Solution Explanation

The problem asks to find the minimum positive startValue such that the cumulative sum of startValue and the elements in nums remains positive throughout the iteration. This can be efficiently solved using prefix sums and a single pass through the array.

Approach:

1. Prefix Sum: The core idea is to calculate the prefix sum of the nums array. The prefix sum at any index i represents the cumulative sum of elements from index 0 to i.

2. Minimum Prefix Sum: We need to find the minimum prefix sum encountered during the iteration. This minimum sum represents the point where the cumulative sum is the lowest.

3. Minimum startValue: To ensure the cumulative sum remains positive, the startValue must be at least 1 - minimumPrefixSum. This guarantees that even at the lowest point (minimum prefix sum), the cumulative sum will be at least 1 (positive).

4. Handling Positive Minimum: If the minimum prefix sum is already non-negative, it implies the cumulative sum is always positive, so startValue can simply be 1.

Time Complexity: O(N), where N is the length of nums. We iterate through nums once to calculate the prefix sums and find the minimum.

Space Complexity: O(1). We use only a few constant extra variables. Although some solutions might use an array for prefix sums, it's not necessary; we can update the minimum in place.

Code Explanation (Python3 - Solution 1):

class Solution:
    def minStartValue(self, nums: List[int]) -> int:
        s, t = 0, float('inf')  # Initialize sum (s) and minimum sum (t)
        for num in nums:
            s += num            # Update cumulative sum
            t = min(t, s)      # Update minimum sum encountered
        return max(1, 1 - t)    # Return the maximum of 1 and the required startValue

The code iterates through nums, keeping track of the cumulative sum (s) and the minimum cumulative sum (t). Finally, it returns max(1, 1 - t), ensuring the result is always at least 1 (a positive integer).

Code Explanation (Python3 - Solution 2):

class Solution:
    def minStartValue(self, nums: List[int]) -> int:
        s = list(accumulate(nums)) # Use accumulate to get prefix sums directly
        return 1 if min(s) >= 0 else abs(min(s)) + 1 #Concisely calculate startValue

This version leverages the accumulate function from the itertools library for a more concise solution. It directly calculates the prefix sums and then determines the startValue based on the minimum prefix sum.

The other languages follow a similar logic, adapting the syntax and data types accordingly. The core algorithmic approach remains the same across all implementations.