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

Dijkstra's algorithm is a greedy algorithm that is used to find the shortest path between a source node and all other nodes in a weighted graph. It works by iteratively selecting the node with the smallest distance from the source and updating the distances of its neighboring nodes.

Here is a step-by-step explanation of how Dijkstra's algorithm works:

1. Initialize the algorithm by setting the distance of the source node to 0 and the distances of all other nodes to infinity. Mark all nodes as unvisited.

2. Select the node with the smallest distance from the source (initially, this will be the source node itself). This node is considered as the current node.

3. For each neighboring node of the current node, calculate the distance from the source through the current node. If this distance is smaller than the previously recorded distance for that node, update the distance.

4. Mark the current node as visited.

5. Repeat steps 2-4 until all nodes have been visited or the destination node has been reached.

6. Once all nodes have been visited or the destination node has been reached, the algorithm terminates. The shortest path from the source to each node can be obtained by backtracking from the destination node to the source node, following the nodes with the smallest distances.

Dijkstra's algorithm guarantees that the shortest path to each node is found as long as the graph does not contain negative weight edges. It is an efficient algorithm with a time complexity of O(V^2), where V is the number of nodes in the graph. However, with the use of a priority queue, the time complexity can be reduced to O((V + E) log V), where E is the number of edges in the graph.