Probabilistic computation of the Smith normal form of a sparse integer matrix

Abstract

We present a new probabilistic algorithm to compute the Smith normal form of a sparse integer matrix A ∈ ℤ^m×n. The algorithm treats A as a “black-box”; A is only used to compute matrix-vector products and we don't access individual entries in A directly. The algorithm requires about O(m ² log ∥A∥) such black-box evaluations reduced modulo word-sized primes p on vectors in ℤ ^n×1_p , plus O(m ² n log ∥A∥) additional bit operations. For sparse matrices this represents a substantial improvement over previously known algorithms. For example, on an n×n integer matrix A with O(n log n) non-zero entries, only about O(n ³ log² ∥A∥) bit operations are required to find the Smith form using standard integer arithmetic. The new algorithm suffers from no “fill-in” or intermediate value explosion, and uses very little additional space. The algorithm is probabilistic of the Monte Carlo type — on any input it returns the correct answer with a controllable, exponentially small probability of error.

Research was supported in part by Natural Sciences and Engineering Research Council of Canada research grant OGP0155376, and University of Manitoba research grant 431-1725-80.