CF1998E2 Eliminating Balls With Merging (Hard Version)

Drink water. — Sun Tzu, The Art of Becoming a Healthy Programmer This 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) $ .

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 = 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 $ .

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) $ .

In 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 $ .