Dijkstra's algorithm is a fundamental graph algorithm that solves the shortest path problem. Given a weighted graph with non-negative edge weights, it finds the shortest distance from a source node to all other nodes. In our example, we want to find the shortest paths from node A to all other nodes B, C, D, and E.
The algorithm starts with initialization. We set the distance to the source node A as zero, and all other distances as infinity. All nodes begin as unvisited. Then we select the unvisited node with the smallest distance, which is initially node A with distance zero, and mark it as visited.
Now we perform edge relaxation. For the current node A, we examine all its neighbors B and D. We calculate the distance to B through A as zero plus four equals four, which is less than infinity, so we update B's distance to four. Similarly, the distance to D through A is zero plus two equals two, so we update D's distance to two.
We continue the algorithm by selecting the unvisited node with minimum distance. Next is D with distance 2. From D, we can reach E with total distance 3, updating E. Then we select E with distance 3, reaching C with total distance 5. Finally, we select B with distance 4, but its path to C gives distance 7, which is greater than 5, so no update occurs. The algorithm completes with all shortest distances found.
To summarize what we've learned about Dijkstra's algorithm: It efficiently finds shortest paths from a source node to all other nodes using a greedy approach. The algorithm repeatedly selects the unvisited node with minimum distance and performs edge relaxation to update distances. It has excellent time complexity and is widely used in network routing and GPS navigation systems.