<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) {
        const int n = nums.size(), m = n/k;
        const vector<pair<int, int>> bmap = getMap(nums, m);

        vector<int> dp(1<<n, -1);
        dp[0] = 0;
        for (int b = 0; b < (1<<n); ++b) {
            if (__builtin_popcount(b) % m != 0) continue;
            for (const auto& [tb, v] : bmap) {
                if (tb > b) break; // Only try m-digit combinations smaller than b.
                if ((tb & b) != tb) continue;
                if (dp[b-tb] == -1) continue;
                if (dp[b] == -1) dp[b] = dp[b-tb]+v;
                else dp[b] = min(dp[b], dp[b-tb]+v);
            }
        }
        return dp.back();
    }

private:
    // Pre-generates all valid subsets and corresponding incompatibilities.
    vector<pair<int, int>> getMap(const vector<int>& nums, int m) {
	    const int n = nums.size();
        vector<pair<int, int>> ans;
        for (int b = 0; b < (1<<n); ++b) {
            if (__builtin_popcount(b) != m) continue;

            int vis = 0;
            int mx = 0, mn = 16;
            bool is_good = true;
            for (int i = 0; i < n; ++i) {
                if ((b & (1<<i)) == 0) continue;
                if (vis & (1<<nums[i])) {
                    is_good = false;
                    break;
                }
                vis |= (1<<nums[i]);
                mx = max(mx, nums[i]);
                mn = min(mn, nums[i]);
            }
            if (!is_good) continue;
            
            ans.emplace_back(b, mx-mn);
        }
        return ans;
    }
};
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