Summary
In the first part of this paper we give an introduction to the contraction method for the analysis of additive recursive sequences of divide and conquer type. Recently some general limit theorems have been obtained by this method based on a general transfer theorem. This allows to conclude from the recursive structure and the asymptotics of first moment(s) the limiting distribution. In the second part we extend the contraction method to max-recursive sequences. We obtain a general existence and uniqueness result for solutions of stochastic equations including maxima and sum terms. We finally derive a general limit theorem for max-recursive sequences of the divide and conquer type.
Research supported by an Emmy Noether Fellowship of the DFG.
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
Michael Cramer. Stochastic analysis of Merge-Sort algorithm. Random Structures Algorithms, 11:81–96, 1997.
Hsien-Kuei Hwang and Ralph Neininger. Phase change of limit laws in the quicksort recurrence under varying toll functions. SIAM Journal on Computating, 31:1687–1722, 2002.
P Jagers and Uwe Rösler. Fixed points of max-recursive sequences. Preprint, 2002.
Donald E. Knuth. The Art of Computer Programming, volume 3: Sorting and Searching. Addison-Wesley Publishing Co., Reading, 1973.
Hosam M Mahmoud. Sorting. Wiley-Interscience Series in Discrete Mathemstics and Optomization. Wiley-Interscience, New York, 2000.
Ralph Neininger. Limit Laws for Random Recursive Structures and Algorithms. Dissertation, University of Freiburg, 1999.
Ralph Neininger. On a multivariate contraction method for random recursive structures with applications to Quicksort. Random Structures and Algorithms, 19:498–524, 2001.
Ralph Neininger and Ludger Rüschendorf. Rates of convergence for Quicksort. Journal of Algorithms, 44:52–62, 2002.
Ralph Neininger and Ludger Rüschendorf. A general limit theorem for recursive algorithms and combinatorial structures. to appear in: Annals of Applied Probability, 2003.
Ralph Neininger and Ludger Rüschendorf. On the contraction method with degenerate limit equation. to appear, 2003.
Svetlozar T. Rachev. Probability Metrics and the Stability of Stochastic Models. Wiley, 1991.
Svetlozar T. Rachev and Ludger Rüschendorf. Rate of convergene for sums and maxima and doubly ideal metrics. Theory Prob. Appl., 37:276–289, 1992.
Svetlozar T. Rachev and Ludger Rüschendorf. Probability metrics and recursive algorithms. Advances Applied Probability, 27:770–799, 1995.
Mireille Régnier. A limiting distribution for quicksort. RAIRO, Informatique Théoriqué et Appl., 33:335–343, 1989.
Uwe Rösler. A limit theorem for Quicksort. RAIRO, Informatique Théoriqué et Appl., 25:85–100, 1991.
Uwe Rösler. A fixed point theorem for distribution. Stochastic Processes Applications, 42:195–214, 1992.
Uwe Rösler. On the analysis of stochastic divide and conquer algorithms. Algorithmica, 29:238–261, 2001.
Uwe Rösler and Ludger Rüschendorf. The contraction method for recursive algorithms. Algorithmica, 29:3–33, 2001.
Vladimir M. Zolotarev. Modern Theory of Summation of Random Variables. VSP, Utrecht, 1997.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Neininger, R., Rüschendorf, L. (2005). Analysis of Algorithms by the Contraction Method: Additive and Max-recursive Sequences. In: Deuschel, JD., Greven, A. (eds) Interacting Stochastic Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-27110-4_20
Download citation
DOI: https://doi.org/10.1007/3-540-27110-4_20
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-23033-5
Online ISBN: 978-3-540-27110-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)