1644C · Increase Subarray Sums

Problem: 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.

Input Format: 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 Format: For each testcase, print $n + 1$ integers — the maximum value of $f(k)$ for all $k$ from $0$ to $n$ independently.

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 $[1, 3]$.