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4th Edition

Richard N. Aufmann + 3 others

Publisher: Cengage Learning

ISBN: 9781305965584

Chapter 5.2, Problem 19ES

Textbook Problem

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Travel A company representative lives in Louisville, Kentucky, and needs to visit offices in five different Indiana cities over the next few days. The representative wants to drive between cities and return to Louisville at the end of the trip. The estimated driving times, in hours, between cities are given in the table below. Represent the driving times by a weighted graph. Use the greedy algorithm to design an efficient route for the representative to follow.

To determine

To represent the driving times by a weighted graph and to determine an efficient route for the representative to follow using the greedy algorithm.

**Given information:**

Given, a company representative lives in Louisville, Kentucky, and needs to visit offices in five different Indiana cities over the next few days.

The representative wants to drive between cities and return to Louisville at the end of the trip.

The estimated driving times, in hours, between cities are given in the table below:

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**Calculation:**

Given, there are six cities out of which the representative lives in Louisville, Kentucky.

Hence six vertices corresponding to the six 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 time to travel between Louisville and Bloomington is 3.6. Hence the edge drawn between them has a weight of 3.6

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 representative 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 Louisville is selected.

Vertex representing Louisville is of degree 5 and the weights of the connected edges are 3.6, 6.4, 3.2, 4.9, 3.1

Hence, as per the Greedy Algorithm the edge with weight 3.1 is selected i.e. path reaches vertex representing Evansville.

Vertex representing Evansville is of degree 5 and the weights of the connected edges are 3...

Mathematical Excursions (MindTap Course List)

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