<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>

class Solution {
private:
    int source, dest;
    int n;
    vector<vector<int>> edges;
    int totalUnknowns = 0;
    void init(int _n, vector<vector<int>>& _edges, int _source, int _dest){
        n = _n;
        edges = _edges;
        source = _source;
        dest = _dest;
        totalUnknowns = 0;
        for(vector<int>& edge:edges){
            if(edge[2] == -1)
                totalUnknowns++;
        }
    }

    vector<vector<int>> edgelist;
    vector<vector<pair<int,int>>> adjlist;
    long long dijk(){
        priority_queue<pair<long long,int>> pq;
        pq.push({0, source});
        vector<bool> visited(n, false);
        while(!pq.empty()){
            int x = pq.top().second;
            long long d = -pq.top().first;
            pq.pop();
            if(visited[x]){
                continue;
            }
            visited[x]=true;
            if(x == dest){
                return d;
            }
            for(pair<int,int> p : adjlist[x]){
                int y = p.first;
                int weight = p.second;
                pq.push({-(d + weight), y});
            }
        }
        // no possible path
        return -1;
    }
    long long f(long long w){
        edgelist.clear();
        adjlist.resize(n);
        for(int i=0;i<n;i++)adjlist[i].clear();
        int lower = 0, upper = 0, lower_count = 0;
        if(totalUnknowns != 0){
            lower = w/totalUnknowns;
            upper = lower + 1;
            lower_count = (int)(((long long)upper)*totalUnknowns - w);
        }
        int unknown_counter = 0;
        for(vector<int>& edge:edges){
            int a=edge[0];
            int b=edge[1];
            int w=edge[2];
            if (w == -1){
                w = 1 + ((unknown_counter < lower_count) ? lower : upper);
                unknown_counter++;
            }
            adjlist[a].push_back(make_pair(b, w));
            adjlist[b].push_back(make_pair(a, w));
            edgelist.push_back({a,b,w});
        }
        return dijk();
    }
public:
    vector<vector<int>> modifiedGraphEdges(int _n, vector<vector<int>>& _edges, int _source, int _destination, int target) {
        init(_n, _edges, _source, _destination);
        long long lower = 0;
        long long upper = ((long long)totalUnknowns)*target;
        if(f(lower) == -1 || f(lower) > target || f(upper) < target) {
            return {};
        }
        while(lower <= upper){
            long long m = (lower + upper) / 2;
            long long cur_target = f(m);
            if(cur_target == target){
                return edgelist;
            }
            if(cur_target < target){
                lower = m+1;
            }else{
                upper = m-1;
            }
        }
        assert(false);
        return {};
    }
};
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