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.