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Given an array of integers nums, you start with an initial value startValue = 0. On each element of the array, you can modify startValue using the following process:

• Change the startValue to startValue + nums[i] for each element in the array nums from left to right.

However, we need to ensure that startValue remains positive for each step.

Write a function minStartValue(nums) that finds the minimum initial value of startValue such that when it is processed through the given array, it remains positive at every step.

Clarifying Questions

1. What is the range of the integer values that can be in nums?
   • The problem typically allows for standard constraints such as -100 <= nums[i] <= 100.
2. What is the length of nums?
   • Usually, it can be between 1 and 10000.
3. Will nums contain both positive and negative integers?
   • Yes, nums can contain both positive and negative integers including zero.
4. Can we have an empty array?
   • No, as per the constraints, the array will have at least one element.

Strategy

To ensure that the startValue remains positive after each step, we will:

1. Initialize a cumulative_sum value starting from 0.
2. Traverse through each element in the array and update the cumulative_sum.
3. Track the minimum value of cumulative_sum encountered during this traversal.
4. To guarantee all values of the cumulative_sum are positive, startValue must be greater than or equal to the absolute minimum cumulative_sum observed plus 1.

Code

Here is the Python code to implement this solution:

def minStartValue(nums):
    cumulative_sum = 0
    min_cumulative_sum = 0

    for num in nums:
        cumulative_sum += num
        min_cumulative_sum = min(min_cumulative_sum, cumulative_sum)
    
    # The minimum startValue must be 1 - the minimum cumulative sum encountered
    return 1 - min_cumulative_sum

Explanation

1. Initialize cumulative_sum to 0 and min_cumulative_sum to 0.
2. Iterate through each element in the array nums. For each element:
   • Add the element to cumulative_sum.
   • Track the minimum value of cumulative_sum encountered so far.
3. The required startValue must be at least 1 - min_cumulative_sum to ensure that the running total never drops below 1.

Time Complexity

• The solution traverses the list once, resulting in a time complexity of O(n) where n is the length of the array nums. This ensures the solution is efficient even for larger input sizes.

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