Question

Computer Science

Posted 5 months ago

```
Using Prim's Algorithm, starting at vertex A, find the
Minimal Spanning Tree (MST) for the weighted graph
G in figure 4.2 below.
A
10
6
D
7
3
B
C
E
6
3
4
5
4
1
2
9
H
1
F
G
```

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Answer from Sia

Posted 5 months ago

java

Explanation

This Java code implements Prim's Algorithm to find the Minimal Spanning Tree (MST) of a given weighted graph. The "primMST" method constructs the MST using an adjacency matrix representation of the graph. The "minKey" method is used to find the vertex with the minimum key value that is not already included in the MST. The "printMST" method prints the edges included in the MST along with their weights. The "main" method provides a sample graph to test the algorithm.

Step-by-step Instruction

▪

Define the `primMST` method to construct the MST using the adjacency matrix representation of the graph

▪

Initialize all keys as infinite and set the first vertex's key to 0 to start the MST from the first vertex

▪

Use a loop to find and add the minimum key vertex to the MST set and update the keys of its adjacent vertices

▪

Define the `printMST` method to print the constructed MST showing the edges and their weights

▪

In the `main` method, create a sample graph and call the `primMST` method to print the MST

Time Complexity

The time complexity of Prim's Algorithm is O(V^2) where V is the number of vertices in the graph, as we are using an adjacency matrix representation and searching for the minimum key vertex in every iteration.

Space Complexity

The space complexity is O(V) as we are maintaining arrays to store the keys, the MST set, and the parent of each vertex in the MST.

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