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

03/19/2019

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by   James Allen Fill, et al.

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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 [1 - F(x)] that matches conjectured asymptotics of Knessl and Szpankowski (1999) through terms of order ( x)^2; the corresponding order for the Janson (2015) bound is the lead order, x 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).