Graph Expansion and Communication Costs of Fast Matrix Multiplication

The communication cost of algorithms (also known as I/O-complexity) is shown to be closely related to the expansion properties of the corresponding computation graphs. We demonstrate this on Strassen's and other fast matrix multiplication algorithms, and obtain the first lower bounds on their communication costs. In the sequential case, where the processor has a fast memory of size M, too small to store three n-by-n matrices, the lower bound on the number of words moved between fast and slow memory is, for a large class of matrix multiplication algorithms, Omega((n/root M)(omega 0) . M), where omega(0) is the exponent in the arithmetic count (e.g., omega(0) = lg 7 for Strassen, and omega(0) = 3 for conventional matrix multiplication). With p parallel processors, each with fast memory of size M, the lower bound is asymptotically lower by a factor of p. These bounds are attainable both for sequential and for parallel algorithms and hence optimal.