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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.