Decision Maths

1
Decision Maths

• Lesson 3 - Networks

2
Networks

• Maps are examples of a real life networks.
• In the map below each town is a vertex (node) and
  each road is an edge (arc).

3
Networks

• In real life we often want to know what is the
  shortest path between two places.
• In the past you used to look on a map and plan
  the route yourself.
• These days there are websites that will do this
  for you.
• http//www.theaa.com/travelwatch/planner_main.jsp
• The computer cannot decide which route is the
  best, so it uses an algorithm to do so.
• One such Algorithm is Dijkstras.

4
Dijkstras Algorithm.

• Throughout this algorithm you will be required to
  fill in the grid below.
• It is important that you have a key in your work
  so that the examiner will understand your
  workings.

5
Dijkstras Algorithm

• Consider the network below. We are going to apply
  an algorithm to find the shortest route from S to
  T.
• The solution to this should be obvious but it is
  important to learn the algorithm so we can apply
  it in more complex situations.
• As you work through the algorithm try to
  understand why it works.

6
Dijkstras Algorithm - Step 1

• Give the start node a permanent label of 0.
• The 1 indicates that this is the first node to
  receive a permanent label.

7
Dijkstras Algorithm - Step 1

• Look at all the nodes which can be reached from
  the start node in one edge.
• Thats A, D and F.
• Give them a temporary label of their distance
  from the start node.

8
Dijkstras Algorithm - Step 2

• Make the lowest temporary node permanent.
• This is node A in this case.
• If there had been more than one, then you could
  choose any.

9
Dijkstras Algorithm - Step 3

• Node A has just received the permanent label 3.
  Look at all the nodes you can reach from A
  without a permanent label.
• Give such nodes a temporary label of 3 their
  distance from A.

10
Dijkstras Algorithm - Step 4

• Make the lowest temporary node permanent.
• This is node D in this case.

11
Dijkstras Algorithm - Step 5

• Repeat step 4, only this time with node D.
• Label all nodes from D with temporary label (4
  distance) from D.
• If a node has a temporary label, replace it, if
  (4 distance) is less than the temporary label.

12
Dijkstras Algorithm - Step 6

• Make the lowest temporary node permanent.
• F now gets a permanent label.
• Node G already has a temporary label which does
  not change.

13
Dijkstras Algorithm - Step 7

• Both B and E have the same temporary label.
• It makes no difference which we pick, so B is
  selected.
• B is the 5th permanent label. C needs a temporary
  label.

14
Dijkstras Algorithm - Step 8

• E gets a permanent label, as it has the lowest
  temporary label.
• Cs temporary label must change to 9 as you can
  go S,D,E,C which has length 4 3 2 9 as this
  is less than the existing label.
• T is given a temporary label of 11.

15
Dijkstras Algorithm - Step 9

• Again there are two vertices that can be assigned
  a permanent label. G is chosen.
• No adjustments need to be made.

16
Dijkstras Algorithm - Step 10

• C is assigned a permanent label.
• Ts temporary label can be adjusted as 4 3 2
  1 10.

17
Dijkstras Algorithm - Step 11

• T now gets a permanent label of 10.
• This tells us that the shortest route from S to T
  has a length 10.

18
Dijkstras Algorithm - Step 12

• The shortest path will be marked with a red line.
• To find it you work backwards from T to S
  identifying whose length is the same as the
  difference between the permanent labels at either
  end.

19
Dijkstras Algorithm - Step 13

• The weights on the arcs can represent different
  things that might lead to alternate routes.
• Example They could change to represent the
  time taken to travel a stretch of road rather
  than distance.

20
Dijkstras Algorithm

• Can you explain how the Algorithm works?
• What do the temporary and permanent labels mean?
• They are values for the quickest route to that
  particular node.

21
Dijkstras Algorithm

• Why do we assign temporary labels?
• The algorithm systematically searches for the
  quickest route to every node. We assign a
  temporary label as that stands for the current
  quickest route to that node. Later in the
  algorithm an alternative route may be found so we
  replace the temporary label with a new temporary
  label.

22
Dijkstras Algorithm

• Why do we make the node with the lowest temporary
  label permanent?
• All routes up to a certain point have been
  covered, so the node with the lowest temporary
  label must be the next nearest node to the start
  point.

23
Dijkstras Algorithm

• What happens after this?
• You are certain that there is no shorter way of
  reaching the node you are currently at. So you
  can assign neighbouring nodes a temporary label.
• Why does the first node get assigned permanent
  label zero?
• It is the start point , you have not travelled
  anywhere.

24
Dijkstras Algorithm

• Why does the method for finding the route at the
  end work?
• At the end of the algorithm each node has a
  permanent label that represents the shortest
  distance to that node. If two nodes are joined
  then the difference in their permanent labels
  will tell you the shortest distance between them.
  If the arc joining them matches this distance
  then it must be the quickest route.

25
Ex 3d q1i Shortest route from S to T
7
6
7
9
7
2
2
9
2
0
1
10
8
4
4
10
10
15
6
4
15
3
3
10
9
3
6
5
10
6
26
Ex 3d q1i Shortest route from S to T
7
6
7
9
7
2
2
9
2
0
1
10
8
4
4
10
10
15
6
4
15
3
3
10
9
3
6
5
10
6