Question

Given an undirected weighted graph G with n nodes and m edges, and we have used Prim's algorithm to construct a minimum spanning tree T.

Suppose the weight of one of the tree edge ((u,v)∈T) is changed from w to w′, design an algorithm to verify whether T is still a minimum spanning tree.
Your algorithm should run in O(m) time, and explain why your algorithm is correct. You can assume all the weights are distinct.
(Hint: When an edge is removed, nodes of T will break into two groups. Which edge should we choose in the cut of these two groups?)

Answer 1

To verify if a modified edge weight affects the MST status of a tree created by Prim's algorithm, check if there's a lighter edge across the cut which requires O(m) time.

Step-by-step explanation:

When modifying the weight of an edge in a minimum spanning tree (MST) created by Prim's algorithm, we want to check if the updated tree is still an MST. Assume the edge weight is changed from w to w'. If w' > w, we must check if there is another edge across the cut created by removing (u,v) that has a weight less than w' and connects the two sets of nodes. If such an edge exists, the tree is not an MST. If w' < w, the tree remains an MST because the uniqueness of edge weights implies that this edge had to be the lightest edge crossing the cut originally when Prim's algorithm was run. To perform this check efficiently, we iterate through all edges in the graph, an operation that takes O(m) time.

Answer 2

To verify whether the modified tree T is still a minimum spanning tree after changing the weight of one of its edges, we can use the following algorithm:

1. Identify the edge (u, v) in the tree T that had its weight changed from w to w'.

2. Remove the edge (u, v) from T, creating two disjoint sets of nodes.

3. Choose the edge with the minimum weight that crosses the cut between the two sets.

4. If the weight of the chosen edge is less than w', then T is still a minimum spanning tree.

Step-by-step explanation:

The algorithm works by simulating the process of adding the modified edge back to the minimum spanning tree T. After removing the modified edge, the two sets of nodes created represent the two disjoint subsets in the original tree. The algorithm then selects the minimum weight edge that crosses the cut between these sets.

If the weight of the chosen edge is less than the modified weight w', adding it back to the tree will result in a new minimum spanning tree with a total weight less than or equal to the modified tree. This is because the chosen edge, being the minimum weight across the cut, is the optimal choice for connecting the two subsets.

The algorithm's time complexity is O(m), where m is the number of edges in the graph. This is because we only need to iterate through the edges once to identify the modified edge and find the minimum weight edge crossing the cut. The correctness of the algorithm lies in its adherence to the fundamental property of minimum spanning trees—connecting nodes with the minimum possible total edge weight.