Minimum Value to Get Positive Step by Step Sum - Solution & Explanation

Problem Overview: You are given an integer array nums. Starting with an unknown positive value startValue, you add each element in order and keep a running sum. The requirement is simple: the running sum must never drop below 1. The task is to compute the smallest possible starting value that guarantees this condition.

Approach 1: Calculate Minimum Start Value using Cumulative Sum (O(n) time, O(1) space)

Traverse the array while maintaining a running cumulative sum. Track the minimum value that this cumulative sum reaches during the traversal. If the lowest prefix sum becomes negative, the starting value must offset that dip so the total never falls below 1. Mathematically, the required start value is 1 - minPrefixSum. If the minimum prefix sum is already positive, the answer is simply 1. This works because every running total equals startValue + prefixSum, so ensuring the smallest prefix keeps the total ≥ 1 guarantees the rest will also be valid. The algorithm scans the array once and uses constant extra memory.

This approach relies on tracking prefix accumulation, which is a common technique in array problems and especially those involving prefix sum reasoning. You only store the current running sum and the minimum value encountered.

Approach 2: Prefix Sum with Constant Adjustment (O(n) time, O(1) space)

Another way to think about the problem is to explicitly compute the prefix sums and adjust them so the smallest prefix becomes 1. Iterate through nums, accumulate a prefix sum, and track the minimum prefix encountered. Instead of adjusting the prefix array itself, compute a constant shift that raises the minimum prefix to 1. That constant shift is the starting value. This interpretation emphasizes the idea that every prefix sum can be translated upward by the same constant to satisfy the constraint.

The logic is nearly identical to the cumulative-sum approach but framed as a transformation of the prefix sequence. It helps build intuition for similar problems where you normalize prefix values or maintain constraints across running totals. Because only two variables are maintained (current prefix and minimum prefix), the space complexity remains constant.

Recommended for interviews: The cumulative sum solution is the expected answer. It demonstrates that you recognize the prefix-sum pattern and can derive the minimum offset required to keep the running total positive. Explaining the minimum-prefix insight clearly usually matters more than the code itself. A brute-force simulation with increasing start values shows basic understanding, but interviewers typically expect the single-pass O(n) prefix analysis.