CF1784C.Monsters (hard version)

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:

1. Deal $1$ damage to any single alive monster of your choice.
2. 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.

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.