Rainbow Triples

2000ms

262144K

Description:

You are given a sequence $$$a_1, a_2, \ldots, a_n$$$ of non-negative integers.

You need to find the largest number $$$m$$$ of triples $$$(i_1, j_1, k_1)$$$, $$$(i_2, j_2, k_2)$$$, ..., $$$(i_m, j_m, k_m)$$$ such that:

$$$1 \leq i_p < j_p < k_p \leq n$$$ for each $$$p$$$ in $$$1, 2, \ldots, m$$$;
$$$a_{i_p} = a_{k_p} = 0$$$, $$$a_{j_p} \neq 0$$$;
all $$$a_{j_1}, a_{j_2}, \ldots, a_{j_m}$$$ are different;
all $$$i_1, j_1, k_1, i_2, j_2, k_2, \ldots, i_m, j_m, k_m$$$ are different.

Input:

The first line of input contains one integer $$$t$$$ ($$$1 \leq t \leq 500\,000$$$): the number of test cases.

The first line of each test case contains one integer $$$n$$$ ($$$1 \leq n \leq 500\,000$$$).

The second line contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$0 \leq a_i \leq n$$$).

The total sum of $$$n$$$ is at most $$$500\,000$$$.

Output:

For each test case, print one integer $$$m$$$: the largest number of proper triples that you can find.

Sample Input:

8
1
1
2
0 0
3
0 1 0
6
0 0 1 2 0 0
6
0 1 0 0 1 0
6
0 1 3 2 0 0
6
0 0 0 0 5 0
12
0 1 0 2 2 2 0 0 3 3 4 0

Sample Output:

Note:

In the first two test cases, there are not enough elements even for a single triple, so the answer is $$$0$$$.

In the third test case we can select one triple $$$(1, 2, 3)$$$.

In the fourth test case we can select two triples $$$(1, 3, 5)$$$ and $$$(2, 4, 6)$$$.

In the fifth test case we can select one triple $$$(1, 2, 3)$$$. We can't select two triples $$$(1, 2, 3)$$$ and $$$(4, 5, 6)$$$, because $$$a_2 = a_5$$$.

Informação

Provedor Codeforces

Código CF1408H