Skip to main content

Advertisement

Springer Nature Link
Log in

Estimating parallel runtimes for randomized algorithms in constraint solving

  • Published:
Journal of Heuristics Aims and scope Submit manuscript

Abstract

This paper presents a detailed analysis of the scalability and parallelization of Local Search algorithms for constraint-based and SAT (Boolean satisfiability) solvers. We propose a framework to estimate the parallel performance of a given algorithm by analyzing the runtime behavior of its sequential version. Indeed, by approximating the runtime distribution of the sequential process with statistical methods, the runtime behavior of the parallel process can be predicted by a model based on order statistics. We apply this approach to study the parallel performance of a constraint-based Local Search solver (Adaptive Search), two SAT Local Search solvers (namely Sparrow and CCASAT), and a propagation-based constraint solver (Gecode, with a random labeling heuristic). We compare the performance predicted by our model to actual parallel implementations of those methods using up to 384 processes. We show that the model is accurate and predicts performance close to the empirical data. Moreover, as we study different types of problems, we observe that the experimented solvers exhibit different behaviors and that their runtime distributions can be approximated by two types of distributions: exponential (shifted and non-shifted) and lognormal. Our results show that the proposed framework estimates the runtime of the parallel algorithm with an average discrepancy of 21 % w.r.t. the empirical data across all the experiments with the maximum allowed number of processors for each technique.

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

Access this article

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

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

Notes

  1. The SAT community usually refer to the multi-walk framework as portfolio algorithms.

  2. http://www.csplib.org.

  3. The Notation is the same as in Sect. 3.

  4. We also experimented with industrial instances but the tested LS solvers performed poorly and a very limited number of instances could be solved with a reasonable time limit.

  5. http://baldur.iti.kit.edu/SAT-Challenge-2012/results.html.

  6. www.gecode.org.

  7. We use AFC, the Gecode implementation of wdeg.

  8. We use the exponential distribution in lieu of the shifted exponential because the min value is negligible w.r.t. to the mean \(0.008\) versus \(58.08\).

  9. For experiments with the Adaptive Search we use report values with 32, 64, 128, and 256 cores.

  10. For empirical we mean the empirical runtime distribution computed with the actual empirical results of the experiments with 48 and 192 processors.

  11. We have performed the same analysis with other instances and observed a similar tendency, i.e., increasing the number of processors degrades the performance of the empirical estimated probability distribution.

References

  • Aida, K., Osumi, T.: A case study in running a parallel branch and bound application on the grid. In: SAINT ’05: Proceedings of the The 2005 Symposium on Applications and the Internet, pp. 164–173. IEEE Computer Society, Washington, DC (2005)

  • Aiex, R., Resende, M., Ribeiro, C.: TTT plots: a perl program to create time-to-target plots. Optim. Lett. 1, 355–366 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Aiex, R.M., Resende, M.G.C., Ribeiro, C.C.: Probability distribution of solution time in grasp: an experimental investigation. J. Heuristics 8(3), 343–373 (2002)

  • Alba, E.: Special issue on new advances on parallel meta-heuristics for complex problems. J. Heuristics 10(3), 239–380 (2004)

    Article  Google Scholar 

  • Ansótegui, C., Sellmann, M., Tierney, K.: A gender-based genetic algorithm for the automatic configuration of algorithms. In: Gent, I.P. (ed.) 15th International Conference on Principles and Practice of Constraint Programming, LNCS, vol. 5732, pp. 142–157. Springer, Lisbon (2009)

  • Arbelaez, A., Hamadi, Y.: Improving parallel local search for SAT. In: Coello, C.A.C. (ed.) Learning and Intelligent Optimization, 5th International Conference, LION’11, LNCS, vol. 6683, pp. 46–60. Springer, Lisbon (2011)

  • Arbelaez, A., Codognet, P.: Massivelly parallel local search for SAT. In: ICTAI’12, pp. 57–64. IEEE Computer Society, Athens (2012)

  • Arbelaez. A., Codognet, P.: From sequential to parallel local search for SAT. In: 13th European Conference on Evolutionary Computation in Combinatorial Optimisation (EvoCOP’13) (2013)

  • Arbelaez, A., Truchet, C., Codognet, P.: Using sequential runtime distributions for the parallel speedup prediction of SAT local search. J. Theory Pract. Log. Program. 13, 625–639 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Babai, L.: Monte-Carlo algorithms in graph isomorphism testing. Research Report D.M.S. No. 79–10, Université de Montréal (1979)

  • Balint, A., Fröhlich, A.: Improving stochastic local search for SAT with a new probability distribution. In: Strichman, O., Szeider, S. (eds.) SAT’10, LNCS, vol. 6175, pp. 10–15. Springer, Edinburgh (2010)

    Google Scholar 

  • Bolze, R., et al.: Grid’5000: a large scale and highly reconfigurable experimental grid testbed. Int. J. High Perform. Comput. Appl. 20(4), 481–494 (2006)

    Article  Google Scholar 

  • Bordeaux, L., Hamadi, Y., Samulowitz, H.: Experiments with massively parallel constraint solving. In: Boutilier, C. (ed.) Proceedings of IJCAI 2009, 21st International Joint Conference on Artificial Intelligence, pp. 443–448 (2009)

  • Boussemart, F., Hemery, F., Lecoutre, C., Sais, L.: Boosting systematic search by weighting constraints. In: ECAI’ 2004, pp. 146–150 (2004)

  • Cai, S., Luo, C., Su, K.: CCASAT: solver description. In: SAT Challenge 2012: Solver and Benchmark Descriptions, vol. B-2012-2. Department of Computer Science Series of Publications B, University of Helsinki, pp. 13–14 (2012)

  • Caniou, Y., Diaz, D., Richoux, F., Codognet, P., Abreu, S.: Performance analysis of parallel constraint-based local search. In: PPoPP 2012, 17th ACM SIGPLAN Symposium on Principles and Practice of Parallel Programming, ACM Press, New Orleans, Poster Paper (2012)

  • Caromel. D., di Costanzo, A., Baduel, L., Matsuoka, S.: Grid’BnB: a parallel branch and bound framework for grids. In: Proceedings of HiPC’07, 14th International Conference on High Performance Computing, pp, 566–579. Springer, Berlin (2007)

  • Chu, G., Schulte, C., Stuckey, P.J.: Confidence-based work stealing in parallel constraint programming. In: Gent, I.P. (ed.) CP 2009, 15th International Conference on Principles and Practice of Constraint Programming, Springer, Lecture Notes in Computer Science, vol. 5732, pp. 226–241 (2009)

  • Codognet, P., Diaz, D.: Yet another local search method for constraint solving. In: Proceedings of SAGA’01, pp. 73–90. Springer, New York (2001)

  • Codognet, P., Diaz, D.: An efficient library for solving CSP with local search. In: Ibaraki, T. (ed.) MIC’03, 5th International Conference on Metaheuristics (2003)

  • Crainic, T., Toulouse, M.: Special Issue on parallel meta-heuristics. J. Heuristics 8(3), 247–388 (2002)

    Article  Google Scholar 

  • Crainic, T.G., Gendreau, M., Hansen, P., Mladenovic, N.: Cooperative parallel variable neighborhood search for the p-median. J. Heuristics 10(3), 293–314 (2004)

    Article  Google Scholar 

  • David, H., Nagaraja, H.: Order Statistics. Wiley Series in Probability and Mathematical Statistics. Probability and Mathematical Statistics. Wiley, New York (2003)

    Google Scholar 

  • de Kergommeaux, J.C., Codognet, P.: Parallel logic programming systems. ACM Comput. Surv. 26(3), 295–336 (1994)

    Article  Google Scholar 

  • Diaz, D., Richoux, F., Caniou, Y., Codognet, P., Abreu, S.: Parallel local search for the costas array problem. In: IEEE Workshop on New Trends in Parallel Computing and Optimization (PC012), in Conjunction with IPDPS 2012, IEEE Press, Shanghai (2012)

  • Eadie, W.: Statistical Methods in Experimental Physics. North-Holland Publishing Co., Amsterdam (1971)

    MATH  Google Scholar 

  • Gendron, B., Crainic, T.: Parallel branch-and-bound algorithms: survey and synthesis. Oper. Res. 42(6), 1042–1066 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  • Gent, I. P., Walsh, T.: The SAT phase transition. In: ECAI’94, pp. 105–109 (1994)

  • Gomes, C.P., Selman, B.: Algorithm portfolios. Artif. Intell. 126(1–2), 43–62 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Gomes, C.P., Selman, B., Crato, N., Kautz, H.A.: Heavy-tailed phenomena in satisfiability and constraint satisfaction problems. J. Autom. Reason. 24(1/2), 67–100 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  • Gonzalez, T. (ed.): Handbook of Approximation Algorithms and Metaheuristics. Chapman and Hall/CRC, Boca Raton (2007)

    MATH  Google Scholar 

  • Haralick, R.M., Elliott, G.L.: Increasing tree search efficiency for constraint satisfaction problems. In: IJCAI, pp. 356–364 (1979)

  • Hoos, H., Stütze, T.: Stochastic Local Search: Foundations and Applications. Morgan Kaufmann, San Francisco (2005)

    Google Scholar 

  • Hoos, H.H., Stützle, T.: Towards a characterisation of the behaviour of stochastic local search algorithms for SAT. Artif. Intell. 112(1–2), 213–232 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • Hutter, F., Hoos, H.H., Leyton-Brown, K., Stützle, T.: ParamILS: an automatic algorithm configuration framework. J. Artif. Intell. Res. 36, 267–306 (2009)

    MATH  Google Scholar 

  • Ibaraki, T., Nonobe, K., Yagiura, M. (eds.): Metaheuristics: Progress as Real Problem Solvers. Springer, New York (2005)

    Google Scholar 

  • Kroc, L., Sabharwal, A., Selman, B.: An empirical study of optimal noise and runtime distributions in local search. In: Strichman, O., Szeider, S. (eds.) SAT’10, LNCS, vol. 6175, pp. 346–351. Springer, Edinburgh (2010)

  • Luby, M., Sinclair, A., Zuckerman, D.: Optimal speedup of Las Vegas algorithms. Inf. Process. Lett. 47, 173–180 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  • Michel, L., See, A., Van Hentenryck, P.: Distributed constraint-based local search. In: Benhamou, F. (ed.) CP’06, 12th International Conference on Principles and Practice of Constraint Programming, Lecture Notes in Computer Science, pp. 344–358. Springer, New York (2006)

  • Michel, L., See, A., Van Hentenryck, P.: Parallelizing constraint programs transparently. In: Bessiere, C. (ed.) CP’07, 13th International Conference on Principles and Practice of Constraint Programming, Lecture Notes in Computer Science, pp. 514–528. Springer, New York (2007)

  • Michel, L., See, A., Van Hentenryck, P.: Parallel and distributed local search in COMET. Comput. Oper. Res. 36, 2357–2375 (2009)

    Article  MATH  Google Scholar 

  • Moisan, T., Gaudreault, J., Quimper, C.: Parallel discrepancy-based search. In: Proceedings of the Principles and Practice of Constraint Programming—19th International Conference, CP 2013, Uppsala, Sweden, September 16–20, 2013, pp. 30–46 (2013). doi:10.1007/978-3-642-40627-0_6

  • Nadarajah, S.: Explicit expressions for moments of order statistics. Stat. Probab. Lett. 78(2), 196–205 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Pardalos, P.M., Pitsoulis, L.S., Mavridou, T.D., Resende, M.G.C.: Parallel search for combinatorial optimization: genetic algorithms, simulated annealing, tabu search and GRASP. In: Parallel Algorithms for Irregularly Structured Problems (IRREGULAR), pp. 317–331 (1995)

  • Pardalos, P.M., Pitsoulis, L.S., Resende, M.G.C.: A parallel grasp for MAX-SAT problems. In: Wasniewski, J., Dongarra, J., Madsen, K., Olesen, D. (eds.) 3rd International Workshop on Applied Parallel Computing, Industrial Computation and Optimization, LNCS. Springer, Lyngby (1996)

  • Perron, L.: Search procedures and parallelism in constraint programming. In: CP’99, 5th International Conference on Principles and Practice of Constraint Programming, Lecture Notes in Computer Science, pp. 346–360. Springer, Berlin (1999)

  • Pham, D.N., Gretton, C.: gNovelty+. In: Solver Description, SAT Competition 2007 (2007)

  • Régin, J.C., Rezgui, M., Malapert, A.: Embarrassingly parallel search. In: Schulte, C. (ed.) Proceedings of CP’2013, 19th International Conference on Principles and Practice of Constraint Programming, Lecture Notes in Computer Science, vol. 8124, pp. 596–610. Springer, Berlin (2013)

  • Ribeiro, C., Rosseti, I., Vallejos, R.: Exploiting run time distributions to compare sequential and parallel stochastic local search algorithms. J. Glob. Optim. 54, 405–429 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Shylo, O.V., Middelkoop, T., Pardalos, P.M.: Restart strategies in optimization: parallel and serial cases. Parallel Comput. 37(1), 60–68 (2011). doi:10.1016/j.parco.2010.08.004

    Article  MathSciNet  MATH  Google Scholar 

  • Truchet, C., Richoux, F., Codognet, P.: Prediction of parallel speed-ups for las vegas algorithms. In: Dongarra, J., Robert, Y. (eds.) Proceedings of ICPP-2013, 42nd International Conference on Parallel Processing. IEEE Press, Lyon (2013)

  • Van Hentenryck, P.: Parallel constraint satisfaction in logic programming: preliminary results of CHIP within PEPsys. In: ICLP’89, International Conference on Logic Programming, MIT Press, Lisbon, pp. 165–180 (1989)

  • Van Hentenryck, P., Michel, L.: Constraint-Based Local Search. The MIT Press, Cambridge (2005)

    MATH  Google Scholar 

  • Van Luong, T., Melab, N., Talbi, E.G.: Local search algorithms on graphics processing units. In: Evolutionary Computation in Combinatorial Optimization, LNCS 6022, pp. 264–275. Springer, Heidelberg (2010)

  • Verhoeven, M., Aarts, E.: Parallel local search. J. Heuristics 1(1), 43–65 (1995)

    Article  MATH  Google Scholar 

  • Wolfram, S.: The Mathematica Book, 5th edn. Wolfram Media, Champaign (2003). URL http://reference.wolfram.com

  • Xie, F., Davenport, A.J.: Massively parallel constraint programming for supercomputers: challenges and initial results. In: CPAIOR’10, 7th International Conference on the Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems, Lecture Notes in Computer Science, vol. 6140, pp. 334–338. Springer, Berlin (2010)

Download references

Acknowledgments

We acknowledge that some results in this paper have been achieved using the Grid’5000 experimental testbed, being developed under the INRIA ALADDIN development action with support from CNRS, RENATER and several universities as well as other funding bodies. We acknowledge that some other results in this paper have been achieved using the PRACE Research Infrastructure resource JUGENE based in Germany at Jülich Supercomputing Centre. The authors would like to thank the anonymous reviewers for their comments and suggestions that helped to improve the paper. We would like to thank the anonymous reviewers for suggesting to compare our method versus the empirical method depicted in Sect. 7.

Author information

Authors and Affiliations

  1. LINA, UMR 6241, University of Nantes, Nantes, France

    Charlotte Truchet & Florian Richoux

  2. INSIGHT Centre for Data Analytics, University College Cork, Cork, Ireland

    Alejandro Arbelaez

  3. JFLI - CNRS / UPMC, University of Tokyo, Tokyo, Japan

    Philippe Codognet

Authors
  1. Charlotte Truchet
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Alejandro Arbelaez
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Florian Richoux
    View author publications

    You can also search for this author inPubMed Google Scholar

  4. Philippe Codognet
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Charlotte Truchet.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Truchet, C., Arbelaez, A., Richoux, F. et al. Estimating parallel runtimes for randomized algorithms in constraint solving. J Heuristics 22, 613–648 (2016). https://doi.org/10.1007/s10732-015-9292-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10732-015-9292-3

Keywords