Understanding Dijkstra's Algorithm for Shortest Path Finding

What is Dijkstra’s Algorithm?

Dijkstra’s Algorithm is a graph search algorithm that solves the shortest path problem for a graph with non-negative edge weights. It efficiently finds the shortest path from a starting node to all other nodes in the graph.

How Dijkstra’s Algorithm Works

1. Initialization: Start with a graph represented as a set of nodes (vertices) connected by edges with weights (distances).
2. Set Distances: Assign an initial distance of 0 to the starting node and infinity to all other nodes.
3. Visit Nodes: Use a priority queue to visit nodes in order of their distance, updating the distances of neighboring nodes.
4. Update Distances: For each neighboring node, if the calculated distance through the current node is less than the known distance, update it.
5. Repeat: Continue visiting nodes until all nodes have been processed.

Example Graph

Consider the following weighted graph:

    A
   / \
  1   4
 /     \
B---2---C
 \     /
  3   5
   \ /
    D

Graph Representation:

• Nodes: A, B, C, D
• Edges with weights:
  • A to B: 1
  • A to C: 4
  • B to C: 2
  • B to D: 3
  • C to D: 5

Step-by-Step Application of Dijkstra’s Algorithm

1. Initialization:

   • Start Node: A
   • Distances: A=0, B=∞, C=∞, D=∞
   • Priority Queue: [(0, A)]

2. Visit A:

   • Update distances for B and C:
     • B: min(∞, 0 + 1) = 1
     • C: min(∞, 0 + 4) = 4
   • New Distances: A=0, B=1, C=4, D=∞
   • Priority Queue: [(1, B), (4, C)]

3. Visit B:

   • Update distances for C and D:
     • C: min(4, 1 + 2) = 3
     • D: min(∞, 1 + 3) = 4
   • New Distances: A=0, B=1, C=3, D=4
   • Priority Queue: [(3, C), (4, D)]

4. Visit C:

   • Update distance for D:
     • D: min(4, 3 + 5) = 4 (no change)
   • New Distances: A=0, B=1, C=3, D=4
   • Priority Queue: [(4, D)]

5. Visit D:

   • All nodes visited.

Final Shortest Path Distances from A

• A to A: 0
• A to B: 1
• A to C: 3
• A to D: 4

Conclusion

Dijkstra’s Algorithm efficiently finds the shortest paths from a starting node to all other nodes in a graph. By following the steps outlined above, you can apply this algorithm to a variety of problems in graph theory, such as mapping routes or optimizing network paths.