How does the Dijkstra's algorithm work for finding the shortest path in a weighted graph?

Dijkstra's algorithm is a greedy algorithm that is used to find the shortest path in a weighted graph. It works by maintaining a set of vertices for which the shortest path has already been determined. Initially, the distance to all vertices except the source vertex is set to infinity.

The algorithm starts by selecting the source vertex and setting its distance to 0. Then, it iteratively selects the vertex with the minimum distance from the set of vertices whose shortest path has not been determined yet. This vertex is added to the set of vertices for which the shortest path has been determined.

For each selected vertex, the algorithm updates the distances of its neighboring vertices. It calculates the distance from the source vertex to each neighboring vertex through the selected vertex and compares it with the current distance of the neighboring vertex. If the newly calculated distance is smaller, it updates the distance.

This process continues until all vertices have been added to the set of vertices for which the shortest path has been determined or until the destination vertex is reached. At the end of the algorithm, the shortest path from the source vertex to each vertex in the graph is determined.

To reconstruct the actual shortest path, the algorithm also maintains a predecessor array that keeps track of the previous vertex on the shortest path. By backtracking from the destination vertex to the source vertex using the predecessor array, the shortest path can be obtained.

Overall, Dijkstra's algorithm works by iteratively selecting the vertex with the minimum distance, updating the distances of its neighboring vertices, and keeping track of the shortest path using the predecessor array. This process ensures that the algorithm finds the shortest path from the source vertex to all other vertices in the weighted graph.