Books Exchange (hard version)
Contest: Codeforces Round 595 (Div. 3)
Difficulty rating: 1300
Tags: dfs and similar, dsu, math

The only difference between easy and hard versions is constraints.

There are $$$n$$$ kids, each of them is reading a unique book. At the end of any day, the $$$i$$$-th kid will give his book to the $$$p_i$$$-th kid (in case of $$$i = p_i$$$ the kid will give his book to himself). It is guaranteed that all values of $$$p_i$$$ are distinct integers from $$$1$$$ to $$$n$$$ (i.e. $$$p$$$ is a permutation). The sequence $$$p$$$ doesn't change from day to day, it is fixed.

For example, if $$$n=6$$$ and $$$p=[4, 6, 1, 3, 5, 2]$$$ then at the end of the first day the book of the $$$1$$$-st kid will belong to the $$$4$$$-th kid, the $$$2$$$-nd kid will belong to the $$$6$$$-th kid and so on. At the end of the second day the book of the $$$1$$$-st kid will belong to the $$$3$$$-th kid, the $$$2$$$-nd kid will belong to the $$$2$$$-th kid and so on.

Your task is to determine the number of the day the book of the $$$i$$$-th child is returned back to him for the first time for every $$$i$$$ from $$$1$$$ to $$$n$$$.

Consider the following example: $$$p = [5, 1, 2, 4, 3]$$$. The book of the $$$1$$$-st kid will be passed to the following kids:

- after the $$$1$$$-st day it will belong to the $$$5$$$-th kid,
- after the $$$2$$$-nd day it will belong to the $$$3$$$-rd kid,
- after the $$$3$$$-rd day it will belong to the $$$2$$$-nd kid,
- after the $$$4$$$-th day it will belong to the $$$1$$$-st kid.

So after the fourth day, the book of the first kid will return to its owner. The book of the fourth kid will return to him for the first time after exactly one day.

You have to answer $$$q$$$ independent queries.

Input format:
The first line of the input contains one integer $$$q$$$ ($$$1 \le q \le 1000$$$) — the number of queries. Then $$$q$$$ queries follow.

The first line of the query contains one integer $$$n$$$ ($$$1 \le n \le 2 \cdot 10^5$$$) — the number of kids in the query. The second line of the query contains $$$n$$$ integers $$$p_1, p_2, \dots, p_n$$$ ($$$1 \le p_i \le n$$$, all $$$p_i$$$ are distinct, i.e. $$$p$$$ is a permutation), where $$$p_i$$$ is the kid which will get the book of the $$$i$$$-th kid.

It is guaranteed that $$$\sum n \le 2 \cdot 10^5$$$ (sum of $$$n$$$ over all queries does not exceed $$$2 \cdot 10^5$$$).

Output format:
For each query, print the answer on it: $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$, where $$$a_i$$$ is the number of the day the book of the $$$i$$$-th child is returned back to him for the first time in this query.

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

Example input:
6
5
1 2 3 4 5
3
2 3 1
6
4 6 2 1 5 3
1
1
4
3 4 1 2
5
5 1 2 4 3

Example output:
1 1 1 1 1 
3 3 3 
2 3 3 2 1 3 
1 
2 2 2 2 
4 4 4 1 4

Editorial:
In this problem, we can notice that when we calculate the answer for the kid $$$i$$$, we also calculate the answer for kids $$$p_i$$$, $$$p_{p_i}$$$ and so on. So we can a little bit modify the pseudocode from the easy version to calculate answers faster:
And of course, we don't need to run this while for all elements for which we already calculated the answer. Total time complexity is $$$O(n)$$$ because you'll process each element exactly once.