1. Csc 2720 Instructor: Zhuojun Duan
Data structure Graph
Csc Instructor: Zhuojun Duan

2. Outline

3. Outline Definition of graph Variations of graph Implementation
Adjacency matrix
Adjacency list
Traversal of graph

4. Definition

5. Terminology G = {V, E} A graph G consists of two sets A subgraph
A set V of vertices, or nodes
A set E of edges
A subgraph
Consists of a subset of a graph’s vertices and a subset of its edges
Adjacent vertices
Two vertices that are joined by an edge

6. Terminology A path between two vertices A simple path A cycle
A sequence of edges that begins at one vertex and ends at another vertex
May pass through the same vertex more than once
A simple path
A path that passes through a vertex only once
A cycle
A path that begins and ends at the same vertex
A simple cycle
A cycle that does not pass through a vertex more than once

7. Graph variations A connected graph A disconnected graph
A graph that has a path between each pair of distinct vertices
A disconnected graph
A graph that has at least one pair of vertices without a path between them
A complete graph
A graph that has an edge between each pair of distinct vertices

8. Graph variations Multigraph Not a graph
Allows multiple edges between vertices

9. Graph variations Weighted graph
A graph whose edges have numeric labels

10. Graph variations Undirected graph Edges do not indicate a direction
Directed graph, or diagraph
Each edge is a directed edge
Can have two edges between a pair of vertices, one in each direction
Directed path
A sequence of directed edges between two vertices
Vertex y is adjacent to vertex x if
There is a directed edge from x to y

11. Outline Definition of graph Variations of graph Implementation
Adjacency matrix
Adjacency list
Traversal of graph

12. Operations in a graph Graphs can be used as abstract data types
Two options for defining graphs
Vertices contain values
Vertices do not contain values
Operations of the ADT graph
Create an empty graph
Determine whether a graph is empty
Determine the number of vertices in a graph
Determine the number of edges in a graph

13. Operations in a graph Operations of the ADT graph (Continued)
Determine whether an edge exists between two given vertices; for weighted graphs, return weight value
Insert a vertex in a graph whose vertices have distinct search keys that differ from the new vertex’s search key
Insert an edge between two given vertices in a graph
Delete a particular vertex from a graph and any edges between the vertex and other vertices
Delete the edge between two given vertices in a graph
Retrieve from a graph the vertex that contains a given search key

14. Implementing Graphs Most common implementations of a graph
Adjacency matrix
Adjacency list
Adjacency matrix for a graph with n vertices numbered 0, 1, …, n – 1
An n by n array matrix such that matrix[i][j] is
1 (or true) if there is an edge from vertex i to vertex j
0 (or false) if there is no edge from vertex i to vertex j

15. Implementing Graphs
a) A directed graph and b) its adjacency matrix

16. Implementing Graphs
Adjacency matrix for a weighted graph with n vertices numbered 0, 1, …, n – 1
An n by n array matrix such that matrix[i][j] is
The weight that labels the edge from vertex i to vertex j if there is an edge from i to j
 if there is no edge from vertex i to vertex j
Diagonal
Symmetric matrix
a) A weighted undirected graph and b) its adjacency matrix

17. Outline Definition of graph Variations of graph Implementation
Adjacency matrix
Adjacency list
Traversal of graph

18. Implementing Graphs Adjacency list
An adjacency list for a graph with n vertices numbered 0, 1, …, n – 1
Consists of n linked lists
The ith linked list has a node for vertex j if and only if the graph contains an edge from vertex i to vertex j
This node can contain either
Vertex j’s value, if any
An indication of vertex j’s identity

19. Implementing Graphs a) A directed graph and b) its adjacency list
© 2011 Pearson Addison-Wesley. All rights reserved
Implementing Graphs
a) A directed graph and b) its adjacency list

20. Implementing Graphs Adjacency list for an undirected graph
Treats each edge as if it were two directed edges in opposite directions
a) A weighted undirected graph and b) its adjacency list

21. Implementing Graphs Adjacency matrix compared with adjacency list
Two common operations on graphs
1- Determine whether there is an edge from vertex i to vertex j
2- Find all vertices adjacent to a given vertex i
Adjacency matrix
Supports operation 1 more efficiently
Adjacency list
Supports operation 2 more efficiently
Often requires less space than an adjacency matrix

22. Outline Definition of graph Variations of graph Implementation
Adjacency matrix
Adjacency list
Traversal of graph

23. Graph Traversals A graph-traversal algorithm
Visits all the vertices that it can reach
Visits all vertices of the graph if and only if the graph is connected
A connected component
The subset of vertices visited during a traversal that begins at a given vertex
Can loop indefinitely if a graph contains a loop
To prevent this, the algorithm must
Mark each vertex during a visit, and
Never visit a vertex more than once

24. Graph Traversals Search vs Traversal Search: Look for a given node
stop when node found, even if not all nodes were visited
Traversal: Always visit all nodes
Two way of Traversal for graph
Depth-first Search (DFS)
Breadth-first Search (BFS)

25. Graph Traversals if unvisited
Pseudo-Code for Depth-First Search
depth-first-search
mark vertex x as visited
for each adjacent vertex of x
if unvisited
do a depth-first search on adjacent vertex

26. First example of DFS

27. Depth-First Search for undirected graphB C D E F G

28. Depth-First Search for undirected graphv A B C D E F G A

29. Depth-First Search for undirected graphv A B C D E F G A

30. Depth-First Search for undirected graphv A v B C D E F G
A B

31. Depth-First Search for undirected graphv A v B C D E F G
A B

32. Depth-First Search for undirected graphv A v B C D E F G
A B

33. Depth-First Search for undirected graphv A v B C v D E F G
A B D

34. Depth-First Search for undirected graphv A v B C v D E F G
A B D

35. Depth-First Search for undirected graphv A v B C v D E F G
A B D

36. Depth-First Search for undirected graphv A v B C v D E v F G
A B D E

37. Depth-First Search for undirected graphv A v B C v D E v F G
A B D E

38. Depth-First Search for undirected graphv A v B C v D E v F G
A B D E

39. Depth-First Search for undirected graphv A v B C v D E F G
A B D E

40. Depth-First Search for undirected graphv A v B C v D E v F G
A B D E

41. Depth-First Search for undirected graphv A v B C v D E v F G
A B D E

42. Depth-First Search for undirected graphv A v B C v D E v F G
A B D E

43. Depth-First Search for undirected graphv A v B C v D E v F G v
A B D E F

44. Depth-First Search for undirected graphv A v B C v D E v F G v
A B D E F

45. Depth-First Search for undirected graphv A v B C v D E v F G v
A B D E F

46. Depth-First Search for undirected graphv A v v B C v D E v F G v
A B D E F C

47. Depth-First Search for undirected graphv A v v B C v D E v F G v
A B D E F C

48. Depth-First Search for undirected graphv A v v B C v D E v F G v
A B D E F C

49. Depth-First Search for undirected graphv A v v B C v D E v F G v
A B D E F C

50. Depth-First Search for undirected graphv A v v B C v v D E v F G v
A B D E F C G

51. Depth-First Search for undirected graphv A v v B C v v D E v F G v
A B D E F C G

52. Depth-First Search for undirected graphv A v v B C v v D E v F G v
A B D E F C G

53. Depth-First Search for undirected graphv A v v B C v v D E v F G v
A B D E F C G

54. Depth-First Search for undirected graphv A v v B C v v D E v F G v
A B D E F C G

55. Depth-First Search for undirected graphv A v v B C v v D E v F G v
A B D E F C G

56. Depth-First Search for undirected graphv A v v B C v v D E v F G v
A B D E F C G

57. Depth-First Search for undirected graphv A v v B C v v D E v F G v
A B D E F C G

58. Depth-First Search for undirected graphB C D E F G
A B D E F C G

59. Second example of DFS

60. Task: Conduct a depth-first search of the graph starting with node D
DFS for directed graph by Stack
Visited Array F C A B C D E F G H A B D H G E
Task: Conduct a depth-first search of the graph starting with node D

61. DFS for directed graph by Stack
Visited Array F C A B C D √ E F G H D A B D H G E
The order nodes are visited: D
Visit D

62. DFS for directed graph by Stack
Visited Array F C A B C D √ E F G H D A B D H G E
The order nodes are visited: D
Consider nodes adjacent to D, decide to visit C first (Rule: visit adjacent nodes in alphabetical order)

63. DFS for directed graph by Stack
Visited Array F C A B C √ D E F G H C D A B D H G E
The order nodes are visited:
D, C
Visit C

64. DFS for directed graph by Stack
Visited Array F C A B C √ D E F G H C D A B D H G E
The order nodes are visited:
D, C
No nodes adjacent to C; cannot continue  backtrack, i.e., pop stack and restore previous state

65. Back to D – C has been visited, decide to visit E next
DFS for directed graph by Stack
Visited Array F C A B C √ D E F G H D A B D H G E
The order nodes are visited:
D, C
Back to D – C has been visited, decide to visit E next

66. Back to D – C has been visited, decide to visit E next
DFS for directed graph by Stack
Visited Array F C A B C √ D E F G H E D A B D H G E
The order nodes are visited:
D, C, E
Back to D – C has been visited, decide to visit E next

67. DFS for directed graph by Stack
Visited Array F C A B C √ D E F G H E D A B D H G E
The order nodes are visited:
D, C, E
Only G is adjacent to E

68. DFS for directed graph by Stack
Visited Array F C A B C √ D E F G H G E D A B D H G E
The order nodes are visited:
D, C, E, G
Visit G

69. DFS for directed graph by Stack
Visited Array F C A B C √ D E F G H G E D A B D H G E
The order nodes are visited:
D, C, E, G
Nodes D and H are adjacent to G. D has already been visited. Decide to visit H.

70. DFS for directed graph by Stack
Visited Array F C A B C √ D E F G H H G E D A B D H G E
The order nodes are visited:
D, C, E, G, H
Visit H

71. Nodes A and B are adjacent to F. Decide to visit A next.
DFS for directed graph by Stack
Visited Array F C A B C √ D E F G H H G E D A B D H G E
The order nodes are visited:
D, C, E, G, H
Nodes A and B are adjacent to F. Decide to visit A next.

72. DFS for directed graph by Stack
Visited Array F C A √ B C D E F G H A H G E D A B D H G E
The order nodes are visited:
D, C, E, G, H, A
Visit A

73. Only Node B is adjacent to A. Decide to visit B next.
DFS for directed graph by Stack
Visited Array F C A √ B C D E F G H A H G E D A B D H G E
The order nodes are visited:
D, C, E, G, H, A
Only Node B is adjacent to A. Decide to visit B next.

74. DFS for directed graph by Stack
Visited Array F C A √ B C D E F G H B A H G E D A B D H G E
The order nodes are visited:
D, C, E, G, H, A, B
Visit B

75. No unvisited nodes adjacent to B. Backtrack (pop the stack).
DFS for directed graph
Visited Array F C A √ B C D E F G H A H G E D A B D H G E
The order nodes are visited:
D, C, E, G, H, A, B
No unvisited nodes adjacent to B. Backtrack (pop the stack).

76. No unvisited nodes adjacent to A. Backtrack (pop the stack).
DFS for directed graph
Visited Array F C A √ B C D E F G H H G E D A B D H G E
The order nodes are visited:
D, C, E, G, H, A, B
No unvisited nodes adjacent to A. Backtrack (pop the stack).

77. No unvisited nodes adjacent to H. Backtrack (pop the stack).
DFS for directed graph
Visited Array F C A √ B C D E F G H G E D A B D H G E
The order nodes are visited:
D, C, E, G, H, A, B
No unvisited nodes adjacent to H. Backtrack (pop the stack).

78. No unvisited nodes adjacent to G. Backtrack (pop the stack).
DFS for directed graph
Visited Array F C A √ B C D E F G H E D A B D H G E
The order nodes are visited:
D, C, E, G, H, A, B
No unvisited nodes adjacent to G. Backtrack (pop the stack).

79. No unvisited nodes adjacent to E. Backtrack (pop the stack).
DFS for directed graph
Visited Array F C A √ B C D E F G H D A B D H G E
The order nodes are visited:
D, C, E, G, H, A, B
No unvisited nodes adjacent to E. Backtrack (pop the stack).

80. F is unvisited and is adjacent to D. Decide to visit F next.
DFS for directed graph
Visited Array F C A √ B C D E F G H D A B D H G E
The order nodes are visited:
D, C, E, G, H, A, B
F is unvisited and is adjacent to D. Decide to visit F next.

81. DFS for directed graph The order nodes are visited:
Visited Array F C A √ B C D E F G H F D A B D H G E
The order nodes are visited:
D, C, E, G, H, A, B, F
Visit F

82. No unvisited nodes adjacent to F. Backtrack.
DFS for directed graph
Visited Array F C A √ B C D E F G H D A B D H G E
The order nodes are visited:
D, C, E, G, H, A, B, F
No unvisited nodes adjacent to F. Backtrack.

83. No unvisited nodes adjacent to D. Backtrack.
DFS for directed graph
Visited Array F C A √ B C D E F G H A B D H G E
The order nodes are visited:
D, C, E, G, H, A, B, F
No unvisited nodes adjacent to D. Backtrack.

84. Stack is empty. Depth-first traversal is done.
DFS for directed graph
Visited Array F C A √ B C D E F G H A B D H G E
The order nodes are visited:
D, C, E, G, H, A, B, F
Stack is empty. Depth-first traversal is done.

85. Graph Traversals put root node onto a queue
Pseudo-Code for Breadth-First Search (BFS)
breadth-first-traversal
put root node onto a queue
while the queue is not empty
dequeue a node from the queue
visit the node x e.g., print value
enqueue the children of x

86. First example of BFS

87. Breadth-First Search for undirected graph
Queue: A B C
Current: D E F G
A B C D E F G

88. Breadth-First Search for undirected graph
Queue: A A B C
Current: D E F G

89. Breadth-First Search for undirected graph
Queue: A A B C
Current: D E F G A

90. Breadth-First Search for undirected graph
Queue: A B C
Current: D E F G A A

91. Breadth-First Search for undirected graph
Queue: A B B C
Current: D E F G A A

92. Breadth-First Search for undirected graph
Queue: A C B B C
Current: D E F G A A

93. Breadth-First Search for undirected graph
Queue: A C B B C
Current: D E F G B A

94. Breadth-First Search for undirected graph
Queue: A C B C
Current: D E F G B
A B

95. Breadth-First Search for undirected graph
Queue: A D C B C
Current: D E F G B
A B

96. Breadth-First Search for undirected graph
Queue: A E D C B C
Current: D E F G B
A B

97. Breadth-First Search for undirected graph
Queue: A E D C B C
Current: D E F G C
A B

98. Breadth-First Search
Queue: A E D B C
Current: D E F G C
A B C

99. Breadth-First Search for undirected graph
Queue: A F E D B C
Current: D E F G C
A B C

100. Breadth-First Search for undirected graph
Queue: A G F E D B C D E F G
Current: C
A B C

101. Breadth-First Search for undirected graph
Queue: A G F E D B C D E F G
Current: D
A B C

102. Breadth-First Search for undirected graph
Queue: A G F E B C D E F G
Current: D
A B C D

103. Breadth-First Search for undirected graph
Queue: A G F E B C D E F G
Current: E
A B C D

104. Breadth-First Search for undirected graph
Queue: A G F B C D E F G
Current: E
A B C D E

105. Breadth-First Search for undirected graph
Queue: A G F B C D E F G
Current: F
A B C D E

106. Breadth-First Search for undirected graph
Queue: A G B C D E F G
Current: F
A B C D E F

107. Breadth-First Search for undirected graph
Queue: A G B C D E F G
Current: G
A B C D E F

108. Breadth-First Search for undirected graph
Queue: A B C D E F G
Current: G
A B C D E F G

109. Breadth-First Search for undirected graph
A B C D E F G

110. Second example of BFS

111. Breadth-First Search for directed graph
Breadth-first search starts with given node F C A B D H G E
Task: Conduct a breadth-first search of the graph starting with node D

112. Breadth-First Search for undirected graph
Breadth-first search starts with given node
Then visits nodes adjacent in some specified order (e.g., alphabetical)
Like ripples in a pond F C A B D H G E 1
Nodes visited: D

113. Breadth-First Search for undirected graph
Breadth-first search starts with given node
Then visits nodes adjacent in some specified order (e.g., alphabetical)
Like ripples in a pond F C A B D H G E 1
Nodes visited: D, C

114. Breadth-First Search for undirected graph
Breadth-first search starts with given node
Then visits nodes adjacent in some specified order (e.g., alphabetical)
Like ripples in a pond F C A B D H G E 1
Nodes visited: D, C, E

115. Breadth-First Search for undirected graph
Breadth-first search starts with given node
Then visits nodes adjacent in some specified order (e.g., alphabetical)
Like ripples in a pond F C A B D H G E 1
Nodes visited: D, C, E, F

116. Breadth-First Search for undirected graph
When all nodes in ripple are visited, visit nodes in next ripples F C A B D H G E 2 1
Nodes visited: D, C, E, F, G

117. Breadth-First Search for undirected graph
When all nodes in ripple are visited, visit nodes in next ripples F C A B D H 3 G E 2 1
Nodes visited: D, C, E, F, G, H

118. Nodes visited: D, C, E, F, G, H, A
Breadth-First Search for undirected graph
When all nodes in ripple are visited, visit nodes in next ripples 4 F C A B D H 3 G E 2 1
Nodes visited: D, C, E, F, G, H, A

119. Nodes visited: D, C, E, F, G, H, A, B
Breadth-First Search for undirected graph
When all nodes in ripple are visited, visit nodes in next ripples 4 F C A B D H 3 G E 2 1
Nodes visited: D, C, E, F, G, H, A, B

120. © 2011 Pearson Addison-Wesley. All rights reserved
Graph Traversals
Visitation order for a) a depth-first search; b) a breadth-first search