codeforces#P1736D. Equal Binary Subsequences
Equal Binary Subsequences
Description
Everool has a binary string $s$ of length $2n$. Note that a binary string is a string consisting of only characters $0$ and $1$. He wants to partition $s$ into two disjoint equal subsequences. He needs your help to do it.
You are allowed to do the following operation exactly once.
- You can choose any subsequence (possibly empty) of $s$ and rotate it right by one position.
In other words, you can select a sequence of indices $b_1, b_2, \ldots, b_m$, where $1 \le b_1 < b_2 < \ldots < b_m \le 2n$. After that you simultaneously set $$s_{b_1} := s_{b_m},$$ $$s_{b_2} := s_{b_1},$$ $$\ldots,$$ $$s_{b_m} := s_{b_{m-1}}.$$
Can you partition $s$ into two disjoint equal subsequences after performing the allowed operation exactly once?
A partition of $s$ into two disjoint equal subsequences $s^p$ and $s^q$ is two increasing arrays of indices $p_1, p_2, \ldots, p_n$ and $q_1, q_2, \ldots, q_n$, such that each integer from $1$ to $2n$ is encountered in either $p$ or $q$ exactly once, $s^p = s_{p_1} s_{p_2} \ldots s_{p_n}$, $s^q = s_{q_1} s_{q_2} \ldots s_{q_n}$, and $s^p = s^q$.
If it is not possible to partition after performing any kind of operation, report $-1$.
If it is possible to do the operation and partition $s$ into two disjoint subsequences $s^p$ and $s^q$, such that $s^p = s^q$, print elements of $b$ and indices of $s^p$, i. e. the values $p_1, p_2, \ldots, p_n$.
Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^5$). Description of the test cases follows.
The first line of each test case contains a single integer $n$ ($1 \le n \le 10^5$), where $2n$ is the length of the binary string.
The second line of each test case contains the binary string $s$ of length $2n$.
It is guaranteed that the sum of $n$ over all test cases does not exceed $10^5$.
For each test case, follow the following output format.
If there is no solution, print $-1$.
Otherwise,
- In the first line, print an integer $m$ ($0 \leq m \leq 2n$), followed by $m$ distinct indices $b_1$, $b_2$, ..., $b_m$(in increasing order).
- In the second line, print $n$ distinct indices $p_1$, $p_2$, ..., $p_n$ (in increasing order).
If there are multiple solutions, print any.
Input
Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^5$). Description of the test cases follows.
The first line of each test case contains a single integer $n$ ($1 \le n \le 10^5$), where $2n$ is the length of the binary string.
The second line of each test case contains the binary string $s$ of length $2n$.
It is guaranteed that the sum of $n$ over all test cases does not exceed $10^5$.
Output
For each test case, follow the following output format.
If there is no solution, print $-1$.
Otherwise,
- In the first line, print an integer $m$ ($0 \leq m \leq 2n$), followed by $m$ distinct indices $b_1$, $b_2$, ..., $b_m$(in increasing order).
- In the second line, print $n$ distinct indices $p_1$, $p_2$, ..., $p_n$ (in increasing order).
If there are multiple solutions, print any.
4
2
1010
3
100010
2
1111
2
1110
0
1 2
2 3 5
1 2 5
3 2 3 4
1 4
-1
Note
In the first test case, $b$ is empty. So string $s$ is not changed. Now $s^p = s_1 s_2 = \mathtt{10}$, and $s^q = s_3s_4 = \mathtt{10}$.
In the second test case, $b=[3,5]$. Initially $s_3=\mathtt{0}$, and $s_5=\mathtt{1}$. On performing the operation, we simultaneously set $s_3=\mathtt{1}$, and $s_5=\mathtt{0}$.
So $s$ is updated to 101000 on performing the operation.
Now if we take characters at indices $[1,2,5]$ in $s^p$, we get $s_1=\mathtt{100}$. Also characters at indices $[3,4,6]$ are in $s^q$. Thus $s^q=100$. We are done as $s^p=s^q$.
In fourth test case, it can be proved that it is not possible to partition the string after performing any operation.