Minimum Value to Get Positive Step by Step Sum

2 min readUpdated 18 June 2025

Problem Statement

Given an array of integers nums, you begin with an initial positive value called startValue. For each element in nums, you incrementally add this element to the startValue cumulatively from left to right of the array.

The challenge is to determine the smallest initial value (startValue) such that the cumulative sum never drops below 1 at any point during the addition process. The startValue must itself be a positive number, and every intermediate total must also be at least 1.

Examples

Example 1

Input:

EXAMPLE_ENTITY

nums = [-3,2,-3,4,2]

Output:

EXAMPLE_ENTITY

Explanation:

EXAMPLE_ENTITY

Trying startValue = 4:
Step 1: 4 + (-3) = 1  
Step 2: 1 + 2 = 3  
Step 3: 3 + (-3) = 0 (drops below 1)  

Trying startValue = 5:
Step 1: 5 + (-3) = 2  
Step 2: 2 + 2 = 4  
Step 3: 4 + (-3) = 1  
Step 4: 1 + 4 = 5  
Step 5: 5 + 2 = 7  

The minimum `startValue` that keeps the sum ≥ 1 at every step is 5.

Example 2

Input:

EXAMPLE_ENTITY

nums = [1,2]

Output:

EXAMPLE_ENTITY

Explanation:

EXAMPLE_ENTITY

Starting at 1:
Step 1: 1 + 1 = 2  
Step 2: 2 + 2 = 4  

The cumulative sum never drops below 1. Minimum startValue is 1.

Example 3

Input:

EXAMPLE_ENTITY

nums = [1,-2,-3]

Output:

EXAMPLE_ENTITY

Explanation:

EXAMPLE_ENTITY

Try startValue = 5:
Step 1: 5 + 1 = 6  
Step 2: 6 + (-2) = 4  
Step 3: 4 + (-3) = 1  

StartValue = 5 keeps all cumulative sums ≥ 1. Any value < 5 fails.

Constraints

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

Approach and Intuition

To ensure the cumulative sum never falls below 1, we simulate the running total and track the minimum value reached:

Start with total = 0 and min_total = 0.
Traverse the array and compute the running sum at each step.
Track the minimum value of the running sum (min_total) across all steps.
To ensure the total never drops below 1, choose startValue = 1 - min_total.

This guarantees that even the lowest point in the running sum stays at or above 1.

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

This method is optimal and handles all edge cases within the given constraints efficiently.

Solutions

CPP
class Solution {
public:
    int minInitialValue(vector<int>& numbers) {
        int leastTotal = 0;
        int cumulativeSum = 0;
            
        for (int value : numbers) {
            cumulativeSum += value;
            leastTotal = min(leastTotal, cumulativeSum);
        }
            
        return -leastTotal + 1;
    }
};
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The provided C++ solution addresses the problem of finding the minimum initial value such that the cumulative sum of an array remains positive when incremented step by step. Here’s an outline of how the solution efficiently solves the problem:

A function named minInitialValue is defined, which receives a vector of integers numbers.
Two integer variables, leastTotal and cumulativeSum, are initialized to zero. These variables track the minimum sum encountered during the sequence and the ongoing cumulative sum, respectively.
The function iterates through each element in the numbers vector, updating the cumulativeSum with the current value and subsequently adjusting leastTotal to capture the lesser of its current value or the new cumulativeSum.
The smallest starting value required to keep the cumulative sum positive throughout the sequence is calculated by returning the negation of leastTotal plus one. This effectively ensures that even with the lowest cumulative sum, by starting at this value, all subsequent sums remain non-negative.
The core operation of finding the minimum in each step ensures efficiency, handling each number in the array only once, resulting in a linear time complexity relative to the size of the input.

This approach leverages straightforward logic and efficient iteration to accurately determine the minimum value needed for the desired condition.

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Minimum Value to Get Positive Step by Step Sum

2 min readUpdated 18 June 2025

Problem Statement

Examples

Example 1

Input:

EXAMPLE_ENTITY

nums = [-3,2,-3,4,2]

Output:

EXAMPLE_ENTITY

Explanation:

EXAMPLE_ENTITY

Trying startValue = 4:
Step 1: 4 + (-3) = 1  
Step 2: 1 + 2 = 3  
Step 3: 3 + (-3) = 0 (drops below 1)  

Trying startValue = 5:
Step 1: 5 + (-3) = 2  
Step 2: 2 + 2 = 4  
Step 3: 4 + (-3) = 1  
Step 4: 1 + 4 = 5  
Step 5: 5 + 2 = 7  

The minimum `startValue` that keeps the sum ≥ 1 at every step is 5.

Example 2

Input:

EXAMPLE_ENTITY

nums = [1,2]

Output:

EXAMPLE_ENTITY

Explanation:

EXAMPLE_ENTITY

Starting at 1:
Step 1: 1 + 1 = 2  
Step 2: 2 + 2 = 4  

The cumulative sum never drops below 1. Minimum startValue is 1.

Example 3

Input:

EXAMPLE_ENTITY

nums = [1,-2,-3]

Output:

EXAMPLE_ENTITY

Explanation:

EXAMPLE_ENTITY

Try startValue = 5:
Step 1: 5 + 1 = 6  
Step 2: 6 + (-2) = 4  
Step 3: 4 + (-3) = 1  

StartValue = 5 keeps all cumulative sums ≥ 1. Any value < 5 fails.

Constraints

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

Approach and Intuition

To ensure the cumulative sum never falls below 1, we simulate the running total and track the minimum value reached:

Start with total = 0 and min_total = 0.
Traverse the array and compute the running sum at each step.
Track the minimum value of the running sum (min_total) across all steps.
To ensure the total never drops below 1, choose startValue = 1 - min_total.

This guarantees that even the lowest point in the running sum stays at or above 1.

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

This method is optimal and handles all edge cases within the given constraints efficiently.

Solutions

CPP
class Solution {
public:
    int minInitialValue(vector<int>& numbers) {
        int leastTotal = 0;
        int cumulativeSum = 0;
            
        for (int value : numbers) {
            cumulativeSum += value;
            leastTotal = min(leastTotal, cumulativeSum);
        }
            
        return -leastTotal + 1;
    }
};
Powered by Vultr AgentBeta

A function named minInitialValue is defined, which receives a vector of integers numbers.
Two integer variables, leastTotal and cumulativeSum, are initialized to zero. These variables track the minimum sum encountered during the sequence and the ongoing cumulative sum, respectively.
The function iterates through each element in the numbers vector, updating the cumulativeSum with the current value and subsequently adjusting leastTotal to capture the lesser of its current value or the new cumulativeSum.
The smallest starting value required to keep the cumulative sum positive throughout the sequence is calculated by returning the negation of leastTotal plus one. This effectively ensures that even with the lowest cumulative sum, by starting at this value, all subsequent sums remain non-negative.
The core operation of finding the minimum in each step ensures efficiency, handling each number in the array only once, resulting in a linear time complexity relative to the size of the input.

This approach leverages straightforward logic and efficient iteration to accurately determine the minimum value needed for the desired condition.

More to Explore

Tech Talk: Running a Self-Improving AI Agent with Hermes on Vultr

Tech Talk: Implementing Fine-Grained Access Control with Vultr IAM

Tech Talk: Getting Started with Vultr IAM

Tech Talk: Simplifying Zero-Trust Networking with NetBird on Vultr Marketplace

0 Comments

Start the conversation

Be the first to comment and share your perspective with the community.