Introduction

Prim's Algorithm: Prim's Algorithm is a greedy algorithm used to find the Minimum Spanning Tree (MST) of a connected, weighted graph. It ensures that the total weight of the tree is minimized while connecting all vertices.
This algorithm starts with a single vertex and grows the MST by adding the smallest edge that connects a vertex in the tree to a vertex outside the tree.

How does it work?

• Start with an arbitrary vertex and mark it as part of the MST.
• Find the smallest edge that connects a vertex in the MST to a vertex outside the MST.
• Add the selected edge and the connected vertex to the MST.
• Repeat the process until all vertices are included in the MST.

Important Observations

• The graph must be connected for the algorithm to work.
• The algorithm guarantees the MST for weighted graphs.
• It uses a priority queue to efficiently select the smallest edge.

Key Characteristics:

• Greedy algorithm for MST construction.
• Efficient for dense graphs.
• Can handle both directed and undirected graphs.

Advantages:

• Accurate and reliable for MST construction.
• Works well for dense graphs.
• Time complexity is manageable for medium-sized graphs.

Disadvantages:

• Not suitable for disconnected graphs.
• Can be slow for very large graphs.

Time Complexity:

• Using a simple array: O(V²), where V is the number of vertices.
• Using a priority queue: O((V + E) log V), where E is the number of edges.