codeforces#P1794C. Scoring Subsequences
Scoring Subsequences
Description
The score of a sequence $[s_1, s_2, \ldots, s_d]$ is defined as $\displaystyle \frac{s_1\cdot s_2\cdot \ldots \cdot s_d}{d!}$, where $d!=1\cdot 2\cdot \ldots \cdot d$. In particular, the score of an empty sequence is $1$.
For a sequence $[s_1, s_2, \ldots, s_d]$, let $m$ be the maximum score among all its subsequences. Its cost is defined as the maximum length of a subsequence with a score of $m$.
You are given a non-decreasing sequence $[a_1, a_2, \ldots, a_n]$ of integers of length $n$. In other words, the condition $a_1 \leq a_2 \leq \ldots \leq a_n$ is satisfied. For each $k=1, 2, \ldots , n$, find the cost of the sequence $[a_1, a_2, \ldots , a_k]$.
A sequence $x$ is a subsequence of a sequence $y$ if $x$ can be obtained from $y$ by deletion of several (possibly, zero or all) elements.
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.
The first line of each test case contains an integer $n$ ($1\le n\le 10^5$) — the length of the given sequence.
The second line of each test case contains $n$ integers $a_1,a_2,\ldots,a_n$ ($1\le a_i\leq n$) — the given sequence. It is guaranteed that its elements are in non-decreasing order.
It is guaranteed that the sum of $n$ over all test cases does not exceed $5\cdot 10^5$.
For each test case, output $n$ integers — the costs of sequences $[a_1, a_2, \ldots , a_k]$ in ascending order of $k$.
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.
The first line of each test case contains an integer $n$ ($1\le n\le 10^5$) — the length of the given sequence.
The second line of each test case contains $n$ integers $a_1,a_2,\ldots,a_n$ ($1\le a_i\leq n$) — the given sequence. It is guaranteed that its elements are in non-decreasing order.
It is guaranteed that the sum of $n$ over all test cases does not exceed $5\cdot 10^5$.
Output
For each test case, output $n$ integers — the costs of sequences $[a_1, a_2, \ldots , a_k]$ in ascending order of $k$.
3
3
1 2 3
2
1 1
5
5 5 5 5 5
1 1 2
1 1
1 2 3 4 5
Note
In the first test case:
- The maximum score among the subsequences of $[1]$ is $1$. The subsequences $[1]$ and $[]$ (the empty sequence) are the only ones with this score. Thus, the cost of $[1]$ is $1$.
- The maximum score among the subsequences of $[1, 2]$ is $2$. The only subsequence with this score is $[2]$. Thus, the cost of $[1, 2]$ is $1$.
- The maximum score among the subsequences of $[1, 2, 3]$ is $3$. The subsequences $[2, 3]$ and $[3]$ are the only ones with this score. Thus, the cost of $[1, 2, 3]$ is $2$.