<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>

from typing import List
from collections import defaultdict

class Solution:
    def minimumIncompatibility(self, nums: List[int], k: int) -> int:
        n = len(nums)
        subset_size = n // k
        
        # Base case: If subset size is 1, every item is its own subset of size k
        if subset_size == 1:
            return 0

        # Precompute valid subset incompatibilities
        incompatibilities = {}
        
        # Try all subsets using bitmasking
        for mask in range(1, 1 << n):
            if bin(mask).count('1') == subset_size:
                elements = [nums[i] for i in range(n) if (mask & (1 << i) > 0)]
                if len(elements) == len(set(elements)):  # Must be all unique
                    incompatibilities[mask] = max(elements) - min(elements)
        
        # Initialize dp array
        dp = [float('inf')] * (1 << n)
        dp[0] = 0

        # Iterate through states
        for mask in range(1 << n):
            if dp[mask] < float('inf'):
                # Try to add a valid subset on top of current dp state
                remaining = ((1 << n) - 1) ^ mask
                submask = remaining
                while submask > 0:
                    if submask in incompatibilities:
                        dp[mask | submask] = min(dp[mask | submask], dp[mask] + incompatibilities[submask])
                    submask = (submask - 1) & remaining

        # Return minimum sum of incompatibilities for the full set (all elements used)
        return dp[-1] if dp[-1] < float('inf') else -1

# Example usage:
# sol = Solution()
# print(sol.minimumIncompatibility([1,2,1,4], 2)) # Output: 4
# print(sol.minimumIncompatibility([6,3,8,1,3,1,2,2], 4)) # Output: 6
# print(sol.minimumIncompatibility([5,3,3,6,3,3], 3)) # Output: -1
17300