433 research outputs found
Quicksort algorithm again revisited
We consider the standard Quicksort algorithm that sorts n distinct keys with all possible n! orderings of keys being equally likely. Equivalently, we analyze the total path length L(n) in a randomly built \emphbinary search tree. Obtaining the limiting distribution of L(n) is still an outstanding open problem. In this paper, we establish an integral equation for the probability density of the number of comparisons L(n). Then, we investigate the large deviations of L(n). We shall show that the left tail of the limiting distribution is much ''thinner'' (i.e., double exponential) than the right tail (which is only exponential). Our results contain some constants that must be determined numerically. We use formal asymptotic methods of applied mathematics such as the WKB method and matched asymptotics
A characterization of the set of fixed points of the Quicksort transformation
The limiting distribution \mu of the normalized number of key comparisons
required by the Quicksort sorting algorithm is known to be the unique fixed
point of a certain distributional transformation T -- unique, that is, subject
to the constraints of zero mean and finite variance. We show that a
distribution is a fixed point of T if and only if it is the convolution of \mu
with a Cauchy distribution of arbitrary center and scale. In particular,
therefore, \mu is the unique fixed point of T having zero mean.Comment: 9 pages. See also http://www.mts.jhu.edu/~fill/ and
http://www.math.uu.se/~svante/papers . Submitted for publication in May,200
On the tails of the limiting Quicksort distribution
We give asymptotics for the left and right tails of the limiting Quicksort
distribution. The results agree with, but are less precise than, earlier
non-rigorous results by Knessl and Spankowski.Comment: 8 pages. v2: Typos corrected and some formulations improve
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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