CF1463D.Pairs

You have $2n$ integers $1, 2, \dots, 2n$ . You have to redistribute these $2n$ elements into $n$ pairs. After that, you choose $x$ pairs and take minimum elements from them, and from the other $n - x$ pairs, you take maximum elements.

Your goal is to obtain the set of numbers $\{b_1, b_2, \dots, b_n\}$ as the result of taking elements from the pairs.

What is the number of different $x$ -s ( $0 \le x \le n$ ) such that it's possible to obtain the set $b$ if for each $x$ you can choose how to distribute numbers into pairs and from which $x$ pairs choose minimum elements?

The first line contains a single integer $t$ ( $1 \le t \le 1000$ ) — the number of test cases.

The first line of each test case contains the integer $n$ ( $1 \le n \le 2 \cdot 10^5$ ).

The second line of each test case contains $n$ integers $b_1, b_2, \dots, b_n$ ( $1 \le b_1 < b_2 < \dots < b_n \le 2n$ ) — the set you'd like to get.

It's guaranteed that the sum of $n$ over test cases doesn't exceed $2 \cdot 10^5$ .

For each test case, print one number — the number of different $x$ -s such that it's possible to obtain the set $b$ .

In the first test case, $x = 1$ is the only option: you have one pair $(1, 2)$ and choose the minimum from this pair.

In the second test case, there are three possible $x$ -s. If $x = 1$ , then you can form the following pairs: $(1, 6)$ , $(2, 4)$ , $(3, 5)$ , $(7, 9)$ , $(8, 10)$ . You can take minimum from $(1, 6)$ (equal to $1$ ) and the maximum elements from all other pairs to get set $b$ .

If $x = 2$ , you can form pairs $(1, 2)$ , $(3, 4)$ , $(5, 6)$ , $(7, 9)$ , $(8, 10)$ and take the minimum elements from $(1, 2)$ , $(5, 6)$ and the maximum elements from the other pairs.

If $x = 3$ , you can form pairs $(1, 3)$ , $(4, 6)$ , $(5, 7)$ , $(2, 9)$ , $(8, 10)$ and take the minimum elements from $(1, 3)$ , $(4, 6)$ , $(5, 7)$ .

In the third test case, $x = 0$ is the only option: you can form pairs $(1, 3)$ , $(2, 4)$ and take the maximum elements from both of them.