Math Problem Statement
Solution
The problem asks to apply the repeated nearest neighbor algorithm to the given weighted graph, starting and ending at vertex .
The repeated nearest neighbor algorithm works as follows:
- Start at the initial vertex (in this case ).
- From the current vertex, choose the edge with the smallest weight to an unvisited vertex.
- Repeat step 2 until all vertices are visited.
- Return to the starting vertex to complete the tour.
Step-by-step solution:
-
Start at vertex :
- The edges from are:
- with weight 14
- with weight 10
- with weight 12
- The smallest weight is 10, so the next vertex is .
- The edges from are:
-
Move to vertex :
- The edges from are:
- with weight 10 (already visited)
- with weight 3
- with weight 15
- with weight 10
- The smallest weight is 3, so the next vertex is .
- The edges from are:
-
Move to vertex :
- The edges from are:
- with weight 12 (already visited)
- with weight 3 (already visited)
- with weight 8
- with weight 15
- The smallest weight is 8, so the next vertex is .
- The edges from are:
-
Move to vertex :
- The edges from are:
- with weight 5
- with weight 11
- with weight 8 (already visited)
- with weight 15 (already visited)
- The smallest weight is 5, so the next vertex is .
- The edges from are:
Since we are back at vertex and have visited all vertices, the tour is complete.
Final Tour:
The path is , with a total distance of .
Do you have any questions, or would you like further details on any step?
Related Questions:
- How does the nearest neighbor algorithm compare to other algorithms like the Christofides or Held-Karp?
- What is the total cost of the tour using this algorithm?
- Is the nearest neighbor algorithm guaranteed to give the optimal solution?
- How does the choice of the starting vertex affect the solution?
- How can we adapt this algorithm to work for larger graphs?
Tip: The nearest neighbor algorithm is a greedy algorithm and doesn't always guarantee the optimal solution but is often used due to its simplicity and speed.
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Math Problem Analysis
Mathematical Concepts
Graph Theory
Greedy Algorithm
Nearest Neighbor Algorithm
Weighted Graphs
Formulas
Nearest Neighbor Algorithm: Choose the edge with the smallest weight to an unvisited vertex.
Theorems
Nearest Neighbor Heuristic
Suitable Grade Level
Undergraduate or advanced high school students (Grade 11-12)
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