Math Problem Statement

Apply the repeated nearest neighbor algorithm to the graph above. Give your answer as a list of vertices, starting and ending at vertex A.

Solution

The problem asks to apply the repeated nearest neighbor algorithm to the given weighted graph, starting and ending at vertex AA.

The repeated nearest neighbor algorithm works as follows:

  1. Start at the initial vertex (in this case AA).
  2. From the current vertex, choose the edge with the smallest weight to an unvisited vertex.
  3. Repeat step 2 until all vertices are visited.
  4. Return to the starting vertex to complete the tour.

Step-by-step solution:

  1. Start at vertex AA:

    • The edges from AA are:
      • ABA \to B with weight 14
      • AEA \to E with weight 10
      • ADA \to D with weight 12
    • The smallest weight is 10, so the next vertex is EE.
  2. Move to vertex EE:

    • The edges from EE are:
      • EAE \to A with weight 10 (already visited)
      • EDE \to D with weight 3
      • ECE \to C with weight 15
      • EBE \to B with weight 10
    • The smallest weight is 3, so the next vertex is DD.
  3. Move to vertex DD:

    • The edges from DD are:
      • DAD \to A with weight 12 (already visited)
      • DED \to E with weight 3 (already visited)
      • DCD \to C with weight 8
      • DBD \to B with weight 15
    • The smallest weight is 8, so the next vertex is CC.
  4. Move to vertex CC:

    • The edges from CC are:
      • CAC \to A with weight 5
      • CBC \to B with weight 11
      • CDC \to D with weight 8 (already visited)
      • CEC \to E with weight 15 (already visited)
    • The smallest weight is 5, so the next vertex is AA.

Since we are back at vertex AA and have visited all vertices, the tour is complete.

Final Tour:

The path is AEDCAA \to E \to D \to C \to A, with a total distance of 10+3+8+5=2610 + 3 + 8 + 5 = 26.

Do you have any questions, or would you like further details on any step?

Related Questions:

  1. How does the nearest neighbor algorithm compare to other algorithms like the Christofides or Held-Karp?
  2. What is the total cost of the tour using this algorithm?
  3. Is the nearest neighbor algorithm guaranteed to give the optimal solution?
  4. How does the choice of the starting vertex affect the solution?
  5. 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)