Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

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The categories of this paper are: nucl-ex
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The categories of this paper are: atom-ph
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The categories of this paper are: q-fin
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The categories of this paper are: math-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: solv-int
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The categories of this paper are: plasm-ph
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The categories of this paper are: adap-org
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The categories of this paper are: alg-geom
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The categories of this paper are: q-alg
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The categories of this paper are: mtrl-th
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The categories of this paper are: nucl-th
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The categories of this paper are: physics
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The categories of this paper are: patt-sol
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The categories of this paper are: funct-an
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The categories of this paper are: cmp-lg