Prim’s algorithm is also a Greedy algorithm. It starts with an empty spanning tree. The idea is to maintain two sets of vertices. The first set contains the vertices already included in the MST, the other set contains the vertices not yet included. At every step, it considers all the edges that connect the two sets and picks the minimum weight edge from these edges. After picking the edge, it moves the other endpoint of the edge to the set containing MST. A group of edges that connects two sets of vertices in a graph is called cut in graph theory. So, at every step of Prim’s algorithm, we find a cut (of two sets, one contains the vertices already included in MST and the other contains the rest of the vertices), pick the minimum weight edge from the cut, and include this vertex to MST Set (the set that contains already included vertices). Algorithm 1) Create a set mstSet that keeps track of vertices already included in MST. 2) Assign a key value to all vertices in the input graph. Initialize all key values as INFINITE. Assign key value as 0 for the first vertex so that it is picked first. 3) While mstSet doesn’t include all vertices ….a) Pick a vertex u which is not there in mstSet and has minimum key value. ….b) Include u to mstSet. ….c) Update the key value of all adjacent vertices of u. To update the key values, iterate through all adjacent vertices. For every adjacent vertex v, if the weight of edge u-v is less than the previous key value of v, update the key value as the weight of u-v The idea of using key values is to pick the minimum weight edge from the cut. The key values are used only for vertices that are not yet included in MST, the key value for these vertices indicates the minimum weight edges connecting them to the set of vertices included in MST.

1. Minimum
Spanning Tree
Using
Prims’s Algorithm

2. Design & Analysis
of Algorithm
Greedy Technique
Algorithm Design
Technique :
Problem :
Prim's Algorithm
Name: Mrunal Sanjay Patil
Roll No: 64
Year: FYMCA

3.  Greedy Method
• Greedy Method finds out of many options, but you have to choose
the best option.
• In this method we focused on the first stage and decide the output.

4.  Minimum Spanning Tree
A connected subgraph ‘M’ of graph G(V,E) is said to be spanning
Iff,
 ‘M’ should contain all vertices of ‘G’.
 ‘M’ should contain (|V|-1) edges.
𝑉1
𝑉2 𝑉3
𝑉4
𝐸1
𝐸2 𝐸4
𝐸3

5. Connected
Graph
Spanning Trees
𝑉1
𝑉2 𝑉3
𝑉4
𝐸1
𝐸2 𝐸4
𝐸3

6.  Minimum Cost Spanning Tree
• The cost of the spanning tree is the sum of the weights of all
the edges in the tree.
• Minimum spanning tree is the spanning tree where the cost is
minimum among all the spanning trees.
Undirected
Graph
4
1
3 2
5
A B
C D
Spanning Tree
Cost = 4+5+2
= 11
4
1
3 2
5
A B
C D
Minimum Cost Spanning Tree
Cost = 4+1+2
= 7
1
4 3 2
5
A B
C D

7.  We can find the minimum cost spanning
tree using 2 methods:
Prim’s Algorithms
Kruskal’s Algorithms

8.  Prim’s Algorithms
• Prim’s algorithm is a famous greedy algorithm.
• It is used for finding the Minimum Spanning Tree of a given graph.
• To apply prims algorithm, the given graph must be weighted,
connected and undirected.

9.  Explanation :
B
A
C
G
F
H
I
D
1
E
5 10
6
7
6
3
2
8
12
3
7
8

10. B
A D
1
5
C
F
3
2
E
7
H
8
G
7
I
3
Minimum Cost Spanning Tree

11. = Sum of all edge weight
 Cost of Minimum Spanning Tree
= 1 + 5 + 2 + 3 + 7 + 8 + 7 + 3
= 36

13. Program

14. If the input graph is presented using adjacency list , then the
time complexity of Prim’s algorithm can be reduced to O(E log V)
with the help of binary heap.
Time Complexity of the above program is
O( )
𝑉2

15. Thank You…