Context: Take a directed graph $G$ with a specified subset of source vertices $S$ and target vertices $T$. We say a subset $I\subseteq T$ of size $r$ is independent if there exist $r$ distinct vertices in $S$ which can be connected to distinct vertices of $I$ via a collection of $r$ vertex-disjoint paths in $G$. In other words, there is a flow of size $r$ from $S$ to $I$.
It turns out that a graph together with independent subsets of vertices defined in this way forms a structure known as a gammoid, which is itself a special case of a structure known as a matroid. Many algorithms that work with matroids take as input a linear representation of the matroid. In the context of this problem, this representation is a matrix $M$ whose column vectors $\vec{v}_u$ are indexed by vertices $u$ in $G$, with the property that a subset $I$ of vertices is independent if and only if the corresponding list of column vectors $\{\vec{v}_u\}_{u\in I}$ is linearly independent.
Several sources (see here and here for example) observe that is well-known that given $G$, $S$, and $T$, one can build a matrix representation of the associated gammoid in randomized polynomial time. However, I have not been able to find any reference to the precise running time of this algorithm. This motivates the following question.
Question: Given a graph $G$ with source set $S$ and target set $T$, how quickly can we build a matrix representation of the associated gammoid? Randomness is allowed.
I'm mainly interested in the dependence on the the number of vertices $n$ in the graph, but it would be good to know the dependence on the number of source vertices $|S| = k$ as well.
