Algebraic Conditions on One-Step Breadth-First Search

The GraphBLAS community has demonstrated the power of linear algebra-leveraged graph algorithms, such as matrix-vector products for breadth-first search (BFS) traversals. This paper investigates the algebraic conditions needed for such computations when working with directed hypergraphs, represented by incidence arrays with entries from an arbitrary value set with binary addition and multiplication operations. Our results show the one-step BFS traversal is equivalent to requiring specific algebraic properties of those operations. Assuming identity elements 0, 1 for operations, we show that the two operations must be zero-sum-free, zero-divisor-free, and 0 must be an annihilator under multiplication. Additionally, associativity and commutativity are shown to be necessary and sufficient for independence of the one-step BFS computation from several arbitrary conventions. These results aid in application and algorithm development by determining the efficacy of a value set in computations.