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June 2013 Distributional convergence for the number of symbol comparisons used by QuickSort
James Allen Fill
Ann. Appl. Probab. 23(3): 1129-1147 (June 2013). DOI: 10.1214/12-AAP866


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 $n$.


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James Allen Fill. "Distributional convergence for the number of symbol comparisons used by QuickSort." Ann. Appl. Probab. 23 (3) 1129 - 1147, June 2013.


Published: June 2013
First available in Project Euclid: 7 March 2013

zbMATH: 1272.68482
MathSciNet: MR3076680
Digital Object Identifier: 10.1214/12-AAP866

Primary: 60F20
Secondary: 68W40

Keywords: $L^{p}$-convergence , de-Poissonization , key comparisons , limit distribution , natural coupling , probabilistic source , QuickSort , symbol comparisons , tameness

Rights: Copyright © 2013 Institute of Mathematical Statistics


Vol.23 • No. 3 • June 2013
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