1. Networks Kruskal’s Algorithm
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 = 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. Example Consider the Network below.
It helps to rank the arcs in increasing order.

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

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

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

11. Applying the Algorithm.
Next select CF.

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

16. Solution – Ex 3a pg 66 q1

17. Solution – Ex 3a pg 66 q1

18. Solution – Ex 3a pg 66 q1

19. Solution – Ex 3a pg 66 q1

20. Solution – Ex 3a pg 66 q1

21. Solution – Ex 3a pg 66 q1

22. Solution – Ex 3a pg 66 q4

23. Solution – Ex 3a pg 66 q4

24. Solution – Ex 3a pg 66 q4

25. Solution – Ex 3a pg 66 q4

26. Solution – Ex 3a pg 66 q4

27. Solution – Ex 3a pg 66 q4

28. Solution – Ex 3a pg 66 q4

29. Solution – Ex 3a pg 66 q4

30. Solution – Ex 3a pg 66 q4

31. Solution – Ex 3a pg 66 q4

32. Solution – Ex 3a pg 66 q4

33. Solution – Ex 3a pg 66 q4

34. Solution – Ex 3a pg 66 q4

35. Solution – Ex 3a pg 66 q4

36. Solution – Ex 3a pg 66 q4

37. Solution – Ex 3a pg 66 q4