Inverse Transformation
A permutation scientist is studying a self-transforming permutation $$$a$$$ consisting of $$$n$$$ elements $$$a_1,a_2,\ldots,a_n$$$.A permutation is a sequence of integers from $$$1$$$ to $$$n$$$ of length $$$n$$$ containing each number exactly once. For example, $$$[1]$$$, $$$[4, 3, 5, 1, 2]$$$ are permutations, while $$$[1, 1]$$$, $$$[4, 3, 1]$$$ are not.The permutation transforms day by day. On each day, each element $$$x$$$ becomes $$$a_x$$$, that is, $$$a_x$$$ becomes $$$a_{a_x}$$$. Specifically:   on the first day, the permutation becomes $$$b$$$, where $$$b_x = a_{a_x}$$$;  on the second day, the permutation becomes $$$c$$$, where $$$c_x = b_{b_x}$$$;  $$$\ldots$$$  For example, consider permutation $$$a = [2,3,1]$$$. On the first day, it becomes $$$[3,1,2]$$$. On the second day, it becomes $$$[2,3,1]$$$.You're given the permutation $$$a'$$$ on the $$$k$$$-th day.Define $$$\sigma(x) = a_x$$$, and define $$$f(x)$$$ as the minimal positive integer $$$m$$$ such that $$$\sigma^m(x) = x$$$, where $$$\sigma^m(x)$$$ denotes $$$\underbrace{\sigma(\sigma(\ldots \sigma}_{m \text{ times}}(x) \ldots))$$$.For example, if $$$a = [2,3,1]$$$, then $$$\sigma(1) = 2$$$, $$$\sigma^2(1) = \sigma(\sigma(1)) = \sigma(2) = 3$$$, $$$\sigma^3(1) = \sigma(\sigma(\sigma(1))) = \sigma(3) = 1$$$, so $$$f(1) = 3$$$. And if $$$a=[4,2,1,3]$$$, $$$\sigma(2) = 2$$$ so $$$f(2) = 1$$$; $$$\sigma(3) = 1$$$, $$$\sigma^2(3) = 4$$$, $$$\sigma^3(3) = 3$$$ so $$$f(3) = 3$$$.Find the initial permutation $$$a$$$ such that $$$\sum\limits^n_{i=1} \dfrac{1}{f(i)}$$$ is minimum possible.
In the second test case, the initial permutation can be $$$a = [6,2,5,7,1,3,4]$$$, which becomes $$$[3,2,1,4,6,5,7]$$$ on the first day and $$$a' = [1,2,3,4,5,6,7]$$$ on the second day (the $$$k$$$-th day). Also, among all the permutations satisfying that, it has the minimum $$$\sum\limits^n_{i=1} \dfrac{1}{f(i)}$$$, which is $$$\dfrac{1}{4}+\dfrac{1}{1}+\dfrac{1}{4}+\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{4}+\dfrac{1}{2}=3$$$.In the fifth test case, the initial permutation can be $$$a = [4,2,1,3]$$$, which becomes $$$[3,2,4,1]$$$ on the first day, $$$[4,2,1,3]$$$ on the second day, and so on. So it finally becomes $$$a' = [4,2,1,3]$$$ on the $$$8$$$-th day (the $$$k$$$-th day). And it has the minimum $$$\sum\limits^n_{i=1} \dfrac{1}{f(i)} = \dfrac{1}{3} + \dfrac{1}{1} + \dfrac{1}{3} + \dfrac{1}{3} = 2$$$.
Each test contains multiple test cases. The first line contains an integer $$$t$$$ ($$$1 \le t \le 10^4$$$) — the number of test cases.The first line of each test case contains two integers $$$n$$$ and $$$k$$$ ($$$1 \le n \le 2 \cdot 10^5$$$; $$$1 \le k \le 10^9$$$) — the length of $$$a$$$, and the last day.The second line contains $$$n$$$ integers $$$a'_1,a'_2,\ldots,a'_n$$$ ($$$1 \le a'_i \le n$$$) — the permutation on the $$$k$$$-th day.It's guaranteed that the sum of $$$n$$$ does not exceed $$$2 \cdot 10^5$$$.
For each test case, if at least one initial $$$a$$$ consistent with the given $$$a'$$$ exists, print "YES", then print $$$n$$$ integers $$$a_1,a_2,\ldots,a_n$$$ — the initial permutation with the smallest sum $$$\sum\limits^n_{i=1} \dfrac{1}{f(i)}$$$. If there are multiple answers with the smallest sum, print any.If there are no valid permutations, print "NO".
import sys
from collections import Counter

def input_general():
    return sys.stdin.readline().rstrip('\r\n')

def input_num():
    return int(sys.stdin.readline().rstrip('\r\n'))

def input_multi(x=int):
    return map(x, sys.stdin.readline().rstrip('\r\n').split())

def input_list(x=int):
    return list(input_multi(x))

def power(p, x, e):
    base = x
    answer = 1
    while e:
        if e & 1:
            answer = answer * base % p
        base = base * base % p
        e >>= 1
    return answer

def main():
    pre = {2 ** i: i for i in range(20)}
    (n, k) = input_multi()
    arr = input_list()
    arr = [x - 1 for x in arr]
    count = {}
    cycle = []
    seen = [0] * n
    for x in arr:
        if seen[x] == 0:
            loop = [x]
            l = 1
            ptr = arr[x]
            seen[x] = 1
            while ptr != x:
                l += 1
                seen[ptr] = 1
                loop.append(ptr)
                ptr = arr[ptr]
            cycle.append(loop)
            if l not in count:
                count[l] = [loop]
            else:
                count[l].append(loop)
    for l in count.keys():
        if l % 2 == 0:
            if k >= 20 or len(count[l]) % 2 ** k != 0:
                print('NO')
                return
    answer = [0] * n
    batch = 2 ** min(k, 20)
    for (l, loops) in count.items():
        avail = len(loops)
        idx = 0
        while avail:
            size = min(avail & -avail, batch)
            avail -= size
            goal = loops[idx:idx + size]
            idx += size
            if l % 2 == 0:
                group = []
                for i in range(l):
                    for j in range(size):
                        group.append(goal[j][i])
                for (i, j) in zip(group, group[1:] + [group[0]]):
                    answer[i] = j + 1
            else:
                inv_l = (l + 1) // 2
                mix_days = pre[size]
                hop = power(l, inv_l, k - mix_days)
                new = [[g[hop * i % l] for i in range(l)] for g in goal]
                group = []
                for i in range(l):
                    for j in range(size):
                        group.append(new[j][i])
                for (i, j) in zip(group, group[1:] + [group[0]]):
                    answer[i] = j + 1
    print('YES')
    print(' '.join([str(x) for x in answer]))
if __name__ == '__main__':
    cases = input_num()
    for _ in range(cases):
        main()
import sys
from collections import Counter

def input_general():
    return sys.stdin.readline().rstrip('\r\n')

def input_num():
    return int(sys.stdin.readline().rstrip('\r\n'))

def input_multi(x=int):
    return map(x, sys.stdin.readline().rstrip('\r\n').split())

def input_list(x=int):
    return list(input_multi(x))

def power(p, x, e):
    base = x
    answer = 1
    while e:
        if e & 1:
            answer = answer * base % p
        base = base * base % p
        e >>= 1
    return answer

def main():
    pre = {2 ** i: i for i in range(20)}
    (n, k) = input_multi()
    arr = input_list()
    arr = [x - 1 for x in arr]
    count = {}
    cycle = []
    seen = [0] * n
    for x in arr:
        if seen[x] == 0:
            loop = [x]
            l = 1
            ptr = arr[x]
            seen[x] = 1
            while ptr != x:
                l += 1
                seen[ptr] = 1
                loop.append(ptr)
                ptr = arr[ptr]
            cycle.append(loop)
            if l not in count:
                count[l] = [loop]
            else:
                count[l].append(loop)
    for l in count.keys():
        if l % 2 == 0:
            if k >= 20 or len(count[l]) % 2 ** k != 0:
                print('NO')
                return
    answer = [0] * n
    batch = 2 ** min(k, 20)
    for (l, loops) in count.items():
        avail = len(loops)
        idx = 0
        while avail:
            size = min(avail & -avail, batch)
            avail -= size
            goal = loops[idx:idx + size]
            idx += size
            if l % 2 == 0:
                group = []
                for i in range(l):
                    for j in range(size):
                        group.append(goal[j][i])
                for (i, j) in zip(group, group[1:] + [group[0]]):
                    answer[i] = j + 1
            else:
                inv_l = (l + 1) // 2
                mix_days = pre[size]
                hop = power(l, inv_l, k - mix_days)
                new = [[g[hop * i % l] for i in range(l)] for g in goal]
                group = []
                for i in range(l):
                    for j in range(size):
                        group.append(new[j][i])
                for (i, j) in zip(group, group[1:] + [group[0]]):
                    answer[i] = j + 1
    print('YES')
    print(' '.join([str(x) for x in answer]))
if __name__ == '__main__':
    cases = input_num()
    for _ in range(cases):
        main()
import sys
from collections import Counter

def input_general():
    return sys.stdin.readline().rstrip('\r\n')

def input_num():
    return int(sys.stdin.readline().rstrip('\r\n'))

def input_multi(x=int):
    return map(x, sys.stdin.readline().rstrip('\r\n').split())

def input_list(x=int):
    return list(input_multi(x))

def power(p, x, e):
    base = x
    answer = 1
    while e:
        if e & 1:
            answer = answer * base % p
        base = base * base % p
        e >>= 1
    return answer

def main():
    pre = {2 ** i: i for i in range(20)}
    (n, k) = input_multi()
    arr = input_list()
    arr = [x - 1 for x in arr]
    count = {}
    cycle = []
    seen = [0] * n
    for x in arr:
        if seen[x] == 0:
            loop = [x]
            l = 1
            ptr = arr[x]
            seen[x] = 1
            while ptr != x:
                l += 1
                seen[ptr] = 1
                loop.append(ptr)
                ptr = arr[ptr]
            cycle.append(loop)
            if l not in count:
                count[l] = [loop]
            else:
                count[l].append(loop)
    for l in count.keys():
        if l % 2 == 0:
            if k >= 20 or len(count[l]) % 2 ** k != 0:
                print('NO')
                return
    answer = [0] * n
    batch = 2 ** min(k, 20)
    for (l, loops) in count.items():
        avail = len(loops)
        idx = 0
        while avail:
            size = min(avail & -avail, batch)
            avail -= size
            goal = loops[idx:idx + size]
            idx += size
            if l % 2 == 0:
                group = []
                for i in range(l):
                    for j in range(size):
                        group.append(goal[j][i])
                for (i, j) in zip(group, group[1:] + [group[0]]):
                    answer[i] = j + 1
            else:
                inv_l = (l + 1) // 2
                mix_days = pre[size]
                hop = power(l, inv_l, k - mix_days)
                new = [[g[hop * i % l] for i in range(l)] for g in goal]
                group = []
                for i in range(l):
                    for j in range(size):
                        group.append(new[j][i])
                for (i, j) in zip(group, group[1:] + [group[0]]):
                    answer[i] = j + 1
    print('YES')
    print(' '.join([str(x) for x in answer]))
if __name__ == '__main__':
    cases = input_num()
    for _ in range(cases):
        main()