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# Optimal Sorting with Persistent Comparison Errors

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LIPIcs.ESA.2019.49.pdf
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## Acknowledgements

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

## Cite As

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

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## References

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