A Provably Asymptotically Fast Version of the Generalized Jensen Algorithm for Non-dominated Sorting

  • Maxim Buzdalov
  • Anatoly Shalyto
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8672)


The non-dominated sorting algorithm by Jensen, generalized by Fortin et al to handle the cases of equal objective values, has the running time complexity of O(N log K − 1 N) in the general case. Here N is the number of points, K is the number of objectives and K is thought to be a constant when N varies. However, the complexity was not proven to be the same in the worst case.

A slightly modified version of the algorithm is presented, for which it is proven that its worst-case running time complexity is O(N log K − 1 N).


Non-dominated sorting worst-case running time complexity multi-objective optimization 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Source code for the implementation (a part of this paper),
  2. 2.
    Abbass, H.A., Sarker, R., Newton, C.: PDE: A Pareto Frontier Differential Evolution Approach for Multiobjective Optimization Problems. In: Proceedings of the Congress on Evolutionary Computation, pp. 971–978. IEEE Press (2001)Google Scholar
  3. 3.
    Bentley, J.L.: Multidimensional Divide-and-conquer. Communications of ACM 23(4), 214–229 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Corne, D.W., Jerram, N.R., Knowles, J.D., Oates, M.J.: PESA-II: Region-based Selection in Evolutionary Multiobjective Optimization. In: Proceedings of Genetic and Evolutionary Computation Conference, pp. 283–290. Morgan Kaufmann Publishers (2001)Google Scholar
  5. 5.
    Corne, D.W., Knowles, J.D., Oates, M.J.: The Pareto Envelope-based Selection Algorithm for Multiobjective Optimization. In: Deb, K., Rudolph, G., Lutton, E., Merelo, J.J., Schoenauer, M., Schwefel, H.-P., Yao, X. (eds.) PPSN VI. LNCS, vol. 1917, pp. 839–848. Springer, Heidelberg (2000)Google Scholar
  6. 6.
    Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A Fast Elitist Multi-Objective Genetic Algorithm: NSGA-II. Transactions on Evolutionary Computation 6, 182–197 (2000)CrossRefGoogle Scholar
  7. 7.
    Fortin, F.A., Grenier, S., Parizeau, M.: Generalizing the Improved Run-time Complexity Algorithm for Non-dominated Sorting. In: Proceeding of the Fifteenth Annual Conference on Genetic and Evolutionary Computation Conference, GECCO 2013, pp. 615–622. ACM (2013)Google Scholar
  8. 8.
    Jensen, M.T.: Reducing the Run-time Complexity of Multiobjective EAs: The NSGA-II and Other Algorithms. Transactions on Evolutionary Computation 7(5), 503–515 (2003)CrossRefGoogle Scholar
  9. 9.
    Knowles, J.D., Corne, D.W.: Approximating the Nondominated Front Using the Pareto Archived Evolution Strategy. Evolutionary Computation 8(2), 149–172 (2000)CrossRefGoogle Scholar
  10. 10.
    Kung, H.T., Luccio, F., Preparata, F.P.: On finding the maxima of a set of vectors. Journal of ACM 22(4), 469–476 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Zitzler, E., Laumanns, M., Thiele, L.: SPEA2: Improving the Strength Pareto Evolutionary Algorithm for Multiobjective Optimization. In: Proceedings of the EUROGEN 2001 Conference, pp. 95–100 (2001)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Maxim Buzdalov
    • 1
  • Anatoly Shalyto
    • 1
  1. 1.ITMO UniversitySaint-PetersburgRussia

Personalised recommendations