311255: CF1956F. Nene and the Passing Game

Nene is training her team as a basketball coach. Nene's team consists of $n$ players, numbered from $1$ to $n$. The $i$-th player has an arm interval $[l_i,r_i]$. Two players $i$ and $j$ ($i \neq j$) can pass the ball to each other if and only if $|i-j|\in[l_i+l_j,r_i+r_j]$ (here, $|x|$ denotes the absolute value of $x$).

Nene wants to test the cooperation ability of these players. In order to do this, she will hold several rounds of assessment.

• In each round, Nene will select a sequence of players $p_1,p_2,\ldots,p_m$ such that players $p_i$ and $p_{i+1}$ can pass the ball to each other for all $1 \le i < m$. The length of the sequence $m$ can be chosen by Nene. Each player can appear in the sequence $p_1,p_2,\ldots,p_m$ multiple times or not appear in it at all.
• Then, Nene will throw a ball to player $p_1$, player $p_1$ will pass the ball to player $p_2$ and so on... Player $p_m$ will throw a ball away from the basketball court so it can no longer be used.

As a coach, Nene wants each of $n$ players to appear in at least one round of assessment. Since Nene has to go on a date after school, Nene wants you to calculate the minimum number of rounds of assessment needed to complete the task.

Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 2\cdot 10^5$). The description of test cases follows.

The first line contains a single integer $n$ ($1 \le n \le 2\cdot 10^6$) — the number of players.

The $i$-th of the next $n$ lines contains two integers $l_i$ and $r_i$ ($1\leq l_i\leq r_i\leq n$) — the arm interval of the $i$-th player.

It is guaranteed that the sum of $n$ over all test cases does not exceed $2\cdot 10^6$.

For each test case, output one integer — the minimum number of rounds of assessment Nene needs to complete her work.

5
2
1 1
1 1
2
1 1
2 2
3
1 3
1 3
1 3
5
1 1
2 2
1 5
2 2
1 1
6
1 2
5 5
2 3
2 3
2 2
1 2

2
2
2
1
3

In the first two test cases, Nene can host two rounds of assessment: one with $p=[1]$ and one with $p=[2]$. It can be shown that hosting one round of assessment is not enough, so the answer is $2$.

In the third test case, Nene can host two rounds of assessment: one with $p=[1,3]$ and one with $p=[2]$. Player $1$ can pass the ball to player $3$ as $|3-1|=2 \in [1+1,3+3]$. It can be shown that hosting one round of assessment is not enough, so the answer is $2$.