Gaussian Laws for the Main Parameters of the Euclid Algorithms
We provide sharp estimates for the probabilistic behaviour of the main parameters of the Euclid Algorithms, both on polynomials and on integer numbers. We study in particular the distribution of the bit-complexity which involves two main parameters: digit-costs and length of remainders. We first sho...
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| Published in: | Algorithmica Vol. 50; no. 4; pp. 497 - 554 |
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| Main Authors: | , |
| Format: | Journal Article Conference Proceeding |
| Language: | English |
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01.04.2008
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| ISSN: | 0178-4617, 1432-0541 |
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| Abstract | We provide sharp estimates for the probabilistic behaviour of the main parameters of the Euclid Algorithms, both on polynomials and on integer numbers. We study in particular the distribution of the bit-complexity which involves two main parameters: digit-costs and length of remainders. We first show here that an asymptotic Gaussian law holds for the length of remainders at a fraction of the execution, which exhibits a deep regularity phenomenon. Then, we study in each framework—polynomials (
P
) and integer numbers (
I
)—two gcd algorithms, the standard one (
S
) which only computes the gcd, and the extended one (
E
) which also computes the Bezout pair, and is widely used for computing modular inverses.
The extended algorithm is more regular than the standard one, and this explains that our results are more precise for the Extended algorithm: we exhibit an asymptotic Gaussian law for the bit-complexity of the extended algorithm, in both cases (
P
) and (
I
). We also prove that an asymptotic Gaussian law for the bit-complexity of the standard gcd in case (
P
), but we do not succeed obtaining a similar result in case (
I
).
The integer study is more involved than the polynomial study, as it is usually the case. In the polynomial case, we deal with the central tools of the distributional analysis of algorithms, namely bivariate generating functions. In the integer case, we are led to dynamical methods, which heavily use the dynamical system underlying the number Euclidean algorithm, and its transfer operator. Baladi and Vallée (J. Number Theory 110(2):331–386,
2005
) have recently designed a general framework for “distributional dynamical analysis”, where they have exhibited asymptotic Gaussian laws for a large family of parameters. However, this family does not contain neither the bit-complexity cost nor the size of remainders, and we have to extend their methods for obtaining our results. Even if these dynamical methods are not necessary in case (
P
), we explain how the polynomial dynamical system can be also used for proving our results. This provides a common framework for both analyses, which well explains the similarities and the differences between the two cases (
P
) and (
I
), for the algorithms themselves, and also for their analysis. An extended abstract of this paper can be found in Lhote and Vallée (Proceedings of LATIN’06, Lecture Notes in Computer Science, vol. 3887, pp. 689–702,
2006
). |
|---|---|
| AbstractList | We provide sharp estimates for the probabilistic behaviour of the main parameters of the Euclid Algorithms, both on polynomials and on integer numbers. We study in particular the distribution of the bit-complexity which involves two main parameters: digit-costs and length of remainders. We first show here that an asymptotic Gaussian law holds for the length of remainders at a fraction of the execution, which exhibits a deep regularity phenomenon. Then, we study in each framework—polynomials (
P
) and integer numbers (
I
)—two gcd algorithms, the standard one (
S
) which only computes the gcd, and the extended one (
E
) which also computes the Bezout pair, and is widely used for computing modular inverses.
The extended algorithm is more regular than the standard one, and this explains that our results are more precise for the Extended algorithm: we exhibit an asymptotic Gaussian law for the bit-complexity of the extended algorithm, in both cases (
P
) and (
I
). We also prove that an asymptotic Gaussian law for the bit-complexity of the standard gcd in case (
P
), but we do not succeed obtaining a similar result in case (
I
).
The integer study is more involved than the polynomial study, as it is usually the case. In the polynomial case, we deal with the central tools of the distributional analysis of algorithms, namely bivariate generating functions. In the integer case, we are led to dynamical methods, which heavily use the dynamical system underlying the number Euclidean algorithm, and its transfer operator. Baladi and Vallée (J. Number Theory 110(2):331–386,
2005
) have recently designed a general framework for “distributional dynamical analysis”, where they have exhibited asymptotic Gaussian laws for a large family of parameters. However, this family does not contain neither the bit-complexity cost nor the size of remainders, and we have to extend their methods for obtaining our results. Even if these dynamical methods are not necessary in case (
P
), we explain how the polynomial dynamical system can be also used for proving our results. This provides a common framework for both analyses, which well explains the similarities and the differences between the two cases (
P
) and (
I
), for the algorithms themselves, and also for their analysis. An extended abstract of this paper can be found in Lhote and Vallée (Proceedings of LATIN’06, Lecture Notes in Computer Science, vol. 3887, pp. 689–702,
2006
). |
| Author | Lhote, Loïck Vallée, Brigitte |
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| Cites_doi | 10.3934/dcds.2006.15.281 10.1016/j.jnt.2004.08.008 10.1016/S0304-3975(02)00652-7 10.1006/jnth.1994.1088 10.2307/2302607 10.1112/S002557930001528X 10.1090/S0002-9939-96-03394-1 10.1215/S0012-7094-70-03756-7 10.2307/121012 10.1016/0022-314X(70)90044-2 10.1007/978-1-4615-4819-5_7 10.1007/11682462_63 10.1080/00029890.1938.11990797 10.1007/BF00289520 10.4064/aa-81-2-101-144 10.5802/jtnb.296 10.1007/3-540-45022-X_32 10.1017/S0963548304006261 |
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| Keywords | Dynamical analysis of algorithms Tauberian theorems Average-case analysis Transfer operator Perron’s formula Analysis of algorithms Dynamical systems Euclid’s algorithms Distributional analysis Asymptotic Gaussian laws Probabilistic approach Generating function Similarity Euclidean theory Distributional analysis· Dynamical systems Gaussian distribution Perron's formula .Dynamical analysis of algorithms Polynomial method Dynamical system Modeling Euclid's algorithms· Analysis of algorithms· Average-case analysis Bézout idendity Greatest commun divisor Asymptotic approximation Regularity Algorithm analysis Number theory |
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| References | Vallée (CR29) 2006; 15 Dixon (CR8) 1970; 2 CR17 Philipp (CR22) 1970; 37 Vallée (CR27) 2000; 12 CR12 CR11 Heilbronn, Turan (CR15) 1969 Friesen, Hensley (CR14) 1996; 124 Akhavi, Vallée (CR1) 2000 Vallée (CR26) 1997; 81 Delange (CR7) 1954; 71 Ruelle (CR23) 1978 CR4 CR6 CR5 Knopfmacher, Knopfmacher (CR18) 1988; 35 Finch (CR13) 2003 CR24 Baladi, Vallée (CR2) 2005; 110 Lehmer (CR19) 1938; 45 Dolgopyat (CR9) 1998; 147 Hensley (CR16) 1994; 49 CR20 Lhote, Vallée (CR21) 2006 Vallée (CR28) 2003; 297 Berthé, Nakada (CR3) 2000; 18 Von Zur Gathen, Gerhard (CR30) 1999 Tenenbaum (CR25) 1990 Ellison, Ellison (CR10) 1985 9009_CR4 9009_CR5 D. Dolgopyat (9009_CR9) 1998; 147 D. Ruelle (9009_CR23) 1978 B. Vallée (9009_CR27) 2000; 12 J. Knopfmacher (9009_CR18) 1988; 35 D.H. Lehmer (9009_CR19) 1938; 45 9009_CR6 9009_CR24 G. Tenenbaum (9009_CR25) 1990 S.R. Finch (9009_CR13) 2003 9009_CR20 H. Delange (9009_CR7) 1954; 71 V. Berthé (9009_CR3) 2000; 18 J. Zur Gathen Von (9009_CR30) 1999 V. Baladi (9009_CR2) 2005; 110 D. Hensley (9009_CR16) 1994; 49 C. Friesen (9009_CR14) 1996; 124 9009_CR17 W. Ellison (9009_CR10) 1985 9009_CR11 A. Akhavi (9009_CR1) 2000 9009_CR12 H. Heilbronn (9009_CR15) 1969 B. Vallée (9009_CR26) 1997; 81 L. Lhote (9009_CR21) 2006 B. Vallée (9009_CR28) 2003; 297 W. Philipp (9009_CR22) 1970; 37 B. Vallée (9009_CR29) 2006; 15 J.D. Dixon (9009_CR8) 1970; 2 |
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| SubjectTerms | Algorithm Analysis and Problem Complexity Algorithmics. Computability. Computer arithmetics Algorithms Applied sciences Computer Science Computer science; control theory; systems Computer Systems Organization and Communication Networks Data Structures and Information Theory Exact sciences and technology Mathematics of Computing Theoretical computing Theory of Computation |
| Title | Gaussian Laws for the Main Parameters of the Euclid Algorithms |
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