MHB Prim's algorithm for minimal spanning tree

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Prim's algorithm is a method for finding the minimum spanning tree of a connected, undirected graph. To implement the algorithm, start with a single vertex and repeatedly add the smallest edge connecting a vertex in the tree to a vertex outside the tree until all vertices are included. Documentation of each step can include noting the vertices added, the edges selected, and the current state of the tree after each addition. The algorithm can still be applied to any connected graph, ensuring that it maintains the properties of a spanning tree. Understanding the definition of a connected spanning subgraph is crucial for correctly applying Prim's algorithm.
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Quite stuck on this how do i do this and how do i document each step?
 

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There is a nice little variation of the problem. The host says, after you have chosen the door, that you can change your guess, but to sweeten the deal, he says you can choose the two other doors, if you wish. This proposition is a no brainer, however before you are quick enough to accept it, the host opens one of the two doors and it is empty. In this version you really want to change your pick, but at the same time ask yourself is the host impartial and does that change anything. The host...

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