1. The Minimum Cost Spanning Tree Problem
Lecture 8
The Minimum Cost Spanning Tree Problem
Obtain a network, and use the same network to illustrate the shortest path problem for communication networks, the max flow problem, the minimum cost flow problem, and the multicommodity flow problem. This will be a very efficient way of introducing the four problems. (Perhaps under 10 minutes of class time.)

10. a

19. The Greedy Algorithm
animation

20. The Greedy Algorithm in Action35 10 30 15 25 40 20 17 8 11 21 1 2 3 4 5 6 7 2 4 6 1 3 5 7

21. The Greedy Algorithm in Action10 10 8 8 2 4 4 6 2 6 35 35 15 15 1 1 25 25 20 20 30 30 17 17 21 21 40 40 11 3 5 7 3 5 7 15 15 11

22. The Greedy Algorithm in Action10 10 8 8 2 4 4 6 2 6 35 35 15 15 1 1 25 25 20 20 30 30 17 17 21 21 40 40 11 3 5 7 3 5 7 15 15 11

23. The Greedy Algorithm in Action10 10 8 8 2 4 4 6 2 6 35 35 15 15 1 1 25 25 20 20 30 30 17 17 21 21 40 40 11 3 5 7 3 5 7 15 15 11

24. The Greedy Algorithm in Action10 10 8 8 2 4 4 6 2 6 35 35 15 15 1 1 25 25 20 20 30 30 17 17 21 21 40 40 11 3 5 7 3 5 7 15 15 11

25. The Greedy Algorithm in Action10 10 8 8 2 4 4 6 2 6 35 35 15 15 1 1 25 25 20 20 30 30 17 17 21 21 40 40 11 3 5 7 3 5 7 15 15 11

26. The Greedy Algorithm in Action10 10 8 8 2 4 4 6 2 6 35 35 15 15 1 1 25 25 20 20 30 30 17 17 21 21 40 40 11 3 5 7 3 5 7 15 15 11

27. The Greedy Algorithm in Action10 10 8 8 2 4 4 6 2 6 35 35 15 15 1 1 25 25 20 20 30 30 17 17 21 21 40 40 11 3 5 7 3 5 7 15 15 11

28. The Greedy Algorithm in Action10 10 8 8 2 4 4 6 2 6 35 35 15 15 1 1 25 25 20 20 30 30 17 17 21 21 40 40 11 3 5 7 3 5 7 15 15 11

29. The Greedy Algorithm in Action10 10 8 8 2 4 4 6 2 6 35 35 15 15 1 1 25 25 20 20 30 30 17 17 21 21 40 40 11 3 5 7 3 5 7 15 15 11

30. The Greedy Algorithm in Action10 10 8 8 2 4 4 6 2 6 35 35 15 15 1 1 25 25 20 20 30 30 17 17 21 21 40 40 11 3 5 7 3 5 7 15 15 11

31. The Greedy Algorithm in Action10 10 8 8 2 4 4 6 2 6 35 35 15 15 1 1 25 25 20 20 30 30 17 17 21 21 40 40 11 3 5 7 3 5 7 15 15 11

32. The Greedy Algorithm in Action10 10 8 8 2 4 4 6 2 6 35 35 15 15 1 1 25 25 20 20 30 30 17 17 21 21 40 40 11 3 5 7 3 5 7 15 15 11

33. Correctness

34. Exchange Property

35. Self-Reducibility

38. Data Structures “Disjoint Sets”

39. Kruskal Algorithm with Data Structure

40. The Greedy Algorithm in Action
Node
First 10 10 8 8 2 4 2 4 6 6 6 35 35 15 15 1 1 25 25 20 20 30 30 17 17 21 21 40 40 11 3 5 7 3 5 7 15 15 11
root node 40

41. The Greedy Algorithm in Action
Node
First 10 10 8 8 2 2 4 2 4 6 6 35 35 15 15 1 1 25 25 20 20 30 30 17 17 21 21 40 40 11 3 5 7 3 5 7 15 15 11 41

42. The Greedy Algorithm in Action
Node
First 10 10 8 8 2 4 2 4 6 6 35 35 15 15 1 1 25 25 20 20 30 30 17 17 21 21 40 40 11 3 5 7 7 3 5 7 15 15 11 42

43. The Greedy Algorithm in Action
Node
First 10 10 8 8 2 4 6 2 4 6 35 35 15 15 1 1 25 25 20 20 30 30 17 17 21 21 40 40 11 3 3 5 7 7 3 5 7 15 15 11 43

44. The Greedy Algorithm in Action
Node
First 10 10 8 8 2 4 6 2 4 6 35 35 15 15 1 1 25 25 20 20 30 30 17 17 21 21 40 40 11 3 5 7 3 3 5 5 7 7 3 5 7 15 15 11 44

45. The Greedy Algorithm in Action
Node
First 10 10 8 8 2 4 6 2 4 6 35 35 15 15 1 1 1 25 25 20 20 30 30 17 17 21 21 40 40 11 3 3 5 7 5 7 3 5 7 15 15 11 45

47. Prim’s algorithm
animation

49. Correctness
The cut-optimality Condition

50. Prim’s Algorithm in Action10 8 2 2 4 4 6 6 35 15 1 1 25 20 30 17 21 40 11 3 5 7 3 5 7 15
The minimum cost arc from yellow nodes to green nodes can be found by placing arc values in a priority queue.

51. Prim’s Algorithm in Action25 10 10 8 2 2 2 4 4 6 6 35 35 15 1 1 25 20 30 17 21 40 11 3 5 7 3 5 7 15

52. Prim’s Algorithm in Action8 20 30 21 8 10 10 10 2 2 2 4 4 4 6 6 35 35 15 1 1 25 25 30 20 17 21 40 11 3 5 7 3 5 7 15

53. Prim’s Algorithm in Action8 8 8 10 10 10 2 2 2 4 4 4 6 6 6 35 35 17 15 15 1 1 25 25 30 30 17 20 20 21 21 40 11 3 5 7 3 5 7 15

54. Prim’s Algorithm in Action8 8 8 10 10 10 2 2 2 4 4 4 6 6 6 35 35 15 15 15 1 1 25 25 30 30 17 17 20 20 21 21 40 15 11 11 3 5 5 7 3 5 7 15

55. Prim’s Algorithm in Action8 8 8 10 10 10 2 2 2 4 4 4 6 6 6 35 35 15 15 15 1 1 25 25 30 30 17 17 20 20 21 21 40 15 11 11 3 5 5 7 3 5 7 15

56. Prim’s Algorithm in Action8 8 8 10 10 10 2 2 2 4 4 4 6 6 6 35 35 15 15 15 1 1 25 25 30 30 17 17 20 20 21 21 40 11 11 11 3 5 5 7 7 3 5 7 15 15

57. Prim’s Algorithm in Action8 8 8 10 10 10 2 2 2 4 4 4 6 6 6 35 35 15 15 15 1 1 25 25 30 30 17 17 20 20 21 21 40 11 11 11 3 3 5 5 7 7 3 5 7 15 15 15

58. Prim’s Algorithm in Action8 8 8 10 10 10 2 2 2 4 4 4 6 6 6 35 35 15 15 15 1 1 25 25 30 30 17 17 20 20 21 21 40 11 11 11 3 3 5 5 7 7 3 5 7 15 15 15

59. Prim’s Algorithm in Action8 8 8 10 10 10 2 2 2 4 4 4 6 6 6 35 35 15 15 15 1 1 25 25 30 30 17 17 20 20 21 21 40 11 11 11 3 3 5 5 7 7 3 5 7 15 15 15

61. Sollin’s Algorithm
animation

63. Correctness
Cut-optimality Condition

64. Sollin’s Algorithm in Action10 8 2 2 2 4 4 4 6 6 6 35 15 1 1 1 25 20 30 17 21 40 3 3 5 5 7 7 3 5 11 7 15
Treat all nodes as singleton components, and then select the min cost arc leaving the component.

65. Sollin’s Algorithm in Action10 8 2 2 4 4 6 6 6 35 15 1 1 25 20 30 17 21 40 3 3 5 5 7 7 3 5 15 11 7
Find the min cost edge out of each component

67. Appendix: Priority Queue

68. Priority Queue
A priority queue is a data structure for maintaining a set of elements, each with an associated value, called a key.
A min-priority queue supports the following operations: Insert(S,x), Minimum(S),
Extract-Min(S), Increase-Key(S,x,k).
Min-Heap can be used for implementing
min-priority queue.