Given a weighted connected undirected graph $G=(V, E)$ with node set V and edge set E, a subgraph $M$ of $G$ is called a spanning tree of $G$ if $M = (V, T)$, $T \subseteq E$, and $|T|=|V|-1$. $M$ called a minimal spanning tree (MST) of $G$ if $M$ is a spanning tree of $G$ with the minimal total weight. Below is the Prim's MST algorithm to output the MST of a given graph $G$.

Algorithm: Prim's MST algorithm Input: A weighted connected undirected graph $G=(V, E)$, where $|V|=n$ Output: The MST $M=(V, T)$ with the minimal total weight, where $|T|=n-1$

1: $T \leftarrow \phi$ 2: $X \leftarrow \{w\}$, where w is an arbitrary node in V 3: while $|T| < n-1$ do 4: select an edge $(u, v) \in E$ with the minimal weight such that $u \in X$ and $v \in (V-X)$ 5: $T \leftarrow T \cup \{(u, v)\}$ 6: $X \leftarrow X \cup \{v\}$ 7: return $M=(V, T)$

Please follow the form of the Prim's MST algorithm to (1) write the well-known Kruskal's MST algorithm (9%) and (2) analyze the worst-case time complexity of the Kruskal's MST algorithm (6%). Note that you should strictly follow the form of the Prim's MST algorithm to write the Kruskal's MST algorithm; otherwise, you will lose some points.