1. Lecture 4
Graph Search

6. 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.)

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

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

18. 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

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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

29. 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

30. 5

44. Determine the connected components of a network;
Summary
Find the shortest path from s to each other node path length = number of arcs on path;
Determine the connected components of a network;
Determine breadth first search.

45. Next Determine depth first search;
Shows up in other algorithms as well.
Determine topological sort;
Running time is O(n+m) using simple data structures and algorithms.
Very important for preprocessing.

47. Depth 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.)

48. 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

49. 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

50. 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

51. 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

52. 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

53. 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

54. 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

55. 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

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

57. 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

58. 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

59. 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

60. 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

61. 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

62. 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

63. 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

64. 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

65. 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

66. 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

67. 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

68. 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

69. 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

70. 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

71. 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

72. 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

73. 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

74. 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

75. 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

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

80. Topological Ordering 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.)

81. 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

82. 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

83. 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

84. 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

85. 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

86. 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

87. 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

88. 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

89. 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