Minimum Value to Get Positive Step by Step Sum

Description

Given an array of integers nums, you start with an initial positive value startValue. In each iteration, you calculate the step by step sum of startValue plus elements in nums (from left to right). Return the minimum positive value of startValue such that the step by step sum is never less than 1.

Example 1:
Input: nums = [-3, 2, -3, 4, 2]
Output: 5
Explanation: If you choose startValue = 4, in the third iteration your step by step sum is less than 1.
step by step sum:
startValue = 4 | startValue = 5
(4 - 3 = 1) | (5 - 3 = 2)
(1 + 2 = 3) | (2 + 2 = 4)
(3 - 3 = 0) | (4 - 3 = 1)
(0 + 4 = 4) | (1 + 4 = 5)
(4 + 2 = 6) | (5 + 2 = 7)
With startValue = 5, step sum is always >= 1.

Example 2:
Input: nums = [1, 2]
Output: 1
Explanation: Minimum start value should be positive, and step sums are 1+1=2, 2+2=4 (always >= 1).

Example 3:
Input: nums = [1, -2, -3]
Output: 5
Explanation: Step by step sums with startValue = 5: 6, 4, 1 (all >= 1).

Constraints

1 <= nums.length <= 100
-100 <= nums[i] <= 100

Signature

def solution(nums: list[int]) -> int: ...

Complexity

Time: O(n)
Space: O(1)

Hints

Pattern: Prefix sum to find minimum running sum
Approach: Find minimum prefix sum. If min_prefix < 1, startValue = 1 - min_prefix. Otherwise startValue = 1.
Complexity: Single pass to find minimum prefix sum

Solutions

Python

def solution(nums):
    sums = [nums[0]]
    for i in range(1, len(nums)):
        sums.append(nums[i] + sums[-1])

    ans = 1
    for i in range(len(sums)):
        if sums[i] < 0:
            ans = max(ans, abs(sums[i]) + 1)

    return ans