leetcode-problemset/leetcode-cn/originData/minimum-edge-weight-equilibrium-queries-in-a-tree.json

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            "categoryTitle": "Algorithms",
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            "title": "Minimum Edge Weight Equilibrium Queries in a Tree",
            "titleSlug": "minimum-edge-weight-equilibrium-queries-in-a-tree",
            "content": "<p>There is an undirected tree with <code>n</code> nodes labeled from <code>0</code> to <code>n - 1</code>. You are given the integer <code>n</code> and a 2D integer array <code>edges</code> of length <code>n - 1</code>, where <code>edges[i] = [u<sub>i</sub>, v<sub>i</sub>, w<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> with weight <code>w<sub>i</sub></code> in the tree.</p>\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, find the <strong>minimum number of operations</strong> required to make the weight of every edge on the path from <code>a<sub>i</sub></code> to <code>b<sub>i</sub></code> equal. In one operation, you can choose any edge of the tree and change its weight to any value.</p>\n\n<p><strong>Note</strong> that:</p>\n\n<ul>\n\t<li>Queries are <strong>independent</strong> of each other, meaning that the tree returns to its <strong>initial state</strong> on each new query.</li>\n\t<li>The path from <code>a<sub>i</sub></code> to <code>b<sub>i</sub></code> is a sequence of <strong>distinct</strong> nodes starting with node <code>a<sub>i</sub></code> and ending with node <code>b<sub>i</sub></code> such that every two adjacent nodes in the sequence share an edge in the tree.</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/2023/08/11/graph-6-1.png\" style=\"width: 339px; height: 344px;\" />\n<pre>\n<strong>Input:</strong> n = 7, edges = [[0,1,1],[1,2,1],[2,3,1],[3,4,2],[4,5,2],[5,6,2]], queries = [[0,3],[3,6],[2,6],[0,6]]\n<strong>Output:</strong> [0,0,1,3]\n<strong>Explanation:</strong> In the first query, all the edges in the path from 0 to 3 have a weight of 1. Hence, the answer is 0.\nIn the second query, all the edges in the path from 3 to 6 have a weight of 2. Hence, the answer is 0.\nIn the third query, we change the weight of edge [2,3] to 2. After this operation, all the edges in the path from 2 to 6 have a weight of 2. Hence, the answer is 1.\nIn the fourth query, we change the weights of edges [0,1], [1,2] and [2,3] to 2. After these operations, all the edges in the path from 0 to 6 have a weight of 2. Hence, the answer is 3.\nFor each queries[i], it can be shown that answer[i] is the minimum number of operations needed to equalize all the edge weights in the path from a<sub>i</sub> to b<sub>i</sub>.\n</pre>\n\n<p><strong class=\"example\">Example 2:</strong></p>\n<img alt=\"\" src=\"https://assets.leetcode.com/uploads/2023/08/11/graph-9-1.png\" style=\"width: 472px; height: 370px;\" />\n<pre>\n<strong>Input:</strong> n = 8, edges = [[1,2,6],[1,3,4],[2,4,6],[2,5,3],[3,6,6],[3,0,8],[7,0,2]], queries = [[4,6],[0,4],[6,5],[7,4]]\n<strong>Output:</strong> [1,2,2,3]\n<strong>Explanation:</strong> In the first query, we change the weight of edge [1,3] to 6. After this operation, all the edges in the path from 4 to 6 have a weight of 6. Hence, the answer is 1.\nIn the second query, we change the weight of edges [0,3] and [3,1] to 6. After these operations, all the edges in the path from 0 to 4 have a weight of 6. Hence, the answer is 2.\nIn the third query, we change the weight of edges [1,3] and [5,2] to 6. After these operations, all the edges in the path from 6 to 5 have a weight of 6. Hence, the answer is 2.\nIn the fourth query, we change the weights of edges [0,7], [0,3] and [1,3] to 6. After these operations, all the edges in the path from 7 to 4 have a weight of 6. Hence, the answer is 3.\nFor each queries[i], it can be shown that answer[i] is the minimum number of operations needed to equalize all the edge weights in the path from a<sub>i</sub> to b<sub>i</sub>.\n</pre>\n\n<p>&nbsp;</p>\n<p><strong>Constraints:</strong></p>\n\n<ul>\n\t<li><code>1 &lt;= n &lt;= 10<sup>4</sup></code></li>\n\t<li><code>edges.length == n - 1</code></li>\n\t<li><code>edges[i].length == 3</code></li>\n\t<li><code>0 &lt;= u<sub>i</sub>, v<sub>i</sub> &lt; n</code></li>\n\t<li><code>1 &lt;= w<sub>i</sub> &lt;= 26</code></li>\n\t<li>The input is generated such that <code>edges</code> represents a valid tree.</li>\n\t<li><code>1 &lt;= queries.length == m &lt;= 2 * 10<sup>4</sup></code></li>\n\t<li><code>queries[i].length == 2</code></li>\n\t<li><code>0 &lt;= a<sub>i</sub>, b<sub>i</sub> &lt; n</code></li>\n</ul>\n",
            "translatedTitle": "边权重均等查询",
            "translatedContent": "<p>现有一棵由 <code>n</code> 个节点组成的无向树，节点按从 <code>0</code> 到 <code>n - 1</code> 编号。给你一个整数 <code>n</code> 和一个长度为 <code>n - 1</code> 的二维整数数组 <code>edges</code> ，其中 <code>edges[i] = [u<sub>i</sub>, v<sub>i</sub>, w<sub>i</sub>]</code> 表示树中存在一条位于节点 <code>u<sub>i</sub></code> 和节点 <code>v<sub>i</sub></code> 之间、权重为 <code>w<sub>i</sub></code> 的边。</p>\n\n<p>另给你一个长度为 <code>m</code> 的二维整数数组 <code>queries</code> ，其中 <code>queries[i] = [a<sub>i</sub>, b<sub>i</sub>]</code> 。对于每条查询，请你找出使从 <code>a<sub>i</sub></code> 到 <code>b<sub>i</sub></code> 路径上每条边的权重相等所需的 <strong>最小操作次数</strong> 。在一次操作中，你可以选择树上的任意一条边，并将其权重更改为任意值。</p>\n\n<p><strong>注意：</strong></p>\n\n<ul>\n\t<li>查询之间 <strong>相互独立</strong> 的，这意味着每条新的查询时，树都会回到 <strong>初始状态</strong> 。</li>\n\t<li>从 <code>a<sub>i</sub></code> 到 <code>b<sub>i</sub></code>的路径是一个由 <strong>不同</strong> 节点组成的序列，从节点 <code>a<sub>i</sub></code> 开始，到节点 <code>b<sub>i</sub></code> 结束，且序列中相邻的两个节点在树中共享一条边。</li>\n</ul>\n\n<p>返回一个长度为 <code>m</code> 的数组 <code>answer</code> ，其中 <code>answer[i]</code> 是第 <code>i</code> 条查询的答案。</p>\n\n<p>&nbsp;</p>\n\n<p><strong class=\"example\">示例 1：</strong></p>\n<img alt=\"\" src=\"https://assets.leetcode.com/uploads/2023/08/11/graph-6-1.png\" style=\"width: 339px; height: 344px;\" />\n<pre>\n<strong>输入：</strong>n = 7, edges = [[0,1,1],[1,2,1],[2,3,1],[3,4,2],[4,5,2],[5,6,2]], queries = [[0,3],[3,6],[2,6],[0,6]]\n<strong>输出：</strong>[0,0,1,3]\n<strong>解释：</strong>第 1 条查询，从节点 0 到节点 3 的路径中的所有边的权重都是 1 。因此，答案为 0 。\n第 2 条查询，从节点 3 到节点 6 的路径中的所有边的权重都是 2 。因此，答案为 0 。\n第 3 条查询，将边 [2,3] 的权重变更为 2 。在这次操作之后，从节点 2 到节点 6 的路径中的所有边的权重都是 2 。因此，答案为 1 。\n第 4 条查询，将边 [0,1]、[1,2]、[2,3] 的权重变更为 2 。在这次操作之后，从节点 0 到节点 6 的路径中的所有边的权重都是 2 。因此，答案为 3 。\n对于每条查询 queries[i] ，可以证明 answer[i] 是使从 a<sub>i</sub> 到 b<sub>i</sub> 的路径中的所有边的权重相等的最小操作次数。\n</pre>\n\n<p><strong class=\"example\">示例 2：</strong></p>\n<img alt=\"\" src=\"https://assets.leetcode.com/uploads/2023/08/11/graph-9-1.png\" style=\"width: 472px; height: 370px;\" />\n<pre>\n<strong>输入：</strong>n = 8, edges = [[1,2,6],[1,3,4],[2,4,6],[2,5,3],[3,6,6],[3,0,8],[7,0,2]], queries = [[4,6],[0,4],[6,5],[7,4]]\n<strong>输出：</strong>[1,2,2,3]\n<strong>解释：</strong>第 1 条查询，将边 [1,3] 的权重变更为 6 。在这次操作之后，从节点 4 到节点 6 的路径中的所有边的权重都是 6 。因此，答案为 1 。\n第 2 条查询，将边 [0,3]、[3,1] 的权重变更为 6 。在这次操作之后，从节点 0 到节点 4 的路径中的所有边的权重都是 6 。因此，答案为 2 。\n第 3 条查询，将边 [1,3]、[5,2] 的权重变更为 6 。在这次操作之后，从节点 6 到节点 5 的路径中的所有边的权重都是 6 。因此，答案为 2 。\n第 4 条查询，将边 [0,7]、[0,3]、[1,3] 的权重变更为 6 。在这次操作之后，从节点 7 到节点 4 的路径中的所有边的权重都是 6 。因此，答案为 3 。\n对于每条查询 queries[i] ，可以证明 answer[i] 是使从 a<sub>i</sub> 到 b<sub>i</sub> 的路径中的所有边的权重相等的最小操作次数。 \n</pre>\n\n<p>&nbsp;</p>\n\n<p><strong>提示：</strong></p>\n\n<ul>\n\t<li><code>1 &lt;= n &lt;= 10<sup>4</sup></code></li>\n\t<li><code>edges.length == n - 1</code></li>\n\t<li><code>edges[i].length == 3</code></li>\n\t<li><code>0 &lt;= u<sub>i</sub>, v<sub>i</sub> &lt; n</code></li>\n\t<li><code>1 &lt;= w<sub>i</sub> &lt;= 26</code></li>\n\t<li>生成的输入满足 <code>edges</code> 表示一棵有效的树</li>\n\t<li><code>1 &lt;= queries.length == m &lt;= 2 * 10<sup>4</sup></code></li>\n\t<li><code>queries[i].length == 2</code></li>\n\t<li><code>0 &lt;= a<sub>i</sub>, b<sub>i</sub> &lt; n</code></li>\n</ul>\n",
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