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Decision Maths Dijkstra s Algorithm Networks Maps are

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Decision Maths Dijkstra s Algorithm 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). – PowerPoint PPT presentation

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Title: Decision Maths Dijkstra s Algorithm Networks Maps are


1
Decision Maths
  • Dijkstras Algorithm

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

11
Dijkstras Algorithm
  • 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
  • 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
  • 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
  • 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
  • Again there are two vertices that can be assigned
    a permanent label. G is chosen.
  • No adjustments need to be made.

16
Dijkstras Algorithm
  • C is assigned a permanent label.
  • Ts temporary label can be adjusted as 4 3 2
    1 10.

17
Dijkstras Algorithm
  • 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
  • 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
  • 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
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4
4
10
10
15
6
4
15
3
3
10
9
3
6
5
10
6
26
Ex 3d q1i
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
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