Increase Subarray Sums
Description
You are given an array $a_1, a_2, \dots, a_n$, consisting of $n$ integers. You are also given an integer value $x$.
Let $f(k)$ be the maximum sum of a contiguous subarray of $a$ after applying the following operation: add $x$ to the elements on exactly $k$ distinct positions. An empty subarray should also be considered, it has sum $0$.
Note that the subarray doesn't have to include all of the increased elements.
Calculate the maximum value of $f(k)$ for all $k$ from $0$ to $n$ independently.
The first line contains a single integer $t$ ($1 \le t \le 5000$) — the number of testcases.
The first line of the testcase contains two integers $n$ and $x$ ($1 \le n \le 5000$; $0 \le x \le 10^5$) — the number of elements in the array and the value to add.
The second line contains $n$ integers $a_1, a_2, \dots, a_n$ ($-10^5 \le a_i \le 10^5$).
The sum of $n$ over all testcases doesn't exceed $5000$.
For each testcase, print $n + 1$ integers — the maximum value of $f(k)$ for all $k$ from $0$ to $n$ independently.
Input
The first line contains a single integer $t$ ($1 \le t \le 5000$) — the number of testcases.
The first line of the testcase contains two integers $n$ and $x$ ($1 \le n \le 5000$; $0 \le x \le 10^5$) — the number of elements in the array and the value to add.
The second line contains $n$ integers $a_1, a_2, \dots, a_n$ ($-10^5 \le a_i \le 10^5$).
The sum of $n$ over all testcases doesn't exceed $5000$.
Output
For each testcase, print $n + 1$ integers — the maximum value of $f(k)$ for all $k$ from $0$ to $n$ independently.
Samples
3
4 2
4 1 3 2
3 5
-2 -7 -1
10 2
-6 -1 -2 4 -6 -1 -4 4 -5 -4
10 12 14 16 18
0 4 4 5
4 6 6 7 7 7 7 8 8 8 8
Note
In the first testcase, it doesn't matter which elements you add $x$ to. The subarray with the maximum sum will always be the entire array. If you increase $k$ elements by $x$, $k \cdot x$ will be added to the sum.
In the second testcase:
- For $k = 0$, the empty subarray is the best option.
- For $k = 1$, it's optimal to increase the element at position $3$. The best sum becomes $-1 + 5 = 4$ for a subarray $[3, 3]$.
- For $k = 2$, it's optimal to increase the element at position $3$ and any other element. The best sum is still $4$ for a subarray $[3, 3]$.
- For $k = 3$, you have to increase all elements. The best sum becomes $(-2 + 5) + (-7 + 5) + (-1 + 5) = 5$ for a subarray $[1, 3]$.