feature/Building-Roads

4_graph_algorithms/Building_Roads.cpp

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+/*
+Problem: Building Roads
+Category: Graph Algorithms (Undirected Graph)
+Difficulty: Medium
+Time Complexity: O(n + m)
+Space Complexity: O(n + m)
+
+Approach:
+Use graph traversal to find connected components. Pick one representative node from each
+component; the minimum number of roads required to connect all components is (components - 1).
+Connect the representatives in a simple chain: (rep[0], rep[1]), (rep[1], rep[2]), ...
+
+Key Insights:
+- Any connected component can be represented by any one of its vertices.
+- Connecting k components requires exactly k-1 edges; connecting representatives in a chain
+  achieves this minimum.
+- Use iterative DFS (explicit stack) to avoid recursion-depth issues on large inputs.
+- Handle edge cases: n = 1, m = 0, already connected graphs.
+
+Notes:
+- Nodes are 1-indexed (as per CSES problem statement).
+- Fast I/O is used for competitive programming.
+*/
+
+#include <bits/stdc++.h>
+using namespace std;
+
+int main() {
+    ios::sync_with_stdio(false);
+    cin.tie(nullptr);
+
+    // Read number of cities (nodes) and existing roads (edges)
+    int num_nodes, num_edges;
+    if (!(cin >> num_nodes >> num_edges)) return 0;
+
+    // Adjacency list for the undirected graph (1-indexed)
+    vector<vector<int>> adjacency(num_nodes + 1);
+    for (int i = 0; i < num_edges; ++i) {
+        int u, v;
+        cin >> u >> v;
+        // add both directions because graph is undirected
+        adjacency[u].push_back(v);
+        adjacency[v].push_back(u);
+    }
+
+    // visited[i] == true means node i has been visited in component discovery
+    vector<char> visited(num_nodes + 1, false);
+    // representatives: one node per connected component
+    vector<int> representatives;
+    representatives.reserve(num_nodes);
+
+    // Iterative DFS function using an explicit stack to avoid recursion depth limits
+    auto iterative_dfs = [&](int start) {
+        // stack for DFS traversal
+        std::stack<int> dfs_stack;
+        dfs_stack.push(start);
+        visited[start] = true;
+
+        while (!dfs_stack.empty()) {
+            int node = dfs_stack.top();
+            dfs_stack.pop();
+
+            // visit all neighbors
+            for (int neighbor : adjacency[node]) {
+                if (!visited[neighbor]) {
+                    visited[neighbor] = true;
+                    dfs_stack.push(neighbor);
+                }
+            }
+        }
+    };
+
+    // Find connected components by scanning all nodes
+    for (int node = 1; node <= num_nodes; ++node) {
+        if (!visited[node]) {
+            // node is a new component representative
+            representatives.push_back(node);
+            // mark all nodes in this component as visited
+            iterative_dfs(node);
+        }
+    }
+
+    // Number of components
+    int component_count = static_cast<int>(representatives.size());
+
+    // Minimum roads required to connect all components = components - 1
+    int roads_to_add = max(0, component_count - 1);
+    cout << roads_to_add << '\n';
+
+    // Output the roads to add: connect consecutive representatives
+    for (int i = 0; i + 1 < component_count; ++i) {
+        cout << representatives[i] << ' ' << representatives[i + 1] << '\n';
+    }
+
+    return 0;
+}