On the Probabilistic Analysis of Normal Form Computation of a Sparse Matrix

Abstract

An $(s, t)$-sparse matrix has $s$ non-zero entries per column and $t$ per row. $(s, t)$-sparse integer matrices arise in the computation of integral homology. In this paper, a probabilistic analysis is given for diagonalizing an integer $(s, t)$-sparse matrix into normal formal. By normal form of a matrix, we mean the diagonalization of the matrix over the ring of integers. We prove that under high probability the expected running time can be achieved with probability very close $(s, t)$-sparse matrix, i.e. this expected running time can be achieved with probability very close to 1 when $(s, t)\ll n$.