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Adaptive Drift Analysis

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Abstract

We show that, for any c>0, the (1+1) evolutionary algorithm using an arbitrary mutation rate p n =c/n finds the optimum of a linear objective function over bit strings of length n in expected time Θ(nlogn). Previously, this was only known for c≤1. Since previous work also shows that universal drift functions cannot exist for c larger than a certain constant, we instead define drift functions which depend crucially on the relevant objective functions (and also on c itself). Using these carefully-constructed drift functions, we prove that the expected optimisation time is Θ(nlogn). By giving an alternative proof of the multiplicative drift theorem, we also show that our optimisation-time bound holds with high probability.

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Authors and Affiliations

  1. Max Planck Institute for Computer Science, Campus E1 4, 66123, Saarbrücken, Germany

    Benjamin Doerr

  2. Department of Computer Science, University of Liverpool, Ashton Bldg, Liverpool, L69 3BX, UK

    Leslie Ann Goldberg

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  1. Benjamin Doerr
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  2. Leslie Ann Goldberg
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Correspondence to Leslie Ann Goldberg.

Additional information

This work was begun while both authors were visiting the “Centre de Recerca Matemática de Catalunya”. It profited greatly from this ideal environment for collaboration.

A preliminary announcement of the result (without proofs) appeared in [3].

The work described in this paper was partly supported by EPSRC Research Grant (refs EP/I011528/1) “Computational Counting”.

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Doerr, B., Goldberg, L.A. Adaptive Drift Analysis. Algorithmica 65, 224–250 (2013). https://doi.org/10.1007/s00453-011-9585-3

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