1. Networks & Minimum Spanning Trees

2. Cambridge is installing fibre optic cabling between the surrounding villages. All the villages must be connected. We need to minimise the amount of cable that is used. Networks and Minimum Spanning Trees

4. Cambridge Milton Bottisham Fulbourn Great Shelford Grantchester Coton Girton Impington 6 miles 5 miles 4 miles 3 miles 6 miles 3 miles 4 miles 7 miles 6 miles 5 miles 3 miles 4 miles 5 miles

5. Cambridge Milton Bottisham Fulbourn Great Shelford Grantchester Coton Girton Impington 6 miles 5 miles 4 miles 3 miles 6 miles 3 miles 4 miles 7 miles 6 miles 5 miles 3 miles 4 miles 5 miles

6. C M B F S G O T I 6 6 5 4 3 6 3 3 4 7 5 3 4 5 5 6 This network can now be used to model our situation.

7. Networks and Minimum Spanning Trees A network is a type of graph used in Decision Maths. Points (called nodes or vertices) are connected by lines (called arcs or edges). The lines (edges) have values attached to them, these may represent a cost or a distance.

8. Networks and Minimum Spanning Trees A minimum spanning tree is a section of the network. All points (nodes) must be included but you should not have any cycles. A cycle links points together in a closed ring. The minimum spanning tree links the points (nodes) using the lines (edges) that have the smallest values. Cycle ✖ ✔

9. Networks and Minimum Spanning Trees Algorithms have been created to help us decide which edges should be used to create the minimum spanning tree. We are going to use Kruskal’s Algorithm.

10. Kruskal's’s Algorithm 1.Write down all the edges in size order 1.Select the shortest edge in a network 2.Select the next shortest edge which does not create a cycle 3.Repeat step 3 until all vertices have been connected

11. C M B F S G O T I 6 6 5 4 3 6 3 3 4 7 5 3 4 5 5 6 TI – 3 IM – 3 GC – 3 GO – 3 MC – 4 SG – 4 OC – 4 IC – 5 TC – 5 SC – 5 SF – 5 TO – 6 CF – 6 BF – 6 BM – 6 CB – 7

12. C M B F S G O T I 6 6 5 4 3 6 3 3 4 7 5 3 4 5 5 6 TI – 3 ✔ IM – 3 GC – 3 GO – 3 MC – 4 SG – 4 OC – 4 IC – 5 TC – 5 SC – 5 SF – 5 TO – 6 CF – 6 BF – 6 BM – 6 CB – 7

13. C M B F S G O T I 6 6 5 4 3 6 3 3 4 7 5 3 4 5 5 6 TI – 3 ✔ IM – 3 GC – 3 GO – 3 MC – 4 SG – 4 OC – 4 IC – 5 TC – 5 SC – 5 SF – 5 TO – 6 CF – 6 BF – 6 BM – 6 CB – 7

14. C M B F S G O T I 6 6 5 4 3 6 3 3 4 7 5 3 4 5 5 6 TI – 3 ✔ IM – 3 ✔ GC – 3 GO – 3 MC – 4 SG – 4 OC – 4 IC – 5 TC – 5 SC – 5 SF – 5 TO – 6 CF – 6 BF – 6 BM – 6 CB – 7

15. C M B F S G O T I 6 6 5 4 3 6 3 3 4 7 5 3 4 5 5 6 TI – 3 ✔ IM – 3 ✔ GC – 3 GO – 3 MC – 4 SG – 4 OC – 4 IC – 5 TC – 5 SC – 5 SF – 5 TO – 6 CF – 6 BF – 6 BM – 6 CB – 7

16. C M B F S G O T I 6 6 5 4 3 6 3 3 4 7 5 3 4 5 5 6 TI – 3 ✔ IM – 3 ✔ GC – 3 ✔ GO – 3 MC – 4 SG – 4 OC – 4 IC – 5 TC – 5 SC – 5 SF – 5 TO – 6 CF – 6 BF – 6 BM – 6 CB – 7

17. C M B F S G O T I 6 6 5 4 3 6 3 3 4 7 5 3 4 5 5 6 TI – 3 ✔ IM – 3 ✔ GC – 3 ✔ GO – 3 MC – 4 SG – 4 OC – 4 IC – 5 TC – 5 SC – 5 SF – 5 TO – 6 CF – 6 BF – 6 BM – 6 CB – 7

18. C M B F S G O T I 6 6 5 4 3 6 3 3 4 7 5 3 4 5 5 6 TI – 3 ✔ IM – 3 ✔ GC – 3 ✔ GO – 3 ✔ MC – 4 SG – 4 OC – 4 IC – 5 TC – 5 SC – 5 SF – 5 TO – 6 CF – 6 BF – 6 BM – 6 CB – 7

19. C M B F S G O T I 6 6 5 4 3 6 3 3 4 7 5 3 4 5 5 6 TI – 3 ✔ IM – 3 ✔ GC – 3 ✔ GO – 3 ✔ MC – 4 SG – 4 OC – 4 IC – 5 TC – 5 SC – 5 SF – 5 TO – 6 CF – 6 BF – 6 BM – 6 CB – 7

20. C M B F S G O T I 6 6 5 4 3 6 3 3 4 7 5 3 4 5 5 6 TI – 3 ✔ IM – 3 ✔ GC – 3 ✔ GO – 3 ✔ MC – 4 ✔ SG – 4 OC – 4 IC – 5 TC – 5 SC – 5 SF – 5 TO – 6 CF – 6 BF – 6 BM – 6 CB – 7

21. C M B F S G O T I 6 6 5 4 3 6 3 3 4 7 5 3 4 5 5 6 TI – 3 ✔ IM – 3 ✔ GC – 3 ✔ GO – 3 ✔ MC – 4 ✔ SG – 4 OC – 4 IC – 5 TC – 5 SC – 5 SF – 5 TO – 6 CF – 6 BF – 6 BM – 6 CB – 7

22. C M B F S G O T I 6 6 5 4 3 6 3 3 4 7 5 3 4 5 5 6 TI – 3 ✔ IM – 3 ✔ GC – 3 ✔ GO – 3 ✔ MC – 4 ✔ SG – 4 ✔ OC – 4 IC – 5 TC – 5 SC – 5 SF – 5 TO – 6 CF – 6 BF – 6 BM – 6 CB – 7

23. C M B F S G O T I 6 6 5 4 3 6 3 3 4 7 5 3 4 5 5 6 TI – 3 ✔ IM – 3 ✔ GC – 3 ✔ GO – 3 ✔ MC – 4 ✔ SG – 4 ✔ OC – 4 IC – 5 TC – 5 SC – 5 SF – 5 TO – 6 CF – 6 BF – 6 BM – 6 CB – 7

24. C M B F S G O T I 6 6 5 4 3 6 3 3 4 7 5 3 4 5 5 6 TI – 3 ✔ IM – 3 ✔ GC – 3 ✔ GO – 3 ✔ MC – 4 ✔ SG – 4 ✔ OC – 4 ✖ (cycle) IC – 5 TC – 5 SC – 5 SF – 5 TO – 6 CF – 6 BF – 6 BM – 6 CB – 7

25. C M B F S G O T I 6 6 5 4 3 6 3 3 4 7 5 3 4 5 5 6 TI – 3 ✔ IM – 3 ✔ GC – 3 ✔ GO – 3 ✔ MC – 4 ✔ SG – 4 ✔ OC – 4 ✖ (cycle) IC – 5 ✖ (cycle) TC – 5 SC – 5 SF – 5 TO – 6 CF – 6 BF – 6 BM – 6 CB – 7

26. C M B F S G O T I 6 6 5 4 3 6 3 3 4 7 5 3 4 5 5 6 TI – 3 ✔ IM – 3 ✔ GC – 3 ✔ GO – 3 ✔ MC – 4 ✔ SG – 4 ✔ OC – 4 ✖ (cycle) IC – 5 ✖ (cycle) TC – 5 ✖ (cycle) SC – 5 SF – 5 TO – 6 CF – 6 BF – 6 BM – 6 CB – 7

27. C M B F S G O T I 6 6 5 4 3 6 3 3 4 7 5 3 4 5 5 6 TI – 3 ✔ IM – 3 ✔ GC – 3 ✔ GO – 3 ✔ MC – 4 ✔ SG – 4 ✔ OC – 4 ✖ (cycle) IC – 5 ✖ (cycle) TC – 5 ✖ (cycle) SC – 5 ✖ (cycle) SF – 5 TO – 6 CF – 6 BF – 6 BM – 6 CB – 7

28. C M B F S G O T I 6 6 5 4 3 6 3 3 4 7 5 3 4 5 5 6 TI – 3 ✔ IM – 3 ✔ GC – 3 ✔ GO – 3 ✔ MC – 4 ✔ SG – 4 ✔ OC – 4 ✖ (cycle) IC – 5 ✖ (cycle) TC – 5 ✖ (cycle) SC – 5 ✖ (cycle) SF – 5 ✔ TO – 6 CF – 6 BF – 6 BM – 6 CB – 7

29. C M B F S G O T I 6 6 5 4 3 6 3 3 4 7 5 3 4 5 5 6 TI – 3 ✔ IM – 3 ✔ GC – 3 ✔ GO – 3 ✔ MC – 4 ✔ SG – 4 ✔ OC – 4 ✖ (cycle) IC – 5 ✖ (cycle) TC – 5 ✖ (cycle) SC – 5 ✖ (cycle) SF – 5 ✔ TO – 6 CF – 6 BF – 6 BM – 6 CB – 7

30. C M B F S G O T I 6 6 5 4 3 6 3 3 4 7 5 3 4 5 5 6 TI – 3 ✔ IM – 3 ✔ GC – 3 ✔ GO – 3 ✔ MC – 4 ✔ SG – 4 ✔ OC – 4 ✖ (cycle) IC – 5 ✖ (cycle) TC – 5 ✖ (cycle) SC – 5 ✖ (cycle) SF – 5 ✔ TO – 6 ✖ (cycle) CF – 6 BF – 6 BM – 6 CB – 7

31. C M B F S G O T I 6 6 5 4 3 6 3 3 4 7 5 3 4 5 5 6 TI – 3 ✔ IM – 3 ✔ GC – 3 ✔ GO – 3 ✔ MC – 4 ✔ SG – 4 ✔ OC – 4 ✖ (cycle) IC – 5 ✖ (cycle) TC – 5 ✖ (cycle) SC – 5 ✖ (cycle) SF – 5 ✔ TO – 6 ✖ (cycle) CF – 6 ✖ (cycle) BF – 6 BM – 6 CB – 7

32. C M B F S G O T I 6 6 5 4 3 6 3 3 4 7 5 3 4 5 5 6 TI – 3 ✔ IM – 3 ✔ GC – 3 ✔ GO – 3 ✔ MC – 4 ✔ SG – 4 ✔ OC – 4 ✖ (cycle) IC – 5 ✖ (cycle) TC – 5 ✖ (cycle) SC – 5 ✖ (cycle) SF – 5 ✔ TO – 6 ✖ (cycle) CF – 6 ✖ (cycle) BF – 6 ✔ BM – 6 CB – 7

33. C M B F S G O T I 6 6 5 4 3 6 3 3 4 7 5 3 4 5 5 6 TI – 3 ✔ IM – 3 ✔ GC – 3 ✔ GO – 3 ✔ MC – 4 ✔ SG – 4 ✔ OC – 4 ✖ (cycle) IC – 5 ✖ (cycle) TC – 5 ✖ (cycle) SC – 5 ✖ (cycle) SF – 5 ✔ TO – 6 ✖ (cycle) CF – 6 ✖ (cycle) BF – 6 ✔ BM – 6 CB – 7

34. C M B F S G O T I 6 6 5 4 3 6 3 3 4 7 5 3 4 5 5 6 TI – 3 ✔ IM – 3 ✔ GC – 3 ✔ GO – 3 ✔ MC – 4 ✔ SG – 4 ✔ OC – 4 ✖ (cycle) IC – 5 ✖ (cycle) TC – 5 ✖ (cycle) SC – 5 ✖ (cycle) SF – 5 ✔ TO – 6 ✖ (cycle) CF – 6 ✖ (cycle) BF – 6 ✔ BM – 6 ✖ (cycle) CB – 7

35. C M B F S G O T I 6 6 5 4 3 6 3 3 4 7 5 3 4 5 5 6 TI – 3 ✔ IM – 3 ✔ GC – 3 ✔ GO – 3 ✔ MC – 4 ✔ SG – 4 ✔ OC – 4 ✖ (cycle) IC – 5 ✖ (cycle) TC – 5 ✖ (cycle) SC – 5 ✖ (cycle) SF – 5 ✔ TO – 6 ✖ (cycle) CF – 6 ✖ (cycle) BF – 6 ✔ BM – 6 ✖ (cycle) CB – 7 ✖ (cycle)

36. C M B F S G O T I 6 5 4 3 3 3 4 3 The minimum spanning tree contains only the edges we have selected. The total weight of the tree comes from the summation of our distances. 3 + 3 + 4 + 3 + 3 + 4 + 5 + 6 = 31 So, the city needs a total of 31 km in fibre optic cabling!

37. Tunneling Students In an attempt to get out of the rain and away from the tourists, the students of the University of Cambridge are setting up some underground tunnels between the university buildings! Suggest a network of tunnels they could use, minimising the amount of tunneling required.

39. Cl S Q Pb K T SJ J E 0.2 0.5 0.7 0.5 0.2 0.1 0.3 0.5 0.4 0.2

40. Cl S Q Pb K T SJ J E 0.2 0.5 0.7 0.5 0.2 0.1 0.3 0.5 0.4 0.2

41. Cl S Q Pb K T SJ J E 0.2 0.5 0.7 0.5 0.2 0.1 0.3 0.5 0.4 0.2 KPb – 0.1 KQ – 0.2 KT – 0.2 TSj – 0.2 PbE – 0.3 TE – 0.4 ClS – 0.5 SQ – 0.5 EJ – 0.5 TCl – 0.5 SjJ – 0.5 SjCl – 0.7

42. Cl S Q Pb K T SJ J E 0.2 0.5 0.7 0.5 0.2 0.1 0.3 0.5 0.4 0.2 KPb – 0.1 ✔ KQ – 0.2 KT – 0.2 TSj – 0.2 PbE – 0.3 TE – 0.4 ClS – 0.5 SQ – 0.5 EJ – 0.5 TCl – 0.5 SjJ – 0.5 SjCl – 0.7

43. Cl S Q Pb K T SJ J E 0.2 0.5 0.7 0.5 0.2 0.1 0.3 0.5 0.4 0.2 KPb – 0.1 ✔ KQ – 0.2 KT – 0.2 TSj – 0.2 PbE – 0.3 TE – 0.4 ClS – 0.5 SQ – 0.5 EJ – 0.5 TCl – 0.5 SjJ – 0.5 SjCl – 0.7

44. Cl S Q Pb K T SJ J E 0.2 0.5 0.7 0.5 0.2 0.1 0.3 0.5 0.4 0.2 KPb – 0.1 ✔ KQ – 0.2 ✔ KT – 0.2 TSj – 0.2 PbE – 0.3 TE – 0.4 ClS – 0.5 SQ – 0.5 EJ – 0.5 TCl – 0.5 SjJ – 0.5 SjCl – 0.7

45. Cl S Q Pb K T SJ J E 0.2 0.5 0.7 0.5 0.2 0.1 0.3 0.5 0.4 0.2 KPb – 0.1 ✔ KQ – 0.2 ✔ KT – 0.2 TSj – 0.2 PbE – 0.3 TE – 0.4 ClS – 0.5 SQ – 0.5 EJ – 0.5 TCl – 0.5 SjJ – 0.5 SjCl – 0.7

46. Cl S Q Pb K T SJ J E 0.2 0.5 0.7 0.5 0.2 0.1 0.3 0.5 0.4 0.2 KPb – 0.1 ✔ KQ – 0.2 ✔ KT – 0.2 ✔ TSj – 0.2 PbE – 0.3 TE – 0.4 ClS – 0.5 SQ – 0.5 EJ – 0.5 TCl – 0.5 SjJ – 0.5 SjCl – 0.7

47. Cl S Q Pb K T SJ J E 0.2 0.5 0.7 0.5 0.2 0.1 0.3 0.5 0.4 0.2 KPb – 0.1 ✔ KQ – 0.2 ✔ KT – 0.2 ✔ TSj – 0.2 PbE – 0.3 TE – 0.4 ClS – 0.5 SQ – 0.5 EJ – 0.5 TCl – 0.5 SjJ – 0.5 SjCl – 0.7

48. Cl S Q Pb K T SJ J E 0.2 0.5 0.7 0.5 0.2 0.1 0.3 0.5 0.4 0.2 KPb – 0.1 ✔ KQ – 0.2 ✔ KT – 0.2 ✔ TSj – 0.2 ✔ PbE – 0.3 TE – 0.4 ClS – 0.5 SQ – 0.5 EJ – 0.5 TCl – 0.5 SjJ – 0.5 SjCl – 0.7

49. Cl S Q Pb K T SJ J E 0.2 0.5 0.7 0.5 0.2 0.1 0.3 0.5 0.4 0.2 KPb – 0.1 ✔ KQ – 0.2 ✔ KT – 0.2 ✔ TSj – 0.2 ✔ PbE – 0.3 TE – 0.4 ClS – 0.5 SQ – 0.5 EJ – 0.5 TCl – 0.5 SjJ – 0.5 SjCl – 0.7

50. Cl S Q Pb K T SJ J E 0.2 0.5 0.7 0.5 0.2 0.1 0.3 0.5 0.4 0.2 KPb – 0.1 ✔ KQ – 0.2 ✔ KT – 0.2 ✔ TSj – 0.2 ✔ PbE – 0.3 ✔ TE – 0.4 ClS – 0.5 SQ – 0.5 EJ – 0.5 TCl – 0.5 SjJ – 0.5 SjCl – 0.7

51. Cl S Q Pb K T SJ J E 0.2 0.5 0.7 0.5 0.2 0.1 0.3 0.5 0.4 0.2 KPb – 0.1 ✔ KQ – 0.2 ✔ KT – 0.2 ✔ TSj – 0.2 ✔ PbE – 0.3 ✔ TE – 0.4 ClS – 0.5 SQ – 0.5 EJ – 0.5 TCl – 0.5 SjJ – 0.5 SjCl – 0.7

52. Cl S Q Pb K T SJ J E 0.2 0.5 0.7 0.5 0.2 0.1 0.3 0.5 0.4 0.2 KPb – 0.1 ✔ KQ – 0.2 ✔ KT – 0.2 ✔ TSj – 0.2 ✔ PbE – 0.3 ✔ TE – 0.4 ✖ (cycle) ClS – 0.5 SQ – 0.5 EJ – 0.5 TCl – 0.5 SjJ – 0.5 SjCl – 0.7

53. Cl S Q Pb K T SJ J E 0.2 0.5 0.7 0.5 0.2 0.1 0.3 0.5 0.4 0.2 KPb – 0.1 ✔ KQ – 0.2 ✔ KT – 0.2 ✔ TSj – 0.2 ✔ PbE – 0.3 ✔ TE – 0.4 ✖ (cycle) ClS – 0.5 ✔ SQ – 0.5 EJ – 0.5 TCl – 0.5 SjJ – 0.5 SjCl – 0.7

54. Cl S Q Pb K T SJ J E 0.2 0.5 0.7 0.5 0.2 0.1 0.3 0.5 0.4 0.2 KPb – 0.1 ✔ KQ – 0.2 ✔ KT – 0.2 ✔ TSj – 0.2 ✔ PbE – 0.3 ✔ TE – 0.4 ✖ (cycle) ClS – 0.5 ✔ SQ – 0.5 EJ – 0.5 TCl – 0.5 SjJ – 0.5 SjCl – 0.7

55. Cl S Q Pb K T SJ J E 0.2 0.5 0.7 0.5 0.2 0.1 0.3 0.5 0.4 0.2 KPb – 0.1 ✔ KQ – 0.2 ✔ KT – 0.2 ✔ TSj – 0.2 ✔ PbE – 0.3 ✔ TE – 0.4 ✖ (cycle) ClS – 0.5 ✔ SQ – 0.5 ✔ EJ – 0.5 TCl – 0.5 SjJ – 0.5 SjCl – 0.7

56. Cl S Q Pb K T SJ J E 0.2 0.5 0.7 0.5 0.2 0.1 0.3 0.5 0.4 0.2 KPb – 0.1 ✔ KQ – 0.2 ✔ KT – 0.2 ✔ TSj – 0.2 ✔ PbE – 0.3 ✔ TE – 0.4 ✖ (cycle) ClS – 0.5 ✔ SQ – 0.5 ✔ EJ – 0.5 TCl – 0.5 SjJ – 0.5 SjCl – 0.7

57. Cl S Q Pb K T SJ J E 0.2 0.5 0.7 0.5 0.2 0.1 0.3 0.5 0.4 0.2 KPb – 0.1 ✔ KQ – 0.2 ✔ KT – 0.2 ✔ TSj – 0.2 ✔ PbE – 0.3 ✔ TE – 0.4 ✖ (cycle) ClS – 0.5 ✔ SQ – 0.5 ✔ EJ – 0.5 ✔ TCl – 0.5 SjJ – 0.5 SjCl – 0.7

58. Cl S Q Pb K T SJ J E 0.2 0.5 0.7 0.5 0.2 0.1 0.3 0.5 0.4 0.2 KPb – 0.1 ✔ KQ – 0.2 ✔ KT – 0.2 ✔ TSj – 0.2 ✔ PbE – 0.3 ✔ TE – 0.4 ✖ (cycle) ClS – 0.5 ✔ SQ – 0.5 ✔ EJ – 0.5 ✔ TCl – 0.5 SjJ – 0.5 SjCl – 0.7

59. Cl S Q Pb K T SJ J E 0.2 0.5 0.7 0.5 0.2 0.1 0.3 0.5 0.4 0.2 KPb – 0.1 ✔ KQ – 0.2 ✔ KT – 0.2 ✔ TSj – 0.2 ✔ PbE – 0.3 ✔ TE – 0.4 ✖ (cycle) ClS – 0.5 ✔ SQ – 0.5 ✔ EJ – 0.5 ✔ TCl – 0.5 ✖ (cycle) SjJ – 0.5 SjCl – 0.7

60. Cl S Q Pb K T SJ J E 0.2 0.5 0.7 0.5 0.2 0.1 0.3 0.5 0.4 0.2 KPb – 0.1 ✔ KQ – 0.2 ✔ KT – 0.2 ✔ TSj – 0.2 ✔ PbE – 0.3 ✔ TE – 0.4 ✖ (cycle) ClS – 0.5 ✔ SQ – 0.5 ✔ EJ – 0.5 ✔ TCl – 0.5 ✖ (cycle) SjJ – 0.5 ✖ (cycle) SjCl – 0.7

61. Cl S Q Pb K T SJ J E 0.2 0.5 0.7 0.5 0.2 0.1 0.3 0.5 0.4 0.2 KPb – 0.1 ✔ KQ – 0.2 ✔ KT – 0.2 ✔ TSj – 0.2 ✔ PbE – 0.3 ✔ TE – 0.4 ✖ (cycle) ClS – 0.5 ✔ SQ – 0.5 ✔ EJ – 0.5 ✔ TCl – 0.5 ✖ (cycle) SjJ – 0.5 ✖ (cycle) SjCl – 0.7 ✖ (cycle)

62. Cl S Q Pb K T SJ J E 0.2 0.5 0.2 0.1 0.3 0.5 0.2 The students should tunnel between the university buildings as shown. The total length of tunneling needed is 2.5 miles.