<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 customLess {
public:
    bool operator() (pair<int, long long> a, pair<int, long long> b) {
        // vector has 2 ints --> (node, dist)
        return a.second < b.second;
    }
};

struct pair_hash {
    template <class T1, class T2>
    std::size_t operator () (const std::pair<T1,T2> &p) const {
        auto h1 = std::hash<T1>{}(p.first);
        auto h2 = std::hash<T2>{}(p.second);

        // Mainly for demonstration purposes, i.e. works but is overly simple
        // In the real world, use sth. like boost.hash_combine
        return (h1 << 32) ^ h2;
    }
};

class Solution {
public:
    const int BIG = 2e9;
    long long shortestPath(int n, int source, int destination,
        const vector<vector<pair<int, int>>>& graph) {
        vector<vector<long long>> edge_map(n, vector<long long>(n, BIG));
        vector<long long> min_dist(n, BIG);
        vector<bool> visited(n, false);
        priority_queue<pair<int, int>, vector<pair<int, int>>, greater<>> pq;
        min_dist[source] = 0;
        pq.emplace(0, source);

        while (!pq.empty()) {
            auto [d, u] = pq.top();
            pq.pop();
            if (d > min_dist[u]) continue;
            for (const auto& [v, w] : graph[u]) {
                if (d + w < min_dist[v]) {
                    min_dist[v] = d + w;
                    pq.emplace(min_dist[v], v);
                }
            }
        }
        return min_dist[destination];
    }
    vector<vector<int>> modifiedGraphEdges(int n, vector<vector<int>>& edges, int source, int destination, int target) {
        // first, run shortest path for a graph without any negative edges
        // we will have 4 scenarios:
        // (1) no path between source/dest --> proceed normally
        // (2) path exists, < target --> return []
        // (3) path exists, == target --> all negative edges can be assigned a value == target and return
        // (4) path exists, > target --> assign 1 to all negative edges first, run shortest path, then re-visit
        //                               to update edges to reach target for shortest distance
        // edge_map, edge_map[i][j] = weight

        vector<vector<pair<int, int>>> graph(n);
        for (const auto& edge : edges) {
            if (edge[2] != -1) {
                graph[edge[0]].emplace_back(edge[1], edge[2]);
                graph[edge[1]].emplace_back(edge[0], edge[2]);
            }
        }

        // find a first shortest distance, without negative edges setting to positive values
        long long shortest = shortestPath(n, source, destination, graph);
        if (shortest < target) return {};
        
        bool matched = (shortest == target);
        for (vector<int>& edge : edges) {
            if (edge[2] > 0) continue;
            // if we have already find a solution, set all negative edges to BIG
            // otherwise set it to 1 and try to find a shortest path
            edge[2] = matched ? BIG : 1;
            graph[edge[0]].emplace_back(edge[1], edge[2]);
            graph[edge[1]].emplace_back(edge[0], edge[2]);

            if (!matched) {
                long long new_shortest = shortestPath(n, source, destination, graph);
                if (new_shortest > 0 && new_shortest <= target) {
                    matched = true;
                    edge[2] += target - new_shortest;
                }
            }
        }
        return matched ? edges : vector<vector<int>> {};
    }
};
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