Biggest issue with this language. But... fairly trivial to implement codegen with gleam/glance[0]. No good libraries do this well right now (e.g. support for discriminated unions).
> [2] Perhaps choosing a better two functions could more closely explain the growth dynamics of network effects, which could be more exponential. I think the analogy diminishes in value if you try to directly numerically match it to some growth metric.
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.
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?
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.
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.
You learn about it in real analysis, but it's worth noting that in analysis you pretty much always use little-O, which is the one that makes the guarantee you need in that context.
Most founders - especially vc backed founders - only care about whether the optics look good enough for an acquisition or an IPO. They could care less if it fails after that.
Experience != talent. Perhaps a better way to phrase it is that they hire the minimum needing to succeed in a well defined role. Startups aren't afforded this comfort as the roles are not well defined.
I use business, startup, and company interchangeably. Generally I tried to use business for O(n), startup for O(n^2), but I guess I wasn't strict with my usage...
Perhaps they are harder to start, but they are also vastly more likely to succeed to their O(n^2), and this is not only due to the increased barrier to entry.
That's what I mean when I say, founders are more likely to succeed at O(n) companies.
Unfortunately, many businesses are arguably a bit worse than that: success this year means you put even more at risk next year.
(For example, you get successful enough that you need a bit of office space. Well, your little business is not going to persuade anyone with nice office space to lease to it alone... Landlords will instead demand the owners personally guarantee the lease, i.e. commit to paying it or go bankrupt trying, even if the business shrinks and no longer needs or can afford the space. So you thought, great, we had a couple years of strong growth. Now you get to commit personally to five years of a huge expense. Congratulations.)
Where are you seeing commercial office space require 5 year personal guarantees from the founders? Terms like that would have everyone laughing as they hung up the phone around here.
In the US, that sounds rather normal for normal triple-net commercial space for a normal new business run by people without established commercial relationships.
Typically, new businesses don’t get very much credit on favorable terms without established commercial relationships.
Dealing with failing or failed businesses is just not worth the hassle for most established businesses working in the established business market segment.
Silicon Valley is probably different because the business relationships are different. And of course month to month and short lease real estate are an entirely different market segment than triple-net.
Unless you want working plumbing and leaky roofs taken care of.
A hungry landlord is a landlord who is in dire financial straights.
Good landlords can afford to wait because they are using real-estate to park wealth, not as a means of putting food on the table. Think insurance company reserves and House of Windsor. It is a long play.
[0] https://hexdocs.pm/glance/glance.html