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• W2952203735 abstract "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 $tilde{O} left(tfrac{N}{B} minleft(sqrt{tfrac{Z}{M}},tfrac{N}{M}right) right)$ 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 $Omega left(tfrac{N}{B} minleft(sqrt{tfrac{Z}{M}},tfrac{N}{M}right) right)$ 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." @default.
• W2952203735 created "2019-06-27" @default.
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• W2952203735 date "2014-03-14" @default.
• W2952203735 modified "2023-09-27" @default.
• W2952203735 title "The Input/Output Complexity of Sparse Matrix Multiplication" @default.
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