1. Data Structures and Algorithms
Objectives
In this session, you will learn to:
Implement a graph
Apply graphs to solve programming problems
Ver. 1.0 Session 17

2. Data Structures and Algorithms
Representing a Graph
To implement a graph, you need to first represent the given
information in the form of a graph.
The two most commonly used ways of representing a graph
are as follows:
Adjacency Matrix
Adjacency List
Ver. 1.0 Session 17

3. Data Structures and Algorithms
Adjacency Matrix
Consider the following Adjacency Matrix Representation
graph:
v1 v2 v3 v4
v1 0 1 0 0
v2 0 0 1 0
v3 0 0 0 0
v4 1 0 1 0
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4. Data Structures and Algorithms
Adjacency List
Consider the following Adjacency List Representation
graph:
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5. Data Structures and Algorithms
Traversing a Graph
Traversing a graph means visiting all the vertices in a
graph.
You can traverse a graph with the help of the following two
methods:
Depth First Search (DFS)
Breadth First Search (BFS)
Ver. 1.0 Session 17

6. Data Structures and Algorithms
DFS
Algorithm: DFS(v)
1. Push the starting vertex, v into the stack.
2. Repeat until the stack becomes empty:
a. Pop a vertex from the stack.
b. Visit the popped vertex.
c. Push all the unvisited vertices adjacent to the popped vertex
into the stack.
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7. Data Structures and Algorithms
DFS (Contd.)
Push the starting vertex, v1 into the stack
v1
Ver. 1.0 Session 17

8. Data Structures and Algorithms
DFS (Contd.)
Pop a vertex, v1 from the stack
Visit v1
Push all unvisited vertices adjacent to v1 into the stack
v1
Visited:
v1
Ver. 1.0 Session 17

9. Data Structures and Algorithms
DFS (Contd.)
Pop a vertex, v1 from the stack
Visit v1
Push all unvisited vertices adjacent to v1 into the stack
v2
v4
Visited:
v1
Ver. 1.0 Session 17

10. Data Structures and Algorithms
DFS (Contd.)
Pop a vertex, v2 from the stack
Visit v2
Push all unvisited vertices adjacent to v2 into the stack
v2
v4
Visited:
v1 v2
Ver. 1.0 Session 17

11. Data Structures and Algorithms
DFS (Contd.)
Pop a vertex, v2 from the stack
Visit v2
Push all unvisited vertices adjacent to v2 into the stack
v6
v3
v4
Visited:
v1 v2
Ver. 1.0 Session 17

12. Data Structures and Algorithms
DFS (Contd.)
Pop a vertex, v6 from the stack
Visit v6
Push all unvisited vertices adjacent to v6 into the stack
v6
v3
v4
There are no unvisited vertices
adjacent to v6
Visited:
v1 v2 v6
Ver. 1.0 Session 17

13. Data Structures and Algorithms
DFS (Contd.)
Pop a vertex, v3 from the stack
Visit v3
Push all unvisited vertices adjacent to v3 into the stack
v3
v4
Visited:
v1 v2 v6 v3
Ver. 1.0 Session 17

14. Data Structures and Algorithms
DFS (Contd.)
Pop a vertex, v3 from the stack
Visit v3
Push all unvisited vertices adjacent to v3 into the stack
v5
v4
Visited:
v1 v2 v6 v3
Ver. 1.0 Session 17

15. Data Structures and Algorithms
DFS (Contd.)
Pop a vertex, v5 from the stack
Visit v5
Push all unvisited vertices adjacent to v5 into the stack
v5
v4
There are no unvisited vertices
adjacent to v5
Visited:
v1 v2 v6 v3 v5
Ver. 1.0 Session 17

16. Data Structures and Algorithms
DFS (Contd.)
Pop a vertex, v4 from the stack
Visit v4
Push all unvisited vertices adjacent to v4 into the stack
v4
There are no unvisited vertices
adjacent to v4
Visited:
v1 v2 v6 v3 v5 v4
Ver. 1.0 Session 17

17. Data Structures and Algorithms
DFS (Contd.)
The stack is now empty
Therefore, traversal is complete
Visited:
v1 v2 v6 v3 v5 v4
Ver. 1.0 Session 17

18. Data Structures and Algorithms
DFS (Contd.)
• Although the preceding algorithm provides a simple and
convenient method to traverse a graph, the algorithm will
not work correctly if the graph is not connected.
• In such a case, you will not be able to traverse all the
vertices from one single starting vertex.
Ver. 1.0 Session 17

19. Data Structures and Algorithms
DFS (Contd.)
To solve this problem, you need to 1. Repeat step 2 for each
vertex, v in the graph
execute the preceding algorithm
2. If v is not visited:
repeatedly for all unvisited vertices in a. Call DFS(v)
the graph.
Ver. 1.0 Session 17

20. Data Structures and Algorithms
BFS
Algorithm: BFS(v)
1. Visit the starting vertex, v and insert it into a queue.
2. Repeat step 3 until the queue becomes empty.
3. Delete the front vertex from the queue, visit all its unvisited
adjacent vertices, and insert them into the queue.
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21. Data Structures and Algorithms
BFS (Contd.)
Visit v1
Insert v1 into the queue
v1
v1
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22. Data Structures and Algorithms
BFS (Contd.)
• Remove a vertex v1 from the queue
• Visit all unvisited vertices adjacent to v1 and insert them in
the queue
v1
Visited:
v1
Ver. 1.0 Session 17

23. Data Structures and Algorithms
BFS (Contd.)
• Remove a vertex v1 from the queue
• Visit all unvisited vertices adjacent to v1 and insert them in
the queue
v2 v4
v1 v2 v4
Ver. 1.0 Session 17

24. Data Structures and Algorithms
BFS (Contd.)
• Remove a vertex v2 from the queue
• Visit all unvisited vertices adjacent to v2 and insert them in
the queue
v2 v4
v1 v2 v4
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25. Data Structures and Algorithms
BFS (Contd.)
• Remove a vertex v2 from the queue
• Visit all unvisited vertices adjacent to v2 and insert them in
the queue
v4 v3 v6
v1 v2 v4 v3 v6
Ver. 1.0 Session 17

26. Data Structures and Algorithms
BFS (Contd.)
• Remove a vertex v4 from the queue
• Visit all unvisited vertices adjacent to v4 and insert them in
the queue
v4 v3 v6 v5
v1 v2 v4 v3 v6 v5
Ver. 1.0 Session 17

27. Data Structures and Algorithms
BFS (Contd.)
Remove a vertex v3 from the queue
Visit all unvisited vertices adjacent to v3 and insert them in
the queue
v3 v6 v5
v3 does not have any
unvisited adjacent vertices
v1 v2 v4 v3 v6 v5
Ver. 1.0 Session 17

28. Data Structures and Algorithms
BFS (Contd.)
• Remove a vertex v6 from the queue
• Visit all unvisited vertices adjacent to v6 and insert them in
the queue
v6 v5
v3 does not have any
unvisited adjacent vertices
v1 v2 v4 v3 v6 v5
Ver. 1.0 Session 17

29. Data Structures and Algorithms
BFS (Contd.)
• Remove a vertex v6 from the queue
• Visit all unvisited vertices adjacent to v6 and insert them in
the queue
v5
v6 does not have any
unvisited adjacent vertices
v1 v2 v4 v3 v6 v5
Ver. 1.0 Session 17

30. Data Structures and Algorithms
BFS (Contd.)
• Remove a vertex v5 from the queue
• Visit all unvisited vertices adjacent to v5 and insert them in
the queue
v5
v6 does not have any
unvisited adjacent vertices
v1 v2 v4 v3 v6 v5
Ver. 1.0 Session 17

31. Data Structures and Algorithms
BFS (Contd.)
• Remove a vertex v5 from the queue
• Visit all unvisited vertices adjacent to v5 and insert them in
the queue
v5 does not have any
unvisited adjacent vertices
v1 v2 v4 v3 v6 v5
Ver. 1.0 Session 17

32. Data Structures and Algorithms
BFS (Contd.)
The queue is now empty
Therefore, traversal is complete
v5 does not have any
unvisited adjacent vertices
v1 v2 v4 v3 v6 v5
Ver. 1.0 Session 17

33. Data Structures and Algorithms
BFS (Contd.)
Although the preceding algorithm provides a simple and
convenient method to traverse a graph, the algorithm will
not work correctly if the graph is not connected.
In such a case, you will not be able to traverse all the
vertices from one single starting vertex.
Ver. 1.0 Session 17

34. Data Structures and Algorithms
BFS (Contd.)
To solve this problem, you need to 1. Repeat step 2 for each
vertex, v in the graph
execute the preceding algorithm
2. If v is not visited:
repeatedly for all unvisited vertices in a. Call BFS(v)
the graph.
Ver. 1.0 Session 17

35. Data Structures and Algorithms
Activity: Implementing a Graph by Using Adjacency Matrix Representation
Problem Statement:
You have to represent a set of cities and the distances
between them in the form of a graph. Write a program to
represent the graph in the form of an adjacency matrix.
Ver. 1.0 Session 17

36. Data Structures and Algorithms
Applications of Graphs
Many problems can be easily solved by reducing them in
the form of a graph
Graph theory has been instrumental in analyzing and
solving problems in areas as diverse as computer network
design, urban planning, finding shortest paths and
molecular biology.
Ver. 1.0 Session 17

37. Data Structures and Algorithms
Solving the Shortest Path Problem
The shortest path problem can be solved by applying the
Dijkstra’s algorithm on a graph
The Dijkstra’s algorithm is based on the greedy approach
The steps in the Dijkstra’s algorithm are as follows:
1. Choose vertex v corresponding to the smallest distance
recorded in the DISTANCE array such that v is not already in
FINAL.
2. Add v to FINAL.
3. Repeat for each vertex w in the graph that is not in FINAL:
a. If the path from v1 to w via v is shorter than the previously
recorded distance from v1 to w (If ((DISTANCE[v] + weight of
edge(v,w)) < DISTANCE[w])):
i. Set DISTANCE[w]=DISTANCE[v] + weight of edge(v,w).
4. If FINAL does not contain all the vertices, go to step 1.
Ver. 1.0 Session 17

38. Data Structures and Algorithms
Solving the Shortest Path Problem (Contd.)
5 Suppose you need to find the
shortest distance of all the
vertices from vertex v1.
3 4 6
Add v1 to the FINAL array.
2
3
6 3
v1 v2 v3 v4 v5 v6
DISTANCE 0 5 ∞ 3 ∞ ∞
FINAL v1
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39. Data Structures and Algorithms
Solving the Shortest Path Problem (Contd.)
5 In the DISTANCE array, vertex
v4 has the shortest distance
from vertex v1.
3 4 6 Therefore, v4 is added to the
FINAL array.
2
3
6 3
v1 v2 v3 v4 v5 v6
DISTANCE 0 5 ∞ 3 ∞ ∞
FINAL v1 v4
Ver. 1.0 Session 17

40. Data Structures and Algorithms
Solving the Shortest Path Problem (Contd.)
5
v1 → v2 = 5
v1 → v4 → v2 = 3 + ∞ = ∞
3 4 6
∞>5
Therefore, no change is
2
3 made.
6 3
v1 v2 v3 v4 v5 v6
DISTANCE 0 5 ∞ 3 ∞ ∞
FINAL v1 v4
Ver. 1.0 Session 17

41. Data Structures and Algorithms
Solving the Shortest Path Problem (Contd.)
5
v1 → v3 = ∞
v1 → v4 → v3 = 3 + 2 = 5
3 4 6
5<∞
Therefore, the entry
2
3 corresponding to v3 in the
DISTANCE array is changed
6 3
to 5.
v1 v2 v3 v4 v5 v6
DISTANCE 0 5 ∞
5 3 ∞ ∞
FINAL v1 v4
Ver. 1.0 Session 17

42. Data Structures and Algorithms
Solving the Shortest Path Problem (Contd.)
5
v1 → v5 = ∞
v1 → v4 → v5 = 3 + 6 = 9
3 4 6
9<∞
Therefore, the entry
2
3 corresponding to v5 in the
DISTANCE array is changed
6 3
to 9.
v1 v2 v3 v4 v5 v6
DISTANCE 0 5 5 3 ∞
9 ∞
FINAL v1 v4
Ver. 1.0 Session 17

43. Data Structures and Algorithms
Solving the Shortest Path Problem (Contd.)
5
v1 → v6 = ∞
v1 → v4 → v6 = 3 + ∞ = ∞
3 4 6
Both the values are equal.
2
3 Therefore, no change is made.
6 3
PASS 1 complete
v1 v2 v3 v4 v5 v6
DISTANCE 0 5 5 3 9 ∞
FINAL v1 v4
Ver. 1.0 Session 17

44. Data Structures and Algorithms
Solving the Shortest Path Problem (Contd.)
5
From the DISTANCE array,
select the vertex with the
shortest distance from v1, such
that the selected vertex is not
3 4 6
in the FINAL array.
2
v2 and v3 have the shortest
3 and the same distance from v1.
6 3 Let us select v2 and add it to
the FINAL array.
v1 v2 v3 v4 v5 v6
DISTANCE 0 5 5 3 9 ∞
FINAL v1 v4 v2
Ver. 1.0 Session 17

45. Data Structures and Algorithms
Solving the Shortest Path Problem (Contd.)
5
v1 → v3 = 5
v1 → v2 → v3 = 5 + 4 = 9
3 4 6
9>5
Therefore, no change is
2
3 made.
6 3
v1 v2 v3 v4 v5 v6
DISTANCE 0 5 5 3 9 ∞
FINAL v1 v4 v2
Ver. 1.0 Session 17

46. Data Structures and Algorithms
Solving the Shortest Path Problem (Contd.)
5
v1 → v5 = 9
v1 → v2 → v5 = 5 + ∞ = ∞
3 4 6
∞>9
Therefore, no change is
2
3 made.
6 3
v1 v2 v3 v4 v5 v6
DISTANCE 0 5 5 3 9 ∞
FINAL v1 v4 v2
Ver. 1.0 Session 17

47. Data Structures and Algorithms
Solving the Shortest Path Problem (Contd.)
5
v1 → v6 = ∞
v1 → v2 → v6 = 5 + 6 = 11
3 4 6
11 < ∞
Therefore, the entry
2
3 corresponding to v6 in the
DISTANCE array is changed
6 3
to 11.
Pass 2 complete
v1 v2 v3 v4 v5 v6
DISTANCE 0 5 5 3 9 ∞
11
FINAL v1 v4 v2
Ver. 1.0 Session 17

48. Data Structures and Algorithms
Solving the Shortest Path Problem (Contd.)
5
From the DISTANCE array,
select the vertex with the
shortest distance from v1, such
that the selected vertex is not
3 4 6
in the FINAL array.
2 Let us select v3 and add it to
3
the FINAL array.
6 3
v1 v2 v3 v4 v5 v6
DISTANCE 0 5 5 3 9 11
FINAL v1 v4 v2 v3
Ver. 1.0 Session 17

49. Data Structures and Algorithms
Solving the Shortest Path Problem (Contd.)
5
v1 → v5 = 9
v1 → v3 → v5 = 5 + 3 = 8
3 4 6
8<9
Therefore, the entry
2
3 corresponding to v5 in the
DISTANCE array is changed
6 3
to 8.
v1 v2 v3 v4 v5 v6
DISTANCE 0 5 5 3 9
8 11
FINAL v1 v4 v2 v3
Ver. 1.0 Session 17

50. Data Structures and Algorithms
Solving the Shortest Path Problem (Contd.)
5
v1 → v6 = 11
v1 → v3 → v6 = 5 + 3 = 8
3 4 6
8 < 11
Therefore, the entry
2
3 corresponding to v6 in the
DISTANCE array is changed
6 3
to 8.
Pass 3 complete
v1 v2 v3 v4 v5 v6
DISTANCE 0 5 5 3 8 11
8
FINAL v1 v4 v2 v3
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51. Data Structures and Algorithms
Solving the Shortest Path Problem (Contd.)
5
From the DISTANCE array,
select the vertex with the
shortest distance from v1, such
that the selected vertex is not
3 4 6
in the FINAL array.
2 Let us select v5 and add it to
3
the FINAL array.
6 3
v1 v2 v3 v4 v5 v6
DISTANCE 0 5 5 3 8 8
FINAL v1 v4 v2 v3 v5
Ver. 1.0 Session 17

52. Data Structures and Algorithms
Solving the Shortest Path Problem (Contd.)
5
v1 → v6 = 8
v1 → v5 → v6 = 8 + ∞ = ∞
3 4 6
∞>8
Therefore, no change is
2
3 made.
6 3
Pass 4 complete
v1 v2 v3 v4 v5 v6
DISTANCE 0 5 5 3 8 8
FINAL v1 v4 v2 v3 v5
Ver. 1.0 Session 17

53. Data Structures and Algorithms
Solving the Shortest Path Problem (Contd.)
5
Now add the only remaining
vertex, v6 to the FINAL
array.
3 4 6 All vertices have been
added to the FINAL array.
2
This means that the
3
DISTANCE array now
6 3
contains the shortest
distances from vertex v1 to
all other vertices.
v1 v2 v3 v4 v5 v6
DISTANCE 0 5 5 3 8 8
FINAL v1 v4 v2 v3 v5 v6
Ver. 1.0 Session 17

54. Data Structures and Algorithms
Activity: Solving the Shortest Path Problem
Problem Statement:
In the previous activity, you created a program to represent a
set of cities and the distances between them in the form of a
graph. Extend the program to include the functionality for
finding the shortest path from a given city to all the other cities.
Ver. 1.0 Session 17

55. Data Structures and Algorithms
Summary
In this session, you learned that:
The two most commonly used ways of representing a graph
are as follows:
– Adjacency matrix
– Adjacency list
Traversing a graph means visiting all the vertices in the graph.
In a graph, there is no special vertex designated as the starting
vertex. Therefore, traversal of the graph may start from any
vertex.
You can traverse a graph with the help of the following two
methods:
DFS
BFS
Ver. 1.0 Session 17

56. Data Structures and Algorithms
Summary (Contd.)
Graph theory has been instrumental in analyzing and solving
problems in areas as diverse as computer network design,
urban planning, finding shortest paths and molecular biology.
Ver. 1.0 Session 17