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class__12- Shortest Path-Dijkstra's Algorithm/Shortest Path Problem-Theory Dijkstra's Algorithm-Python.md

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+# Shortest Path Problem: Theory, Mathematical Formulation, and Dijkstra's Algorithm
+
+## 1. Theoretical Explanation
+
+The **Shortest Path Problem** (also called the minimum path problem) seeks to find the path between two nodes in a network that minimizes the total distance, cost, or travel time.  
+- The network consists of a single supply node (origin) and a single demand node (destination), with all other nodes being transshipment nodes (zero supply and demand).
+- The problem can be solved using mathematical programming or graph algorithms such as Dijkstra's algorithm.
+
+---
+
+## 2. Mathematical Formulation
+
+Let:
+
+- **Parameters:**  
+  \( c_{ij} \) = distance, cost, or time from node \( i \) to node \( j \).
+
+- **Decision Variables:**  
+  \( x_{ij} = \begin{cases} 
+      1 & \text{if arc } (i, j) \text{ is included in the shortest path} \\
+      0 & \text{otherwise}
+    \end{cases} \)
+
+- **Objective Function:**  
+  Minimize the total cost of the path from the origin to the destination:
+  \[
+  \min z = \sum_{(i,j)} c_{ij} x_{ij}
+  \]
+
+- **Constraints:**  
+  - The total flow out of the origin is 1.
+  - The total flow into the destination is 1.
+  - For all intermediate nodes, the flow in equals the flow out.
+
+---
+
+## 3. Dijkstra's Algorithm: Step-by-Step Procedure
+
+**Initialization:**
+- Let \( R \) be the set of labeled (closed) nodes, initially empty.
+- Let \( NR \) be the set of unlabeled (open) nodes, initially all nodes.
+- Assign distance 0 to the source node and \( \infty \) to all other nodes.
+
+**Algorithm Steps:**
+1. While \( NR \) is not empty:
+   - Select the node \( k \) in \( NR \) with the smallest tentative distance.
+   - Move \( k \) from \( NR \) to \( R \).
+   - For each unlabeled successor \( j \) of \( k \):
+     - If \( \text{distance}(k) + c_{kj} < \text{distance}(j) \), update \( \text{distance}(j) \) and set \( k \) as the predecessor of \( j \).
+
+---
+
+## 4. Example 3: Oil Company Logistics Network
+
+**Problem:**  
+An oil company analyzes the flow of its products in a logistics network from node **A** (origin) to node **E** (destination), passing through intermediate nodes **B, C, D**. Arc weights represent flow time in seconds.
+
+### **Network Structure**
+
+- **A → B:** 25
+- **A → C:** 28
+- **B → D:** 22
+- **C:** No outgoing arcs
+- **D → E:** 18
+
+### **Dijkstra's Algorithm Tableaus**
+
+#### **Tableau 1: Initialization**
+
+| Node | Distance | Predecessor | Status     |
+|------|----------|-------------|------------|
+| A    | 0        | -           | Permanent  |
+| B    | ∞        | -           | Temporary  |
+| C    | ∞        | -           | Temporary  |
+| D    | ∞        | -           | Temporary  |
+| E    | ∞        | -           | Temporary  |
+
+#### **Tableau 2: After Visiting 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**
+
+| 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**
+
+| 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**
+
+| 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 Total Time**
+
+- **Path:** A → B → D → E
+- **Total Time:** 25 + 22 + 18 = **65 seconds**
+
+```
+
+graph LR
+A --25--> B --22--> D --18--> E
+
+```
+
+---
+
+## 5. Python Implementation Example
+
+```
+
+import heapq
+
+def dijkstra(graph, start):
+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)
+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))
+return distances, predecessors
+
+# Example 3 Graph
+
+graph = {
+'A': {'B': 25, 'C': 28},
+'B': {'D': 22},
+'C': {},
+'D': {'E': 18},
+'E': {}
+}
+
+distances, predecessors = dijkstra(graph, 'A')
+print("Distances:", distances)
+print("Predecessors:", predecessors)
+
+```
+
+---
+
+## 6. References
+
+- [Class Material PDF: Caminho Mínimo e LP - PUC-SP](https://ppl-ai-file-upload.s3.amazonaws.com/web/direct-files/attachments/27709701/1f6878e6-1eb2-41b6-835a-7bb02c00cc10/class_12-Caminho-Minimoe-LP.pdf)
+
+---
+
+**This README provides a comprehensive overview, mathematical formulation, algorithmic steps, tableaus, and a worked example for the shortest path problem as presented in your course material.**
+
+