Math Problem Statement

C 13 14 9 D E F [Graphs generated by this script: setBorder(60,30,60,30);initPicture(-10,10,2.5,7.5); fontsize=14;fontstyle='none';marker=null;line([-10,2.5],[10,2.5]);line([-10,2.5],[10,5]);line([-10,2.5],[10,7.5]);line([-10,5],[10,2.5]);line([-10,5],[10,5]);line([-10,5],[10,7.5]);line([-10,7.5],[10,2.5]);line([-10,7.5],[10,5]);line([-10,7.5],[10,7.5]);fontbackground='white';fontfill='blue';text([-10,2.5],'A','belowleft');dot([-10,2.5]);fontfill='red';text([-6,2.5],'15');fontfill='red';text([6,4.5],'6');fontfill='red';text([-6,3.5],'11');fontfill='blue';text([-10,5],'B','belowleft');dot([-10,5]);fontfill='red';text([6,3],'10');fontfill='red';text([-6,5],'5');fontfill='red';text([6,7],'1');fontfill='blue';text([-10,7.5],'C','aboveleft');dot([-10,7.5]);fontfill='red';text([-6,6.5],'13');fontfill='red';text([6,5.5],'14');fontfill='red';text([-6,7.5],'9');fontfill='blue';text([10,2.5],'D','belowright');dot([10,2.5]);fontfill='blue';text([10,5],'E','belowright');dot([10,5]);fontfill='blue';text([10,7.5],'F','aboveright');dot([10,7.5]);]

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

Solution

The nearest neighbor algorithm for the graph involves starting at a given vertex (here, AA), moving to the nearest (shortest distance) unvisited vertex, and repeating until all vertices are visited before returning to the starting vertex.

Steps for Nearest Neighbor Algorithm:

  1. Start at vertex AA.
  2. Move to the nearest unvisited vertex.
  3. Repeat step 2 until all vertices are visited.
  4. Return to vertex AA.

Distance between vertices:

Here are the distances from the given graph's data:

  • From AA (−10,2.5-10, 2.5):
    • BB: 2.52.5 (straight vertical distance)
    • CC: 5.05.0
    • DD: 20.020.0
    • EE: 22.522.5
    • FF: 25.025.0
  • From BB (−10,5-10, 5):
    • CC: 2.52.5
    • DD: 17.517.5
    • EE: 20.020.0
    • FF: 22.522.5
  • From CC (−10,7.5-10, 7.5):
    • DD: 15.015.0
    • EE: 17.517.5
    • FF: 20.020.0
  • From DD (10,2.510, 2.5):
    • EE: 2.52.5
    • FF: 5.05.0
  • From EE (10,510, 5):
    • FF: 2.52.5

Application of the Algorithm:

  1. Start at AA.

    • Nearest neighbor: BB (distance 2.52.5).
    • Path so far: A→BA \to B.

  2. At BB:

    • Nearest unvisited neighbor: CC (distance 2.52.5).
    • Path so far: A→B→CA \to B \to C.

  3. At CC:

    • Nearest unvisited neighbor: DD (distance 15.015.0).
    • Path so far: A→B→C→DA \to B \to C \to D.

  4. At DD:

    • Nearest unvisited neighbor: EE (distance 2.52.5).
    • Path so far: A→B→C→D→EA \to B \to C \to D \to E.

  5. At EE:

    • Nearest unvisited neighbor: FF (distance 2.52.5).
    • Path so far: A→B→C→D→E→FA \to B \to C \to D \to E \to F.

  6. At FF:

    • Return to AA (distance 25.025.0).
    • Path: A→B→C→D→E→F→AA \to B \to C \to D \to E \to F \to A.

Final Path:

ABCDEF→A\boxed{ABCDEF \to A}

Let me know if you need further explanation or details about the computation!

Here are 5 related questions:

  1. How do you calculate the total distance traveled in this path?
  2. What happens if we start from a different vertex instead of AA?
  3. How does this algorithm perform in terms of optimality compared to other TSP heuristics?
  4. Can we modify the algorithm to visit vertices in a weighted graph?
  5. What is the complexity of the nearest neighbor algorithm?

Tip: Visualizing the graph while solving such problems helps ensure the accuracy of your path!

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Math Problem Analysis

Mathematical Concepts

Graph Theory
Algorithms
Traveling Salesman Problem (TSP)

Formulas

Distance formula: √((x2 - x1)² + (y2 - y1)²)
Nearest Neighbor Algorithm

Theorems

Triangle Inequality (used in TSP)
Graph traversal techniques

Suitable Grade Level

Grades 10-12

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