<p>Given an integer array <code>nums</code> and an integer <code>k</code>, return <code>true</code> if it is possible to divide this array into <code>k</code> non-empty subsets whose sums are all equal.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> nums = [4,3,2,3,5,2,1], k = 4
<strong>Output:</strong> true
<strong>Explanation:</strong> It is possible to divide it into 4 subsets (5), (1, 4), (2,3), (2,3) with equal sums.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> nums = [1,2,3,4], k = 3
<strong>Output:</strong> false
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= k &lt;= nums.length &lt;= 16</code></li>
	<li><code>1 &lt;= nums[i] &lt;= 10<sup>4</sup></code></li>
	<li>The frequency of each element is in the range <code>[1, 4]</code>.</li>
</ul>

class Solution {
private:
    bool backtrack(vector<int>& nums, int index, int count, int currSum, int k, int subSum, std::string& taken, std::unordered_map<std::string, bool>& memo){
        int numsSize = nums.size();
        if(count == k - 1){
            return true;
        }
        if(currSum > subSum){
            return false;
        }
        if(memo.find(taken) != memo.end()){
            return memo[taken];
        }
        if(currSum == subSum){
            return backtrack(nums, 0, count + 1, 0 , k, subSum, taken, memo);
        }
        for(int j = index; j < numsSize; j++){
            if(taken[j] == '0'){
                taken[j] = '1';
                if(backtrack(nums, j + 1, count, currSum + nums[j], k, subSum, taken, memo)){
                    return true;
                }
                taken[j] = '0';
            }

        }
        return memo[taken] = false;
    }
public:
    bool canPartitionKSubsets(vector<int>& nums, int k) {
        int sum = 0;
        int numsSize = nums.size();

        for(int i = 0; i < numsSize; i++){
            sum += nums[i];
        }
        if(sum % k){
            return false;
        }
        int subSum = sum / k;

        // sort the nums in decreasing order
        std::sort(nums.begin(), nums.end(), std::greater<int>());

        std::string taken(numsSize, '0');
        std::unordered_map<std::string, bool> memo;
        return backtrack(nums, 0, 0 ,0, k, subSum, taken, memo);
        
    }
};
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