<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>& arr, int k) {

        int n = arr.size();
        sort( arr.begin(), arr.end());

        if(k==n) {
            return 0;
        }
        
        if(n%k!=0) {
            // cout << " case 1 " << endl;
            return -1;
        }

        int limit = n/k;
        unordered_map<int,int> fmp;

        for( int val: arr) {
            fmp[val] += 1;
            if(fmp[val]>k) {
                // cout << " case 2 " << endl;
                return -1;
            }
        }

        if(k==1) {
            return arr.back() - arr[0];
        }

        int K = 1<<k;
        vector< vector<int> > bc(K+1, vector<int>());

        for( int mask=0; mask<K; mask++) {
            int c = __builtin_popcount(mask);
            bc[c].push_back(mask);
        }

        auto getString = []( vector<int> & nums)->string {
            string t = "";
            for( int val: nums) {
                t.append(to_string(val));
                t.append(",");
            }
            return t;
        };

        unordered_map<string,int> mp;
        queue<string> q;

        vector<int> count( k, 0);
        string key = getString(count);

        q.push(key);
        mp[key] = 0;

        while(!q.empty()) {

            string s = q.front();
            q.pop();

            int currVal = mp[s];
            // cout << " currKey: " << s << endl;

            for( int i=0, p=0; i<s.length(); i++, p++) {
                int j = i;
                while((j+1<s.length()) && isdigit(s[j+1])) {
                    j += 1;
                }
                int v = stoi(s.substr( i, j-i+1));
                count[p] = v;
                i = j+1;
            }

            int ne = 0;
            for( int val: count) {
                ne += val;
            }

            if(ne==n) {
                break;
            }

            int currCount = fmp[arr[ne]];

            for( int mask: bc[currCount]) {
                bool val = true;
                for( int i=0; i<k; i++) {
                    if(mask&(1<<i)) {
                        if(count[i]==limit) {
                            val = false;
                            break;
                        }
                    }
                }
                if(!val) {
                    continue;
                }
                int tot = currVal;
                for( int i=0; i<k; i++) {
                    if(mask&(1<<i)) {
                        count[i] += 1;
                        if(count[i]==1) {
                            tot -= arr[ne];
                        } else if(count[i]==limit) {
                            tot += arr[ne];
                        }
                    }
                }
                key = getString(count);
                // cout << " mask: " << mask << " gen key: " << key << endl;
                for( int i=0; i<k; i++) {
                    if(mask&(1<<i)) {
                        count[i] -= 1;
                    }
                }
                if(mp.find(key)==mp.end()) {
                    q.push(key);
                    mp[key] = tot;
                } else {
                    mp[key] = min( mp[key], tot);
                }
            }
        }

        // final ans computation
        for( int i=0; i<k; i++) {
            count[i] = limit;
        }
        key = getString(count);
        if(mp.find(key)!=mp.end()) {
            return mp[key];
        } else {
            cout << " case 3 " << endl;
            return -1;
        }
    }
};
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