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Worst-Case Complexity of the Optimal LLL Algorithm

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1776))

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Abstract

In this paper, we consider the open problem of the complexity of the LLL algorithm in the case when the approximation parameter of the algorithm has its extreme value 1. This case is of interest because the output is then the strongest Lovász-reduced basis. Experiments reported by Lagarias and Odlyzko [13] seem to show that the algorithm remain polynomial in average. However no bound better than a naive exponential order one is established for the worst-case complexity of the optimal LLL algorithm, even for fixed small dimension (higher than 2). Here we prove that, for any fixed dimensionn, the number of iterations of the LLL algorithm is linear with respect to the size of the input. It is easy to deduce from [17] that the linear order is optimal. Moreover in 3 dimensions, we give a tight bound for the maximum number of iterations and we characterize precisely the output basis. Our bound also improves the known one for the usual (non-optimal) LLL algorithm.

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Authors and Affiliations

  1. GREYC – Université de Caen, F-14032, Caen Cedex, France

    Ali Akhavi

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  1. Ali Akhavi
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Editors and Affiliations

  1. ETH Zurich, Institute of Computational Science, CH-8092 Zurich and, Swiss Institute of Bioinformatics, Switzerland

    Gaston H. Gonnet

  2. Pedeciba Informática, Casilla de Correo 16120, Distrito 6, Montevideo, Uruguay

    Alfredo Viola

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© 2000 Springer-Verlag Berlin Heidelberg

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Akhavi, A. (2000). Worst-Case Complexity of the Optimal LLL Algorithm. In: Gonnet, G.H., Viola, A. (eds) LATIN 2000: Theoretical Informatics. LATIN 2000. Lecture Notes in Computer Science, vol 1776. Springer, Berlin, Heidelberg. https://doi.org/10.1007/10719839_35

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  • DOI: https://doi.org/10.1007/10719839_35

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-67306-4

  • Online ISBN: 978-3-540-46415-0

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