<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 {
public:
    // TC = log(E)*(E log(V)) 

    #define F first
    #define S second
    #define pb push_back
    #define pii pair<int,int>
    #define vi vector<int>

    typedef long long int ll;
    const int INF = 2e9;

    vi parent, dist; // used in dijkstra
    vector<pii> neg_edges; 
    vector<vector<pii>> G; //adj list
    
    void Dijkstra(int n, int src){ //O(ElogV)
        dist.clear(); dist.resize(n,INF);
        parent.clear(); parent.resize(n,-1);
        set<pii> st;
        st.insert({0,src});
        vi vis(n);
        dist[src]=0; 
        while(!st.empty()){
            int dist_so_far = st.begin()->F;
            int node = st.begin()->S;
            st.erase(st.begin());
            if(vis[node])continue;
            vis[node]=1;
            for(auto [cur_node, w] : G[node]){
                if(dist[cur_node]>dist_so_far+w){
                    dist[cur_node]=dist_so_far+w;
                    parent[cur_node]=node;
                    st.insert({dist[cur_node],cur_node});
                }
            }
        }
    }

    //O(E)+O(ElogV)=O(ElogV)
    void ModifyG(int k, int src, int n, vector<vector<int>>& edges){
        G.clear(); G.resize(n);
        //reconstrcut G with first k neg_edges included with weight 1
        for(int i=0;i<k;i++){
            G[neg_edges[i].F].pb({neg_edges[i].S,1});
            G[neg_edges[i].S].pb({neg_edges[i].F,1});
        }
        //and the pos_edges included as normal
        for(auto e:edges){
            if(e[2]<0)continue;
            G[e[0]].pb({e[1],e[2]});
            G[e[1]].pb({e[0],e[2]});
        }

        //run Dijkstra on this modified G
        Dijkstra(n, src);
    }   

    //O(E)
    void Finalize(vector<vector<int>>& edges, int k, int target, int d){
        for(int i=0;i<edges.size();i++){
            if(edges[i][2]>0) continue;
            if(k==0){
                edges[i][2]=INF;
                continue;
            }
            if(k==1) 
                edges[i][2]=target-d+1;
            else edges[i][2]=1;
            
            k--;
        }
    }

    void print_path(int src, int dst){
        int node=dst;
        while(node!=-1){
            cout<<node<<"<-";
            node=parent[node];   
        }
        cout<<endl;
    }

    vector<vector<int>> modifiedGraphEdges(int n, vector<vector<int>>& edges, int source, int destination, int target) {
        G.clear(); G.resize(n);
        //form neg_edges list, and pos_edge weights Graph
        for(auto e : edges){ //O(E)
            if(e[2] == -1){
                neg_edges.pb({e[0],e[1]});
            }
            else{
                G[e[0]].pb({e[1],e[2]});
                G[e[1]].pb({e[0],e[2]});
            }
        }

        //edge-case1: min dist with pos edges < target
        Dijkstra(n, source); //O(ElogV)
        if(dist[destination] < target) return {};
        
        //binary search over the 'number' of neg_edges
        int l=0, r=neg_edges.size();
        while(l<r){ //bin_search complexity: O(logE). inside O(ElogV) TC
            int mid=(l+r)>>1;
            //add 'mid' number of negative edges to G with w=1
            ModifyG(mid, source, n, edges);
            //now if dist[destination] becomes <= target
            //narrow down to the smallest set: r=mid, and continue
            if(dist[destination] <= target) r=mid;
            //else dist[destination] > target:
            //two cases: 
            //1)not connected to dst yet, so dist[dst] = INF
            //2)the '1' weight edges occur such that sum > target
            else l=mid+1;
        }
        // //check for edge-case:2 
        ModifyG(r, source, n, edges);
        if(dist[destination] > target) return {};
        
        Finalize(edges, l, target, dist[destination]);
        return edges;
    }
};
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