| In preparation for his final exam, Ethan is doing his fourth programming |
| assignment: finding the subarray with the maximum sum in an array of integers. |
|
|
| Given an array of **N** integers **A1..N**, Ethan's task is to find the |
| maximum sum of any (possibly empty) contiguous subarray of **A**. Ethan has |
| implemented an algorithm to solve this problem, described by the following |
| pseudocode: |
|
|
| * 1\. Set **s** and **m** to both be equal to 0. |
| * 2\. Iterate _i_ upwards from 1 to **N**: |
| * 2a. If **Ai** ≥ 0, increment **s** by **Ai**, otherwise set **s** to be equal to 0. |
| * 2b. If **s** > **m**, set **m** to be equal to **s**. |
| * 3\. Output **m**. |
|
|
| Is there any hope for Ethan? With exasperation, you set out in vain to teach |
| another lesson. |
|
|
| The professor of the class has once again left you with some half-written test |
| cases. You're given an initial array **B1..M**, such that the absolute value |
| of each element is at most **K**. You'd like to insert **M** \- 1 more |
| integers into the array, one between each pair of adjacent elements in the |
| original array, to construct a new array **A1..N** where **N** = 2**M** \- 1. |
| Each of the inserted elements must likewise have an absolute value of at most |
| **K**. You'll then feed the new array **A** into Ethan's algorithm. Your goal |
| is to maximize the absolute difference between the final array's correct |
| maximum subarray sum and the output of Ethan's algorithm. |
|
|
| ### Input |
|
|
| Input begins with an integer **T**, the number of test cases. For each test |
| case, there is first a line containing the space-separated integers **M** and |
| **K**. Then one more line follows containing the **M** space-separated |
| integers **B1** through **BM**. |
|
|
| ### Output |
|
|
| For the _i_th test case, output a line containing "Case #_i_: " followed by |
| the maximum possible absolute difference between the correct maximum subarray |
| sum and the output of Ethan's algorithm. |
|
|
| ### Constraints |
|
|
| 1 ≤ **T** ≤ 60 |
| 1 ≤ **M** ≤ 50 |
| 1 ≤ **K** ≤ 50 |
| -**K** ≤ **Ai** ≤ **K** |
|
|
| ### Explanation of Sample |
|
|
| In the first case, **A** = [3], and both Ethan's answer and the correct answer |
| are equal to 3. |
|
|
| In the second case, **A** = [-3], and both Ethan's answer and the correct |
| answer are equal to 0. |
|
|
| In the third case, one value will be inserted into **B**, and you should |
| choose to insert -1 to yield **A** = [2, -1, 2]. This results in Ethan's |
| answer being 2 and the correct answer being 3, yielding an absolute answer |
| difference of 1. |
|
|
| In the fourth case, there are multiple choices of inserted elements which |
| result in an absolute answer difference of 3. For example, it's possible for |
| Ethan's answer equal to be made to equal 3 while the correct answer equals 6. |
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|
