Large Deviations for Quicksort

Abstract

Let Q n be the random number of comparisons made by quicksort in sorting n distinct keys, when we assume that all n! possible orderings are equally likely. Known results concerning moments for Q n do not show how rare it is for Q n to make large deviations from its mean. Here we give a good approximation to the probability of such a large deviation, and find that this probability is quite small. As well as the basic quicksort we consider the variant in which the partitioning key is chosen as the median of (2t + 1) keys.

On the number of iterations required by Von Neumann addition

“…The latter could be seen as a large deviation result; see McDiarmid [13] for general comments and McDiarmid and Hayward [14] for a similar result in the Quicksort context.…”

“…the discussion in Grübel [8]. Chassaing et al [3] discuss related optimality concepts; concentration of mass in connection with stochastic algorithms is also discussed in McDiarmid and Hayward [14] and McDiarmid [13].…”

On martingale tail sums for the path length in random trees

“…Together with (28), it follows (β + m) −2 α 2 n E X 2 n |F n−1 = Var(D n ) + M n−1 (1) + S n−1 − S 2 n−1 .…”

On a multivariate contraction method for random recursive structures with applications to Quicksort

“…Then the conditions of the limit law might be still satisfied but the unique solution in (14) is the degenerated one X = 0. With σ r (n) growing too slow we typically cannot satisfy b (n) → b * as in (11).…”

“…For C n a huge body of probabilistic results is available even for the median-of-(2t + 1) version of Quicksort. These include in particular asymptotic expressions for the means and variances, as well as limit laws for the scaled quantities, and large deviation inequalities, see Hennequin [22,23], Régnier [42], Rösler [43,45], McDiarmid and Hayward [11], Bruhn [3], and for a detailed survey the book of Mahmoud [28]. For the number of exchanges B n the mean and variance were for general t ∈ N 0 studied in Hennequin [23], Chern and Hwang [5] refined the analysis of the mean, and Hwang and Neininger [25] gave a limit law for the standard case t = 0.…”