Can this formula be calculated online in a single pass, with a constant number of computation variables (for example, the memory used is $ O (1) $ as opposed to $ O (j) $). I take pairs $ (x, y) $:

$$ sum_ {i} left | sum_ {j} x_ {i, j} – sum_ {j} y_ {i, j} right | $$

The absolute value can be replaced by a square if that makes things easier. In fact, any monotonous replacement for $ | x | $ work.

**My minds:**

I know you can reduce the medium term to $ sum_j x_ {i, j} -y_ {i, j} $. So that you can accumulate $ j temporary values with this summation; then $ sum_i | mathrm {temp} _i | $. The problem is, the size of $ i $ pushes very fast, so I want to see if the elimination of the temporary accumulation chart is possible. It's been a while since I fiddle with the formula to try to find something, but nothing works. Is there something in the absolute value that seems to make it insoluble? I thought maybe it had something to do with Simpson's paradox.

Another idea was to summarize some basic statistics on $ x $ and $ y $, and maybe the calculation is equivalent to a calculation on statistics. However, I do not have much luck there; perhaps another application of Simpson's paradox, I'm not sure.

Maybe the integration of Monte Carlo could be applied here?