Score : $500$ points

问题描述

我们有一个由 $(1,2,\ldots,N)$ 形成的排列 $P=(P_1,P_2,\ldots,P_N)$。

有 $M$ 种操作可供选择。第 $i$ 种操作会交换 $P$ 中的第 $a_i$ 项和第 $b_i$ 项。

是否存在一种可能，通过以任意顺序执行总共最多 $5\times 10^5$ 次操作，将排列 $P$ 按升序排序？

如果可能，请给出这样一组操作序列。否则，请报告无法完成。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

We have a permutation $P=(P_1,P_2,\ldots,P_N)$ of $(1,2,\ldots,N)$.

There are $M$ kinds of operations available to you. Operation $i$ swaps the $a_i$-th and $b_i$-th elements of $P$.

Is it possible to sort $P$ in ascending order by doing at most $5\times 10^5$ operations in total in any order?

If it is possible, give one such sequence of operations. Otherwise, report so.

Constraints

• $2\leq N \leq 1000$
• $P$ is a permutation of $(1,2,\ldots,N)$.
• $1\leq M \leq \min(2\times 10^5, \frac{N(N-1)}{2})$
• $1\leq a_i \lt b_i\leq N$
• $(a_i,b_i)\neq (a_j,b_j)$ if $i\neq j$.
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$

$P_1$ $P_2$ $\ldots$ $P_N$

$M$

$a_1$ $b_1$

$a_2$ $b_2$

$\vdots$

$a_M$ $b_M$

Output

If it is possible to sort $P$ in ascending order, print the following:

$K$

$c_1$ $c_2$ $\ldots$ $c_K$

Here, $K$ represents the number of operations to do, and $c_i$ $(1\leq i \leq K)$ means the $i$-th operation to do is Operation $c_i$.

Note that $0\leq K \leq 5\times 10^5$ must hold.

If it is impossible to sort $P$ in ascending order, print -1.

Sample Input 1

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

Sample Output 1

3
4 2 1

$P$ changes as follows: $(5,3,2,4,6,1)\to (5,2,3,4,6,1)\to (5,2,3,4,1,6)\to (1,2,3,4,5,6)$.

Sample Input 2

5
3 4 1 2 5
2
1 3
2 5

Sample Output 2

-1

We cannot sort $P$ in ascending order.

Sample Input 3

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

Sample Output 3

0

$P$ may already be sorted in ascending order.

Additionally, here is another accepted output:

4
5 5 5 5

Note that it is not required to minimize the number of operations.

update @ 2024/3/10 10:09:11