1. 15.082 and 6.855J Depth First Search
Get ahold of a network, and use the same network to illustrate the shortest path problem for communication newtorks, 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. Initialize 1 2 4 5 3 6 9 7 8 1 2 4 5 3 6 9 7 8 1 1
pred(1) = 0
next := 1 order(next) = 1
LIST:= {1}
Unmark all nodes in N;
Mark node s
LIST 1
next 1

3. Select a node i in LIST 1 2 4 5 3 6 9 7 8 1 1 1
In depth first search, i is the last node in LIST
LIST 1
next 1

4. If node i is incident to an admissible arc…2 4 2 2 8 1 1 1 1 5 7
Next := Next + 1
order(j) := next
add j to LIST
Mark Node j
pred(j) := i
Select an admissible arc (i,j) 9 3 6
LIST 1 2 2 1
next

5. Select the last node on LIST2 4 2 2 2 8 1 1 1 1 1 5 7 9 3 6
Node 2 gets selected
LIST 1 2 2 1
next

6. If node i is incident to an admissible arc…2 4 4 2 2 2 3 8 1 1 1 1 1 5 7
Select an admissible arc (i,j)
Next := Next + 1
order(j) := next
add j to LIST
Mark Node j
pred(j) := i 9 3 6
LIST 1 2 4 3 2 1
next

7. Select 2 4 4 4 2 2 2 2 3 8 1 1 1 1 1 5 7 Select the last node on LIST9 3 6
LIST 1 2 4 2 3 1
next

8. If node i is incident to an admissible arc…2 4 4 4 2 2 2 2 3 8 8 4 1 1 1 1 1 5 7
Mark Node j
pred(j) := i
Select an admissible arc (i,j)
Next := Next + 1
order(j) := next
add j to LIST 9 3 6
LIST 1 2 4 8 3 4 2 1
next

9. Select 2 4 4 4 2 2 2 2 3 8 8 8 4 1 1 1 1 1 5 7
Select the last node on LIST 9 3 6
LIST 1 2 4 8 3 2 1 4
next

10. If node i is not incident to an admissible arc…2 4 4 4 2 2 2 2 3 8 8 8 8 4 1 1 1 1 1 5 7
Delete node i from LIST 9 3 6
LIST 1 2 4 8 1 3 2 4
next

11. Select 2 4 4 4 4 2 2 2 2 3 8 8 8 8 4 1 1 1 1 1 5 7
Select the last node on LIST 9 3 6
LIST 1 2 4 8 1 4 2 3
next

12. If node i is incident to an admissible arc…2 4 4 4 4 2 2 2 2 3 8 8 8 8 4 5 1 1 1 1 1 5 5 7
Mark Node j
pred(j) := i
Next := Next + 1
order(j) := next
add j to LIST
Select an admissible arc (i,j) 9 3 6
LIST 1 2 4 5 8 1 5 2 4 3
next

13. Select 2 4 4 4 4 4 2 2 2 2 3 8 8 8 8 4 5 1 1 1 1 1 5 5 5 7
Select the last node on LIST 9 3 6
LIST 1 2 4 8 5 1 2 5 3 4
next

14. If node i is incident to an admissible arc…2 4 4 4 4 4 2 2 2 2 3 8 8 8 8 4 5 1 1 1 1 1 5 5 5 7
Select an admissible arc (i,j)
Next := Next + 1
order(j) := next
add j to LIST
Mark Node j
pred(j) := i 9 3 6 6 6
LIST 1 2 4 8 5 6 6 3 4 2 5 1
next

15. Select the last node on LIST2 4 4 4 4 4 2 2 2 2 3 8 8 8 8 4 5 1 1 1 1 1 5 5 5 5 7
Select node 6 9 3 6 6 6 6
LIST 1 2 4 8 5 6 3 4 6 5 1 2
next

16. If node i is incident to an admissible arc…2 4 4 4 4 4 2 2 2 2 3 8 8 8 8 4 5 1 1 1 1 1 5 5 5 5 7
Mark Node j
pred(j) := i
Select an admissible arc (i,j)
Next := Next + 1
order(j) := next
add j to LIST 9 9 3 6 6 6 7 6
LIST 1 2 4 5 8 6 9 1 7 2 4 5 3 6
next

17. Select the last node on LIST2 4 4 4 4 4 2 2 2 2 3 8 8 8 8 4 5 1 1 1 1 1 5 5 5 5 7
Select node 9 9 9 9 3 6 6 6 6 7 6
LIST 1 2 4 5 8 6 9 7 1 2 4 6 5 3
next

18. If node i is incident to an admissible arc…2 4 4 4 4 4 2 2 2 2 3 8 8 8 8 4 5 1 8 1 1 1 1 5 5 5 5 7 7
Select an admissible arc (i,j)
Next := Next + 1
order(j) := next
add j to LIST
Mark Node j
pred(j) := i 9 9 9 3 6 6 6 6 7 6
LIST 1 2 4 5 8 6 9 7 8 4 3 1 6 2 5 7
next

19. Select the last node on LIST2 4 4 4 4 4 2 2 2 2 3 8 8 8 8 4 5 1 8 1 1 1 1 5 5 5 5 7 7 7
Select node 7 9 9 9 9 3 6 6 6 6 7 6
LIST 1 2 4 5 8 6 9 7 4 8 1 6 2 3 5 7
next

20. If node i is not incident to an admissible arc…2 4 4 4 4 4 2 2 2 2 3 8 8 8 8 4 5 1 8 1 1 1 1 5 5 5 5 7 7 7 7
Delete node 7 from LIST 9 9 9 9 3 6 6 6 6 7 6
LIST 1 2 4 8 5 6 9 7 1 2 8 4 6 3 7 5
next

21. Select node 9 2 4 4 4 4 4 2 2 2 2 3 8 8 8 8 4 5 1 8 1 1 1 1 5 5 5 5 7 7 7 7
Delete node 9 from LIST
But node 9 is not incident to an admissible arc. 9 9 9 9 9 9 3 6 6 6 6 7 6
LIST 1 2 4 8 5 6 9 7 8 4 2 1 7 3 5 6
next

22. Select node 6 2 4 4 4 4 4 2 2 2 2 3 8 8 8 8 4 5 1 8 1 1 1 1 5 5 5 5 7 7 7 7
But node 6 is not incident to an admissible arc.
Delete node 6 from LIST 9 9 9 9 9 9 3 6 6 6 6 6 6 7 6
LIST 1 2 4 5 8 6 9 7 4 3 2 6 7 8 1 5
next

23. Select node 5 2 4 4 4 4 4 2 2 2 2 3 8 8 8 8 4 5 1 8 1 1 1 1 5 5 5 5 5 5 7 7 7 7
But node 5 is not incident to an admissible arc.
Delete node 5 from LIST 9 9 9 9 9 9 3 6 6 6 6 6 6 7 6
LIST 1 2 4 8 5 6 9 7 3 2 1 4 5 6 8 7
next

24. Select node 4 2 4 4 4 4 4 4 4 2 2 2 2 3 8 8 8 8 4 5 1 8 1 1 1 1 5 5 5 5 5 5 7 7 7 7
Delete node 4 from LIST
But node 4 is not incident to an admissible arc. 9 9 9 9 9 9 3 6 6 6 6 6 6 7 6
LIST 1 2 4 5 8 6 9 7 7 1 2 5 8 4 6 3
next

25. Select node 2 2 4 4 4 4 4 4 4 2 2 2 2 2 2 3 8 8 8 8 4 5 1 8 1 1 1 1 5 5 5 5 5 5 7 7 7 7
But node 2 is not incident to an admissible arc.
Delete node 2 from LIST 9 9 9 9 9 9 3 6 6 6 6 6 6 7 6
LIST 1 2 4 5 8 6 9 7 6 3 1 4 5 7 2 8
next

26. Select node 1 2 4 4 4 4 4 4 4 2 2 2 2 2 2 3 8 8 8 8 4 5 1 8 1 1 1 1 1 5 5 5 5 5 5 7 7 7 7
Next := Next + 1
order(j) := next
add j to LIST
Mark Node j
pred(j) := i
Select an admissible arc (i,j) 9 9 9 9 9 9 3 3 6 6 6 6 6 6 7 9 6
LIST 1 3 2 4 8 5 6 9 7 2 6 9 7 3 8 5 1 4
next

27. Select node 3 2 4 4 4 4 4 4 4 2 2 2 2 2 2 3 8 8 8 8 4 5 1 8 1 1 1 1 1 1 5 5 5 5 5 5 7 7 7 7
Delete node 3 from LIST
But node 3 is not incident to an admissible arc. 9 9 9 9 9 9 3 3 3 3 6 6 6 6 6 6 7 9 6
LIST 1 3 2 4 5 8 6 9 7 9 8 5 3 4 2 1 7 6
next

28. Select node 1 2 4 4 4 4 4 4 4 2 2 2 2 2 2 3 8 8 8 8 4 5 1 8 1 1 1 1 1 1 1 1 5 5 5 5 5 5 7 7 7 7
Delete node 1 from LIST
But node 1 is not incident to an admissible arc. 9 9 9 9 9 9 3 3 3 3 6 6 6 6 6 6 7 9 6
LIST 1 2 3 4 8 5 6 9 7 9 5 2 1 3 4 7 6 8
next

29. LIST is empty 2 4 4 4 4 4 4 4 2 2 2 2 2 2 3 8 8 8 8 4 5 1 8 1 1 1 1 1 1 1 1 5 5 5 5 5 5 7 7 7 7
The algorithm ends! 9 9 9 9 9 9 3 3 3 3 6 6 6 6 6 6 7 9 6
LIST 1 2 3 4 5 8 6 9 7 9 4 5 3 6 2 1 7 8
next

30. The depth first search tree1 3 2 9 8 7 5 4 6
Note that each induced subtree has consecutively labeled nodes