Problem $2.(20$ points total$)$ Boruvka's algorithm is another greedy algorithm for finding a minimum spanning tree of a weighted undirected graph $G=(V,E).$ It operates via the following steps:

Initialize a forest $F$ to $(V,E^{\text{'}})$ where $E^{\text{'}}=\frac{O}{?}.$ Equivalently, each node defines its own $($edgeless$)$ component $C$

At each iteration

For each component $C$ in $F,$ find a minimum weight edge that crosses from $C$ to another component $($call these boundary edges$)$

Add all of these minimum$-$weight edges to the forest $F$

Terminate where all nodes have been merged into one component.