  Google Scholar 

Download references

Acknowledgements

We want to thank Anna Blume Jakobsen for her help implementing the use of i-level cuts in Adiar and Kristoffer Arnsfelt Hansen for his input on the computational complexity of these cuts. Finally, thanks to the Centre for Scientific Computing, Aarhus, (phys.au.dk/forskning/cscaa/) for running our benchmarks.

Author information

Authors and Affiliations

  1. Aarhus University, Aarhus, Denmark

    Steffan Christ Sølvsten & Jaco van de Pol

Authors
  1. Steffan Christ Sølvsten
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jaco van de Pol
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Steffan Christ Sølvsten .

Editor information

Editors and Affiliations

  1. Université Sorbonne Paris Nord, Villetaneuse, France

    Étienne André

  2. Singapore Management University, Singapore, Singapore

    Jun Sun

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Sølvsten, S.C., van de Pol, J. (2023). Predicting Memory Demands of BDD Operations Using Maximum Graph Cuts. In: André, É., Sun, J. (eds) Automated Technology for Verification and Analysis. ATVA 2023. Lecture Notes in Computer Science, vol 14216. Springer, Cham. https://doi.org/10.1007/978-3-031-45332-8_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-45332-8_4

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-45331-1

  • Online ISBN: 978-3-031-45332-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation