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4th Edition

Richard N. Aufmann + 3 others

Publisher: Cengage Learning

ISBN: 9781305965584

Chapter 5.2, Problem 29ES

Textbook Problem

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Scheduling A research company has a large supercomputer that is used by different teams for a variety of computational tasks. In between each task, the software must be reconfigured. The time required depends on which tasks follow which, because some settings are shared by different tasks. The times (in minutes) required to reconfigure the machine from one task to another are given in the table below. Use the greedy algorithm and the edge-picking algorithm to find time-efficient sequences in which to assign the tasks to the computer. The software configuration must start and end in the home state.

To determine

To find time-efficient sequences in which to assign the tasks to the computer, with the software configuration starting and ending in the home state, using the greedy algorithm and the edge-picking algorithm.

**Given information:**

Given, a research company has a large supercomputer that is used by different teams for a variety of computational tasks.

In between each task, the software must be reconfigured.

The time required depends on which tasks follow which, because some settings are shared by different tasks.

The times (in minutes) required to reconfigure the machine from one task to another are given in the table below:

The software configuration must start and end in the home state.

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**Calculation:**

Given, there are four tasks to perform which starts and ends at Home state.

Hence five vertices corresponding to the five 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 tasks.

The time required to reconfigure the machine between Home State and Task A is 35. Hence the edge drawn between them has a weight of 35.

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 Home State is selected.

Vertex representing Home State is of degree 4 and the weights of the connected edges are 35, 15, 40, 27

Hence, as per the Greedy Algorithm the edge with weight 15 is selected i.e. path reaches vertex representing Task B.

Vertex representing Task B is of degree 4 and the weights of the connected edges are 15, 30, 35, 16

The least of the weights is 15 but it leads to vertex representing Home State which is already visited.

Hence, as per the Greedy Algorithm the edge with weight 16 is selected i.e. path reaches vertex representing Task D.

Vertex representing Task D is of degree 4 and the weights of the connected edges are 27, 25, 16, 32

The least of the weights is 16 but it leads to vertex representing Task B which is already visited. Hence, as per the Greedy Algorithm the edge with weight 25 is selected i.e. path reaches vertex representing Task A.

Vertex representing Task A is of degree 4 and the weights of the connected edges are 35, 30, 18, 25

Hence, as per the Greedy Algorithm the edge with weight 18 is selected i.e. path reaches vertex representing Task C.

Vertex representing Task C is of degree 4 and the weights of the connected edges are 40, 18, 35, 32

The least of the weights is 18 but it leads to vertex representing Task A which is already visited...

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