, Volume 80, Issue 5, pp 1579–1603 | Cite as

The \((1+1)\) Elitist Black-Box Complexity of LeadingOnes

  • Carola Doerr
  • Johannes Lengler
Part of the following topical collections:
  1. Special Issue on Genetic and Evolutionary Computation


One important goal of black-box complexity theory is the development of complexity models allowing to derive meaningful lower bounds for whole classes of randomized search heuristics. Complementing classical runtime analysis, black-box models help us to understand how algorithmic choices such as the population size, the variation operators, or the selection rules influence the optimization time. One example for such a result is the \(\varOmega (n \log n)\) lower bound for unary unbiased algorithms on functions with a unique global optimum (Lehre and Witt in Algorithmica 64:623–642, 2012), which tells us that higher arity operators or biased sampling strategies are needed when trying to beat this bound. In lack of analyzing techniques, such non-trivial lower bounds are very rare in the existing literature on black-box optimization and therefore remain to be one of the main challenges in black-box complexity theory. With this paper we contribute to our technical toolbox for lower bound computations by proposing a new type of information-theoretic argument. We regard the permutation- and bit-invariant version of LeadingOnes and prove that its \((1+1)\) elitist black-box complexity is \(\varOmega (n^2)\), a bound that is matched by \((1+1)\)-type evolutionary algorithms. The \((1+1)\) elitist complexity of LeadingOnes is thus considerably larger than its unrestricted one, which is known to be of order \(n\log \log n\) (Afshani et al. in Lecture notes in computer science, vol 8066, pp 1–11. Springer, New York, 2013). The \(\varOmega (n^2)\) lower bound does not rely on the fact that elitist black-box algorithms are not allowed to make use of absolute fitness values. In contrast, we show that even if absolute fitness values are revealed to the otherwise elitist algorithm, it cannot significantly profit from this additional information. Our result thus shows that for LeadingOnes the memory-restriction, together with the selection requirement, has a substantial impact on the best possible performance.


Black-box complexity Query complexity LeadingOnes Elitist algorithm Memory restriction Truncation selection Evolutionary algorithms 



This research benefited from the support of the “FMJH Program Gaspard Monge in optimization and operation research”, and from the support to this program from EDF (Électricité de France).


  1. 1.
    Afshani, P., Agrawal, M., Doerr, B., Doerr, C., Larsen, K.G., Mehlhorn, K.: The query complexity of finding a hidden permutation. Space-Efficient Data Structures, Streams, and Algorithms-Papers in Honor of J. Ian Munro on the Occasion of His 66th Birthday. Lecture Notes in Computer Science, vol. 8066, pp. 1–11. Springer, New York (2013)CrossRefGoogle Scholar
  2. 2.
    Böttcher, S., Doerr, B., Neumann, F.: Optimal fixed and adaptive mutation rates for the LeadingOnes problem. In: Proceedings of the 11th International Conference on Parallel Problem Solving from Nature (PPSN’10), pp. 1–10. Springer (2010)Google Scholar
  3. 3.
    Doerr, B., Doerr, C., Ebel, F.: Lessons from the black-box: Fast crossover-based genetic algorithms. In: Proceedings of Genetic and Evolutionary Computation Conference (GECCO’13), pp. 781–788. ACM (2013)Google Scholar
  4. 4.
    Doerr, B., Johannsen, D., Kötzing, T., Lehre, P.K., Wagner, M., Winzen, C.: Faster black-box algorithms through higher arity operators. In: Proceedings of Foundations of Genetic Algorithms (FOGA’11), pp. 163–172. ACM (2011)Google Scholar
  5. 5.
    Doerr, B., Winzen, C.: Black-box complexity: breaking the \(O(n \log n)\) barrier of LeadingOnes. Artificial Evolution (EA’11), Revised Selected Papers. Lecture Notes in Computer Science, vol. 7401, pp. 205–216. Springer, New York (2012)Google Scholar
  6. 6.
    Doerr, B., Winzen, C.: Playing mastermind with constant-size memory. Theory Comput. Syst. 55, 658–684 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Doerr, B., Winzen, C.: Ranking-based black-box complexity. Algorithmica 68, 571–609 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Doerr, C., Lengler, J.: Elitist black-box models: analyzing the impact of elitist selection on the performance of evolutionary algorithms. In: Proceedings of Genetic and Evolutionary Computation Conference (GECCO’15), pp. 839–846. ACM (2015)Google Scholar
  9. 9.
    Doerr, C., Lengler, J.: OneMax in black-box models with several restrictions. In: Proceedings of Genetic and Evolutionary Computation Conference (GECCO’15), pp. 1431–1438. ACM (2015)Google Scholar
  10. 10.
    Doerr, C., Lengler, J.: The \((1+1)\) elitist black-box complexity of LeadingOnes. In: Proceedings of Genetic and Evolutionary Computation Conference (GECCO’16), pp. 1131–1138. ACM (2016)Google Scholar
  11. 11.
    Droste, S., Jansen, T., Wegener, I.: On the analysis of the \((1+1)\) evolutionary algorithm. Theor. Comput. Sci. 276, 51–81 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Droste, S., Jansen, T., Wegener, I.: Upper and lower bounds for randomized search heuristics in black-box optimization. Theory Comput. Syst. 39, 525–544 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ladret, V.: Asymptotic hitting time for a simple evolutionary model of protein folding. J. Appl. Prob. 42, 39–51 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lehre, P.K., Witt, C.: Black-box search by unbiased variation. Algorithmica 64, 623–642 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press, Cambridge (1995)CrossRefzbMATHGoogle Scholar
  16. 16.
    Mühlenbein, H.: How genetic algorithms really work: mutation and hillclimbing. In: Proceedings of the 2nd International Conference on Parallel Problem Solving from Nature (PPSN’92), pp. 15–26. Elsevier (1992)Google Scholar
  17. 17.
    Rudolph, G.: Convergence Properties of Evolutionary Algorithms. Verlag Dr. Kovač, Hamburg (1997)zbMATHGoogle Scholar
  18. 18.
    Sudholt, D.: A new method for lower bounds on the running time of evolutionary algorithms. IEEE Trans. Evol. Comput. 17, 418–435 (2013)CrossRefGoogle Scholar
  19. 19.
    Yao, A.C.C.: Probabilistic computations: toward a unified measure of complexity. In: Proceedings of Foundations of Computer Science (FOCS’77), pp. 222–227. IEEE (1977)Google Scholar

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© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.CNRS and Sorbonne Universités, UPMC Univ Paris 06, CNRS, LIP6 UMR 7606ParisFrance
  2. 2.Institute for Theoretical Computer ScienceETH ZürichZürichSwitzerland

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