Increase Subarray Sums

2000ms

262144K

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.

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.

Sample Input:

3
4 2
4 1 3 2
3 5
-2 -7 -1
10 2
-6 -1 -2 4 -6 -1 -4 4 -5 -4

Sample Output:

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 $$$[1, 3]$$$.

Informação

Provedor Codeforces

Código CF1644C