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Multiplicative Drift Analysis

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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.

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Notes

  1. By ℕ:={0,1,2,…} we denote the set of non-negative integers and by ℝ we denote the set of real numbers.

  2. We might as well perform our analysis of g as defined in (11). However, our choice of g does not make the somewhat artificial binary distinction between bits with high and low indices and, thus, seems to be more natural.

  3. In [23], Neumann and Wegener actually analyze the MST problem on the larger search space consisting of all subgraphs of G. In this section, we restrict the problem to the sub-problem of finding a MST when starting with an arbitrary spanning tree. It is shown in [23] that this sub-problem is the bottleneck in the runtime analysis of the (1+1) EA optimizing the MST problem.

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Acknowledgements

We like to thank Dirk Sudholt for pointing out that Theorem 11 holds for all pseudo-Boolean function with a unique global optimum rather than only for all linear functions.

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Authors and Affiliations

  1. Max-Planck-Institut für Informatik, Campus E1 4, 66123, Saarbrücken, Germany

    Benjamin Doerr & Carola Winzen

  2. School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv, 69978, Israel

    Daniel Johannsen

Authors
  1. Benjamin Doerr
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  2. Daniel Johannsen
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  3. Carola Winzen
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Correspondence to Daniel Johannsen.

Additional information

Daniel Johannsen is supported by a fellowship within the Postdoc-Programme of the German Academic Exchange Service (DAAD).

Carola Winzen is a recipient of the Google Europe Fellowship in Randomized Algorithms, and this work is supported in part by this Google Fellowship.

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Doerr, B., Johannsen, D. & Winzen, C. Multiplicative Drift Analysis. Algorithmica 64, 673–697 (2012). https://doi.org/10.1007/s00453-012-9622-x

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