<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>

struct GValue {
    int d, v;
    
    GValue(int dist) {
        if (dist == -1) {
            d = 0;
            v = 1;
        } else {
            d = dist;
            v = 0;
        }
    }
    
    GValue() : GValue(INT32_MAX) {}
    
    bool operator==(const GValue& other) const {
        return d == other.d && v == other.v;
    }
    
    bool operator<(const GValue& other) const {
        if (d + v == other.d + other.v) return d < other.d;
        return d + v < other.d + other.v;
    }
    
    GValue operator+(const GValue& other) const {
        GValue res = GValue(d + other.d);
        res.v = v + other.v;
        return res;
    }
};

class Solution {
    vector<int> g;
    vector<bool> closed;
    set<pair<int, int>> open;
public:
    vector<vector<int>> modifiedGraphEdges(int n, vector<vector<int>>& edges, int source, int destination, int target) {
        g.resize(n);
        closed.resize(n);
        
        vector<vector<pair<int, GValue>>> es(n);
        vector<unordered_map<int, int>> adj(n);
        for (auto& e : edges) {
            es[e[0]].emplace_back(e[1], GValue(e[2]));
            es[e[1]].emplace_back(e[0], GValue(e[2]));
            if (e[2] != -1) {
                adj[e[0]][e[1]] = max(e[2], 1);
                adj[e[1]][e[0]] = max(e[2], 1);
            }
        }
        vector<int> path;
        GValue res = find(source, destination, es, path);
        if (res == GValue()) return {};
        // for (auto& e : edges)
        //     if (e[2] == -1)
        //         e[2] = 1;
        int stable_sp = shortest_path(source, destination, adj);
        if (stable_sp < target) {
            return {};
        }
        if (res.d + res.v > target) return {};
        // cout << "ps: " << path.size() << endl;
        // for (int pv : path) cout << " " << pv;
        // cout << endl;
        // cout << "target=" << target << endl;
        bool found = stable_sp == target;
        for (auto& e : edges) {
            if (e[2] == -1) {
                // cout << "(" << e[0] << ", " << e[1] << ")" << endl;
                int d1 = min(shortest_path(source, e[0], adj), shortest_path(source, e[1], adj));
                if (d1 == INT32_MAX) {
                    adj[e[0]][e[1]] = adj[e[1]][e[0]] = 1;
                    continue;
                }
                // cout << "d1: " << d1 << endl;
                int d2 = min(shortest_path(e[0], destination, adj), shortest_path(e[1], destination, adj));
                // cout << "d2: " << d2 << endl;
                if (d2 == INT32_MAX) {
                    adj[e[0]][e[1]] = adj[e[1]][e[0]] = 1;
                    continue;
                }
                
                if (d1 + d2 < target) {
                    adj[e[0]][e[1]] = adj[e[1]][e[0]] = target - d1 - d2;
                    found = true;
                    break;
                } else {
                    adj[e[0]][e[1]] = adj[e[1]][e[0]] = 1;
                }
            }
        }
        
        if (!found) return {};
        
        for (auto& e : edges) 
            if (e[2] == -1 && adj[e[0]][e[1]] == 0) {
                adj[e[1]][e[0]] = adj[e[0]][e[1]] = target + 1;
            }
        edges.clear();
        for (int s = 0; s < adj.size(); ++s)
            for (auto& p : adj[s]) 
                if (s < p.first)
                    edges.push_back({s, p.first, p.second});
        return edges;
    }
    
    int shortest_path(int start, int goal, vector<unordered_map<int, int>>& es) {
        g.assign(g.size(), INT32_MAX);
        closed.assign(closed.size(), false);
        open.clear();
        
        open.insert({0, start});
        g[start] = 0;
        
        while (!open.empty()) {
            int gv = open.begin()->first;
            int v = open.begin()->second;
            closed[v] = true;
            if (v == goal) break;
            open.erase(open.begin());
            for (auto& np : es[v]) {
                if (!closed[np.first] && gv + np.second < g[np.first]) {
                    auto it = open.find({g[np.first], np.first});
                    if (it != open.end()) open.erase(it);
                    g[np.first] = gv + np.second;
                    open.insert({g[np.first], np.first});
                }
            }
        }
        return g[goal];
    }
    
    GValue find(int start, int goal, vector<vector<pair<int, GValue>>>& es, vector<int>& path) {
        vector<GValue> g(es.size());
        vector<int> bps(es.size(), -1);
        vector<bool> closed(es.size(), false);
        set<pair<GValue, int>> open; 
        
        open.insert({GValue(0), start});
        bps[start] = start;
        
        while (!open.empty()) {
            GValue gv = open.begin()->first;
            int v = open.begin()->second;
            closed[v] = true;
            // cout << v << " gv: " << gv.d << " " << gv.v << endl;
            if (v == goal) {
                int x = goal;
                while (bps[x] != x) {
                    path.push_back(x);
                    x = bps[x];
                }
                path.push_back(start);
                reverse(path.begin(), path.end());
                break;
            }
            open.erase(open.begin());
            for (auto& np : es[v]) {
                if (!closed[np.first] && gv + np.second < g[np.first]) {
                    bps[np.first] = v;
                    auto it = open.find({g[np.first], np.first});
                    if (it != open.end()) open.erase(it);
                    g[np.first] = gv + np.second;
                    open.insert({g[np.first], np.first});
                }
            }
        }
        return g[goal];
    }
};
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