Create Multistage-graph.md

docs/dynamic-programming/Multistage-graph.md

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+---
+id: MULTISTAGE GRAPH
+title: MULTISTAGE GRAPH
+sidebar_label: SHORTEST PATH IN MULTISTAGE GRAPH
+tags: [Dynamic Programming,Graph, DSA]
+description: The multistage graph problem is finding the path with minimum cost from source to sink.
+---
+
+# SHORTEST PATH IN MULTISTAGE GRAPH
+
+### Description
+This program calculates the shortest path from a given source node to a target node in a directed graph using a dynamic programming approach. The graph is represented as an adjacency matrix, where the user provides input for the number of vertices and the adjacency matrix values.
+### Problem Definition
+- **Input**: The adjacency Cost Matrix.
+- **Output**: Minimum cost path and Minimum Cost.
+
+### Example
+- **Input**: 
+  - Enter the number of vertices in the graph: 12                              
+    Enter the adjacency matrix (use 99 for INF):                            
+    99 9 7 3 2 99 99 99 99 99 99 99                       
+    99 99 99 99 99 4 2 1 99 99 99 99                         
+    99 99 99 99 99 2 7 99 99 99 99 99                        
+    99 99 99 99 99 99 99 11 99 99 99 99                     
+    99 99 99 99 99 99 11 8 99 99 99 99                        
+    99 99 99 99 99 99 99 99 6 5 99 99                                            
+    99 99 99 99 99 99 99 99 4 3 99 99                                            
+    99 99 99 99 99 99 99 99 99 5 6 99                    
+    99 99 99 99 99 99 99 99 99 99 99 4                       
+    99 99 99 99 99 99 99 99 99 99 99 2                       
+    99 99 99 99 99 99 99 99 99 99 99 5                        
+    99 99 99 99 99 99 99 99 99 99 99 0                     
+
+    Enter the source node: 1                       
+    Enter the target node: 12                                  
+
+- **Output**: 
+  - The shortest path distance from node 1 to node 12 is: 16                       
+    Path: 1 -> 2 -> 7 -> 10 -> 12.                             
+### Diagrammatic represntation of the above example
+  ![Screenshot 2024-10-22 184618](https://github.com/user-attachments/assets/b4a70ddf-332e-4885-b367-38b20362d31f)
+
+### Algorithm Overview
+1. **Initialization:**                  
+   The distance array `dist[]` is initialized with INF for all nodes except the target node, which has a distance of `0`. This array will store the shortest known distances from each node to the target.
+   The path array `path[]` is initialized to store the next node on the shortest path for each node.
+2. **Dynamic Programming Loop:**                       
+   The function iterates from the target node backward to the source, evaluating each node's potential distance to the target by checking its connection to subsequent nodes.
+   If the current distance from node `i` to the target is greater than the distance from node `i` to `j` plus `dist[j]` (where j is a neighboring node), it updates `dist[i]` and sets `path[i]` to `j`.
+3. **Path Construction:**                 
+   If a valid path exists, the function prints the shortest distance from the source to the target.
+   It then traces the path from the source to the target using the `path[]` array and prints each node in the order they are visited.
+
+### Time Complexity
+- O(n^2) - where `n` is the total number of nodes in the multistage graph
+
+### C++ Implementation
+
+```cpp
+#include <iostream>
+#include <vector>
+#include <climits>
+
+#define INF 99
+
+void shortestDist(const std::vector<std::vector<int>>& graph, int n, int source, int target) {
+    std::vector<int> dist(n + 1, INF), path(n + 1, -1);
+
+    dist[target] = 0;
+    path[target] = target;
+
+    for (int i = target - 1; i >= 1; i--) {
+        for (int j = i + 1; j <= n; j++) {
+            if (graph[i][j] != INF && dist[i] > graph[i][j] + dist[j]) {
+                dist[i] = graph[i][j] + dist[j];
+                path[i] = j;
+            }
+        }
+    }
+
+    if (dist[source] == INF) {
+        std::cout << "There is no path from node " << source << " to node " << target << "\n";
+        return;
+    }
+
+    std::cout << "The shortest path distance from node " << source << " to node " << target << " is: " << dist[source] << "\n";
+
+    std::cout << "Path: " << source;
+    int current = source;
+    while (current != target) {
+        current = path[current];
+        std::cout << " -> " << current;
+    }
+    std::cout << "\n";
+}
+
+int main() {
+    int n, source, target;
+
+    std::cout << "Enter the number of vertices in the graph: ";
+    std::cin >> n;
+
+    std::vector<std::vector<int>> graph(n + 1, std::vector<int>(n + 1));
+
+    std::cout << "Enter the adjacency matrix (use " << INF << " for INF):\n";
+    for (int i = 1; i <= n; i++) {
+        for (int j = 1; j <= n; j++) {
+            std::cin >> graph[i][j];
+        }
+    }
+
+    std::cout << "Enter the source node: ";
+    std::cin >> source;
+    std::cout << "Enter the target node: ";
+    std::cin >> target;
+
+    shortestDist(graph, n, source, target);
+
+    return 0;
+}
+```