Minimum Operations to Reduce X to Zero

class Solution {
public:
    int minimumOperations(vector<int>& nums, int target) {
        int sum = accumulate(nums.begin(), nums.end(), 0);
        int size = nums.size();
        int smallest = INT_MAX;
        int start = 0;

        for (int end = 0; end < size; end++) {
            sum -= nums[end];

            while (sum < target && start <= end) {
                sum += nums[start];
                start++;
            }

            if (sum == target) {
                smallest = min(smallest, (size - 1 - end) + start);
            }
        }
        return smallest != INT_MAX ? smallest : -1;
    }
};

The solution in C++ addresses the problem of finding the minimum number of operations required to reduce the sum of integers to exactly zero from a given array nums by removing a subarray. The targeted sum to be removed, target, must equate the sum of the removed subarray.

• Calculate the total sum of the array and store it.
• Initialize variables to keep track of the smallest subarray length (smallest), the start index (start), and the current sum reduction (sum). The variable smallest begins with a value of INT_MAX to capture the minimum length found.
• Iterate over the array using a for-loop, subtracting each value from the total sum. Adjust the start of the subarray as necessary if the current sum exceeds or equals the target.
• When the current sum matches the target, update the smallest variable to store the lesser of the current smallest subarray length or the new subarray length found.
• Continue this process until the end of the array to ensure all possible subarrays are considered.
• If a valid subarray is found, return its length. If no valid subarray is found, return -1.

This approach utilizes a sliding window technique to efficiently find the smallest subarray sum that can be subtracted to achieve the result. By adjusting the start and end pointers based on the current sum's relation to the target, the algorithm efficiently narrows down the possible solutions. The complexity is controlled by the length of the array as it ensures each element is considered in the summation only once through the sliding window mechanics.

class Solution {
    public int minSteps(int[] arr, int target) {
        int sum = 0;
        for (int value : arr) {
            sum += value;
        }
        int length = arr.length;
        int result = Integer.MAX_VALUE;
        int start = 0;

        for (int end = 0; end < length; end++) {
            sum -= arr[end];
            while (sum < target && start <= end) {
                sum += arr[start];
                start++;
            }
            if (sum == target) {
                result = Math.min(result, (length-end-1)+start);
            }
        }
        return result != Integer.MAX_VALUE ? result : -1;
    }
}

The given Java solution is designed to find the minimum number of operations required to reduce the sum of an array to exactly x by removing the elements either from the beginning or from the end of the array. This approach leverages a sliding window technique:

• Start by calculating the sum of all array elements. Initialize the result to hold the minimum steps required, setting it to the maximum integer value.
• Iterate through the array using two pointers, start and end, while adjusting the sum by subtracting the current element (moving the end pointer).
• If after adjustments, the sum is less than the target, increase the start pointer to reduce the subarray sum from the start.
• Check if the sum equals the target after each adjustment. If it does, calculate the operations needed by adding the number of elements removed from the start and from the end.
• At the end of the iteration, compare the number of operations required in each iteration and update the result with the minimum value.
• Return the minimum number of steps, or -1 if it’s not possible to achieve the target value.

This method ensures efficient calculation by iteratively adjusting the range considered while keeping track of the minimum operations needed. The primary strength lies in its ability to dynamically resize the subarray based on the current sum in relation to the target, ensuring optimal performance.

class Solution:
    def minimumOperations(self, numbers: List[int], target: int) -> int:
        total = sum(numbers)
        size = len(numbers)
        smallest = inf
        start = 0

        for end in range(size):
            total -= numbers[end]
            while total < target and start <= end:
                total += numbers[start]
                start += 1
            if total == target:
                smallest = min(smallest, (size-1-end)+start)

        return smallest if smallest != inf else -1

This Python solution involves finding the minimum number of operations to adjust the sum of an array of numbers to exactly reduce a given target number to zero. The approach utilizes a two-pointer technique, involving start and end pointers facilitated with a sliding window concept:

1. Compute the total sum of the array.
2. Initiate start at 0, iterate through the array using end pointer.
3. Adjust the total by subtracting each element pointed by end. This is akin to decreasing the target you have to reach.
4. If total becomes less than target, increment the start pointer to the right whilst adding the numbers back to total until the total is at least target. This operation ensures that you're compressing your array window to the smallest possible size that sums up to or exceeds the target.
5. Each time the total equals the target, calculate the total number of operations or changes (which translates as array elements not included between start and end).
6. Update the smallest variable to record the minimal number of operations required whenever a new minimum is found by comparing current operations count with previously stored smallest.
7. Finally, the function returns the smallest number of operations or -1 if it's impossible to achieve the target total from the provided array.

This method efficiently ensures that each possible window that could mimic the required sum to drop to the target is checked while optimizing to find the smallest set of changes (or operations). The approach leverages the benefits of both reducing runtime complexity and intuitive logical processing using the sliding window technique.