Prim's algorithm is a popular greedy algorithm used to find a minimum spanning tree (MST) in a weighted undirected graph. It starts with an arbitrary vertex and repeatedly grows the MST by adding the lightest edge that connects a vertex in the MST to a vertex outside the MST..

Here is the step-by-step explanation of Prim's algorithm:

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.

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.

5. Return the MST.

Here is a pseudocode representation of Prim's algorithm:

Prim's Algorithm (G):
    Initialize an empty MST
    Initialize a set of visited vertices

    Choose an arbitrary starting vertex 'v'
    Mark 'v' as visited

    Repeat until all vertices are visited:
        Find the minimum-weight edge (u, v) such that u is visited and v is not visited
        Add edge (u, v) to the MST
        Mark 'v' as visited

    Return the MST

The time complexity of Prim's algorithm using an adjacency matrix representation is O(V^2), where V is the number of vertices in the graph. However, by using a more efficient data structure like a binary heap or Fibonacci heap, the time complexity can be improved to O(E log V), where E is the number of edges in the graph.

Note that Prim's algorithm assumes the input graph is connected, meaning there is a path between any pair of vertices. If the graph is disconnected, you may need to modify the algorithm to handle multiple connected components.

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