Abstract
Most prior work in leader election algorithms deals with univariate statistics. We consider multivariate issues in a broad class of fair leader election algorithms. We investigate the joint distribution of the duration of two competing candidates. Under rather mild conditions on the splitting protocol, we prove the convergence of the joint distribution of the duration of any two contestants to a limit via convergence of distance (to 0) in a metric space on distributions. We then show that the limiting distribution is a Marshall-Olkin bivariate geometric distribution. Under the classic binomial splitting we are able to say a few more precise words about the exact joint distribution and exact covariance, and to explore (via Rice’s integral method) the oscillatory behavior of the diminishing covariance. We discuss several practical examples and present empirical observations on the rate of convergence of the covariance.
Similar content being viewed by others
References
Fill J, Mahmoud H, Szpankowski W (1996) The distribution for the duration of a randomized leader election algorithm. Ann Appl Probab 6:1260–1283
Flajolet P, Sedgewick R (1995) Mellin transforms and asymptotics: Finite differences and Rice’s integrals. Theor Comput Sci 144:101–124
Janson S, Szpankowski W (1997) Analysis of an asymmetric leader election algorithm. Electron J Comb 4(R17):1–16
Janson S, Lavault C, Louchard G (2008) Convergence of some leader election algorithms. Discret Math Theor Comput Sci 10:171–196
Kalpathy R, Mahmoud H (2014) Perpetuities in fair leader election algorithms. Adv App Prob 46
Kalpathy R, Ward M (2014) On a leader election algorithm: Truncated geometric case study. Stat Probab Lett 87:40–47
Kalpathy R, Mahmoud H, Rosenkrantz W (2013) Survivors in Leader Election Algorithms. Stat Probab Lett 83:2743–2749
Kalpathy R, Mahmoud H, Ward M (2011) Asymptotic properties of a leader election algorithm. J Appl Probab 48:569–575
Louchard G, Prodinger H (2009) The asymmetric leader election algorithm: Another approach. Ann of Comb 12:449–478
Louchard G, Martínez C, Prodinger H (2011) The Swedish leader election protocol: Analysis and variations. In: Flajolet P, Panario D (eds) Proceedings of the Eighth ACM-SIAM Workshop on Analytic Algorithmics and Combinatorics (ANALCO) San Francisco, pp. 127–134
Louchard G, Prodinger H, Ward M (2012) Number of survivors in the presence of a demon. Period Math Hung 64:101–117
Marshall A, Olkin I (1985) A family of bivariate distributions generated by the Bernoulli distribution. J Publ Am Stat Assoc 80:332–338
Neininger R (2001) On a multivariate contraction method for random recursive structures with applications to Quicksort. Random Struct Algorithm 19:498–524
Prodinger H (1993) How to select a loser. Discret Math 120:149–159
Saralees N (2008) Marshall and Olkin’s Distributions. Acta Applicandae Math 103:87–100
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhang, C., Mahmoud, H. Bivariate Issues in Leader Election Algorithms with Marshall-Olkin Limit Distribution. Methodol Comput Appl Probab 18, 401–418 (2016). https://doi.org/10.1007/s11009-014-9428-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11009-014-9428-1
Keywords
- Randomized algorithm
- Stochastic recurrence
- Marshall-Olkin distribution
- Weak convergence
- Rice’s integral method