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Sorting by Swaps with Noisy Comparisons

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Abstract

We study sorting of permutations by random swaps if each comparison gives the wrong result with some fixed probability \(p<1/2\). We use this process as prototype for the behaviour of randomized, comparison-based optimization heuristics in the presence of noisy comparisons. As quality measure, we compute the expected fitness of the stationary distribution. To measure the runtime, we compute the minimal number of steps after which the average fitness approximates the expected fitness of the stationary distribution. We study the process where in each round a random pair of elements at distance at most r are compared. We give theoretical results for the extreme cases \(r=1\) and \(r=n\), and experimental results for the intermediate cases. We find a trade-off between faster convergence (for large r) and better quality of the solution after convergence (for small r).

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Notes

  1. Using W or D as the energy leads to different, less local error probability functions.

  2. Generally not a sorting network since the resulting networks are generally not sorting every input.

  3. That is with probability at least \(1-1/n\).

  4. Observe that the decrease of the total number of inversions is \((1+2m)\).

  5. That is with probability at least \(1-1/n\).

  6. By Lemma 7, \( W(\pi ) = \sum _{i\in \{1,\dots ,n\}}i(i-\pi (i))\). Furthermore, since \(\pi \) is a permutation of the numbers \(\{1,\dots ,n\}\), \(\sum _{i\in \{1,\dots ,n\}}i^2 = \sum _{i\in \{1,\dots ,n\}}\pi (i)^2\).

  7. https://github.com/gavento/swap-sorting-experiments.

  8. Experiments show that the convergence behavior of \(W(\pi )\) is very similar and the convergence would differ by less than \(5\%\).

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Acknowledgements

We would like to thank all the anonymous reviewers for their careful and attentive reading, as well as their numerous helpful comments to improve this paper. Tomáš Gavenčiak was supported by the Czech Science Foundation (GAČR) Project 17-10090Y “Network optimization”. Barbara Geissmann was supported by the Swiss National Science Foundation (SNSF), Project Number 200021_165524.

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Authors and Affiliations

  1. Department of Applied Mathematics, Charles University, Prague, Czech Republic

    Tomáš Gavenčiak

  2. Department of Computer Science, FEE, CTU, Prague, Czech Republic

    Tomáš Gavenčiak

  3. Department of Computer Science, ETH Zurich, Zurich, Switzerland

    Barbara Geissmann & Johannes Lengler

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  1. Tomáš Gavenčiak
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Correspondence to Johannes Lengler.

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An extended abstract of this paper has been presented at Genetic and Evolutionary Computation Conference (GECCO 2017).

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Gavenčiak, T., Geissmann, B. & Lengler, J. Sorting by Swaps with Noisy Comparisons. Algorithmica 81, 796–827 (2019). https://doi.org/10.1007/s00453-018-0429-2

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