<p>You are given an <strong>undirected weighted</strong> <strong>connected</strong> graph containing <code>n</code> nodes labeled from <code>0</code> to <code>n - 1</code>, and an integer array <code>edges</code> where <code>edges[i] = [a<sub>i</sub>, b<sub>i</sub>, w<sub>i</sub>]</code> indicates that there is an edge between nodes <code>a<sub>i</sub></code> and <code>b<sub>i</sub></code> with weight <code>w<sub>i</sub></code>.</p>

<p>Some edges have a weight of <code>-1</code> (<code>w<sub>i</sub> = -1</code>), while others have a <strong>positive</strong> weight (<code>w<sub>i</sub> &gt; 0</code>).</p>

<p>Your task is to modify <strong>all edges</strong> with a weight of <code>-1</code> by assigning them <strong>positive integer values </strong>in the range <code>[1, 2 * 10<sup>9</sup>]</code> so that the <strong>shortest distance</strong> between the nodes <code>source</code> and <code>destination</code> becomes equal to an integer <code>target</code>. If there are <strong>multiple</strong> <strong>modifications</strong> that make the shortest distance between <code>source</code> and <code>destination</code> equal to <code>target</code>, any of them will be considered correct.</p>

<p>Return <em>an array containing all edges (even unmodified ones) in any order if it is possible to make the shortest distance from </em><code>source</code><em> to </em><code>destination</code><em> equal to </em><code>target</code><em>, or an <strong>empty array</strong> if it&#39;s impossible.</em></p>

<p><strong>Note:</strong> You are not allowed to modify the weights of edges with initial positive weights.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<p><strong class="example"><img alt="" src="https://assets.leetcode.com/uploads/2023/04/18/graph.png" style="width: 300px; height: 300px;" /></strong></p>

<pre>
<strong>Input:</strong> n = 5, edges = [[4,1,-1],[2,0,-1],[0,3,-1],[4,3,-1]], source = 0, destination = 1, target = 5
<strong>Output:</strong> [[4,1,1],[2,0,1],[0,3,3],[4,3,1]]
<strong>Explanation:</strong> The graph above shows a possible modification to the edges, making the distance from 0 to 1 equal to 5.
</pre>

<p><strong class="example">Example 2:</strong></p>

<p><strong class="example"><img alt="" src="https://assets.leetcode.com/uploads/2023/04/18/graph-2.png" style="width: 300px; height: 300px;" /></strong></p>

<pre>
<strong>Input:</strong> n = 3, edges = [[0,1,-1],[0,2,5]], source = 0, destination = 2, target = 6
<strong>Output:</strong> []
<strong>Explanation:</strong> The graph above contains the initial edges. It is not possible to make the distance from 0 to 2 equal to 6 by modifying the edge with weight -1. So, an empty array is returned.
</pre>

<p><strong class="example">Example 3:</strong></p>

<p><strong class="example"><img alt="" src="https://assets.leetcode.com/uploads/2023/04/19/graph-3.png" style="width: 300px; height: 300px;" /></strong></p>

<pre>
<strong>Input:</strong> n = 4, edges = [[1,0,4],[1,2,3],[2,3,5],[0,3,-1]], source = 0, destination = 2, target = 6
<strong>Output:</strong> [[1,0,4],[1,2,3],[2,3,5],[0,3,1]]
<strong>Explanation:</strong> The graph above shows a modified graph having the shortest distance from 0 to 2 as 6.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= n &lt;= 100</code></li>
	<li><code><font face="monospace">1 &lt;= edges.length &lt;= n * (n - 1) / 2</font></code></li>
	<li><code>edges[i].length == 3</code></li>
	<li><code>0 &lt;= a<sub>i</sub>, b<sub>i&nbsp;</sub>&lt;&nbsp;n</code></li>
	<li><code><font face="monospace">w<sub>i</sub>&nbsp;= -1&nbsp;</font></code>or <code><font face="monospace">1 &lt;= w<sub>i&nbsp;</sub>&lt;= 10<sup><span style="font-size: 10.8333px;">7</span></sup></font></code></li>
	<li><code>a<sub>i&nbsp;</sub>!=&nbsp;b<sub>i</sub></code></li>
	<li><code>0 &lt;= source, destination &lt; n</code></li>
	<li><code>source != destination</code></li>
	<li><code><font face="monospace">1 &lt;= target &lt;= 10<sup>9</sup></font></code></li>
	<li>The graph is connected, and there are no self-loops or repeated edges</li>
</ul>

// static const auto fast = []() {
//     std::ios::sync_with_stdio(false);
//     std::cin.tie(nullptr);
//     std::cout.tie(nullptr);
//     return 0;
// }();

class Solution {
private:
    // static constexpr int HV = 1e9 + 1;

    int find_sd(int source, int dest, vector<int> &sd, const vector<vector<pair<int,int>>> &adj, int limit) {
        priority_queue<pair<int,int>, vector<pair<int,int>>, greater<>> pq;
            pq.emplace(sd[source], source);

        while(!pq.empty()) {
            auto x = pq.top(); pq.pop();
            if (sd[x.second] < x.first) continue;
            if (x.second == dest || x.first > limit) return x.first;
            sd[x.second] = x.first;
            for (const auto &e: adj[x.second]) {
                int d = x.first+e.first;
                if (sd[e.second] < d || d > limit) continue;
                pq.emplace(d, e.second);
            }
        }
        return limit+1;
    }

    
public:
    vector<vector<int>> modifiedGraphEdges(int n, vector<vector<int>>& edges, int source, int destination, int target) {
        vector<vector<pair<int,int>>> adj(n);
        for (const auto &e : edges) {
            if (e[2] != -1) {
                adj[e[0]].emplace_back(e[2], e[1]);
                adj[e[1]].emplace_back(e[2], e[0]);
            }
        }
        vector<int> sd(n, target+1);
        sd[source] = 0;
        int cur_sd = find_sd(source, destination, sd, adj, target);
        // cout << "1: " << cur_sd << '\n';
        if (cur_sd < target) return {};
        bool matchFound = (cur_sd == target);

        for (auto &e: edges) {
            if(e[2] != -1) continue;
            if (matchFound) {
                e[2] = target + 1;
                continue;
            }
            e[2] = 1;
            adj[e[0]].emplace_back(e[2], e[1]);
            adj[e[1]].emplace_back(e[2], e[0]);
            
            if (sd[e[0]] == sd[e[1]]) continue;
            int node = (sd[e[0]] < sd[e[1]]) ? e[0] : e[1];
            cur_sd = find_sd(node, destination, sd, adj, target);
            // cout << "2: " << cur_sd << '\n';
            if (cur_sd <= target) {
                e[2] += target - cur_sd;
                matchFound = true;
            }
        }
        
        return matchFound ? edges : vector<vector<int>>{};
    }
};




// class Solution {
// public:
//     const int INF = 2e9;

//     vector<vector<int>> modifiedGraphEdges(int n, vector<vector<int>>& edges,
//                                            int source, int destination,
//                                            int target) {
//         vector<vector<pair<int, int>>> graph(n);

//         // Step 1: Build the graph, excluding edges with -1 weights
//         for (const auto& edge : edges) {
//             if (edge[2] != -1) {
//                 graph[edge[0]].emplace_back(edge[1], edge[2]);
//                 graph[edge[1]].emplace_back(edge[0], edge[2]);
//             }
//         }

//         // Step 2: Compute the initial shortest distance from source to
//         // destination
//         int currentShortestDistance =
//             runDijkstra(n, source, destination, graph);
//         if (currentShortestDistance < target) {
//             return vector<vector<int>>();
//         }

//         bool matchesTarget = (currentShortestDistance == target);

//         // Step 3: Iterate through each edge to adjust its weight if necessary
//         for (auto& edge : edges) {
//             if (edge[2] != -1)
//                 continue;  // Skip edges with already known weights

//             // Set edge weight to a large value if current distance matches
//             // target, else set to 1
//             edge[2] = matchesTarget ? INF : 1;
//             graph[edge[0]].emplace_back(edge[1], edge[2]);
//             graph[edge[1]].emplace_back(edge[0], edge[2]);

//             // Step 4: If current shortest distance does not match target
//             if (!matchesTarget) {
//                 // Compute the new shortest distance with the updated edge
//                 // weight
//                 int newDistance = runDijkstra(n, source, destination, graph);
//                 // If the new distance is within the target range, update edge
//                 // weight to match target
//                 if (newDistance <= target) {
//                     matchesTarget = true;
//                     edge[2] += target - newDistance;
//                 }
//             }
//         }

//         // Return modified edges if the target distance is achieved, otherwise
//         // return an empty result
//         return matchesTarget ? edges : vector<vector<int>>();
//     }

// private:
//     int runDijkstra(int n, int source, int destination,
//                     const vector<vector<pair<int, int>>>& graph) {
//         vector<int> minDistance(n, INF);
//         priority_queue<pair<int, int>, vector<pair<int, int>>, greater<>>
//             minHeap;

//         minDistance[source] = 0;
//         minHeap.emplace(0, source);

//         while (!minHeap.empty()) {
//             auto [d, u] = minHeap.top();
//             minHeap.pop();

//             if (d > minDistance[u]) continue;

//             for (const auto& [v, weight] : graph[u]) {
//                 if (d + weight < minDistance[v]) {
//                     minDistance[v] = d + weight;
//                     minHeap.emplace(minDistance[v], v);
//                 }
//             }
//         }

//         return minDistance[destination];
//     }
// };


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