$8.$As shown in the following figures, Prim's algorithm is usually used to find the minimum spanning tree, in which the intermediate set of edges $x$ always forms a subtree, and $S$ is chosen to be the set of this tree's vertices. On each iteration, the subtree defined by $x$ grows by one edge, namely the lightest edge between a vertex in $S$ and a vertex outside $S.(1)$ Draw a table to show the changes of cost and prev values, and construct the final Minimum Spanning Tree. $(12\text{'})(2)$ What is the main difference between the Dijkstra's algorithm and the Prim's algorithm. $(3\text{'})$

procedure prim $(G,w)$

Input: A connected undirected graph $G=(V,E)$ with edge weights $w_e$

output: A minimum spanning tree defined $by$ the array prev

for all uinV :

$cost(u)=$

prev$(u)=n11$

pick any initial node $wo$

$cost(u_0)=0$

$cost(u_0)=0$

$H=$ makequeue $(V)(priority$ queue, using cost$-$values $as$ keys$)$

while $His$ not empty:

$v=$deletemin$(H)$

for each $\{v,z\}inE$:

$ifcost(z)>w(v,2)$:

$cost(z)=w(v,z)$

prev$(z)=v$

decreasekey $(H,z)$