#CF1784C. Monsters (hard version)
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ID: 7324
Type: RemoteJudge
4000ms
512MiB
Tried: 0
Accepted: 0
Difficulty: (None)
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Monsters (hard version)
Description
This is the hard version of the problem. In this version, you need to find the answer for every prefix of the monster array.
In a computer game, you are fighting against $n$ monsters. Monster number $i$ has $a_i$ health points, all $a_i$ are integers. A monster is alive while it has at least $1$ health point.
You can cast spells of two types:
- Deal $1$ damage to any single alive monster of your choice.
- Deal $1$ damage to all alive monsters. If at least one monster dies (ends up with $0$ health points) as a result of this action, then repeat it (and keep repeating while at least one monster dies every time).
Dealing $1$ damage to a monster reduces its health by $1$.
Spells of type 1 can be cast any number of times, while a spell of type 2 can be cast at most once during the game.
For every $k = 1, 2, \ldots, n$, answer the following question. Suppose that only the first $k$ monsters, with numbers $1, 2, \ldots, k$, are present in the game. What is the smallest number of times you need to cast spells of type 1 to kill all $k$ monsters?
Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^4$). The description of the test cases follows.
Each test case consists of two lines. The first line contains a single integer $n$ ($1 \le n \le 2 \cdot 10^5$) — the number of monsters.
The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le n$) — monsters' health points.
It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$.
For each test case, print $n$ integers. The $k$-th of these integers must be equal to the smallest number of times you need to cast spells of type 1 to kill all $k$ monsters, if only monsters with numbers $1, 2, \ldots, k$ are present in the game.
Input
Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^4$). The description of the test cases follows.
Each test case consists of two lines. The first line contains a single integer $n$ ($1 \le n \le 2 \cdot 10^5$) — the number of monsters.
The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le n$) — monsters' health points.
It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$.
Output
For each test case, print $n$ integers. The $k$-th of these integers must be equal to the smallest number of times you need to cast spells of type 1 to kill all $k$ monsters, if only monsters with numbers $1, 2, \ldots, k$ are present in the game.
2
3
3 1 2
6
4 1 5 4 1 1
2 1 0
3 2 4 4 4 4
Note
In the first test case, for $k = n$, the initial health points of the monsters are $[3, 1, 2]$. It is enough to cast a spell of type 2:
- Monsters' health points change to $[2, 0, 1]$. Since monster number $2$ dies, the spell is repeated.
- Monsters' health points change to $[1, 0, 0]$. Since monster number $3$ dies, the spell is repeated.
- Monsters' health points change to $[0, 0, 0]$. Since monster number $1$ dies, the spell is repeated.
- Monsters' health points change to $[0, 0, 0]$.
Since it is possible to use no spells of type 1 at all, the answer is $0$.
In the second test case, for $k = n$, the initial health points of the monsters are $[4, 1, 5, 4, 1, 1]$. Here is one of the optimal action sequences:
- Using a spell of type 1, deal $1$ damage to monster number $1$. Monsters' health points change to $[3, 1, 5, 4, 1, 1]$.
- Using a spell of type 1, deal $1$ damage to monster number $4$. Monsters' health points change to $[3, 1, 5, 3, 1, 1]$.
- Using a spell of type 1, deal $1$ damage to monster number $4$ again. Monsters' health points change to $[3, 1, 5, 2, 1, 1]$.
- Use a spell of type 2:
- Monsters' health points change to $[2, 0, 4, 1, 0, 0]$. Since monsters number $2$, $5$, and $6$ die, the spell is repeated.
- Monsters' health points change to $[1, 0, 3, 0, 0, 0]$. Since monster number $4$ dies, the spell is repeated.
- Monsters' health points change to $[0, 0, 2, 0, 0, 0]$. Since monster number $1$ dies, the spell is repeated.
- Monsters' health points change to $[0, 0, 1, 0, 0, 0]$.
- Using a spell of type 1, deal $1$ damage to monster number $3$. Monsters' health points change to $[0, 0, 0, 0, 0, 0]$.
Spells of type 1 are cast $4$ times in total. It can be shown that this is the smallest possible number.