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class__12- Shortest Path-Dijkstra's Algorithm/Exercise_1-Dijkstra's Algorithm Step-by-Step Solution/Exercise_1 - Dijkstra's Algorithm.md

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+# Exercise 1: Dijkstra's Algorithm – Step-by-Step Solution
+
+## Problem Statement
+
+A US transportation company delivers packages daily in New York City from origin node 1 (Queens) to destination node 6 (Manhattan), using different possible routes as shown in the graph below. The flow on the arcs represents the cost to transport the required demand between neighborhoods.  
+**Task:** Determine the best route using Dijkstra's algorithm.
+
+![Graph and Tableaus](https://pplx-res.cloudinary.com/image/private/user_uploads/27709701/a31aabec-aeef-42c0-b4cd-9024927871af/Screenshot-2025-05-14-at-13.24.26.jpg)
+
+<br>
+
+## Step-by-Step Solution with Tableaus
+
+### Initialization
+
+Assign value 0 to the source node (1) and ∞ (infinity) to all other nodes.
+
+| 1* | 2* | 3* | 4* | 5* | 6* |
+|----|----|----|----|----|----|
+| 0  | ∞  | ∞  | ∞  | ∞  | ∞  |
+
+---
+
+### Iteration 1: Visit Node 1 (current minimum: 0)
+
+- Update 2: 0 + 6 = 6, predecessor 1
+- Update 3: 0 + 9 = 9, predecessor 1
+
+| 1* | 2*    | 3*    | 4* | 5* | 6* |
+|----|-------|-------|----|----|----|
+| 0  | (1,6) | (1,9) | ∞  | ∞  | ∞  |
+
+---
+
+### Iteration 2: Visit Node 2 (current minimum: 6)
+
+- Update 4: 6 + 4 = 10, predecessor 2
+- Update 5: 6 + 7 = 13, predecessor 2
+
+| 1* | 2*    | 3*    | 4*     | 5*     | 6* |
+|----|-------|-------|--------|--------|----|
+| 0  | 6     | (1,9) | (2,10) | (2,13) | ∞  |
+
+---
+
+### Iteration 3: Visit Node 3 (current minimum: 9)
+
+(No better paths found from 3.)
+
+| 1* | 2*    | 3*    | 4*     | 5*     | 6* |
+|----|-------|-------|--------|--------|----|
+| 0  | 6     | 9     | (2,10) | (2,13) | ∞  |
+
+---
+
+### Iteration 4: Visit Node 4 (current minimum: 10)
+
+- Update 5: 10 + 2 = 12, predecessor 4 (improves from 13 to 12)
+- Update 6: 10 + 7 = 17, predecessor 4
+
+| 1* | 2*    | 3*    | 4* | 5*     | 6*     |
+|----|-------|-------|----|--------|--------|
+| 0  | 6     | 9     | 10 | (4,12) | (4,17) |
+
+---
+
+### Iteration 5: Visit Node 5 (current minimum: 12)
+
+- Update 6: 12 + 3 = 15, predecessor 5 (improves from 17 to 15)
+
+| 1* | 2*    | 3*    | 4* | 5* | 6*     |
+|----|-------|-------|----|----|--------|
+| 0  | 6     | 9     | 10 | 12 | (5,15) |
+
+---
+
+### Iteration 6: Visit Node 6 (current minimum: 15)
+
+| 1* | 2*    | 3*    | 4* | 5* | 6* |
+|----|-------|-------|----|----|----|
+| 0  | 6     | 9     | 10 | 12 | 15 |
+
+---
+
+## Minimum Path and Cost
+
+- **Path:** 1 → 2 → 4 → 5 → 6
+- **Total Cost:** 15
+
+---
+
+## Explanation of Tableaus
+
+- Each cell (X, Y) indicates the predecessor node (X) and the cumulative cost to reach the node (Y).
+- At each step, only the best (lowest cost) paths are kept and updated.
+- The algorithm stops when the destination node (6) receives a permanent label.
+
+---
+
+## Python Implementation
+
+```
+
+import heapq
+
+def dijkstra(graph, start, end):
+distances = {node: float('inf') for node in graph}
+predecessors = {node: None for node in graph}
+distances[start] = 0
+queue = [(0, start)]
+while queue:
+curr_dist, curr_node = heapq.heappop(queue)
+if curr_node == end:
+break
+for neighbor, weight in graph[curr_node].items():
+distance = curr_dist + weight
+if distance < distances[neighbor]:
+distances[neighbor] = distance
+predecessors[neighbor] = curr_node
+heapq.heappush(queue, (distance, neighbor))
+\# Reconstruct path
+path = []
+node = end
+while node is not None:
+path.append(node)
+node = predecessors[node]
+path.reverse()
+return distances[end], path
+
+# Graph from the example
+
+graph = {
+1: {2: 6, 3: 9},
+2: {4: 4, 5: 7},
+3: {},
+4: {5: 2, 6: 7},
+5: {6: 3},
+6: {}
+}
+
+cost, path = dijkstra(graph, 1, 6)
+print(f"Minimum cost: {cost}, Path: {path}")
+
+```
+
+---
+
+## Excel Solver Step-by-Step
+
+1. **Model the Network:**
+    - Create a table with nodes as rows and columns, filling in arc costs (use a large number or blank for non-existent arcs).
+
+2. **Define Decision Variables:**
+    - For each arc (i, j), create a binary variable \( x_{ij} \) (1 if the arc is used, 0 otherwise).
+
+3. **Objective Function:**
+    - Minimize the total cost:  
+      \[
+      \text{Minimize} \quad Z = \sum_{(i,j)} c_{ij} x_{ij}
+      \]
+
+4. **Constraints:**
+    - **Flow conservation:**  
+      - For the source node: Outflow - Inflow = 1  
+      - For the destination node: Inflow - Outflow = 1  
+      - For all other nodes: Inflow - Outflow = 0
+
+5. **Set Up Solver:**
+    - Set the objective cell to the sum of selected arc costs.
+    - Add constraints for flow conservation and binary variables.
+    - Run Solver to find the minimum cost path.
+
+---
+
+## References
+
+- [PUC-SP Class Material](https://pplx-res.cloudinary.com/image/private/user_uploads/27709701/a31aabec-aeef-42c0-b4cd-9024927871af/Screenshot-2025-05-14-at-13.24.26.jpg)
+- Dijkstra, E. W. (1959). A note on two problems in connexion with graphs.
+
+#
+
+This README provides a complete, clear, and actionable guide for solving Example 1 with Dijkstra’s algorithm, including all tableaus, explanations, and step-by-step solutions in both Python and Excel Solver.
+
+
+