Factorization and pseudofactorization of weighted graphs

For unweighted graphs, finding isometric embeddings of a graph $G$ is closely related to decompositions of $G$ into Cartesian products of smaller graphs. When $G$ is isomorphic to a Cartesian graph product, we call the factors of this product a factorization of $G$. When $G$ is isomorphic to an isometric subgraph of a Cartesian graph product, we call those factors a pseudofactorization of $G$. Prior work has shown that an unweighted graph's pseudofactorization can be used to generate a canonical isometric embedding into a product of the smallest possible pseudofactors. However, for arbitrary weighted graphs, which represent a richer variety of metric spaces, methods for finding isometric embeddings or determining their existence remain elusive, and indeed pseudofactorization and factorization have not previously been extended to this context. In this work, we address the problem of finding the factorization and pseudofactorization of a weighted graph $G$, where $G$ satisfies the property that every edge constitutes a shortest path between its endpoints. We term such graphs minimal graphs, noting that every graph can be made minimal by removing edges not affecting its path metric. We generalize pseudofactorization and factorization to minimal graphs and develop new proof techniques that extend the previously proposed algorithms due to Graham and Winkler [Graham and Winkler, '85] and Feder [Feder, '92] for pseudofactorization and factorization of unweighted graphs. We show that any $n$-vertex, $m$-edge graph with positive integer edge weights can be factored in $O\left(m^2\right)$ time, plus the time to find all pairs shortest paths (APSP) distances in a weighted graph, resulting in an overall running time of $O\left(m^2+n^2\text{\hspace{0.17em}log\hspace{0.17em}log\hspace{0.17em}}n\right)$ time. We also show that a pseudofactorization for such a graph can be computed in $O\left(mn\right)$ time, plus the time to solve APSP, resulting in an $O\left(mn+n^2\text{\hspace{0.17em}log\hspace{0.17em}log\hspace{0.17em}}n\right)$ running time.