Skip to main content

Parallelization of B&BT Algorithms

  • Chapter
  • First Online:
  • 340 Accesses

Part of the book series: Studies in Computational Intelligence ((SCI,volume 805))

Abstract

This chapter presents implementation details of B&BT algorithms. We consider several tools and features: data structures, memory management and, most notably, parallelization. Both shared-memory and distributed-memory environments are considered. Issues of both cases are discussed. Promising future directions are outlined, also.

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

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover 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

Learn about institutional subscriptions

References

  1. GNU Linear Programming Kit (2014). http://www.gnu.org/software/glpk/

  2. C++ eXtended Scientific Computing library (2015). http://www.xsc.de

  3. Intel Threading Building Blocks (2017). http://www.threadingbuildingblocks.org

  4. Alexandrescu, A.: Modern C++ Design: Generic Programming and Design Patterns Applied. Addison-Wesley (2001)

    Google Scholar 

  5. Baboulin, M., Donfack, S., Dongarra, J., Grigori, L., Rémy, A., Tomov, S.: A class of communication-avoiding algorithms for solving general dense linear systems on CPU/GPU parallel machines. Procedia Comput. Sci. 9, 17–26 (2012)

    Google Scholar 

  6. Beck, P.D., Nehmeier, M.: Parallel interval Newton method on CUDA. In: PARA 2012 Proceedings. Lecture Notes in Computer Science, vol. 7782, pp. 454–464 (2013)

    Google Scholar 

  7. Berner, S.: Parallel methods for verified global optimization practice and theory. J. Glob. Optim. 9(1), 1–22 (1996)

    Google Scholar 

  8. Borkowski, T.: Comparison of existence tests of zeros of equations systems in a given region: tests of Miranda, Borsuk and Newton. Master’s thesis, ICCE WUT (2013). (under supervision of Bartłomiej J. Kubica). (in Polish)

    Google Scholar 

  9. CXSC-MPI: MPI extension for the use of C-XSC in parallel environments (2015). http://www2.math.uni-wuppertal.de/~xsc/xsc/cxsc_software.html#cxsc_mpi

  10. Dąbrowski, R., Kubica, B.J.: Comparison of interval C/C++ libraries in global optimization. Prace Naukowe Politechniki Warszawskiej. Elektronika 169, 51–56 (2009)

    Google Scholar 

  11. Dąbrowski, R., Kubica, B.J.: Cache-oblivious algorithms and matrix formats for computations on interval matrices. Lecture Notes in Computer Science, vol. 7134, pp. 269–279 (2012)

    Google Scholar 

  12. Franek, P., Krčál, M.: Robust satisfiability of systems of equations. In: Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 193–203. SIAM (2014)

    Google Scholar 

  13. Franek, P., Krčál, M.: Cohomotopy groups capture robust properties of zero sets (2015). arXiv preprint arXiv:1507.04310

  14. Franek, P., Ratschan, S.: Effective topological degree computation based on interval arithmetic. Math. Comput. 84(293), 1265–1290 (2015)

    Google Scholar 

  15. Frommer, A., Hoxha, F., Lang, B.: Proving the existence of zeros using the topological degree and interval arithmetic. J. Comput. Appl. Math. 199(2), 397–402 (2007)

    Google Scholar 

  16. Gau, C.Y., Stadtherr, M.A.: Dynamic load balancing for parallel interval-Newton using message passing. Comput. Chem. Eng. 26(6), 811–825 (2002)

    Google Scholar 

  17. Herlihy, M., Shavit, N.: The Art of Multiprocessor Programming, Revised Reprint. Elsevier (2012)

    Google Scholar 

  18. Hoffler, T.: Advanced MPI: new features of MPI-3 (2016). http://htor.inf.ethz.ch/teaching/mpi_tutorials/speedup15/hoefler-advanced-mpi-speedup15.pdf

  19. Jaulin, L., Kieffer, M., Didrit, O., Walter, É.: Applied Interval Analysis. Springer, London (2001)

    Google Scholar 

  20. Kearfott, R.B.: Rigorous Global Search: Continuous Problems. Kluwer, Dordrecht (1996)

    Google Scholar 

  21. Kolev, L.V.: Some ideas towards global optimization of improved efficiency. In: GICOLAG Workshop, Wien, Austria, pp. 4–8 (2006)

    Google Scholar 

  22. Kozikowski, G.: Implementation of an OpenCL library for automatic differentiation. Bachelor’s thesis, ICCE WUT (2011). (under supervision of Bartłomiej J. Kubica). (in Polish)

    Google Scholar 

  23. Kozikowski, G., Papamanousakis, G., Yang, J.: Potential future exposure, modelling and accelerating on GPU and FPGA. In: Proceedings of the 8th Workshop on High Performance Computational Finance, WHPCF 2015, pp. 4:1–4:8. ACM, New York, NY, USA (2015). https://doi.org/10.1145/2830556.2830560

  24. Kreinovich, V., Bernat, A.: Parallel algorithms for interval computations: an introduction. Interval Comput. 3, 6–62 (1994)

    Google Scholar 

  25. Kubica, B.J.: Intel TBB as a tool for parallelization of an interval solver of nonlinear equations systems. Technical Report 09-02, ICCE WUT (2009)

    Google Scholar 

  26. Kubica, B.J.: Shared-memory parallelization of an interval equations systems solver—comparison of toos. Prace Naukowe Politechniki Warszawskiej. Elektronika 169, 121–128 (2009). KAEiOG 2009 (Konferencja Algorytmy Ewolucyjne i Optymalizacja Globalna) Proceedings

    Google Scholar 

  27. Kubica, B.J.: A class of problems that can be solved using interval algorithms. Computing 94, 271–280 (2012). SCAN 2010 (14th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics) Proceedings

    Google Scholar 

  28. Kubica, B.J.: Tuning the multithreaded interval method for solving underdetermined systems of nonlinear equations. In: PPAM 2011 (9th International Conference on Parallel Processing and Applied Mathematics) Proceedings. Lecture Notes in Computer Science, vol. 7204, pp. 467–476 (2012)

    Google Scholar 

  29. Kubica, B.J.: Presentation of a highly tuned multithreaded interval solver for underdetermined and well-determined nonlinear systems. Numer. Algorithms 70(4), 929–963 (2015). https://doi.org/10.1007/s11075-015-9980-y

  30. Kubica, B.J.: Preliminary experiments with an interval Model-Predictive-Control solver. In: PPAM 2015 Proceedings. Lecture Notes in Computer Science, vol. 9574, pp. 464–473 (2016)

    Google Scholar 

  31. Kubica, B.J.: Parallelization of a bound-consistency enforcing procedure and its application in solving nonlinear systems. J. Parallel Distrib. Comput. 107, 57–66 (2017). https://doi.org/10.1016/j.jpdc.2017.03.009

  32. Kubica, B.J.: Advanced interval tools for computing solutions of continuous games. Vychislennyie Tiehnologii (Computational Technologies) 23(1), 3–18 (2018)

    Google Scholar 

  33. Kubica, B.J., Woźniak, A.: An interval method for seeking the Nash equilibria of non-cooperative games. In: PPAM 2009 Proceedings. Lecture Notes in Computer Science, vol. 6068, pp. 446–455 (2010)

    Google Scholar 

  34. Kubica, B.J., Woźniak, A.: A multi-threaded interval algorithm for the Pareto-front computation in a multi-core environment. In: PARA 2008 Proceedings. Lecture Notes in Computer Science, vol. 6126/6127. Accepted for Publication (2010)

    Google Scholar 

  35. Kubica, B.J., Woźniak, A.: Optimization of the multi-threaded interval algorithm for the Pareto-set computation. J. Telecommun. Inf. Technol. 1, 70–75 (2010)

    Google Scholar 

  36. Kubica, B.J., Woźniak, A.: Interval methods for computing strong Nash equilibria of continuous games. Decis. Mak. Manuf. Serv. 9(1), 63–78 (2015). SING10 Proceedings

    Google Scholar 

  37. Loki: Loki C++ template library (2015). http://loki-lib.sourceforge.net

  38. Lyudvin, D.Y., Shary, S.P.: Testing implementations of pps-methods for interval linear systems. Reliab. Comput. 19(2), 176–196 (2013). SCAN 2012 Proceedings

    Google Scholar 

  39. Matocha, J., Camp, T.: A taxonomy of distributed termination detection algorithms. J. Syst. Softw. 43(3), 207–221 (1998)

    Google Scholar 

  40. Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. SIAM, Philadelphia (2009)

    Google Scholar 

  41. MPI: Message Passing Interface (2017). http://www.mpi-forum.org

  42. Nguyen, H.D.: Efficient implementation of interval matrix multiplication. Lecture Notes in Computer Science, vol. 7134, pp. 179–188 (2012)

    Google Scholar 

  43. Owczarek, B.: Parallel algorithms for computing eigenvalues of interval matrices. Master’s thesis, ICCE WUT (2015). (under supervision of Bartłomiej J. Kubica). (in Polish)

    Google Scholar 

  44. Skalna, I., Duda, J.: A study on vectorisation and paralellisation of the monotonicity approach. Lecture Notes in Computer Science (2016). Submitted

    Google Scholar 

  45. Szustak, Ł., Halbiniak, K., Kuczyński, Ł., Wróbel, J., Kulawik, A.: Porting and optimization of solidification application for CPU-MIC hybrid platforms. Int. J. High Perform. Comput. Appl. (2016). https://doi.org/10.1177/1094342016677740

    Google Scholar 

  46. Szustak, Ł., Rojek, K., Olas, T., Kuczynski, Ł., Halbiniak, K., Gepner, P.: Adaptation of MPDATA heterogeneous stencil computation to Intel Xeon Phi coprocessor. Sci. Program. 2015 (2015). https://doi.org/10.1155/2015/642705

  47. Szustak, Ł., Rojek, K., Wyrzykowski, R., Gepner, P.: Toward efficient distribution of MPDATA stencil computation on Intel MIC architecture. In: Proceedings of the 1st International Workshop on High-Performance Stencil Computations, pp. 51–56 (2014)

    Google Scholar 

  48. Ueberholz, P., Willems, P., Bull, M., Lang, B.: Non-blocking load balancing for branch-and-bound-type algorithms. In: PARA 2008 Proceedings. Lecture Notes in Computer Science. Accepted for Publication (2013)

    Google Scholar 

  49. Vyukov, D.: Non-intrusive MPSC node-based queue. http://www.1024cores.net/home/lock-free-algorithms/queues/non-intrusive-mpsc-node-based-queue. (web page; Accessed 2017)

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Bartłomiej Jacek Kubica .

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Kubica, B.J. (2019). Parallelization of B&BT Algorithms. In: Interval Methods for Solving Nonlinear Constraint Satisfaction, Optimization and Similar Problems. Studies in Computational Intelligence, vol 805. Springer, Cham. https://doi.org/10.1007/978-3-030-13795-3_7

Download citation

Publish with us

Policies and ethics