Dijkstra's algorithm is a fundamental graph algorithm used to find the shortest path from a source node to all other nodes in a weighted graph. It works efficiently with graphs that have non-negative edge weights. The algorithm is widely used in network routing, GPS navigation, and many other applications where finding optimal paths is crucial.
The algorithm starts by initializing distances. We set the distance to the source node A to zero, and all other nodes to infinity. All nodes are initially marked as unvisited. The main loop then repeatedly selects the unvisited node with the minimum distance, marks it as visited, and updates the distances to its neighbors if a shorter path is found.
Let's trace through the algorithm execution. We start at node A with distance zero. In the first iteration, we update the distances to A's neighbors: B gets distance 4 and D gets distance 2. Next, we select the unvisited node with minimum distance, which is D with distance 2. From D, we can update the distance to E, giving it a distance of 5.
After completing all iterations, we have found the shortest distances from source node A to all other nodes. The final distances are: A to A is 0, A to B is 4, A to C is 7, A to D is 2, A to E is 5, and A to F is 7. The red edges show the shortest path tree, which represents the optimal paths from A to each destination node.
To summarize what we've learned about Dijkstra's algorithm: It efficiently finds shortest paths from a single source to all other nodes. The algorithm only works with non-negative weights and uses a greedy approach that guarantees optimal solutions. With a time complexity of O of V plus E log V, it's widely used in real-world applications like GPS navigation and network routing protocols.