codeforces#P1798E. Multitest Generator
Multitest Generator
Description
Let's call an array $b_1, b_2, \ldots, b_m$ a test if $b_1 = m - 1$.
Let's call an array $b_1, b_2, \ldots, b_m$ a multitest if the array $b_2, b_3, \ldots, b_m$ can be split into $b_1$ non-empty subarrays so that each of these subarrays is a test. Note that each element of the array must be included in exactly one subarray, and the subarrays must consist of consecutive elements.
Let's define the function $f$ from the array $b_1, b_2, \ldots, b_m$ as the minimum number of operations of the form "Replace any $b_i$ with any non-negative integer $x$", which needs to be done so that the array $b_1, b_2, \ldots, b_m$ becomes a multitest.
You are given an array of positive integers $a_1, a_2, \ldots, a_n$. For each $i$ from $1$ to $n - 1$, find $f([a_i, a_{i+1}, \ldots, a_n])$.
Below are some examples of tests and multitests.
- Tests: $[\underline{1}, 5]$, $[\underline{2}, 2, 2]$, $[\underline{3}, 4, 1, 1]$, $[\underline{5}, 0, 0, 0, 0, 0]$, $[\underline{7}, 1, 2, 3, 4, 5, 6, 7]$, $[\underline{0}]$. These arrays are tests since their first element (underlined) is equal to the length of the array minus one.
- Multitests: $[1, \underline{\underline{1}, 1}]$, $[2, \underline{\underline{3}, 0, 0, 1}, \underline{\underline{1}, 12}]$, $[3, \underline{\underline{2}, 2, 7}, \underline{\underline{1}, 1}, \underline{\underline{3}, 4, 4, 4}]$, $[4, \underline{\underline{0}}, \underline{\underline{3}, 1, 7, 9}, \underline{\underline{4}, 2, 0, 0, 9}, \underline{\underline{1}, 777}]$. Underlined are the subarrays after the split, and double underlined are the first elements of each subarray.
Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 300\,000$). The description of the test cases follows.
The first line of each test case contains a single integer $n$ ($2 \le n \le 300\,000$) — the length of the array $a$.
The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 300\,000$) — elements of the array $a$.
It is guaranteed that the sum of $n$ over all test cases does not exceed $300\,000$.
For each test case print $n - 1$ numbers — $f([a_i, a_{i+1}, \ldots, a_n])$ for each $i$ from $1$ to $n - 1$.
Input
Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 300\,000$). The description of the test cases follows.
The first line of each test case contains a single integer $n$ ($2 \le n \le 300\,000$) — the length of the array $a$.
The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 300\,000$) — elements of the array $a$.
It is guaranteed that the sum of $n$ over all test cases does not exceed $300\,000$.
Output
For each test case print $n - 1$ numbers — $f([a_i, a_{i+1}, \ldots, a_n])$ for each $i$ from $1$ to $n - 1$.
3
4
1 2 1 7
7
3 1 3 1 2 1 1
4
2 7 1 1
1
19
3 4 1 2 1 7 7 3 1 3 1 2 1 1 4 2 7 1 1
0 1 1
0 1 1 0 1 1
1 1 1
0 0 1 1 1 1 1 1 1 0 1 0 1 0 2 1 1 1
Note
In the first test case of the first test the array $[1, 2, 1, 7]$ is a multitest since the array $[2, 1, 7]$ is a test. The array $[2, 1, 7]$ is not a multitest, but after replacing the first number with $1$, an array $[1, 1, 7]$ is obtained, which is a multitest. The array $[1, 7]$ is also not a multitest, but the array $[1, 0]$ is, so $f([1, 7]) = 1$.
In the second test case of first test, for $i = 2$, $f([a_i, a_{i+1}, \ldots, a_n]) = f([1, 3, 1, 2, 1, 1]) = 1$, since the array itself is not a multitest, but after replacing the second element with $4$ you get multitest.
In the third test case of first test, for $i = 1$, $f([a_i, a_{i+1}, \ldots, a_n]) = f([2, 7, 1, 1]) = 1$, since the array itself is not a multitest, but after replacing the second element with $0$ you get multitest.
The second test is an array composed of all the numbers of the first test. Therefore $f([a_1, a_2, \ldots, a_n])$ naturally equals to $0$.