https://file.aiquickdraw.com/mathbot/file/1728766649718zmsmpprr.jpg

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.

Find the shortest path from vertex A to vertex L, using a sequence of vertices.

Learn how to find the shortest path from vertex A to vertex L in a weighted graph using Dijkstra's algorithm. This example demonstrates key graph theory concepts, including updating distances and prioritizing the shortest known distances between vertices.

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