<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 {
public:
    bool backtrack(vector<int>& nums, vector<bool>& used, int startIndex, int k, int currentSum, int targetSum, unordered_map<int, bool>& memo) {
        int state = 0;
        for (int i = 0; i < used.size(); ++i) {
            if (used[i]) state |= (1 << i);
        }
        
        // Check if this state has been computed before
        if (memo.find(state) != memo.end()) {
            return memo[state];
        }
        
        if (k == 0) return true;
        if (currentSum > targetSum) return false;
        
        if (currentSum == targetSum) {
            bool result = backtrack(nums, used, 0, k - 1, 0, targetSum, memo);
            memo[state] = result;
            return result;
        }
        
        for (int i = startIndex; i < nums.size(); ++i) {
            if (used[i] || currentSum + nums[i] > targetSum) continue;
            
            used[i] = true;
            if (backtrack(nums, used, i + 1, k, currentSum + nums[i], targetSum, memo)) {
                memo[state] = true;
                return true;
            }
            used[i] = false;
        }
        
        memo[state] = false;
        return false;
    }
    
    bool canPartitionKSubsets(vector<int>& nums, int k) {
        int sum = 0;
        for (int num : nums) sum += num;
        
        if (sum % k != 0) return false;
        
        int targetSum = sum / k;
        vector<bool> used(nums.size(), false);
        unordered_map<int, bool> memo;
        
        return backtrack(nums, used, 0, k, 0, targetSum, memo);
    }
};

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