Make the Prefix Sum Non-negative

class Solution {
public:
    int calculateMinOperations(vector<int>& elements) {
        int totalOperations = 0;
        long cumulativeSum = 0;
        priority_queue<int, vector<int>, greater<int>> minHeap;

        for (int element : elements) {
            if (element < 0) {
                minHeap.push(element);
            }

            cumulativeSum += element;
            if (cumulativeSum < 0) {
                cumulativeSum -= minHeap.top();
                minHeap.pop();
                totalOperations++;
            }
        }

        return totalOperations;
    }
};

Title: Make the Prefix Sum Non-negative

Programming Language: C++

Code Analysis:

The provided C++ function calculateMinOperations() operates on an input vector elements to compute the minimum number of operations required to make the prefix sum of array elements non-negative at all points. The method uses a few specific techniques as outlined:

• Initialization:

  • totalOperations starts at 0, representing the count of operations performed.
  • cumulativeSum, initialized at 0, stores the current sum of elements as the loop progresses.
  • minHeap is used to efficiently keep track of the negative elements in order to easily get the smallest (most negative) element when needed.

• Iterative processing of the vector elements:

  • Traverse each element of the input vector:
    • Push negative elements into minHeap.
    • Update cumulativeSum by adding the current element.
    • If cumulativeSum becomes negative, adjust it by subtracting the smallest element (i.e., the top of the priority queue minHeap) and increase the totalOperations count by one.
    • Remove the used element from minHeap.

• Returning the result:

  • After completing the iteration over all elements, the function returns the value of totalOperations, which reflects the minimum number of items that need to be removed to ensure no prefix sum is negative.

This approach effectively ensures that the prefix sum remains non-negative, all while attempting to minimize the disturbance caused to the original array. The use of a minimum heap helps in efficiently accessing the smallest element, which is essential for correcting the negative sum promptly. By the end, the return value states how many elements had to be adjusted (removed from the sum in this case) to maintain non-negative prefix sums throughout the initial sequence.

class Solution {

    public int minimizeNegativePrefix(int[] values) {
        int countOps = 0;
        long sumPrefix = 0;
        PriorityQueue<Integer> minHeap = new PriorityQueue<>();

        for (int value : values) {
            if (value < 0) {
                minHeap.offer(value);
            }

            sumPrefix += value;
            if (sumPrefix < 0) {
                sumPrefix -= minHeap.poll();
                countOps++;
            }
        }

        return countOps;
    }
}

This Java solution addresses the problem of ensuring that the prefix sum of an array remains non-negative by performing a minimal number of operations. Each operation involves removing the smallest negative element from the prefix sum when it becomes negative.

• Implementation Overview:

  • The function minimizeNegativePrefix takes an integer array as input.
  • Two variables are initialized: countOps, which tracks the number of operations performed, and sumPrefix, which keeps the running total of the prefix sum.
  • A PriorityQueue, minHeap, is used to store negative numbers efficiently. Each negative element encountered is added to this priority queue.
  • As values are added to sumPrefix, the algorithm checks if sumPrefix becomes negative.
  • If sumPrefix is negative, the smallest value (which is negative) is removed from minHeap, and sumPrefix is adjusted accordingly. The countOps counter is incremented each time this adjustment is needed.

• Detailed Steps:

  1. Traverse through each element in the input array.
  2. Add negative elements to minHeap.
  3. Update sumPrefix with the current value.
  4. If sumPrefix drops below zero, remove the smallest negative element from minHeap and adjust sumPrefix.
  5. Increment countOps each time an element is removed from the heap.

• Expected Output:

  • The function returns the count of operations performed (countOps), which is the minimum number needed to ensure the prefix sum never becomes negative.

This approach leverages a min-heap to efficiently remove the smallest negative numbers from the calculation, ensuring the prefix sum remains non-negative with minimal adjustments. This solution is optimal for scenarios where the sequence of numbers is unpredictable and could cause frequent drops below zero in cumulative sums.

class Solution:
    def stabilizePrefixSum(self, values):
        count_operations = 0
        cumulative_sum = 0
        min_heap = []

        for value in values:
            if value < 0:
                heapq.heappush(min_heap, value)

            cumulative_sum += value
            if cumulative_sum < 0:
                cumulative_sum -= heapq.heappop(min_heap)
                count_operations += 1

        return count_operations

The provided Python solution addresses the problem of making the prefix sum of an array non-negative. The function stabilizePrefixSum in the Solution class operates as follows:

• Initialize a variable count_operations to keep track of the number of operations needed.
• Use cumulative_sum to track the running total of the array values.
• A min-heap min_heap is employed to efficiently manage and retrieve minimum values when required.

The algorithm processes each element in the input list values:

• If an element is negative, it is added to the min_heap.
• The element is then added to cumulative_sum.
• If cumulative_sum drops below zero, the smallest element is removed from min_heap and subtracted from cumulative_sum, and count_operations is incremented.

In summary, the code efficiently ensures the prefix sum of a list remains non-negative, using a min-heap to handle negative contributions and minimize their impact efficiently. The solution's effectiveness is achieved through careful management of the prefix sum and utilizing operations on the min-heap, leading to an optimal result in terms of operation count.