A note on the multiplication of sparse matrices

Abstract

We present a practical algorithm for multiplication of two sparse matrices. In fact if A and B are two matrices of size n with m ₁ and m ₂ non-zero elements respectively, then our algorithm performs O(min{m ₁ n, m ₂ n, m ₁ m ₂}) multiplications and O(k) additions where k is the number of non-zero elements in the tiny matrices that are obtained by the columns times rows matrix multiplication method. Note that in the useful case, k ≤ m ₂ n. However, in Proposition 3.3 and Proposition 3.4 we obtain tight upper bounds for the complexity of additions. We also study the complexity of multiplication in a practical case where non-zero elements of A (resp. B) are distributed independently with uniform distribution among columns (resp. rows) of them and show that the expected number of multiplications is O(m ₁ m ₂/n). Finally a comparison of number of required multiplications in the naïve matrix multiplication, Strassen’s method and our algorithm is given.