Sparse matrix-matrix multiplication on modern architectures

摘要

Sparse matrix-sparse/dense matrix multiplications, spgemm and csrmm, respectively, among other applications find usage in various matrix formulations of graph problems. Considering the difficulties in executing graph problems and the duality between graphs and matrices, computations such as spgemm and csrmm have recently caught the attention of HPC community. These computations pose challenges such as load balancing, irregular nature of the computation, and difficulty in predicting the output size. It is even more challenging when combined with the GPU architectural constraints such as memory accesses, limited shared memory, strict SIMD and thread execution. To address these challenges on a GPU, we evaluate three possible variations of matrix multiplication (Row-Column, Column-Row, Row-Row) and perform suitable optimizations targeted at sparse matrices. Our experiments indicate that the Row-Row formulation, which mostly outperforms the other formulations, is 3.5x faster on average compared to an optimized multi-core implementation in the Intel MKL library. We extend the Row-Row formulation to a CPU+GPU hybrid algorithm that simultaneously utilizes the CPU also. In this direction, we present heuristics to find the right amount of work division between the CPU and the GPU. Our hybrid row-row formulation of the spgemm operation performs 5.5x faster on average when compared to the optimized multi-core implementation in the Intel MKL library. Our experience indicates that it is difficult to identify right amount of work division between the CPU and the GPU. We therefore investigate a subclass of sparse matrices, band matrices, and present an analytical method to identify a good work division when multiplying two band matrices. Our GPU csrmm operation performs 2.5x faster on average when compared to a corresponding implementation in the cusparse library, which outperforms the Intel MKL library implementation.