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class__12- Shortest Path-Dijkstra's Algorithm/Exerc_3-Applying Dijkstra's Algorithm to the Shortest Path Problem/Exercise_3-Applying Dijkstra's Algorithm to the Shortest Path Problem.md

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+## Exercise 3: Step-by-Step Tableau for Dijkstra's Algorithm
+
+Below are the tableaus (tabulações) representing each iteration of Dijkstra's algorithm for Example 3, as described in the PDF[^1].
+
+### **Network Data**
+
+- Nodes: A (source), B, C, D, E (destination)
+- Arc weights (in seconds):  
+  - A→B: 25  
+  - A→C: 28  
+  - B→D: 22  
+  - D→E: 18  
+
+### **Tableau 1: Initialization**
+
+| Node | Distance | Predecessor | Status   |
+|------|----------|-------------|----------|
+| A    | 0        | -           | Permanent|
+| B    | ∞        | -           | Temporary|
+| C    | ∞        | -           | Temporary|
+| D    | ∞        | -           | Temporary|
+| E    | ∞        | -           | Temporary|
+
+### **Tableau 2: After Visiting A**
+
+- Update B: 0 + 25 = 25 (Predecessor: A)
+- Update C: 0 + 28 = 28 (Predecessor: A)
+
+| Node | Distance | Predecessor | Status   |
+|------|----------|-------------|----------|
+| A    | 0        | -           | Permanent|
+| B    | 25       | A           | Temporary|
+| C    | 28       | A           | Temporary|
+| D    | ∞        | -           | Temporary|
+| E    | ∞        | -           | Temporary|
+
+### **Tableau 3: After Visiting B**
+
+- Update D: 25 + 22 = 47 (Predecessor: B)
+
+| Node | Distance | Predecessor | Status   |
+|------|----------|-------------|----------|
+| A    | 0        | -           | Permanent|
+| B    | 25       | A           | Permanent|
+| C    | 28       | A           | Temporary|
+| D    | 47       | B           | Temporary|
+| E    | ∞        | -           | Temporary|
+
+### **Tableau 4: After Visiting C**
+
+- No updates (C has no outgoing arcs).
+
+| Node | Distance | Predecessor | Status   |
+|------|----------|-------------|----------|
+| A    | 0        | -           | Permanent|
+| B    | 25       | A           | Permanent|
+| C    | 28       | A           | Permanent|
+| D    | 47       | B           | Temporary|
+| E    | ∞        | -           | Temporary|
+
+### **Tableau 5: After Visiting D**
+
+- Update E: 47 + 18 = 65 (Predecessor: D)
+
+| Node | Distance | Predecessor | Status   |
+|------|----------|-------------|----------|
+| A    | 0        | -           | Permanent|
+| B    | 25       | A           | Permanent|
+| C    | 28       | A           | Permanent|
+| D    | 47       | B           | Permanent|
+| E    | 65       | D           | Temporary|
+
+### **Tableau 6: After Visiting E (Final)**
+
+| Node | Distance | Predecessor | Status   |
+|------|----------|-------------|----------|
+| A    | 0        | -           | Permanent|
+| B    | 25       | A           | Permanent|
+| C    | 28       | A           | Permanent|
+| D    | 47       | B           | Permanent|
+| E    | 65       | D           | Permanent|
+
+#
+
+### **Optimal Path and Cost**
+
+- **Path:** A → B → D → E
+- **Total Time:** 65 seconds
+
+These tableaus follow the standard Dijkstra's procedure, allowing step-by-step verification and understanding of the shortest path calculation in the network.
+
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