1. 6.4: The Brute-Force Algorithms
List all of the possible Hamilton circuits
Calculate the total weights for each of the possible Hamilton circuit
Choose an optimal circuit (cheapest, shortest, or fastest…)

2. The Brute-Force Algorithms
A,B,C,D,E,A = 812
A,B,C,E,D,A = 777
A,B,D,C,E,A = 762
A,B,D,E,C,A = 773
A,B,E,C,D,A = 831
A,B,E,D,C,A = 877
A,C,B,D,E,A = 722
A,C,B,E,D,A = 791
A,C,D,B,E,A = 776
A,C,E,B,D,A = 741
A,D,B,C,E,A = 676
A,D,C,B,E,A = 780
There are 5 vertices so we have (5-1)! = 24 Hamilton circuits
Optimal route
Plus 12 mirror images

3. The Brute-Force Algorithms
Start at C
C,A,B,D,C = =150
C,A,D,B,C = =125
C,B,A,D,C = =115
C,B,D,A,C =125
C,D,A,B,C =115
C,D,B,A,C =150
Therefore: optimal circuits are C,B,A,D,C or C,D,A,B,C
If the problem asks to start at A then the optimal circuits are
A,D,C,B,A or A,B,C,D,A A 20 70 30 C 15 B
Mirror images 50 10 D

4. 6.5: The Nearest-Neighbor Algorithm
From the starting vertex, go to its nearest neighbor (the edge has the smallest weight)
Repeating step 1 with the next vertex and the next vertex until there is no more vertex to visit
From the last vertex, return to the starting point

5. The Nearest-Neighbor Algorithm
Start at A, pick AC = 119
From C, pick CE = 120
From E, pick ED = 199
From D, pick DB = 150
No more vertex left so connect B and A, with BA=185
Therefore the trip is: A,C,E,D,B,A = 773 dollars

6. The Nearest-Neighbor Algorithm
Start at C, pick CB = 15
From B, pick BD = 10
From D, pick DA = 30
No more vertex left, so connect A and C, with AC = 70
Therefore, the route is
C,B,D,A,C = 125

7. Practice with the Nearest Neighbor

8. Apply the Nearest Neighbor Algorithm to find the way to go to the bank, dry cleaner, post office, and Wegmans starting and ending at home. The mileage chart is shown below. D W B P
HOME

9. 6.7 The repetitive Nearest-Neighbor Algorithm

10. The Brute-Force Algorithm: optimal but inefficient algorithm
The Nearest-Neighbor Algorithm and Repetitive Nearest-Neighbor: efficient but not optimal algorithms.

11. The Repetitive Nearest-Neighbor Algorithm
Apply the Nearest Neighbor Algorithm to each of the vertices to calculate the total weights.
Choose the best circuit. If there is a designated starting vertex, rewrite the circuit using that vertex as the reference point

12. The Repetitive Nearest-Neighbor Algorithm
From A:
A, C, E, D, B, A = 773
From B:
B, C, A, E, D, B = 722
From C:
C, A, E, D, B, C = 722
From D:
D, B, C, A, E, D = 722
From E:
E, C, A, D, B, E = 741
Therefore, one of the best routes is B, C, A, E, D, B so we rewrite it so the starting point is at A (his home): A, E, D, B, C, A

13. The Repetitive Nearest-Neighbor Algorithm
From A:
A, B, D, C, A = = 150
From B:
B, D, A, C, B = = 125
From C:
C, B, D, A, C = = 125
From D:
D, B, C, A, D = = 125
One of the best routes is C,B,D,A,C = 125
or if re-write with starting point A:
A,C,B,D,A