Flight routes graph algorithms solution added

4_graph_algorithms/flight_routes.cpp

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+/*
+Problem: Flight Routes (CSES Problem Set 1196)
+Category: Graph Algorithms
+Difficulty: Medium
+Time Complexity: O(m * log(n * k))
+Space Complexity: O(n * k)
+
+Approach:
+This is a variant of Dijkstra’s algorithm where we need the k shortest paths
+from the source (1) to destination (n), allowing repeated nodes in routes.
+
+We maintain for each node a max-heap of up to k shortest distances found so far.
+We process nodes in increasing order of distance (via a min-heap).
+If a new smaller distance is found, we push it into that node's heap.
+If the heap exceeds k, we remove the largest (keeping only k smallest).
+
+This avoids sorting every time (unlike a vector-based approach),
+resulting in much faster performance for large graphs.
+
+Key Insights:
+- Standard Dijkstra only tracks one best distance per node; here we track up to k.
+- Using a max-heap per node allows O(log k) insertion/removal.
+- We only store k distances per node → O(n * k) memory.
+- Since k ≤ 10, the algorithm is very efficient.
+
+Edge Cases:
+- Multiple edges between the same nodes.
+- Repeated visits to the same node with different costs.
+- Large weights (use long long).
+*/
+
+#include <bits/stdc++.h>
+using namespace std;
+
+#define int long long
+#define pii pair<long long, int>
+
+int32_t main() {
+    ios::sync_with_stdio(false);
+    cin.tie(nullptr);
+
+    int n, m, k;
+    cin >> n >> m >> k;
+
+    vector<vector<pair<int,int>>> adj(n + 1);
+    for (int i = 0; i < m; i++) {
+        int a, b, c;
+        cin >> a >> b >> c;
+        adj[a].push_back({b, c});
+    }
+
+    // Max-heaps to store up to k shortest distances per node
+    vector<priority_queue<long long>> dist(n + 1);
+
+    // Min-heap for Dijkstra: {distance, node}
+    priority_queue<pair<long long,int>, vector<pair<long long,int>>, greater<pair<long long,int>>> pq;
+
+    pq.push({0, 1});
+    dist[1].push(0);
+
+    while (!pq.empty()) {
+        auto [currDist, node] = pq.top();
+        pq.pop();
+
+        // Skip if this path is not among the k smallest for this node
+        if (dist[node].top() < currDist && (int)dist[node].size() == k)
+            continue;
+
+        for (auto &[next, wt] : adj[node]) {
+            long long newDist = currDist + wt;
+
+            // If less than k distances stored or this one is smaller than the largest
+            if ((int)dist[next].size() < k || newDist < dist[next].top()) {
+                dist[next].push(newDist);
+                pq.push({newDist, next});
+                if ((int)dist[next].size() > k)
+                    dist[next].pop(); // Keep only k smallest distances
+            }
+        }
+    }
+
+    // Collect all distances from max-heap of destination node and sort ascending
+    vector<long long> ans;
+    while (!dist[n].empty()) {
+        ans.push_back(dist[n].top());
+        dist[n].pop();
+    }
+    sort(ans.begin(), ans.end());
+
+    for (auto &d : ans)
+        cout << d << " ";
+    cout << "\n";
+
+    return 0;
+}