prism algo

Author: JitinSaxenaa

CPP/data_structures/graphs/graph/prism.cpp

@@ -0,0 +1,89 @@
+#include <bits/stdc++.h>
+using namespace std;
+
+/*
+======================================================
+Prim's Algorithm: Minimum Spanning Tree (MST)
+======================================================
+Description:
+Prim's algorithm finds the Minimum Spanning Tree (MST) of a weighted 
+undirected graph. The MST is a subset of edges connecting all vertices 
+with the minimum total edge weight, without forming cycles.
+
+Approach:
+1. Start with a single vertex (we start with vertex 0).
+2. Pick the smallest edge connecting a vertex in MST to a vertex outside MST.
+3. Include that edge and repeat until all vertices are included in MST.
+
+Time Complexity:
+- O(V^2), where V is the number of vertices (using adjacency matrix).
+- Using priority queues (min-heap) can reduce it to O(E log V).
+
+Space Complexity:
+- O(V^2) for adjacency matrix.
+======================================================
+*/
+
+const int INF = 1e9; // Infinity value for initialization
+
+int main() {
+    int n; // Number of vertices
+    cout << "Enter number of vertices: ";
+    cin >> n;
+
+    // Input adjacency matrix
+    vector<vector<int>> graph(n, vector<int>(n));
+    cout << "Enter adjacency matrix (0 for no edge):\n";
+    for (int i = 0; i < n; i++) {
+        for (int j = 0; j < n; j++) {
+            cin >> graph[i][j];
+        }
+    }
+
+    // key[i] -> Minimum weight edge to include vertex i in MST
+    vector<int> key(n, INF);
+
+    // inMST[i] -> True if vertex i is included in MST
+    vector<bool> inMST(n, false);
+
+    // parent[i] -> Stores the parent of vertex i in MST
+    vector<int> parent(n, -1);
+
+    // Start from vertex 0
+    key[0] = 0;
+
+    // MST will have n vertices
+    for (int count = 0; count < n - 1; count++) {
+        // Step 1: Pick the minimum key vertex from the set of vertices not yet included in MST
+        int u = -1;
+        int minKey = INF;
+
+        for (int v = 0; v < n; v++) {
+            if (!inMST[v] && key[v] < minKey) {
+                minKey = key[v];
+                u = v;
+            }
+        }
+
+        // Step 2: Include this vertex in MST
+        inMST[u] = true;
+
+        // Step 3: Update key and parent for adjacent vertices
+        for (int v = 0; v < n; v++) {
+            // If there is an edge u-v and v is not in MST and weight is smaller
+            if (graph[u][v] != 0 && !inMST[v] && graph[u][v] < key[v]) {
+                key[v] = graph[u][v];
+                parent[v] = u;
+            }
+        }
+    }
+
+    // Print the MST
+    cout << "\nEdges in the Minimum Spanning Tree (MST):\n";
+    cout << "Edge \tWeight\n";
+    for (int i = 1; i < n; i++) {
+        cout << parent[i] << " - " << i << "\t" << graph[i][parent[i]] << "\n";
+    }
+
+    return 0;
+}