Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

O(n) vs. O(n^2) Startups

rohan.ga

Post reply on HN
Print view View original

71–80 of 92 posts

Re: O(n) vs. O(n^2) Startups

#71

Post by nine_k »

fooker wrote: post #67

There's an interesting misunderstanding in this article. The argument for O(n) is well formed here. O(n^2) is not, the core argument is that these grow faster because of compounding. Compounding is fundamentally an exponential process, far larger asymptotically than a quadratic.

True exponential growth is possible, but I suspect is rare, because the expenses can also compound, so a polynomial growth may be an acceptable approximation. It's also important to remember that "every exponential growth curve is a sigmoid in real life" (can't remember the source of the quotation).

Re: O(n) vs. O(n^2) Startups

#72

Post by scarface_74 »

ocean_moist wrote: post #51
dvt wrote: post #46

> I think many prospective founders, if their goal is money, should optimize for O(n) businesses from day 1. Honestly, I don't think anyone "picks" the kind of business they want to run. You just kind of go with the flow. If you raise VC money, you follow their lead, if you're running a small bakery, you'll do whatever makes sense there. So while this is a fun intellectual exercise, it's an exercise in hindsight. In…

I think any prospecting founder should be able to answer the question "will it always take a fixed amount of work to get each new customer?". Generally if you have some sort of idea of what you want to do, you'll be more successful at it.

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.

YC backed companies are no exception

https://medium.com/@kazeemibrahim18/the-post-ipo-performance...

Re: O(n) vs. O(n^2) Startups

#73

Post by robocat »

fooker wrote: post #67

There's an interesting misunderstanding in this article. The argument for O(n) is well formed here. O(n^2) is not, the core argument is that these grow faster because of compounding. Compounding is fundamentally an exponential process, far larger asymptotically than a quadratic.

The article clearly isn't meant to be mathematically correct: you are being over-rigorous in your criticism. From the article:

  Businesses generally grow following a few patterns. They generally have some TAM to saturate and saturate the TAM at some rate. This rate can be vaguely linear, what I call O(n), or vaguely superlinear, what I call O(n²). The reason I borrow the asymptotic notation is because it implies the growth rate is an upper bound (best case scenario) and generalizes away specific constant factors and sums. The analogy breaks down when you force n or n² imply something numerically specific about your growth rate, or introduce functions with different growth rates like logs or exponentials. For now we will (somewhat unprincipledly) stick with two sole classes.
Article would be better using something like O(linear) and O(≫linear). The big O notation is a useful and memorable metaphor, but the n squared is really confusing. The article also doesn't use Unicode for the notation - which fucks usability (e.g. I used screenshot OCR and reedited).

Re: O(n) vs. O(n^2) Startups

#74

Post by wavemode »

thaumasiotes wrote: post #25
wavemode wrote: post #21

> An O(n) startup grows its key metric (revenue, users, etc.) roughly linearly with time—double the time, double the metric. An O(n^2) startup accelerates, with growth compounding super-linearly over time. Kind of a strange formulation to have n represent the key metric. In algorithm analysis, we would typically have n represent time (or some other cost). So we would say that the startup whose key metrics accelerate…

>> An O(n) startup grows its key metric (revenue, users, etc.) roughly linearly with time—double the time, double the metric. An O(n^2) startup accelerates, with growth compounding super-linearly over time. > Kind of a strange formulation to have n represent the key metric. In algorithm analysis, we would typically have n represent time In the quote you pulled, n is time. If n were the key metric, everything would be…

> In the quote you pulled, n is time.

It's definitely not. If their usage of O(n) has n as time, then they wouldn't say an O(n^2) startup has accelerated growth of the key metric. You'd be squaring the time, which means slowing down growth of the key metric.

When they say O(n^2) startup they clearly mean a startup which achieves n^2 results in n time. Which is the opposite of how the notation would typically be used.

> No, you don't know how the notation is used.

No, you're confidently wrong.

Re: O(n) vs. O(n^2) Startups

#75

Post by ocean_moist »

fooker wrote: post #67

There's an interesting misunderstanding in this article. The argument for O(n) is well formed here. O(n^2) is not, the core argument is that these grow faster because of compounding. Compounding is fundamentally an exponential process, far larger asymptotically than a quadratic.

Addressed in the footnotes:

> [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.

Re: O(n) vs. O(n^2) Startups

#76

Post by wavemode »

mitthrowaway2 wrote: post #40
wavemode wrote: post #17

Earlier quoted context omitted.

N is customers not employees

In the article it's apparently "time since launch". Kind of like how an O(n^2) sorting algorithm sorts n^2 elements in time n. Right?

No, it's the opposite. An O(n^2) algorithm sorts n items in n^2 time. So O(n^2) is worse than O(n).

(should really be θ rather than O but you get my point)

Re: O(n) vs. O(n^2) Startups

#77

Post by jxjnskkzxxhx »

nine_k wrote: post #71
fooker wrote: post #67

There's an interesting misunderstanding in this article. The argument for O(n) is well formed here. O(n^2) is not, the core argument is that these grow faster because of compounding. Compounding is fundamentally an exponential process, far larger asymptotically than a quadratic.

True exponential growth is possible, but I suspect is rare, because the expenses can also compound, so a polynomial growth may be an acceptable approximation. It's also important to remember that "every exponential growth curve is a sigmoid in real life" (can't remember the source of the quotation).

> True exponential growth is possible, but I suspect is rare, because the expenses can also compound

If you have an exponential (revenues) and another exponential of smaller rate (expenses; assume you have profit) then the difference is still an exponential.

Re: O(n) vs. O(n^2) Startups

#78

Post by ocean_moist »

curiousgibbon wrote: post #63
ocean_moist wrote: post #54

Earlier quoted context omitted.

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…

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)…

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.

Re: O(n) vs. O(n^2) Startups

#79

Post by godelski »

nis251413 wrote: post #64
godelski wrote: post #55

Earlier quoted context omitted.

I took the author's use of O(n) vs O(n^2) as a framing point rather than a literal model. It just seems to be missing the forest for the trees. Besides, we can approximate sigmoids with linear or quadratic functions when windowing them. Considering startup as context I think we know what part of the graph we're talking about... Do we see that exponential explosion or is the sigmoid much more flat. Replace the x in yo…

Big-O notation is about asymptotics. You have to approach something, and typically there is some infinity involved because if it is not, then you can just compute things instead of giving asymptotic approximations, or else you have 10^10*n and 0.001*n^2-10^20*n and the big-O asymptotics at infinity are useless for smaller numbers. I understand what OP tries to say but that's not really a good framing point for many r…

  > Big-O notation is about asymptotics.
Excuse me, but what made you believe I do not understand this? I'm not sure what comment you're responding to, but it sure isn't mine.

  >> I took the author's use of O(n) vs O(n^2) as a framing point rather than a literal model.
Honestly, the reason I said this is because from the article

  | The reason I borrow the asymptotic notation is because it implies the growth rate is an upper bound (best case scenario) and generalizes away specific constant factors and sums. The analogy breaks down when you force n or n^2 imply something numerically specific about your growth rate, or introduce functions with different growth rates like logs or exponentials. For now we will (somewhat unprincipledly) stick with two sole classes: O(n) and O(n^2).
Along with their emphasis on "vaguely". I can forgive the author for bad verbage. It is a personal blog where they're not trying to sell anyone on a fully fledged out idea and appear to be trying to spur conversations. Especially considering it looks like they are an undergrad. Frankly, I can understand them despite the wrong words. Given this, it would require me to operate in bad faith by rejecting the main thing they are attempting to communicate by focusing on the details that ultimately don't matter to their claim.

  > If you want to talk about a finite period of time, use a regression model, not asymptotics
They did. They said "startup". The whole time they bound the conversation to early businesses. I even directly stated this

  >> Considering startup as context I think we know what part of the graph we're talking about...
Maybe you're referring to the preceding line

  >> Besides, we can approximate sigmoids with linear or quadratic functions when windowing them.
Which again, same thing.

My point is: you're derailing the conversation

You are technically right, but you're derailing the conversation in an effort to prove your intellectual capabilities to a person who were not questioning them in the first place. You're flexing to the wrong group. You just responded to something my comment was never about.

  >  I understand what OP tries to say but that's not really a good framing point for many reasons.
So address what the OP tries to say, *and while doing so* you can add additional technical correctness. *This does not derail the conversation.* It continues the conversation and enhances it! You can do both! But as your comment stands (and bee_rider's), you just are moving the conversation away from what OP wanted to discuss and instead hyper-fixating on what they themselves said is not the best language.

Re: O(n) vs. O(n^2) Startups

#80

Post by brudgers »

Aurornis wrote: post #16
kylecordes wrote: post #13

Earlier quoted context omitted.

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 bankr…

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.

Post reply on HN