Greedy Algorithms Questions Medium
Prim's algorithm is a greedy algorithm used to find a minimum spanning tree (MST) in a weighted undirected graph. The MST is a tree that connects all the vertices of the graph with the minimum total weight.
The algorithm starts by selecting an arbitrary vertex as the starting point. It then repeatedly adds 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 Prim's algorithm:
1. Initialize an empty MST and a set of visited vertices.
2. Choose an arbitrary vertex as the starting point and add it to the MST.
3. Mark the chosen vertex as visited.
4. Repeat the following steps until all vertices are visited:
a. For each visited vertex, consider all its adjacent vertices that are not yet visited.
b. Select the edge with the minimum weight among all the edges connecting the visited vertex to an unvisited vertex.
c. Add the selected edge and the unvisited vertex to the MST.
d. Mark the unvisited vertex as visited.
5. Once all vertices are visited, the MST is complete.
Prim's algorithm guarantees that the resulting tree will be a minimum spanning tree. This is because at each step, it selects the edge with the minimum weight, ensuring that the total weight of the MST is minimized.
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 conclusion, Prim's algorithm is a greedy approach that efficiently finds a minimum spanning tree by iteratively selecting the edge with the minimum weight.