1. Introduction to Algorithms
LECTURE 16 (Chap. 22)
Elementary Graph Algorithm
• 22.2 Breath-first search
• 22.3 Depth-first search
• 22.4 Topological sort
Breadth first search animation

2. Breadth first search animation
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.)
Breadth first search animation

3. 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

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

5. 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

6. If node i is incident to an admissible arc…2 4 2 2 8 1 1 1 1 5 5 7 3
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 5 3 2
next

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

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

9. Select Node i 2 4 2 2 2 8 1 1 1 1 1 5 5 7 3 9
The first node on LIST becomes node i 3 3 6 4
LIST 1 2 5 3 2 4 3
next

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

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

12. Select a node 5 2 4 4 2 2 2 8 1 1 1 5 5 5 7 3
The first node on LIST becomes node i 9 3 3 6 4
LIST 1 2 5 3 4 5 2 3 4
next

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

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

15. Select node 3 5 2 4 4 2 2 2 8 1 1 1 5 5 5 5 7 3
node 3 is not incident to any admissible arcs
delete node 3 from LIST 9 3 3 3 3 6 6 4 6
LIST 1 2 5 3 4 6 2 3 5 4 6
next

16. Select a node 5 2 4 4 4 2 2 8 1 1 1 5 5 7 3
i : = 4 9 3 3 6 6 4 6
LIST 1 2 5 3 4 6 6 2 3 4 5
next

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

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

19. Select node i 5 2 4 4 7 2 2 8 8 1 1 1 5 5 7 3
i := 6 9 3 3 6 6 6 4 6
LIST 1 2 5 3 4 6 8 7 3 2 6 4 5
next

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

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

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

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

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

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

26. Breadth first search animation
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.)
Breadth first search animation

27. 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

28. 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

29. 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

30. 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

31. 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

32. 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

33. 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

34. 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

35. 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

36. 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

37. 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

38. 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

39. 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

40. 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

41. 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

42. 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

43. 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

44. 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

45. 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

46. 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

47. 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

48. 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

49. 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

50. 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

51. 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

52. 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

53. 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

54. 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

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

56. Breadth first search animation
Topological Ordering
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.)
Breadth first search animation

57. Preliminary to Topological Sorting
LEMMA. If each node has at least one arc going out, then the first inadmissible arc of a depth first search determines a directed cycle.
COROLLARY 1. If G has no directed cycle, then there is a node in G with no arcs going. And there is at least one node in G with no arcs coming in.
COROLLARY 2. If G has no directed cycle, then one can relabel the nodes so that for each arc (i,j), i < j. 1 4 6 7 3

58. Initialization 6 1 Determine the indegree of each node
LIST is the set of nodes with indegree of 0.
“Next” will be the label of nodes in the topological order. 5 2 8 3 7 4
next 1 2 3 4 5 6 7 8
Node
Indegree
LIST 7

59. Select a node from LIST 6 1 next := next +1 order(i) := next;
update indegrees
update LIST
Select a node from LIST and delete it. 5 2 8 3 7 7 4 1 1
next 1 2 3 4 5 6 7 8
Node
LIST
Indegree 2 2 3 2 1 1 1 2 7 5

60. Select a node from LIST 6 1 next := next +1 order(i) := next;
update indegrees
update LIST
Select a node from LIST and delete it. 2 5 5 2 8 3 7 7 4 1 1 2
next 1 2 3 4 5 6 7 8
Node
LIST
Indegree 2 2 1 3 2 1 1 1 2 4 5 7 6

61. Select a node from LIST 3 6 6 1 next := next +1 order(i) := next;
update indegrees
update LIST
Select a node from LIST and delete it. 2 5 5 2 8 3 7 7 4 1 2 1 3
next 1 2 3 4 5 6 7 8
Node
LIST
Indegree 1 2 2 1 3 1 2 1 1 2 4 5 7 2 6

62. Select a node from LIST 3 6 6 1 next := next +1 order(i) := next;
update indegrees
update LIST
Select a node from LIST and delete it. 2 5 5 2 2 8 3 4 7 7 4 1 4 3 2 1
next 1 2 3 4 5 6 7 8
Node
LIST
Indegree 1 2 2 1 3 2 1 1 1 2 4 7 5 6 2 1

63. Select a node from LIST 5 3 6 6 1 1 next := next +1 order(i) := next;
update indegrees
update LIST
Select a node from LIST and delete it. 2 5 5 2 2 8 3 4 7 7 4 1 3 5 2 4 1
next 1 2 3 4 5 6 7 8
Node
LIST
Indegree 1 2 1 2 2 3 2 1 1 1 2 1 4 7 5 1 2 6

64. Select a node from LIST 5 3 6 6 1 1 next := next +1 order(i) := next;
update indegrees
update LIST
Select a node from LIST and delete it. 2 5 5 2 2 8 3 4 7 7 4 4 1 6 6 3 5 4 2 1
next 1 2 3 4 5 6 7 8
Node
LIST
Indegree 1 2 2 1 3 2 1 1 2 1 1 2 1 2 1 4 6 7 5 8

65. Select a node from LIST 5 3 6 6 1 1 next := next +1 order(i) := next;
update indegrees
update LIST
Select a node from LIST and delete it. 7 2 5 5 2 2 8 8 3 4 7 7 4 4 1 6 2 7 4 5 3 6 1
next 1 2 3 4 5 6 7 8
Node
LIST
Indegree 1 2 1 2 2 3 1 1 2 1 1 1 2 8 3

66. Select a node from LIST 5 3 6 6 1 1 next := next +1 order(i) := next;
update indegrees
update LIST
Select a node from LIST and delete it. 7 8 2 5 5 2 2 8 8 3 3 4 7 7 4 4 1 6 2 5 8 6 7 3 1 4
next
List is empty.
The algorithm terminates with a topological order of the nodes 1 2 3 4 5 6 7 8
Node
LIST
Indegree 1 2 2 1 2 1 3 1 2 1 1 2 1 3