GraphBLAS

Graph algorithms have long taken advantage of the idea that a graph can be represented as a matrix, and graph operations can be performed as linear transformations and other linear algebraic operations on sparse matrices. For example, matrix-vector multiplication can be used to perform a step in a breadth-first search. the GraphBLAS standard has also begun to interest people outside the graph community, including researchers in machine learning, and bioinformatics. GraphBLAS implementations have also been used in high-performance graph database applications such as FalkorDB formerly RedisGraph. Specification The GraphBLAS specification has been in development since 2013, and has reached version 2.1.0 as of December 2023. While formally a specification for the C programming language, a variety of programming languages have been used to develop implementations in the spirit of GraphBLAS, including C++, Java, and Nvidia CUDA. Compliant implementations and language bindings There are currently two fully-compliant reference implementations of the GraphBLAS specification. Bindings assuming a compliant specification exist for the Python, MATLAB, and Julia programming languages. ==Linear algebraic foundations==

The mathematical foundations of GraphBLAS are based in linear algebra and the duality between matrices and graphs. Each graph operation in GraphBLAS operates on a semiring, which is made up of the following elements: • A scalar addition operator (\oplus) • A scalar multiplication operator (\otimes) • A set (or domain) Note that the zero element (i.e. the element that represents the absence of an edge in the graph) can also be reinterpreted. For example, the following algebras can be implemented in GraphBLAS: All the examples above satisfy the following two conditions in their respective domains: • Additive identity, a \oplus 0 = a • Multiplicative annihilation, a \otimes 0 = 0 For instance, a user can specify the min-plus algebra over the domain of double-precision floating point numbers with GrB_Semiring_new(&min_plus_semiring, GrB_MIN_FP64, GrB_PLUS_FP64). ==Functionality==

While the GraphBLAS specification generally allows significant flexibility in implementation, some functionality and implementation details are explicitly described: • GraphBLAS objects, including matrices and vectors, are opaque data structures. • Non-blocking execution mode, which permits lazy or asynchronous evaluation of certain operations. • Masked assignment, denoted A\langle M \rangle = B, which assigns elements of matrix B to matrix A only in positions where the mask matrix M is non-zero. The GraphBLAS specification also prescribes that library implementations be thread-safe. ==Example code==

The following is a GraphBLAS 2.1-compliant example of a breadth-first search in the C programming language. • include • include • include • include • include "GraphBLAS.h" /* * Given a boolean n x n adjacency matrix A and a source vertex s, performs a BFS traversal * of the graph and sets v[i] to the level in which vertex i is visited (v[s] == 1). * If i is not reachable from s, then v[i] = 0 does not have a stored element. * Vector v should be uninitialized on input. */ GrB_Info BFS(GrB_Vector *v, GrB_Matrix A, GrB_Index s) { GrB_Index n; GrB_Matrix_nrows(&n,A); // n = # of rows of A GrB_Vector_new(v,GrB_INT32,n); // Vector v(n) GrB_Vector q; // vertices visited in each level GrB_Vector_new(&q, GrB_BOOL, n); // Vector q(n) GrB_Vector_setElement(q, (bool)true, s); // q[s] = true, false everywhere else /* * BFS traversal and label the vertices. */ int32_t level = 0; // level = depth in BFS traversal GrB_Index nvals; do { ++level; // next level (start with 1) GrB_apply(*v, GrB_NULL, GrB_PLUS_INT32, GrB_SECOND_INT32, q, level, GrB_NULL); // v[q] = level GrB_vxm(q, *v, GrB_NULL, GrB_LOR_LAND_SEMIRING_BOOL, q, A, GrB_DESC_RC); // q[!v] = q ||.&& A; finds all the // unvisited successors from current q GrB_Vector_nvals(&nvals, q); } while (nvals); // if there is no successor in q, we are done. GrB_free(&q); // q vector no longer needed return GrB_SUCCESS; } == See also ==