Highly Parallel Linear Forest Extraction from a Weighted Graph on GPUs

Abstract

For graph matching, each vertex is allowed to match with exactly one other vertex, such that the spanning subgraph of the matching has a maximum degree of one, i.e., the subgraph is a [0,1]-factor. In this work, we provide a highly parallel algorithm to extract a spanning subgraph with a maximum degree of n (the subgraph is a [0,n]-factor) and demonstrate the efficiency of our GPU implementation for n=1,2,3,4 by expressing the algorithm in terms of generalized sparse matrix-vector products. Moreover, from the [0,2]-factor, we compute a maximum linear forest (union of disjoint paths) by breaking up cycles and permuting the subgraph with respect to the vertex order within the paths. Both tasks execute efficiently on the GPU because of our novel parallel scan implementation, which does not require a random access iterator. As an application of linear forests, we demonstrate the algebraic creation of enhanced tridiagonal preconditioners for various large matrices from the Sparse Matrix Collection and report runtimes in the order of milliseconds for graphs with millions of edges and vertices on an RTX 2080 Ti.