Algorithms · Dynamic programming and state design
Prim’s algorithm starts at an arbitrary root and repeatedly adds a minimum-weight edge…
Problem
Prim’s algorithm starts at an arbitrary root and repeatedly adds a minimum-weight edge with exactly one endpoint in the current tree. State and prove an invariant connecting the current tree to some minimum spanning tree. Use it to prove correctness for graphs with repeated and negative edge weights, and explain formally why the choice of starting root cannot change the minimum possible output weight.
Hint
** Let \(A\) be the selected edges and \(S\) the reached vertices.
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