<p>Given an array of integers, return the maximum sum for a <strong>non-empty</strong>&nbsp;subarray (contiguous elements) with at most one element deletion.&nbsp;In other words, you want to choose a subarray and optionally delete one element from it so that there is still at least one element left and the&nbsp;sum of the remaining elements is maximum possible.</p>

<p>Note that the subarray needs to be <strong>non-empty</strong> after deleting one element.</p>

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

<pre>
<strong>Input:</strong> arr = [1,-2,0,3]
<strong>Output:</strong> 4
<strong>Explanation: </strong>Because we can choose [1, -2, 0, 3] and drop -2, thus the subarray [1, 0, 3] becomes the maximum value.</pre>

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

<pre>
<strong>Input:</strong> arr = [1,-2,-2,3]
<strong>Output:</strong> 3
<strong>Explanation: </strong>We just choose [3] and it&#39;s the maximum sum.
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> arr = [-1,-1,-1,-1]
<strong>Output:</strong> -1
<strong>Explanation:</strong>&nbsp;The final subarray needs to be non-empty. You can&#39;t choose [-1] and delete -1 from it, then get an empty subarray to make the sum equals to 0.
</pre>

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

<ul>
	<li><code>1 &lt;= arr.length &lt;= 10<sup>5</sup></code></li>
	<li><code>-10<sup>4</sup> &lt;= arr[i] &lt;= 10<sup>4</sup></code></li>
</ul>

class Solution:
    def maximumSum(self, arr: List[int]) -> int:
        prefix_max = []

        Sum = 0
        Max = -inf
        for i in range(len(arr)):
            Sum += arr[i]
            Max = max(Max, Sum)
            prefix_max.append(Sum)
            if Sum < 0:
                Sum = 0

        suffix_max = [0 for i in range(len(arr))]

        Sum = 0
        for i in range(len(arr)-1, -1, -1):
            Sum += arr[i]
            Max = max(Max, Sum)

            suffix_max[i] = Sum

            if Sum < 0:
                Sum = 0

        # print(suffix_max, prefix_max)

        for i in range(len(arr)):
            if i == 0:
                Max = max(Max, suffix_max[min(len(arr)-1, i+1)])

            elif i == len(arr) - 1:
                Max = max(Max, prefix_max[max(0, i-1)])
            else:
                Max = max(Max, prefix_max[max(0, i-1)] + suffix_max[min(len(arr)-1, i+1)])

        return Max



        
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