Let's solve the shortest path from node **O** to node **T** using **Dijkstra's algorithm** step-by-step. --- ## **Step 1: Initialization** Let’s label the nodes: O, A, B, C, D, E, T Set the initial distances: - O: 0 (start node) - All others: ∞ (infinity) Unvisited nodes: {O, A, B, C, D, E, T} Previous node for each: None --- ## **Step 2: Table Initialization** | Node | Distance | Previous | |------|----------|----------| | O | 0 | - | | A | ∞ | - | | B | ∞ | - | | C | ∞ | - | | D | ∞ | - | | E | ∞ | - | | T | ∞ | - | --- ## **Step 3: Visit Node O (current node with smallest distance)** Neighbors: A (2), B (5), C (4) Update their distances: - A: min(∞, 0+2) = 2 → Previous: O - B: min(∞, 0+5) = 5 → Previous: O - C: min(∞, 0+4) = 4 → Previous: O | Node | Distance | Previous | |------|----------|----------| | O | 0 | - | | A | 2 | O | | B | 5 | O | | C | 4 | O | | D | ∞ | - | | E | ∞ | - | | T | ∞ | - | Mark O as visited. --- ## **Step 4: Visit Node A (next smallest distance, 2)** Neighbors: B (2), D (7) Update their distances: - B: min(5, 2+2) = 4 → Previous: A - D: min(∞, 2+7) = 9 → Previous: A | Node | Distance | Previous | |------|----------|----------| | O | 0 | - | | A | 2 | O | | B | 4 | A | | C | 4 | O | | D | 9 | A | | E | ∞ | - | | T | ∞ | - | Mark A as visited. --- ## **Step 5: Visit Node B (next smallest, 4)** Neighbors: D (4), E (3), C (1) Update their distances: - D: min(9, 4+4) = 8 → Previous: B - E: min(∞, 4+3) = 7 → Previous: B - C: min(4, 4+1) = 4 (no change) | Node | Distance | Previous | |------|----------|----------| | O | 0 | - | | A | 2 | O | | B | 4 | A | | C | 4 | O | | D | 8 | B | | E | 7 | B | | T | ∞ | - | Mark B as visited. --- ## **Step 6: Visit Node C (next smallest, 4)** Neighbors: E (4) - E: min(7, 4+4) = 7 (no change) No other updates. Mark C as visited. --- ## **Step 7: Visit Node E (next smallest, 7)** Neighbors: D (1), T (7) - D: min(8, 7+1) = 8 (no change) - T: min(∞, 7+7) = 14 → Previous: E | Node | Distance | Previous | |------|----------|----------| | O | 0 | - | | A | 2 | O | | B | 4 | A | | C | 4 | O | | D | 8 | B | | E | 7 | B | | T | 14 | E | Mark E as visited. --- ## **Step 8: Visit Node D (next smallest, 8)** Neighbors: T (5), E (1) - T: min(14, 8+5) = 13 → Previous: D | Node | Distance | Previous | |------|----------|----------| | O | 0 | - | | A | 2 | O | | B | 4 | A | | C | 4 | O | | D | 8 | B | | E | 7 | B | | T | 13 | D | Mark D as visited. --- ## **Step 9: Visit Node T (distance 13)** Reached the destination. --- ## **Step 10: Trace the Shortest Path** Start from T, follow the "Previous" pointers: - T ← D ← B ← A ← O So, the path is: **O → A → B → D → T** Total distance: **13** --- ### **Final Answer** **Shortest path from O to T using Dijkstra's algorithm:** > **O → A → B → D → T** > > **Total distance: 13**

Certainly! Let's walk through the detailed step-by-step calculation of the shortest path from node **O** to node **T** using Dijkstra's algorithm, based on the diagram provided. --- ### **Initial Setup:** - The start node: **O** - All other nodes: A, B, C, D, E, T - Distances to nodes initially: - Distance to **O**: 0 (since starting point) - Distance to all others: ∞ (infinity) - The set of unvisited nodes includes all nodes at the start: {O, A, B, C, D, E, T}. --- ### **Step 1: Visit Node O** - **Neighbors of O**: - **A** with edge weight **2** - **B** with edge weight **5** - **C** with edge weight **4** - **Update distances**: - **A**: min(∞, 0 + 2) = **2** - **B**: min(∞, 0 + 5) = **5** - **C**: min(∞, 0 + 4) = **4** - Mark **O** as visited (finalized). --- ### **Step 2: Visit Node A (smallest unvisited distance = 2)** - **Neighbors of A**: - **B** with edge weight **2** - **D** with edge weight **7** - **Update distances**: - **B**: min(5, 2 + 2) = **4** (improved from 5 to 4) - **D**: min(∞, 2 + 7) = **9** - Mark **A** as visited. --- ### **Step 3: Visit Node B (next smallest distance = 4)** - **Neighbors of B**: - **D** with edge weight **4** - **E** with edge weight **3** - **Update distances**: - **D**: min(9, 4 + 4) = **8** (improved from 9 to 8) - **E**: min(∞, 4 + 3) = **7** - **Note**: The alternative path to **D** via B is shorter (8 vs. 9). - Mark **B** as visited. --- ### **Step 4: Visit Node C (next smallest distance = 4)** - **Neighbors of C**: - **E** with edge weight **4** - **Update distances**: - **E**: min(7, 4 + 4) = **7** (no change, already 7) - Mark **C** as visited. --- ### **Step 5: Visit Node E (next smallest distance = 7)** - **Neighbors of E**: - **D** with edge weight **1** - **T** with edge weight **7** - **Update distances**: - **D**: min(8, 7 + 1) = **8** (no change) - **T**: min(∞, 7 + 7) = **14** - **Note**: The shortest path to **T** so far is via E with total distance 14. - Mark **E** as visited. --- ### **Step 6: Visit Node D (next smallest distance = 8)** - **Neighbors of D**: - **T** with edge weight **5** - **Update distances**: - **T**: min(14, 8 + 5) = **13** (improved from 14 to 13) - **Note**: Now, the shortest path to **T** is of length **13** via D. - Mark **D** as visited. --- ### **Step 7: Visit Node T (distance = 13)** - **Reached destination**. The shortest distance from **O** to **T** is **13**. --- ### **Path Reconstruction:** - Starting from **T**, trace back via the "Previous" nodes: - **T** came via **D** (distance 13) - **D** came via **B** (distance 4 + 4 + 5 = 13) - **B** came via **A** (distance 2 + 2 = 4) - **A** came directly from **O** (distance 0 + 2) **Therefore, the shortest path is:** **O → A → B → D → T** **Total shortest path distance: 13** --- ### **Final answer:** **The shortest path from node O to node T is:** **O → A → B → D → T** **Total distance: 13**