Ex - Difference of Distance

AtCoder

IDabc301_h

Time5000ms

Memory256MB

Difficulty—

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: $N$ $M$ $U_1$ $V_1$ $W_1$ $\vdots$ $U_M$ $V_M$ $W_M$ $Q$ $A_1$ $S_1$ $T_1$ $\vdots$ $A_Q$ $S_Q$ $T_Q$ [samples]

Samples

Input #1

Output #1

0
0
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1

![image](https://img.atcoder.jp/abc301/00609898872063e16f2e7b43c6436c6d.png)
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 $5$.

Input #2

Output #2

0

The given graph may contain multi-edges.

API Response (JSON)

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      "content": "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 $",
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      "content": "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 $...",
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