Abstract
The worst-case complexity of an implementation of Quicksort depends on the random source that is used to select the pivot elements. In this paper we estimate the expected number of comparisons of Quicksort as a function of the entropy of the random source. We give upper and lower bounds and show that the expected number of comparisons increases from nlog n to n 2, if the entropy of the random source is bounded. As examples we show explicit bounds for distributions with bounded min-entropy and the geometrical distribution, as well as an upper bound when using a δ-random source.
Keywords
Work supported by DFG research grant Scho 302/6-1
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Alon, N., Rabin, M.O.: Biased coins and randomized algorithms. In: Preparata, F.P., Micali, S. (eds.) Advances in Computing Research 5, pp. 499–507. JAI Press (1989)
Ash, R.B.: Information Theory. Dover, New York (1965)
Devroye, L.: On the probabilistic worst-case time of “FIND”. Algorithmica 31, 291–303 (2001)
Hoare, C.A.R.: Quicksort. Computer Journal 5(1), 10–15 (1962)
Karloff, H.J., Raghavan, P.: Randomized algorithms and pseudorandom numbers. Journal of the Association for Computing Machinery 40, 454–476 (1993)
Knuth, D.: The Art of Computer Programming. Sorting and Searching, vol. 3. Addison-Wesley, Reading (1973)
List, B.: Probabilistische Algorithmen und schlechte Zufallszahlen. PhD thesis, Universität Ulm (1999)
Luby, M.: Pseudorandomness and Cryptographic Applications. Princeton University Press, Princeton (1996)
Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1994)
Robert Sedgewick, R., Flajolet, P.: Analysis of Algorithms. Addison-Wesley, Reading (1994)
Santha, M., Vazirani, U.V.: Generating quasi-random sequences from slightly random sources. In: Proceedings of the 25th IEEE (1984)
Tompa, M.: Probabilistic Algorithms and Pseudorandom Generators. Lecture Notes (1991)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
List, B., Maucher, M., Schöning, U., Schuler, R. (2005). Randomized Quicksort and the Entropy of the Random Source. In: Wang, L. (eds) Computing and Combinatorics. COCOON 2005. Lecture Notes in Computer Science, vol 3595. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11533719_46
Download citation
DOI: https://doi.org/10.1007/11533719_46
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-28061-3
Online ISBN: 978-3-540-31806-4
eBook Packages: Computer ScienceComputer Science (R0)