Fast algorithms for anti-distance matrices as a generalization of Boolean matrices

We show that Boolean matrix multiplication, computed as a sum of products of column vectors with row vectors, is essentially the same as Warshalls algorithm for computing the transitive closure matrix of a graph from its adjacency matrix. Warshalls algorithm can be generalized to Floyds algorithm for computing the distance matrix of a graph with weighted edges. We will generalize Boolean matrices in the same way, keeping matrix multiplication essentially equivalent to the Floyd-Warshall algorithm. This way, we get matrices over a semiring, which are similar to the so-called funny matrices. We discuss our implementation of operations on Boolean matrices and on their generalization, which make use of vector instructions.