Algorithmica

, Volume 59, Issue 3, pp 369–386 | Cite as

Simplified Drift Analysis for Proving Lower Bounds in Evolutionary Computation

Article

Abstract

Drift analysis is a powerful tool used to bound the optimization time of evolutionary algorithms (EAs). Various previous works apply a drift theorem going back to Hajek in order to show exponential lower bounds on the optimization time of EAs. However, this drift theorem is tedious to read and to apply since it requires two bounds on the moment-generating (exponential) function of the drift. A recent work identifies a specialization of this drift theorem that is much easier to apply. Nevertheless, it is not as simple and not as general as possible. The present paper picks up Hajek’s line of thought to prove a drift theorem that is very easy to use in evolutionary computation. Only two conditions have to be verified, one of which holds for virtually all EAs with standard mutation. The other condition is a bound on what is really relevant, the drift. Applications show how previous analyses involving the complicated theorem can be redone in a much simpler and clearer way. In some cases even improved results may be achieved. Therefore, the simplified theorem is also a didactical contribution to the runtime analysis of EAs.

Keywords

Randomized search heuristics Evolutionary algorithms Computational complexity Runtime analysis Drift analysis 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Droste, S., Jansen, T., Wegener, I.: Upper and lower bounds for randomized search heuristics in black-box optimization. Theory Comput. Syst. 39(4), 525–544 (2006) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Friedrich, T., Oliveto, P.S., Sudholt, D., Witt, C.: Theoretical analysis of diversity mechanisms for global exploration. In: Proc. of GECCO ’08, pp. 945–952. ACM, New York (2008) CrossRefGoogle Scholar
  3. 3.
    Garnier, J., Kallel, L., Schoenauer, M.: Rigorous hitting times for binary mutations. Evol. Comput. 7(2), 173–203 (1999) CrossRefGoogle Scholar
  4. 4.
    Giel, O., Wegener, I.: Evolutionary algorithms and the maximum matching problem. In: Proc. of STACS ’03, pp. 415–426. Springer, Berlin (2003) CrossRefGoogle Scholar
  5. 5.
    Hajek, B.: Hitting-time and occupation-time bounds implied by drift analysis with applications. Adv. Appl. Probab. 13(3), 502–525 (1982) CrossRefMathSciNetGoogle Scholar
  6. 6.
    Happ, E., Johannsen, D., Klein, C., Neumann, F.: Rigorous analyses of fitness-proportional selection for optimizing linear functions. In: Proc. of GECCO ’08, pp. 953–960. ACM, New York (2008) CrossRefGoogle Scholar
  7. 7.
    He, J., Yao, X.: Drift analysis and average time complexity of evolutionary algorithms. Artif. Intell. 127(1), 57–85 (2001) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    He, J., Yao, X.: A study of drift analysis for estimating computation time of evolutionary algorithms. Nat. Comput. 3(1), 21–35 (2004) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Neumann, F., Oliveto, P.S., Witt, C.: Theoretical analysis of fitness-proportional selection: landscapes and efficiency. In: Proc. of GECCO ’09, pp. 835–842. ACM, New York (2009) CrossRefGoogle Scholar
  10. 10.
    Oliveto, P.S., Witt, C.: Simplified drift analysis for proving lower bounds in evolutionary computation. In: Proc. of PPSN’08. LNCS, vol. 5199, pp. 82–91. Springer, Berlin (2008) Google Scholar
  11. 11.
    Oliveto, P.S., He, J., Yao, X.: Evolutionary algorithms and the vertex cover problem. In: Proc. of CEC ’07, pp. 1430–1438 (2007) Google Scholar
  12. 12.
    Oliveto, P.S., He, J., Yao, X.: Time complexity of evolutionary algorithms for combinatorial optimization: a decade of results. Int. J. Autom. Comput. 4(3), 281–293 (2007) CrossRefGoogle Scholar
  13. 13.
    Sasaki, G.H., Hajek, B.: The time complexity of maximum matching by simulated annealing. J. Assoc. Comput. Mach. 35(2), 387–403 (1988) MathSciNetGoogle Scholar
  14. 14.
    Wegener, I.: Methods for the analysis of evolutionary algorithms on pseudo-Boolean functions. In: Sarker, R., Mohammadian, M., Yao, X. (eds.) Evolutionary Optimization. Kluwer Academic, Dordrecht (2001) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Centre of Excellence for Research in Computational Intelligence and Applications (CERCIA), School of Computer ScienceUniversity of BirminghamBirminghamUK
  2. 2.DTU InformaticsTechnical University of DenmarkKgs. LyngbyDenmark

Personalised recommendations