Minimum Weighted Subgraph With the Required Paths
<p>You are given an integer <code>n</code> denoting the number of nodes of a <strong>weighted directed</strong> graph. The nodes are numbered from <code>0</code> to <code>n - 1</code>.</p>

<p>You are also given a 2D integer array <code>edges</code> where <code>edges[i] = [from<sub>i</sub>, to<sub>i</sub>, weight<sub>i</sub>]</code> denotes that there exists a <strong>directed</strong> edge from <code>from<sub>i</sub></code> to <code>to<sub>i</sub></code> with weight <code>weight<sub>i</sub></code>.</p>

<p>Lastly, you are given three <strong>distinct</strong> integers <code>src1</code>, <code>src2</code>, and <code>dest</code> denoting three distinct nodes of the graph.</p>

<p>Return <em>the <strong>minimum weight</strong> of a subgraph of the graph such that it is <strong>possible</strong> to reach</em> <code>dest</code> <em>from both</em> <code>src1</code> <em>and</em> <code>src2</code> <em>via a set of edges of this subgraph</em>. In case such a subgraph does not exist, return <code>-1</code>.</p>

<p>A <strong>subgraph</strong> is a graph whose vertices and edges are subsets of the original graph. The <strong>weight</strong> of a subgraph is the sum of weights of its constituent edges.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://fastly.jsdelivr.net/gh/doocs/leetcode@main/solution/2200-2299/2203.Minimum%20Weighted%20Subgraph%20With%20the%20Required%20Paths/images/example1drawio.png" style="width: 263px; height: 250px;" />
<pre>
<strong>Input:</strong> n = 6, edges = [[0,2,2],[0,5,6],[1,0,3],[1,4,5],[2,1,1],[2,3,3],[2,3,4],[3,4,2],[4,5,1]], src1 = 0, src2 = 1, dest = 5
<strong>Output:</strong> 9
<strong>Explanation:</strong>
The above figure represents the input graph.
The blue edges represent one of the subgraphs that yield the optimal answer.
Note that the subgraph [[1,0,3],[0,5,6]] also yields the optimal answer. It is not possible to get a subgraph with less weight satisfying all the constraints.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://fastly.jsdelivr.net/gh/doocs/leetcode@main/solution/2200-2299/2203.Minimum%20Weighted%20Subgraph%20With%20the%20Required%20Paths/images/example2-1drawio.png" style="width: 350px; height: 51px;" />
<pre>
<strong>Input:</strong> n = 3, edges = [[0,1,1],[2,1,1]], src1 = 0, src2 = 1, dest = 2
<strong>Output:</strong> -1
<strong>Explanation:</strong>
The above figure represents the input graph.
It can be seen that there does not exist any path from node 1 to node 2, hence there are no subgraphs satisfying all the constraints.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>3 &lt;= n &lt;= 10<sup>5</sup></code></li>
	<li><code>0 &lt;= edges.length &lt;= 10<sup>5</sup></code></li>
	<li><code>edges[i].length == 3</code></li>
	<li><code>0 &lt;= from<sub>i</sub>, to<sub>i</sub>, src1, src2, dest &lt;= n - 1</code></li>
	<li><code>from<sub>i</sub> != to<sub>i</sub></code></li>
	<li><code>src1</code>, <code>src2</code>, and <code>dest</code> are pairwise distinct.</li>
	<li><code>1 &lt;= weight[i] &lt;= 10<sup>5</sup></code></li>
</ul>
Graph; Shortest Path
class Solution {
    private static final Long INF = Long.MAX_VALUE;

    public long minimumWeight(int n, int[][] edges, int src1, int src2, int dest) {
        List<Pair<Integer, Long>>[] g = new List[n];
        List<Pair<Integer, Long>>[] rg = new List[n];
        for (int i = 0; i < n; ++i) {
            g[i] = new ArrayList<>();
            rg[i] = new ArrayList<>();
        }
        for (int[] e : edges) {
            int f = e[0], t = e[1];
            long w = e[2];
            g[f].add(new Pair<>(t, w));
            rg[t].add(new Pair<>(f, w));
        }
        long[] d1 = dijkstra(g, src1);
        long[] d2 = dijkstra(g, src2);
        long[] d3 = dijkstra(rg, dest);
        long ans = -1;
        for (int i = 0; i < n; ++i) {
            if (d1[i] == INF || d2[i] == INF || d3[i] == INF) {
                continue;
            }
            long t = d1[i] + d2[i] + d3[i];
            if (ans == -1 || ans > t) {
                ans = t;
            }
        }
        return ans;
    }

    private long[] dijkstra(List<Pair<Integer, Long>>[] g, int u) {
        int n = g.length;
        long[] dist = new long[n];
        Arrays.fill(dist, INF);
        dist[u] = 0;
        PriorityQueue<Pair<Long, Integer>> q
            = new PriorityQueue<>(Comparator.comparingLong(Pair::getKey));
        q.offer(new Pair<>(0L, u));
        while (!q.isEmpty()) {
            Pair<Long, Integer> p = q.poll();
            long d = p.getKey();
            u = p.getValue();
            if (d > dist[u]) {
                continue;
            }
            for (Pair<Integer, Long> e : g[u]) {
                int v = e.getKey();
                long w = e.getValue();
                if (dist[v] > dist[u] + w) {
                    dist[v] = dist[u] + w;
                    q.offer(new Pair<>(dist[v], v));
                }
            }
        }
        return dist;
    }
}