Graph Theory Questions Medium
Prim's algorithm is a greedy algorithm used to find the minimum spanning tree in a graph. It starts with an arbitrary vertex and repeatedly adds the edge with the minimum weight that connects a vertex in the current tree to a vertex outside the tree. The algorithm continues until all vertices are included in the minimum spanning tree.
Here is a step-by-step explanation of Prim's algorithm:
1. Initialize an empty set called the minimum spanning tree (MST) and a priority queue to store the edges.
2. Choose an arbitrary vertex to start the algorithm.
3. Add all the edges connected to the chosen vertex to the priority queue.
4. While the priority queue is not empty, do the following steps:
a. Remove the edge with the minimum weight from the priority queue.
b. If the removed edge connects a vertex that is already in the MST to a vertex outside the MST, ignore the edge.
c. Otherwise, add the edge to the MST and add all the edges connected to the newly added vertex to the priority queue.
5. Repeat steps 4 until all vertices are included in the MST.
6. The resulting MST is the minimum spanning tree of the graph.
Prim's algorithm guarantees that the resulting tree will have the minimum total weight among all possible spanning trees in the graph. The time complexity of Prim's algorithm is O(E log V), where E is the number of edges and V is the number of vertices in the graph.