Build a spanning tree subset by subset

Question

Suppose a general graph $G=(V,E)$ and let $E$ be partitioned into $E_1$, $E_2$, ..., $E_n$. I would like to obtain a spanning tree of $G$ such that a forest $F_1$ is built from $E_1$ first, then the components of $F_1$ are linked by edges of $E_2$ yielding the forest $F_2$, then the components of $F_2$ are linked by edges of $E_3$, leading to the forest $F_3$, etc. What is the speediest way to construct such a spanning tree?

Answer 1

Find the minimum-weight spanning tree where the weight of an edge $e \in E_k$ is $k$.