Computing the Hermite Normal Form of an n × n integer matrix using the best current algorithms typically requires &Ogr;(n3 log M) space, where M is a bound on the entries of the input matrix. Although polynomial in the input size (which is &Ogr;(n2 log M)), this space blow-up can easily become a serious issue in practice when working on big integer matrices. In this paper we present a new algorithm for computing the Hermite Normal Form which uses only &Ogr;(n2 log M) space (i.e., essentially the same as the input size). When implemented using standard algorithms for integer and matrix multiplication, our algorithm has the same time complexity of the asymptotically fastest (but space inefficient) algorithms. We also present a heuristic algorithm for HNF that achieves a substantial speedup when run on randomly generated input matrices.
使用最佳当前算法计算 n I>× n I>整数矩阵的Hermite范式通常需要&Ogr; I>( n I> 3 SUP>记录 M I>)空间,其中<I> M I>是输入矩阵项上的边界。尽管输入大小为多项式(&Ogr; I>( n I> 2 SUP> log M I>)),但这种空间打击实际上,在处理大整数矩阵时,up很容易成为一个严重的问题。在本文中,我们提出了一种仅使用&Ogr; I>( n I> 2 SUP> log M I>)空间(即,与输入大小基本相同)。当使用用于整数和矩阵乘法的标准算法实现时,我们的算法具有与渐近最快(但空间效率低)算法相同的时间复杂度。我们还提出了一种针对HNF的启发式算法,当在随机生成的输入矩阵上运行时,该算法可显着提高速度。 P>
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