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

  • Mark Giesbrecht
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1122)

Abstract

We present a new probabilistic algorithm to compute the Smith normal form of a sparse integer matrix A ∈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(m2 log ∥A∥) such black-box evaluations reduced modulo word-sized primes p on vectors in ℤ p n×1 , plus O(m2n 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(n3 log2A∥) 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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Mark Giesbrecht
    • 1
  1. 1.Department of Computer ScienceUniversity of ManitobaWinnipegCanada

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