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4. Minimum spanning trees

5. 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.

6. 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

7. 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 4 5 8 6 4 5 3 8 Example

8. ABCDEF A-3--47 B3-5--8 C-5-4-6 D--4-28 E4--2-5 F78685- Prim’s algorithm in tabular form First put the information from the network into a distance matrix

10. ABCDEF A-3--47 B3-5--8 C-5-4-6 D--4-28 E4--2-5 F78685- 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

11. ABCDEF A-3--47 B3-5--8 C-5-4-6 D--4-28 E4--2-5 F78685- 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

12. ABCDEF A-3--47 B3-5--8 C-5-4-6 D--4-28 E4--2-5 F78685- 12 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

13. ABCDEF A-3--47 B3-5--8 C-5-4-6 D--4-28 E4--2-5 F78685- 123 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

14. ABCDEF A-3--47 B3-5--8 C-5-4-6 D--4-28 E4--2-5 F78685- 123 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 4 Fingley 5

15. ABCDEF A-3--47 B3-5--8 C-5-4-6 D--4-28 E4--2-5 F78685- 12345 FINALLY Label column F “6” Delete row F 6 Avenford Brinleigh 3 Edan 4 Donster 2 Cornwell 4 Fingley 5

16. ABCDEF A-3--47 B3-5--8 C-5-4-6 D--4-28 E4--2-5 F78685- 12345 FINALLY 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 4 Fingley 5

17. 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.

19. Exam Questions Extension and independent study

26. Plenary: A, B, C, D cards (no E )

30. 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.