Optimal algebraic Breadth-First Search for sparse graphs
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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. These are useful in practice because many graph algorithms are amenable to sparse matrix multiplication. An exemplar for these approaches is Breadth-First Search (BFS). Despite many redundant operations over nonzeros that ultimately lead to suboptimal performance, the algebraic BFS is appealing for practical implementations because it is simple and embarrassingly parallel. 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 recent 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. Our method multiplies progressively smaller submatrices of the adjacency matrix at each step. The matrix remains unchanged, rather we are masking the rows and columns in the matrix. The input vector in each step is also effectively masked, thus our method ultiplies a sparse submatrix by a sparse vector in decreasing size each step. Our algebraic BFS algorithm takes O(m) algebraic operations on a sparse graph with O(m) edges as opposed to O(mn) operations of other sparse matrix approaches.
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