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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.
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Abstract

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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.

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 

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