Score : $625$ points

问题描述

我们有一个包含 $N$ 个顶点的连通无向图，顶点编号从 $1$ 到 $N$，并有 $M$ 条边，编号从 $1$ 到 $M$。第 $i$ 条边连接顶点 $U_i$ 和顶点 $V_i$，具有整数权重 $W_i$。对于 $1\leq s,t \leq N$ 且 $s\neq t$，让我们定义 $d(s,t)$ 如下：

• 对于每条连接顶点 $s$ 和顶点 $t$ 的路径，考虑该路径上最大边权重。$d(s,t)$ 是这个权重的最小值。

解答 $Q$ 个查询。第 $j$ 个查询如下：

• 给定 $A_j,S_j,T_j$，当边 $A_j$ 的权重增加 $1$ 时，$d(S_j,T_j)$ 将增加多少？

注意：每个查询实际上并不会改变边的权重。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

We have a connected undirected graph with $N$ vertices numbered $1$ to $N$ and $M$ edges numbered $1$ to $M$. Edge $i$ connects vertex $U_i$ and vertex $V_i$, and has an integer weight of $W_i$. For $1\leq s,t \leq N,\ s\neq t$, let us define $d(s,t)$ as follows.

• For every path connecting vertex $s$ and vertex $t$, consider the maximum weight of an edge along that path. $d(s,t)$ is the minimum value of this weight.

Answer $Q$ queries. The $j$-th query is as follows.

• You are given $A_j,S_j,T_j$. By what value will $d(S_j,T_j)$ increase when the weight of edge $A_j$ is increased by $1$?

Note that each query does not actually change the weight of the edge.

Constraints

• $2\leq N \leq 2\times 10^5$
• $N-1\leq M \leq 2\times 10^5$
• $1 \leq U_i,V_i \leq N$
• $U_i \neq V_i$
• $1 \leq W_i \leq M$
• The given graph is connected.
• $1\leq Q \leq 2\times 10^5$
• $1 \leq A_j \leq M$
• $1 \leq S_j,T_j \leq N$
• $S_j\neq T_j$
• All values in the input are integers.

Input

The input is given from Standard Input in the following format:

$N$ $M$

$U_1$ $V_1$ $W_1$

$\vdots$

$U_M$ $V_M$ $W_M$

$Q$

$A_1$ $S_1$ $T_1$

$\vdots$

$A_Q$ $S_Q$ $T_Q$

Output

Print $Q$ lines. The $j$-th line $(1\leq j \leq Q)$ should contain the answer to the $j$-th query.

Sample Input 1

6 6
1 2 1
3 1 5
4 1 5
3 4 3
5 6 4
2 6 5
7
1 4 6
2 4 6
3 4 6
4 4 6
5 4 6
6 4 6
5 6 5

Sample Output 1

0
0
0
0
0
1
1

The above figure shows the edge numbers in black and the edge weights in blue.

Let us explain the first through sixth queries.

First, consider $d(4,6)$ in the given graph. The maximum weight of an edge along the path $4 \rightarrow 1 \rightarrow 2 \rightarrow 6$ is $5$, which is the minimum over all paths connecting vertex $4$ and vertex $6$, so $d(4,6)=5$.

Next, consider the increment of $d(4,6)$ when the weight of edge $x\ (1 \leq x \leq 6)$ increases by $1$. When $x=6$, we have $d(4,6)=6$, and the increment is $1$. On the other hand, when $x \neq 6$, we have $d(4,6)=5$, and the increment is $0$. For instance, when $x=3$, the maximum weight of an edge along the path $4 \rightarrow 1 \rightarrow 2 \rightarrow 6$ is $6$, but the maximum weight of an edge along the path $4 \rightarrow 3 \rightarrow 1 \rightarrow 2 \rightarrow 6$ is $5$, so $d(4,6)$ is still $5$.

Sample Input 2

2 2
1 2 1
1 2 1
1
1 1 2

Sample Output 2

0

The given graph may contain multi-edges.

update @ 2024/3/10 08:28:28