Prim's Algorithm

01 Oct 2023

Prim’s Algorithm is an efficient algorithm used to find the minimum spanning tree (MST) of a connected weighted undirected graph. MST is a tree that spans all the vertices of the graph with the minimum total weight.

How does Prim’s Algorithm work?

The algorithm starts with an arbitrary vertex and gradually grows the MST by adding the edge with the minimum weight that connects a vertex already in the MST to a vertex outside the MST. This process continues until all vertices are included in the MST.

Here are the steps to implement Prim’s Algorithm:

Create a Min Heap to store the edges of the graph. Each edge is associated with a weight.
Create a boolean array to keep track of vertices included in the MST.
Initialize the first vertex as the starting point.
While there are vertices not yet included in the MST:
- Extract the minimum weight edge from the Min Heap.
- Include the extracted edge to the MST.
- Mark the vertex as included in the MST.
- Update the weights of adjacent vertices if they are less than the current weights.
- Insert the updated vertices to the Min Heap if they are not included in MST.

Pseudocode for Prim’s Algorithm:

Here is a simple implementation of Prim’s Algorithm using pseudocode:

initialize:
    - Create a Min Heap and insert the first vertex with weight 0.
    - Create a boolean array to keep track of included vertices.
    
while Min Heap is not empty:
    - Extract the edge with the minimum weight from the Min Heap.
    - If the extracted edge connects a vertex not yet included in the MST:
        - Add the edge to the MST.
        - Mark the vertex as included.
        - Update the weights of adjacent vertices if they are less than the current weights.
        - Insert the updated vertices to the Min Heap if they are not included.

Complexity Analysis:

The time complexity of Prim’s Algorithm is O(V^2) when implemented using an adjacency matrix, where V is the number of vertices in the graph. However, with the help of a binary heap data structure, the time complexity can be reduced to O(E log V), where E is the number of edges in the graph.

Conclusion:

Prim’s Algorithm is a powerful technique to find the minimum spanning tree of a connected weighted graph. It ensures that all vertices are connected with the minimum total weight possible. By understanding and implementing this algorithm, you can efficiently solve problems related to network layout planning, electrical power grid design, and more.

#graphtheory #algorithms