Minimum Incompatibility
<p>You are given an integer array <code>nums</code>​​​ and an integer <code>k</code>. You are asked to distribute this array into <code>k</code> subsets of <strong>equal size</strong> such that there are no two equal elements in the same subset.</p>

<p>A subset&#39;s <strong>incompatibility</strong> is the difference between the maximum and minimum elements in that array.</p>

<p>Return <em>the <strong>minimum possible sum of incompatibilities</strong> of the </em><code>k</code> <em>subsets after distributing the array optimally, or return </em><code>-1</code><em> if it is not possible.</em></p>

<p>A subset is a group integers that appear in the array with no particular order.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> nums = [1,2,1,4], k = 2
<strong>Output:</strong> 4
<strong>Explanation:</strong> The optimal distribution of subsets is [1,2] and [1,4].
The incompatibility is (2-1) + (4-1) = 4.
Note that [1,1] and [2,4] would result in a smaller sum, but the first subset contains 2 equal elements.</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> nums = [6,3,8,1,3,1,2,2], k = 4
<strong>Output:</strong> 6
<strong>Explanation:</strong> The optimal distribution of subsets is [1,2], [2,3], [6,8], and [1,3].
The incompatibility is (2-1) + (3-2) + (8-6) + (3-1) = 6.
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> nums = [5,3,3,6,3,3], k = 3
<strong>Output:</strong> -1
<strong>Explanation:</strong> It is impossible to distribute nums into 3 subsets where no two elements are equal in the same subset.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= k &lt;= nums.length &lt;= 16</code></li>
	<li><code>nums.length</code> is divisible by <code>k</code></li>
	<li><code>1 &lt;= nums[i] &lt;= nums.length</code></li>
</ul>
Bit Manipulation; Array; Dynamic Programming; Bitmask
class Solution:
    def minimumIncompatibility(self, nums: List[int], k: int) -> int:
        n = len(nums)
        m = n // k
        g = [-1] * (1 << n)
        for i in range(1, 1 << n):
            if i.bit_count() != m:
                continue
            s = set()
            mi, mx = 20, 0
            for j, x in enumerate(nums):
                if i >> j & 1:
                    if x in s:
                        break
                    s.add(x)
                    mi = min(mi, x)
                    mx = max(mx, x)
            if len(s) == m:
                g[i] = mx - mi
        f = [inf] * (1 << n)
        f[0] = 0
        for i in range(1 << n):
            if f[i] == inf:
                continue
            s = set()
            mask = 0
            for j, x in enumerate(nums):
                if (i >> j & 1) == 0 and x not in s:
                    s.add(x)
                    mask |= 1 << j
            if len(s) < m:
                continue
            j = mask
            while j:
                if g[j] != -1:
                    f[i | j] = min(f[i | j], f[i] + g[j])
                j = (j - 1) & mask
        return f[-1] if f[-1] != inf else -1
