Finding the shortest paths from a single source vertex to all other vertices in a graph with non-negative edge weights.
Dijkstra's algorithm is a greedy algorithm that finds the shortest paths from a single source vertex to all other vertices in a weighted graph, provided that the edge weights are non-negative. It works by maintaining a set of vertices whose shortest distance from the source is already known. Initially, this set is empty. The algorithm iteratively selects the vertex `u` not yet in the set that has the smallest known distance from the source. It then adds `u` to the set and updates the distances for all neighbors of `u`. This 'update' step, often called 'relaxation', checks if the path through `u` to a neighbor `v` is shorter than the currently known shortest path to `v`. The process continues until all vertices have been added to the set. To efficiently find the vertex with the minimum distance at each step, a min-priority queue is typically used. With a binary heap implementation of the priority queue, Dijkstra's algorithm has a time complexity of O(E log V), where V is the number of vertices and E is the number of edges. This algorithm is widely used in network routing protocols, GPS systems for finding the fastest route, and other applications where finding the shortest path in a non-negative weighted graph is required.