<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>

// Approach DP + Bitmasks
class Solution {
    int recur(int mask,int subsetLength,vector<int> &dp,vector<int> &candidates, unordered_map<int,int> &incompatibility){
        if(mask==0) return 0;
        
        if(dp[mask]!=-1) return dp[mask];
        
        int curr=200;
        for(int &submask:candidates){
            if((submask&mask)==submask){
                curr=min(curr,incompatibility[submask] + recur(mask^submask,subsetLength,dp,candidates,incompatibility));
            }
        }
        
        return dp[mask]=curr;
    }
public:
    int minimumIncompatibility(vector<int>& nums, int k) {
        int n=nums.size();
        if(n==k) return 0;
        int subsetLength=(n/k), mask=(1<<n)-1;
        // if (freq of any ele > no of subsets k) return -1;
        vector<int> freq(n+1,0);
        for(int &num:nums){     // O(n)
            if(++freq[num]>k) return -1;
        }
        vector<int> dp((1<<n),-1), candidates;
        unordered_map<int,int> incompatibility;
        for(int submask=mask;submask;submask=mask&(submask-1)){
            if(__builtin_popcount(submask)==subsetLength){
                bool good=true;
                unordered_set<int> hset;
                int maxEle=0, minEle=20;
                for(int i=0;i<n;i++){
                    if((1<<i)&submask){
                        if(hset.find(nums[i])!=hset.end()){
                            good=false;
                            break;
                        }
                        hset.insert(nums[i]);
                        maxEle=max(maxEle,nums[i]);
                        minEle=min(minEle,nums[i]);
                    }
                }
                if(good){
                    candidates.push_back(submask);
                    incompatibility[submask]=maxEle-minEle;
                }
            }
        }
        return recur(mask,subsetLength,dp,candidates,incompatibility);
    }
};
/*
// Initial Approach - Wrong Answer IDK Why
class Solution {
    int recur(int unique,vector<int> &freq,int k,int &subsetLength,vector<vector<int>> &dp){
        if(k==0) return 0;
        
        // cout<<k<<" "<<unique<<endl;
        if(dp[k][unique]!=-1) return dp[k][unique];
        int curr=200;
        for(int submask=unique;submask;submask=unique&(submask-1)){     // O(10^5)
            if(__builtin_popcount(submask)==subsetLength){
                int maxEle=0, minEle=17, currMask=submask, i=0, newUnique=unique;
                while(currMask){
                    if(currMask&1){
                        maxEle=max(i+1,maxEle);
                        minEle=min(i+1,minEle);
                        if(--freq[i+1]==0) newUnique^=(1<<i);
                        // cout<<newUnique<<endl;
                    }
                    i++;
                    currMask>>=1;
                }
                curr=min(curr,(maxEle-minEle) + recur(newUnique,freq,k-1,subsetLength,dp));
                currMask=submask;
                i=0;
                while(currMask){
                    if(currMask&1){
                        ++freq[i+1];
                    }
                    i++;
                    currMask>>=1;
                }
            }
        }
        return dp[k][unique]=curr;
    }
public:
    int minimumIncompatibility(vector<int>& nums, int k) {
        int n=nums.size(), subsetLength=(n/k), unique=0;
        if(n==k) return 0;
        // if (freq of any ele > no of subsets k) return -1;
        vector<int> freq(n+1,0);
        for(int &num:nums){     // O(n)
            if(++freq[num]>k) return -1;
            unique|=(1<<(num-1));
        }
        vector<vector<int>> dp(k+1,vector<int>(100001,-1));
        return recur(unique,freq,k,subsetLength,dp);
    }
};
*/

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