Optimal Sorting with Persistent Comparison Errors

Authors Barbara Geissmann , Stefano Leucci , Chih-Hung Liu , Paolo Penna



PDF
Thumbnail PDF

File

LIPIcs.ESA.2019.49.pdf
  • Filesize: 0.66 MB
  • 14 pages

Document Identifiers

Author Details

Barbara Geissmann
  • Department of Computer Science, ETH Zürich, Switzerland
Stefano Leucci
  • Department of Algorithms and Complexity, Max Planck Institute for Informatics, Germany
Chih-Hung Liu
  • Department of Computer Science, ETH Zürich, Switzerland
Paolo Penna
  • Department of Computer Science, ETH Zürich, Switzerland

Acknowledgements

The authors wish to thank Peter Widmayer for many insightful discussions.

Cite As Get BibTex

Barbara Geissmann, Stefano Leucci, Chih-Hung Liu, and Paolo Penna. Optimal Sorting with Persistent Comparison Errors. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 49:1-49:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ESA.2019.49

Abstract

We consider the problem of sorting n elements in the case of persistent comparison errors. In this problem, each comparison between two elements can be wrong with some fixed (small) probability p, and comparisons cannot be repeated (Braverman and Mossel, SODA'08). Sorting perfectly in this model is impossible, and the objective is to minimize the dislocation of each element in the output sequence, that is, the difference between its true rank and its position. Existing lower bounds for this problem show that no algorithm can guarantee, with high probability, maximum dislocation and total dislocation better than Omega(log n) and Omega(n), respectively, regardless of its running time.
In this paper, we present the first O(n log n)-time sorting algorithm that guarantees both O(log n) maximum dislocation and O(n) total dislocation with high probability. This settles the time complexity of this problem and shows that comparison errors do not increase its computational difficulty: a sequence with the best possible dislocation can be obtained in O(n log n) time and, even without comparison errors, Omega(n log n) time is necessary to guarantee such dislocation bounds. 
In order to achieve this optimality result, we solve two sub-problems in the persistent error comparisons model, and the respective methods have their own merits for further application. One is how to locate a position in which to insert an element in an almost-sorted sequence having O(log n) maximum dislocation in such a way that the dislocation of the resulting sequence will still be O(log n). The other is how to simultaneously insert m elements into an almost sorted sequence of m different elements, such that the resulting sequence of 2m elements remains almost sorted.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • approximate sorting
  • comparison errors
  • persistent errors

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Miklós Ajtai, Vitaly Feldman, Avinatan Hassidim, and Jelani Nelson. Sorting and selection with imprecise comparisons. ACM Transactions on Algorithms, 12(2):19, 2016. Google Scholar
  2. Laurent Alonso, Philippe Chassaing, Florent Gillet, Svante Janson, Edward M Reingold, and René Schott. Quicksort with unreliable comparisons: a probabilistic analysis. Combinatorics, Probability and Computing, 13(4-5):419-449, 2004. Google Scholar
  3. Michael Ben-Or and Avinatan Hassidim. The Bayesian Learner is Optimal for Noisy Binary Search (and Pretty Good for Quantum as Well). In 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008, October 25-28, 2008, Philadelphia, PA, USA, pages 221-230, 2008. URL: https://doi.org/10.1109/FOCS.2008.58.
  4. Mark Braverman, Jieming Mao, and S Matthew Weinberg. Parallel algorithms for select and partition with noisy comparisons. In Proc. of the 48th Annual ACM Symposium on Theory of Computing (STOC), pages 851-862. ACM, 2016. Google Scholar
  5. Mark Braverman and Elchanan Mossel. Noisy Sorting Without Resampling. In Proceedings of the 19th Annual Symposium on Discrete Algorithms, pages 268-276, 2008. URL: http://arxiv.org/abs/0707.1051.
  6. Ferdinando Cicalese. Fault-Tolerant Search Algorithms - Reliable Computation with Unreliable Information. Monographs in Theoretical Computer Science. An EATCS Series. Springer, 2013. Google Scholar
  7. Yuval Dagan, Yuval Filmus, Daniel Kane, and Shay Moran. The entropy of lies: playing twenty questions with a liar. CoRR, abs/1811.02177, 2018. URL: http://arxiv.org/abs/1811.02177.
  8. Peter Damaschke. The Solution Space of Sorting with Recurring Comparison Faults. In Combinatorial Algorithms - 27th International Workshop, IWOCA 2016, Helsinki, Finland, August 17-19, 2016, Proceedings, pages 397-408, 2016. Google Scholar
  9. Uriel Feige, Prabhakar Raghavan, David Peleg, and Eli Upfal. Computing with Noisy Information. SIAM Journal on Computing, 23(5):1001-1018, 1994. URL: https://doi.org/10.1137/S0097539791195877.
  10. Barbara Geissmann, Stefano Leucci, Chih-Hung Liu, and Paolo Penna. Sorting with Recurrent Comparison Errors. In Yoshio Okamoto and Takeshi Tokuyama, editors, 28th International Symposium on Algorithms and Computation (ISAAC 2017), volume 92 of Leibniz International Proceedings in Informatics (LIPIcs), pages 38:1-38:12, Dagstuhl, Germany, 2017. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2017.38.
  11. Barbara Geissmann, Stefano Leucci, Chih-Hung Liu, and Paolo Penna. Optimal Dislocation with Persistent Errors in Subquadratic Time. In Proc. of the 35th Symposium on Theoretical Aspects of Computer Science (STACS), volume 96 of LIPIcs, pages 36:1-36:13. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.STACS.2018.36.
  12. Petros Hadjicostas and KB Lakshmanan. Recursive merge sort with erroneous comparisons. Discrete Applied Mathematics, 159(14):1398-1417, 2011. Google Scholar
  13. Rolf Klein, Rainer Penninger, Christian Sohler, and David P. Woodruff. Tolerant Algorithms. In Proc. of the 19th Annual European Symposium on Algorithm (ESA), pages 736 - -747, 2011. Google Scholar
  14. Andrzej Pelc. Searching games with errors - fifty years of coping with liars. Theor. Comput. Sci., 270(1-2):71-109, 2002. Google Scholar
  15. Alfréd Rényi. On a problem of information theory. MTA Mat. Kut. Int. Kozl. B, 6:505-516, 1961. Google Scholar
  16. Ronald L. Rivest, Albert R. Meyer, Daniel J. Kleitman, Karl Winklmann, and Joel Spencer. Coping with Errors in Binary Search Procedures. J. Comput. Syst. Sci., 20(3):396-404, 1980. URL: https://doi.org/10.1016/0022-0000(80)90014-8.
  17. Aviad Rubinstein and Shai Vardi. Sorting from Noisier Samples. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 960-972, 2017. URL: https://doi.org/10.1137/1.9781611974782.60.
  18. Stanislav M. Ulam. Adventures of a Mathematician, 1976. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail