#P1998E1. Eliminating Balls With Merging (Easy Version)
Eliminating Balls With Merging (Easy Version)
Description
This is the easy version of the problem. The only difference is that $x=n$ in this version. You must solve both versions to be able to hack.
You are given two integers $n$ and $x$ ($x=n$). There are $n$ balls lined up in a row, numbered from $1$ to $n$ from left to right. Initially, there is a value $a_i$ written on the $i$-th ball.
For each integer $i$ from $1$ to $n$, we define a function $f(i)$ as follows:
- Suppose you have a set $S = \{1, 2, \ldots, i\}$.
- In each operation, you have to select an integer $l$ ($1 \leq l < i$) from $S$ such that $l$ is not the largest element of $S$. Suppose $r$ is the smallest element in $S$ which is greater than $l$.
- If $a_l > a_r$, you set $a_l = a_l + a_r$ and remove $r$ from $S$.
- If $a_l < a_r$, you set $a_r = a_l + a_r$ and remove $l$ from $S$.
- If $a_l = a_r$, you choose either the integer $l$ or $r$ to remove from $S$:
- If you choose to remove $l$ from $S$, you set $a_r = a_l + a_r$ and remove $l$ from $S$.
- If you choose to remove $r$ from $S$, you set $a_l = a_l + a_r$ and remove $r$ from $S$.
- $f(i)$ denotes the number of integers $j$ ($1 \le j \le i$) such that it is possible to obtain $S = \{j\}$ after performing the above operations exactly $i - 1$ times.
For each integer $i$ from $x$ to $n$, you need to find $f(i)$.
The first line contains $t$ ($1 \leq t \leq 10^4$) — the number of test cases.
The first line of each test case contains two integers $n$ and $x$ ($1 \leq n \leq 2 \cdot 10^5; x = n$) — the number of balls and the smallest index $i$ for which you need to find $f(i)$.
The second line of each test case contains $a_1, a_2, \ldots, a_n$ ($1 \leq a_i \leq 10^9$) — the initial number written on each ball.
It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$.
For each test case, output $n-x+1$ space separated integers on a new line, where the $j$-th integer should represent $f(x+j-1)$.
Input
The first line contains $t$ ($1 \leq t \leq 10^4$) — the number of test cases.
The first line of each test case contains two integers $n$ and $x$ ($1 \leq n \leq 2 \cdot 10^5; x = n$) — the number of balls and the smallest index $i$ for which you need to find $f(i)$.
The second line of each test case contains $a_1, a_2, \ldots, a_n$ ($1 \leq a_i \leq 10^9$) — the initial number written on each ball.
It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$.
Output
For each test case, output $n-x+1$ space separated integers on a new line, where the $j$-th integer should represent $f(x+j-1)$.
3
5 5
1 2 3 2 1
7 7
4 5 1 2 1 4 5
11 11
1 2 3 1 1 9 3 2 4 1 3
3
4
4
Note
In the first test case, you are required to calculate $f(5)$. It can be shown that after $4$ operations, $S$ can contain $2$, $3$, or $4$. The following shows the operations required to make $S = \{4\}$.
- Initially, $S = \{1, 2, 3, 4, 5\}$ and $a = [1, 2, 3, 2, 1]$.
- Choose $l = 1$. Naturally, $r = 2$. Since $a_1 < a_2$, we set $a_2 = 1 + 2$ and remove $1$ from $S$. Now, $S = \{2, 3, 4, 5\}$ and $a = [1, 3, 3, 2, 1]$.
- Choose $l = 4$. Naturally, $r = 5$. Since $a_4 > a_5$, we set $a_4 = 2 + 1$ and remove $5$ from $S$. Now, $S = \{2, 3, 4\}$ and $a = [1, 3, 3, 3, 1]$.
- Choose $l = 3$. Naturally, $r = 4$. Since $a_3 = a_4$, we have a choice whether to remove $3$ or $4$. Since we want to preserve $4$, let's remove $3$. So, set $a_4 = 3 + 3$ and remove $3$ from $S$. Now, $S = \{2, 4\}$ and $a = [1, 3, 3, 6, 1]$.
- Choose $l = 2$. Naturally, $r = 4$. Since $a_2 < a_4$, we set $a_4 = 3 + 6$ and remove $2$ from $S$. Finally, $S = \{4\}$ and $a = [1, 3, 3, 9, 1]$.
In the second test case, you are required to calculate $f(7)$. It can be shown that after $6$ operations, $S$ can contain $2$, $4$, $6$, or $7$.