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On the Upper Bound of the Complexity of Sorting

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Abstract

It is shown that \({{\log}_{2}}(n!) + o(n)\) pairwise comparisons is sufficient for sorting any subset of \(n\) elements of a linearly ordered set.

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Notes

  1. In [7], the strongest form of the result is represented graphically.

  2. The Hasse diagram—the edges connect the pairs of elements the order of which is known (for details see, e.g., [2]).

  3. Since the strategy works with noninteger interval lengths, by an element in this paragraph we mean a conditional point of the interval partition.

  4. By the specific complexity of inserting an individual element, we mean the specific complexity of the insertion that goes to this element when the pair is inserted.

  5. Here and below, an element usually means a partition point of an interval.

  6. It is hardly possible to design an efficient insertion algorithm without using estimates of type \(Q(a)\), which is seen in the examples of algorithms described in [10, 11] and the simple algorithm described in Section 3.

  7. The symbol \( \asymp \) denotes the equality of the growth orders.

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Funding

This work was supported by the Russian Foundation for Basic Research, project no. 19-01-00294а.

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Authors and Affiliations

  1. Research Institute Kvant, 125438, Moscow, Russia

    I. S. Sergeev

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  1. I. S. Sergeev
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Correspondence to I. S. Sergeev.

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Translated by A. Klimontovich

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Sergeev, I.S. On the Upper Bound of the Complexity of Sorting. Comput. Math. and Math. Phys. 61, 329–346 (2021). https://doi.org/10.1134/S0965542521020111

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