mirror of
https://gitee.com/coder-xiaomo/leetcode-problemset
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196 lines
29 KiB
JSON
196 lines
29 KiB
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"title": "Most Profitable Path in a Tree",
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"content": "<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>\n\n<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>\n\n<ul>\n\t<li>the price needed to open the gate at node <code>i</code>, if <code>amount[i]</code> is negative, or,</li>\n\t<li>the cash reward obtained on opening the gate at node <code>i</code>, otherwise.</li>\n</ul>\n\n<p>The game goes on as follows:</p>\n\n<ul>\n\t<li>Initially, Alice is at node <code>0</code> and Bob is at node <code>bob</code>.</li>\n\t<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>\n\t<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:\n\t<ul>\n\t\t<li>If the gate is <strong>already open</strong>, no price will be required, nor will there be any cash reward.</li>\n\t\t<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 <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>\n\t</ul>\n\t</li>\n\t<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>\n</ul>\n\n<p>Return<em> the <strong>maximum</strong> net income Alice can have if she travels towards the optimal leaf node.</em></p>\n\n<p> </p>\n<p><strong class=\"example\">Example 1:</strong></p>\n<img alt=\"\" src=\"https://assets.leetcode.com/uploads/2022/10/29/eg1.png\" style=\"width: 275px; height: 275px;\" />\n<pre>\n<strong>Input:</strong> edges = [[0,1],[1,2],[1,3],[3,4]], bob = 3, amount = [-2,4,2,-4,6]\n<strong>Output:</strong> 6\n<strong>Explanation:</strong> \nThe above diagram represents the given tree. The game goes as follows:\n- Alice is initially on node 0, Bob on node 3. They open the gates of their respective nodes.\n Alice's net income is now -2.\n- Both Alice and Bob move to node 1. \n Since they reach here simultaneously, they open the gate together and share the reward.\n Alice's net income becomes -2 + (4 / 2) = 0.\n- Alice moves on to node 3. Since Bob already opened its gate, Alice's income remains unchanged.\n Bob moves on to node 0, and stops moving.\n- Alice moves on to node 4 and opens the gate there. Her net income becomes 0 + 6 = 6.\nNow, neither Alice nor Bob can make any further moves, and the game ends.\nIt is not possible for Alice to get a higher net income.\n</pre>\n\n<p><strong class=\"example\">Example 2:</strong></p>\n<img alt=\"\" src=\"https://assets.leetcode.com/uploads/2022/10/29/eg2.png\" style=\"width: 250px; height: 78px;\" />\n<pre>\n<strong>Input:</strong> edges = [[0,1]], bob = 1, amount = [-7280,2350]\n<strong>Output:</strong> -7280\n<strong>Explanation:</strong> \nAlice follows the path 0->1 whereas Bob follows the path 1->0.\nThus, Alice opens the gate at node 0 only. Hence, her net income is -7280. \n</pre>\n\n<p> </p>\n<p><strong>Constraints:</strong></p>\n\n<ul>\n\t<li><code>2 <= n <= 10<sup>5</sup></code></li>\n\t<li><code>edges.length == n - 1</code></li>\n\t<li><code>edges[i].length == 2</code></li>\n\t<li><code>0 <= a<sub>i</sub>, b<sub>i</sub> < n</code></li>\n\t<li><code>a<sub>i</sub> != b<sub>i</sub></code></li>\n\t<li><code>edges</code> represents a valid tree.</li>\n\t<li><code>1 <= bob < n</code></li>\n\t<li><code>amount.length == n</code></li>\n\t<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>\n</ul>\n",
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"translatedTitle": "树上最大得分和路径",
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"translatedContent": "<p>一个 <code>n</code> 个节点的无向树，节点编号为 <code>0</code> 到 <code>n - 1</code> ，树的根结点是 <code>0</code> 号节点。给你一个长度为 <code>n - 1</code> 的二维整数数组 <code>edges</code> ，其中 <code>edges[i] = [a<sub>i</sub>, b<sub>i</sub>]</code> ，表示节点 <code>a<sub>i</sub></code> 和 <code>b<sub>i</sub></code> 在树中有一条边。</p>\n\n<p>在每一个节点 <code>i</code> 处有一扇门。同时给你一个都是偶数的数组 <code>amount</code> ，其中 <code>amount[i]</code> 表示：</p>\n\n<ul>\n\t<li>如果 <code>amount[i]</code> 的值是负数，那么它表示打开节点 <code>i</code> 处门扣除的分数。</li>\n\t<li>如果 <code>amount[i]</code> 的值是正数，那么它表示打开节点 <code>i</code> 处门加上的分数。</li>\n</ul>\n\n<p>游戏按照如下规则进行：</p>\n\n<ul>\n\t<li>一开始，Alice 在节点 <code>0</code> 处，Bob 在节点 <code>bob</code> 处。</li>\n\t<li>每一秒钟，Alice 和 Bob <strong>分别</strong> 移动到相邻的节点。Alice 朝着某个 <strong>叶子结点</strong> 移动，Bob 朝着节点 <code>0</code> 移动。</li>\n\t<li>对于他们之间路径上的 <strong>每一个</strong> 节点，Alice 和 Bob 要么打开门并扣分，要么打开门并加分。注意：\n\t<ul>\n\t\t<li>如果门 <strong>已经打开</strong> （被另一个人打开），不会有额外加分也不会扣分。</li>\n\t\t<li>如果 Alice 和 Bob <strong>同时</strong> 到达一个节点，他们会共享这个节点的加分或者扣分。换言之，如果打开这扇门扣 <code>c</code> 分，那么 Alice 和 Bob 分别扣 <code>c / 2</code> 分。如果这扇门的加分为 <code>c</code> ，那么他们分别加 <code>c / 2</code> 分。</li>\n\t</ul>\n\t</li>\n\t<li>如果 Alice 到达了一个叶子结点，她会停止移动。类似的，如果 Bob 到达了节点 <code>0</code> ，他也会停止移动。注意这些事件互相 <strong>独立</strong> ，不会影响另一方移动。</li>\n</ul>\n\n<p>请你返回 Alice 朝最优叶子结点移动的 <strong>最大</strong> 净得分。</p>\n\n<p> </p>\n\n<p><strong>示例 1：</strong></p>\n\n<p><img alt=\"\" src=\"https://assets.leetcode.com/uploads/2022/10/29/eg1.png\" style=\"width: 275px; height: 275px;\"></p>\n\n<pre><b>输入：</b>edges = [[0,1],[1,2],[1,3],[3,4]], bob = 3, amount = [-2,4,2,-4,6]\n<b>输出：</b>6\n<b>解释：</b>\n上图展示了输入给出的一棵树。游戏进行如下：\n- Alice 一开始在节点 0 处，Bob 在节点 3 处。他们分别打开所在节点的门。\n Alice 得分为 -2 。\n- Alice 和 Bob 都移动到节点 1 。\n 因为他们同时到达这个节点，他们一起打开门并平分得分。\n Alice 的得分变为 -2 + (4 / 2) = 0 。\n- Alice 移动到节点 3 。因为 Bob 已经打开了这扇门，Alice 得分不变。\n Bob 移动到节点 0 ，并停止移动。\n- Alice 移动到节点 4 并打开这个节点的门，她得分变为 0 + 6 = 6 。\n现在，Alice 和 Bob 都不能进行任何移动了，所以游戏结束。\nAlice 无法得到更高分数。\n</pre>\n\n<p><strong>示例 2：</strong></p>\n\n<p><img alt=\"\" src=\"https://assets.leetcode.com/uploads/2022/10/29/eg2.png\" style=\"width: 250px; height: 78px;\"></p>\n\n<pre><b>输入：</b>edges = [[0,1]], bob = 1, amount = [-7280,2350]\n<b>输出：</b>-7280\n<b>解释：</b>\nAlice 按照路径 0->1 移动，同时 Bob 按照路径 1->0 移动。\n所以 Alice 只打开节点 0 处的门，她的得分为 -7280 。\n</pre>\n\n<p> </p>\n\n<p><strong>提示：</strong></p>\n\n<ul>\n\t<li><code>2 <= n <= 10<sup>5</sup></code></li>\n\t<li><code>edges.length == n - 1</code></li>\n\t<li><code>edges[i].length == 2</code></li>\n\t<li><code>0 <= a<sub>i</sub>, b<sub>i</sub> < n</code></li>\n\t<li><code>a<sub>i</sub> != b<sub>i</sub></code></li>\n\t<li><code>edges</code> 表示一棵有效的树。</li>\n\t<li><code>1 <= bob < n</code></li>\n\t<li><code>amount.length == n</code></li>\n\t<li><code>amount[i]</code> 是范围 <code>[-10<sup>4</sup>, 10<sup>4</sup>]</code> 之间的一个 <strong>偶数</strong> 。</li>\n</ul>\n",
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