310419: CF1830E. Bully Sort

E. Bully Sorttime limit per test10 secondsmemory limit per test1024 megabytesinputstandard inputoutputstandard output

On a permutation $p$ of length $n$, we define a bully swap as follows:

• Let $i$ be the index of the largest element $p_i$ such that $p_i \neq i$.
• Let $j$ be the index of the smallest element $p_j$ such that $i < j$.
• Swap $p_i$ and $p_j$.

We define $f(p)$ as the number of bully swaps we need to perform until $p$ becomes sorted. Note that if $p$ is the identity permutation, $f(p)=0$.

You are given $n$ and a permutation $p$ of length $n$. You need to process the following $q$ updates.

In each update, you are given two integers $x$ and $y$. You will swap $p_x$ and $p_y$ and then find the value of $f(p)$.

Note that the updates are persistent. Changes made to the permutation $p$ will apply when processing future updates.

Input

The first line of the input contains two integers $n$ and $q$ ($2 \le n \le 5 \cdot 10^5$, $1 \le q \le 5 \cdot 10^4$) — the length of the permutation and the number of updates.

The second line of input contains $n$ integer $p_1,p_2,\ldots,p_n$ ($1 \leq p_i \leq n$) — the permutation $p$. All elements of $p$ are distinct.

The $i$-th of the next $q$ lines of input contains two integers $x_i$ and $y_i$ ($1 \le x_i < y_i \le n$) — describing the $i$-th update.

Output

After each update, output $f(p)$.

ExampleInput

8 5
6 2 1 5 3 4 7 8
1 8
2 3
4 7
7 8
3 6

Output

5
6
9
8
7

Note

After the first update, we have $f(p)=5$. The $5$ bully swaps are illustrated below.

• $[\mathbf{1}, 2, \mathbf{8}, 5, 3, 4, 7, 6]$,
• $[1, 2, \mathbf{3}, 5, \mathbf{8}, 4, 7, 6]$,
• $[1, 2, 3, 5, \mathbf{4}, \mathbf{8}, 7, 6]$,
• $[1, 2, 3, 5, 4, \mathbf{6}, 7, \mathbf{8}]$,
• $[1, 2, 3, \mathbf{4}, \mathbf{5}, 6, 7, 8]$.

题意翻译

对于一个非升序的排列 $\{p_i\}$，定义一次操作为按顺序执行以下步骤： 1. 令 $p_i$ 为排列中最大的满足 $p_i\neq i$ 的数。 2. 令 $p_j$ 为排列中最小的满足 $i<j$ 的数。 3. 交换 $p_i$ 和 $p_j$ 可以证明在排列非升序的前提下总能找到满足条件的 $i$ 和 $j$。 定义一个排列的权值为将其升序排序所需的操作次数，可以证明总能在有限次操作内将其升序排序，注意如果排列本身就是升序的那么其权值为零。 现在给定一个排列和若干次修改操作，求出每次修改后排列的权值，每个修改操作为交换排列中指定的两个数。 注意修改是永久的。