Algorithmica

, Volume 64, Issue 4, pp 673–697 | Cite as

Multiplicative Drift Analysis

Article

Abstract

We introduce multiplicative drift analysis as a suitable way to analyze the runtime of randomized search heuristics such as evolutionary algorithms. Our multiplicative version of the classical drift theorem allows easier analyses in the often encountered situation that the optimization progress is roughly proportional to the current distance to the optimum.

To display the strength of this tool, we regard the classical problem of how the (1+1) Evolutionary Algorithm optimizes an arbitrary linear pseudo-Boolean function. Here, we first give a relatively simple proof for the fact that any linear function is optimized in expected time O(nlogn), where n is the length of the bit string. Afterwards, we show that in fact any such function is optimized in expected time at most (1+o(1))1.39enlnn, again using multiplicative drift analysis. We also prove a corresponding lower bound of (1−o(1))enlnn which actually holds for all functions with a unique global optimum.

We further demonstrate how our drift theorem immediately gives natural proofs (with better constants) for the best known runtime bounds for the (1+1) Evolutionary Algorithm on combinatorial problems like finding minimum spanning trees, shortest paths, or Euler tours in graphs.

Keywords

Evolutionary algorithms Randomized search heuristics Runtime analysis Drift analysis 

References

  1. 1.
    Auger, A., Doerr, B. (eds.): Theory of Randomized Search Heuristics. Series on Theoretical Computer Science, vol. 1. World Scientific, Singapore (2011) MATHGoogle Scholar
  2. 2.
    Baswana, S., Biswas, S., Doerr, B., Friedrich, T., Kurur, P., Neumann, F.: Computing single source shortest paths using single-objective fitness. In: FOGA ’09: Proceedings of the 10th ACM Workshop on Foundations of Genetic Algorithms, pp. 59–66. ACM, New York (2009) CrossRefGoogle Scholar
  3. 3.
    Doerr, B., Goldberg, L.A.: Drift analysis with tail bounds. In: PPSN ’10: Proceedings of the 11th International Conference on Parallel Problem Solving from Nature, pp. 174–183. Springer, Berlin (2010) CrossRefGoogle Scholar
  4. 4.
    Doerr, B., Goldberg, L.A.: Adaptive drift analysis. In: PPSN ’10: Proceedings of the 11th International Conference on Parallel Problem Solving from Nature, pp. 32–41. Springer, Berlin (2010) CrossRefGoogle Scholar
  5. 5.
    Doerr, B., Johannsen, D.: Adjacency list matchings—an ideal genotype for cycle covers. In: GECCO ’07: Proceedings of the 9th Annual Genetic and Evolutionary Computation Conference, pp. 1203–1210. ACM, New York (2007) CrossRefGoogle Scholar
  6. 6.
    Doerr, B., Johannsen, D.: Edge-based representation beats vertex-based representation in shortest path problems. In: GECCO ’10: Proceedings of the 12th Annual Genetic and Evolutionary Computation Conference, pp. 759–766. ACM, New York (2010) CrossRefGoogle Scholar
  7. 7.
    Doerr, B., Fouz, M., Witt, C.: Quasirandom evolutionary algorithms. In: GECCO ’10: Proceedings of the 12th Annual Genetic and Evolutionary Computation Conference, pp. 1457–1464. ACM, New York (2010) CrossRefGoogle Scholar
  8. 8.
    Doerr, B., Jansen, T., Sudholt, D., Winzen, C., Zarges, C.: Optimizing monotone functions can be difficult. In: PPSN ’10: Proceedings of the 11th International Conference on Parallel Problem Solving from Nature, pp. 42–51. Springer, Berlin (2010) CrossRefGoogle Scholar
  9. 9.
    Doerr, B., Johannsen, D., Winzen, C.: Drift analysis and linear functions revisited. In: CEC ’10: Proceedings of the 2010 IEEE Congress on Evolutionary Computation, pp. 1967–1974. IEEE, New York (2010) Google Scholar
  10. 10.
    Doerr, B., Johannsen, D., Winzen, C.: Multiplicative drift analysis. In: GECCO ’10: Proceedings of the 12th Annual Genetic and Evolutionary Computation Conference, pp. 1449–1456. ACM, New York (2010) CrossRefGoogle Scholar
  11. 11.
    Droste, S., Jansen, T., Wegener, I.: On the analysis of the (1+1) evolutionary algorithm. Theor. Comput. Sci. 276(1–2), 51–81 (2002) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Giel, O., Lehre, P.K.: On the effect of populations in evolutionary multi-objective optimization. In: GECCO ’06: Proceedings of the 8th Annual Genetic and Evolutionary Computation Conference, pp. 651–658. ACM, New York (2006) CrossRefGoogle Scholar
  13. 13.
    Giel, O., Wegener, I.: Evolutionary algorithms and the maximum matching problem. In: STACS ’03: Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science. Lecture Notes in Computer Science, vol. 2607, pp. 415–426. Springer, Berlin (2003) Google Scholar
  14. 14.
    Hajek, B.: Hitting-time and occupation-time bounds implied by drift analysis with applications. Adv. Appl. Probab. 14(3), 387–403 (1982) CrossRefGoogle Scholar
  15. 15.
    Happ, E., Johannsen, D., Klein, C., Neumann, F.: Rigorous analyses of fitness-proportional selection for optimizing linear functions. In: GECCO ’08: Proceedings of the 10th Annual Genetic and Evolutionary Computation Conference, pp. 953–960. ACM, New York (2008) CrossRefGoogle Scholar
  16. 16.
    He, J., Yao, X.: Drift analysis and average time complexity of evolutionary algorithms. Acta Inform. 127(1), 51–81 (2001) MathSciNetGoogle Scholar
  17. 17.
    He, J., Yao, X.: Erratum to: Drift analysis and average time complexity of evolutionary algorithms [Artificial Intelligence 127 (2001) 57–85]. Acta Inform. 140(1–2), 245–248 (2002) MathSciNetMATHGoogle Scholar
  18. 18.
    He, J., Yao, X.: A study of drift analysis for estimating computation time of evolutionary algorithms. Nat. Comput. 3(1), 21–35 (2004) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Jägersküpper, J.: A blend of Markov-chain and drift analysis. In: PPSN ’08: Proceedings of the 10th International Conference on Parallel Problem Solving from Nature, pp. 41–51. Springer, Berlin (2008) CrossRefGoogle Scholar
  20. 20.
    Jansen, T., Oliveto, P.S., Zarges, C.: On the analysis of the immune-inspired B-Cell Algorithm for the vertex cover problem. In: ICARIS ’11: Proceedings of the 10th International Conference on Artificial Immune Systems. Lecture Notes in Computer Science, vol. 6825, pp. 117–131. Springer, Berlin (2011) Google Scholar
  21. 21.
    Kano, M.: Maximum and kth maximal spanning trees of a weighted graph. Combinatorica 7(2), 205–214 (1987) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Mühlenbein, H.: How genetic algorithms really work: mutation and hill-climbing. In: PPSN ’92: Proceedings of the 2nd International Conference on Parallel Problem Solving from Nature, pp. 15–25 (1992) Google Scholar
  23. 23.
    Neumann, F., Wegener, I.: Randomized local search, evolutionary algorithms, and the minimum spanning tree problem. Theor. Comput. Sci. 378(1), 32–40 (2007) MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Neumann, F., Oliveto, P.S., Witt, C.: Theoretical analysis of fitness-proportional selection: landscapes and efficiency. In: GECCO ’09: Proceedings of the 11th Annual Genetic and Evolutionary Computation Conference, pp. 835–842. ACM, New York (2009) CrossRefGoogle Scholar
  25. 25.
    Oliveto, P.S., Witt, C.: Simplified drift analysis for proving lower bounds in evolutionary computation. Algorithmica 59(3), 369–386 (2011) MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Scheder, D., Welzl, E.: Private communications (2008) Google Scholar
  27. 27.
    Witt, C.: Tight bounds on the optimization time of the (1+1) EA on linear functions. In: STACS ’12: Proceedings of the 29th Annual Symposium on Theoretical Aspects of Computer Science (2012, to appear) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Benjamin Doerr
    • 1
  • Daniel Johannsen
    • 2
  • Carola Winzen
    • 1
  1. 1.Max-Planck-Institut für InformatikSaarbrückenGermany
  2. 2.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael

Personalised recommendations