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© Nuffield Foundation 2011 Nuffield Free-Standing Mathematics Activity Cable TV

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

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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?

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Kruskals Algorithm Step 1List 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.

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Peel to St Johns2.7 Edges in order of increasing length: Ballaugh to Kirk Michael2.8 Bride to Ramsey4.6 Castletown to Port Erin4.7 Ballaugh to Ramsey6.5 Kirk Michael to Peel6.8 Douglas to Laxey7.7 Kirk Michael to St Johns7.4 Douglas to St Johns 8.2 Ballaugh to Bride Castletown to St Johns 9.3 Laxey to Ramsey Castletown to Douglas10.2 Peel to Port Erin13.9 Douglas to Ramsey15.4

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

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Prims Algorithm The method can be carried out using an adjacency matrix. 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.

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Writing the network as an adjacency network: B1B1 B1B1 B2B2 C D K L P1P1 P2P2 R S B2B2 CDKLP1P1 P2P2 RS - - - - - - - - - - 9.3 2.8 6.5 4.6 10.2 7.7 15.4 8.2 6.8 13.9 2.7 4.7 - - - - -- - - -- -- - - - - -- - - ---- - ---- - -- - - -- --- - - - - - - -- - -- - -- -- - -- -- 9.3 7.4 9.3

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Starting from any vertex – say B 1 : B1B1 B1B1 B2B2 C D K L P1P1 P2P2 R S B2B2 CDKLP1P1 P2P2 RS - - - - - - - - - - 9.3 2.8 6.5 4.6 10.2 7.7 15.4 8.2 6.8 13.9 2.7 - - - - -- - - ---- - - -- -- ---- 9.37.4 - -- - - -- - 9.3 -- - - - - 4.7 - - - -- - - - - -- - -- - -- -- - - 9.3 - -- 123456 7 8 9 10

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Prims Algorithm results in the same minimum spanning tree, but the order the edges are added to the tree is different Ballaugh to Kirk Michael2.8 Ballaugh to Ramsey6.5 Ramsey to Bride4.6 Kirk Michael to Peel6.8 Peel to St Johns2.7 St Johns to Douglas8.2 Douglas to Laxey7.7 St Johns to Castletown9.3 Castletown to Port Erin4.7

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

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© Nuffield Foundation 2011 Cable TV Reflect on your work What are the advantages or disadvantages of Kruskals and Prims algorithms? What other sorts of things might need to be considered in a real situation? How many edges will there be in the minimum spanning tree of a network with n vertices?

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