Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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### Prim's Algorithm and Minimum Spanning Tree
To find a minimum spanning tree for a weighted undirected graph using Prim's algorithm, you can follow these steps:

1. **Start from the selected vertex (v0)**:
    - In this case, vertex \( v_0 \) is the starting point.

2. **Grow the spanning tree**:
    - Always choose the smallest weight edge that connects a vertex in the growing spanning tree to a vertex outside of it but is connected to it.

The provided graph and algorithm details are as follows:

#### Graph Breakdown
Vertices: \( v_0, v_1, v_2, v_3, v_4, v_5, v_6, v_7 \)

Edges with Weights:
- \( v_0 - v_5 \) with weight 4
- \( v_0 - v_1 \) with weight 12
- \( v_1 - v_3 \) with weight 7
- \( v_1 - v_2 \) with weight 20
- \( v_2 - v_3 \) with weight 2
- \( v_3 - v_4 \) with weight 15
- \( v_3 - v_7 \) with weight 18
- \( v_4 - v_5 \) with weight 10
- \( v_4 - v_7 \) with weight 15
- \( v_6 - v_7 \) with weight 13
- \( v_5 - v_6 \) with weight 8

#### Steps to Find the Minimum Spanning Tree (MST):
1. Start with vertex \( v_0 \).
2. Select the edge with the smallest weight connected to it (in this case, \( (v_0, v_5) \) with weight 4).
3. Now consider vertices \( v_0, v_5 \).
4. Choose the edge \( (v_5, v_6) \) with weight 8.
5. Now consider vertices \( v_0, v_5, v_6 \).
6. Select the edge \( (v_4, v_5) \) with weight 10.
7. Now consider vertices \( v_0, v_4, v_5, v_6 \).
8. Select the edge \( (v
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Transcribed Image Text:### Prim's Algorithm and Minimum Spanning Tree To find a minimum spanning tree for a weighted undirected graph using Prim's algorithm, you can follow these steps: 1. **Start from the selected vertex (v0)**: - In this case, vertex \( v_0 \) is the starting point. 2. **Grow the spanning tree**: - Always choose the smallest weight edge that connects a vertex in the growing spanning tree to a vertex outside of it but is connected to it. The provided graph and algorithm details are as follows: #### Graph Breakdown Vertices: \( v_0, v_1, v_2, v_3, v_4, v_5, v_6, v_7 \) Edges with Weights: - \( v_0 - v_5 \) with weight 4 - \( v_0 - v_1 \) with weight 12 - \( v_1 - v_3 \) with weight 7 - \( v_1 - v_2 \) with weight 20 - \( v_2 - v_3 \) with weight 2 - \( v_3 - v_4 \) with weight 15 - \( v_3 - v_7 \) with weight 18 - \( v_4 - v_5 \) with weight 10 - \( v_4 - v_7 \) with weight 15 - \( v_6 - v_7 \) with weight 13 - \( v_5 - v_6 \) with weight 8 #### Steps to Find the Minimum Spanning Tree (MST): 1. Start with vertex \( v_0 \). 2. Select the edge with the smallest weight connected to it (in this case, \( (v_0, v_5) \) with weight 4). 3. Now consider vertices \( v_0, v_5 \). 4. Choose the edge \( (v_5, v_6) \) with weight 8. 5. Now consider vertices \( v_0, v_5, v_6 \). 6. Select the edge \( (v_4, v_5) \) with weight 10. 7. Now consider vertices \( v_0, v_4, v_5, v_6 \). 8. Select the edge \( (v
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