To solve this problem using Dijkstra's Algorithm, we can approach it step by step. Since the graph has labeled vertices and edge weights, we need to find the shortest path from one vertex to another by exploring all possible paths.

Part 1: Order in which vertices become "current"

Dijkstra’s algorithm starts by initializing distances from the source vertex and then picking the smallest distance vertex, marking it as "current," and exploring its neighbors.

We can start with A or B as the source vertex. Let’s begin by picking A and applying Dijkstra’s Algorithm.

Part 2: Shortest Path from A to B

We will trace the shortest path from vertex A to vertex B, following the algorithm’s logic for selecting the minimum edge weights.

Let me analyze this visually and compute the path for you.### Dijkstra's Algorithm: Step-by-Step Application

Let's assume we are starting at vertex A and applying Dijkstra's Algorithm to find both the order in which the vertices become "current" and the shortest path from A to B.

Step-by-Step Execution:

1. Start at A (Initial node: A, Distance = 0)

   • Neighbor C: Distance = 8 (A → C)
   • Neighbor H: Distance = 5 (A → H)
   • Mark A as "current" (visited).

2. Move to H (H has the smallest distance, 5)

   • Neighbor F: Distance = 5 + 2 = 7 (A → H → F)
   • Mark H as "current".

3. Move to F (F now has the smallest distance, 7)

   • Neighbor G: Distance = 7 + 4 = 11 (A → H → F → G)
   • Neighbor D: Distance = 7 + 2 = 9 (A → H → F → D)
   • Mark F as "current".

4. Move to C (C has the next smallest distance, 8)

   • Neighbor D: Distance = 8 + 2 = 10 (This is higher than the previous calculated distance to D, so no update)
   • Mark C as "current".

5. Move to D (D has the next smallest distance, 9)

   • Neighbor G: Distance = 9 + 3 = 12 (This is higher than the previous calculated distance to G, so no update)
   • Neighbor E: Distance = 9 + 2 = 11 (A → H → F → D → E)
   • Mark D as "current".

6. Move to G (G has the next smallest distance, 11)

   • Neighbor E: Distance = 11 + 5 = 16 (Higher than the already calculated distance to E, so no update)
   • Mark G as "current".

7. Move to E (E has the next smallest distance, 11)

   • Neighbor B: Distance = 11 + 1 = 12 (A → H → F → D → E → B)
   • Mark E as "current".

8. Finally, move to B (B now has the smallest distance, 12)

   • Mark B as "current".

Answer to the Questions:

1. Order of "current" vertices:
   The vertices are visited in the following order: A H F C D G E B.

2. Shortest Path from A to B:
   The shortest path from A to B is: A → H → F → D → E → B with a total distance of 12.

Would you like any more details or explanations?

Here are five follow-up questions that can further your understanding:

1. How does Dijkstra's Algorithm ensure that we find the shortest path?
2. Can Dijkstra's Algorithm be applied to graphs with negative edge weights?
3. How would the path differ if we started the algorithm from vertex B instead of A?
4. What are the time complexity implications of Dijkstra's Algorithm with different data structures (e.g., priority queues)?
5. How does Dijkstra's Algorithm differ from other shortest-path algorithms, such as Bellman-Ford?

Tip: When implementing Dijkstra's Algorithm, using a priority queue or min-heap can significantly optimize performance.