CF1904C.Array Game

You are given an array $a$ of $n$ positive integers. In one operation, you must pick some $(i, j)$ such that $1\leq i < j\leq |a|$ and append $|a_i - a_j|$ to the end of the $a$ (i.e. increase $n$ by $1$ and set $a_n$ to $|a_i - a_j|$ ). Your task is to minimize and print the minimum value of $a$ after performing $k$ operations.

Each test contains multiple test cases. The first line contains an integer $t$ ( $1 \leq t \leq 1000$ ) — the number of test cases. The description of the test cases follows.

The first line of each test case contains two integers $n$ and $k$ ( $2\le n\le 2\cdot 10^3$ , $1\le k\le 10^9$ ) — the length of the array and the number of operations you should perform.

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ( $1\le a_i\le 10^{18}$ ) — the elements of the array $a$ .

It is guaranteed that the sum of $n^2$ over all test cases does not exceed $4\cdot 10^6$ .

For each test case, print a single integer — the smallest possible value of the minimum of array $a$ after performing $k$ operations.

In the first test case, after any $k=2$ operations, the minimum value of $a$ will be $1$ .

In the second test case, an optimal strategy is to first pick $i=1, j=2$ and append $|a_1 - a_2| = 3$ to the end of $a$ , creating $a=[7, 4, 15, 12, 3]$ . Then, pick $i=3, j=4$ and append $|a_3 - a_4| = 3$ to the end of $a$ , creating $a=[7, 4, 15, 12, 3, 3]$ . In the final operation, pick $i=5, j=6$ and append $|a_5 - a_6| = 0$ to the end of $a$ . Then the minimum value of $a$ will be $0$ .

In the third test case, an optimal strategy is to first pick $i=2, j=3$ to append $|a_2 - a_3| = 3$ to the end of $a$ . Any second operation will still not make the minimum value of $a$ be less than $3$ .