<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:
    def canPartitionKSubsets(self, nums: List[int], k: int) -> bool:
        if k < 0 or not nums: return False

        if k > len(nums): return False

        sum_ = sum(nums)

        if sum_ % k != 0: return False

        target = sum_ // k

        memo = {}

        used = [0] * len(nums)

        return self.backtrack(k, 0, 0, target, 0, used, memo, nums)

    def backtrack(self, bucketcount: int, bucketdone: int, bucketsum: int, target: int, start: int, used: List[int], memo: Dict, nums: List[int]) -> bool:

        if bucketdone == bucketcount:
            return True
        
        state = ''.join(map(str, used))

        if state in memo:
            return memo[state]

        found = False

        for i in range(start, len(nums)):
            if found:
                break

            if used[i]:
                continue
            
            if bucketsum + nums[i] > target:
                continue
            
            used[i] = 1
            bucketsum += nums[i]
            if bucketsum == target:
                # done with current bucket
                found = found or self.backtrack(bucketcount, bucketdone + 1, 0, target, 0, used, memo, nums)
            else:
                found = found or self.backtrack(bucketcount, bucketdone, bucketsum, target, i + 1, used, memo, nums)
            used[i] = 0
            bucketsum -= nums[i]
        
        # update state
        # state = ''.join(map(str, used))
        memo[state] = found
        return found

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