# The Roskind-Tarjan Algorithm

I am going through the paper https://pubsonline.informs.org/doi/abs/10.1287/moor.10.4.701 which is A Note on Finding Minimum-Cost Edge-Disjoint Spanning Trees and the authors are James Roskind and Robert E. Tarjan. In it, there is an algorithm to generate an augmenting sequence used in updating edge-disjoint forests.

If $$F$$ is a forest and $$e = \{v, w\}$$ is an edge such that $$v$$ and $$w$$ are in the same tree of $$F$$, we define $$F(e)$$ to be the unique path in $$F$$ joining $$v$$ and $$w$$. If $$i$$ is any integer, we define $$i+ = (i$$ mod $$k) + 1$$. For a given set of edge-disjoint forests $$F_1$$, $$F_2$$, $$\ldots$$, $$F_k$$ and edges $$e_0$$ and $$e_l$$, a swap sequence from $$e_0$$ to $$e_l$$ is a sequence of edges $$e_0$$, $$e_1$$, . . , $$e_l$$ such that, for $$j \in [0 \ldots l- 1]$$, $$e_{j+1} \in F_{j+} (e_j)$$. The swap sequence is augmenting if $$e_0$$ is in none of the forests $$F_1$$, $$F_2$$, $$\ldots$$, $$F_k$$, the endpoints of $$e_l$$, are in different trees of $$F_{l+}$$, and the swap sequence is minimal in the sense that there are no two edges $$e_{j'}$$ and $$e_j$$, such that $$j' > j + 1$$ and $$e_j' \in F_{j+}$$

The definition for an augmenting swap sequence seems very arbitrary to me at the moment, as I am unable to visualize it properly. I do understand what it is saying, but I am not understanding how this definition is well-motivated. An example would be very great in clarifying this