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Inversions from Sorting with Distance-Based Errors

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SOFSEM 2018: Theory and Practice of Computer Science (SOFSEM 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10706))

Abstract

We study the number of inversions after running the Insertion Sort or Quicksort algorithm, when errors in the comparisons occur with some probability. We investigate the case in which probabilities depend on the difference between the two numbers to be compared and only differences up to some threshold \(\tau \) are prone to errors. We give upper bounds for this model and show that for constant \(\tau \), the expected number of inversions is linear in the number of elements to be sorted. For Insertion Sort, we also yield an upper bound on the expected number of runs, i.e., the number of consecutive increasing subsequences.

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Notes

  1. 1.

    The number of inversions of a sequence \(\sigma =(\sigma _1,\ldots ,\sigma _n)\) is the number of pairs (i, j) with \(i<j\) such that \(\sigma _i> \sigma _j\).

  2. 2.

    We find the correct position of an element by linear search not by binary search.

  3. 3.

    The analysis on the number of inversions appearing after one run of Quicksort holds for arbitrarily chosen pivots. In particular, it also holds for random pivots.

  4. 4.

    See [7] for a analysis of the time complexity of Quicksort with errors.

  5. 5.

    This modification of marking the elements is purely imaginary; the Quicksort algorithm does not know the correctness of a comparison.

  6. 6.

    This is true if and only if the sequence contains at least one non-trivial block.

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Acknowledgements

This research has been supported by the Swiss National Science Foundation (SNFS project 200021_165524).

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Authors and Affiliations

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

    Barbara Geissmann & Paolo Penna

Authors
  1. Barbara Geissmann
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  2. Paolo Penna
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Corresponding author

Correspondence to Barbara Geissmann .

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Editors and Affiliations

  1. Vienna University of Technology , Vienna, Austria

    A Min Tjoa

  2. ISAE-ENSMA, Chasseneuil-du-Poitou, France

    Ladjel Bellatreche

  3. Vienna University of Technology, Vienna, Austria

    Stefan Biffl

  4. Utrecht University, Utrecht, The Netherlands

    Jan van Leeuwen

  5. Academy of Sciences, Prague, Czech Republic

    Jiří Wiedermann

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Geissmann, B., Penna, P. (2018). Inversions from Sorting with Distance-Based Errors. In: Tjoa, A., Bellatreche, L., Biffl, S., van Leeuwen, J., Wiedermann, J. (eds) SOFSEM 2018: Theory and Practice of Computer Science. SOFSEM 2018. Lecture Notes in Computer Science(), vol 10706. Edizioni della Normale, Cham. https://doi.org/10.1007/978-3-319-73117-9_36

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  • DOI: https://doi.org/10.1007/978-3-319-73117-9_36

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  • Publisher Name: Edizioni della Normale, Cham

  • Print ISBN: 978-3-319-73116-2

  • Online ISBN: 978-3-319-73117-9

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