Revisiting Hypergraph Models for Sparse Matrix Decomposition

We provide an exposition of the hypergraph models for parallel sparse matrix-vector multiplies based on one-dimensional (1D) matrix partitioning. Our aim is to emphasize the expressive power of the hypergraph models. We first set forth an elementary hypergraph model in which vertices represent the data elements of a matrix-vector multiply operation and nets encode data dependencies. We then apply a recently proposed hypergraph transformation operation to devise models for 1D sparse matrix decomposition. The resulting 1D partitioning models are equivalent to the previously proposed computational hypergraph models and are not meant to be replacements for them. Nevertheless, the new models give us insights into the previous ones and help us explain a subtle requirement, known as the consistency condition, of the hypergraph partitioning models. We also demonstrate the flexibility of the elementary model on a few partitioning problems that are hard to solve using the previously proposed models.