Advances in Applied Probability

Perpetuities in fair leader election algorithms

Ravi Kalpathy and Hosam Mahmoud

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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.

Article information

Adv. in Appl. Probab., Volume 46, Number 1 (2014), 203-216.

First available in Project Euclid: 1 April 2014

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Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability
Secondary: 60F05: Central limit and other weak theorems 68W40: Analysis of algorithms [See also 68Q25]

Leader election recurrence functional equation fixed point contraction method metric space perpetuity weak convergence


Kalpathy, Ravi; Mahmoud, Hosam. Perpetuities in fair leader election algorithms. Adv. in Appl. Probab. 46 (2014), no. 1, 203--216. doi:10.1239/aap/1396360110.

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