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Minimum spanning trees
Aims:
To know the terms: tree, spanning tree, minimum spanning tree.
To understand that a minimum spanning tree connects a network using the lowest possible
total weight of arcs.
To apply Prim’s Algorithm from a table to finding minimum connectors.
Minimum Spanning tree
Prim’s algorithm
1. Select any vertex
2. Select the shortest edge connected to that vertex
3. Select the shortest edge which connects a previously chosen vertex to a new vertex
4. Repeat step 3 until all vertices have been connected
A cable company want to connect five villages to their network which currently extends to the market town of Avenford. What is the minimum length of cable needed?
Avenford Fingley
Brinleigh Cornwell
Donster
Edan
2
7
45
8 6 4
5
3
8
Example
A B C D E FA - 3 - - 4 7B 3 - 5 - - 8C - 5 - 4 - 6D - - 4 - 2 8E 4 - - 2 - 5F 7 8 6 8 5 -
Prim’s algorithm in tabular form
First put the information from the network into a distance matrix
A B C D E FA - 3 - - 4 7B 3 - 5 - - 8C - 5 - 4 - 6D - - 4 - 2 8E 4 - - 2 - 5F 7 8 6 8 5 -
• Start at vertex A. Label column A “1” .
• Delete row A
• Select the smallest entry in column A (AB, length 3)
1
Avenford
Brinleigh
3
A B C D E FA - 3 - - 4 7B 3 - 5 - - 8C - 5 - 4 - 6D - - 4 - 2 8E 4 - - 2 - 5F 7 8 6 8 5 -
1• Label column B “2”
• Delete row B
• Select the smallest uncovered entry in either column A or column B (AE, length 4)
2
Avenford
Brinleigh
3
Edan
4
A B C D E FA - 3 - - 4 7B 3 - 5 - - 8C - 5 - 4 - 6D - - 4 - 2 8E 4 - - 2 - 5F 7 8 6 8 5 -
1 2• Label column E “3”
• Delete row E
• Select the smallest uncovered entry in either column A, B or E (ED, length 2)
3
Avenford
Brinleigh
3
Edan
4
Donster
2
A B C D E FA - 3 - - 4 7B 3 - 5 - - 8C - 5 - 4 - 6D - - 4 - 2 8E 4 - - 2 - 5F 7 8 6 8 5 -
1 2 3• Label column D “4”
• Delete row D
• Select the smallest uncovered entry in either column A, B, D or E (DC, length 4)
4
Avenford
Brinleigh
3
Edan
4
Donster
2
Cornwell
4
A B C D E FA - 3 - - 4 7B 3 - 5 - - 8C - 5 - 4 - 6D - - 4 - 2 8E 4 - - 2 - 5F 7 8 6 8 5 -
1 2 3• Label column C “5”
• Delete row C
• Select the smallest uncovered entry in either column A, B, D, E or C (EF, length 5)
45
Avenford
Brinleigh
3
Edan
4
Donster
2
Cornwell
4Fingley
5
A B C D E FA - 3 - - 4 7B 3 - 5 - - 8C - 5 - 4 - 6D - - 4 - 2 8E 4 - - 2 - 5F 7 8 6 8 5 -
1 2 345FINALLY
• Label column F “6”
• Delete row F
6
Avenford
Brinleigh
3
Edan
4
Donster
2
Cornwell
4Fingley
5
A B C D E FA - 3 - - 4 7B 3 - 5 - - 8C - 5 - 4 - 6D - - 4 - 2 8E 4 - - 2 - 5F 7 8 6 8 5 -
1 2 345FINALLY
• Label column F “6”
• Delete row F
6
The spanning tree is shown in the diagram
Length 3 + 4 + 4 + 2 + 5 = 18Km
Avenford
Brinleigh
3
Edan
4
Donster
2
Cornwell
4Fingley
5
Your turn
Prim’s Method Matrix Questions
Be Careful!
These questions need reading carefully. Make sure you understand each part of
the question and read through the whole question before you start.
Exam Questions
Extension and independent study
Plenary: A, B, C, D cards (no E )
For next lesson
2 videos on EDpuzzle
with quiz
Independent study:
Excel worksheet to practise method on moodle.
Further questions, including past exam questions on worksheets.