The document defines and explains several key graph concepts and algorithms, including: - Graph representations like adjacency matrix and adjacency list. - Graph traversal algorithms like breadth-first search (BFS) and depth-first search (DFS). - Minimum spanning tree algorithms like Prim's algorithm. - Single-source shortest path algorithms like Dijkstra's algorithm and Floyd's algorithm. Pseudocode and examples are provided to illustrate how BFS, DFS, Prim's, Dijkstra's and Floyd's algorithms work on graphs. Key properties of minimum spanning trees and shortest path problems are also defined.

1. Graph
Prepared By : Ketaki Pattani
Enroll. No. : 130210107039
1
4
76
3 5
2
1 2
3

2. Some related terms:
• Graph: A graph G is a pair (V,E), where V is a finite
set of points called vertices and E is a finite set of
edges.
• Path: A path from a vertex v to a vertex u is a
sequence <v0,v1,v2,…,vk> of vertices where v0 ,v1,
v2… are the set of vertices followed to move from a
source to destination.
• Pathlength:The length of a path is defined as the
number of edges in the path.

3. Related terms: Degree of vertex
• Degree of vertex: It represents
the total number of edges linked
to a particular vertex.
• Indegree: It is the number of
incomming edges to a vertex.
•Outdegree: It is the number of
outgoing edges from a vertex.
•Source: A vertex having indegree
zero is a source.
•Sink: A vertex having outdegree
zero is a sink.

4. Some related terms:
• Undirected Graph: An
undirected graph is the
graph consisting of
unordered pair of vertices .
For eg :figure (a)
• Directed Graph : In a
directed graph, the edge e is
an ordered pair (u,v) which
is incident from vertex u and
is incident to vertex v. For
eg : fig(b)

5. Some related terms: Weighted graph
• A weighted graph is a graph
where each edge is assigned
a particular value.
• This may be the distance or
the cost.
• Here, the graph G(V,E) may
be directed or undirected
graph.
• Also the weight may be
represented by ‘W’.

6. Graph Representation:
Adjacency Matrix:
• Here, a two dimensional
square matrix can be used
to store a graph.
• Adjacency matrix for
undirected graph is always
symmetric .
Adjacency List
• Adjacency List creates a
seperste linked list for each
adjacent vertex.
• Adjacency graph for n nodes
is represented by an Array
of Pointers.
a
dc
b a
b
c
d
b
a
d
d c
c
a b
a c
a
dc
b
1 2
3 4
1 2 3 4
1 0 1 1 1
2 1 0 1 0
3 1 1 0 1
4 1 0 1 0

7. Graph Traversal:
• Most of the applications of graphs require the
traversal of graphs.
• Traversal means visiting each node in a graph
exactly once.
• It also stands for that there must be no cycles in
the traversal.
• The two commonly used techniques are:
(1) Depth First Search(DFS)
(2) Breadth First Search(BFS)

8. Breadth First Search:
• Strategy
 choose a starting vertex,
distance d = 0
 vertices are visited in order
of increasing distance from
the starting vertex,
 examine all edges leading
from vertices (at distance
d) to adjacent vertices (at
distance d+1)
 then, examine all edges
leading from vertices at
distance d+1 to distance
d+2, and so on,
 until no new vertex is
discovered

9. Algorithm for BFS:
1. for each vertex u in V[G] – {s}
2 do color[u]  white
3 d[u]  
4 [u]  nil
5 color[s]  gray
6 d[s]  0
7 [s]  nil
8 Q  
9 enqueue(Q,s)
10 while Q  
11 do u  dequeue(Q)
12 for each v in Adj[u]
13 do if color[v] = white
14 then color[v]  gray
15 d[v]  d[u] + 1
16 [v]  u
17 enqueue(Q,v)
18 color[u]  black

10. Example (BFS)
 0
  
 

r s t u
v w x y
Q: s
0
(Courtesy of Prof. Jim Anderson)

11. Example (BFS)
1 0
1  
 

r s t u
v w x y
Q: w r
1 1

12. Example (BFS)
1 0
1 2 
2 

r s t u
v w x y
Q: r t x
1 2 2

13. Example (BFS)
1 0
1 2 
2 
2
r s t u
v w x y
Q: t x v
2 2 2

14. Example (BFS)
1 0
1 2 
2 3
2
r s t u
v w x y
Q: x v u
2 2 3

15. Example (BFS)
1 0
1 2 3
2 3
2
r s t u
v w x y
Q: v u y
2 3 3

16. Example (BFS)
1 0
1 2 3
2 3
2
r s t u
v w x y
Q: u y
3 3

17. Example (BFS)
1 0
1 2 3
2 3
2
r s t u
v w x y
Q: y
3

18. Example (BFS)
1 0
1 2 3
2 3
2
r s t u
v w x y
Q: 

19. Use of a queue in BFS Graph:
• It is very common to use a queue to keep
track of:
– nodes to be visited next, or
– nodes that we have already visited.
• Typically, use of a queue leads to a breadth-
first visit order.
• Breadth-first visit order is “cautious” in the
sense that it examines every path of length i
before going on to paths of length i+1.

20. Depth First Search:
• Strategy
– choose a starting vertex,
distance d = 0
– examine One edges leading
from vertices (at distance d)
to adjacent vertices (at
distance d+1)
– then, examine One edges
leading from vertices at
distance d+1 to distance d+2,
and so on,
– until no new vertex is
discovered, or dead end
– then, backtrack one distance
back up, and try other edges,
and so on.

21. Algorithm for DFS:
.

22. Example (DFS)
1/
u v w
x y z

23. Example (DFS)
1/ 2/
u v w
x y z

24. Example (DFS)
1/
3/
2/
u v w
x y z

25. Example (DFS)
1/
4/ 3/
2/
u v w
x y z

26. Example (DFS)
1/
4/ 3/
2/
u v w
x y z
B

27. Example (DFS)
1/
4/5 3/
2/
u v w
x y z
B

28. Example (DFS)
1/
4/5 3/6
2/
u v w
x y z
B

29. Example (DFS)
1/
4/5 3/6
2/7
u v w
x y z
B

30. Example (DFS)
1/
4/5 3/6
2/7
u v w
x y z
BF

31. Example (DFS)
1/8
4/5 3/6
2/7
u v w
x y z
BF

32. Example (DFS)
1/8
4/5 3/6
2/7 9/
u v w
x y z
BF

33. Example (DFS)
1/8
4/5 3/6
2/7 9/
u v w
x y z
BF C

34. Example (DFS)
1/8
4/5 3/6 10/
2/7 9/
u v w
x y z
BF C

35. Example (DFS)
1/8
4/5 3/6 10/
2/7 9/
u v w
x y z
BF C
B

36. Example (DFS)
1/8
4/5 3/6 10/11
2/7 9/
u v w
x y z
BF C
B

37. Example (DFS)
1/8
4/5 3/6 10/11
2/7 9/12
u v w
x y z
BF C
B

38. Use of a stack in DFS Graph
• It is very common to use a stack to keep track
of:
– nodes to be visited next, or
– nodes that we have already visited.
• Typically, use of a stack leads to a depth-first
visit order.
• Depth-first visit order is “aggressive” in the
sense that it examines complete paths.

39. 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
A minimum spanning tree is a subgraph of an
undirected weighted graph G as shown, such that

40. Minimum Spanning Tree: Prim's
Algorithm
• Prim's algorithm for finding an MST is a greedy
algorithm.
• Start by selecting an arbitrary vertex, include
it into the current MST.
• Grow the current MST by inserting into it the
vertex closest to one of the vertices already in
current MST.

41. Minimum Spanning Tree: Prim's Algorithm
.

42. Minimum Spanning Tree: Prim's
Algorithm
.

43. Dijkstra's algorithm
• For a weighted graph G = (V,E,w), the single-
source shortest paths problem is to find the
shortest paths from a vertex v ∈ V to all other
vertices in V.
• Dijkstra's algorithm is similar to Prim's algorithm.
– maintains a set of nodes for which the shortest paths
are known.
– grow this set based on the node closest to source
using one of the nodes in the current shortest path
set.

44. Single-Source Shortest Paths: Dijkstra's
Algorithm
.

45. Floyd's Algorithm
• For any pair of vertices vi, vj ∈ V, consider all paths
from vi to vj whose intermediate vertices belong to
the set {v1,v2,…,vk}.
• Let pi
(
,
k
j
) (of weight di
(
,
k
j
)) be the minimum-weight
path among them.
• If vertex vk is not in the shortest path from vi to vj,
then pi
(
,
k
j
) is the same as pi
(
,
k
j
-1).
• If f vk is in pi
(
,
k
j
), then we can break pi
(
,
k
j
) into two
paths - one from vi to vk and one from vk to vj . Each
of these paths uses vertices from {v1,v2,…,vk-1}.

46. Floyd's Algorithm
From our observations, the following recurrence
relation follows:
This equation must be computed for each pair of nodes and
for k = 1, n. The serial complexity is O(n3).

47. Floyd's Algorithm
Floyd's all-pairs shortest paths algorithm.
This program computes the all-pairs shortest paths
of the graph G = (V,E) with adjacency matrix A.