A Compressed, Divide and Conquer Algorithm for Scalable Distributed Matrix-Matrix Multiplication

Matrix-matrix multiplication (GEMM) is a widely used linear algebra primitive common in scientific computing and data sciences. While several highly-tuned libraries and implementations exist, these typically target either sparse or dense matrices. The performance of these tuned implementations on unsupported types can be poor, and this is critical in cases where the structure of the computations is associated with varying degrees of sparsity. One such example is Algebraic Multigrid (AMG), a popular solver and preconditioner for large sparse linear systems. In this work, we present a new divide and conquer sparse GEMM, that is also highly performant and scalable when the matrix becomes dense, as in the case of AMG matrix hierarchies. In addition, we implement a lossless data compression method to reduce the communication cost. We combine this with an efficient communication pattern during distributed-memory GEMM to provide 2.24 times (on average) better performance than the state-of-the-art library PETSc. Additionally, we show that the performance and scalability of our method surpass PETSc even more when the density of the matrix increases. We demonstrate the efficacy of our methods by comparing our GEMM with PETSc on a wide range of matrices.