1684E · MEX vs DIFF

Problem: You are given an array $a$ of $n$ non-negative integers. In one operation you can change any number in the array to any other non-negative integer.

Let's define the cost of the array as $\operatorname{DIFF}(a) - \operatorname{MEX}(a)$, where $\operatorname{MEX}$ of a set of non-negative integers is the smallest non-negative integer not present in the set, and $\operatorname{DIFF}$ is the number of different numbers in the array.

For example, $\operatorname{MEX}(\{1, 2, 3\}) = 0$, $\operatorname{MEX}(\{0, 1, 2, 4, 5\}) = 3$.

You should find the minimal cost of the array $a$ if you are allowed to make at most $k$ operations.

Input Format: The input consists of multiple test cases. The first line contains a single integer $t$ ($1 \le t \le 10^4$) — the number of test cases. Description of the test cases follows.

The first line of each test case contains two integers $n$ and $k$ ($1 \le n \le 10^5$, $0 \le k \le 10^5$) — the length of the array $a$ and the number of operations that you are allowed to make.

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($0 \le a_i \le 10^9$) — the elements of the array $a$.

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

Output Format: For each test case output a single integer — minimal cost that it is possible to get making at most $k$ operations.

Note: In the first test case no operations are needed to minimize the value of $\operatorname{DIFF} - \operatorname{MEX}$.

In the second test case it is possible to replace $5$ by $1$. After that the array $a$ is $[0,\, 2,\, 4,\, 1]$, $\operatorname{DIFF} = 4$, $\operatorname{MEX} = \operatorname{MEX}(\{0, 1, 2, 4\}) = 3$, so the answer is $1$.

In the third test case one possible array $a$ is $[4,\, 13,\, 0,\, 0,\, 13,\, 1,\, 2]$, $\operatorname{DIFF} = 5$, $\operatorname{MEX} = 3$.

In the fourth test case one possible array $a$ is $[1,\, 2,\, 3,\, 0,\, 0,\, 0]$.