A16301 | Local Deletions

For an array $b_1, b_2, \ldots, b_m$ , for some $i$ ( $1 < i < m$ ), element $b_i$ is said to be a local minimum if $b_i < b_{i-1}$ and $b_i < b_{i+1}$ . Element $b_1$ is said to be a local minimum if $b_1 < b_2$ . Element $b_m$ is said to be a local minimum if $b_m < b_{m-1}$ .

For an array $b_1, b_2, \ldots, b_m$ , for some $i$ ( $1 < i < m$ ), element $b_i$ is said to be a local maximum if $b_i > b_{i-1}$ and $b_i > b_{i+1}$ . Element $b_1$ is said to be a local maximum if $b_1 > b_2$ . Element $b_m$ is said to be a local maximum if $b_m > b_{m-1}$ .

Let $x$ be an array of distinct elements. We define two operations on it:

- $1$ — delete all elements from $x$ that are not local minima.
- $2$ — delete all elements from $x$ that are not local maxima.

Define $f(x)$ as follows. Repeat operations $1, 2, 1, 2, \ldots$ in that order until you get only one element left in the array. Return that element.

For example, take an array $[1,3,2]$ . We will first do type $1$ operation and get $[1, 2]$ . Then we will perform type $2$ operation and get $[2]$ . Therefore, $f([1,3,2]) = 2$ .

You are given a permutation $^\dagger$ $a$ of size $n$ and $q$ queries. Each query consists of two integers $l$ and $r$ such that $1 \le l \le r \le n$ . The query asks you to compute $f([a_l, a_{l+1}, \ldots, a_r])$ .

$^\dagger$ A permutation of length $n$ is an array 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).

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

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ( $1 \le a_i \le n$ ) — the elements of permutation $a$ .

The $i$ -th of the next $q$ lines contains two integers $l_i$ and $r_i$ ( $1 \le l_i \le r_i \le n$ ) — the description of $i$ -th query.

For each query, output a single integer — the answer to that query.