scholarly journals Asymptotic Properties of a Leader Election Algorithm

2011 ◽  
Vol 48 (02) ◽  
pp. 569-575 ◽  
Author(s):  
Ravi Kalpathy ◽  
Hosam M. Mahmoud ◽  
Mark Daniel Ward
Keyword(s):  
Geometric Distribution ◽  
Asymptotic Properties ◽  
Random Variable ◽  
Leader Election ◽  
Wasserstein Metric ◽  
Coin Tossing ◽  
Mean And Variance ◽  
The Mean ◽  
Election Algorithm ◽  
Leader Election Algorithm

We consider a serialized coin-tossing leader election algorithm that proceeds in rounds until a winner is chosen, or all contestants are eliminated. The analysis allows for either biased or fair coins. We find the exact distribution for the duration of any fixed contestant; asymptotically, it turns out to be a geometric distribution. Rice's method (an analytic technique) shows that the moments of the duration contain oscillations, which we give explicitly for the mean and variance. We also use convergence in the Wasserstein metric space to show that the distribution of the total number of coin flips (among all participants), suitably normalized, approaches a normal limiting random variable.

Asymptotic Properties of a Leader Election Algorithm

2011 ◽  
Vol 48 (2) ◽  
pp. 569-575 ◽  
Author(s):  
Ravi Kalpathy ◽  
Hosam M. Mahmoud ◽  
Mark Daniel Ward
Keyword(s):  
Geometric Distribution ◽  
Asymptotic Properties ◽  
Random Variable ◽  
Leader Election ◽  
Wasserstein Metric ◽  
Coin Tossing ◽  
Mean And Variance ◽  
The Mean ◽  
Election Algorithm ◽  
Leader Election Algorithm

We consider a serialized coin-tossing leader election algorithm that proceeds in rounds until a winner is chosen, or all contestants are eliminated. The analysis allows for either biased or fair coins. We find the exact distribution for the duration of any fixed contestant; asymptotically, it turns out to be a geometric distribution. Rice's method (an analytic technique) shows that the moments of the duration contain oscillations, which we give explicitly for the mean and variance. We also use convergence in the Wasserstein metric space to show that the distribution of the total number of coin flips (among all participants), suitably normalized, approaches a normal limiting random variable.

Asymptotic properties of the number of replications of a paired comparison

1974 ◽  
Vol 11 (1) ◽  
pp. 43-52 ◽  
Author(s):  
V. R. R. Uppuluri ◽  
W. J. Blot
Keyword(s):  
Paired Comparison ◽  
Beta Function ◽  
Asymptotic Properties ◽  
Random Variable ◽  
Incomplete Beta Function ◽  
Discrete Random Variable ◽  
World Series ◽  
Mean And Variance ◽  
The Mean ◽  
Made In

A discrete random variable describing the number of comparisons made in a sequence of comparisons between two opponents which terminates as soon as one opponent wins m comparisons is studied. By equating two different expressions for the mean of the variable, a closed form for the incomplete beta function with equal arguments is obtained. This expression is used in deriving asymptotic (m-large) expressions for the mean and variance. The standardized variate is shown to converge to the Gaussian distribution as m→ ∞. A result corresponding to the DeMoivre-Laplace limit theorem is proved. Finally applications are made to the genetic code problem, to Banach's Match Box Problem, and to the World Series of baseball.

Asymptotic properties of the number of replications of a paired comparison

1974 ◽  
Vol 11 (01) ◽  
pp. 43-52 ◽  
Author(s):  
V. R. R. Uppuluri ◽  
W. J. Blot
Keyword(s):  
Paired Comparison ◽  
Beta Function ◽  
Asymptotic Properties ◽  
Random Variable ◽  
Incomplete Beta Function ◽  
Discrete Random Variable ◽  
World Series ◽  
Mean And Variance ◽  
The Mean ◽  
Made In

A discrete random variable describing the number of comparisons made in a sequence of comparisons between two opponents which terminates as soon as one opponent wins m comparisons is studied. By equating two different expressions for the mean of the variable, a closed form for the incomplete beta function with equal arguments is obtained. This expression is used in deriving asymptotic (m-large) expressions for the mean and variance. The standardized variate is shown to converge to the Gaussian distribution as m→ ∞. A result corresponding to the DeMoivre-Laplace limit theorem is proved. Finally applications are made to the genetic code problem, to Banach's Match Box Problem, and to the World Series of baseball.

scholarly journals Convergence of some leader election algorithms

2008 ◽  
Vol Vol. 10 no. 3 (Analysis of Algorithms) ◽  
Author(s):  
Svante Janson ◽  
Christian Lavault ◽  
Guy Louchard
Keyword(s):  
Probability Distributions ◽  
Random Variable ◽  
Wasserstein Distance ◽  
Leader Election ◽  
Stochastic Monotonicity ◽  
Fixed Proportion ◽  
Variation Distance ◽  
International Audience ◽  
Election Algorithm ◽  
Leader Election Algorithm

International audience We start with a set of $n$ players. With some probability $P(n,k)$, we kill $n-k$ players; the other ones stay alive, and we repeat with them. What is the distribution of the number $X_n$ of \emph{phases} (or rounds) before getting only one player? We present a probabilistic analysis of this algorithm under some conditions on the probability distributions $P(n,k)$, including stochastic monotonicity and the assumption that roughly a fixed proportion $\al$ of the players survive in each round. We prove a kind of convergence in distribution for $X_n - \log_{1/\!\alpha}(n)$; as in many other similar problems there are oscillations and no true limit distribution, but suitable subsequences converge, and there is an absolutely continuous random variable $Z$ such that $d\l(X_n, \lceil Z + \log_{1/\!\alpha} (n)\rceil\r) \to 0$, where $d$ is either the total variation distance or the Wasserstein distance. Applications of the general result include the leader election algorithm where players are eliminated by independent coin tosses and a variation of the leader election algorithm proposed by W.R. Franklin. We study the latter algorithm further, including numerical results.

A Novel Hypergraph-Based Leader Election Algorithm for Distributed Systems

2020 ◽  
pp. 437-445
Author(s):  
E. R. S. Subramanian ◽  
B. Sri Gurubaran ◽  
A. S. Sayee Shruthi ◽  
V. Aishwarya ◽  
N. Balaji ◽  
...  
Keyword(s):  
Distributed Systems ◽  
Leader Election ◽  
Election Algorithm ◽  
Leader Election Algorithm

Implementing Leader Election Algorithm after Evaluation of Node Trust in MANETs

2017 ◽  
pp. 282-311
Author(s):  
Jayanta Das ◽  
Abhijit Das
Keyword(s):  
Leader Election ◽  
Election Algorithm ◽  
Leader Election Algorithm

Security and trust are two inevitable concepts for secure Manet. There are various systems used for ensuring security and trust in case of Manet. These systems have several advantages as well as several disadvantages in terms high communication and computation overhead. In this proposed trust based system, trust of node is evaluated on the basis of ratio of signal sent and acknowledgement received. After that, priority of each node is calculated and at last Leader Election algorithm is applied to select node leader.

scholarly journals Limit distribution of distances in biased random tries

2006 ◽  
Vol 43 (02) ◽  
pp. 377-390 ◽  
Author(s):  
Rafik Aguech ◽  
Nabil Lasmar ◽  
Hosam Mahmoud
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Limit Distribution ◽  
Analytical Techniques ◽  
Random Variable ◽  
Wasserstein Metric ◽  
Contraction Method ◽  
Point Solution ◽  
Mean And Variance

Thetrieis a sort of digital tree. Ideally, to achieve balance, the trie should grow from an unbiased source generating keys of bits with equal likelihoods. In practice, the lack of bias is not always guaranteed. We investigate the distance between randomly selected pairs of nodes among the keys in a biased trie. This research complements that of Christophi and Mahmoud (2005); however, the results and some of the methodology are strikingly different. Analytical techniques are still useful for moments calculation. Both mean and variance are of polynomial order. It is demonstrated that the standardized distance approaches a normal limiting random variable. This is proved by the contraction method, whereby the limit distribution is shown to approach the fixed-point solution of a distributional equation in the Wasserstein metric space.

The Asymmetric Leader Election Algorithm: Another Approach

2009 ◽  
Vol 12 (4) ◽  
pp. 449-478 ◽  
Author(s):  
Guy Louchard ◽  
Helmut Prodinger
Keyword(s):  
Leader Election ◽  
Election Algorithm ◽  
Leader Election Algorithm
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