  We propose COSMA: a parallel matrix-matrix multiplication algorithm that is
near communication-optimal for all combinations of matrix dimensions, processor
counts, and memory sizes. The key idea behind COSMA is to derive an optimal (up
to a factor of 0.03\% for 10MB of fast memory) sequential schedule and then
parallelize it, preserving I/O optimality. To achieve this, we use the red-blue
pebble game to precisely model MMM dependencies and derive a constructive and
tight sequential and parallel I/O lower bound proofs. Compared to 2D or 3D
algorithms, which fix processor decomposition upfront and then map it to the
matrix dimensions, it reduces communication volume by up to $\sqrt{3}$ times.
COSMA outperforms the established ScaLAPACK, CARMA, and CTF algorithms in all
scenarios up to 12.8x (2.2x on average), achieving up to 88\% of Piz Daint's
peak performance. Our work does not require any hand tuning and is maintained
as an open source implementation.
