<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<typename T>
bool
set_min(T& result, T candidate) {
    bool condition = candidate < result;
    if (condition) {
        result = candidate;
    }
    return condition;
}

class Solution {
public:
    vector<vector<int>> modifiedGraphEdges(int n, vector<vector<int>>& edges, int source, int destination, int target) {
        static constexpr const int not_in_heap = numeric_limits<int>::max();
        static constexpr const long invalid_cost = numeric_limits<long>::max();
        static constexpr const int max_edge_cost = 2000000000;
        struct Path {
            long cost;
            int to;
            int index;

            Path(int to) : cost(invalid_cost), to(to), index(not_in_heap) {
            }
        };
        vector<pair<int, int>> missings;
        vector<vector<pair<int, int>>> connected(n);
        vector<bool> order(n * n, false);
        for (const vector<int>& edge : edges) {
            int from = edge[0];
            int to = edge[1];
            int cost = edge[2];
            order[from * n + to] = !(from < to);
            order[to * n + from] = !(from < to);
            if (cost != -1) {
                connected[from].emplace_back(to, cost);
                connected[to].emplace_back(from, cost);
            } else {
                missings.emplace_back(from, to);
            }
        }
        vector<Path> paths;
        paths.reserve(n);
        for (int i = 0; i < n; ++i) {
            paths.emplace_back(i);
        }
        vector<Path*> heap;
        heap.reserve(n);
        auto sift_down = [&heap](int i){
            for (int j = 2 * i + 1, je = heap.size(); j < je; ) {
                j += (j + 1 < je && heap[j + 1]->cost < heap[j]->cost);
                if (heap[j]->cost < heap[i]->cost) {
                    swap(heap[j]->index, heap[i]->index);
                    swap(heap[j], heap[i]);
                    i = j;
                    j = 2 * i + 1;
                } else {
                    break;
                }
            }
        };
        auto heapify = [&heap, &sift_down](){
            for (int i = heap.size(); i != 0; ) {
                --i;
                sift_down(i);
            }
        };
        auto sift_up = [&heap](int i){
            while (i != 0) {
                int j = (i - 1) / 2;
                if (heap[i]->cost < heap[j]->cost) {
                    swap(heap[i]->index, heap[j]->index);
                    swap(heap[i], heap[j]);
                    i = j;
                } else {
                    break;
                }
            }
        };
        {
            Path* path = &paths[source];
            path->cost = 0L;
            path->index = 0;
            heap.emplace_back(path);
            sift_up(0);
        }
        auto dijkstra = [&connected, &paths, &heap, &sift_down, &sift_up](){
            while (!heap.empty()) {
                Path* first = heap[0];
                first->index = not_in_heap;
                if (int j = heap.size() - 1; j != 0) {
                    heap[j]->index = 0;
                    heap[0] = heap[j];
                    heap.pop_back();
                    sift_down(0);
                } else {
                    heap.pop_back();
                }
                long cost = first->cost;
                for (const pair<int, int>& p : connected[first->to]) {
                    if (set_min(paths[p.first].cost, cost + p.second)) {
                        int& i = paths[p.first].index;
                        if (i != not_in_heap) {
                        } else {
                            i = heap.size();
                            heap.emplace_back(&paths[p.first]);
                        }
                        sift_up(i);
                    }
                }
            }
        };
        dijkstra();
        vector<vector<int>> result;
        if (paths[destination].cost >= target) {
            if (paths[destination].cost > target) {
                while (!missings.empty()) {
                    pair<int, int> last = missings.back();
                    missings.pop_back();
                    connected[last.first].emplace_back(last.second, 1);
                    connected[last.second].emplace_back(last.first, 1);
                    if (paths[last.first].cost != invalid_cost) {
                        int j = heap.size();
                        paths[last.first].index = j;
                        heap.emplace_back(&paths[last.first]);
                        sift_up(j);
                    }
                    if (paths[last.second].cost != invalid_cost) {
                        int j = heap.size();
                        paths[last.second].index = j;
                        heap.emplace_back(&paths[last.second]);
                        sift_up(j);
                    }
                    dijkstra();
                    //cout << last.first << ", " << last.second << '\n';
                    //cout << paths[destination].cost << '\n';
                    if (paths[destination].cost > target) {
                    } else {
                        long d = target - paths[destination].cost;
                        connected[last.first].back().second += d;
                        connected[last.second].back().second += d;
                        break;
                    }
                }
            }
            if (paths[destination].cost <= target) {
                for (int i = 0; i < n; ++i) {
                    for (const pair<int, int>& p : connected[i]) {
                        if ((i < p.first) ^ order[i * n + p.first]) {
                            result.emplace_back(vector<int>{i, p.first, p.second});
                        }
                    }
                }
                for (const pair<int, int>& missing : missings) {
                    result.emplace_back(vector<int>{missing.first, missing.second, max_edge_cost});
                }
            }
        }
        return result;
    }
};
240817