leetcode-problemset/leetcode-cn/originData/cycle-length-queries-in-a-tree.json

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        "question": {
            "questionId": "2597",
            "questionFrontendId": "2509",
            "categoryTitle": "Algorithms",
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            "title": "Cycle Length Queries in a Tree",
            "titleSlug": "cycle-length-queries-in-a-tree",
            "content": "<p>You are given an integer <code>n</code>. There is a <strong>complete binary tree</strong> with <code>2<sup>n</sup> - 1</code> nodes. The root of that tree is the node with the value <code>1</code>, and every node with a value <code>val</code> in the range <code>[1, 2<sup>n - 1</sup> - 1]</code> has two children where:</p>\n\n<ul>\n\t<li>The left node has the value <code>2 * val</code>, and</li>\n\t<li>The right node has the value <code>2 * val + 1</code>.</li>\n</ul>\n\n<p>You are also given a 2D integer array <code>queries</code> of length <code>m</code>, where <code>queries[i] = [a<sub>i</sub>, b<sub>i</sub>]</code>. For each query, solve the following problem:</p>\n\n<ol>\n\t<li>Add an edge between the nodes with values <code>a<sub>i</sub></code> and <code>b<sub>i</sub></code>.</li>\n\t<li>Find the length of the cycle in the graph.</li>\n\t<li>Remove the added edge between nodes with values <code>a<sub>i</sub></code> and <code>b<sub>i</sub></code>.</li>\n</ol>\n\n<p><strong>Note</strong> that:</p>\n\n<ul>\n\t<li>A <strong>cycle</strong> is a path that starts and ends at the same node, and each edge in the path is visited only once.</li>\n\t<li>The length of a cycle is the number of edges visited in the cycle.</li>\n\t<li>There could be multiple edges between two nodes in the tree after adding the edge of the query.</li>\n</ul>\n\n<p>Return <em>an array </em><code>answer</code><em> of length </em><code>m</code><em> where</em> <code>answer[i]</code> <em>is the answer to the</em> <code>i<sup>th</sup></code> <em>query.</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/2022/10/25/bexample1.png\" style=\"width: 647px; height: 128px;\" />\n<pre>\n<strong>Input:</strong> n = 3, queries = [[5,3],[4,7],[2,3]]\n<strong>Output:</strong> [4,5,3]\n<strong>Explanation:</strong> The diagrams above show the tree of 2<sup>3</sup> - 1 nodes. Nodes colored in red describe the nodes in the cycle after adding the edge.\n- After adding the edge between nodes 3 and 5, the graph contains a cycle of nodes [5,2,1,3]. Thus answer to the first query is 4. We delete the added edge and process the next query.\n- After adding the edge between nodes 4 and 7, the graph contains a cycle of nodes [4,2,1,3,7]. Thus answer to the second query is 5. We delete the added edge and process the next query.\n- After adding the edge between nodes 2 and 3, the graph contains a cycle of nodes [2,1,3]. Thus answer to the third query is 3. We delete the added edge.\n</pre>\n\n<p><strong class=\"example\">Example 2:</strong></p>\n<img alt=\"\" src=\"https://assets.leetcode.com/uploads/2022/10/25/aexample2.png\" style=\"width: 146px; height: 71px;\" />\n<pre>\n<strong>Input:</strong> n = 2, queries = [[1,2]]\n<strong>Output:</strong> [2]\n<strong>Explanation:</strong> The diagram above shows the tree of 2<sup>2</sup> - 1 nodes. Nodes colored in red describe the nodes in the cycle after adding the edge.\n- After adding the edge between nodes 1 and 2, the graph contains a cycle of nodes [2,1]. Thus answer for the first query is 2. We delete the added edge.\n</pre>\n\n<p>&nbsp;</p>\n<p><strong>Constraints:</strong></p>\n\n<ul>\n\t<li><code>2 &lt;= n &lt;= 30</code></li>\n\t<li><code>m == queries.length</code></li>\n\t<li><code>1 &lt;= m &lt;= 10<sup>5</sup></code></li>\n\t<li><code>queries[i].length == 2</code></li>\n\t<li><code>1 &lt;= a<sub>i</sub>, b<sub>i</sub> &lt;= 2<sup>n</sup> - 1</code></li>\n\t<li><code>a<sub>i</sub> != b<sub>i</sub></code></li>\n</ul>\n",
            "translatedTitle": "查询树中环的长度",
            "translatedContent": "<p>给你一个整数&nbsp;<code>n</code>&nbsp;，表示你有一棵含有&nbsp;<code>2<sup>n</sup> - 1</code>&nbsp;个节点的 <strong>完全二叉树</strong>&nbsp;。根节点的编号是&nbsp;<code>1</code>&nbsp;，树中编号在<code>[1, 2<sup>n - 1</sup> - 1]</code>&nbsp;之间，编号为&nbsp;<code>val</code>&nbsp;的节点都有两个子节点，满足：</p>\n\n<ul>\n\t<li>左子节点的编号为&nbsp;<code>2 * val</code></li>\n\t<li>右子节点的编号为&nbsp;<code>2 * val + 1</code></li>\n</ul>\n\n<p>给你一个长度为 <code>m</code>&nbsp;的查询数组 <code>queries</code>&nbsp;，它是一个二维整数数组，其中&nbsp;<code>queries[i] = [a<sub>i</sub>, b<sub>i</sub>]</code>&nbsp;。对于每个查询，求出以下问题的解：</p>\n\n<ol>\n\t<li>在节点编号为&nbsp;<code>a<sub>i</sub></code> 和&nbsp;<code>b<sub>i</sub></code>&nbsp;之间添加一条边。</li>\n\t<li>求出图中环的长度。</li>\n\t<li>删除节点编号为&nbsp;<code>a<sub>i</sub></code> 和&nbsp;<code>b<sub>i</sub></code>&nbsp;之间新添加的边。</li>\n</ol>\n\n<p><strong>注意：</strong></p>\n\n<ul>\n\t<li><strong>环</strong> 是开始和结束于同一节点的一条路径，路径中每条边都只会被访问一次。</li>\n\t<li>环的长度是环中边的数目。</li>\n\t<li>在树中添加额外的边后，两个点之间可能会有多条边。</li>\n</ul>\n\n<p>请你返回一个长度为 <code>m</code>&nbsp;的数组<em>&nbsp;</em><code>answer</code>&nbsp;，其中&nbsp;<code>answer[i]</code>&nbsp;是第&nbsp;<code>i</code>&nbsp;个查询的结果<i>。</i></p>\n\n<p>&nbsp;</p>\n\n<p><strong>示例 1：</strong></p>\n\n<p><img alt=\"\" src=\"https://assets.leetcode.com/uploads/2022/10/25/bexample1.png\" style=\"width: 647px; height: 128px;\" /></p>\n\n<pre>\n<b>输入：</b>n = 3, queries = [[5,3],[4,7],[2,3]]\n<b>输出：</b>[4,5,3]\n<b>解释：</b>上图是一棵有 2<sup>3</sup> - 1 个节点的树。红色节点表示添加额外边后形成环的节点。\n- 在节点 3 和节点 5 之间添加边后，环为 [5,2,1,3] ，所以第一个查询的结果是 4 。删掉添加的边后处理下一个查询。\n- 在节点 4 和节点 7 之间添加边后，环为 [4,2,1,3,7] ，所以第二个查询的结果是 5 。删掉添加的边后处理下一个查询。\n- 在节点 2 和节点 3 之间添加边后，环为 [2,1,3] ，所以第三个查询的结果是 3 。删掉添加的边。\n</pre>\n\n<p><strong>示例 2：</strong></p>\n\n<p><img alt=\"\" src=\"https://assets.leetcode.com/uploads/2022/10/25/aexample2.png\" style=\"width: 146px; height: 71px;\" /></p>\n\n<pre>\n<b>输入：</b>n = 2, queries = [[1,2]]\n<b>输出：</b>[2]\n<b>解释：</b>上图是一棵有 2<sup>2</sup> - 1 个节点的树。红色节点表示添加额外边后形成环的节点。\n- 在节点 1 和节点 2 之间添加边后，环为 [2,1] ，所以第一个查询的结果是 2 。删掉添加的边。\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 &lt;= 30</code></li>\n\t<li><code>m == queries.length</code></li>\n\t<li><code>1 &lt;= m &lt;= 10<sup>5</sup></code></li>\n\t<li><code>queries[i].length == 2</code></li>\n\t<li><code>1 &lt;= a<sub>i</sub>, b<sub>i</sub> &lt;= 2<sup>n</sup> - 1</code></li>\n\t<li><code>a<sub>i</sub> != b<sub>i</sub></code></li>\n</ul>\n",
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href=\\\"https:\\/\\/godoc.org\\/github.com\\/emirpasic\\/gods\\\" target=\\\"_blank\\\">https:\\/\\/godoc.org\\/github.com\\/emirpasic\\/gods<\\/a> \\u7b2c\\u4e09\\u65b9\\u5e93\\u3002<\\/p>\"],\"python3\":[\"Python3\",\"<p>\\u7248\\u672c\\uff1a<code>Python 3.10<\\/code><\\/p>\\r\\n\\r\\n<p>\\u4e3a\\u4e86\\u65b9\\u4fbf\\u8d77\\u89c1\\uff0c\\u5927\\u90e8\\u5206\\u5e38\\u7528\\u5e93\\u5df2\\u7ecf\\u88ab\\u81ea\\u52a8 \\u5bfc\\u5165\\uff0c\\u5982<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>\\u3002 \\u5982\\u679c\\u60a8\\u9700\\u8981\\u4f7f\\u7528\\u5176\\u4ed6\\u5e93\\u51fd\\u6570\\uff0c\\u8bf7\\u81ea\\u884c\\u5bfc\\u5165\\u3002<\\/p>\\r\\n\\r\\n<p>\\u5982\\u9700\\u4f7f\\u7528 Map\\/TreeMap 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