Explain the concept of relaxation in the Dijkstra Algorithm.

In the Dijkstra Algorithm, relaxation is a fundamental concept that is used to update the distance values of vertices in order to find the shortest path from a source vertex to all other vertices in a weighted graph.

The algorithm maintains a set of vertices for which the shortest path has already been determined, and a set of vertices for which the shortest path is yet to be determined. Initially, the distance value of the source vertex is set to 0, and the distance values of all other vertices are set to infinity.

During each iteration of the algorithm, the vertex with the minimum distance value from the set of vertices yet to be determined is selected. This vertex is then marked as visited and its distance value is considered as the shortest path to that vertex.

Relaxation is the process of updating the distance values of the neighboring vertices of the currently selected vertex. For each neighboring vertex, the algorithm checks if the distance value can be improved by considering the current vertex as an intermediate vertex. If the distance value can be reduced, it means that a shorter path to that neighboring vertex has been found.

To perform relaxation, the algorithm compares the current distance value of the neighboring vertex with the sum of the distance value of the current vertex and the weight of the edge connecting them. If the sum is smaller than the current distance value, the distance value is updated to the new smaller value. Additionally, the algorithm also updates the predecessor of the neighboring vertex to be the current vertex.

This process of relaxation is repeated for all the neighboring vertices of the currently selected vertex. By continuously performing relaxation, the algorithm gradually determines the shortest path from the source vertex to all other vertices in the graph.

The relaxation process ensures that the distance values of the vertices are always updated to the shortest possible values. It guarantees that the algorithm explores all possible paths and eventually finds the shortest path from the source vertex to all other vertices in the graph.