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

## TL;DR

This paper generalizes Boolean matrices to matrices over a semiring, extending the connection between Boolean matrix multiplication and graph algorithms like Warshall's and Floyd-Warshall, with efficient vectorized implementations.

## Contribution

It introduces a generalized matrix framework over a semiring that extends Boolean matrices and maintains algorithmic equivalence with Floyd-Warshall, along with optimized implementation techniques.

## Key findings

- Generalized matrices preserve Floyd-Warshall algorithm properties.
- Efficient vectorized operations improve computational performance.
- Framework unifies Boolean and weighted graph matrix computations.

## Abstract

We show that Boolean matrix multiplication, computed as a sum of products of column vectors with row vectors, is essentially the same as Warshall's algorithm for computing the transitive closure matrix of a graph from its adjacency matrix. Warshall's algorithm can be generalized to Floyd's 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.