Mathematical Foundations of the GraphBLAS

TL;DR

The paper introduces the mathematical principles of the GraphBLAS standard, emphasizing matrix operations for graph algorithms, and demonstrates its low performance overhead in implementations.

Contribution

It provides a mathematical foundation for GraphBLAS, detailing how matrix operations can efficiently implement diverse graph algorithms across programming environments.

Findings

Matrix operations enable wide range of graph algorithms

Prototype implementations show low performance overhead

GraphBLAS facilitates composable and efficient graph processing

Abstract

The GraphBLAS standard (GraphBlas.org) is being developed to bring the potential of matrix based graph algorithms to the broadest possible audience. Mathematically the Graph- BLAS defines a core set of matrix-based graph operations that can be used to implement a wide class of graph algorithms in a wide range of programming environments. This paper provides an introduction to the mathematics of the GraphBLAS. Graphs represent connections between vertices with edges. Matrices can represent a wide range of graphs using adjacency matrices or incidence matrices. Adjacency matrices are often easier to analyze while incidence matrices are often better for representing data. Fortunately, the two are easily connected by matrix mul- tiplication. A key feature of matrix mathematics is that a very small number of matrix operations can be used to manipulate a very wide range of graphs. This composability of small number of operations is the foundation of the GraphBLAS. A standard such as the GraphBLAS can only be effective if it has low performance overhead. Performance measurements of prototype GraphBLAS implementations indicate that the overhead is low.