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Decision Maths

Decision Maths. Networks Kruskal’s Algorithm. Networks. A Network is a weighted graph, which just means there is a number associated with each edge. The numbers can represent distances, costs, times in real world applications.

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Decision Maths

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  1. Decision Maths Networks Kruskal’s Algorithm

  2. Networks • A Network is a weighted graph, which just means there is a number associated with each edge. • The numbers can represent distances, costs, times in real world applications. • Obvious examples include maps and similar geographical networks.

  3. Networks

  4. Minimum Connector Problem • Basically you need to travel to every node using the least total length. • Consider 4 houses in a Network shown in the diagram below. The weight on each arc represents the distance between each house. • An Electricity company wants to supply every house by using as little cable as possible. • Clearly the shortest possible route is to go from A to B to C and then to D. • So 4 + 3 + 3 = 10, there is no shorter way of supplying every house.

  5. Algorithms • The previous example was a simple one and the solution was very easy to spot. • For more complicated examples you will need to use an algorithm. • An Algorithm is simply a list of instructions that solve a particular problem. • (You will cover Algorithms in more depth later on in the course)

  6. Kruskal`s Algorithm • There are 3 steps to follow in Kruskal`s Algorithm. • Step 1 – Select the shortest arc in the network. • Step 2 – Select the shortest arc from those which are remaining. Ensure that you do not create a cycle. If you do ignore and move on to the next shortest arc. • Step 3 – If all the vertices are connected then stop. If not return to step 2.

  7. Consider the Network below. It helps to rank the arcs in increasing order. Example

  8. 1 – Start by selecting the smallest arc, AB or DE, it makes no difference. Select AB. Applying the Algorithm

  9. 2 – Now select the next smallest, which is DE. Applying the Algorithm

  10. 3 – Next we can select CF or ` DF, again it makes no difference. Lets pick DF. Applying The Algorithm

  11. Next select CF. Applying the Algorithm.

  12. Applying the Algorithm • The next smallest length is EF. However there is already a route from E to F, so this arc is not required.

  13. Applying the Algorithm • Adding CD will again create a loop so the last arc to add is AF. All vertices are now joined so the problem is complete.

  14. Question – Ex 3a pg 66 q1 • Find the minimal spanning tree and associated shortest distance for the network below:

  15. Solution – Ex 3a pg 66 q1

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