Permutation and Queries (Codeforces 1619H)
Tags: Others, divide and conquer

You are given a permutation $$$p$$$ of $$$n$$$ elements. A permutation of $$$n$$$ elements is an array of length $$$n$$$ containing each integer from $$$1$$$ to $$$n$$$ exactly once. For example, $$$[1, 2, 3]$$$ and $$$[4, 3, 5, 1, 2]$$$ are permutations, but $$$[1, 2, 4]$$$ and $$$[4, 3, 2, 1, 2]$$$ are not permutations. You should perform $$$q$$$ queries.There are two types of queries:  $$$1$$$ $$$x$$$ $$$y$$$ — swap $$$p_x$$$ and $$$p_y$$$.  $$$2$$$ $$$i$$$ $$$k$$$ — print the number that $$$i$$$ will become if we assign $$$i = p_i$$$ $$$k$$$ times. 

Input: The first line contains two integers $$$n$$$ and $$$q$$$ ($$$1 \le n, q \le 10^5$$$).The second line contains $$$n$$$ integers $$$p_1, p_2, \dots, p_n$$$.Each of the next $$$q$$$ lines contains three integers. The first integer is $$$t$$$ ($$$1 \le t \le 2$$$) — type of query. If $$$t = 1$$$, then the next two integers are $$$x$$$ and $$$y$$$ ($$$1 \le x, y \le n$$$; $$$x \ne y$$$) — first-type query. If $$$t = 2$$$, then the next two integers are $$$i$$$ and $$$k$$$ ($$$1 \le i, k \le n$$$) — second-type query.It is guaranteed that there is at least one second-type query.

Output: For every second-type query, print one integer in a new line — answer to this query.

Note: In the first example $$$p = \{5, 3, 4, 2, 1\}$$$. The first query is to print $$$p_3$$$. The answer is $$$4$$$.The second query is to print $$$p_{p_1}$$$. The answer is $$$1$$$.The third query is to swap $$$p_1$$$ and $$$p_3$$$. Now $$$p = \{4, 3, 5, 2, 1\}$$$.The fourth query is to print $$$p_{p_1}$$$. The answer is $$$2$$$.

Sample Inputs:
5 4
5 3 4 2 1
2 3 1
2 1 2
1 1 3
2 1 2

5 9
2 3 5 1 4
2 3 5
2 5 5
2 5 1
2 5 3
2 5 4
1 5 4
2 5 3
2 2 5
2 5 1

Sample Outputs:
4
1
2

3
5
4
2
3
3
3
1

Solution (Python):
import io, os
import sys
import math
import array as arr
from sys import stdin
input = sys.stdin.buffer.readline

def work():
    ans = 0
    return

def main():
    test = 1
    for idt in range(test):
        (n, q) = map(int, input().split())
        p = arr.array('i', list(map(int, input().split())))
        for i in range(n):
            p[i] -= 1
        r = arr.array('i', [0] * n)
        for i in range(n):
            r[p[i]] = i
        a = arr.array('i', [0] * n)
        Q = 100
        for i in range(n):
            x = i
            for j in range(Q):
                x = p[x]
            a[i] = x
        for _ in range(q):
            (t, x, y) = map(int, input().split())
            if t == 2:
                x -= 1
                while y >= Q:
                    y -= Q
                    x = a[x]
                while y:
                    x = p[x]
                    y -= 1
                print(x + 1)
            else:
                (x, y) = (x - 1, y - 1)
                (r[p[x]], r[p[y]]) = (r[p[y]], r[p[x]])
                (p[x], p[y]) = (p[y], p[x])
                ax = x
                for i in range(Q):
                    ax = p[ax]
                x2 = x
                for i in range(Q):
                    a[x] = ax
                    x = r[x]
                    ax = r[ax]
                (x, y) = (y, x)
                ax = x
                for i in range(Q):
                    ax = p[ax]
                x2 = x
                for i in range(Q):
                    a[x] = ax
                    x = r[x]
                    ax = r[ax]
    return
if __name__ == '__main__':
    main()

Solution (Python):
import io, os
import sys
from sys import stdin
from bisect import bisect_left, bisect_right
from collections import defaultdict, deque, namedtuple
from heapq import heappush, heappop
from math import gcd, ceil, floor, factorial, sqrt
from itertools import combinations, permutations
input = sys.stdin.buffer.readline

def work():
    ans = 0
    return

def main():
    test = 1
    for idt in range(test):
        (n, q) = map(int, input().split())
        p = list(map(int, input().split()))
        for i in range(n):
            p[i] -= 1
        r = [0] * n
        for i in range(n):
            r[p[i]] = i
        a = [0] * n
        Q = 100
        for i in range(n):
            x = i
            for j in range(Q):
                x = p[x]
            a[i] = x
        for _ in range(q):
            (t, x, y) = map(int, input().split())
            if t == 2:
                x -= 1
                while y >= Q:
                    y -= Q
                    x = a[x]
                while y:
                    x = p[x]
                    y -= 1
                print(x + 1)
            else:
                (x, y) = (x - 1, y - 1)
                (r[p[x]], r[p[y]]) = (r[p[y]], r[p[x]])
                (p[x], p[y]) = (p[y], p[x])
                ax = x
                for i in range(Q):
                    ax = p[ax]
                x2 = x
                for i in range(Q):
                    a[x] = ax
                    x = r[x]
                    ax = r[ax]
                (x, y) = (y, x)
                ax = x
                for i in range(Q):
                    ax = p[ax]
                x2 = x
                for i in range(Q):
                    a[x] = ax
                    x = r[x]
                    ax = r[ax]
    return
if __name__ == '__main__':
    main()

Solution (Python):
import io, os
import sys
import math
from sys import stdin
input = sys.stdin.buffer.readline

def work():
    ans = 0
    return

def main():
    test = 1
    for idt in range(test):
        (n, q) = map(int, input().split())
        p = list(map(int, input().split()))
        for i in range(n):
            p[i] -= 1
        r = [0] * n
        for i in range(n):
            r[p[i]] = i
        a = [0] * n
        Q = int(0.4 * math.sqrt(n)) + 1
        for i in range(n):
            x = i
            for j in range(Q):
                x = p[x]
            a[i] = x
        for _ in range(q):
            (t, x, y) = map(int, input().split())
            if t == 2:
                x -= 1
                while y >= Q:
                    y -= Q
                    x = a[x]
                while y:
                    x = p[x]
                    y -= 1
                print(x + 1)
            else:
                (x, y) = (x - 1, y - 1)
                (r[p[x]], r[p[y]]) = (r[p[y]], r[p[x]])
                (p[x], p[y]) = (p[y], p[x])
                ax = x
                for i in range(Q):
                    ax = p[ax]
                x2 = x
                for i in range(Q):
                    a[x] = ax
                    x = r[x]
                    ax = r[ax]
                (x, y) = (y, x)
                ax = x
                for i in range(Q):
                    ax = p[ax]
                x2 = x
                for i in range(Q):
                    a[x] = ax
                    x = r[x]
                    ax = r[ax]
    return
if __name__ == '__main__':
    main()