Assigning to Classes
Contest: Codeforces Round 618 (Div. 2)
Difficulty rating: 1000
Tags: greedy, implementation, sortings

Reminder: the median of the array $$$[a_1, a_2, \dots, a_{2k+1}]$$$ of odd number of elements is defined as follows: let $$$[b_1, b_2, \dots, b_{2k+1}]$$$ be the elements of the array in the sorted order. Then median of this array is equal to $$$b_{k+1}$$$.

There are $$$2n$$$ students, the $$$i$$$-th student has skill level $$$a_i$$$. It's not guaranteed that all skill levels are distinct.

Let's define skill level of a class as the median of skill levels of students of the class.

As a principal of the school, you would like to assign each student to one of the $$$2$$$ classes such that each class has odd number of students (not divisible by $$$2$$$). The number of students in the classes may be equal or different, by your choice. Every student has to be assigned to exactly one class. Among such partitions, you want to choose one in which the absolute difference between skill levels of the classes is minimized.

What is the minimum possible absolute difference you can achieve?

Input format:
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 10^4$$$). The description of the test cases follows.

The first line of each test case contains a single integer $$$n$$$ ($$$1 \le n \le 10^5$$$) — the number of students halved.

The second line of each test case contains $$$2n$$$ integers $$$a_1, a_2, \dots, a_{2 n}$$$ ($$$1 \le a_i \le 10^9$$$) — skill levels of students.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$10^5$$$.

Output format:
For each test case, output a single integer, the minimum possible absolute difference between skill levels of two classes of odd sizes.

Constraints:
Time limit: 2.0 s
Memory limit: 256.0 MB

Note:
In the first test, there is only one way to partition students — one in each class. The absolute difference of the skill levels will be $$$|1 - 1| = 0$$$.

In the second test, one of the possible partitions is to make the first class of students with skill levels $$$[6, 4, 2]$$$, so that the skill level of the first class will be $$$4$$$, and second with $$$[5, 1, 3]$$$, so that the skill level of the second class will be $$$3$$$. Absolute difference will be $$$|4 - 3| = 1$$$.

Note that you can't assign like $$$[2, 3]$$$, $$$[6, 5, 4, 1]$$$ or $$$[]$$$, $$$[6, 5, 4, 1, 2, 3]$$$ because classes have even number of students.

$$$[2]$$$, $$$[1, 3, 4]$$$ is also not possible because students with skills $$$5$$$ and $$$6$$$ aren't assigned to a class.

In the third test you can assign the students in the following way: $$$[3, 4, 13, 13, 20], [2, 5, 8, 16, 17]$$$ or $$$[3, 8, 17], [2, 4, 5, 13, 13, 16, 20]$$$. Both divisions give minimal possible absolute difference.

Example input:
3
1
1 1
3
6 5 4 1 2 3
5
13 4 20 13 2 5 8 3 17 16

Example output:
0
1
5

Editorial:
Let's sort the array. From now on, $$$a_1 \le a_2 \dots \le a_{2n}$$$.
Consider any partition. Suppose that the first class has $$$2k+1$$$ students, and the skill level of this class is $$$a_i$$$, and the second class had $$$2l+1$$$ students, and the skill level of this class is $$$a_j$$$, where $$$(2k + 1) + (2l + 1) = 2n \implies k + l = n-1$$$. Without losing generality, $$$i<j$$$.
At least $$$n+1$$$ students have skill level at least $$$a_i$$$. Indeed, as $$$a_i$$$ is a median of his class, he and $$$k$$$ other students have skill level at least $$$a_i$$$. As $$$a_j\ge a_i$$$ and at least $$$l$$$ other students of the second class have skill level at least $$$a_j\ge a_i$$$, we get at least $$$k + l + 2 = n + 1$$$ students (including $$$a_i$$$) with skill level at least $$$a_i$$$. Therefore, $$$a_i\le a_n$$$.
Similarly, we get that at least $$$n + 1$$$ students have skill level at most $$$a_j$$$. Therefore, $$$a_j\ge a_{n+1}$$$.
So, $$$|a_j - a_i| \ge a_{n+1} - a_n$$$. However, $$$a_{n+1} - a_n$$$ is achievable. Let's put student $$$a_n$$$ into the class alone, and all other students into other class. $$$a_{n+1}$$$ will be the median skill level of that class, so the absolute difference will be exactly $$$a_{n+1} - a_n$$$.
Therefore, it's enough to sort the array and to output the difference between two middle elements.