A Linear Space Algorithm for Computing the Hermite NormalFormDaniele

Computing the Hermite Normal Form of an n n matrix using the best current algorithms typically requires O(n 3 log M) space, where M is a bound on the length of the columns of the input matrix. Although polynomial in the input size (which is O(n 2 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 O(n 2 log M) space (i.e., essentially the same as the input size). When implemented using standard integer arithmetic, our algorithm has the same time complexity of the asymptotically fastest (but space ineecient) algorithms. We also suggest simple heuristics that when incorporated in our algorithm result in essentially the same asymptotic running time of the theoretically fastest solutions, still maintaining our algorithm extremely practical.