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**Networks Prim’s Algorithm**

Decision Maths Networks Prim’s Algorithm

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Prim`s Algorithm In Lesson 1 we learnt about Kruskal`s algorithm, which was used to solve minimum connector problems. Another method that can be used is Prim`s algorithm. Step 1 – Select any node Step 2 – Connect it to the nearest node Step 3 – Connect one node already selected to the nearest unconnected node. Step 4 – Repeat 3 until all nodes are connected.

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**Prim`s Algorithm Consider the example we looked at last lesson.**

Select any node you like. Lets select F.

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**Prim`s Algorithm Connect it to the nearest node.**

C and D are both 3 away so we can choose either. Lets select C.

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Prim`s Algorithm The nearest node to either of F or C is D, which is only 3 away from F. So connect D to F.

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**Prim`s Algorithm The nearest to D, F or C is E which is 2 from D.**

So connect E to D.

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Prim`s Algorithm The nearest to any of these four nodes is A which is 5 away from F. Connect A to F.

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**Prim`s Algorithm We now need to connect the last node, B.**

The shortest arc is AB, which is 2. Connect B to A.

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Prim`s Algorithm All the nodes are now connected so this is the minimum connector or minimal spanning tree.

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**Distance Table The Network can also be represented as a table.**

The infinity symbol (∞) means there is no edge between the two nodes.

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**Prim`s on a Distance Table**

We are going to apply Prim`s algorithm to the distance table. This demonstrates how a computer could apply the algorithm. Prim`s is more suitable than Kruskal`s as computers have a problem recognising loops. As you go through the algorithm, see if you can relate the procedure to the last example.

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**Prim`s on a Distance Table**

Step 1 – Select any arbitrary node. Step 2 – Delete the row and loop the column that correspond to the node selected. Step 3 – Choose the smallest number in the loop. Step 4 – Delete the row that this smallest number is in. Step 5 – Loop the column that corresponds to the row just deleted. Step 6 – Choose the smallest number in any loop. Step 7 – Repeat steps 4, 5 and 6 until all rows have been deleted and columns looped.

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**Prim`s on a Distance Table**

Here I have chosen F. Delete the row. Loop the column. Select the smallest number in the loop.

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**Prim`s on a Distance Table**

Delete row C. Loop column C. Select the smallest number in any loop that is not crossed out.

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**Prim`s on a Distance Table**

Delete row D. Loop column D. Select the smallest number in any loop that is not crossed out.

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**Prim`s on a Distance Table**

Delete row E. Loop column E. Select the smallest number in any loop that is not crossed out.

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**Prim`s on a Distance Table**

Delete row A. Loop column A. Select the smallest number in any loop that is not crossed out.

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**Prim`s on a Distance Table**

Delete row B. Loop column B.

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**Prim`s on a Distance Table**

The algorithm is complete when all the columns have been looped and the rows crossed out. The circles show the edges in the minimum connector.

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**Prim`s on a Distance Table**

In this case they are AB, DE, AF, CF, DF Can you explain why this procedure is exactly the same as applying Prim`s algorithm?

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Chapter 5: Transportation, Assignment and Network Models © 2007 Pearson Education.

Chapter 5: Transportation, Assignment and Network Models © 2007 Pearson Education.

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