Swap and Reverse
Contest: Harbour.Space Scholarship Contest 2023-2024 (Div. 1 + Div. 2)
Difficulty rating: 1100
Tags: constructive algorithms, greedy, sortings, strings

You are given a string $$$s$$$ of length $$$n$$$ consisting of lowercase English letters, and an integer $$$k$$$. In one step you can perform any one of the two operations below:

- Pick an index $$$i$$$ ($$$1 \le i \le n - 2$$$) and swap $$$s_{i}$$$ and $$$s_{i+2}$$$.
- Pick an index $$$i$$$ ($$$1 \le i \le n-k+1$$$) and reverse the order of letters formed by the range $$$[i,i+k-1]$$$ of the string. Formally, if the string currently is equal to $$$s_1\ldots s_{i-1}s_is_{i+1}\ldots s_{i+k-2}s_{i+k-1}s_{i+k}\ldots s_{n-1}s_n$$$, change it to $$$s_1\ldots s_{i-1}s_{i+k-1}s_{i+k-2}\ldots s_{i+1}s_is_{i+k}\ldots s_{n-1}s_n$$$.

You can make as many steps as you want (possibly, zero). Your task is to find the lexicographically smallest string you can obtain after some number of steps.

A string $$$a$$$ is lexicographically smaller than a string $$$b$$$ of the same length if and only if the following holds:

- in the first position where $$$a$$$ and $$$b$$$ differ, the string $$$a$$$ has a letter that appears earlier in the alphabet than the corresponding letter in $$$b$$$.

Input format:
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 10^4$$$). The description of the test cases follows.

The first line of each test case contains two integers $$$n$$$ and $$$k$$$ ($$$1 \le k < n \le 10^5$$$).

The second line of each test case contains the string $$$s$$$ of length $$$n$$$ consisting of lowercase English letters.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$10^5$$$.

Output format:
For each test case, print the lexicographically smallest string after doing some (possibly, zero) steps.

Constraints:
Time limit: 1.0 s
Memory limit: 256.0 MB

Note:
In the first test case, we can obtain the string "aimn" using the following operations:

1. Reverse the range $$$[3,4]$$$. The string turns into "niam".
2. Swap $$$s_1$$$ and $$$s_3$$$. The string turns into "ainm".
3. Reverse the range $$$[3,4]$$$. The string turns into "aimn".

It can be proven that we cannot obtain any string lexicographically smaller than "aimn". Therefore, "aimn" is the answer.

In the second test case, we can obtain the string "aandp" using the following operations:

1. Swap $$$s_3$$$ and $$$s_5$$$. The string turns into "paadn".
2. Swap $$$s_1$$$ and $$$s_3$$$. The string turns into "aapdn".
3. Swap $$$s_3$$$ and $$$s_5$$$. The string turns into "aandp".

It can be proven that we cannot obtain any string lexicographically smaller than "aandp". Therefore, "aandp" is the answer.

Example input:
5
4 2
nima
5 3
panda
9 2
theforces
7 3
amirfar
6 4
rounds

Example output:
aimn
aandp
ceefhorst
aafmirr
dnorsu

Editorial:
By the first kind of operation, we already know that every odd index (same for the even ones) can be swapped with each other freely. Therefore, let us write down the values of the indices modulo $$$2$$$. For example, if $$$n$$$ is $$$10$$$, the indices modulo $$$2$$$ are $$$[1,0,1,0,1,0,1,0,1,0]$$$. Now, we consider the two cases.
1. When $$$k$$$ is odd.We can find out that after reversing any subarray of length $$$k$$$, the indices modulo $$$2$$$ do not change. So in this case, any series of the second operation is identical to some series of the first operation. Therefore, you should sort the odd indices and the even indices separately, and output the result of merging them into one string.
2. When $$$k$$$ is even.Let us observe how we can swap two adjacent indices in this case. First, reverse $$$[i,i+k-1]$$$, and then reverse $$$[i+1,i+k]$$$. If we do this on $$$[1,0,1,0,1,0,1,0,1,0]$$$, assuming $$$i=1$$$ and $$$k=6$$$, the indices modulo $$$2$$$ turn into $$$[0,1,0,1,0,1,1,0,1,0]$$$, and then $$$[0,1,1,0,1,0,1,0,1,0]$$$. Using these two steps and some series of the first operation, we can see that we can swap any two adjacent indices as a result. And such a series of operation is always possible, as $$$k<n$$$. Therefore, we can sort the entire string, and output the result.