A16089. Split Sort

You are given a permutation $^{\dagger}$ $p_1, p_2, \ldots, p_n$ of integers $1$ to $n$ .

You can change the current permutation by applying the following operation several (possibly, zero) times:

- choose some $x$ ( $2 \le x \le n$ );
- create a new permutation by:
- first, writing down all elements of $p$ that are less than $x$ , without changing their order;
- second, writing down all elements of $p$ that are greater than or equal to $x$ , without changing their order;
- replace $p$ with the newly created permutation.

For example, if the permutation used to be $[6, 4, 3, 5, 2, 1]$ and you choose $x = 4$ , then you will first write down $[3, 2, 1]$ , then append this with $[6, 4, 5]$ . So the initial permutation will be replaced by $[3, 2, 1, 6, 4, 5]$ .

Find the minimum number of operations you need to achieve $p_i = i$ for $i = 1, 2, \ldots, n$ . We can show that it is always possible to do so.

$^{\dagger}$ A permutation of length $n$ is an array consisting of $n$ distinct integers from $1$ to $n$ in arbitrary order. For example, $[2,3,1,5,4]$ is a permutation, but $[1,2,2]$ is not a permutation ( $2$ appears twice in the array), and $[1,3,4]$ is also not a permutation ( $n=3$ but there is $4$ in the array).

Each test contains multiple test cases. The first line contains the number of test cases $t$ ( $1 \le t \le 1000$ ). The description of the test cases follows.

The first line of each test case contains one integer $n$ ( $1 \le n \le 100\,000$ ).

The second line of each test case contains $n$ integers $p_1, p_2, \ldots, p_n$ ( $1 \le p_i \le n$ ). It is guaranteed that $p_1, p_2, \ldots, p_n$ is a permutation.

It is guaranteed that the sum of $n$ over all test cases does not exceed $100\,000$ .

For each test case, output the answer on a separate line.

In the first test case, $n = 1$ and $p_1 = 1$ , so there is nothing left to do.

In the second test case, we can choose $x = 2$ and we immediately obtain $p_1 = 1$ , $p_2 = 2$ .

In the third test case, we can achieve the minimum number of operations in the following way:

1. $x = 4$ : $[6, 4, 3, 5, 2, 1] \rightarrow [3, 2, 1, 6, 4, 5]$ ;
2. $x = 6$ : $[3, 2, 1, 6, 4, 5] \rightarrow [3, 2, 1, 4, 5, 6]$ ;
3. $x = 3$ : $[3, 2, 1, 4, 5, 6] \rightarrow [2, 1, 3, 4, 5, 6]$ ;
4. $x = 2$ : $[2, 1, 3, 4, 5, 6] \rightarrow [1, 2, 3, 4, 5, 6]$ .