Explain the concept of the minimum spanning tree problem with weighted edges and how it can be solved using a greedy algorithm.

The minimum spanning tree (MST) problem is a well-known optimization problem in graph theory. Given a connected, undirected graph with weighted edges, the goal is to find a spanning tree with the minimum total weight.

A spanning tree of a graph is a subgraph that includes all the vertices of the original graph and forms a tree (i.e., it is acyclic and connected). The total weight of a spanning tree is the sum of the weights of its edges.

To solve the minimum spanning tree problem, a popular approach is to use a greedy algorithm called Kruskal's algorithm or Prim's algorithm.

Kruskal's algorithm:
1. Sort all the edges of the graph in non-decreasing order of their weights.
2. Initialize an empty set to store the edges of the MST.
3. Iterate through the sorted edges, starting from the smallest weight:
a. If adding the current edge to the MST does not create a cycle, add it to the MST set.
b. Otherwise, discard the edge.
4. Repeat step 3 until all vertices are included in the MST or there are no more edges left.

Prim's algorithm:
1. Choose an arbitrary vertex as the starting point.
2. Initialize an empty set to store the vertices of the MST and a priority queue to store the edges.
3. Add the starting vertex to the MST set.
4. Add all the edges connected to the starting vertex to the priority queue.
5. Repeat the following steps until all vertices are included in the MST:
a. Extract the edge with the minimum weight from the priority queue.
b. If the edge connects a vertex not yet in the MST set, add it to the MST set and add all its adjacent edges to the priority queue.
c. Discard the edge if it connects two vertices already in the MST set.
6. Return the MST set.

Both Kruskal's and Prim's algorithms guarantee the construction of a minimum spanning tree. Kruskal's algorithm is based on sorting the edges and checking for cycles, while Prim's algorithm is based on selecting the minimum weight edge at each step.

It is important to note that the choice of the starting vertex in Prim's algorithm can affect the resulting MST, but the overall weight will remain the same. Additionally, these algorithms have a time complexity of O(E log E) or O(E log V), depending on the implementation, where E is the number of edges and V is the number of vertices in the graph.