Probabilistic analyses of the plain multiple gcd algorithm

Among multiple gcd algorithms on polynomials as on integers, one of the most natural ones performs a sequence of ℓ−1 phases (ℓ is the number of inputs), with each of them being the Euclid algorithm on two entries. We present here a complete probabilistic analysis of this algorithm, by providing both...

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Vydané v:Journal of symbolic computation Ročník 74; s. 425 - 474
Hlavní autori: Berthé, Valérie, Lhote, Loïck, Vallée, Brigitte
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: Elsevier Ltd 01.05.2016
Elsevier
Predmet:
Analysis of algorithms
Analytic combinatorics
Beta law
Computer Arithmetic
Computer Science
Discrete Mathematics
Dynamical analysis
Gcd algorithms
Generating functions
Landau Theorem
Limit laws
Perron Formula
Transfer operator
Gcd algorithms
Limit laws
Landau Theorem
Transfer operator
Generating functions
Dynamical analysis
Perron Formula
Analysis of algorithms
Analytic combinatorics
Beta law
Limit distributions
GCD algorithms
Dynamical analysis of Algorithms
ISSN:0747-7171, 1095-855X
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Popis
Shrnutí:Among multiple gcd algorithms on polynomials as on integers, one of the most natural ones performs a sequence of ℓ−1 phases (ℓ is the number of inputs), with each of them being the Euclid algorithm on two entries. We present here a complete probabilistic analysis of this algorithm, by providing both the average-case and the distributional analysis, and by handling in parallel the integer and the polynomial cases, for polynomials with coefficients in a finite field. The main parameters under consideration are the number of iterations in each phase and the evolution of the size of the current gcd along the execution. Three phenomena are clearly emphasized through this analysis: the fact that almost all the computations are performed during the first phase, the great difference between the probabilistic behavior of the first phase compared to subsequent phases, and, as can be expected, the great similarity between the integer and the polynomial cases.
ISSN:0747-7171
1095-855X
DOI:10.1016/j.jsc.2015.08.007