Multiple GCDs. probabilistic analysis of the plain algorithm
Pages 37 - 44
Abstract
This paper provides a probabilistic analysis of an algorithm which computes the gcd of ℓ inputs (with ℓ ≥ 2), with a succession of ℓ - 1 phases, each of them being the Euclid algorithm on two entries. This algorithm is both basic and natural, and two kinds of inputs are studied: polynomials over the finite field Fq and integers. The analysis exhibits the precise probabilistic behaviour of the main parameters, namely the number of iterations in each phase and the evolution of the length of the current gcd along the execution. We first provide an average-case analysis. Then we make it even more precise by a distributional analysis. Our results rigorously exhibit two phenomena: (i) there is a strong difference between the first phase, where most of the computations are done and the remaining phases; (ii) there is a strong similarity between the polynomial and integer cases, as can be expected.
References
[1]
V. Baladi and B. Vallée. Euclidean algorithms are Gaussian. Journal of Number Theory, 110:331--386, 2006.
[2]
V. Berthé and H. Nakada. On continued fraction expansions in positive characteristic: Equivalence relations and some metric properties. Expositiones Mathematicae, 18:257--284, 2000.
[3]
H. Daudé, P. Flajolet, and B. Vallée. An average-case analysis of the Gaussian Algorithm for Lattice Reduction. Combinatorics, Probability & Computing, 6(4):397--433, 1997.
[4]
H. Delange. Généralisation du théorème d'Ikehara. Ann. Sc. ENS, 71:213--422, 1954.
[5]
J. D. Dixon. The number of steps in the Euclidean algorithm. Journal of Number Theory, 2:414--422, 1970.
[6]
D. Dolgopyat. On decay of correlations in Anosov flows. Ann. of Math., 147(2):357--390, 1998.
[7]
H. Edwards. Riemann's Zeta Function. Dover Publications, 2001.
[8]
P. Flajolet and R. Sedgewick. Analytic Combinatorics. Cambridge University Press, 2009.
[9]
P. Flajolet and B. Vallée. Continued fraction algorithms, functional operators, and structure constants. Theor. Comput. Sci., 194(1-2):1--34, 1998.
[10]
C. Friesen and D. Hensley. The statistics of continued fractions for polynomials over a finite field. Proc. Amer. Math. Soc, 124:2661--2673, 1996.
[11]
H. Heilbronn. On the average length of a class of continued fractions. In P. Turan, editor, Number Theory and Analysis, pages 87--96, 1969.
[12]
D. Hensley. The number of steps in the Euclidean algorithm. Journal of Number Theory, 2(49):149--182, 1994.
[13]
A. Knopfmacher and J. Knopfmacher. The exact length of the Euclidean algorithm in Fq{X}. Mathematika, 35:297--304, 1988.
[14]
D. E. Knuth. Seminumerical Algorithms, volume 2 of The Art of Computer Programming. Addison-Wesley, 3rd edition, 1998.
[15]
L. Lhote and B. Vallée. Gaussian laws for the main parameters of the Euclid algorithms. Algorithmica, 50(4):497--554, 2008.
[16]
K. Ma and J. von zur Gathen. Analysis of Euclidean algorithms for polynomials over finite fields. Journal of Symbolic Computation, 9(4):429--455, 1990.
[17]
D. Ruelle. Thermodynamic Formalism. Cambridge University Press, 2004.
[18]
B. Vallée. Euclidean dynamics. Discrete and Continuous Dynamical Systems, 1(15):281--352, 2006.
[19]
J. von zur Gathen and J. Gerhard. Modern Computer Algebra. Cambridge University Press, 2003.
[20]
J. von zur Gathen, M. Mignotte, and I. Shparlinski. Approximate polynomial gcd: Small degree and small height perturbations. Journal of Symbolic Computation, 45(8):879--886, 2010.
Index Terms
- Multiple GCDs. probabilistic analysis of the plain algorithm
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Published In
June 2013
400 pages
ISBN:9781450320597
DOI:10.1145/2465506
- General Chairs:
- Michael Monagan,
- Gene Cooperman,
- Program Chair:
- Mark Giesbrecht
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Published: 26 June 2013
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