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Make Evolutionary Multiobjective Algorithms Scale Better with Advanced Data Structures: Van Emde Boas Tree for Non-dominated Sorting

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Evolutionary Multi-Criterion Optimization (EMO 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11411))

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Abstract

We improve the worst-case time complexity of non-dominated sorting, an operation frequently used in evolutionary multiobjective algorithms, to \(O(n \cdot (\log n)^{k-2} \log \log n)\), where n is the number of solutions, k is the number of objectives, and the random-access memory computation model is assumed. This improvement was possible thanks to the van Emde Boas tree, an “advanced” data structure which stores a set of non-negative integers less than n and supports many queries in \(O(\log \log n)\). This is not only a theoretical improvement, as we also provide an efficient implementation of the van Emde Boas tree, which resulted in a competitive algorithm that scales better than other algorithms when n grows, at least for small numbers of objectives greater than two.

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Notes

  1. 1.

    Available at https://github.com/mbuzdalov/non-dominated-sorting/tree/v0.2.

  2. 2.

    https://github.com/mbuzdalov/non-dominated-sorting/tree/v0.2/implementations/src/main/java/ru/ifmo/nds/util/veb.

References

  1. Bouter, A., Alderliesten, T., Witteveen, C., Bosman, P.A.N.: Exploiting linkage information in real-valued optimization with the real-valued gene-pool optimal mixing evolutionary algorithm. In: Proceedings of Genetic and Evolutionary Computation Conference, pp. 705–712 (2017)

    Google Scholar 

  2. Bringmann, K., Friedrich, T.: Approximating the least hypervolume contributor: NP-hard in general, but fast in practice. In: Ehrgott, M., Fonseca, C.M., Gandibleux, X., Hao, J.-K., Sevaux, M. (eds.) EMO 2009. LNCS, vol. 5467, pp. 6–20. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-01020-0_6

    Chapter  Google Scholar 

  3. Buchsbaum, A.L., Goodrich, M.T.: Three-dimensional layers of maxima. Algorithmica 39, 275–286 (2004)

    Article  MathSciNet  Google Scholar 

  4. Buzdalov, M.: Generalized offline orthant search: one code for many problems in multiobjective optimization. In: Proceedings of Genetic and Evolutionary Computation Conference, pp. 593–600 (2018)

    Google Scholar 

  5. Buzdalov, M., Shalyto, A.: A provably asymptotically fast version of the generalized jensen algorithm for non-dominated sorting. In: Bartz-Beielstein, T., Branke, J., Filipič, B., Smith, J. (eds.) PPSN 2014. LNCS, vol. 8672, pp. 528–537. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-10762-2_52

    Chapter  Google Scholar 

  6. Chicano, F., Whitley, D., Ochoa, G., Tinós, R.: Optimizing one million variable NK landscapes by hybridizing deterministic recombination and local search. In: Proceedings of Genetic and Evolutionary Computation Conference, pp. 753–760 (2017)

    Google Scholar 

  7. Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn. MIT Press, Cambridge (2001)

    MATH  Google Scholar 

  8. Deb, K., Jain, H.: An evolutionary many-objective optimization algorithm using reference-point-based nondominated sorting approach, part I: solving problems with box constraints. IEEE Trans. Evol. Comput. 18(4), 577–601 (2013)

    Article  Google Scholar 

  9. Deb, K., Myburgh, C.: A population-based fast algorithm for a billion-dimensional resource allocation problem with integer variables. Eur. J. Oper. Res. 261(2), 460–474 (2017)

    Article  MathSciNet  Google Scholar 

  10. Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A fast and elitist multi-objective genetic algorithm: NSGA-II. IEEE Trans. Evol. Comput. 6(2), 182–197 (2002)

    Article  Google Scholar 

  11. Drozdik, M., Akimoto, Y., Aguirre, H., Tanaka, K.: Computational cost reduction of nondominated sorting using the m-front. IEEE Trans. Evol. Comput. 19(5), 659–678 (2015)

    Article  Google Scholar 

  12. van Emde Boas, P., Kaas, R., Zijlstra, E.: Design and implementation of an efficient priority queue. Math. Syst. Theory 10, 99–127 (1976)

    Article  MathSciNet  Google Scholar 

  13. van Emde Boas, P.: Preserving order in a forest in less than logarithmic time. In: Proceedings of the Annual Symposium on Foundations of Computer Science, pp. 75–84 (1975)

    Google Scholar 

  14. Fortin, F.A., Grenier, S., Parizeau, M.: Generalizing the improved run-time complexity algorithm for non-dominated sorting. In: Proceedings of Genetic and Evolutionary Computation Conference, pp. 615–622. ACM (2013)

    Google Scholar 

  15. Glasmachers, T.: A fast incremental BSP tree archive for non-dominated points. In: Trautmann, H., et al. (eds.) EMO 2017. LNCS, vol. 10173, pp. 252–266. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-54157-0_18

    Chapter  Google Scholar 

  16. Gupta, S., Tan, G.: A scalable parallel implementation of evolutionary algorithms for multi-objective optimization on GPUs. In: Proceedings of IEEE Congress on Evolutionary Computation, pp. 1567–1574 (2015)

    Google Scholar 

  17. Gustavsson, P., Syberfeldt, A.: A new algorithm using the non-dominated tree to improve non-dominated sorting. Evol. Comput. 26(1), 89–116 (2018)

    Article  Google Scholar 

  18. Jensen, M.T.: Reducing the run-time complexity of multiobjective EAs: the NSGA-II and other algorithms. IEEE Trans. Evol. Comput. 7(5), 503–515 (2003)

    Article  Google Scholar 

  19. Kung, H.T., Luccio, F., Preparata, F.P.: On finding the maxima of a set of vectors. J. ACM 22(4), 469–476 (1975)

    Article  MathSciNet  Google Scholar 

  20. Markina, M., Buzdalov, M.: Towards large-scale multiobjective optimisation with a hybrid algorithm for non-dominated sorting. In: Auger, A., Fonseca, C.M., Lourenço, N., Machado, P., Paquete, L., Whitley, D. (eds.) PPSN 2018. LNCS, vol. 11101, pp. 347–358. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-99253-2_28

    Chapter  Google Scholar 

  21. Nekrich, Y.: A fast algorithm for three-dimensional layers of maxima problem. In: Dehne, F., Iacono, J., Sack, J.-R. (eds.) WADS 2011. LNCS, vol. 6844, pp. 607–618. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22300-6_51

    Chapter  Google Scholar 

  22. Roy, P.C., Deb, K., Islam, M.M.: An efficient nondominated sorting algorithm for large number of fronts. IEEE Trans. Cybern. (2018). https://doi.org/10.1109/TCYB.2017.2789158

  23. Roy, P.C., Islam, M.M., Deb, K.: Best order sort: a new algorithm to non-dominated sorting for evolutionary multi-objective optimization. In: Proceedings of Genetic and Evolutionary Computation Conference Companion, pp. 1113–1120 (2016)

    Google Scholar 

  24. Srinivas, N., Deb, K.: Multiobjective optimization using nondominated sorting in genetic algorithms. Evol. Comput. 2(3), 221–248 (1994)

    Article  Google Scholar 

  25. Zhang, X., Tian, Y., Cheng, R., Jin, Y.: An efficient approach to nondominated sorting for evolutionary multiobjective optimization. IEEE Trans. Evol. Comput. 19(2), 201–213 (2015)

    Article  Google Scholar 

  26. Zitzler, E., Thiele, L.: Multiobjective evolutionary algorithms: a comparative case study and the Strength Pareto approach. IEEE Trans. Evol. Comput. 3(4), 257–271 (1999)

    Article  Google Scholar 

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Acknowledgment

The research is financially supported by The Russian Science Foundation, Agreement No. 17-71-30029 with co-financing of Bank Saint Petersburg.

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Authors and Affiliations

  1. ITMO University, 49 Kronverkskiy Avenue, Saint Petersburg, 197101, Russia

    Maxim Buzdalov

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  1. Maxim Buzdalov
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Correspondence to Maxim Buzdalov .

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Editors and Affiliations

  1. Michigan State University, East Lansing, MI, USA

    Kalyanmoy Deb

  2. Michigan State University, East Lansing, MI, USA

    Erik Goodman

  3. CINVESTAV-IPN, Mexico, Mexico

    Carlos A. Coello Coello

  4. University of Wuppertal, Wuppertal, Germany

    Kathrin Klamroth

  5. University of Jyvaskyla, Jyväskylä, Finland

    Kaisa Miettinen

  6. Otto von Guericke University Magdeburg, Magdeburg, Germany

    Sanaz Mostaghim

  7. Cornell University, Ithaca, NY, USA

    Patrick Reed

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Buzdalov, M. (2019). Make Evolutionary Multiobjective Algorithms Scale Better with Advanced Data Structures: Van Emde Boas Tree for Non-dominated Sorting. In: Deb, K., et al. Evolutionary Multi-Criterion Optimization. EMO 2019. Lecture Notes in Computer Science(), vol 11411. Springer, Cham. https://doi.org/10.1007/978-3-030-12598-1_6

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  • DOI: https://doi.org/10.1007/978-3-030-12598-1_6

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-12597-4

  • Online ISBN: 978-3-030-12598-1

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