Identify the positions of all black balls ('1's) in the string. Let's say there are `k` black balls at positions `p1, p2, ..., pk`.

The optimal grouping of black balls will be centered around the median of their positions. Calculate the median position `m` from the list of black ball positions.

For each black ball at position `pi`, calculate the number of swaps needed to move it to the median position `m`. The total minimum swaps is the sum of absolute differences between each `pi` and `m`, divided by 2 (since each swap moves a ball one position).