On the distribution for the duration of a randomized leader election algorithm

Fill, James Allen; Mahmoud, Hosam M.; Szpankowski, Wojciech

doi:10.1214/aoap/1035463332

Cited by 66 publications

(65 citation statements)

References 27 publications

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“…This is closely related to the rich and intensively studied topic of randomized leader elections (see, e.g., [6,12,22,23,28]), where a computer network comprised of n processors attempts to single out a leader (in charge of communication, etc.) by means of a distributed randomized process generating the hierarchic tree.…”

Section: Related Workmentioning

confidence: 99%

Stochastic coalescence in logarithmic time

Loh

Lubetzky²

2013

Ann. Appl. Probab.

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The following distributed coalescence protocol was introduced by Dahlia Malkhi in 2006 motivated by applications in social networking. Initially there are n agents wishing to coalesce into one cluster via a decentralized stochastic process, where each round is as follows: every cluster flips a fair coin to dictate whether it is to issue or accept requests in this round. Issuing a request amounts to contacting a cluster randomly chosen proportionally to its size. A cluster accepting requests is to select an incoming one uniformly (if there are such) and merge with that cluster. Empirical results by Fernandess and Malkhi suggested the protocol concludes in O(log n) rounds with high probability, whereas numerical estimates by Oded Schramm, based on an ingenious analytic approximation, suggested that the coalescence time should be super-logarithmic.Our contribution is a rigorous study of the stochastic coalescence process with two consequences. First, we confirm that the above process indeed requires super-logarithmic time w.h.p., where the inefficient rounds are due to oversized clusters that occasionally develop. Second, we remedy this by showing that a simple modification produces an essentially optimal distributed protocol; if clusters favor their smallest incoming merge request then the process does terminate in O(log n) rounds w.h.p., and simulations show that the new protocol readily outperforms the original one. Our upper bound hinges on a potential function involving the logarithm of the number of clusters and the cluster-susceptibility, carefully chosen to form a supermartingale. The analysis of the lower bound builds upon the novel approach of Schramm which may find additional applications: rather than seeking a single parameter that controls the system behavior, instead one approximates the system by the Laplace transform of the entire cluster-size distribution. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Applied Probability, 2013, Vol. 23, No. 2, 492-528. This reprint differs from the original in pagination and typographic detail. 1 2 P.-S. LOH AND E. LUBETZKY 1. Introduction. The following stochastic distributed coalescence protocol was proposed by Malkhi in 2006, motivated by applications in social networking and the reliable formation of peer-to-peer networks (see [11] for more on these applications). The objective is to coalesce n participating agents into a single hierarchal cluster reliably and efficiently. To do so without relying on a centralized authority, the protocol first identifies each agent as a cluster (a singleton), and then proceeds in rounds as follows:

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Section: Related Workmentioning

confidence: 99%

Stochastic coalescence in logarithmic time

Loh

Lubetzky²

2013

Ann. Appl. Probab.

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“…(Technically, X * turns out to be the fixed point of the contraction given implicitly by the distributional equation (3). The contraction method was introduced by Rösler [14] to analyze the Quick Sort algorithm.…”

Section: Asymptotic Properties Of a Leader Election Algorithmmentioning

confidence: 99%

“…The possibly winnerless algorithm is a variation of an algorithm discussed in [3] and [12], the difference being when all the contestants flip tails in the possibly winnerless algorithms, they get eliminated, whereas in [3] and [12] the contest is reset and starts afresh with all the 570 R. KALPATHY ET AL.…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic Properties of a Leader Election Algorithm

2011

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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.

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“…Without being exhaustive, let us mention the following related papers: Prodinger [19], Fill, Mahmoud, Szpankowski [1], Janson, Szpankowski [5], Knessel [10], Lavault, Louchard [11], Janson, Lavault, Louchard [4], Louchard, Prodinger [15], Kalpathy, Mahmoud, Ward [9], Louchard, Martinez, Prodinger [12], Louchard, Prodinger [16], Louchard, Prodinger, Ward [17], Kalpathy [6], Kalpathy [7].…”

Section: Introductionmentioning

confidence: 99%

The asymmetric leader election algorithm: Number of survivors near the end of the game

Louchard

2015

Quaestiones Mathematicae

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The following classical asymmetric leader election algorithm has obtained quite a bit of attention lately. Starting with n players, each one throws a coin, and the k of them which have thrown head (with probability q) go on, and the leader will be found amongst them, using the same strategy. Should nobody advance, will the party repeat the procedure. One of the most interesting parameter here is the number J(n) of rounds until a leader has been identified. In this paper we investigate, in the classical leader election algorithm, what happens near the end of the game, namely we fix an integer κ and we study the behaviour of the number of survivors L at level J(n) − κ. In our asymptotic analysis (for n → ∞) we are focusing on the limiting distribution functions. We also investigate what happens, if the parameter p = 1 − q gets small (p → 0) or large (p → 1). We use three efficient tools: an urn model, a Mellin-Laplace technique for Harmonic sums and some asymptotic distributions related to one of the extreme-value distributions: the Gumbel law. This study was motivated by by a recent paper by Kalpathy, Mahmoud and Rosenkrantz, where they consider the number of survivors S n,t , after t election rounds, in a broad class of fair leader election algorithms starting with n candidates.Keywords: Asymmetric leader election algorithm, number of rounds until a leader has been identified, number of survivors near the end of the game, limiting distribution functions, urn model,

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On the distribution for the duration of a randomized leader election algorithm

Cited by 66 publications

References 27 publications

Stochastic coalescence in logarithmic time

Stochastic coalescence in logarithmic time

Asymptotic Properties of a Leader Election Algorithm

The asymmetric leader election algorithm: Number of survivors near the end of the game

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