Graphs I

1
Graphs I

• Kruse and Ryba
• Chapter 12

2
Undirected Graphs

• An undirected graph is a set of nodes and a set
  of edges between nodes e.g.
• The graph is undirected because its edges do not
  have a particular direction.

3
Edges

• We sometimes need to refer to the edges ei of a
  graph

e0
e5
e1
e2
e4
e3
4
Vertices

• We also need to refer to the vertices vj of a
  graph

e0
e5
v0
v4
v1
e1
e2
e4
v2
e3
v3
5
Undirected Graph Definition

• An undirected graph G (V, E) is a finite set of
  vertices V, together with a finite set of edges
  E.
• Both sets might be empty (no vertices and no
  edges), which is called the empty graph.
• Each edge has associated with it two vertices.
• If two vertices are connected by an edge, they
  are said to be adjacent.

6
Undirected Graph Example Subway Map
7
Another Undirected Graph vBNS (fast internet
backbone)
8
Directed Graphs

• In a directed graph each vertex has an associated
  direction e.g.
• Each edge is associated with two vertices its
  source vertex and target vertex.

9
Directed Graph Examples

• Flow graph vertices are pumping stations and
  edges are pipelines connecting them.
• Task network each project has a number of
  component activities called tasks. Each task has
  a duration (amount of time needed to complete
  task).

10
Loops

• A loop is an edge that connects a vertex with
  itself for example
• Loops can occur in directed graphs or undirected
  graphs.

11
Multiple Edges

• In principle, a graph may have two or more edges
  connecting the same two vertices in the same
  direction
• These are called multiple edges. In a graph
  diagram, each edge is drawn separately.

12
Simple Graphs

• A graph is said to be a simple graph, if
• the graph has no loops, and
• no multiple edges

13
Path

• A path in a graph is a sequence of vertices, v0,
  v1, vm, such that each adjacent pair of
  vertices in the list are connected by an edge.

v0
v4
v1
v2
v3
14
Path

• A path in a graph is a sequence of vertices, v0,
  v1, vm, such that each adjacent pair of
  vertices in the list are connected by an edge.

Example path v0, v3, v4
v0
v4
v1
v2
v3
15
Path

• A path in a graph is a sequence of vertices, v0,
  v1, vm, such that each adjacent pair of
  vertices in the list are connected by an edge.

Example path v0, v3, v4 v0, v3, v2, v1, v4
v0
v4
v1
v2
v3
16
Path

• In a directed graph, the connection must go from
  source vi to target vi1.

Is there a path from v0 to v4? v3 to v1? v1 to
v2? v2 to v1?
v0
v4
v1
v2
v3
17
Example Task Network
Investigate neighborhood schools
Find a realtor
Look at homes
3
Investigate termite and soil reports
5
9
Start
Present offer
Choose a home to buy
10
2
Apply for financing and establish price limit
0
2
Prepare offer
15
20
How long will it take to buy a house? (The
duration of each task is shown below each task).
18
Critical Path
Investigate neighborhood schools
Find a realtor
Look at homes
3
Investigate termite and soil reports
5
9
Start
Present offer
Choose a home to buy
10
2
Apply for financing and establish price limit
0
2
Prepare offer
15
20
The task will take as long as the maximum
duration path from start to finish.
19
Cycles

• A cycle in a graph is a path that leads back to
  the first node in the path.

Example cycles v1, v2, v3, v2, v1
v0
v4
v1
v2
v3
20
Cycles

• A cycle in a graph is a path that leads back to
  the first node in the path.

Example cycles v1, v2, v3, v2, v1 v0, v3, v4,
v1, v0
v0
v4
v1
v2
v3
21
Does this graph have any cycles?
22
Cycles in Directed Graphs

• A cycle in a directed graph must follow edge
  directions e.g.

23
DAG Directed Acyclic Graph

• A directed acyclic graph (DAG) must have no
  cycles e.g.
• In other words, there is no path that starts and
  ends at the same node.

24
Complete Graph

• An undirected graph which has an edge between
  every pair of vertices.

25
Connected Graph

• An undirected graph which has a path between
  every pair of vertices.

1
2
3
0
26
Strongly Connected Graph

• A directed graph which has a path between every
  pair of vertices.

27
Weakly Connected Graph

• A directed graph which is connected, but not not
  strongly connected.

28
A DAG that is not Connected
29
Connected Components
30
Review of Terms

• Undirected vs. directed graphs
• Multiple edges, loops
• Simple graph
• Path
• Cycle
• Directed, acyclic graph (DAG)
• Connected, strongly connected, and weakly
  connected graphs
• Connected component

31
Graph Implementations
32
Adjacency Matrix
0 1 2 3
0 1 2 3




2
0
1
3
33
Adjacency Matrix
0 1 2 3
0 1 2 3
1



2
0
1
3
34
Adjacency Matrix
0 1 2 3
0 1 2 3
1 0



2
0
1
3
35
Adjacency Matrix
0 1 2 3
0 1 2 3
1 0 1



2
0
1
3
36
Adjacency Matrix
0 1 2 3
0 1 2 3
1 0 1 1



2
0
1
3
37
Adjacency Matrix
0 1 2 3
0 1 2 3
1 0 1 1
0 0 0 1


2
0
1
3
38
Adjacency Matrix
0 1 2 3
0 1 2 3
1 0 1 1
0 0 0 1
1 0 0 0

2
0
1
3
39
Adjacency Matrix
0 1 2 3
0 1 2 3
1 0 1 1
0 0 0 1
1 0 0 0
0 1 0 0
2
0
1
3
40
C Adjacency Matrix
0 1 2 3
0 1 2 3
1 0 1 1
0 0 0 1
1 0 0 0
0 1 0 0
2
0
1
3
bool graph_adjacency44
41
Adjacency Matrix Weighted Edges
6.0
0 1 2 3
2.5
2
5.8 0 6.0 1.5
0 0 0 2.5
2.5 0 0 0
0 4.2 0 0
0 1 2 3
5.8
0
1
2.5
1.5
3
4.2
double graph_adjacency44
42
Weighted Edges An Example
Flights. Whats the cheapest way to fly San
Fransisco ? Boston?
Boston
San Fransisco
452
245
320
Chicago
60
180
Dallas
43
Weighted Edges An Example
Mileage for various hops from San Fransisco ?
Boston
Boston
San Fransisco
2800
1200
1800
Chicago
2300
1600
Dallas
Mileage is not real, but for sake of example.
44
Weighted Edges An Example
Available internet bandwidth on various paths
between an ADSL provider and customers.
Pat
Fran
100Kbs
1Gbs
500Kbs
100Kbs
ADSL provider
This is a simplication for sake of example,
and bit rates are fictional.
45
Adjacency Matrix Undirected Graph
0 1 2 3
0 1 2 3




2
0
1
3
bool graph_adjacency44
46
Adjacency Matrix Undirected Graph
0 1 2 3
0 1 2 3
0
0
0
0
2
0
1
3
No loops ? zero diagonal.
47
Adjacency Matrix Undirected Graph
0 1 2 3
0 1 2 3
0 1 1 1
0
0
0
2
0
1
3
48
Adjacency Matrix Undirected Graph
0 1 2 3
0 1 2 3
0 1 1 1
1 0 0 1
0
0
2
0
1
3
49
Adjacency Matrix Undirected Graph
0 1 2 3
0 1 2 3
0 1 1 1
1 0 0 1
1 0 0 0
0
2
0
1
3
50
Adjacency Matrix Undirected Graph
0 1 2 3
0 1 2 3
0 1 1 1
1 0 0 1
1 0 0 0
1 1 0 0
2
0
1
3
51
Symmetry of Adjacency Matrix of Undirected Graph
0 1 2 3
0 1 2 3
0 1 1 1
1 0 0 1
1 0 0 0
1 1 0 0
2
0
1
3
aij aji
52
Non-symmetric Adjacency Matrix for Directed Graph
0 1 2 3
0 1 2 3
1 0 1 1
0 0 0 1
1 0 0 0
0 1 0 0
2
0
1
3
aij? aji
53
Symmetric Adjacency Matrix

• An undirected graph will always have a symmetric
  adjacency matrix.
• A directed graph may not have a symmetric
  adjacency matrix.
• If a directed graph has a symmetric matrix, what
  does that mean?

54
Non-symmetric Adjacency Matrix Unweighted
Directed Graph
0 1 2 3
0 1 2 3
1 0 1 1
0 0 0 1
1 0 0 0
0 1 0 0
2
0
1
3
aij? aji
55
Add an Edge
0 1 2 3
0 1 2 3
1 0 1 1
0 0 0 1
1 0 0 0
1 1 0 0
2
0
1
3
aijaji
56
Adjacency Matrix for Weighted Directed Graph
3
0 1 2 3
0 1 2 3
5 0 3 6
0 0 0 4
3 0 0 0
6 4 0 0
2
3
2
0
5
1
6
4
3
6
4
aijaji
57
Time Complexity Analysis

• Given N we have vertices in a graph, and we use
  the adjacency matrix representation.
• What is the worst case and average complexity of
  inserting an edge in the graph?
• What is the worst case and average complexity of
  removing an edge in the graph?
• What is the complexity of retrieving a list of
  all nodes that share and edge with a particular
  vertex?

58
Representing Graphs with Edge Lists
2
Edge list for vertex 0
Edge list for vertex 1
Edge list for vertex 2
Edge list for vertex 3
0
1
Null
2
0
1
3
Null
Null
3
Null
59
Time Complexity Analysis

• Given N we have vertices in a graph, and we use
  the edge list representation.
• What is the worst case and average complexity of
  inserting an edge in the graph?
• What is the worst case and average complexity of
  removing an edge in the graph?
• What is the complexity of retrieving a list of
  all nodes that share and edge with a particular
  vertex?

60
Representing Graphs with Edge Sets
Assume a Set template class is available Declare
array of sets, one per node in graph Setltintgt
connectionsN Each nodes set contains the
indices of all vertices that it is connected to.
2
0
1
3
61
Time Complexity Analysis

• Given N we have vertices in a graph, and we use
  the edge set representation.
• What is the worst case and average complexity of
  inserting an edge in the graph?
• What is the worst case and average complexity of
  removing an edge in the graph?
• What is the complexity of retrieving a list of
  all nodes that share and edge with a particular
  vertex?

62
Graph Implementations Summary

• Adjacency matrix
• Edge lists
• Edge sets

63
Graph Searching
64
Graph Search

• Choice of container
• If a stack is used as the container for adjacent
  vertices, we get depth first search.
• If a list is used as the container adjacent
  vertices, we get breadth first search.

65
Breadth First Algorithm

• Given graph G(V,E) and source vertex s ? V
• Create a queue Q
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• Q? s
• While Q ? ?
• u ? headQ
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Enqueue(Q,v)
• Dequeue(Q)
• coloru ? black

1
4
0
2
5
3
66
Breadth First Algorithm

• Given graph G(V,E) and source vertex s ? V
• Create a queue Q
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• Q? s
• While Q ? ?
• u ? headQ
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Enqueue(Q,v)
• Dequeue(Q)
• coloru ? black

1
4
s
0
2
5
3
Q ?
67
Breadth First Algorithm

• Given graph G(V,E) and source vertex s ? V
• Create a queue Q
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• Q? s
• While Q ? ?
• u ? headQ
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Enqueue(Q,v)
• Dequeue(Q)
• coloru ? black

1
4
s
0
2
5
3
Q ?
68
Breadth First Algorithm

• Given graph G(V,E) and source vertex s ? V
• Create a queue Q
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• Q? s
• While Q ? ?
• u ? headQ
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Enqueue(Q,v)
• Dequeue(Q)
• coloru ? black

1
4
s
0
2
5
3
Q
0
69
Breadth First Algorithm

• Given graph G(V,E) and source vertex s ? V
• Create a queue Q
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• Q? s
• While Q ? ?
• u ? headQ
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Enqueue(Q,v)
• Dequeue(Q)
• coloru ? black

1
4
0
2
5
3
u
Q
0
70
Breadth First Algorithm

• Given graph G(V,E) and source vertex s ? V
• Create a queue Q
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• Q? s
• While Q ? ?
• u ? headQ
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Enqueue(Q,v)
• Dequeue(Q)
• coloru ? black

1
4
0
2
5
3
u
Q
0
1
71
Breadth First Algorithm

• Given graph G(V,E) and source vertex s ? V
• Create a queue Q
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• Q? s
• While Q ? ?
• u ? headQ
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Enqueue(Q,v)
• Dequeue(Q)
• coloru ? black

1
4
0
2
5
3
u
Q
0
1
2
72
Breadth First Algorithm

• Given graph G(V,E) and source vertex s ? V
• Create a queue Q
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• Q? s
• While Q ? ?
• u ? headQ
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Enqueue(Q,v)
• Dequeue(Q)
• coloru ? black

1
4
0
2
5
3
u
Q
0
1
2
3
73
Breadth First Algorithm

• Given graph G(V,E) and source vertex s ? V
• Create a queue Q
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• Q? s
• While Q ? ?
• u ? headQ
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Enqueue(Q,v)
• Dequeue(Q)
• coloru ? black

1
4
0
2
5
3
u
Q
0
1
2
3
74
Breadth First Algorithm
u

• Given graph G(V,E) and source vertex s ? V
• Create a queue Q
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• Q? s
• While Q ? ?
• u ? headQ
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Enqueue(Q,v)
• Dequeue(Q)
• coloru ? black

1
4
0
2
5
3
Q
0
1
2
3
75
Breadth First Algorithm
u

• Given graph G(V,E) and source vertex s ? V
• Create a queue Q
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• Q? s
• While Q ? ?
• u ? headQ
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Enqueue(Q,v)
• Dequeue(Q)
• coloru ? black

1
4
0
2
5
3
Q
0
1
2
3
4
76
Breadth First Algorithm
u

• Given graph G(V,E) and source vertex s ? V
• Create a queue Q
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• Q? s
• While Q ? ?
• u ? headQ
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Enqueue(Q,v)
• Dequeue(Q)
• coloru ? black

1
4
0
2
5
3
Q
0
1
2
3
4
77
Breadth First Algorithm
u

• Given graph G(V,E) and source vertex s ? V
• Create a queue Q
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• Q? s
• While Q ? ?
• u ? headQ
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Enqueue(Q,v)
• Dequeue(Q)
• coloru ? black

1
4
0
2
5
3
Q
0
1
2
3
4
78
Breadth First Algorithm
u

• Given graph G(V,E) and source vertex s ? V
• Create a queue Q
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• Q? s
• While Q ? ?
• u ? headQ
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Enqueue(Q,v)
• Dequeue(Q)
• coloru ? black

1
4
0
2
5
3
Q
0
1
2
3
4
79
Breadth First Algorithm
u

• Given graph G(V,E) and source vertex s ? V
• Create a queue Q
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• Q? s
• While Q ? ?
• u ? headQ
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Enqueue(Q,v)
• Dequeue(Q)
• coloru ? black

1
4
0
2
5
3
Q
0
1
2
3
4
80
Breadth First Algorithm

• Given graph G(V,E) and source vertex s ? V
• Create a queue Q
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• Q? s
• While Q ? ?
• u ? headQ
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Enqueue(Q,v)
• Dequeue(Q)
• coloru ? black

1
4
0
2
5
3
u
Q
0
1
2
3
4
81
Breadth First Algorithm

• Given graph G(V,E) and source vertex s ? V
• Create a queue Q
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• Q? s
• While Q ? ?
• u ? headQ
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Enqueue(Q,v)
• Dequeue(Q)
• coloru ? black

1
4
0
2
5
3
u
Q
0
1
2
3
4
5
82
Breadth First Algorithm

• Given graph G(V,E) and source vertex s ? V
• Create a queue Q
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• Q? s
• While Q ? ?
• u ? headQ
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Enqueue(Q,v)
• Dequeue(Q)
• coloru ? black

1
4
0
2
5
3
u
Q
0
1
2
3
4
5
83
Breadth First Algorithm

• Given graph G(V,E) and source vertex s ? V
• Create a queue Q
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• Q? s
• While Q ? ?
• u ? headQ
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Enqueue(Q,v)
• Dequeue(Q)
• coloru ? black

1
4
0
2
5
3
u
Q
0
1
2
3
4
5
84
Breadth First Algorithm

• Given graph G(V,E) and source vertex s ? V
• Create a queue Q
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• Q? s
• While Q ? ?
• u ? headQ
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Enqueue(Q,v)
• Dequeue(Q)
• coloru ? black

1
4
0
2
5
3
u
Q
0
1
2
3
4
5
85
Breadth First Algorithm

• Given graph G(V,E) and source vertex s ? V
• Create a queue Q
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• Q? s
• While Q ? ?
• u ? headQ
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Enqueue(Q,v)
• Dequeue(Q)
• coloru ? black

1
4
0
2
5
3
Q
u
0
1
2
3
4
5
86
Breadth First Algorithm

• Given graph G(V,E) and source vertex s ? V
• Create a queue Q
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• Q? s
• While Q ? ?
• u ? headQ
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Enqueue(Q,v)
• Dequeue(Q)
• coloru ? black

1
4
0
2
5
3
Q ?
u
87
Graph Search

• Choice of container
• If a stack is used as the container for adjacent
  vertices, we get depth first search.
• If a list is used as the container adjacent
  vertices, we get breadth first search.

88
Depth First Algorithm

• Given graph G(V,E) and source vertex s ? V
• Create a stack S
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• S? s
• While S ? ?
• u Pop(S)
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Push(S,v)
• coloru ? black

1
4
s
0
2
5
3
S ?
89
Depth First Algorithm

• Given graph G(V,E) and source vertex s ? V
• Create a stack S
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• S? s
• While S ? ?
• u Pop(S)
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Push(S,v)
• coloru ? black

1
4
s
0
2
5
3
S ?
90
Depth First Algorithm

• Given graph G(V,E) and source vertex s ? V
• Create a stack S
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• S? s
• While S ? ?
• u Pop(S)
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Push(S,v)
• coloru ? black

1
4
s
0
2
5
3
S ?
0
91
Depth First Algorithm

• Given graph G(V,E) and source vertex s ? V
• Create a stack S
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• S? s
• While S ? ?
• u Pop(S)
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Push(S,v)
• coloru ? black

1
4
0
2
5
u
3
S ?
92
Depth First Algorithm

• Given graph G(V,E) and source vertex s ? V
• Create a stack S
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• S? s
• While S ? ?
• u Pop(S)
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Push(S,v)
• coloru ? black

1
4
0
2
5
u
3
S ?
1
93
Depth First Algorithm

• Given graph G(V,E) and source vertex s ? V
• Create a stack S
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• S? s
• While S ? ?
• u Pop(S)
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Push(S,v)
• coloru ? black

1
4
0
2
5
u
3
S ?
3
2
1
94
Depth First Algorithm

• Given graph G(V,E) and source vertex s ? V
• Create a stack S
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• S? s
• While S ? ?
• u Pop(S)
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Push(S,v)
• coloru ? black

1
4
0
2
5
u
3
S ?
3
2
1
95
Depth First Algorithm

• Given graph G(V,E) and source vertex s ? V
• Create a stack S
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• S? s
• While S ? ?
• u Pop(S)
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Push(S,v)
• coloru ? black

1
4
0
2
5
u
3
S ?
3
2
1
96
Depth First Algorithm

• Given graph G(V,E) and source vertex s ? V
• Create a stack S
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• S? s
• While S ? ?
• u Pop(S)
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Push(S,v)
• coloru ? black

1
4
0
2
5
3
u
S ?
3
2
1
97
Depth First Algorithm

• Given graph G(V,E) and source vertex s ? V
• Create a stack S
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• S? s
• While S ? ?
• u Pop(S)
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Push(S,v)
• coloru ? black

1
4
0
2
5
3
u
S ?
3
2
1
5
98
Depth First Algorithm

• Given graph G(V,E) and source vertex s ? V
• Create a stack S
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• S? s
• While S ? ?
• u Pop(S)
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Push(S,v)
• coloru ? black

1
4
0
2
5
3
u
S ?
3
2
1
5
99
Depth First Algorithm

• Given graph G(V,E) and source vertex s ? V
• Create a stack S
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• S? s
• While S ? ?
• u Pop(S)
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Push(S,v)
• coloru ? black

1
4
0
2
5
3
u
S ?
3
5
2
1
100
Depth First Algorithm

• Given graph G(V,E) and source vertex s ? V
• Create a stack S
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• S? s
• While S ? ?
• u Pop(S)
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Push(S,v)
• coloru ? black

1
4
0
2
5
3
u
S ?
3
5
2
1
4
101
Depth First Algorithm

• Given graph G(V,E) and source vertex s ? V
• Create a stack S
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• S? s
• While S ? ?
• u Pop(S)
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Push(S,v)
• coloru ? black

1
4
0
2
5
3
u
S ?
3
5
2
1
4
102
Depth First Algorithm

• Given graph G(V,E) and source vertex s ? V
• Create a stack S
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• S? s
• While S ? ?
• u Pop(S)
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Push(S,v)
• coloru ? black

1
4
0
2
u
5
3
S ?
4
3
5
2
1
103
Depth First Algorithm

• Given graph G(V,E) and source vertex s ? V
• Create a stack S
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• S? s
• While S ? ?
• u Pop(S)
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Push(S,v)
• coloru ? black

1
4
0
2
u
5
3
S ?
4
3
5
2
1
104
Depth First Algorithm

• Given graph G(V,E) and source vertex s ? V
• Create a stack S
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• S? s
• While S ? ?
• u Pop(S)
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Push(S,v)
• coloru ? black

1
4
0
2
u
5
3
S ?
4
3
5
2
1
105
Depth First Algorithm

• Given graph G(V,E) and source vertex s ? V
• Create a stack S
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• S? s
• While S ? ?
• u Pop(S)
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Push(S,v)
• coloru ? black

1
4
0
2
u
5
3
S ?
4
3
5
2
1
106
Depth First Algorithm

• Given graph G(V,E) and source vertex s ? V
• Create a stack S
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• S? s
• While S ? ?
• u Pop(S)
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Push(S,v)
• coloru ? black

1
4
0
2
u
5
3
S ?
107
Depth First Algorithm

• Given graph G(V,E) and source vertex s ? V
• Create a stack S
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• S? s
• While S ? ?
• u Pop(S)
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Push(S,v)
• coloru ? black

1
4
0
2
u
5
3
S ?
108
Graph Search

• Choice of container
• If a stack is used as the container for adjacent
  vertices, we get depth first search.
• If a list is used as the container adjacent
  vertices, we get breadth first search.

109
Depth First Algorithm

• Given graph G(V,E) and source vertex s ? V
• Create a stack S
• For each vertex u ? V s
• coloru ? white
• colors ? gray
• S? s
• While S ? ?
• u Pop(S)
• for each v ? Adjacentu
• if colorv white
• colorv ? gray
• Push(S,v)
• coloru ? black

1
4
0
2
u
5
3
S ?