Understanding the Context

What Is Dijkstra's Algorithm?

Dijkstra's algorithm is a greedy shortest-path algorithm that computes the shortest path from a single source node to all other nodes in a weighted, directed or undirected graph with non-negative edge weights. It guarantees the optimal (minimum cost) path, provided all edge weights are non-negative.

Key Insights

How Does Dijkstra's Algorithm Work?

While the full internal logic is algorithmically rich, here’s a high-level overview:

1. Initialization:
   Start by assigning a tentative distance value to each vertex—set the source node’s distance to zero, and all others to infinity. Keep track of visited nodes and maintain a priority queue (min-heap) sorting nodes by smallest tentative distance.

2. Visit the Closest Node:
   Extract the node with the smallest tentative distance from the priority queue.

3. Relaxation Step:
   For each neighboring node, check if going through the current node offers a shorter path. If so, update its distance.

Final Thoughts

4. Repeat:
   Continue this process until all nodes are visited or the target node is reached.

This process efficiently updates path costs using a greedy strategy: always expanding the closest unvisited node.

Key Features of Dijkstra’s Algorithm

• ✅ Optimal for non-negative weights: It guarantees the shortest path only when weights are ≥ 0.
  
• ✅ Efficient and scalable: With a min-heap/priority queue, runtime is typically O((V + E) log V), where V is the number of vertices and E is the number of edges.
  
• ✅ Versatile: Works on both directed and undirected graphs.
  
• ⚠️ Not suitable for graphs with negative weights: Algorithms like Bellman-Ford are needed in such cases.