132 research outputs found
Distributional convergence for the number of symbol comparisons used by QuickSort
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.Comment: Published in at http://dx.doi.org/10.1214/12-AAP866 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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