<p>Given an integer array <code>nums</code>, return <code>true</code> <em>if you can partition the array into two subsets such that the sum of the elements in both subsets is equal or </em><code>false</code><em> otherwise</em>.</p>

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

<pre>
<strong>Input:</strong> nums = [1,5,11,5]
<strong>Output:</strong> true
<strong>Explanation:</strong> The array can be partitioned as [1, 5, 5] and [11].
</pre>

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

<pre>
<strong>Input:</strong> nums = [1,2,3,5]
<strong>Output:</strong> false
<strong>Explanation:</strong> The array cannot be partitioned into equal sum subsets.
</pre>

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

<ul>
	<li><code>1 &lt;= nums.length &lt;= 200</code></li>
	<li><code>1 &lt;= nums[i] &lt;= 100</code></li>
</ul>

# 1. Clarification
# duplicates
# range 
# length

# time: O(N*total sum , N*logN)
# space: O(N)

class Solution:
    def canPartition(self, nums: List[int]) -> bool:
        # 1
        total_sum = sum(nums)
        if total_sum % 2 == 1:
            return False

        # 2
        nums.sort()

        # 3 
        half_sum = total_sum // 2
        def dfs(index, curr_sum):
            nonlocal nums

            if curr_sum == half_sum:
                return True
            if curr_sum > half_sum:
                return False
            if index == len(nums):
                return False
            if (index, curr_sum) in mem:
                return mem[(index, curr_sum)]

            if dfs(index + 1, curr_sum):
                return True
            if dfs(index + 1, curr_sum + nums[index]):
                return True
            mem[(index, curr_sum)] = False
            return mem[(index, curr_sum)] 

        mem = {} 
        return dfs(0, 0)
        
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