{ "data": { "question": { "questionId": "3079", "questionFrontendId": "2846", "categoryTitle": "Algorithms", "boundTopicId": 2424055, "title": "Minimum Edge Weight Equilibrium Queries in a Tree", "titleSlug": "minimum-edge-weight-equilibrium-queries-in-a-tree", "content": "
There is an undirected tree with n
nodes labeled from 0
to n - 1
. You are given the integer n
and a 2D integer array edges
of length n - 1
, where edges[i] = [ui, vi, wi]
indicates that there is an edge between nodes ui
and vi
with weight wi
in the tree.
You are also given a 2D integer array queries
of length m
, where queries[i] = [ai, bi]
. For each query, find the minimum number of operations required to make the weight of every edge on the path from ai
to bi
equal. In one operation, you can choose any edge of the tree and change its weight to any value.
Note that:
\n\nai
to bi
is a sequence of distinct nodes starting with node ai
and ending with node bi
such that every two adjacent nodes in the sequence share an edge in the tree.Return an array answer
of length m
where answer[i]
is the answer to the ith
query.
\n
Example 1:
\n\n\nInput: 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]]\nOutput: [0,0,1,3]\nExplanation: 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 ai to bi.\n\n\n
Example 2:
\n\n\nInput: 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]]\nOutput: [1,2,2,3]\nExplanation: 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 ai to bi.\n\n\n
\n
Constraints:
\n\n1 <= n <= 104
edges.length == n - 1
edges[i].length == 3
0 <= ui, vi < n
1 <= wi <= 26
edges
represents a valid tree.1 <= queries.length == m <= 2 * 104
queries[i].length == 2
0 <= ai, bi < n
现有一棵由 n
个节点组成的无向树,节点按从 0
到 n - 1
编号。给你一个整数 n
和一个长度为 n - 1
的二维整数数组 edges
,其中 edges[i] = [ui, vi, wi]
表示树中存在一条位于节点 ui
和节点 vi
之间、权重为 wi
的边。
另给你一个长度为 m
的二维整数数组 queries
,其中 queries[i] = [ai, bi]
。对于每条查询,请你找出使从 ai
到 bi
路径上每条边的权重相等所需的 最小操作次数 。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。
注意:
\n\nai
到 bi
的路径是一个由 不同 节点组成的序列,从节点 ai
开始,到节点 bi
结束,且序列中相邻的两个节点在树中共享一条边。返回一个长度为 m
的数组 answer
,其中 answer[i]
是第 i
条查询的答案。
\n\n
示例 1:
\n\n\n输入: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输出:[0,0,1,3]\n解释: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 ai 到 bi 的路径中的所有边的权重相等的最小操作次数。\n\n\n
示例 2:
\n\n\n输入: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输出:[1,2,2,3]\n解释: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 ai 到 bi 的路径中的所有边的权重相等的最小操作次数。 \n\n\n
\n\n
提示:
\n\n1 <= n <= 104
edges.length == n - 1
edges[i].length == 3
0 <= ui, vi < n
1 <= wi <= 26
edges
表示一棵有效的树1 <= queries.length == m <= 2 * 104
queries[i].length == 2
0 <= ai, bi < n
freq[node][weight]
which saves the frequency of each edge weight
on the path from the root to each node
.",
"The frequency of edge weight w
on the path from a
to b
is equal to freq[a][w] + freq[b][w] - freq[lca(a,b)][w] * 2
, where lca(a,b)
is the lowest common ancestor of a
and b
in the tree.",
"lca(a,b)
can be calculated using binary lifting algorithm or Tarjan algorithm."
],
"solution": null,
"status": null,
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