<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 canPartitionKSubsets(vector<int>& nums, int k) {
        int sum = accumulate(nums.begin(), nums.end(), 0);
        if (sum % k != 0) return false;
        int target = sum / k;
        if (any_of(nums.begin(), nums.end(), [&target](int x){return x > target;}))
            return false;
        sort(nums.begin(), nums.end());
        // return solve2(nums, target, 0, 0, {});
        return solve(nums, target, 0, 0, 0, {});
    }

    bool solve(vector<int>& nums, int target, int index, int currSum, int mask, unordered_map<string, bool>&& memo) {
        int n = nums.size();
        if (mask == (1<<n) - 1) return currSum == 0;
        if (currSum == 0) index = 0;  // fresh start for a new group
        string key = to_string(mask) + "#" + to_string(currSum);
        if (auto it = memo.find(key); it != memo.end()) {
            return it->second;
        }
        for (int i = index; i < n; i++) {
            if ((mask >> i) & 1) continue;
            if (currSum + nums[i] <= target) {
                if (solve(nums, target, i + 1, (currSum + nums[i]) % target, mask | (1 << i), move(memo)))
                    return memo[key] = true;
            }
        }
        return memo[key] = false;
    }

    bool solve2(vector<int>& nums, int target, int currSum, int mask, unordered_map<string, bool>&& memo) {
        int n = nums.size();
        if (mask == (1<<n) - 1) {
            return currSum == 0;
        }
        string key = to_string(mask) + "#" + to_string(currSum);
        if (auto it = memo.find(key); it != memo.end()) {
            return it->second;
        }
        for (int i = 0; i < n; i++) {
            if ((mask >> i) & 1) continue;
            if (currSum + nums[i] <= target) {
                if (solve2(nums, target, (currSum + nums[i]) % target, mask | (1 << i), move(memo)))
                    return memo[key] = true;
            }
        }
        return memo[key] = false;
    }
};
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