A13717. Fixed Point Removal

Let $a_1, \ldots, a_n$ be an array of $n$ positive integers. In one operation, you can choose an index $i$ such that $a_i = i$ , and remove $a_i$ from the array (after the removal, the remaining parts are concatenated).

The weight of $a$ is defined as the maximum number of elements you can remove.

You must answer $q$ independent queries $(x, y)$ : after replacing the $x$ first elements of $a$ and the $y$ last elements of $a$ by $n+1$ (making them impossible to remove), what would be the weight of $a$ ?

The first line contains two integers $n$ and $q$ ( $1 \le n, q \le 3 \cdot 10^5$ ) — the length of the array and the number of queries.

The second line contains $n$ integers $a_1$ , $a_2$ , ..., $a_n$ ( $1 \leq a_i \leq n$ ) — elements of the array.

The $i$ -th of the next $q$ lines contains two integers $x$ and $y$ ( $x, y \ge 0$ and $x+y < n$ ).

Print $q$ lines, $i$ -th line should contain a single integer — the answer to the $i$ -th query.

Explanation of the first query:

After making first $x = 3$ and last $y = 1$ elements impossible to remove, $a$ becomes $[\times, \times, \times, 9, 5, 4, 6, 5, 7, 8, 3, 11, \times]$ (we represent $14$ as $\times$ for clarity).

Here is a strategy that removes $5$ elements (the element removed is colored in red):

- $[\times, \times, \times, 9, \color{red}{5}, 4, 6, 5, 7, 8, 3, 11, \times]$
- $[\times, \times, \times, 9, 4, 6, 5, 7, 8, 3, \color{red}{11}, \times]$
- $[\times, \times, \times, 9, 4, \color{red}{6}, 5, 7, 8, 3, \times]$
- $[\times, \times, \times, 9, 4, 5, 7, \color{red}{8}, 3, \times]$
- $[\times, \times, \times, 9, 4, 5, \color{red}{7}, 3, \times]$
- $[\times, \times, \times, 9, 4, 5, 3, \times]$ (final state)

It is impossible to remove more than $5$ elements, hence the weight is $5$ .