<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 Node
{
    public:
    int curr;
    int dist;
    int negativeWtEdgeIndex;
    Node(int c, int d, int i) : curr(c), dist(d), negativeWtEdgeIndex(i) {}
};
class GreaterNode
{
    public:
    bool operator() (const Node& p1, const Node& p2) const
    {
        return p1.dist > p2.dist;
    }
};
class Solution {
    typedef priority_queue<Node, vector<Node>, GreaterNode> minPQ;
public:
    int dijkstra(vector<vector<vector<int>>>& adj, int src, int dest, int& resNegEdge)
    {
        int n = adj.size();
        vector<int> visited(n, INT_MAX);
        minPQ pq;
        pq.push(Node(src, 0, -1));
        visited[src] = 0;
        int negEdge1 = -1;
        while(!pq.empty())
        {
            auto& top = pq.top();
            int curr = top.curr;
            int dist = top.dist;
            negEdge1 = top.negativeWtEdgeIndex;
            pq.pop();
            if(dist > visited[curr]) continue;
            if(curr == dest) break;
            for(auto& neighbor : adj[curr])
            {
                int next = neighbor[0];
                int nextDist = neighbor[1];
                if(visited[next] > dist + nextDist)
                {
                    int negEdge = negEdge1;
                    if(neighbor[2] >= 0 && negEdge == -1) negEdge = neighbor[2];
                    visited[next] = dist + nextDist;
                    pq.push(Node(next, visited[next], negEdge));
                }
            }
        }
        resNegEdge = negEdge1;
        return visited[dest];
    }
    void modifyNegativeEdges(vector<vector<int>>& edges, vector<vector<vector<int>>>& adj, unordered_map<int, int>& negEdgeIdxToAdjIterMap1)
    {
        int iter = -1;
        for(auto& e : edges)
        {
            iter++;
            if(e[2] != -1) continue;
            int idx = negEdgeIdxToAdjIterMap1[iter];
            e[2] = adj[e[0]][idx][1];
        }
    }//edges[idx][2] = maxWt;
    bool modifyNegativeEdgesHelper(vector<vector<int>>& edges, vector<vector<vector<int>>>& adj, int src, int dest, int target, unordered_map<int, int>& negEdgeIdxToAdjIterMap1, unordered_map<int, int>& negEdgeIdxToAdjIterMap2, int shortestDist, int& resNegEdge)
    {
        if(shortestDist >= target)
        {
            modifyNegativeEdges(edges, adj, negEdgeIdxToAdjIterMap1);
            return true;
        }
        // add to the negative edges now
        if(resNegEdge == -1) return false;
        int idx = resNegEdge;
        auto& e = edges[idx];
        int idx0 = negEdgeIdxToAdjIterMap1[idx];
        int idx1 = negEdgeIdxToAdjIterMap2[idx];
        int save = adj[e[0]][idx0][1];
        int maxWt = target - shortestDist + save;
        int saveIdx0 = adj[e[0]][idx0][2];
        int saveIdx1 = adj[e[1]][idx1][2];
        adj[e[0]][idx0][1] = maxWt;
        adj[e[1]][idx1][1] = maxWt;
        adj[e[0]][idx0][2] = adj[e[1]][idx1][2] = -1;
        int resNegEdge1 = -1;
        int shortestDist1 = dijkstra(adj, src, dest, resNegEdge1);
        if(modifyNegativeEdgesHelper(edges, adj, src, dest, target, negEdgeIdxToAdjIterMap1, negEdgeIdxToAdjIterMap2, shortestDist1, resNegEdge1)) return true;
        adj[e[0]][idx0][1] = save;
        adj[e[1]][idx1][1] = save;
        adj[e[0]][idx0][2] = saveIdx0;
        adj[e[1]][idx1][2] = saveIdx1;
        return false;
    }
    vector<vector<int>> modifiedGraphEdges(int n, vector<vector<int>>& edges, int src, int dest, int target) {
        // Idea is to first traverse the graph 
        // considering the shortest path including the negative edges
        // by considering the negative edges to be of weight 1 keeping track
        // of the edge indices with negative weights.
        // Then increase the weights to the negative edges in this path
        // such that 1 is assigned to all except one
        // which is assigned the difference remaining.
        // But ensure that it does not decrease the shortest path lesser than target. To maintain this, dijkstra is run multiple times.
        int sz = edges.size();
        vector<vector<vector<int>>> adj(n);
        // adj[i] : vector of vectors whose each element is like
        // adj[i][j][0] : neighbor
        // adj[i][j][1] : neighborWt
        // adj[i][j][2] : edgeIdx for negative wts else -1
        int iter = 0;
        unordered_map<int, int> negEdgeIdxToAdjIterMap1, negEdgeIdxToAdjIterMap2;
        for(auto& e : edges)
        {
            int idx = (e[2] == -1) ? iter : -1;
            int wt = (e[2] == -1) ? 1 : e[2];
            negEdgeIdxToAdjIterMap1[idx] = adj[e[0]].size();
            negEdgeIdxToAdjIterMap2[idx] = adj[e[1]].size();
            adj[e[0]].push_back({e[1], wt, idx});
            adj[e[1]].push_back({e[0], wt, idx});
            iter++;
        }
        int resNegEdge = -1;
        int shortestDist = dijkstra(adj, src, dest, resNegEdge);
        if(shortestDist > target) return {};
        if(modifyNegativeEdgesHelper(edges, adj, src, dest, target, negEdgeIdxToAdjIterMap1, negEdgeIdxToAdjIterMap2, shortestDist, resNegEdge)) return edges;
        return {};
    }
};
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