E2. Eliminating Balls With Merging (Hard Version)time limit per test4 secondsmemory limit per test512 megabytesinputstandard inputoutputstandard outputDrink water.— Sun Tzu, The Art of Becoming a Healthy ProgrammerThis is the hard version of the problem. The only difference is that $$$x=1$$$ in this version. You must solve both versions to be able to hack.You are given two integers $$$n$$$ and $$$x$$$ ($$$x=1$$$). 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 $$$f(i)$$$.InputThe 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 $$$x$$$ ($$$1 \leq n \leq 2 \cdot 10^5; x = 1$$$) — 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 $$$2 \cdot 10^5$$$.OutputFor each test case, output $$$n-x+1$$$ space separated integers on a new line, where the $$$j$$$-th integer should represent $$$f(x+j-1)$$$.ExampleInput35 11 2 3 2 17 14 5 1 2 1 4 511 11 2 3 1 1 9 3 2 4 1 3Output1 1 2 2 3
1 1 1 1 1 3 4
1 1 2 2 2 1 1 1 3 3 4
NoteIn the first test case, below are the possible values of $$$j$$$ for each $$$f(i)$$$ from $$$1$$$ to $$$n$$$. For $$$f(1)$$$, the only possible value of $$$j$$$ is $$$1$$$. For $$$f(2)$$$, the only possible value of $$$j$$$ is $$$2$$$. For $$$f(3)$$$, the possible values of $$$j$$$ are $$$2$$$ and $$$3$$$. For $$$f(4)$$$, the possible values of $$$j$$$ are $$$2$$$ and $$$3$$$. For $$$f(5)$$$, the possible values of $$$j$$$ are $$$2$$$, $$$3$$$, and $$$4$$$.