<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>

template <class T>
struct edge {
    int node;
    T weight;

    edge() {}

    edge(int _node, T _weight) : node(_node), weight(_weight) {}
};

template <class T, bool directed = true>
struct dijkstra {  // 0 or 1-indexed, doesn't matter
    int n;
    vector<vector<edge<T>>> edges;
    const T INF = numeric_limits<T>::max() / 2;

    dijkstra() {}

    dijkstra(int N) {
        n = N;
        edges.resize(n + 1);
    }

    void add_edge(int from, int to, T dist) {
        edges[from].emplace_back(to, dist);
        if (!directed)
            edges[to].emplace_back(from, dist);
    }

    vector<T> dist(vector<int> sources) {
        vector<T> res(n + 1, INF);
        priority_queue<pair<T, int>, vector<pair<T, int>>, greater<pair<T, int>>> pq;
        for (int source : sources) {
            res[source] = 0;
            pq.push({0, source});
        }
        while (!pq.empty()) {
            auto cur = pq.top();
            pq.pop();
            if (cur.first > res[cur.second])
                continue;
            for (auto e : edges[cur.second]) {
                if (res[e.node] > cur.first + e.weight) {
                    res[e.node] = cur.first + e.weight;
                    pq.push({res[e.node], e.node});
                }
            }
        }
        return res;
    }

    T dist(int source, int dest) {
        return dist({source})[dest];
    }

    pair<T, vector<int>> path(int source, int dest) {
        // Return the distance from source to dest, along with a shortest path
        vector<T> res(n + 1, INF);
        vector<int> before(n + 1);
        priority_queue<pair<T, int>, vector<pair<T, int>>, greater<pair<T, int>>> pq;
        res[source] = 0;
        pq.push({0, source});
        while (!pq.empty()) {
            auto cur = pq.top();
            pq.pop();
            if (cur.first > res[cur.second])
                continue;
            for (auto e : edges[cur.second]) {
                if (res[e.node] > cur.first + e.weight) {
                    res[e.node] = cur.first + e.weight;
                    before[e.node] = cur.second;
                    pq.push({res[e.node], e.node});
                }
            }
        }
        if (res[dest] == INF)
            return make_pair(-1, vector<int>());
        vector<int> path = {dest};
        while (path.back() != source)
            path.push_back(before[path.back()]);
        reverse(path.begin(), path.end());
        return make_pair(res[dest], path);
    }
};

class Solution {
public:
    const int mx = 2 * int(1e9);
    vector<vector<int>> modifiedGraphEdges(int n, vector<vector<int>>& edges, int source, int destination, int target) {
        dijkstra<long long, false> graph(n);
        for (auto &e : edges) {
            if (e[2] != -1) {
                graph.add_edge(e[0], e[1], 1LL * e[2]);
            }
        }
        auto distance = graph.dist(source, destination);
        if (distance < target) {
            return {};
        }
        if (distance == 1LL * target) {
            for (auto &e : edges) {
                if (e[2] == -1) {
                    e[2] = mx;
                }
            }
            return edges;
        }
        for (auto &e : edges) {
            if (e[2] == -1) {
                e[2] = 1;
                graph.add_edge(e[0], e[1], e[2]);
                distance = graph.dist(source, destination);
                if (distance <= 1LL * target) {
                    e[2] += target - distance;
                    for (auto &e : edges) {
                        if (e[2] == -1) {
                            e[2] = mx;
                        }
                    }
                    return edges;
                }
            }
        }
        return {};
    }
};
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