To solve the problem of finding the shortest path from vertex A to vertex L in this weighted graph, we can apply Dijkstra's algorithm. I'll walk through the process based on the image provided:

Steps for Dijkstra’s Algorithm:

1. Initialize Distances: Start at vertex A and set the distance to 0. Set the distance to all other vertices as infinity initially.
2. Update Neighbors: For each vertex, update the distances to its neighboring vertices by adding the weight of the connecting edge.
3. Choose the Smallest Distance: Move to the vertex with the smallest known distance and repeat the process until reaching vertex L.

Step-by-Step Application:

1. Start at A:

   • Current known distances:
     A = 0 (starting point),
     D = 19,
     B = 18.
     Other vertices are still infinity.

2. Move to B (smaller distance):

   • Update distances from B:
     E = 19 (B → E via edge 1),
     C = 24 (B → C via edge 6). Current known distances:
     A = 0,
     B = 18,
     D = 19,
     E = 19,
     C = 24.

3. Move to D (smallest known distance):

   • Update distances from D:
     G = 36 (D → G via edge 17).
     No shorter paths found for other vertices. Current known distances:
     A = 0,
     B = 18,
     D = 19,
     E = 19,
     C = 24,
     G = 36.

4. Move to E (next smallest):

   • Update distances from E:
     F = 33 (E → F via edge 14),
     H = 31 (E → H via edge 12).
     Current known distances:
     A = 0,
     B = 18,
     D = 19,
     E = 19,
     F = 33,
     C = 24,
     H = 31,
     G = 36.

5. Move to C:

   • Update distances from C:
     F = 33 (C → F via edge 16).
     No shorter paths found for other vertices.

6. Move to H:

   • Update distances from H:
     I = 61 (H → I via edge 30),
     K = 53 (H → K via edge 22).
     Current known distances:
     A = 0,
     B = 18,
     D = 19,
     E = 19,
     F = 33,
     C = 24,
     H = 31,
     K = 53,
     G = 36,
     I = 61.

7. Move to F:

   • Update distances from F:
     I = 57 (F → I via edge 24),
     L = 59 (F → L via edge 26).
     Current known distances:
     A = 0,
     B = 18,
     D = 19,
     E = 19,
     F = 33,
     C = 24,
     H = 31,
     K = 53,
     G = 36,
     I = 57,
     L = 59.

8. Move to L:
   You have reached the target vertex, L, with a shortest path of 59.

Shortest Path:

The shortest path from A to L is:
A → B → E → F → L.

Would you like more details on the steps, or any further clarification?

Related Questions:

1. How does Dijkstra's algorithm work in graphs with negative weights?
2. What is the difference between Dijkstra's and Bellman-Ford algorithms?
3. Can Dijkstra’s algorithm be used for unweighted graphs?
4. What are the time complexity considerations of Dijkstra's algorithm?
5. How does A* (A-star) algorithm compare to Dijkstra’s in pathfinding?

Tip:

When using Dijkstra’s algorithm, always prioritize vertices with the smallest known distance at each step to ensure you find the shortest path efficiently.