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1. Nuffield Free-Standing Mathematics Activity Cable TV

2. Bride 9.3 4.6 6.5 Ramsey Kirk Michael Ballaugh 2.8 9.3 6.8 15.4 7.4 Peel Laxey St Johns 7.7 2.7 8.2 13.9 9.3 Douglas 10.2 4.7 Port Erin The Isle of Man Castletown Cable TV A Cable TV company wants to connect all the towns shown on the map. The company wants to use the minimum length of cable. This is called a minimum connector problem. This activity is about solving minimum connector problems

3. To solve the problem you need to find a spanning tree of minimum length. A spanning tree is a tree that connects all the vertices together. A minimum spanning tree is a spanning tree of minimum length. Think about How many edges will there be in a minimum spanning tree that connects n vertices?

4. Kruskal’s Algorithm Step 1 List the edges in order of increasing length. Step 2 Start with the shortest edge. Step 3 From the remaining edges, select the shortest edge which does not form a cycle. (If there are 2 shortest edges choose either.) Step 4 Repeat Step 3 until all the vertices are connected.

5. Edges in order of increasing length: Douglas to St Johns 8.2 Peel to St Johns2.7 Ballaugh to Kirk Michael 2.8 Ballaugh to Bride Castletown to St Johns 9.3Laxey to Ramsey Bride to Ramsey 4.6 Castletown to Douglas 10.2 Castletown to Port Erin 4.7 Peel to Port Erin 13.9 Ballaugh to Ramsey 6.5 Douglas to Ramsey 15.4 Kirk Michael to Peel 6.8 Kirk Michael to St Johns 7.4 Douglas to Laxey 7.7

6. 4.6 6.5 2.8 6.8 7.7 2.7 8.2 9.3 4.7 The Isle of Man Bride Peel to St Johns 2.7 Ball to K Michael 2.8 Ramsey Bride to Ramsey 4.6 Kirk Michael Ballaugh Castletown to Port Erin 4.7 Ballaugh to Ramsey 6.5 Peel Laxey St Johns Kirk Michael to Peel 6.8 Douglas to Laxey 7.7 Douglas Douglas to St Johns 8.2 Castletown to St Johns 9.3 Port Erin Castletown Total length = 53.3 miles

7. Prim’s Algorithm Step 1 Starting from any vertex, join it to the nearest adjacent vertex. Step 2 Join the next nearest vertex to those already included, provided this does not form a cycle. Step 3 Repeat Step 2 until all the vertices are included. The method can be carried out using an adjacency matrix.

8. B1 B2 C D K L P1 P2 R S - - - - - - B1 - - - - - - - - B2 - - - - - - - 9.3 4.7 C - - - 7.7 - 15.4 - - 8.2 D - - - - - - - 6.8 7.4 K - - - - - 7.7 - - 9.3 - L - - - - - - 2.7 - 6.8 13.9 P1 - - - 13.9 - - - - - 4.7 P2 - - 15.4 - - - - 9.3 R - - - 9.3 8.2 7.4 2.7 - - - S Writing the network as an adjacency network: - 9.3 2.8 6.5 9.3 4.6 10.2 10.2 2.8 6.5 4.6

9. B1 B2 C D K L P1 P2 R S - - - - - - B1 - - - - - - - - B2 - - - - - - - 4.7 9.3 C - - - 7.7 - 15.4 - - 8.2 D - - - - - - - 6.8 7.4 K - - - - - 7.7 - - - 9.3 L - - - - - - 2.7 - 6.8 13.9 P1 - - - 13.9 - - - - - 4.7 P2 - - 15.4 - - - - 9.3 R - - - 9.3 8.2 7.4 2.7 - - - S Starting from any vertex – say B1: 1 4 2 5 10 3 6 8 9 7 - 9.3 2.8 6.5 9.3 4.6 10.2 10.2 2.8 6.5 4.6

10. Prim’s Algorithm results in the same minimum spanning tree, but the order the edges are added to the tree is different Ballaugh to Kirk Michael 2.8 Ballaugh to Ramsey 6.5 Ramsey to Bride 4.6 Kirk Michael to Peel 6.8 Peel to St Johns 2.7 St Johns to Douglas 8.2 Douglas to Laxey 7.7 St Johns to Castletown 9.3 Castletown to Port Erin 4.7

11. 4.6 6.5 2.8 6.8 7.7 2.7 8.2 9.3 4.7 The Isle of Man Bride Total length = 2.8 + 6.5 + 4.6 + 6.8 + 2.7 + 8.2 + 7.7 + 9.3 + 4.7 Ramsey Kirk Michael Ballaugh = 53.3miles Peel Laxey St Johns Think about Can you explain how the matrix method works?Which method do you prefer? Why? What else would need to be considered in the real situation? Douglas Port Erin Castletown

12. Cable TV Reflect on your work How many edges will there be in the minimum spanning tree of a network with n vertices? What are the advantages or disadvantages of Kruskal’s and Prim’s algorithms? What other sorts of things might need to be considered in a real situation?