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The idea is to simulate the step by step summation of the start value with the elements of the array. We maintain a running sum and adjust the start value such that this sum never drops below 1. At each step, if the running sum is less than 1, we calculate and update the new minimum start value needed to keep the running sum at least 1.
Time Complexity: O(n), where n is the length of nums.
Space Complexity: O(1), as we are using a constant amount of space.
1def min_start_value(nums):
2 min_sum = 0
3 current_sum = 0
4 for num in nums:
5 current_sum += num
6 min_sum = min(min_sum, current_sum)
7 return 1 - min_sum
8
9# Example usage:
10nums = [-3, 2, -3, 4, 2]
11print(min_start_value(nums)) # Output: 5In this Python function, we calculate the running sum of the elements in the list and track the minimum value. The start value is calculated to ensure the cumulative sum never goes below 1, by adjusting with 1 minus the minimum sum.
This approach is similar to the first but conceptualized through using prefix sums and adjusting the needed start value based on the lowest prefix sum reached. We cumulatively add each number and check if the prefix sum dips below a certain threshold, indicating the minimal adjustment needed for the start value.
Time Complexity: O(n).
Space Complexity: O(1).
The prefix sum is maintained over the iteration, comparing against a minimum threshold. The necessary start value ensures this cumulative prefix never dips below one during any point.