<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>

import heapq

class Solution:
    
    def recursive(self, nums, index, k):

        if k==0:
            # for i in range(index, len(nums)):
            #     self.arr[k] += nums[i]
            self.arr[k] = self.prefix_sum[len(nums)] - self.prefix_sum[index]
            self.ans = min(self.ans, max(self.arr))

            return self.arr[k]

        if (k,index) in self.cache:
            return self.cache[(k, index)]

        cur_ans  = 1e11
        self.arr[k] = 0
        for i in range(index, len(nums)-k):
                self.arr[k] += nums[i]            
                if self.arr[k] >= self.ans:
                    break

                self.arr[k-1] = 0
                output = self.recursive(nums, i+1, k-1)
                cur_ans = min(cur_ans, max(output, self.arr[k]))

            
        self.cache[(k, index)] = cur_ans
            
        return cur_ans
            

    
    def splitArray(self, nums: List[int], k: int) -> int:

        self.ans = 1e11
        self.arr = [0]*k
        self.cache = {}

        self.prefix_sum = [0]*(len(nums)+1)
        for i in range(1,len(nums)+1):
            self.prefix_sum[i] = nums[i-1] + self.prefix_sum[i-1]

        return self.recursive(nums, 0, k-1)

        # return self.ans

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