You are given a graph G = (V, E) with positive edge weights, and a minimum spanning tree T = (V, E’) with respect to these weights; you may assume G and T are given as adjacency lists. Now suppose the weight of a particular edge e ∈ E is modified from w(e) to a new value ŵ(e). You wish to quickly update the minimum spanning tree T to reflect this change, without recomputing the entire tree from scratch. There are four cases. In each case give a linear-time algorithm for updating the tree.
(a) e ∉ E’ and ŵ(e) > w(e).
(b) e ∉ E' and ŵ (e) < w(e).
(c) e ∈ E' and ŵ (e) < w(e).
(d) e ∈ E' and ŵ (e) > w (e).
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