Quicksort asymptotics

The number of comparisons Xn used by Quicksort to sort an array of n distinct numbers has mean μn of order n log n and standard deviation of order n. Using different methods, Régnier and Rösler each showed that the normalized variate Yn := (Xn−μn)/n converges in distribution, say to Y ; the distribution of Y can be characterized as the unique fixed point with zero mean of a certain distributional transformation. We provide the first rates of convergence for the distribution of Yn to that of Y , using various metrics. In particular, we establish the bound 2n−1/2 in the d2-metric, and the rate O(nε−(1/2)) for Kolmogorov–Smirnov distance, for any positive ε. AMS 2000 subject classifications. Primary 68W40; secondary 68P10, 60F05, 60E05.