<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:
    static constexpr int INF = 1'000'100'000;
    vector<vector<int>> modifiedGraphEdges(int n, vector<vector<int>>& edges, int source, int destination, int target) {
        g.assign(n, vector<int>(n, INF));
        for (const auto& e : edges) {
            int u = e[0];
            int v = e[1];
            int w = e[2];
            if (w < 0) {
                modify.emplace_back(u, v);
            } else {
                g[u][v] = g[v][u] = w;
            }
        }

        for (auto [u, v] : modify) g[u][v] = g[v][u] = 1;
        if (calc_dist(source, destination) > target) return {};
        for (auto [u, v] : modify) g[u][v] = g[v][u] = INF;
        if (calc_dist(source, destination) < target) return {};
        // f(1) <= target, f(INF) >= target
        int l = 1, r = INF + 1;
        while (r - l > 1) {
            int m = (l + r) >> 1;
            for (auto [u, v] : modify) g[u][v] = g[v][u] = m;
            if (calc_dist(source, destination) <= target) l = m;
            else r = m;
        }
        const int base = l;
        l = 0, r = modify.size() + 1;
        while (r - l > 1) {
            int m = (l + r) >> 1;
            for (int i = 0; i < m; ++i) {
                auto [u, v] = modify[i];
                g[u][v] = g[v][u] = base + 1;
            }
            for (int i = m; i < modify.size(); ++i) {
                auto [u, v] = modify[i];
                g[u][v] = g[v][u] = base;
            }
            if (calc_dist(source, destination) <= target) l = m;
            else r = m;
        }

        for (auto& e : edges) {
            if (e[2] < 0) {
                e[2] = base;
                if (l) {
                    l--;
                    e[2]++;
                }
            }
        }
        return std::move(edges);
    }

    int calc_dist(int from, int to) {
        const int n = g.size();
        dist.assign(n, INF);
        dist[from] = 0;
        for (int iter = 0; iter < n; ++iter) {
            auto* ds = reinterpret_cast<unsigned*>(&dist[0]);
            const int u = std::min_element(ds, ds + n) - ds;
            const int du = dist[u];
            if (du == INF || u == to) break;
            dist[u] = -1;
            for (int v = 0; v < n; ++v) {
                dist[v] = min(dist[v], du + g[u][v]);
            }
        }
        return dist[to];
    }

    vector<vector<int>> g;
    vector<pair<int, int>> modify;
    vector<int> dist;
};
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