I think I have an understanding of why this works, but I'm not completely sure: is that the ratio in each column (in the Alias table) is proportional to the actual size of the original column, so if you were to sum across all alias table columns for x_i, you would have the same proportion as the original column x_i?
Why it's always 2 identities? Couldn't there be more?
Is it guaranteed that you will only need two things per column? What if we had 1 column which was much less than the mean and the rest were only 1 unit above it?