Two Teams Composing
Contest: Codeforces Round 634 (Div. 3)
Difficulty rating: 1100
Tags: binary search, greedy, implementation, sortings

You have $$$n$$$ students under your control and you have to compose exactly two teams consisting of some subset of your students. Each student had his own skill, the $$$i$$$-th student skill is denoted by an integer $$$a_i$$$ (different students can have the same skills).

So, about the teams. Firstly, these two teams should have the same size. Two more constraints:

- The first team should consist of students with distinct skills (i.e. all skills in the first team are unique).
- The second team should consist of students with the same skills (i.e. all skills in the second team are equal).

Note that it is permissible that some student of the first team has the same skill as a student of the second team.

Consider some examples (skills are given):

- $$$[1, 2, 3]$$$, $$$[4, 4]$$$ is not a good pair of teams because sizes should be the same;
- $$$[1, 1, 2]$$$, $$$[3, 3, 3]$$$ is not a good pair of teams because the first team should not contain students with the same skills;
- $$$[1, 2, 3]$$$, $$$[3, 4, 4]$$$ is not a good pair of teams because the second team should contain students with the same skills;
- $$$[1, 2, 3]$$$, $$$[3, 3, 3]$$$ is a good pair of teams;
- $$$[5]$$$, $$$[6]$$$ is a good pair of teams.

Your task is to find the maximum possible size $$$x$$$ for which it is possible to compose a valid pair of teams, where each team size is $$$x$$$ (skills in the first team needed to be unique, skills in the second team should be the same between them). A student cannot be part of more than one team.

You have to answer $$$t$$$ independent test cases.

Input format:
The first line of the input contains one integer $$$t$$$ ($$$1 \le t \le 10^4$$$) — the number of test cases. Then $$$t$$$ test cases follow.

The first line of the test case contains one integer $$$n$$$ ($$$1 \le n \le 2 \cdot 10^5$$$) — the number of students. The second line of the test case contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le n$$$), where $$$a_i$$$ is the skill of the $$$i$$$-th student. Different students can have the same skills.

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

Output format:
For each test case, print the answer — the maximum possible size $$$x$$$ for which it is possible to compose a valid pair of teams, where each team size is $$$x$$$.

Constraints:
Time limit: 2.0 s
Memory limit: 256.0 MB

Note:
In the first test case of the example, it is possible to construct two teams of size $$$3$$$: the first team is $$$[1, 2, 4]$$$ and the second team is $$$[4, 4, 4]$$$. Note, that there are some other ways to construct two valid teams of size $$$3$$$.

Example input:
4
7
4 2 4 1 4 3 4
5
2 1 5 4 3
1
1
4
1 1 1 3

Example output:
3
1
0
2

Editorial:
Let the number of students with the skill $$$i$$$ is $$$cnt_i$$$ and the number of different skills is $$$diff$$$. Then the size of the first team can not exceed $$$diff$$$ and the size of the second team can not exceed $$$maxcnt = max(cnt_1, cnt_2, \dots, cnt_n)$$$. So the answer is not greater than the minimum of these two values. Moreover, let's take a look at the skill with a maximum value of $$$cnt$$$. Then there are two cases: all students with this skill go to the second team then the sizes of teams are at most $$$diff-1$$$ and $$$maxcnt$$$ correspondingly. Otherwise, at least one student with this skill goes to the first team and the sizes are at most $$$diff$$$ and $$$maxcnt - 1$$$ correspondingly. So the answer is $$$max(min(diff - 1, maxcnt), min(diff, maxcnt - 1))$$$.
Time complexity: $$$O(n)$$$.