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4th Edition

Richard N. Aufmann + 3 others

Publisher: Cengage Learning

ISBN: 9781305965584

Chapter 5.2, Problem 28ES

Textbook Problem

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Route Planning A bike messenger needs to deliver packages to five different buildings and return to the courier company. The estimated biking times (in minutes) between the buildings are given in the following table. Use the greedy algorithm and the edge-picking algorithm to find two possible routes for the messenger to follow that Will help minimize the total travel time.

To determine

To find two possible routes, starting and ending at courier company, for the messenger to follow that will minimize his total travel time, using the greedy algorithm and the edge-picking algorithm.

**Given information:**

Given, a bike messenger needs to deliver packages to five different buildings and return to the courier company.

The estimated biking times (in minutes) between the buildings are given in the table below:

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**Calculation:**

Given, there are five locations to deliver and the messenger starts and ends at Courier Company.

Hence six vertices corresponding to the six locations 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 locations.

The biking time between Courier Company and Prudential building is 10. Hence the edge drawn between them has a weight of 10.

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

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Greedy algorithm:

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

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 Courier Company is selected.

Vertex representing Courier Company is of degree 5 and the weights of the connected edges are 10, 8, 15, 12, 17

Hence, as per the Greedy Algorithm the edge with weight 8 is selected i.e. path reaches vertex representing Bank of America Building.

Vertex representing Bank of America Building is of degree 5 and the weights of the connected edges are 8, 10, 7, 18, 20

Hence, as per the Greedy Algorithm the edge with weight 7 is selected i.e. path reaches vertex representing Imperial Bank Building.

Vertex representing Imperial Bank Building is of degree 5 and the weights of the connected edges are 15, 6, 7, 22, 16

Hence, as per the Greedy Algorithm the edge with weight 6 is selected i.e. path reaches vertex representing Prudential Building.

Vertex representing Prudential Building is of degree 5 and the weights of the connected edges are 10, 10, 6, 9, 8

The least of the weights is 6 but it leads to vertex representing Imperial Bank Building which is already visited.

Hence, as per the Greedy Algorithm the edge with weight 8 is selected i.e. path reaches vertex representing Design Center.

Vertex representing Design Center is of degree 5 and the weights of the connected edges are 17, 8, 20, 16, 5

Hence, as per the Greedy Algorithm the edge with weight 5 is selected i.e. path reaches vertex representing GE Tower.

Vertex representing GE Tower is of degree 5 and the weights of the connected edges are 12, 9, 18, 22, 5

The least of the weights is 5 but it leads to vertex representing GE Tower which is already visited. The next least of the weights is 9 but it leads to vertex representing Prudential Building which is already visited.

Hence, as per the Greedy Algorithm the edge with weight 12 is selected i...

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