Local Deletions
Description
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).
题面翻译
对于一个数组 ，对于 ，如果 ，那么称 是一个局部最小值。特别的，如果 ，那么称 是一个局部最小值，如果 ，那么称 是一个局部最小值。 类似的，对于一个数组 ，对于 ，如果 ，那么称 是一个局部最大值。特别的，如果 ，那么称 是一个局部最大值，如果 ，那么称 是一个局部最大值。 现在定义一个无重复元素的数组 ，定义两个操作：
- 删除 中所有不是局部最小值的数字。
- 删除 中所有不是局部最大值的数字。
定义函数 表示对于数组 ，循环进行操作 直到这个数组只剩下一个数字，那么 的值就是最后剩下的这个数字。
举个栗子，对于数组 ，为了计算 ，我们依次进行以下操作：
- 进行操作 ，然后数组变成 。
- 进行操作 ，然后数组变成 ，只剩下一个数字，结束。
- 。
现在给你一个 的排列 和 次询问，每次询问给出 ，要你计算 。 。
Input
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.
Output
For each query, output a single integer — the answer to that query.
7 5
1 4 3 6 2 7 5
1 1
1 2
2 3
1 4
1 7
10 1
1 2 3 4 5 6 7 8 9 10
1 10
1
1
3
3
3
1
Note
In the first query of the first example, the only number in the subarray is $1$, therefore it is the answer.
In the second query of the first example, our subarray initially looks like $[1, 4]$. After performing type $1$ operation we get $[1]$.
In the third query of the first example, our subarray initially looks like $[4, 3]$. After performing type $1$ operation we get $[3]$.
In the fourth query of the first example, our subarray initially looks like $[1, 4, 3, 6]$. After performing type $1$ operation we get $[1, 3]$. Then we perform type $2$ operation and we get $[3]$.
In the fifth query of the first example, our subarray initially looks like $[1, 4, 3, 6, 2, 7, 5]$. After performing type $1$ operation we get $[1,3,2,5]$. After performing type $2$ operation we get $[3,5]$. Then we perform type $1$ operation and we get $[3]$.
In the first and only query of the second example our subarray initially looks like $[1,2,3,4,5,6,7,8,9,10]$. Here $1$ is the only local minimum, so only it is left after performing type $1$ operation.
相关
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