Distributional convergence for the number of symbol comparisons used by QuickSort

Abstract

Most previous studies of the sorting algorithm QuickSort have used the number of key comparisons as a measure of the cost of executing the algorithm. Here we suppose that the $n$ independent and identically distributed (i.i.d.) keys are each represented as a sequence of symbols from a probabilistic source and that QuickSort operates on individual symbols, and we measure the execution cost as the number of symbol comparisons. Assuming only a mild “tameness” condition on the source, we show that there is a limiting distribution for the number of symbol comparisons after normalization: first centering by the mean and then dividing by $n$. Additionally, under a condition that grows more restrictive as $p$ increases, we have convergence of moments of orders $p$ and smaller. In particular, we have convergence in distribution and convergence of moments of every order whenever the source is memoryless, that is, whenever each key is generated as an infinite string of i.i.d. symbols. This is somewhat surprising; even for the classical model that each key is an i.i.d. string of unbiased (“fair”) bits, the mean exhibits periodic fluctuations of order $n$.