[ocean_moist]

I actually passed my discrete math class and final a few days ago and got the big O vs Theta vs Omega question right.

The reality is that companies often underperform their best case possible growth rate. O(n) and O(n^2) are meant to represent the best possible growth rate which may be practically be underperformed.

You may be thinking about algorithmic analysis where the term "worst case" is used for the upper bound, but here, the upper bound represents the best case. Sort of counter-intuitive but the underlying mathematical notation is properly defined.

[curiousgibbon]

It's entirely nonsensical to use O as a lower bound though. You could have two companies no growth whatsoever in value and correctly state that one has O(n) growth rate and the other has O(n^2) because a constant is both O(n) and O(n^2) (and O(n!) and O(exp(n^n)) ...). The author is trying to argue that there's some separation between two hypothetical startups' growth rates and as such an upper bound on one, say O(n), and a lower bound on the other, say Omega(n^2), is warranted. It sounds like you're not entirely an expert despite your recently-passed final. Strange concept, eh?

[ocean_moist]

I am actually the author.

You're right that mathematically, a function with constant (or no) growth is O(n)and also O(n^2), and O(anything_that_grows_faster).

My use of "O(n) startup" and "O(n^2) startup" is intended to classify the type of business based on its *inherent best-case growth potential or ceiling*.

An O(n) startup in my framework is one whose fundamental business model, market, or structure means its growth, even in its best-case scenario, is capped at roughly linear. It cannot achieve sustained super-linear growth; its upper bound is linear.

An O(n^2) startup is one whose model (e.g., strong network effects) has the potential for super-linear (which I've simplified to n^2) growth as its best-case scenario. It might be underperforming (even flat, and thus also technically O(n) in that moment), but its design allows for a fundamentally different, higher growth ceiling. The whole point is illustrate potential withholding implications or conclusions from its current growth rate, which is necessary at a companies inception.

So, yes, a flat-lining "O(n^2) type" startup would currently show growth that is O(c) (and thus also O(n)). But the point of my labels is to say that an "O(n) type" startup, by its very nature, cannot achieve the n^2 best-case that the other type can, even if both are struggling.

The labels describe the class they have, dictating their asymptotic best-case limit, not just any loose upper bound on current, possibly sub-optimal, performance. The separation I'm arguing for is based on that fundamental difference in their potential trajectory’s ceiling.

If I used Omega this would imply the actual growth rate of the startup would have to strictly be better n or n^2.

[curiousgibbon]

I see. But now you're in a weird state where you're saying (something a bit like) "at most n^2" to mean "not at most n". Which isn't particularly precise.