I wanted to clarify something about part b on the final. The question asks if the shortest paths will change. By this, I mean the actual paths themselves, as in the sequence of edges making up the path, not the length/weight of the paths. So if, for example, one of the shortest paths was A->B->C->D with weight 10. The question isn't asking if 10 will change, but if the A->B->C->D will change.
5. Minimum Spanning Trees
Consider an undirected graph G = (V, E) with non-negative edge weights such that we have E. Suppose that you have computed a minimum spanning tree for G as well as the shortest paths to all nodes from a particular node s ∈ V. Decrease the weights of all edges in the graph such that w' = w - 1, where w ∈ E.
a. (10 pts) Does the MST of G change under this weight adjustment? Either provide an example of a graph for which it does change or prove that it cannot change.
b. (10 pts) Do the single-source shortest paths from s change? Either provide an example of a graph for which it does change or prove that they cannot change.