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Leetcode 1413. Minimum Value to Get Positive Step by Step Sum

Problem Statement

You are given an integer array nums. In one operation, you can choose any integer x and add it to any element of the array. Your goal is to make the sum of the array elements positive at each step. Return the minimum value that ensures this.

Example:

Input: nums = [-3, 2, -3, 4, 2]
Output: 5
Explanation: If you start with the minimum positive integer `5`, the running sum array would be [2, 4, 1, 5, 7] which is always positive.

Clarifying Questions

  1. Can the input array nums be empty?
    • No, the problem assumes that nums contains at least one element.
  2. What are the constraints on the elements of nums?
    • Elements of nums are integers and can be positive, negative, or zero.
  3. What should we do if there are multiple valid answers?
    • The problem specifies to return the minimum value that makes the running sum always positive.

Strategy

To solve the problem, our strategy is to ensure that the cumulative sum of the array remains positive through all steps. We’ll do this by finding the minimum cumulative sum in the array and adjusting the starting point accordingly.

  1. Calculate the prefix sums: Compute the cumulative sum of the array as we iterate through it.
  2. Track the minimum cumulative sum: Keep track of the minimum value of these cumulative sums.
  3. Determine the starting offset: Calculate the minimum value x to make all cumulative sums positive.

Approach

  1. Initialize a variable to keep track of the current cumulative sum.
  2. As we iterate through the array, update this cumulative sum and simultaneously keep track of the minimum cumulative sum encountered.
  3. After finding the minimum cumulative sum, determine the smallest positive integer x which makes the starting sum positive. This is obtained by ensuring the lowest cumulative sum, when adjusted by x, is greater than zero.

Code

#include <vector>
#include <algorithm>
#include <iostream>

class Solution {
public:
    int minStartValue(std::vector<int>& nums) {
        int minSum = 0;
        int currentSum = 0;
        
        for (int num : nums) {
            currentSum += num;
            minSum = std::min(minSum, currentSum);
        }
        
        // To make sure all cumulative sums are positive, minSum + starting value should be > 0.
        // Therefore, starting value > -minSum, we need to take ceil to make it positive integer
        return 1 - minSum;
    }
};

int main() {
    Solution solution;
    std::vector<int> nums = {-3, 2, -3, 4, 2};
    std::cout << solution.minStartValue(nums) << std::endl; // Output: 5
    return 0;
}

Time Complexity

This completes the explanation and solution for the problem “Minimum Value to Get Positive Step by Step Sum”.

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