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November 1996 On the distribution for the duration of a randomized leader election algorithm
James Allen Fill, Hosam M. Mahmoud, Wojciech Szpankowski
Ann. Appl. Probab. 6(4): 1260-1283 (November 1996). DOI: 10.1214/aoap/1035463332


We investigate the duration of an elimination process for identifying a winner by coin tossing or, equivalently, the height of a random incomplete trie. Applications of the process include the election of a leader in a computer network. Using direct probabilistic arguments we obtain exact expressions for the discrete distribution and the moments of the height. Elementary approximation techniques then yield asymptotics for the distribution. We show that no limiting distribution exists, as the asymptotic expressions exhibit periodic fluctuations.

In many similar problems associated with digital trees, no such exact expressions can be derived. We therefore outline a powerful general approach, based on the analytic techniques of Mellin transforms, Poissonization and de-Poissonization, from which distributional asymptotics for the height can also be derived. In fact, it was this complex variables approach that led to our original discovery of the exact distribution. Complex analysis methods are indispensable for deriving asymptotic expressions for the mean and variance, which also contain periodic terms of small magnitude.


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James Allen Fill. Hosam M. Mahmoud. Wojciech Szpankowski. "On the distribution for the duration of a randomized leader election algorithm." Ann. Appl. Probab. 6 (4) 1260 - 1283, November 1996.


Published: November 1996
First available in Project Euclid: 24 October 2002

zbMATH: 0870.60018
MathSciNet: MR1422986
Digital Object Identifier: 10.1214/aoap/1035463332

Primary: 05C05 , 60F05
Secondary: 05C80 , 60G70

Keywords: asymptotic distribution , de-Poissonization , distributed computing , Height , Leader election , poissonization , Random trees , tries

Rights: Copyright © 1996 Institute of Mathematical Statistics


Vol.6 • No. 4 • November 1996
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