<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>

class Solution {
 public:
  int minimumIncompatibility(vector<int>& nums, int k) {
    constexpr int kMaxCompatibility = (16 - 1) * (16 / 2);
    const int n = nums.size();
    const int subsetSize = n / k;
    const int maxMask = 1 << n;
    const vector<int> incompatibilities =
        getIncompatibilities(nums, subsetSize);
    // dp[i] := the minimum possible sum of incompatibilities of the subset
    // of numbers represented by the bitmask i
    vector<int> dp(maxMask, kMaxCompatibility);
    dp[0] = 0;

    for (unsigned mask = 1; mask < maxMask; ++mask) {
      // The number of 1s in `mask` isn't a multiple of `subsetSize`.
      if (popcount(mask) % subsetSize != 0)
        continue;
      // https://cp-algorithms.com/algebra/all-submasks.html
      for (int submask = mask; submask > 0; submask = (submask - 1) & mask)
        if (incompatibilities[submask] != -1)  // valid subset
          dp[mask] =
              min(dp[mask], dp[mask - submask] + incompatibilities[submask]);
    }

    return dp.back() == kMaxCompatibility ? -1 : dp.back();
  }

 private:
  static constexpr int kMaxNum = 16;

  // Returns an incompatibilities array where
  // * incompatibilities[i] := the incompatibility of the subset of numbers
  //   represented by the bitmask i
  // * incompatibilities[i] := -1 if the number of 1s in the bitmask i is not
  //   `subsetSize`
  vector<int> getIncompatibilities(const vector<int>& nums, int subsetSize) {
    const int maxMask = 1 << nums.size();
    vector<int> incompatibilities(maxMask, -1);
    for (unsigned mask = 0; mask < maxMask; ++mask)
      if (popcount(mask) == subsetSize && isUnique(nums, mask, subsetSize))
        incompatibilities[mask] = getIncompatibility(nums, mask);
    return incompatibilities;
  }

  // Returns true if the numbers selected by `mask` are unique.
  //
  // e.g. If we call isUnique(0b1010, 2, [1, 2, 1, 4]), `used` variable
  // will be 0b1, which only has one 1 (less than `subsetSize`). In this case,
  // we should return false.
  bool isUnique(const vector<int>& nums, int mask, int subsetSize) {
    unsigned used = 0;
    for (int i = 0; i < nums.size(); ++i)
      if (mask >> i & 1)
        used |= 1 << nums[i];
    return popcount(used) == subsetSize;
  }

  // Returns the incompatibility of the selected numbers represented by the
  // `mask`.
  int getIncompatibility(const vector<int>& nums, int mask) {
    int mn = kMaxNum;
    int mx = 0;
    for (int i = 0; i < nums.size(); ++i)
      if (mask >> i & 1) {
        mx = max(mx, nums[i]);
        mn = min(mn, nums[i]);
      }
    return mx - mn;
  }
};
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