Average Bit-Complexity of Euclidean Algorithms (2000)
| Venue: | Proceedings ICALP’00, Lecture Notes Comp. Science 1853, 373–387 |
| Citations: | 13 - 5 self |
BibTeX
@INPROCEEDINGS{Akhavi00averagebit-complexity,
author = {Ali Akhavi and Brigitte Vallée},
title = {Average Bit-Complexity of Euclidean Algorithms},
booktitle = {Proceedings ICALP’00, Lecture Notes Comp. Science 1853, 373–387},
year = {2000},
pages = {373--387},
publisher = {Springer}
}
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Abstract
We obtain new results regarding the precise average bit-complexity of five algorithms of a broad Euclidean type. We develop a general framework for analysis of algorithms, where the averagecase complexity of an algorithm is seen to be related to the analytic behaviour in the complex plane of the set of elementary transformations determined by the algorithms. The methods rely on properties of transfer operators suitably adapted from dynamical systems theory and provide a unifying framework for the analysis of an entire class of gcd-like algorithms. Keywords: Average-case Analysis of algorithms, Bit-Complexity, Euclidean Algorithms, Dynamical Systems, Ruelle operators, Generating Functions, Dirichlet Series, Tauberian Theorems. 1 Introduction Motivations. Euclid's algorithm was analysed first in the worst case in 1733 by de Lagny, then in the average-case around 1969 independently by Heilbronn [12] and Dixon [6], and finally in distribution by Hensley [13] who proved in 1994 that the Eu...
