The Input/Output Complexity of Sparse Matrix Multiplication

We consider the problem of multiplying sparse matrices (over a semiring) where the number of non-zero entries is larger than main memory. In the classical paper of Hong and Kung (STOC '81) it was shown that to compute a product of dense $U \times U$ matrices, $\Theta \left(U^3 / (B \sqrt{M}) \right)$ I/Os are necessary and sufficient in the I/O model with internal memory size $M$ and memory block size $B$. In this paper we generalize the upper and lower bounds of Hong and Kung to the sparse case. Our bounds depend of the number $N = \mathtt{nnz}(A)+\mathtt{nnz}(C)$ of nonzero entries in $A$ and $C$, as well as the number $Z = \mathtt{nnz}(AC)$ of nonzero entries in $AC$. We show that $AC$ can be computed using $\tilde{O} \left(\tfrac{N}{B} \min\left(\sqrt{\tfrac{Z}{M}},\tfrac{N}{M}\right) \right)$ I/Os, with high probability. This is tight (up to polylogarithmic factors) when only semiring operations are allowed, even for dense rectangular matrices: We show a lower bound of $\Omega \left(\tfrac{N}{B} \min\left(\sqrt{\tfrac{Z}{M}},\tfrac{N}{M}\right) \right)$ I/Os. While our lower bound uses fairly standard techniques, the upper bound makes use of ``compressed matrix multiplication'' sketches, which is new in the context of I/O-efficient algorithms, and a new matrix product size estimation technique that avoids the ``no cancellation'' assumption.