Greedy Algorithms Questions Medium
Prim's algorithm is a greedy algorithm used to find the minimum spanning tree (MST) of a connected 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. For each visited vertex, consider all its adjacent edges that connect to unvisited vertices.
b. Select the edge with the minimum weight among these edges.
c. Add the selected edge to the MST and mark the connected vertex as visited.
5. Once all vertices are visited, the MST is complete.
The algorithm continues to select the edge with the minimum weight at each step, ensuring that the MST is formed by connecting vertices with the minimum total weight. This process guarantees that the resulting tree is a spanning tree and has the minimum possible weight among all possible spanning trees of the graph.
It is important to note that Prim's algorithm assumes the graph is connected, meaning that there is a path between any two vertices. If the graph is not connected, the algorithm can be applied to each connected component separately to find the minimum spanning forest.