3419. Minimize the Maximum Edge Weight of Graph | Binary Search - Ignore Threshold Value ✨✅ 
Optimizing Graph Edge Weights Under Constraints 🛠️IntuitionThe problem asks us to minimize the maximum edge weight in a graph while meeting three constraints:
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# **Optimizing Graph Edge Weights Under Constraints 🛠️**

## **Intuition**
The problem asks us to **minimize the maximum edge weight** in a graph while meeting three constraints:
1. Every node must be able to **reach node 0**.
2. Each node must have at most **`threshold` outgoing edges**.
3. The weight of the maximum edge in the graph should be minimized.

The **key insight** is that the maximum weight allowed (`limit`) directly determines the edges included in the graph. If we repeatedly adjust this `limit` using **binary search**, we can find the smallest `limit` that meets all conditions.

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## **Approach**

### **Key Idea**
- Use **binary search** over the range of possible edge weights.
- For each candidate `limit`, check whether all nodes can reach node 0 using only edges with weights ≤ `limit`.

### **Steps**
1. **Reverse the edges:**  
   To simplify checking if every node can reach node 0, we **reverse all edges** in the graph. This transforms the problem into checking if node 0 can reach every other node.
   
2. **Graph Construction:**  
   Build an adjacency list for the reversed graph, where each edge stores its weight.

3. **Feasibility Check (`help` function):**  
   Using **BFS**, traverse the graph, keeping track of visited nodes. Only use edges with weights ≤ `limit`.  
   If all nodes are reachable, the current `limit` is valid.

4. **Binary Search on Edge Weights:**  
   - Start with the range of edge weights (`[min_edge_weight, max_edge_weight]`).  
   - Use the `help` function to check if a given mid-point (`limit`) is feasible.  
   - Adjust the search range based on the feasibility result.

5. **Threshold Constraint:**  
   The `threshold` constraint can be ignored here because the BFS traversal ensures at most one outgoing edge per node is processed.

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## **Code with Explanation**

```cpp
#define vi vector<int>
#define vvi vector<vi>
#define pi pair<int, int>
#define vpi vector<pi>
#define vvpi vector<vpi>

class Solution {
public:
    // Helper function to check feasibility with a given weight limit.
    bool help(int n, vvi& edges, int limit) {
        // Create a reversed graph adjacency list
        vvpi graph(n);
        for (vi& edge : edges) {
            graph[edge[1]].push_back(make_pair(edge[0], edge[2])); // Reverse the edge
        }

        // BFS to check reachability to node 0 with weight ≤ limit
        queue<int> q;
        q.push(0);
        vector<bool> vis(n, false);
        vis[0] = true; // Mark node 0 as visited

        while (!q.empty()) {
            int node = q.front();
            q.pop();

            for (pi it : graph[node]) { // Check neighbors
                if (!vis[it.first] && it.second <= limit) { // Only visit if weight ≤ limit
                    vis[it.first] = true;
                    q.push(it.first);
                }
            }
        }

        // Check if all nodes are visited
        for (bool x : vis) {
            if (!x)
                return false; // If any node is unreachable, return false
        }

        return true; // All nodes are reachable
    }

    // Function to minimize the maximum edge weight
    int minMaxWeight(int n, vector<vector<int>>& edges, int threshold) {
        // Find the range of edge weights
        int maxe = INT_MIN, mine = INT_MAX;
        for (const vi& edge : edges) {
            maxe = max(maxe, edge[2]); // Maximum weight
            mine = min(mine, edge[2]); // Minimum weight
        }

        // Binary search to find the minimum maximum weight
        int low = mine, high = maxe, ans = -1;
        while (low <= high) {
            int mid = low + (high - low) / 2; // Mid-point
            if (help(n, edges, mid)) { // Check feasibility with current weight limit
                ans = mid; // Update answer
                high = mid - 1; // Try for a smaller maximum weight
            } else {
                low = mid + 1; // Increase the limit
            }
        }

        return ans; // Return the minimum maximum weight
    }
};
```

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## **Time Complexity**
1. **Graph Construction:** \(O(E)\), where \(E\) is the number of edges.
2. **BFS Traversal:** \(O(E + V)\) for each `help` call, where \(V\) is the number of nodes.
3. **Binary Search:** Runs \(O(log(max\_weight - min\_weight))\) iterations.

#### **Total Complexity: O(E+V)⋅O(log(max_weight−min_weight))**  


## **Space Complexity**
- **Graph Storage:** \(O(E + V)\) for adjacency list.
- **Visited Array:** \(O(V)\).  
**Total Space: \(O(E + V)\).**

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# **PLEASE UPVOTE IF YOU LIKED IT ✅**
['C++']