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4th Edition

Richard N. Aufmann + 3 others

Publisher: Cengage Learning

ISBN: 9781305965584

Chapter 5.2, Problem 23ES

Textbook Problem

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Travel Nicole wants to tour Asia. She will start and end her journey in Tokyo and visit Hong Kong, Bangkok, Seoul, and Beijing. The airfares available to her between cities are given in the table. Draw a weighted graph that represents the travel costs between cities and use the greedy algorithm to find a low-cost route.

To determine

To represent the travel costs between the cities by a weighted graph and to find a low-cost route for the tour using the greedy algorithm.

**Given information:**

Given, Nicole wants to tour Asia.

She will start and end her journey in Tokyo and visit Hong Kong, Bangkok, Seoul, and Beijing.

The airfares available to her between cities are given in the table below:

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**Calculation:**

Given, there are five cities out of which Nicole starts and ends at Tokyo.

Hence five vertices corresponding to the five cities are marked with respective names.

And then making the complete graph by marking every possible edge with every edge marked with the weight represented in the table between the cities.

The airfare between Tokyo and Hong Kong is $845. Hence the edge drawn between them has a weight of 845.

Similarly marking all the edges with respective weights, the graph is given as:

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To use the greedy algorithm, the graph must be Hamiltonian.

(Since, the tourist needs to visit each city only once)

A path in a Hamiltonian graph is said to be a Hamiltonian Circuit if it begins and ends at the same vertex and passes through each vertex of a graph exactly once.

The greedy algorithm states to travel along a connected edge that has the smallest weight that is not yet visited. This process continues till all the vertices are visited and returned to the starting vertex.

Here, the vertex representing Tokyo is selected.

Vertex representing Tokyo is of degree 4 and the weights of the connected edges are 845, 1275, 470, 880.

Hence, as per the Greedy Algorithm the edge with weight 470 is selected i...

Mathematical Excursions (MindTap Course List)

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