Let’s calculate the shortest path from node A to all other nodes using Dijkstra’s Algorithm based on the diagram provided. Here are the edges and costs:
- A → B with cost 3
- A → C with cost 2
- A → D with cost 5
- B → D with cost 1
- B → E with cost 4
- C → F with cost 1
- D → F with cost 3
- E → F with cost 2
Step-by-Step Process of Dijkstra’s Algorithm:
Initial Setup:
- Start at node A (source node), with an initial distance of 0 for A and infinity (
∞) for all other nodes. - A table of distances and previous nodes:
| Node | Distance | Previous |
|---|---|---|
| A | 0 | - |
| B | ∞ | - |
| C | ∞ | - |
| D | ∞ | - |
| E | ∞ | - |
| F | ∞ | - |
Step 1: Process Node A
- From A:
- A → B: Distance to B = 0 + 3 = 3.
- A → C: Distance to C = 0 + 2 = 2.
- A → D: Distance to D = 0 + 5 = 5.
Update the table:
| Node | Distance | Previous |
|---|---|---|
| A | 0 | - |
| B | 3 | A |
| C | 2 | A |
| D | 5 | A |
| E | ∞ | - |
| F | ∞ | - |
Choose the node with the smallest distance, which is C (distance = 2).
Step 2: Process Node C
- From C:
- C → F: Distance to F = 2 + 1 = 3.
Update the table:
| Node | Distance | Previous |
|---|---|---|
| A | 0 | - |
| B | 3 | A |
| C | 2 | A |
| D | 5 | A |
| E | ∞ | - |
| F | 3 | C |
Now, choose the node with the smallest distance, which is either B or F (both have a distance of 3). Let’s process B.
Step 3: Process Node B
- From B:
- B → D: Distance to D = 3 + 1 = 4 (this is better than the previous distance of 5, so we update it).
- B → E: Distance to E = 3 + 4 = 7.
Update the table:
| Node | Distance | Previous |
|---|---|---|
| A | 0 | - |
| B | 3 | A |
| C | 2 | A |
| D | 4 | B |
| E | 7 | B |
| F | 3 | C |
Next, choose the node with the smallest distance, which is F (distance = 3).
Step 4: Process Node F
- From F:
- F → D: Distance to D = 3 + 3 = 6, but the current distance to D is 4, so no update is needed.
- F → E: Distance to E = 3 + 2 = 5 (this is better than the previous distance of 7, so we update it).
Update the table:
| Node | Distance | Previous |
|---|---|---|
| A | 0 | - |
| B | 3 | A |
| C | 2 | A |
| D | 4 | B |
| E | 5 | F |
| F | 3 | C |
Now, choose the node with the smallest distance, which is D (distance = 4).
Step 5: Process Node D
- D has no neighbors that can provide a shorter path to other nodes, so no updates are needed.
Next, process E (distance = 5), but there are no further updates necessary.
Final Shortest Paths from A:
| Node | Distance from A | Previous Node |
|---|---|---|
| A | 0 | - |
| B | 3 | A |
| C | 2 | A |
| D | 4 | B |
| E | 5 | F |
| F | 3 | C |
Conclusion:
- The shortest path from A → B is 3 (A → B).
- The shortest path from A → C is 2 (A → C).
- The shortest path from A → D is 4 (A → B → D).
- The shortest path from A → E is 5 (A → C → F → E).
- The shortest path from A → F is 3 (A → C → F).
Let me know if you need further clarifications!