leetcode-problemset/leetcode-cn/originData/find-the-maximum-sum-of-node-values.json

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        "question": {
            "questionId": "3307",
            "questionFrontendId": "3068",
            "categoryTitle": "Algorithms",
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            "title": "Find the Maximum Sum of Node Values",
            "titleSlug": "find-the-maximum-sum-of-node-values",
            "content": "<p>There exists an <strong>undirected</strong> tree with <code>n</code> nodes numbered <code>0</code> to <code>n - 1</code>. You are given a <strong>0-indexed</strong> 2D integer array <code>edges</code> of length <code>n - 1</code>, where <code>edges[i] = [u<sub>i</sub>, v<sub>i</sub>]</code> indicates that there is an edge between nodes <code>u<sub>i</sub></code> and <code>v<sub>i</sub></code> in the tree. You are also given a <strong>positive</strong> integer <code>k</code>, and a <strong>0-indexed</strong> array of <strong>non-negative</strong> integers <code>nums</code> of length <code>n</code>, where <code>nums[i]</code> represents the <strong>value</strong> of the node numbered <code>i</code>.</p>\n\n<p>Alice wants the sum of values of tree nodes to be <strong>maximum</strong>, for which Alice can perform the following operation <strong>any</strong> number of times (<strong>including zero</strong>) on the tree:</p>\n\n<ul>\n\t<li>Choose any edge <code>[u, v]</code> connecting the nodes <code>u</code> and <code>v</code>, and update their values as follows:\n\n\t<ul>\n\t\t<li><code>nums[u] = nums[u] XOR k</code></li>\n\t\t<li><code>nums[v] = nums[v] XOR k</code></li>\n\t</ul>\n\t</li>\n</ul>\n\n<p>Return <em>the <strong>maximum</strong> possible <strong>sum</strong> of the <strong>values</strong> Alice can achieve by performing the operation <strong>any</strong> number of times</em>.</p>\n\n<p>&nbsp;</p>\n<p><strong class=\"example\">Example 1:</strong></p>\n<img alt=\"\" src=\"https://assets.leetcode.com/uploads/2023/11/09/screenshot-2023-11-10-012513.png\" style=\"width: 300px; height: 277px;padding: 10px; background: #fff; border-radius: .5rem;\" />\n<pre>\n<strong>Input:</strong> nums = [1,2,1], k = 3, edges = [[0,1],[0,2]]\n<strong>Output:</strong> 6\n<strong>Explanation:</strong> Alice can achieve the maximum sum of 6 using a single operation:\n- Choose the edge [0,2]. nums[0] and nums[2] become: 1 XOR 3 = 2, and the array nums becomes: [1,2,1] -&gt; [2,2,2].\nThe total sum of values is 2 + 2 + 2 = 6.\nIt can be shown that 6 is the maximum achievable sum of values.\n</pre>\n\n<p><strong class=\"example\">Example 2:</strong></p>\n<img alt=\"\" src=\"https://assets.leetcode.com/uploads/2024/01/09/screenshot-2024-01-09-220017.png\" style=\"padding: 10px; background: rgb(255, 255, 255); border-radius: 0.5rem; width: 300px; height: 239px;\" />\n<pre>\n<strong>Input:</strong> nums = [2,3], k = 7, edges = [[0,1]]\n<strong>Output:</strong> 9\n<strong>Explanation:</strong> Alice can achieve the maximum sum of 9 using a single operation:\n- Choose the edge [0,1]. nums[0] becomes: 2 XOR 7 = 5 and nums[1] become: 3 XOR 7 = 4, and the array nums becomes: [2,3] -&gt; [5,4].\nThe total sum of values is 5 + 4 = 9.\nIt can be shown that 9 is the maximum achievable sum of values.\n</pre>\n\n<p><strong class=\"example\">Example 3:</strong></p>\n<img alt=\"\" src=\"https://assets.leetcode.com/uploads/2023/11/09/screenshot-2023-11-10-012641.png\" style=\"width: 600px; height: 233px;padding: 10px; background: #fff; border-radius: .5rem;\" />\n<pre>\n<strong>Input:</strong> nums = [7,7,7,7,7,7], k = 3, edges = [[0,1],[0,2],[0,3],[0,4],[0,5]]\n<strong>Output:</strong> 42\n<strong>Explanation:</strong> The maximum achievable sum is 42 which can be achieved by Alice performing no operations.\n</pre>\n\n<p>&nbsp;</p>\n<p><strong>Constraints:</strong></p>\n\n<ul>\n\t<li><code>2 &lt;= n == nums.length &lt;= 2 * 10<sup>4</sup></code></li>\n\t<li><code>1 &lt;= k &lt;= 10<sup>9</sup></code></li>\n\t<li><code>0 &lt;= nums[i] &lt;= 10<sup>9</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 &lt;= edges[i][0], edges[i][1] &lt;= n - 1</code></li>\n\t<li>The input is generated such that <code>edges</code> represent&nbsp;a valid tree.</li>\n</ul>\n",
            "translatedTitle": "最大节点价值之和",
            "translatedContent": "<p>给你一棵 <code>n</code>&nbsp;个节点的 <strong>无向</strong>&nbsp;树，节点从 <code>0</code>&nbsp;到 <code>n - 1</code>&nbsp;编号。树以长度为 <code>n - 1</code>&nbsp;下标从 <strong>0</strong>&nbsp;开始的二维整数数组 <code>edges</code>&nbsp;的形式给你，其中&nbsp;<code>edges[i] = [u<sub>i</sub>, v<sub>i</sub>]</code>&nbsp;表示树中节点&nbsp;<code>u<sub>i</sub></code>&nbsp;和&nbsp;<code>v<sub>i</sub></code>&nbsp;之间有一条边。同时给你一个 <strong>正</strong>&nbsp;整数&nbsp;<code>k</code>&nbsp;和一个长度为 <code>n</code>&nbsp;下标从&nbsp;<strong>0</strong>&nbsp;开始的&nbsp;<strong>非负</strong>&nbsp;整数数组&nbsp;<code>nums</code>&nbsp;，其中&nbsp;<code>nums[i]</code>&nbsp;表示节点 <code>i</code>&nbsp;的 <strong>价值</strong>&nbsp;。</p>\n\n<p>Alice&nbsp;想 <strong>最大化</strong>&nbsp;树中所有节点价值之和。为了实现这一目标，Alice 可以执行以下操作 <strong>任意</strong>&nbsp;次（<strong>包括</strong><strong>&nbsp;0 次</strong>）：</p>\n\n<ul>\n\t<li>选择连接节点&nbsp;<code>u</code>&nbsp;和&nbsp;<code>v</code>&nbsp;的边&nbsp;<code>[u, v]</code>&nbsp;，并将它们的值更新为：\n\n\t<ul>\n\t\t<li><code>nums[u] = nums[u] XOR k</code></li>\n\t\t<li><code>nums[v] = nums[v] XOR k</code></li>\n\t</ul>\n\t</li>\n</ul>\n\n<p>请你返回 Alice 通过执行以上操作 <strong>任意次</strong>&nbsp;后，可以得到所有节点 <strong>价值之和</strong>&nbsp;的 <strong>最大值</strong>&nbsp;。</p>\n\n<p>&nbsp;</p>\n\n<p><strong class=\"example\">示例 1：</strong></p>\n\n<p><img alt=\"\" src=\"https://assets.leetcode.com/uploads/2023/11/09/screenshot-2023-11-10-012513.png\" style=\"width: 300px; height: 277px;padding: 10px; background: #fff; border-radius: .5rem;\" /></p>\n\n<pre>\n<b>输入：</b>nums = [1,2,1], k = 3, edges = [[0,1],[0,2]]\n<b>输出：</b>6\n<b>解释：</b>Alice 可以通过一次操作得到最大价值和 6 ：\n- 选择边 [0,2] 。nums[0] 和 nums[2] 都变为：1 XOR 3 = 2 ，数组 nums 变为：[1,2,1] -&gt; [2,2,2] 。\n所有节点价值之和为 2 + 2 + 2 = 6 。\n6 是可以得到最大的价值之和。\n</pre>\n\n<p><strong class=\"example\">示例 2：</strong></p>\n\n<p><img alt=\"\" src=\"https://assets.leetcode.com/uploads/2024/01/09/screenshot-2024-01-09-220017.png\" style=\"padding: 10px; background: rgb(255, 255, 255); border-radius: 0.5rem; width: 300px; height: 239px;\" /></p>\n\n<pre>\n<b>输入：</b>nums = [2,3], k = 7, edges = [[0,1]]\n<b>输出：</b>9\n<b>解释：</b>Alice 可以通过一次操作得到最大和 9 ：\n- 选择边 [0,1] 。nums[0] 变为：2 XOR 7 = 5 ，nums[1] 变为：3 XOR 7 = 4 ，数组 nums 变为：[2,3] -&gt; [5,4] 。\n所有节点价值之和为 5 + 4 = 9 。\n9 是可以得到最大的价值之和。\n</pre>\n\n<p><strong class=\"example\">示例 3：</strong></p>\n\n<p><img alt=\"\" src=\"https://assets.leetcode.com/uploads/2023/11/09/screenshot-2023-11-10-012641.png\" style=\"width: 600px; height: 233px;padding: 10px; background: #fff; border-radius: .5rem;\" /></p>\n\n<pre>\n<b>输入：</b>nums = [7,7,7,7,7,7], k = 3, edges = [[0,1],[0,2],[0,3],[0,4],[0,5]]\n<b>输出：</b>42\n<b>解释：</b>Alice 不需要执行任何操作，就可以得到最大价值之和 42 。\n</pre>\n\n<p>&nbsp;</p>\n\n<p><strong>提示：</strong></p>\n\n<ul>\n\t<li><code>2 &lt;= n == nums.length &lt;= 2 * 10<sup>4</sup></code></li>\n\t<li><code>1 &lt;= k &lt;= 10<sup>9</sup></code></li>\n\t<li><code>0 &lt;= nums[i] &lt;= 10<sup>9</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 &lt;= edges[i][0], edges[i][1] &lt;= n - 1</code></li>\n\t<li>输入保证&nbsp;<code>edges</code>&nbsp;构成一棵合法的树。</li>\n</ul>\n",
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                "Let <code>dp[x][c]</code> be the maximum sum we can get for the subtree rooted at node <code>x</code>, where <code>c</code> is a boolean representing whether the edge between node <code>x</code> and its parent (if any) is selected or not.",
                "<code>dp[x][c] = max(sum(dp[y][cy]) + v(nums[x], sum(cy) + c))</code>\r\nwhere <code>cy</code> is <code>0</code> or <code>1</code>. \r\nWhen <code>sum(cy) + c</code> is odd, <code>v(nums[x], sum(cy) + c) = nums[x] XOR k</code>. \r\nWhen <code>sum(cy) + c</code> is even, <code>v(nums[x], sum(cy) + c) = nums[x]</code>.",
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