| <p> | |
| 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. | |
| </p> | |
| <p> | |
| Given an array of <strong>N</strong> integers <strong>A<sub>1..N</sub></strong>, | |
| Ethan's task is to find the maximum sum of any (possibly empty) contiguous subarray of <strong>A</strong>. | |
| Ethan has implemented an algorithm to solve this problem, described by the following pseudocode: | |
| </p> | |
| <ul> | |
| <li> 1. Set <strong>s</strong> and <strong>m</strong> to both be equal to 0. </li> | |
| <li> 2. Iterate <em>i</em> upwards from 1 to <strong>N</strong>: </li> | |
| <li> 2a. If <strong>A<sub>i</sub></strong> ≥ 0, increment <strong>s</strong> by <strong>A<sub>i</sub></strong>, | |
| otherwise set <strong>s</strong> to be equal to 0. </li> | |
| <li> 2b. If <strong>s</strong> > <strong>m</strong>, set <strong>m</strong> to be equal to <strong>s</strong>. </li> | |
| <li> 3. Output <strong>m</strong>. </li> | |
| </ul> | |
| <p> | |
| Is there any hope for Ethan? With exasperation, you set out in vain to teach another lesson. | |
| </p> | |
| <p> | |
| The professor of the class has once again left you with some half-written test cases. | |
| You're given an initial array <strong>B<sub>1..M</sub></strong>, such that the absolute value of each element is at most <strong>K</strong>. | |
| You'd like to insert <strong>M</strong> - 1 more integers into the array, one between each pair of adjacent elements in the original array, | |
| to construct a new array <strong>A<sub>1..N</sub></strong> where <strong>N</strong> = 2<strong>M</strong> - 1. | |
| Each of the inserted elements must likewise have an absolute value of at most <strong>K</strong>. | |
| You'll then feed the new array <strong>A</strong> 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. | |
| </p> | |
| <h3>Input</h3> | |
| <p> | |
| Input begins with an integer <strong>T</strong>, the number of test cases. | |
| For each test case, there is first a line containing the space-separated integers <strong>M</strong> and <strong>K</strong>. | |
| Then one more line follows containing the <strong>M</strong> space-separated integers <strong>B<sub>1</sub></strong> through <strong>B<sub>M</sub></strong>. | |
| </p> | |
| <h3>Output</h3> | |
| <p> | |
| For the <em>i</em>th test case, output a line containing "Case #<em>i</em>: " | |
| followed by the maximum possible absolute difference between the correct maximum subarray sum and the output of Ethan's algorithm. | |
| </p> | |
| <h3>Constraints</h3> | |
| <p> | |
| 1 ≤ <strong>T</strong> ≤ 60 <br /> | |
| 1 ≤ <strong>M</strong> ≤ 50 <br /> | |
| 1 ≤ <strong>K</strong> ≤ 50 <br /> | |
| -<strong>K</strong> ≤ <strong>A<sub>i</sub></strong> ≤ <strong>K</strong> <br /> | |
| </p> | |
| <h3>Explanation of Sample</h3> | |
| <p> | |
| In the first case, <strong>A</strong> = [3], and both Ethan's answer and the correct answer are equal to 3. | |
| </p> | |
| <p> | |
| In the second case, <strong>A</strong> = [-3], and both Ethan's answer and the correct answer are equal to 0. | |
| </p> | |
| <p> | |
| In the third case, one value will be inserted into <strong>B</strong>, and you should choose to insert -1 to yield <strong>A</strong> = [2, -1, 2]. | |
| This results in Ethan's answer being 2 and the correct answer being 3, yielding an absolute answer difference of 1. | |
| </p> | |
| <p> | |
| 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. | |
| </p> | |