Prims Algorithm

Learn Prims Algorithm for finding the Minimum Spanning Tree (MST) in connected, weighted graphs using the Greedy approach. Understand its steps, time complexity, advantages, limitations, and comparison with Kruskal’s Algorithm.

Prims Algorithm (for Minimum Spanning Tree)

Introduction

Prims Algorithm is a well-known greedy algorithm used to find the Minimum Spanning Tree (MST) of a connected, undirected, and weighted graph. It starts from a single vertex and gradually builds the MST by continuously adding edges with the smallest weights. Consequently, the algorithm ensures that each step contributes to the optimal solution.

Basic Idea

The main principle of Prim’s Algorithm is straightforward:
At each step, select the edge with the lowest weight that connects an existing vertex in the MST to a new vertex.

Key Points:

• Every decision is locally optimal, following the greedy choice principle.
• Previously selected edges remain unchanged.

Steps of Prim’s Algorithm

Prim’s Algorithm follows these steps:

1. Select any vertex as the starting vertex.
2. Include that vertex in the MST.
3. Among all edges of the vertices currently in the MST, select the edge with the lowest weight.
4. If the selected edge does not create a cycle, add it to the MST.
5. Include the new vertex in the MST.
6. Repeat steps 3–5 until all vertices are included in the MST.

Algorithm (Pseudo Code)

Prim(G, V):

Select any starting vertex

Initialize MST = {}

While MST does not contain all vertices:

Select the minimum weight edge (u, v) such that u is in MST and v is not

Add edge (u, v) to MST

Return MST

Example

Consider a graph with vertices A, B, C, D:

• Starting vertex: A
• Least weight edge: A–B
• Next least weight edge: B–C
• Next least weight edge: C–D

Finally, all vertices are connected, forming a Minimum Spanning Tree.

Time Complexity

• Using an adjacency matrix → O(n²)
• Using a priority queue (min-heap) → O(E log V)

Space Complexity

• Requires additional arrays and a priority queue.
• Space Complexity: O(V + E)

Advantages:

• Simple and easy to understand.
• Performs well for dense graphs.
• Guarantees an MST.
• Efficient due to the greedy approach.

Limitations:

• Applicable only to connected graphs.
• Less efficient than Kruskal’s Algorithm for sparse graphs.
• Priority queue implementation can be slightly complex.

Prim vs. Kruskal (Comparison)

Conclusion

Prims Algorithm efficiently connects all vertices at minimal cost. By making locally optimal choices at each step, it ensures a globally optimal Minimum Spanning Tree. Since it performs particularly well in dense graphs, it is widely used in network design, optimization, and graph-based applications.

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