codeforces#P1919C. Grouping Increases
Grouping Increases
Description
You are given an array $a$ of size $n$. You will do the following process to calculate your penalty:
- Split array $a$ into two (possibly empty) subsequences$^\dagger$ $s$ and $t$ such that every element of $a$ is either in $s$ or $t^\ddagger$.
- For an array $b$ of size $m$, define the penalty $p(b)$ of an array $b$ as the number of indices $i$ between $1$ and $m - 1$ where $b_i < b_{i + 1}$.
- The total penalty you will receive is $p(s) + p(t)$.
If you perform the above process optimally, find the minimum possible penalty you will receive.
$^\dagger$ A sequence $x$ is a subsequence of a sequence $y$ if $x$ can be obtained from $y$ by the deletion of several (possibly, zero or all) elements.
$^\ddagger$ Some valid ways to split array $a=[3,1,4,1,5]$ into $(s,t)$ are $([3,4,1,5],[1])$, $([1,1],[3,4,5])$ and $([\,],[3,1,4,1,5])$ while some invalid ways to split $a$ are $([3,4,5],[1])$, $([3,1,4,1],[1,5])$ and $([1,3,4],[5,1])$.
Each test contains multiple test cases. The first line contains a single integer $t$ ($1 \leq t \leq 10^4$) — the number of test cases. The description of the test cases follows.
The first line of each test case contains a single integer $n$ ($1\le n\le 2\cdot 10^5$) — the size of the array $a$.
The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le n$) — the elements of the array $a$.
It is guaranteed that the sum of $n$ over all test cases does not exceed $2\cdot 10^5$.
For each test case, output a single integer representing the minimum possible penalty you will receive.
Input
Each test contains multiple test cases. The first line contains a single integer $t$ ($1 \leq t \leq 10^4$) — the number of test cases. The description of the test cases follows.
The first line of each test case contains a single integer $n$ ($1\le n\le 2\cdot 10^5$) — the size of the array $a$.
The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le n$) — the elements of the array $a$.
It is guaranteed that the sum of $n$ over all test cases does not exceed $2\cdot 10^5$.
Output
For each test case, output a single integer representing the minimum possible penalty you will receive.
5
5
1 2 3 4 5
8
8 2 3 1 1 7 4 3
5
3 3 3 3 3
1
1
2
2 1
3
1
0
0
0
Note
In the first test case, a possible way to split $a$ is $s=[2,4,5]$ and $t=[1,3]$. The penalty is $p(s)+p(t)=2 + 1 =3$.
In the second test case, a possible way to split $a$ is $s=[8,3,1]$ and $t=[2,1,7,4,3]$. The penalty is $p(s)+p(t)=0 + 1 =1$.
In the third test case, a possible way to split $a$ is $s=[\,]$ and $t=[3,3,3,3,3]$. The penalty is $p(s)+p(t)=0 + 0 =0$.