Minimum Number of Operations to Move All Balls to Each Box

Minimum Number of Operations to Move All Balls to Each Box

1. What is this problem about?

This problem involves a series of boxes, some of which contain a ball (represented by '1') and some that are empty (represented by '0'). The task is to calculate the total number of moves required to move every ball in the array to each specific box position. A move is defined as shifting a ball from box i to box j, costing |i - j| moves. You need to return an array where each element represents the total cost for that particular box.

2. Why is this asked in interviews?

Tech giants like Meta, Google, and Amazon favor this question because it tests a candidate's ability to optimize a brute-force solution. While a nested loop (O(n²)) is easy to implement, it becomes inefficient for large datasets. This problem challenges you to find a linear time (O(n)) solution using mathematical insights or prefix sums, which demonstrates high-level algorithmic thinking.

3. Algorithmic pattern used

The optimal approach for the "Minimum Number of Operations to Move All Balls to Each Box" coding problem is the Prefix Sum pattern or a "Two-Pass" approach. By iterating from left to right, you can track the number of balls encountered and the cumulative distance. Then, by iterating from right to left, you can do the same. Combining these two passes gives you the total moves for every box in linear time.

4. Example explanation

Imagine boxes: [1, 1, 0].

• For Box 0: Ball at index 1 moves 1 step. Total = 1.
• For Box 1: Ball at index 0 moves 1 step. Total = 1.
• For Box 2: Ball at index 0 moves 2 steps, Ball at index 1 moves 1 step. Total = 3. The resulting array is [1, 1, 3]. In a large array, the two-pass method ensures we don't re-calculate these distances from scratch for every single box.

5. Common mistakes candidates make

Many candidates settle for the O(n²) solution and fail to recognize the opportunity for optimization. Another frequent error is incorrectly updating the cumulative distance during the passes—forgetting that each ball already moved must contribute an additional step for every new box index reached. Mismanaging the indices during the backward pass is also a common source of bugs.

6. Interview preparation tip

Practice problems that involve calculating distances or sums across an array multiple times. Whenever you see a "sum of distances" requirement, think about how the values change as you move from index i to i+1. Often, you can derive the result for the next index based on the result of the previous one, which is the key to transforming a quadratic solution into a linear one.