<p>Given an integer array <code>nums</code> and an integer <code>k</code>, split <code>nums</code> into <code>k</code> non-empty subarrays such that the largest sum of any subarray is <strong>minimized</strong>.</p>

<p>Return <em>the minimized largest sum of the split</em>.</p>

<p>A <strong>subarray</strong> is a contiguous part of the array.</p>

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

<pre>
<strong>Input:</strong> nums = [7,2,5,10,8], k = 2
<strong>Output:</strong> 18
<strong>Explanation:</strong> There are four ways to split nums into two subarrays.
The best way is to split it into [7,2,5] and [10,8], where the largest sum among the two subarrays is only 18.
</pre>

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

<pre>
<strong>Input:</strong> nums = [1,2,3,4,5], k = 2
<strong>Output:</strong> 9
<strong>Explanation:</strong> There are four ways to split nums into two subarrays.
The best way is to split it into [1,2,3] and [4,5], where the largest sum among the two subarrays is only 9.
</pre>

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

<ul>
	<li><code>1 &lt;= nums.length &lt;= 1000</code></li>
	<li><code>0 &lt;= nums[i] &lt;= 10<sup>6</sup></code></li>
	<li><code>1 &lt;= k &lt;= min(50, nums.length)</code></li>
</ul>

from functools import cache
from itertools import accumulate


class Solution:
    def splitArray(self, nums: List[int], m: int) -> int:
        n = len(nums)
        cumsums = [0] + list(accumulate(nums))

        memo = dict()
        for i in range(n - 1, -1, -1):
            memo[i] = cumsums[n] - cumsums[i]
        
        for k in range(2, m + 1):
            new_memo = dict()
            for start in range(0, n - k + 1):
                min_sum = float("inf")
                for i in range(start, n - k + 1):
                    first_split_sum = cumsums[i + 1] - cumsums[start]
                    large_split_sum = max(first_split_sum, memo[i + 1])
                    min_sum = min(min_sum, large_split_sum)
                    if first_split_sum >= min_sum:
                        break
                new_memo[start] = min_sum
            memo = new_memo
        return memo[0]

        # @cache
        # def recurse(start, k):
        #     if k == 1:
        #         return cumsums[len(nums)] - cumsums[start]
        #     min_sum = float("inf")
        #     for i in range(start, len(nums) - k + 1):
        #         first_split_sum = cumsums[i + 1] - cumsums[start]
        #         large_split_sum = max(first_split_sum, recurse(i + 1, k - 1))
        #         min_sum = min(min_sum, large_split_sum)
        #         if first_split_sum >= min_sum:
        #             break
        #     return min_sum


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