Minimum Moves for Alternating Ball Placement in Buckets
To solve this problem, you need to place the balls in such a way that the distance between each ball is 2. You have to find the minimum number of moves to do this. Here’s a Python code to solve this problem:


Here’s how the code works:
 First, we find all the positions where there is a ball (
B
) and where there is an empty bucket (.
).  Then we loop through each possible starting position for the alternating sequence.
 For each starting position, we calculate the minimum moves required to create an alternating sequence starting at that position.
 We keep track of the minimum moves across all possible starting positions.
This solution has a time complexity of O(n^2) where n is the length of the bucket string, making it a bruteforce approach. However, it works well for small input sizes.
Optimized Solution
An optimized solution can consider only the positions where balls (B
) are located, without considering every possible starting point for the sequence. We can compute the minimum moves based on the distance the balls need to be moved to reach their target positions in an alternating sequence.
Here’s an optimized Python code snippet for the problem:


In this optimized version, we loop only through the positions where balls are located, thus reducing the number of iterations. We calculate the target_positions
based on each starting ball position and then find the total moves needed to move the balls to those target positions. Finally, we find the minimum moves required among all possible sequences.
This version still runs in O(n^2) in the worst case, but it’s faster because it only loops through positions where balls are actually located, not every possible position.