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24. 4. 23. 9. 6. 18. 5. 11. 16. 8. 7. 14. 10. 21. Minimum Spanning Tree. 4. 9. 6. 5. 11. 8. 7. G = (V, E), c(e). T ,  eT c(e) = 50. Kruskal’s Algorithm:. Add cheapest edge that does not create a cycle. 24. 4. 23. 9. 6. 18. 5. 11. 16. 8. 7.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

  33. 1(1) 2(1) 6(1) 3(1) 4(1) 5(1) 8(1) 9(1) 7(1) Components Data Structure Pointer Version Merge(1,2)

  34. 1(1) 6(1) 3(1) 4(1) 5(1) 8(1) 9(1) 7(1) Components Data Structure Pointer Version Merge(1,2) 2

  35. 1(2) 6(1) 3(1) 4(1) 5(1) 8(1) 9(1) 7(1) Components Data Structure Pointer Version Merge(3,4) 2

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

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

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

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