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2019 QuickSort: improved right-tail asymptotics for the limiting distribution, and large deviations
James Allen Fill, Wei-Chun Hung
Electron. J. Probab. 24: 1-13 (2019). DOI: 10.1214/19-EJP331

Abstract

We substantially refine asymptotic logarithmic upper bounds produced by Svante Janson (2015) on the right tail of the limiting QuickSort distribution function $F$ and by Fill and Hung (2018) on the right tails of the corresponding density $f$ and of the absolute derivatives of $f$ of each order. For example, we establish an upper bound on $\log [1 - F(x)]$ that matches conjectured asymptotics of Knessl and Szpankowski (1999) through terms of order $(\log x)^{2}$; the corresponding order for the Janson (2015) bound is the lead order, $x \log x$.

Using the refined asymptotic bounds on $F$, we derive right-tail large deviation (LD) results for the distribution of the number of comparisons required by QuickSort that substantially sharpen the two-sided LD results of McDiarmid and Hayward (1996).

Citation

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James Allen Fill. Wei-Chun Hung. "QuickSort: improved right-tail asymptotics for the limiting distribution, and large deviations." Electron. J. Probab. 24 1 - 13, 2019. https://doi.org/10.1214/19-EJP331

Information

Received: 19 January 2019; Accepted: 7 June 2019; Published: 2019
First available in Project Euclid: 28 June 2019

zbMATH: 07089005
MathSciNet: MR3978217
Digital Object Identifier: 10.1214/19-EJP331

Subjects:
Primary: 68P10
Secondary: 60C05, 60E05

JOURNAL ARTICLE
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