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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)

  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)

  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)

  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)

  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)

  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)

  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)

  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)

  10. 10.

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

  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)

  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)

  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)

  15. 15.

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

  16. 16.

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

  17. 17.

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

  18. 18.

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

  19. 19.

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

  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)

  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 (2020).

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