Prims + Dijkstra's || 100% BEATS 🚀 || Threshold Not Needed Explained 🎯
Simple Example to Understand Why Threshold is IgnoredExample Input:
Number of nodes: ( n = 4 )
Directed edges and weights:

Threshold:
Graph Representation:Goal


### **Simple Example to Understand Why Threshold is Ignored**

---

#### **Example Input:**

- Number of nodes: \( n = 4 \)  
- Directed edges and weights:  

```
edges = [[1, 0, 2], [2, 0, 1], [3, 2, 3], [3, 0, 4]]
```

- Threshold:  
```
threshold = 2
```

#### **Graph Representation:**

```
   2
1 ---> 0

   1
2 ---> 0

   3         4
3 ---> 2 ---> 0
```

---

#### **Goal:**
- Ensure all nodes can reach node \( 0 \).
- Minimize the **maximum edge weight** in the final graph.
- Threshold: Each node can have at most \( \text{threshold} = 2 \) outgoing edges.  

---

#### **Key Insight:**
The **threshold condition can be ignored** because:
1. The algorithm naturally minimizes the number of outgoing edges per node while ensuring connectivity.

2. As long as **each node has one valid path to node 0** (either directly or indirectly), the problem is solved:

   - For example, in the graph above:
     - Node \(1 → 0\) (weight 2) ensures node 1 connects to node 0.
     - Node \(2 → 0\) (weight 1) ensures node 2 connects to node 0.
     - Node \(3 → 2 → 0\) ensures node 3 connects to node 0.

---

#### **Why Ignoring Threshold is Safe:**
Imagine node \( 3 \) has three outgoing edges: \( 3 → 2 \), \( 3 → 0 \), and \( 3 → 4 \).  
- The threshold says it can only have two outgoing edges.
- **Logic:**  
  - Keep the smallest edge weights that maintain connectivity.  
  - For example, keeping \( 3 → 2 \) (weight 3) ensures \( 3 \) can still reach \( 0 \) via \( 2 → 0 \).  
  - Removing other edges (like \( 3 → 0 \)) doesn't break the connectivity requirement.

Thus, explicitly enforcing the threshold is unnecessary because the problem focuses on minimizing the maximum edge weight, which indirectly satisfies the threshold constraint.

---

### **Intuition**

- Focus on connecting all nodes to \( 0 \).
- Prioritize minimizing the maximum edge weight.
- The threshold condition is safely ignored as it doesn't impact the connectivity or weight minimization logic.

---

### **Approach**

1. Use a graph representation (adjacency list) to store edges and weights.
2. Use a priority queue to iteratively pick the smallest weight edges, ensuring all nodes connect to \( 0 \).
3. Track visited nodes to prevent cycles.
4. Return the largest edge weight used in the solution.

---

### **Complexity**

- **Time complexity:** \( O(E \log V) \), where \( E \) is the number of edges, and \( V \) is the number of nodes.
- **Space complexity:** \( O(V + E) \) for the graph and auxiliary data structures.

---

### **Code**

```cpp
class Solution {
public:
    int minMaxWeight(int n, vector<vector<int>>& edges, int threshold) {
        // Build adjacency list
        vector<vector<pair<int, int>>> graph(n);
        for (auto &it : edges)
            graph[it[1]].emplace_back(it[0], it[2]);

        vector<bool> visited(n);          // Tracks visited nodes
        vector<int> key(n, INT_MAX);      // Minimum weights to each node
        key[0] = 0;                       // Start with node 0
        priority_queue<pair<int, int>, vector<pair<int, int>>, greater<pair<int, int>>> pq;
        pq.emplace(0, 0);                 // {weight, node}

        // Process nodes
        for (int i = 0; i < n; i++) {
            while (!pq.empty() && visited[pq.top().second])
                pq.pop();

            if (pq.empty()) return -1;    // If no valid path, return -1

            auto it = pq.top();
            pq.pop();
            int idx = it.second;

            visited[idx] = true;

            // Update neighbors
            for (auto &nbr : graph[idx]) {
                if (!visited[nbr.first] && key[nbr.first] > nbr.second) {
                    key[nbr.first] = nbr.second;
                    pq.emplace(nbr.second, nbr.first);
                }
            }
        }

        // Return the maximum edge weight in the solution
        return *max_element(key.begin(), key.end());
    }
};
```

---
['Greedy', 'Graph', 'Heap (Priority Queue)', 'C++']