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Estimating parallel runtimes for randomized algorithms in constraint solving

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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.

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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.

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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.

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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

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  1. Charlotte Truchet
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Correspondence to Charlotte Truchet.

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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

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