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Lecture 15:
Graph Searching and
Traversing
DATA STRUCTURES AND ALGORITHMS

1. Graph Traversal
Definition:
 A graph traversal means visiting all the vertices (nodes) of the
graph.
- Traversal is the facility to move through a structure visiting each of
the vertices once.
• Two traversal methods for a graph are breadth-first and depth-first.

2. Breadth-First Graph Traversal
• Breadth-first traversal makes use of a queue data structure. The
queue holds a list of vertices which have not been visited yet but
which should be visited soon.
• Since a queue is a first-in first-out structure, vertices are visited in
the order in which they are added to the queue.
• Visiting a vertex involves, for example, outputting the data stored
in that vertex, and also adding its neighbours to the queue.
• Neighbours are not added to the queue if they are already in the
queue, or have already been visited.
• In BFS;
- One node is selected as a start position. It’s visited and marked.
- Then, all unvisited nodes adjacent of the next node are visited and
marked in some sequential order.

2. Breadth-First Search [Example-1] (1/5)
Step-1:
Step-2:
4
A
B
S
C
D
E
F
G H
Queue Status
Output: A
B
S
A
B
S
C
D
E
F
G H
Output: A, B,
Queue Status
S

Step-3:
Step-4:
5
A
B
S
C
D
E
F
G H
Queue Status
Output: A, B, S,
C
G
A
B
S
C
D
E
F
G H
Output: A, B, S, C,
Queue Status
G
D
E
F

Step-5:
Step-6:
6
A
B
S
C
D
E
F
G H
Queue Status
Output: A, B, S, C, G,
D
E
F
H
A
B
S
C
D
E
F
G H
Output: A, B, S, C, G, D,
Queue Status
E
F
H

Step-7:
Step-8:
7
A
B
S
C
D
E
F
G H
Queue Status
Output: A, B, S, C, G, D, E,
F
H
A
B
S
C
D
E
F
G H
Output: A, B, S, C, G, D, E, F
Queue Status
H

Step-9:
8
A
B
S
C
D
E
F
G H
Queue Status
Output: A, B, S, C, G, D, E, F, H [Hence Proved]

•If we start at vertex 1 then a valid breadth-first order
would be:
9
2
5
3
0
6
4
7
1
1,2,3,7,6,0,5,4
2. Breadth-First Search [Example-2]

10
Breadth First Search:
B
C
D
E
F
H
G
A
I
A B D E
A B D E F
A B D E F C H
A B D E F C H G I
As was true with depth-first search, a breadth-first search from some vertices
may fail to locate all the vertices.
A breadth-first search from B:
B F G I H C D
2. Breadth-First Search [Example-3]

A
H
B
F
E
D
C
G
A
B
C
D
E
F
G
H
How is this accomplished? Simply replace the stack with a queue!
Rules: (1) Maintain an enqueued array. (2) Visit node when
dequeued.
Q 

A
H
B
F
E
D
C
G
A
B
C
D √
E
F
G
H
Enqueue D. Notice, D not yet visited.
Q  D
Nodes visited:

A
H
B
F
E
D
C
G
A
B
C √
D √
E √
F √
G
H
Dequeue D. Visit D. Enqueue unenqueued nodes adjacent to D.
Q  C  E  F
Nodes visited: D

A
H
B
F
E
D
C
G
A
B
C √
D √
E √
F √
G
H
Dequeue C. Visit C. Enqueue unenqueued nodes adjacent to C.
Q  E  F
Nodes visited: D, C

A
H
B
F
E
D
C
G
A
B
C √
D √
E √
F √
G
H
Dequeue E. Visit E. Enqueue unenqueued nodes adjacent to
E.
Q  F  G
Nodes visited: D, C, E

A
H
B
F
E
D
C
G
A
B
C √
D √
E √
F √
G √
H
Dequeue F. Visit F. Enqueue unenqueued nodes adjacent to F.
Q  G
Nodes visited: D, C, E, F

A
H
B
F
E
D
C
G
A
B
C √
D √
E √
F √
G √
H √
Dequeue G. Visit G. Enqueue unenqueued nodes adjacent to G.
Q  H
Nodes visited: D, C, E, F, G

A
H
B
F
E
D
C
G
A √
B √
C √
D √
E √
F √
G √
H √
Dequeue H. Visit H. Enqueue unenqueued nodes adjacent to H.
Q  A  B
Nodes visited: D, C, E, F, G, H

A
H
B
F
E
D
C
G
A √
B √
C √
D √
E √
F √
G √
H √
Dequeue A. Visit A. Enqueue unenqueued nodes adjacent to
A.
Q  B
Nodes visited: D, C, E, F, G, H, A

A
H
B
F
E
D
C
G
A √
B √
C √
D √
E √
F √
G √
H √
Dequeue B. Visit B. Enqueue unenqueued nodes adjacent to B.
Q empty
Nodes visited: D, C, E, F, G, H, A, B

A
H
B
F
E
D
C
G
A √
B √
C √
D √
E √
F √
G √
H √
Q empty. Algorithm done.
Q empty
Nodes visited: D, C, E, F, G, H, A, B

1. Visit the start vertex
2. Initialize queue to contain the start vertex
3. While queue not empty do
a. Remove a vertex v from the queue
b. For all vertices w adjacent to v do:
If w has not been visited then:
i. Visit w
ii. Add w to queue
End while
22
2. Breadth-First Search [Algorithm]

3. Depth-First Graph Traversal
23
• In DFS;
- Depth first search (DFS) follows first a path from the starting node to an
ending node.
- Then, another path from the start to the End and so forth until all nodes
have been visited.
• Data stored in DFS as stack (working as Push and Pop).
Steps of DFGT:
- stack works as LIFO (Last In First Out).
• This method visits all the vertices, beginning with a specified start vertex.
• This strategy proceeds along a path from vertex V as deeply into the graph as
possible.
• This means that after visiting V, the algorithm tries to visit any unvisited
vertex adjacent to V.
• When the traversal reaches a vertex which has no adjacent vertex, it back
tracks and visits an unvisited adjacent vertex.

3. Depth-First Search [Example-1] (1/6)
Step-1:
Step-2:
24
A
B
S
C
D
E
F
G H
Stack
Output: A
A
A
B
S
C
D
E
F
G H
Output: A, B,
Stack
B
A
No adjacent. So
“B” become
“Pop”

Step-3:
Step-4:
25
A
B
S
C
D
E
F
G H
Stack
Output: A, B, S,
S
A
A
B
S
C
D
E
F
G H
Output: A, B, S, C,
Stack
C
S
A

Step-5:
Step-6:
26
A
B
S
C
D
E
F
G H
Stack
Output: A, B, S, C, D
A
B
S
C
D
E
F
G H
Output: A, B, S, C, D
Stack
D
C
S
A
D
C
S
A
No adjacent. So
“D” become
“Pop”

Step-7:
Step-8:
27
A
B
S
C
D
E
F
G H
Stack
Output: A, B, S, C, D, E
A
B
S
C
D
E
F
G H
Output: A, B, S, C, D, E, H,
Stack
H
E
C
S
A
E
C
S
A

Step-9:
Step-10:
28
A
B
S
C
D
E
F
G H
Stack
Output: A, B, S, C, D, E, H, G,
A
B
S
C
D
E
F
G H
Output: A, B, S, C, D, E, H,G, F
Stack
G
H
E
C
S
A
F
G
H
E
C
S
A

Step-10:
29
A
B
S
C
D
E
F
G H
Output: A, B, S, C, D, E, H,G, F
Stack
No adjacent. So
“F” become
“Pop”.
No adjacent. So
“G” become
“Pop”.
No adjacent. So
“H” become
“Pop”.
No adjacent. So
“E” become
“Pop”.
No adjacent. So
“C” become
“Pop”.
No adjacent. So
“S” become
“Pop”.
No adjacent. So
“A” become
“Pop”.

•If we start at vertex 1 then a valid depth-first order
would be:
30
2
5
3
0
6
4
7
1
1, 2, 3, 6, 4, 5, 7, 0
3. Depth-First Search [Example-2]

A
D E
B
F
C
A B C F E D
3. Depth-First Search [Example-3]

32
B
C
D
E
F
H
G
A
I
 Design Starting depth first search at start vertices as;
(i) Vertices A
(ii) Vertices C
(iii) Vertices G
3. Depth-First Search [Class Participation]

A
H
B
F
E
D
C
G
A
B
C
D
E
F
G
H
Visited Array
Task: Conduct a depth-first search of the graph
starting with node D
3. Depth-First Search [Class Participation]

1. Shortest Paths
2. Minimum Spanning Trees
58
4. Applications of Graph

4.1 Shortest Paths
• Definition: The shortest path problem is about finding a path
between vertices in a graph such that the total sum of the edges
weights is minimum.
• Problem: Given some vertex s, find the shortest path from s to every
other vertex in G.
• Suppose we have a weighted graph G:
• The cost of traversing edge (i, j) is ci,j .
• The cost of traversing a path is the sum of the edge costs.
59

• Phone/Internet communications: to find the cheapest/fastest path
between two computers.
• Transportation/Travelling: to find the cheapest/fastest path between
two cities.
60
4.1 Shortest Paths [Applications]

• Dijkstra’s algorithm is an algorithm for finding the shortest paths between
nodes in a graph which may represent.
• For example; Road networks.
Steps of Dijkstra’s Algorithm:
1. Initialize distances according to the algorithm.
2. Pick 1st
node and calculate distances to adjacent nodes.
3. Pick next node with minimal distance; repeat adjacent node distance
calculations.
61
4.2 Shortest Paths [Dijkstra’s Algorithm]

• Ex
Step-1: Start with source node (i.e., a). And Initialize it with 0.
Other nodes become infinite (∞).
62
4.2 Dijkstra’s Algorithm [Example-1] (1/4)
a
b
c
d
e
f
4
2
1
5
8
10
6
5
2
a
b
c
d
e
f
4
2
1
5
8
10
6
5
2
0
∞ ∞
∞ ∞
∞

Step-2: Source node called as u. and destination as v. Find shortest path.
Step-3: Applying formula. c=2.
63
a
b
c
d
e
f
4
2
1
5
8
10
6
5
2
0
∞ ∞
∞ ∞
∞
u
v
Formula:
If (d (u) + c (u, v) < d(v)}
d (v) = d(u) + c (u, v)
If (0 + 2 < ∞)
2 < ∞ [True]
d (v) = d(u) + c (u, v)
= 0 + 2 = 2
a
b
c
d
e
f
4
2
1
5
8
10
6
5
2
0
∞ ∞
2 ∞
∞

Step-4: Find shortest path from c. u and v changes.
Step-5: Find shortest path from b. u and v changes.
64
a
b
c
d
e
f
4
2
1
5
8
10
6
5
2
0
3 ∞
2 ∞
∞
u
v
a
b
c
d
e
f
4
2
1
5
8
10
6
5
2
0
3
2 ∞
∞
u
8 v

Step-6: Find shortest path from d. u and v changes.
Step-7: Find shortest path from e. u and v changes.
65
a
b
c
d
e
f
4
2
1
5
8
10
6
5
2
0
3
2
∞
u
8
v
10
a
b
c
d
e
f
4
2
1
5
8
10
6
5
2
0
3
2
15
10
Marked (red line) is
most shortest path to
reach destination.
Total cost 15.

Find the shortest path from vertex A to every
other vertex?
Step-1: Create “Shortest distance from A” and “Previous vertex” columns.

Step-2:

Step-3:

Step-4:

Step-5:

Step-6:

Step-7:

Step-8:

Step-9:

Step-10:

Step-11:

Step-12:

Step-13:

Step-14:

Step-15: Finally; Short distance from “A” to all other vertices as;

10
1
5
2
6
4
9
7
2 3
4.2 Dijkstra’s Algorithm [Class Participation]
Find the shortest path in the given weighted
graph by Dijkstra algorithm ?

91
Applications of Graph
1. Shortest Paths
2. Minimum Spanning Trees
5. Applications of Graph [Minimum Spanning Trees]

92
Problem: Laying Telephone Wire
Central office

93
Wiring: First Approach
Central office
Simple but Expensive!

94
Wiring: Better Approach
Central office
Minimize the total length of wire connecting the customers

95
5.1 Minimum Spanning Tree (MST)
• it is a tree (i.e., it is acyclic)
• it covers all the vertices V
– contains |V| - 1 edges
• the total cost associated with tree edges is the
minimum among all possible spanning trees
• not necessarily unique
A minimum spanning tree is a subgraph of an
undirected weighted graph G, such that

96
• Prim's algorithm is a famous greedy algorithm.
- It is used for finding the Minimum Spanning Tree (MST) of a
given graph.
• To apply Prim’s algorithm,
- the given graph must be weighted, connected and undirected.
Steps of Prim’s Algorithm:
1. Remove all loop edges.
2. Remove all parallel edges.
3. Choose any arbitrary node as root node.
5.2 Minimum Spanning Tree (Prim’s Algorithm)

97
[Example-1]
B
A
C
D
E
F
7
8
3
6
4
3
8
8
2
5
2
Step-1: Remove all loop edges
B
A
C
D
E
F
7
8
3
6
4
3
8
2
5
2
Step-2: Remove all Parallel edges.
If more than edges between 2 nodes;
remove “high weighted nodes”.
B
A
C
D
E
F
7
8
3
6
4
3
2
5
2

98
[Example-1]
Step-3: Choose any arbitrary node as root node. We selected “A” vertices.
B
A
C
D
E
F
7
8
3
6
4
3
8
2
5
2
A
Step-4: Check outgoing edges (A) and select the one with less cost.
B
A
7 Remaining edges: AC(8)
Step-5: Check outgoing edges (B) and select the one with less cost.
B
A
7
C
3
Selected edges: AC(8), BD(6), BC(3)
Remaining edges: AC(8), BD(6)
Selected edges: AC(8), AB(7)

99
[Example-1]
Step-6: Check outgoing edges (C) and select the one with less cost.
B
A
7
C
3
Selected edges: AC(8), BD(6), CD(4), CE(3)
Remaining edges: AC(8), BD(6), CD(4),
E
3
B
A
7
C
3
E
3
Selected edges: AC(8), BD(6), CD(4), ED(2), EF(2)
Remaining edges: AC(8), BD(6), CD(4), EF(2)
Select anyone.
Both are minimum
values.
D
2

100
[Example-1]
Selected edges: AC(8), BD(6), CD(4), EF(2), DF(5)
Remaining edges: AC(8), BD(6), CD(4), DF(5)
EF(2) is selected
because of
minimum value in
queue..
B
A
7
C
3
E
3
D
2 F
2
Total vertices (V)= 6, Edge (E) = | V | - 1
= 6 – 1 = 5 [Hence Proof, Minimum Spanning Tree]
Final conclusion

Kruskal’s Algorithm
101
Initialization
a. Create a ‘forest’ of vertices F
b. Initialize the set of “safe edges” T
comprising the MST to the empty set
c. Sort edges by increasing weight
a
c
e
d
b
2
4
5
9
6
4
5
5
F = {a}, {b}, {c}, {d}, {e}
T = 
E = {(a,d), (c,d), (d,e), (a,c),
(b,e), (c,e), (b,d), (a,b)}

Kruskal’s Algorithm
102
a
c
e
d
b
2
4
5
9
6
4
5
5
2 4 4 5 5 5 6 9
(a,d) (c,d) (d,e) (a,c) (b,e) (c,e) (b,d) (a,b)
E = {(a,d), (c,d), (d,e), (b,e)}

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