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17 D 24 14 18- B pply the nearest neighbor algorithm to the graph above starting at vertex A. Give your answer as a list of ertices, starting and ending at vertex A. Example: ABCDA

17 D 24 14 18- B pply the nearest neighbor algorithm to the graph above starting at vertex A. Give your answer as a list of ertices, starting and ending at vertex A. Example: ABCDA

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### Applying the Nearest Neighbor Algorithm #### Graph and Explanation In the graph shown, we have four vertices labeled A, B, C, and D. These vertices are connected by edges with the following weights: - Vertex A is connected to Vertex B with a weight of 18. - Vertex A is connected to Vertex C with a weight of 24. - Vertex A is connected to Vertex D with a weight of 25. - Vertex B is connected to Vertex C with a weight of 14. - Vertex B is connected to Vertex D with a weight of 3. - Vertex C is connected to Vertex D with a weight of 17. The goal is to apply the nearest neighbor algorithm starting at vertex A and find the shortest path that visits each vertex exactly once and returns back to vertex A. Example Input: `ABCD` #### Steps to follow: 1. **Start at Vertex A**: Begin at the designated starting vertex. 2. **Choose the Nearest Neighbor**: From the current vertex, choose the edge with the smallest weight to an unvisited vertex. 3. **Mark the Vertex as Visited**: Once you travel to a vertex, mark it as visited. 4. **Repeat Until All Vertices are Visited**: Continue this process until all vertices have been visited. 5. **Return to Starting Vertex**: End the path by traveling back to the starting vertex A. #### Example Solution Starting at Vertex A: - From A, the nearest vertex is B with a weight of 18. Path: A -> B - From B, the nearest vertex is D with a weight of 3. Path: A -> B -> D - From D, the nearest vertex is C with a weight of 17. Path: A -> B -> D -> C - Finally, return to A from C with a weight of 24. Path: A -> B -> D -> C -> A So, the order of vertices visited would be: **ABDCA** Using the nearest neighbor algorithm, the final path is AB, BD, DC, and CA, returning to the starting vertex A.
Transcribed Image Text:### Applying the Nearest Neighbor Algorithm #### Graph and Explanation In the graph shown, we have four vertices labeled A, B, C, and D. These vertices are connected by edges with the following weights: - Vertex A is connected to Vertex B with a weight of 18. - Vertex A is connected to Vertex C with a weight of 24. - Vertex A is connected to Vertex D with a weight of 25. - Vertex B is connected to Vertex C with a weight of 14. - Vertex B is connected to Vertex D with a weight of 3. - Vertex C is connected to Vertex D with a weight of 17. The goal is to apply the nearest neighbor algorithm starting at vertex A and find the shortest path that visits each vertex exactly once and returns back to vertex A. Example Input: `ABCD` #### Steps to follow: 1. **Start at Vertex A**: Begin at the designated starting vertex. 2. **Choose the Nearest Neighbor**: From the current vertex, choose the edge with the smallest weight to an unvisited vertex. 3. **Mark the Vertex as Visited**: Once you travel to a vertex, mark it as visited. 4. **Repeat Until All Vertices are Visited**: Continue this process until all vertices have been visited. 5. **Return to Starting Vertex**: End the path by traveling back to the starting vertex A. #### Example Solution Starting at Vertex A: - From A, the nearest vertex is B with a weight of 18. Path: A -> B - From B, the nearest vertex is D with a weight of 3. Path: A -> B -> D - From D, the nearest vertex is C with a weight of 17. Path: A -> B -> D -> C - Finally, return to A from C with a weight of 24. Path: A -> B -> D -> C -> A So, the order of vertices visited would be: **ABDCA** Using the nearest neighbor algorithm, the final path is AB, BD, DC, and CA, returning to the starting vertex A.
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