Quicksort algorithm again revisited
TL;DR: An integral equation for the probability density of the number of comparisons L(n) is established and it is shown that the left tail of the limiting distribution is much ''thinner'' than the right tail (which is only exponential).
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Abstract: 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.
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Citations
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An asymptotic theory for Cauchy---Euler differential equations with applications to the analysis of algorithms
TL;DR: A very general framework for Cauchy-Euler equations is studied and an asymptotic theory of coefficients of functions satisfying such equations is proposed that covers almost all applications where CauCHY- Euler equations appear.
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•Posted Content
Quicksort asymptotics
James Allen Fill,Svante Janson +1 more
- 29 May 2001
TL;DR: The first rates of convergence for the distribution of Yn to that of Y are provided, using various metrics, and the bound 2n-1/2 in the d2-metric is established and the rate O(ne - ( 1/2)) for Kolmogorov-Smirnov distance is established.
56
The number of bit comparisons used by Quicksort: an average-case analysis
James Allen Fill,Svante Janson +1 more
TL;DR: In this article, the authors introduce the prospect of a fair comparison between algorithms of the two types by providing an average-case analysis of the number of bit comparisons required by Quicksort.
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References
The Art in Computer Programming
Andrew Hunt,Dave Thomas +1 more
- 01 Jan 2001
TL;DR: Here the authors haven’t even started the project yet, and already they’re forced to answer many questions: what will this thing be named, what directory will it be in, what type of module is it, how should it be compiled, and so on.
•Book
Advanced mathematical methods for scientists and engineers
Carl M. Bender,Steven A. Orszag +1 more
- 01 Jan 1978
TL;DR: A self-contained presentation of the methods of asymptotics and perturbation theory, methods useful for obtaining approximate analytical solutions to differential and difference equations is given in this paper.
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