The Ultimate LIS Problem

Description:

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 $$$k$$$.

Sample Input:

2 6
1 2 3 4 5
1 5
1 5
4 5
5 4
1 4
2 5

Sample Output:

-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 $$$LIS = 2$$$.