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Mixed distributions in Sattolo's algorithm for cyclic permutations via randomization and derandomization

Part of: Algorithms - Computer Science Combinatorics Combinatorial probability

Published online by Cambridge University Press:  14 July 2016

Hosam M. Mahmoud
Hosam M. Mahmoud*
Affiliation:
George Washington University
*
∗Postal address: Department of Statistics, George Washington University, Washington, DC 20052, USA. Email address: hosam@gwu.edu
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Abstract

We investigate the limit distributions associated with cost measures in Sattolo's algorithm for generating random cyclic permutations. The number of moves made by an element turns out to be a mixture of 1 and 1 plus a geometric distribution with parameter ½, where the mixing probability is the limiting ratio of the rank of the element being moved to the size of the permutation. On the other hand, the raw distance traveled by an element to its final destination does not converge in distribution without norming. Linearly scaled, the distance converges to a mixture of a uniform and a shifted product of a pair of independent uniforms. The results are obtained via randomization as a transform, followed by derandomization as an inverse transform. The work extends analysis by Prodinger.

Keywords

Random permutationsmixture distributions

MSC classification

Primary: 60C05: Combinatorial probability 05A05: Permutations, words, matrices 68W40: Analysis of algorithms
Type
Short Communications
Information
Journal of Applied Probability , Volume 40 , Issue 3 , September 2003 , pp. 790 - 796
Copyright
Copyright © Applied Probability Trust 2003 

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References

Grübel, R. and Rösler, U. (1996). Asymptotic distribution theory for Hoare's selection algorithm. Adv. Appl. Prob. 28, 252–269.Google Scholar
Hwang, H., and Tsai, T. (2002). Quickselect and Dickman function. Combinatorics Prob. Comput. 11, 353–371.Google Scholar
Kirschenhofer, P., and Prodinger, H. (1998). Comparisons in Hoare's Find algorithm. Combinatorics Prob. Comput. 7, 111–120.CrossRefGoogle Scholar
Kirschenhofer, P., Prodinger, H. and Martínez, C. (1997). Analysis of Hoare's Find algorithm with median-of-three partition. Random Structures Algorithms 10, 143–156.Google Scholar
Lent, J., and Mahmoud, H. M. (1996). Average-case analysis of multiple Quickselect: an algorithm for finding order statistics. Statist. Prob. Lett. 28, 299–310.Google Scholar
Mahmoud, H. M., and Smythe, R. T. (1998). Probabilistic analysis of Multiple Quick Select. Algorithmica 22, 569–584.Google Scholar
Mahmoud, H. M., Modarres, R., and Smythe, R. T. (1995). Analysis of Quickselect: an algorithm for order statistics. RAIRO Inf. Théor. Appl. 29, 255–276.Google Scholar
Panholzer, A., and Prodinger, H. (1998). A generating functions approach for the analysis of grand averages for Multiple Quickselect. Random Structures Algorithms 13, 189–209.Google Scholar
Prodinger, H. (1995). Multiple Quickselect-Hoare's Find algorithm for several elements. Inf. Processing Lett. 56, 123–129.Google Scholar
Prodinger, H. (2002). On the analysis of an algorithm to generate a random cyclic permutation. Ars Combinatoria 65, 75–78.Google Scholar
Sattolo, S. (1986). An algorithm to generate a random cyclic permutation. Inf. Processing Lett. 22, 315–317.Google Scholar