Mathematical foundations of the GraphBLAS

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 GraphBLAS 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 multiplication. 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 a 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.

GraphBLAS C API: Ideas for future versions of the specification

“…Figure 1(a) illustrates the mathematical notation for that operation (optional items in brackets). This is similar to what is described in the mathematical specification [5], with some additions supported by the API -write masks and accumulation -which are contained in the signature in Figure 1(b). In the signature for every GraphBLAS operation, the output argument always appears first.…”

“…Like the original BLAS, this library performs operations on matrices and vectors. However, there are significant differences to specifically support graph operations as explained in the mathematical specification [5].…”

“…The API supports the basic operations that were outlined in mathematical specification [5]: inserting and extracting data, matrix multiply, element wise operations, subgraph assignment and extraction, apply, reduction, and transpose. It also provides a number of variants that are useful in a number of graph algorithms.…”

“…The GraphBLAS aims to standardize the mathematical concepts [5] and the application programming interface (API) [2] for performing graph computations in the language of linear algebra [6]. The GraphBLAS C API version 1.0 has been provisionally released in May 2017 [3].…”

Delta-Stepping SSSP: From Vertices and Edges to GraphBLAS Implementations

“…The GraphBLAS [6], [7] is a community effort to provide a standardized application programming interface (API) based on the language of linear algebra for the implementation of graph algorithms. This standardized interface allows for a separation of concerns between algorithm writers and library developers.…”