There has been a rise in the popularity of algebraic methods for graph algorithms given the development of the GraphBLAS library and other sparse matrix methods. An exemplar for these approaches is Breadth-First Search (BFS). The algebraic BFS algorithm is simply a recursion of matrix-vector multiplications with the $n \times n$ adjacency matrix, but the many redundant operations over nonzeros ultimately lead to suboptimal performance. Therefore an optimal algebraic BFS should be of keen interest especially if it is easily integrated with existing matrix methods. Current methods, notably in the GraphBLAS, use a Sparse Matrix Sparse Vector (SpMSpV) multiplication in which the input vector is kept in a sparse representation in each step of the BFS. But simply applying SpMSpV in BFS does not lead to optimal runtime. Each nonzero in the vector must be masked in subsequent steps. This has been an area of recent research in GraphBLAS and other libraries. While in theory these masking methods are asymptotically optimal on sparse graphs, many add work that leads to suboptimal runtime. We give a new optimal, algebraic BFS for sparse graphs that is also a constant factor faster than theoretically optimal SpMSpV methods, closing a gap in the literature. Our method multiplies progressively smaller submatrices of the adjacency matrix at each step, taking $O(m)$ algebraic operations on a sparse graph of $O(m)$ edges as opposed to $O(mn)$ operations of other sparse matrix approaches. Thus for sparse graphs it matches the bounds of the best-known sequential algorithm and on a Parallel Random Access Machine (PRAM) it is work-optimal. Compared to a leading GraphBLAS library our method achieves up to 24x faster sequential time and for parallel computation it can be 17x faster on large graphs and 12x faster on large-diameter graphs.

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