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            "questionId": "2321",
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            "title": "Minimum Weighted Subgraph With the Required Paths",
            "titleSlug": "minimum-weighted-subgraph-with-the-required-paths",
            "content": "<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>\n\n<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>\n\n<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>\n\n<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>\n\n<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>\n\n<p>&nbsp;</p>\n<p><strong>Example 1:</strong></p>\n<img alt=\"\" src=\"https://assets.leetcode.com/uploads/2022/02/17/example1drawio.png\" style=\"width: 263px; height: 250px;\" />\n<pre>\n<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\n<strong>Output:</strong> 9\n<strong>Explanation:</strong>\nThe above figure represents the input graph.\nThe blue edges represent one of the subgraphs that yield the optimal answer.\nNote 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.\n</pre>\n\n<p><strong>Example 2:</strong></p>\n<img alt=\"\" src=\"https://assets.leetcode.com/uploads/2022/02/17/example2-1drawio.png\" style=\"width: 350px; height: 51px;\" />\n<pre>\n<strong>Input:</strong> n = 3, edges = [[0,1,1],[2,1,1]], src1 = 0, src2 = 1, dest = 2\n<strong>Output:</strong> -1\n<strong>Explanation:</strong>\nThe above figure represents the input graph.\nIt 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.\n</pre>\n\n<p>&nbsp;</p>\n<p><strong>Constraints:</strong></p>\n\n<ul>\n\t<li><code>3 &lt;= n &lt;= 10<sup>5</sup></code></li>\n\t<li><code>0 &lt;= edges.length &lt;= 10<sup>5</sup></code></li>\n\t<li><code>edges[i].length == 3</code></li>\n\t<li><code>0 &lt;= from<sub>i</sub>, to<sub>i</sub>, src1, src2, dest &lt;= n - 1</code></li>\n\t<li><code>from<sub>i</sub> != to<sub>i</sub></code></li>\n\t<li><code>src1</code>, <code>src2</code>, and <code>dest</code> are pairwise distinct.</li>\n\t<li><code>1 &lt;= weight[i] &lt;= 10<sup>5</sup></code></li>\n</ul>\n",
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                    "lang": "C++",
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                    "code": "class Solution {\npublic:\n    long long minimumWeight(int n, vector<vector<int>>& edges, int src1, int src2, int dest) {\n        \n    }\n};",
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                    "code": "class Solution {\n    public long minimumWeight(int n, int[][] edges, int src1, int src2, int dest) {\n        \n    }\n}",
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                    "lang": "Python",
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                    "code": "class Solution(object):\n    def minimumWeight(self, n, edges, src1, src2, dest):\n        \"\"\"\n        :type n: int\n        :type edges: List[List[int]]\n        :type src1: int\n        :type src2: int\n        :type dest: int\n        :rtype: int\n        \"\"\"\n        ",
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                    "code": "class Solution:\n    def minimumWeight(self, n: int, edges: List[List[int]], src1: int, src2: int, dest: int) -> int:\n        ",
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                    "code": "public class Solution {\n    public long MinimumWeight(int n, int[][] edges, int src1, int src2, int dest) {\n        \n    }\n}",
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                    "code": "/**\n * @param {number} n\n * @param {number[][]} edges\n * @param {number} src1\n * @param {number} src2\n * @param {number} dest\n * @return {number}\n */\nvar minimumWeight = function(n, edges, src1, src2, dest) {\n    \n};",
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                    "code": "# @param {Integer} n\n# @param {Integer[][]} edges\n# @param {Integer} src1\n# @param {Integer} src2\n# @param {Integer} dest\n# @return {Integer}\ndef minimum_weight(n, edges, src1, src2, dest)\n    \nend",
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                    "code": "class Solution {\n    func minimumWeight(_ n: Int, _ edges: [[Int]], _ src1: Int, _ src2: Int, _ dest: Int) -> Int {\n        \n    }\n}",
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                    "lang": "Go",
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                    "code": "func minimumWeight(n int, edges [][]int, src1 int, src2 int, dest int) int64 {\n    \n}",
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                    "code": "object Solution {\n    def minimumWeight(n: Int, edges: Array[Array[Int]], src1: Int, src2: Int, dest: Int): Long = {\n        \n    }\n}",
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                    "code": "class Solution {\n    fun minimumWeight(n: Int, edges: Array<IntArray>, src1: Int, src2: Int, dest: Int): Long {\n        \n    }\n}",
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                    "code": "impl Solution {\n    pub fn minimum_weight(n: i32, edges: Vec<Vec<i32>>, src1: i32, src2: i32, dest: i32) -> i64 {\n        \n    }\n}",
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                    "code": "class Solution {\n\n    /**\n     * @param Integer $n\n     * @param Integer[][] $edges\n     * @param Integer $src1\n     * @param Integer $src2\n     * @param Integer $dest\n     * @return Integer\n     */\n    function minimumWeight($n, $edges, $src1, $src2, $dest) {\n        \n    }\n}",
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                    "code": "function minimumWeight(n: number, edges: number[][], src1: number, src2: number, dest: number): number {\n\n};",
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                    "code": "(define/contract (minimum-weight n edges src1 src2 dest)\n  (-> exact-integer? (listof (listof exact-integer?)) exact-integer? exact-integer? exact-integer? exact-integer?)\n\n  )",
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                    "lang": "Erlang",
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                    "code": "-spec minimum_weight(N :: integer(), Edges :: [[integer()]], Src1 :: integer(), Src2 :: integer(), Dest :: integer()) -> integer().\nminimum_weight(N, Edges, Src1, Src2, Dest) ->\n  .",
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                    "code": "defmodule Solution do\n  @spec minimum_weight(n :: integer, edges :: [[integer]], src1 :: integer, src2 :: integer, dest :: integer) :: integer\n  def minimum_weight(n, edges, src1, src2, dest) do\n\n  end\nend",
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                "Consider what the paths from src1 to dest and src2 to dest would look like in the optimal solution.",
                "It can be shown that in an optimal solution, the two paths from src1 and src2 will coincide at one node, and the remaining part to dest will be the same for both paths. Now consider how to find the node where the paths will coincide.",
                "How can algorithms for finding the shortest path between two nodes help us?"
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