Monsters (hard version)

4000ms

524288K

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?

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.

Sample Input:

2
3
3 1 2
6
4 1 5 4 1 1

Sample Output:

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.

Informação

Provedor Codeforces

Código CF1784C