#P1685E. The Ultimate LIS Problem
The Ultimate LIS Problem
Description
It turns out that this is exactly the $100$-th problem of mine that appears in some programming competition. So it has to be special! And what can be more special than another problem about LIS...
You are given a permutation $p_1, p_2, \ldots, p_{2n+1}$ of integers from $1$ to $2n+1$. You will have to process $q$ updates, where the $i$-th update consists in swapping $p_{u_i}, p_{v_i}$.
After each update, find any cyclic shift of $p$ with $LIS \le n$, or determine that there is no such shift. (Refer to the output section for details).
Here $LIS(a)$ denotes the length of longest strictly increasing subsequence of $a$.
Hacks are disabled in this problem. Don't ask why.
The first line of the input contains two integers $n, q$ ($2 \le n \le 10^5$, $1 \le q \le 10^5$).
The second line of the input contains $2n+1$ integers $p_1, p_2, \ldots, p_{2n+1}$ ($1 \le p_i \le 2n+1$, all $p_i$ are distinct) — the elements of $p$.
The $i$-th of the next $q$ lines contains two integers $u_i, v_i$ ($1 \le u_i, v_i \le 2n+1$, $u_i \neq v_i$) — indicating that you have to swap elements $p_{u_i}, p_{v_i}$ in the $i$-th update.
After each update, output any $k$ $(0 \le k \le 2n)$, such that the length of the longest increasing subsequence of $(p_{k+1}, p_{k+2}, \ldots, p_{2n+1}, p_1, \ldots, p_k)$ doesn't exceed $n$, or $-1$, if there is no such $k$.
Input
The first line of the input contains two integers $n, q$ ($2 \le n \le 10^5$, $1 \le q \le 10^5$).
The second line of the input contains $2n+1$ integers $p_1, p_2, \ldots, p_{2n+1}$ ($1 \le p_i \le 2n+1$, all $p_i$ are distinct) — the elements of $p$.
The $i$-th of the next $q$ lines contains two integers $u_i, v_i$ ($1 \le u_i, v_i \le 2n+1$, $u_i \neq v_i$) — indicating that you have to swap elements $p_{u_i}, p_{v_i}$ in the $i$-th update.
Output
After each update, output any $k$ $(0 \le k \le 2n)$, such that the length of the longest increasing subsequence of $(p_{k+1}, p_{k+2}, \ldots, p_{2n+1}, p_1, \ldots, p_k)$ doesn't exceed $n$, or $-1$, if there is no such $k$.
Samples
2 6
1 2 3 4 5
1 5
1 5
4 5
5 4
1 4
2 5
-1
-1
2
-1
4
0
Note
After the first update, our permutation becomes $(5, 2, 3, 4, 1)$. We can show that all its cyclic shifts have $LIS \ge 3$.
After the second update, our permutation becomes $(1, 2, 3, 4, 5)$. We can show that all its cyclic shifts have $LIS \ge 3$.
After the third update, our permutation becomes $(1, 2, 3, 5, 4)$. Its shift by $2$ is $(3, 5, 4, 1, 2)$, and its $LIS = 2$.
After the fourth update, our permutation becomes $(1, 2, 3, 4, 5)$. We can show that all its cyclic shifts have $LIS \ge 3$.
After the fifth update, our permutation becomes $(4, 2, 3, 1, 5)$. Its shift by $4$ is $(5, 4, 2, 3, 1)$, and its $LIS = 2$.
After the fifth update, our permutation becomes $(4, 5, 3, 1, 2)$. Its shift by $0$ is $(4, 5, 3, 1, 2)$, and its $LIS = 2$.