Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-23T18:27:17.431Z Has data issue: false hasContentIssue false
  • English
  • Français

Perpetuities in Fair Leader Election Algorithms

Part of: Limit theorems Algorithms - Computer Science Combinatorial probability

Published online by Cambridge University Press:  22 February 2016

Ravi Kalpathy  and
Hosam Mahmoud
Ravi Kalpathy*
Affiliation:
The George Washington University
Hosam Mahmoud*
Affiliation:
The George Washington University
*
∗ Postal address: Department of Statistics, The George Washington University, Washington, DC 20052, USA.
∗ Postal address: Department of Statistics, The George Washington University, Washington, DC 20052, USA.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider a broad class of fair leader election algorithms, and study the duration of contestants (the number of rounds a randomly selected contestant stays in the competition) and the overall cost of the algorithm. We give sufficient conditions for the duration to have a geometric limit distribution (a perpetuity built from Bernoulli random variables), and for the limiting distribution of the total cost (after suitable normalization) to be a perpetuity. For the duration, the proof is established via convergence (to 0) of the first-order Wasserstein distance from the geometric limit. For the normalized overall cost, the method of proof is also convergence of the first-order Wasserstein distance, augmented with an argument based on a contraction mapping in the first-order Wasserstein metric space to show that the limit approaches a unique fixed-point solution of a perpetuity distributional equation. The use of these two steps is commonly referred to as the contraction method.

Keywords

Leader electionrecurrencefunctional equationfixed pointcontraction methodmetric spaceperpetuityweak convergence

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 60F05: Central limit and other weak theorems 68W40: Analysis of algorithms
Type
Research Article
Information
Advances in Applied Probability , Volume 46 , Issue 1 , March 2014 , pp. 203 - 216
Copyright
© Applied Probability Trust 

References

Alsmeyer, G., Iksanov, A. and Rösler, U. (2009). On distributional properties of perpetuities. J. Theoret. Prob. 22, 666–682.CrossRefGoogle Scholar
Bickel, P. J. and Freedman, D. A. (1981). Some asymptotic theory for the bootstrap. Ann. Statist. 9, 1196–1217.CrossRefGoogle Scholar
Elmasry, A. and Mahmoud, H. (2011). Analysis of swaps in radix selection. Adv. Appl. Prob. 43, 524–544.Google Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events. Springer, Berlin.CrossRefGoogle Scholar
Fill, J. A., Mahmoud, H. M. and Szpankowski, W. (1996). On the distribution for the duration of a randomized leader election algorithm. Ann. Appl. Prob. 6, 1260–1283.Google Scholar
Itoh, Y. and Mahmoud, H. M. (2003). One-sided variations on interval trees. J. Appl. Prob. 40, 654–670.Google Scholar
Janson, S. and Szpankowski, W. (1997). Analysis of an asymmetric leader election algorithm. Electron. J. Combin. 4, Research Paper 17.CrossRefGoogle Scholar
Janson, S., Lavault, C. and Louchard, G. (2008). Convergence of some leader election algorithms. Discrete Math. Theoret. Comput. Sci. 10, 171–196.Google Scholar
Kalpathy, R., Mahmoud, H. M. and Ward, M. D. (2011). Asymptotic properties of a leader election algorithm. J. Appl. Prob. 48, 569–575.CrossRefGoogle Scholar
Kirschenhofer, P. and Prodinger, H. (1998). Comparisons in Hoare's Find algorithm. Combin. Prob. Comput. 7, 111–120.Google Scholar
Louchard, G. and Prodinger, H. (2009). The asymmetric leader election algorithm: another approach. Ann. Comb. 12, 449–478.Google Scholar
Louchard, G., Martı´nez, C. and Prodinger, H. (2011). The Swedish leader election protocol: analysis and variations. In ANALCO11—Workshop on Analytic Algorithmics and Combinatorics, SIAM, Philadelphia, PA, pp. 127–134.Google Scholar
Louchard, G., Prodinger, H. and Ward, M. D. (2012). Number of survivors in the presence of a demon. Period. Math. Hungar. 64, 101–117.Google Scholar
Mahmoud, H. M. (2010). Distributional analysis of swaps in Quick Select. Theoret. Comput Sci. 411, 1763–1769.CrossRefGoogle Scholar
Neininger, R. and Rüschendorf, L. (2004). A general limit theorem for recursive algorithms and combinatorial structures. Ann. Appl. Prob. 14, 378–418.CrossRefGoogle Scholar
Prodinger, H. (1993). How to select a loser. Discrete Math. 120, 149–159.Google Scholar
Rachev, S. T. and Rüschendorf, L. (1995). Probability metrics and recursive algorithms. Adv. Appl. Prob. 27, 770–799.Google Scholar
Rösler, U. (1991). A limit theorem for ‘Quicksort’. RAIRO Inform. Théor. Appl. 25, 85–100.Google Scholar
Rösler, U. (2004). QUICKSELECT revisited. J. Iranian Statist. Soc. 3, 271–296.Google Scholar
Rösler, U. and Rüschendorf, L. (2001). The contraction method for recursive algorithms. Algorithmica 29, 3–33.CrossRefGoogle Scholar
Vervaat, W. (1979). On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables. Adv. Appl. Prob. 11, 750–783.Google Scholar