How does Dijkstra's algorithm ensure that the shortest path is found?

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Question: How does Dijkstra\'s algorithm ensure that the shortest path is found?

Options:

Correct Answer: By using a greedy approach

Solution:

Dijkstra\'s algorithm uses a greedy approach, always expanding the least costly node first, ensuring that the shortest path is found.

How does Dijkstra's algorithm ensure that the shortest path is found?

How does Dijkstra's algorithm ensure that the shortest path is found?

Step 1: Start with a graph that has nodes (points) and edges (connections between points).
Step 2: Choose a starting node and set its distance to 0 (since it's the starting point).
Step 3: Set the distance to all other nodes to infinity (unknown distance).
Step 4: Create a list of unvisited nodes that includes all nodes in the graph.
Step 5: While there are unvisited nodes, do the following:
Step 5a: Select the unvisited node with the smallest distance (this is the current node).
Step 5b: For each neighbor of the current node, calculate the distance from the start node to that neighbor through the current node.
Step 5c: If this new distance is less than the previously recorded distance for that neighbor, update the neighbor's distance.
Step 5d: Once all neighbors are checked, mark the current node as visited (it won't be checked again).
Step 6: Repeat Step 5 until all nodes are visited or the shortest path to the target node is found.
Step 7: The shortest path is determined by tracing back from the target node to the starting node using the recorded distances.

Greedy Algorithm – Dijkstra's algorithm employs a greedy strategy by selecting the node with the lowest cumulative cost at each step.
Graph Theory – The algorithm operates on weighted graphs, where edges have associated costs.
Optimal Substructure – Dijkstra's algorithm relies on the principle that the shortest path to a node can be constructed from the shortest paths to its predecessors.