How to select a loser

Prodinger, Helmut

doi:10.1016/0012-365x(93)90572-b

Cited by 68 publications

(69 citation statements)

References 4 publications

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“…This is a classic example and several of its properties have been studied (see [16,6,7,12]). Contestants flip coins, producing Heads with probability p, and producing Tails with probability 1 − p. Those who flip Heads advance to the next round; those who flip Tails are out of contention.…”

Section: Binomial Splittingmentioning

confidence: 99%

“…The classic case, introduced in [16], deals with binomial splitting protocols. It was further researched in [6,7,13,12,14,15], among other references.…”

Section: Introductionmentioning

confidence: 99%

“…Typical parameters in the analysis are the number of rounds till termination, i.e., the height of the underlying incomplete tree (see [6,7]), the total cost till termination (see [16,12,9,10]), and the number of rounds a contestant survives (see [12,9,10]), which is an important measure from the point of view of an individual contestant.…”

Section: Introductionmentioning

confidence: 99%

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Survivors in Leader Election Algorithms

Kalpathy

Mahmoud

Rosenkrantz

2013

2014 Proceedings of the Eleventh Workshop on Analytic Algorithmics and Combinatorics (ANALCO)

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We consider the number of survivors in a broad class of fair leader election algorithms after a number of election rounds. We give sufficient conditions for the number of survivors to converge to a product of independent identically distributed random variables. The number of terms in the product is determined by the round number considered. Each individual term in the product is a limit of a scaled random variable associated with the splitting protocol. The proof is established via convergence (to 0) of the first-order Wasserstein distance from the product limit. In a broader context, the paper is a case study of a class of stochastic recursive equations. We give two illustrative examples, one with binomial splitting protocol (for which we show that a normalized version is asymptotically Gaussian), and one with power distribution splitting protocol. This is an extended abstract of [11].

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Section: Binomial Splittingmentioning

confidence: 99%

“…The classic case, introduced in [16], deals with binomial splitting protocols. It was further researched in [6,7,13,12,14,15], among other references.…”

Section: Introductionmentioning

confidence: 99%

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Survivors in Leader Election Algorithms

Kalpathy

Mahmoud

Rosenkrantz

2013

2014 Proceedings of the Eleventh Workshop on Analytic Algorithmics and Combinatorics (ANALCO)

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“…The number of rounds for instance coincides with the leftmost branch of tree, a parameter easily amenable to the analytic techniques described in Section 2. The complete protocol has been analysed thoroughly by Prodinger [49] and Fill et al [20]. Fluctuations are once more everywhere to be found.…”

Section: Leader Election and Protocolsmentioning

confidence: 99%

The Ubiquitous Digital Tree

Flajolet

2006

Stacs 2006

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Abstract. The digital tree also known as trie made its first appearance as a general-purpose data structure in the late 1950's. Its principle is a recursive partitioning based on successive bits or digits of data items. Under various guises, it has then surfaced in the management of very large data bases, in the design of efficient communication protocols, in quantitative data mining, in the leader election problem of distributed computing, in data compression, as well as in some corners of computational geometry. The algorithms are invariably very simple, easy to implement, and in a number of cases surprisingly efficient. The corresponding quantitative analyses pose challenging mathematical problems and have triggered a flurry of research works. Generating functions and symbolic methods, singularity analysis, the saddle-point method, transfer operators of dynamical systems theory, and the Mellin transform have all been found to have a bearing on the probabilistic behaviour of trie algorithms. We offer here a perspective on the rich algorithmic, analytic, and probabilistic aspects of tries, culminating with a connection between a sorting problem and the Riemann hypothesis.

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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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How to select a loser

Cited by 68 publications

References 4 publications

Survivors in Leader Election Algorithms

Survivors in Leader Election Algorithms

The Ubiquitous Digital Tree

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

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