The scalability analysis of a parallel tree search algorithm


Increasing the number of computational cores is a primary way of achieving the high performance of contemporary supercomputers. However, developing parallel applications capable to harness the enormous amount of cores is a challenging task. It is very important to understand the principle limitations of the scalability of parallel applications imposed by the algorithm’s structure. The tree search addressed in this paper has an irregular structure unknown prior to computations. That is why such algorithms are challenging for parallel implementation especially on distributed memory systems. In this paper, we propose a parallel tree search algorithm aimed at distributed memory parallel computers. For this parallel algorithm, we analyze its scalability and show that it is close to the theoretical maximum.

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  1. 1.

    Baldwin, A., Asaithambi, A.: An efficient method for parallel interval global optimization. In: 2011 International Conference on High Performance Computing and Simulation (HPCS), pp. 317–321. IEEE (2011)

  2. 2.

    Barkalov, K., Gergel, V.: Parallel global optimization on gpu. J. Global Optim. 66(1), 3–20 (2016)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Bhatt, S., Greenberg, D., Leighton, T., Liu, P.: Tight bounds for on-line tree embeddings. SIAM J. Comput. 29(2), 474–491 (1999)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Casado, L.G., Martinez, J.A., García, I., Hendrix, E.M.T.: Branch-and-bound interval global optimization on shared memory multiprocessors. Optim. Methods Softw. 23(5), 689–701 (2008)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Evtushenko, Y., Posypkin, M., Rybak, L., Turkin, A.: Approximating a solution set of nonlinear inequalities. J. Global Optim. 71(1), 129–145 (2018)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Evtushenko, Y., Posypkin, M., Sigal, I.: A framework for parallel large-scale global optimization. Comput. Sci. Res. Dev. 23(3–4), 211–215 (2009)

    Article  Google Scholar 

  7. 7.

    Gergel, V.P., Sergeyev, Y.D.: Sequential and parallel algorithms for global minimizing functions with lipschitzian derivatives. Comput. Math. Appl. 37(4–5), 163–179 (1999)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Gmys, J., Leroy, R., Mezmaz, M., Melab, N., Tuyttens, D.: Work stealing with private integer-vector-matrix data structure for multi-core branch-and-bound algorithms. Concurr. Comput. Pract. Exp. 28(18), 4463–4484 (2016)

    Article  Google Scholar 

  9. 9.

    Karp, R.M., Zhang, Y.: Randomized parallel algorithms for backtrack search and branch-and-bound computation. J. ACM (JACM) 40(3), 765–789 (1993)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Knuth, D.E.: Estimating the efficiency of backtrack programs. Math. Comput. 29(129), 122–136 (1975)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Kolpakov, R., Posypkin, M.: The scalability analysis of a parallel tree search algorithm. In: Optimization and Applications. Proceedings of 9th International Conference OPTIMA 2018, Petrovac, Montenegro, October 1–5, 2018, pp. 186–201. Springer (2019)

  12. 12.

    Kolpakov, R.M., Posypkin, M.A.: Upper and lower bounds for the complexity of the branch and bound method for the knapsack problem. Discrete Math. Appl. 20(1), 95–112 (2010)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Kolpakov, R.M., Posypkin, M.A.: Estimating the computational complexity of one variant of parallel realization of the branch-and-bound method for the knapsack problem. J. Comput. Syst. Sci. Int. 50(5), 756 (2011)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Kolpakov, R.M., Posypkin, M.A., Sigal, I.K.: On a lower bound on the computational complexity of a parallel implementation of the branch-and-bound method. Autom. Remote Control 71(10), 2152–2161 (2010)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Roucairol, C.: A parallel branch and bound algorithm for the quadratic assignment problem. Discrete Appl. Math. 18(2), 211–225 (1987)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Sergeyev, Y., Grishagin, V.: Parallel asynchronous global search and the nested optimization scheme. J. Comput. Anal. Appl. 3(2), 123–145 (2001)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Sergeyev, Y.D., Grishagin, V.A.: Sequential and parallel algorithms for global optimization. Optim. Methods Softw. 3(1–3), 111–124 (1994)

    Article  Google Scholar 

  18. 18.

    Strongin, R., Sergeyev, Y.: Global multidimensional optimization on parallel computer. Parallel Comput. 18(11), 1259–1273 (1992)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Strongin, R., Sergeyev, Y.: Global optimization: fractal approach and non-redundant parallelism. J. Global Optim. 27(1), 25–50 (2003)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Vu, T.T., Derbel, B.: Parallel branch-and-bound in multi-core multi-cpu multi-gpu heterogeneous environments. Future Gener. Comput. Syst. 56, 95–109 (2016)

    Article  Google Scholar 

  21. 21.

    Wu, I.C., Kung, H.T.: Communication complexity for parallel divide-and-conquer. In: Proceedings 32nd Annual Symposium on Foundations of Computer Science, 1991, pp. 151–162. IEEE (1991)

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This paper contains a full proof of results claimed in [11]. This work is partially supported by Russian Foundation for Fundamental Research (Grant 18-07-00566).

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Correspondence to Roman Kolpakov.

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Kolpakov, R., Posypkin, M. The scalability analysis of a parallel tree search algorithm. Optim Lett 14, 2211–2226 (2020).

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  • Parallel scalability
  • Parallel efficiency
  • Complexity analysis
  • Parallel tree search
  • Global optimization