Thinking Mathematically (6th Edition)

Published by Pearson
ISBN 10: 0321867327
ISBN 13: 978-0-32186-732-2

Chapter 14 - Graph Theory - 14.3 Hamilton Paths and Hamilton Circuits - Exercise Set 14.3: 44

Answer

Using the Nearest Neighbor Method, the Hamilton circuit is Home,Bank,Post Office,Dry Cleaners,Hospital,Medical Clinic,Home. The total weight of the circuit is 30. Since each block represents one mile, the total length of this path is 30 miles. This is the same answer as the optimal solution that we found with the Brute Force Method.

Work Step by Step

With the Nearest Neighbor Method, we should choose the edge which has the smallest weight for each step along the path. Let's start at vertex Home. We can travel along 5 possible edges. The weights of these edges are 2, 8, 5, 5, and 5. We should travel along the edge Home-Bank because it has the smallest weight. From vertex Bank, we can travel along 4 possible edges. The weights of these edges are 6, 7, 7, and 7. We should travel along edge Bank-Post Office because it has the smallest weight. From vertex Post Office, we can travel along 3 possible edges. The weights of these edges are 7, 11, and 13. We should travel along edge Post Office-Dry Cleaners because it has the smallest weight. From vertex Dry Cleaners, we can travel along 2 possible edges. The weights of these edges are 4 and 6. We should travel along edge Dry Cleaners-Hospital because it has the smallest weight. From vertex Hospital, the path must travel to vertex Medical Clinic, and then finally back to vertex Home to complete the Hamilton circuit. Using the Nearest Neighbor Method, the Hamilton circuit is Home,Bank,Post Office,Dry Cleaners,Hospital,Medical Clinic,Home. The weights of the edges in this path are 2, 6, 7, 4, 6, and 5. We can find the total weight of this Hamilton circuit. total weight = 2 + 6 + 7 + 4 + 6 + 5 total weight = 30 Using the Nearest Neighbor Method, the Hamilton circuit is Home,Bank,Post Office,Dry Cleaners,Hospital,Medical Clinic,Home. The total weight of the circuit is 30. Since each block represents one mile, the total length of this path is 30 miles. This is the same answer as the optimal solution that we found with the Brute Force Method.
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