FREE E LEARNING PLATFORM
☰ HOMEEXCEPTIONSOOPSJVMINTRO
SYLLABUS TUTORIALS FORUM
 

Prim's Algorithm



Introduction

Prim's Algorithm is a Greedy Algorithm used to construct the Minimum Spanning Tree (MST) of a connected and weighted undirected graph. The algorithm starts from any one vertex and repeatedly selects the edge having the minimum weight that connects a vertex already included in the tree with a vertex that has not yet been selected.

The selected edge is added only if it does not form a cycle. This process continues until all the vertices of the graph become connected. The resulting tree contains the minimum possible total edge weight and is called the Minimum Spanning Tree (MST).

Why Do We Need Prim's Algorithm?

Many real-world problems require connecting different locations with the minimum possible cost. Prim's Algorithm helps in designing such networks efficiently by selecting only those edges that contribute to the minimum overall cost.

Application Purpose
Computer Networks Connecting computers using minimum cable length.
Road Construction Connecting cities with minimum construction cost.
Electric Power Distribution Designing economical transmission networks.
Water Pipeline Networks Reducing pipeline installation cost.
Telecommunication Connecting communication towers efficiently.

Prerequisites

Concept Description
Graph A collection of vertices connected by edges.
Weighted Graph Each edge has an associated weight or cost.
Greedy Method Selects the best available choice at every step.
Minimum Spanning Tree A spanning tree having the minimum total edge weight.

Working Principle

Prim's Algorithm starts from any one vertex of the graph. At every iteration, it selects the smallest weighted edge that connects the already selected vertices with an unselected vertex. The selected edge is added to the Minimum Spanning Tree only if it does not create a cycle.

The process continues until all vertices become part of the Minimum Spanning Tree. Since the smallest available edge is selected at every step, Prim's Algorithm follows the Greedy Strategy.

Illustration


          A
         / \
      4 /   \2
       /     \
      B-------C
       \     /
      6 \   /3
         \ /
          D
          |
          |1
          E

Prim's Selection Order

A → C (2)

C → E (3)

E → D (1)

A → B (4)

Total Cost = 10

Figure 1 : Construction of Minimum Spanning Tree using Prim's Algorithm

Step-by-Step Example

Step Selected Edge Total Cost
1 A → C (2) 2
2 C → E (3) 5
3 E → D (1) 6
4 A → B (4) 10

Prim's Algorithm

Prim(Graph) Step 1 : Select any one vertex as the starting vertex. Step 2 : Find the minimum weighted edge connected to the selected vertices. Step 3 : Add the selected edge and the new vertex to the Minimum Spanning Tree. Step 4 : Ignore any edge that forms a cycle. Step 5 : Repeat Steps 2 to 4 until all vertices are included. Step 6 : Display the Minimum Spanning Tree.

Complexity Analysis

Implementation Time Complexity Remarks
Adjacency Matrix O(V²) Suitable for dense graphs.
Binary Heap + Adjacency List O(E log V) Efficient for sparse graphs.
Fibonacci Heap O(E + V log V) Most efficient theoretical implementation.
Space Complexity O(V) Stores visited vertices and parent information.

Advantages of Prim's Algorithm

  • Produces the Minimum Spanning Tree.
  • Simple and easy to implement.
  • Suitable for dense weighted graphs.
  • Ensures minimum total connection cost.
  • Widely used in real-world network design.

Limitations of Prim's Algorithm

  • Applicable only to connected weighted graphs.
  • Cannot be applied directly to disconnected graphs.
  • Performance depends upon the graph representation.
  • Requires a priority queue for efficient implementation.

Applications of Prim's Algorithm

Application Description
Computer Networks Connecting computers with minimum cable cost.
Road Networks Planning economical road construction.
Power Distribution Designing minimum-cost electrical transmission lines.
Water Pipeline Systems Connecting locations using minimum pipe length.
Telecommunication Building low-cost communication networks.

Prim's Algorithm vs Kruskal's Algorithm

Prim's Algorithm Kruskal's Algorithm
Starts from any vertex. Starts from the smallest edge.
Builds one tree throughout execution. Initially builds multiple small trees.
Best suited for dense graphs. Best suited for sparse graphs.
Uses a Priority Queue. Uses Sorting and Union-Find.
Time Complexity: O(E log V) Time Complexity: O(E log E)

Important Points

  • Prim's Algorithm is a Greedy Algorithm.
  • It is used to construct the Minimum Spanning Tree (MST).
  • At every step, the minimum-weight edge is selected.
  • No cycle is allowed in the MST.
  • The final MST always contains (V − 1) edges.
  • The graph must be connected.

Exam Tip

AKTU examinations frequently ask students to construct the Minimum Spanning Tree using Prim's Algorithm for a weighted graph. Practice selecting the minimum edge at every step without creating a cycle. Also remember the time complexity and the difference between Prim's and Kruskal's algorithms.

Summary

Prim's Algorithm is one of the most popular Greedy Algorithms for constructing a Minimum Spanning Tree. Starting from any vertex, it repeatedly selects the minimum-weight edge that connects a selected vertex to an unselected vertex until all vertices are connected. The algorithm guarantees a minimum-cost spanning tree for connected weighted graphs.







Copyright 2020 by Noidatut. All Rights Reserved.

Noidatut.com