Most Profitable Path in a Tree
<p>There is an undirected tree with <code>n</code> nodes labeled from <code>0</code> to <code>n - 1</code>, rooted at node <code>0</code>. You are given a 2D integer array <code>edges</code> of length <code>n - 1</code> where <code>edges[i] = [a<sub>i</sub>, b<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> in the tree.</p>

<p>At every node <code>i</code>, there is a gate. You are also given an array of even integers <code>amount</code>, where <code>amount[i]</code> represents:</p>

<ul>
	<li>the price needed to open the gate at node <code>i</code>, if <code>amount[i]</code> is negative, or,</li>
	<li>the cash reward obtained on opening the gate at node <code>i</code>, otherwise.</li>
</ul>

<p>The game goes on as follows:</p>

<ul>
	<li>Initially, Alice is at node <code>0</code> and Bob is at node <code>bob</code>.</li>
	<li>At every second, Alice and Bob <b>each</b> move to an adjacent node. Alice moves towards some <strong>leaf node</strong>, while Bob moves towards node <code>0</code>.</li>
	<li>For <strong>every</strong> node along their path, Alice and Bob either spend money to open the gate at that node, or accept the reward. Note that:
	<ul>
		<li>If the gate is <strong>already open</strong>, no price will be required, nor will there be any cash reward.</li>
		<li>If Alice and Bob reach the node <strong>simultaneously</strong>, they share the price/reward for opening the gate there. In other words, if the price to open the gate is <code>c</code>, then both Alice and Bob pay&nbsp;<code>c / 2</code> each. Similarly, if the reward at the gate is <code>c</code>, both of them receive <code>c / 2</code> each.</li>
	</ul>
	</li>
	<li>If Alice reaches a leaf node, she stops moving. Similarly, if Bob reaches node <code>0</code>, he stops moving. Note that these events are <strong>independent</strong> of each other.</li>
</ul>

<p>Return<em> the <strong>maximum</strong> net income Alice can have if she travels towards the optimal leaf node.</em></p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2022/10/29/eg1.png" style="width: 275px; height: 275px;" />
<pre>
<strong>Input:</strong> edges = [[0,1],[1,2],[1,3],[3,4]], bob = 3, amount = [-2,4,2,-4,6]
<strong>Output:</strong> 6
<strong>Explanation:</strong> 
The above diagram represents the given tree. The game goes as follows:
- Alice is initially on node 0, Bob on node 3. They open the gates of their respective nodes.
  Alice&#39;s net income is now -2.
- Both Alice and Bob move to node 1. 
&nbsp; Since they reach here simultaneously, they open the gate together and share the reward.
&nbsp; Alice&#39;s net income becomes -2 + (4 / 2) = 0.
- Alice moves on to node 3. Since Bob already opened its gate, Alice&#39;s income remains unchanged.
&nbsp; Bob moves on to node 0, and stops moving.
- Alice moves on to node 4 and opens the gate there. Her net income becomes 0 + 6 = 6.
Now, neither Alice nor Bob can make any further moves, and the game ends.
It is not possible for Alice to get a higher net income.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2022/10/29/eg2.png" style="width: 250px; height: 78px;" />
<pre>
<strong>Input:</strong> edges = [[0,1]], bob = 1, amount = [-7280,2350]
<strong>Output:</strong> -7280
<strong>Explanation:</strong> 
Alice follows the path 0-&gt;1 whereas Bob follows the path 1-&gt;0.
Thus, Alice opens the gate at node 0 only. Hence, her net income is -7280. 
</pre>

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

<ul>
	<li><code>2 &lt;= n &lt;= 10<sup>5</sup></code></li>
	<li><code>edges.length == n - 1</code></li>
	<li><code>edges[i].length == 2</code></li>
	<li><code>0 &lt;= a<sub>i</sub>, b<sub>i</sub> &lt; n</code></li>
	<li><code>a<sub>i</sub> != b<sub>i</sub></code></li>
	<li><code>edges</code> represents a valid tree.</li>
	<li><code>1 &lt;= bob &lt; n</code></li>
	<li><code>amount.length == n</code></li>
	<li><code>amount[i]</code> is an <strong>even</strong> integer in the range <code>[-10<sup>4</sup>, 10<sup>4</sup>]</code>.</li>
</ul>

['snakes-and-ladders', 'time-taken-to-mark-all-nodes']
[]
[{"value": "cpp", "text": "C++", "defaultCode": "class Solution {\npublic:\n    int mostProfitablePath(vector<vector<int>>& edges, int bob, vector<int>& amount) {\n        \n    }\n};"}, {"value": "java", "text": "Java", "defaultCode": "class Solution {\n    public int mostProfitablePath(int[][] edges, int bob, int[] amount) {\n        \n    }\n}"}, {"value": "python", "text": "Python", "defaultCode": "class Solution(object):\n    def mostProfitablePath(self, edges, bob, amount):\n        \"\"\"\n        :type edges: List[List[int]]\n        :type bob: int\n        :type amount: List[int]\n        :rtype: int\n        \"\"\"\n        "}, {"value": "python3", "text": "Python3", "defaultCode": "class Solution:\n    def mostProfitablePath(self, edges: List[List[int]], bob: int, amount: List[int]) -> int:\n        "}, {"value": "c", "text": "C", "defaultCode": "int mostProfitablePath(int** edges, int edgesSize, int* edgesColSize, int bob, int* amount, int amountSize) {\n    \n}"}, {"value": "csharp", "text": "C#", "defaultCode": "public class Solution {\n    public int MostProfitablePath(int[][] edges, int bob, int[] amount) {\n        \n    }\n}"}, {"value": "javascript", "text": "JavaScript", "defaultCode": "/**\n * @param {number[][]} edges\n * @param {number} bob\n * @param {number[]} amount\n * @return {number}\n */\nvar mostProfitablePath = function(edges, bob, amount) {\n    \n};"}, {"value": "typescript", "text": "TypeScript", "defaultCode": "function mostProfitablePath(edges: number[][], bob: number, amount: number[]): number {\n    \n};"}, {"value": "php", "text": "PHP", "defaultCode": "class Solution {\n\n    /**\n     * @param Integer[][] $edges\n     * @param Integer $bob\n     * @param Integer[] $amount\n     * @return Integer\n     */\n    function mostProfitablePath($edges, $bob, $amount) {\n        \n    }\n}"}, {"value": "swift", "text": "Swift", "defaultCode": "class Solution {\n    func mostProfitablePath(_ edges: [[Int]], _ bob: Int, _ amount: [Int]) -> Int {\n        \n    }\n}"}, {"value": "kotlin", "text": "Kotlin", "defaultCode": "class Solution {\n    fun mostProfitablePath(edges: Array<IntArray>, bob: Int, amount: IntArray): Int {\n        \n    }\n}"}, {"value": "dart", "text": "Dart", "defaultCode": "class Solution {\n  int mostProfitablePath(List<List<int>> edges, int bob, List<int> amount) {\n    \n  }\n}"}, {"value": "golang", "text": "Go", "defaultCode": "func mostProfitablePath(edges [][]int, bob int, amount []int) int {\n    \n}"}, {"value": "ruby", "text": "Ruby", "defaultCode": "# @param {Integer[][]} edges\n# @param {Integer} bob\n# @param {Integer[]} amount\n# @return {Integer}\ndef most_profitable_path(edges, bob, amount)\n    \nend"}, {"value": "scala", "text": "Scala", "defaultCode": "object Solution {\n    def mostProfitablePath(edges: Array[Array[Int]], bob: Int, amount: Array[Int]): Int = {\n        \n    }\n}"}, {"value": "rust", "text": "Rust", "defaultCode": "impl Solution {\n    pub fn most_profitable_path(edges: Vec<Vec<i32>>, bob: i32, amount: Vec<i32>) -> i32 {\n        \n    }\n}"}, {"value": "racket", "text": "Racket", "defaultCode": "(define/contract (most-profitable-path edges bob amount)\n  (-> (listof (listof exact-integer?)) exact-integer? (listof exact-integer?) exact-integer?)\n  )"}, {"value": "erlang", "text": "Erlang", "defaultCode": "-spec most_profitable_path(Edges :: [[integer()]], Bob :: integer(), Amount :: [integer()]) -> integer().\nmost_profitable_path(Edges, Bob, Amount) ->\n  ."}, {"value": "elixir", "text": "Elixir", "defaultCode": "defmodule Solution do\n  @spec most_profitable_path(edges :: [[integer]], bob :: integer, amount :: [integer]) :: integer\n  def most_profitable_path(edges, bob, amount) do\n    \n  end\nend"}]
{
  "name": "mostProfitablePath",
  "params": [
    {
      "name": "edges",
      "type": "integer[][]"
    },
    {
      "type": "integer",
      "name": "bob"
    },
    {
      "type": "integer[]",
      "name": "amount"
    }
  ],
  "return": {
    "type": "integer"
  }
}
{"cpp": ["C++", "<p>Compiled with <code> clang 19 </code> using the latest C++ 23 standard, and <code>libstdc++</code> provided by GCC 14.</p>\r\n\r\n<p>Your code is compiled with level two optimization (<code>-O2</code>). <a href=\"https://github.com/google/sanitizers/wiki/AddressSanitizer\" target=\"_blank\">AddressSanitizer</a> is also enabled to help detect out-of-bounds and use-after-free bugs.</p>\r\n\r\n<p>Most standard library headers are already included automatically for your convenience.</p>"], "java": ["Java", "<p><code>OpenJDK 21</code>. Using compile arguments: <code>--enable-preview --release 21</code></p>\r\n\r\n<p>Most standard library headers are already included automatically for your convenience.</p>\r\n<p>Includes <code>Pair</code> class from https://docs.oracle.com/javase/8/javafx/api/javafx/util/Pair.html.</p>"], "python": ["Python", "<p><code>Python 2.7.18</code>.</p>\r\n\r\n<p>Most libraries are already imported automatically for your convenience, such as <a href=\"https://docs.python.org/2/library/array.html\" target=\"_blank\">array</a>, <a href=\"https://docs.python.org/2/library/bisect.html\" target=\"_blank\">bisect</a>, <a href=\"https://docs.python.org/2/library/collections.html\" target=\"_blank\">collections</a>. If you need more libraries, you can import it yourself.</p>\r\n\r\n<p>For Map/TreeMap data structure, you may use <a href=\"http://www.grantjenks.com/docs/sortedcontainers/\" target=\"_blank\">sortedcontainers</a> library.</p>\r\n\r\n<p>Note that Python 2.7 <a href=\"https://www.python.org/dev/peps/pep-0373/\" target=\"_blank\">is not maintained anymore</a>. For the latest Python, please choose Python3 instead.</p>"], "c": ["C", "<p>Compiled with <code>gcc 14</code> using the gnu11 standard.</p>\r\n\r\n<p>Your code is compiled with level one optimization (<code>-O2</code>). <a href=\"https://github.com/google/sanitizers/wiki/AddressSanitizer\" target=\"_blank\">AddressSanitizer</a> is also enabled to help detect out-of-bounds and use-after-free bugs.</p>\r\n\r\n<p>Most standard library headers are already included automatically for your convenience.</p>\r\n\r\n<p>For hash table operations, you may use <a href=\"https://troydhanson.github.io/uthash/\" target=\"_blank\">uthash</a>. \"uthash.h\" is included by default. Below are some examples:</p>\r\n\r\n<p><b>1. Adding an item to a hash.</b>\r\n<pre>\r\nstruct hash_entry {\r\n    int id;            /* we'll use this field as the key */\r\n    char name[10];\r\n    UT_hash_handle hh; /* makes this structure hashable */\r\n};\r\n\r\nstruct hash_entry *users = NULL;\r\n\r\nvoid add_user(struct hash_entry *s) {\r\n    HASH_ADD_INT(users, id, s);\r\n}\r\n</pre>\r\n</p>\r\n\r\n<p><b>2. Looking up an item in a hash:</b>\r\n<pre>\r\nstruct hash_entry *find_user(int user_id) {\r\n    struct hash_entry *s;\r\n    HASH_FIND_INT(users, &user_id, s);\r\n    return s;\r\n}\r\n</pre>\r\n</p>\r\n\r\n<p><b>3. Deleting an item in a hash:</b>\r\n<pre>\r\nvoid delete_user(struct hash_entry *user) {\r\n    HASH_DEL(users, user);  \r\n}\r\n</pre>\r\n</p>"], "csharp": ["C#", "<p><a href=\"https://learn.microsoft.com/en-us/dotnet/csharp/whats-new/csharp-13\" target=\"_blank\">C# 13</a> with .NET 9 runtime</p>"], "javascript": ["JavaScript", "<p><code>Node.js 22.14.0</code>.</p>\r\n\r\n<p>Your code is run with <code>--harmony</code> flag, enabling <a href=\"http://node.green/\" target=\"_blank\">new ES6 features</a>.</p>\r\n\r\n<p><a href=\"https://lodash.com\" target=\"_blank\">lodash.js</a> library is included by default.</p>\r\n\r\n<p>For Priority Queue / Queue data structures, you may use 6.3.2 version of <a href=\"https://github.com/datastructures-js/priority-queue/tree/v6.3.2\" target=\"_blank\">datastructures-js/priority-queue</a> and 4.2.3 version of <a href=\"https://github.com/datastructures-js/queue/tree/v4.2.3\" target=\"_blank\">datastructures-js/queue</a> and 1.0.4 version of <a href=\"https://github.com/datastructures-js/deque/tree/v1.0.4\" target=\"_blank\">datastructures-js/deque</a>.</p>"], "ruby": ["Ruby", "<p><code>Ruby 3.2</code></p>\r\n\r\n<p>Some common data structure implementations are provided in the Algorithms module: https://www.rubydoc.info/github/kanwei/algorithms/Algorithms</p>"], "swift": ["Swift", "<p><code>Swift 6.0</code>.</p>\r\n\r\n<p>You may use the following packages:<br/>\r\n<a href=\"https://github.com/apple/swift-algorithms/tree/1.2.0\" target=\"_blank\">swift-algorithms 1.2.0</a><br/>\r\n<a href=\"https://github.com/apple/swift-collections/tree/1.1.4\" target=\"_blank\">swift-collections 1.1.4</a><br/>\r\n<a href=\"https://github.com/apple/swift-numerics/tree/1.0.2\" target=\"_blank\">swift-numerics 1.0.2</a></p>"], "golang": ["Go", "<p><code>Go 1.23</code></p>\r\n<p>Support <a href=\"https://pkg.go.dev/github.com/emirpasic/gods@v1.18.1\" target=\"_blank\">https://pkg.go.dev/github.com/emirpasic/gods@v1.18.1</a> and <a href=\"https://pkg.go.dev/github.com/emirpasic/gods/v2@v2.0.0-alpha\" target=\"_blank\">https://pkg.go.dev/github.com/emirpasic/gods/v2@v2.0.0-alpha</a> library.</p>"], "python3": ["Python3", "<p><code>Python 3.11</code>.</p>\r\n\r\n<p>Most libraries are already imported automatically for your convenience, such as <a href=\"https://docs.python.org/3/library/array.html\" target=\"_blank\">array</a>, <a href=\"https://docs.python.org/3/library/bisect.html\" target=\"_blank\">bisect</a>, <a href=\"https://docs.python.org/3/library/collections.html\" target=\"_blank\">collections</a>. If you need more libraries, you can import it yourself.</p>\r\n\r\n<p>For Map/TreeMap data structure, you may use <a href=\"http://www.grantjenks.com/docs/sortedcontainers/\" target=\"_blank\">sortedcontainers</a> library.</p>"], "scala": ["Scala", "<p><code>Scala 3.3.1</code>.</p>"], "kotlin": ["Kotlin", "<p><code>Kotlin 2.1.10</code>.</p>"], "rust": ["Rust", "<p><code>Rust 1.85.0</code>. Your code will be compiled with <code>opt-level</code> 2.</p>\r\n\r\n<p>Supports <a href=\"https://crates.io/crates/rand\" target=\"_blank\">rand v0.8</a> and <a href=\"https://crates.io/crates/regex\" target=\"_blank\">regex\u00a0v1</a> and <a href=\"https://crates.io/crates/itertools\" target=\"_blank\">itertools\u00a0v0.14</a> from crates.io</p>"], "php": ["PHP", "<p><code>PHP 8.2</code>.</p>\r\n<p>With bcmath module</p>"], "typescript": ["Typescript", "<p><code>TypeScript 5.7.3, Node.js 22.14.0</code>.</p>\r\n\r\n<p>Compile Options: <code>--alwaysStrict --strictBindCallApply --strictFunctionTypes --target ES2024</code></p>\r\n\r\n<p><a href=\"https://lodash.com\" target=\"_blank\">lodash.js</a> library is included by default.</p>\r\n\r\n<p>For Priority Queue / Queue data structures, you may use 6.3.2 version of <a href=\"https://github.com/datastructures-js/priority-queue/tree/v6.3.2\" target=\"_blank\">datastructures-js/priority-queue</a> and 4.2.3 version of <a href=\"https://github.com/datastructures-js/queue/tree/v4.2.3\" target=\"_blank\">datastructures-js/queue</a> and 1.0.4 version of <a href=\"https://github.com/datastructures-js/deque/tree/v1.0.4\" target=\"_blank\">datastructures-js/deque</a>.</p>"], "racket": ["Racket", "<p><a href=\"https://docs.racket-lang.org/guide/performance.html#%28tech._c%29\" target=\"_blank\">Racket CS</a> v8.15</p>\r\n\r\n<p>Using <code>#lang racket</code></p>\r\n\r\n<p>Required <code>data/gvector data/queue data/order data/heap</code> automatically for your convenience</p>"], "erlang": ["Erlang", "Erlang/OTP 26"], "elixir": ["Elixir", "Elixir 1.17 with Erlang/OTP 26"], "dart": ["Dart", "<p>Dart 3.2. You may use package <a href=\"https://pub.dev/packages/collection/versions/1.18.0\" target=\"_blank\">collection</a></p>\r\n\r\n<p>Your code will be run directly without compiling</p>"]}
['Array', 'Tree', 'Depth-First Search', 'Breadth-First Search', 'Graph']