Problem 5 Using Kruskal's algorithm, const... [FREE SOLUTION]

Chapter 9: Problem 5

Using Kruskal's algorithm, construct a spanning tree for each graph, starting at \(a\).

Short Answer

Expert verified

To construct a minimum spanning tree using Kruskal's algorithm, follow these steps: 1. List all edges and their weights in ascending order. 2. Initialize an empty set for the tree's edges, a counter variable to track the number of edges added, and set the starting point as vertex \(a\). 3. Iterate through the list of edges, adding each edge to the tree without forming a cycle. 4. The tree is complete when the counter variable equals the total number of vertices minus 1, indicating a spanning tree with \(n-1\) edges. 5. Display the final set of edges in the minimum spanning tree and, if necessary, its total weight.

Step by step solution

List all edges and their weights

In this step, you will note down all the edges in the given graph and their corresponding weights. Make sure to list them in ascending order based on their weights for ease in future steps.

Initialize variables and set the starting point

For this step, create an empty set for the tree's edges and initialize a counter variable to keep track of the number of edges added. Set the starting point as vertex \(a\).

Add edges to the tree in increasing order without forming a cycle

Iterate through the list of edges in ascending order. For each edge, check if adding it would create a cycle in the tree. If not, add the edge to the tree and increase the counter variable by 1. If it forms a cycle, ignore the edge and proceed to the next one. Note: You can check if an edge will form a cycle using the "Union-Find" algorithm for disjoint sets.

Check if the tree is complete

When the counter variable equals the total number of vertices minus 1, it means that the tree is complete, as a spanning tree for a graph with \(n\) vertices will always have \(n-1\) edges.

Display the resulting minimum spanning tree

Display the final set of edges in the minimum spanning tree found using Kruskal's algorithm. If necessary, write down the total weight of the resulting tree.

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91影视

Using Kruskal's algorithm, construct a spanning tree for each graph, starting at \(a\).

Short Answer

Step by step solution

List all edges and their weights

Initialize variables and set the starting point

Add edges to the tree in increasing order without forming a cycle

Check if the tree is complete

Display the resulting minimum spanning tree

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