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To find the shortest path from vertex \( A \) to \( E \) using Dijkstra's algorithm, we'll follow these steps:

### Step-by-Step Solution Using Dijkstra's Algorithm
1. **Initialization**:
   - Set the distance of the starting vertex \( A \) to 0.
   - Set the distance of all other vertices to infinity (∞).
   - Keep a record of visited and unvisited nodes.

2. **Graph Data**:
   The edges and their weights are as follows:
   - \( A \rightarrow C = 2 \)
   - \( A \rightarrow D = 5 \)
   - \( A \rightarrow B = 3 \)
   - \( B \rightarrow D = 1 \)
   - \( C \rightarrow D = 2 \)
   - \( C \rightarrow F = 1 \)
   - \( D \rightarrow E = 3 \)
   - \( E \rightarrow F = 2 \)

3. **Algorithm Execution**:
   - Start with \( A \) (distance = 0).
   - Update distances to all adjacent nodes from \( A \).

#### Iterations
1. **Starting at \( A \)**:
   - Distance to \( C \): \( 0 + 2 = 2 \)
   - Distance to \( D \): \( 0 + 5 = 5 \)
   - Distance to \( B \): \( 0 + 3 = 3 \)
   - Current distances: 
     - \( A = 0 \), \( B = 3 \), \( C = 2 \), \( D = 5 \), \( E = \infty \), \( F = \infty \)

2. **Next, visit \( C \) (smallest unvisited)**:
   - Distance to \( D \): \( 2 + 2 = 4 \) (better than previous 5, so update)
   - Distance to \( F \): \( 2 + 1 = 3 \)
   - Current distances:
     - \( A = 0 \), \( B = 3 \), \( C = 2 \), \( D = 4 \), \( E = \infty \), \( F = 3 \)

3. **Next, visit \( B \)**:
   - Distance to \( D \): \( 3 + 1 = 4 \) (no update needed, as \( D \) already has 4)
   - Current distances remain:
     - \( A = 0 \), \( B = 3 \), \( C = 2 \), \( D = 4 \), \( E = \infty \), \( F = 3 \)

4. **Next, visit \( F \)**:
   - Distance to \( E \): \( 3 + 2 = 5 \)
   - Current distances:
     - \( A = 0 \), \( B = 3 \), \( C = 2 \), \( D = 4 \), \( E = 5 \), \( F = 3 \)

5. **Next, visit \( D \)**:
   - Distance to \( E \): \( 4 + 3 = 7 \) (no update needed, as \( E \) already has 5)
   - Final distances:
     - \( A = 0 \), \( B = 3 \), \( C = 2 \), \( D = 4 \), \( E = 5 \), \( F = 3 \)

6. **Final Step**:
   - The shortest path from \( A \) to \( E \) has a total cost of 5.
   - The path is \( A \rightarrow C \rightarrow F \rightarrow E \).

### Answer
The shortest path from \( A \) to \( E \) is \( A \rightarrow C \rightarrow F \rightarrow E \) with a total distance of **5**.

Would you like more details or have any questions?

### Related Questions
1. How does Dijkstra's algorithm handle negative weights?
2. Can Dijkstra's algorithm be used for directed graphs?
3. What would change if we started from a different vertex, such as \( B \)?
4. How does the choice of data structure impact the efficiency of Dijkstra's algorithm?
5. What is the difference between Dijkstra's algorithm and Bellman-Ford algorithm?

### Tip
Using a priority queue (like a min-heap) can significantly improve the efficiency of Dijkstra's algorithm when dealing with large graphs.

Find the shortest path from vertex A to E in the graph given below using Dijkstra's algorithm.

Discover how to find the shortest path from vertex A to E in a weighted graph using Dijkstra's algorithm. This problem involves graph theory concepts and step-by-step calculations for optimal pathfinding, suitable for high school students in Grades 10-12.

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