1. 5 23 10 21 14 24 16 6 4 18 9 7 11 8 5 6 4 9 7 8 T,  e  T c(e) = 50 G = (V, E), c(e) Minimum Spanning Tree

2. 5 23 10 21 14 24 16 6 4 18 9 7 11 8 Kruskal’s Algorithm: Add cheapest edge that does not create a cycle

3. 4 5 23 10 21 14 24 16 6 18 9 7 11 8 Kruskal’s Algorithm: Add cheapest edge that does not create a cycle

4. 4 5 23 10 21 14 24 16 6 18 9 7 11 8 Kruskal’s Algorithm: Add cheapest edge that does not create a cycle

5. 6 4 5 23 10 21 14 24 16 18 9 7 11 8 Kruskal’s Algorithm: Add cheapest edge that does not create a cycle

6. 7 6 4 5 23 10 21 14 24 16 18 9 11 8 Kruskal’s Algorithm: Add cheapest edge that does not create a cycle

7. 8 7 6 4 5 23 10 21 14 24 16 18 9 11 Kruskal’s Algorithm: Add cheapest edge that does not create a cycle

8. 9 8 7 6 4 5 23 10 21 14 24 16 18 11 Kruskal’s Algorithm: Add cheapest edge that does not create a cycle

9. 9 8 7 6 4 5 23 10 21 14 24 16 18 11 Kruskal’s Algorithm: Add cheapest edge that does not create a cycle ?

10. 9 8 7 6 4 5 23 10 21 14 24 16 18 11 Kruskal’s Algorithm: Add cheapest edge that does not create a cycle Ignore edges inside components

11. 9 8 7 6 4 5 23 10 21 14 24 16 18 11 Kruskal’s Algorithm: Add cheapest edge that does not create a cycle Ignore edges inside components

12. 11 9 8 7 6 4 5 23 10 21 14 24 16 18 Kruskal’s Algorithm: Add cheapest edge that does not create a cycle Ignore edges inside components

13. 11 9 8 7 6 4 5 23 10 21 14 24 16 18 Kruskal’s Algorithm: Add cheapest edge that does not create a cycle Ignore edges inside components

15. 5 23 10 21 14 24 16 6 4 18 9 7 11 8 Prim’s Algorithm: Grow a tree. At each step, add cheapest edge from tree node to non-tree node

16. Prim’s Algorithm: Grow a tree. At each step, add cheapest edge from tree node to non-tree node 5 23 10 21 14 24 16 6 4 18 9 7 11 8

17. Prim’s Algorithm: Grow a tree. At each step, add cheapest edge from tree node to non-tree node 5 23 10 21 14 24 16 6 4 18 9 7 11 8

18. Prim’s Algorithm: Grow a tree. At each step, add cheapest edge from tree node to non-tree node 5 23 10 21 14 24 16 6 4 18 9 7 11 8

19. 6 5 4 8 Prim’s Algorithm: Grow a tree. At each step, add cheapest edge from tree node to non-tree node 23 10 21 14 24 16 18 9 7 11

20. 8 5 4 Prim’s Algorithm: Grow a tree. At each step, add cheapest edge from tree node to non-tree node 23 10 21 14 24 16 6 18 9 7 11 Can remove edges between tree nodes

21. 11 6 5 4 8 Prim’s Algorithm: Grow a tree. At each step, add cheapest edge from tree node to non-tree node 23 10 21 14 24 16 18 9 7

22. 7 11 6 5 4 8 Prim’s Algorithm: Grow a tree. At each step, add cheapest edge from tree node to non-tree node 23 10 21 14 24 16 18 9

23. 9 7 11 6 5 4 8 Prim’s Algorithm: Grow a tree. At each step, add cheapest edge from tree node to non-tree node 23 10 21 14 24 16 18

25. node-comp array node123456789 comp118451188 comp-size array comp123456789 size400110030 Lists of nodes 1 : 1, 2, 6, 7 2 : 3 : 4 : 4 5 : 5 6 : 7 : 8 : 8, 9, 3 9 : Components Data Structure Supporting Find and Merge (Union-Find)

26. Components Data Structure Supporting Find and Merge (Union-Find) node-comp array node123456789 comp123456789 comp-size array comp123456789 size111111111 Lists of nodes 1 : 1 2 : 2 3 : 3 4 : 4 5 : 5 6 : 6 7 : 7 8 : 8 9 : 9

27. Components Data Structure node-comp array node123456789 comp123456789 comp-size array comp123456789 size111111111 Lists of nodes 1 : 1 2 : 2 3 : 3 4 : 4 5 : 5 6 : 6 7 : 7 8 : 8 9 : 9 Merge(1,2)

28. Components Data Structure node-comp array node123456789 comp113456789 comp-size array comp123456789 size201111111 Lists of nodes 1 : 1, 2 2 : 3 : 3 4 : 4 5 : 5 6 : 6 7 : 7 8 : 8 9 : 9 Merge(1,2)

29. Components Data Structure node-comp array node123456789 comp118451188 comp-size array comp123456789 size400110030 Lists of nodes 1 : 1, 2, 6, 7 2 : 3 : 4 : 4 5 : 5 6 : 7 : 8 : 8, 9, 3 9 : Merge(1,8)

30. Components Data Structure node-comp array node123456789 comp118451188 comp-size array comp123456789 size400110030 Lists of nodes 1 : 1, 2, 6, 7 2 : 3 : 4 : 4 5 : 5 6 : 7 : 8 : 8, 9, 3 9 : Merge(1,8)

31. Components Data Structure node-comp array node123456789 comp118451118 comp-size array comp123456789 size400110030 Lists of nodes 1 : 1, 2, 6, 7 2 : 3 : 4 : 4 5 : 5 6 : 7 : 8 : 8, 9, 3 9 : Merge(1,8)

32. Components Data Structure node-comp array node123456789 comp118451111 comp-size array comp123456789 size400110030 Lists of nodes 1 : 1, 2, 6, 7 2 : 3 : 4 : 4 5 : 5 6 : 7 : 8 : 8, 9, 3 9 : Merge(1,8)

33. Components Data Structure node-comp array node123456789 comp111451111 comp-size array comp123456789 size400110030 Lists of nodes 1 : 1, 2, 6, 7 2 : 3 : 4 : 4 5 : 5 6 : 7 : 8 : 8, 9, 3 9 : Merge(1,8)

34. Components Data Structure node-comp array node123456789 comp111451111 comp-size array comp123456789 size400110030 Lists of nodes 1 : 1, 2, 6, 7, 8, 9, 3 2 : 3 : 4 : 4 5 : 5 6 : 7 : 8 : 9 : Merge(1,8)

36. Components Data Structure Pointer Version 1(1)2(1)6(1)3(1)4(1)5(1)8(1)9(1)7(1) Merge(1,2)

37. Components Data Structure Pointer Version 1(1) 2 6(1)3(1)4(1)5(1)8(1)9(1)7(1) Merge(1,2)

38. Components Data Structure Pointer Version 1(2) 2 6(1)3(1)4(1)5(1)8(1)9(1)7(1) Merge(3,4)

39. Components Data Structure Pointer Version 1(2) 2 6(1) 3 4 5(1)8(1)9(1)7(1) Merge(3,4)

40. Components Data Structure Pointer Version 1(1) 2 6(1)3(2) 4 5(1)8(1)9(1)7(1) Merge(1,3)

41. Components Data Structure Pointer Version 1(2) 2 6(1)3(2) 4 5(1)8(1)9(1)7(1) Merge(1,3)