How does the Prim's algorithm work for finding the minimum spanning tree of a weighted graph?

Prim's algorithm is a greedy algorithm used to find the minimum spanning tree (MST) of a weighted graph. The algorithm starts with an arbitrary vertex and gradually expands the MST by adding the edge with the minimum weight that connects a vertex in the MST to a vertex outside the MST. This process continues until all vertices are included in the MST.

Here is a step-by-step explanation of how Prim's algorithm works:

1. Initialize an empty MST and a set of visited vertices.
2. Choose an arbitrary vertex to start the algorithm.
3. Mark the chosen vertex as visited and add it to the MST.
4. Repeat the following steps until all vertices are visited:
a. Find the minimum weight edge that connects a visited vertex to an unvisited vertex.
b. Add the found edge to the MST.
c. Mark the unvisited vertex as visited and add it to the MST.
5. Once all vertices are visited, the MST is complete.

The key idea behind Prim's algorithm is to always select the edge with the minimum weight that connects the visited vertices to the unvisited vertices. This ensures that the MST is built by gradually adding the edges with the lowest weights, resulting in the minimum overall weight for the spanning tree.

The time complexity of Prim's algorithm is O(V^2) for an adjacency matrix representation of the graph, where V is the number of vertices. However, by using a priority queue to efficiently select the minimum weight edge, the time complexity can be reduced to O(E log V), where E is the number of edges.

In summary, Prim's algorithm works by greedily selecting the minimum weight edges to gradually build the minimum spanning tree of a weighted graph.