s/Omega/Theta/, my bad.

It seems that you are refusing to acknowledge some basic reality checks in favor of theoretical purity.

N does not actually get to go to infinity, and neither does your budget of time, money, ram, disk space, network bandwidth, etc.

If you can't imagine when one network request and NlogN cpu cycles (EG: query a central server then figure out the sources from the data patterns) could be better than N network requests and N cpu cycles (EG: asking all the sources directly), then I am truly sorry for you.

Addendum: Consider also that you may want to not just think about, but actually implement both an O(NlogN) algorithm and an O(N²) for the same task. Then choose which one to run with an if (N > $breakEvenPoint) { $result = doNlogN() } else { $result = doNsquared() }