## Electronic Journal of Probability

### QuickSort: improved right-tail asymptotics for the limiting distribution, and large deviations

#### 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).

#### Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 67, 13 pp.

Dates
Accepted: 7 June 2019
First available in Project Euclid: 28 June 2019

https://projecteuclid.org/euclid.ejp/1561687600

Digital Object Identifier
doi:10.1214/19-EJP331

Mathematical Reviews number (MathSciNet)
MR3978217

Subjects
Primary: 68P10: Searching and sorting
Secondary: 60E05: Distributions: general theory 60C05: Combinatorial probability

#### Citation

Fill, James Allen; Hung, Wei-Chun. QuickSort: improved right-tail asymptotics for the limiting distribution, and large deviations. Electron. J. Probab. 24 (2019), paper no. 67, 13 pp. doi:10.1214/19-EJP331. https://projecteuclid.org/euclid.ejp/1561687600

#### References

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