高斯算法的另一种观点
Another view of the Gaussian algorithm
摘要：
本文介绍了一个幺模矩阵群\emph{i.e.}中的重写系统，该矩阵是具有整数元素且行列式等于$\pm 1$的矩阵.我们使用这个重写系统来精确地描述高斯算法的机制，该算法在由任何基给出的二维的格中找到最短向量。结合格点约简算法和重写系统理论，提出了一种新的高斯算法的最坏情况分析方法。由于Vallée \cite {ValGaussRevisit}，对于高斯算法的某些变体，已经存在最优的最坏情况界。她基本上使用了几何考虑。我们的分析推广她的结果的情况下，通常的高斯算法。在我们的工作中的一个有趣的一点是它可能（但不容易）推广到同一个问题在更高的维度，为了表现出一个严格的上限的迭代次数LLL-样减少算法在最坏的情况下。此外，我们的方法似乎可以用于分析其他家族的算法。作为一个例子，排序算法的分析在文章的最后一节简要地发展。
英文摘要：
We introduce here a rewrite system in the group of unimodular matrices, \emph{i.e.}, matrices with integer entries and with determinant equal to $\pm 1$. We use this rewrite system to precisely characterize the mechanism of the Gaussian algorithm, that finds shortest vectors in a two--dimensional lattice given by any basis. Putting together the algorithmic of lattice reduction and the rewrite system theory, we propose a new worst--case analysis of the Gaussian algorithm. There is already an optimal worst--case bound for some variant of the Gaussian algorithm due to Vallée \cite {ValGaussRevisit}. She used essentially geometric considerations. Our analysis generalizes her result to the case of the usual Gaussian algorithm. An interesting point in our work is its possible (but not easy) generalization to the same problem in higher dimensions, in order to exhibit a tight upper-bound for the number of iterations of LLL--like reduction algorithms in the worst case. Moreover, our method seems to work for analyzing other families of algorithms. As an illustration, the analysis of sorting algorithms are briefly developed in the last section of the paper.
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