1. Spanning Tree Algorithms
and 6.855J
Spanning Tree Algorithms
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.)

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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 15

16. 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 16

17. 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 17

18. 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 18

19. 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 19

20. 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 20

21. 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.

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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

29. 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

30. 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