Minimum Number of Operations to Move All Balls to Each Box

class Solution {
public:
    vector<int> performOperations(string boxes) {
        int length = boxes.length();
        vector<int> result(length, 0);

        int leftBalls = 0, leftSteps = 0;
        int rightBalls = 0, rightSteps = 0;

        // Calculate both left and right contributions in a single loop
        for (int i = 0; i < length; i++) {
            // Adding left accumulated steps
            result[i] += leftSteps;
            leftBalls += boxes[i] - '0';
            leftSteps += leftBalls;

            // Adding right accumulated steps
            int j = length - 1 - i;
            result[j] += rightSteps;
            rightBalls += boxes[j] - '0';
            rightSteps += rightBalls;
        }

        return result;
    }
};

This solution focuses on calculating the minimum number of operations required to move all balls to each box in an input string of '0's and '1's, where '1' represents a box containing a ball and '0' represents an empty box. The implementation uses C++ and takes advantage of a single linear pass algorithm to efficiently compute the result.

Understand the core idea behind the solution:

• Two arrays, leftBalls and rightBalls, are tracked to count the number of balls on the left side and right side, respectively, for any given index.
• leftSteps and rightSteps capture the cumulative steps or operations needed to move all respective left or right balls to the current box.
• As you iterate from the start to the end of the boxes string, accumulate the total number of operations needed for balls to the left of the current box (leftSteps). Simultaneously, perform a reversed iteration to accumulate the total number of operations for balls to the right using rightSteps.
• Add these values for each box to get the total operations needed for all balls to be moved to every specific box.

To leverage this solution in your application:

1. Include the above class definition in your C++ project.
2. Create an instance of the Solution class.
3. Call the performOperations method with your specific boxes string.
4. The method will return a vector of integers where each element at index i represents the total number of operations required to move all the balls to the box at index i.

Ensure your string only consists of '0's and '1's, and validate the length of the string as needed, keeping in mind performance considerations for very long strings. This method offers a computationally efficient approach with a time complexity closely aligning with O(n), where n is the length of the string, because each element is processed linearly once from each direction--left and right.

class Solution {

    public int[] minimumOperations(String containers) {
        int length = containers.length();
        int[] result = new int[length];

        int leftBalls = 0, leftSteps = 0;
        int rightBalls = 0, rightSteps = 0;

        // Process left to right and right to left simultaneously
        for (int i = 0; i < length; i++) {
            // Process for left to right
            result[i] += leftSteps;
            leftBalls += Character.getNumericValue(containers.charAt(i));
            leftSteps += leftBalls;

            // Process for right to left
            int j = length - 1 - i;
            result[j] += rightSteps;
            rightBalls += Character.getNumericValue(containers.charAt(j));
            rightSteps += rightBalls;
        }

        return result;
    }
}

This Java method computes the minimum number of operations required to move all balls to each box, based on a given string where each character represents the number of balls in a corresponding box. It processes the string from both ends towards the center to optimize the operation count.

The main steps involved are:

1. Initialize arrays to store results and variables to track the number of steps and balls from both the left and right ends.

2. Use a loop to iterate through the string. For each box:

   • Add the accumulated steps to the result.
   • Update the total count of balls and steps moving from left to right.
   • Simultaneously, adjust the steps and balls counts for the right to left calculation.

3. Return the result array containing the minimal operations for each box.

This approach ensures that the solution covers all potential movements of balls for each position within one pass through the containers, making it efficient and direct.

class Solution:
    def minimumMoves(self, boxes: str) -> List[int]:
        length = len(boxes)
        result = [0] * length

        left_count = 0
        left_steps = 0
        right_count = 0
        right_steps = 0

        for index in range(length):
            # Processing left to right
            result[index] += left_steps
            left_count += int(boxes[index])
            left_steps += left_count

            # Processing right to left
            reverse_index = length - 1 - index
            result[reverse_index] += right_steps
            right_count += int(boxes[reverse_index])
            right_steps += right_count

        return result

The provided Python solution efficiently calculates the minimum number of moves required to move all balls to each box in a given string representation of boxes, where "1" represents a box containing a ball and "0" represents an empty box. The function minimumMoves employs a two-pointer approach from both ends of the string to optimize the accumulation of moves:

• Initialize variables to keep track of the total length of the boxes string, result array initialized to zero, counts of balls, and accumulated steps from the left and from the right.
• Iterate through the string with a loop. For each character:
  • Add the current accumulated steps from the left to the result for the current box.
  • Increase the left ball count and steps if the current box contains a ball.
  • Using the reverse index for handling right-to-left calculation simultaneously:
    • Add the current accumulated steps from the right to the result for the box at the reverse index.
    • Increase the right ball count and steps if the box at the reverse index contains a ball.

By the end of the loops, the result array contains the minimum number of moves required for each box position to gather all balls. This approach ensures that each box position is evaluated in terms of minimal operational steps from both directions, making the calculation efficient.