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

rohan.ga

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

#51

Post by ocean_moist »

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.

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

#52

Post by sfpotter »

brap wrote: post #44
sshine wrote: post #12

Since O(n^2) is used as a proxy for “something superlinear, but don’t get hung up on how much”, you might also choose O(n^(1+ε)), an upper bound characterised by some arbitrarily superlinear function.

Why not O(nlog*(n)) startups

Not that it matters, but O(n log n) is often referred to as "quasilinear", but O(n^(1+eps)) is regarded as "superlinear" (and in fact grows faster than O(n log n) for any eps > 0).

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

#53

Post by ocean_moist »

danjl wrote: post #26
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…

Normally, with big-O notation, the goal is to reduce complexity. The author's wording kinda reverses that assumption only to "surprise" you in the end? A somewhat forced irony.

Only in algorithmic analysis. Big-O generally is used to describe and classify any arbitrary function.

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

#54

Post by ocean_moist »

curiousgibbon wrote: post #27

This is one of the myriad situations where Omega should have been be used, not O. What are they teaching in schools these days?

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.

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

#55

Post by godelski »

bee_rider wrote: post #11

Huh. Not working in the field, I assumed startups went like sigmoids (everything is a sigmoid after all). Exponential at first as word of mouth spreads, then linear as your users start bumping into each other and word of mouth stops working, and then you eventually start leveling off near carrying capacity (you’ve hit your addressable market). I thought the game was to try to get bought by some massive company while…

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 your sigmoid with (ax) and is a =1?

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

#56

Post by sshine »

brap wrote: post #44
sshine wrote: post #12

Since O(n^2) is used as a proxy for “something superlinear, but don’t get hung up on how much”, you might also choose O(n^(1+ε)), an upper bound characterised by some arbitrarily superlinear function.

Why not O(nlog*(n)) startups

Because, as the article states,

> The analogy [between asymptotic growth and company economic growth] breaks down when you force or 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). 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.

So the article specifically tries to be unspecific about what superlinearity we're talking about, and also calls it vaguely superlinear. Since O(n^(1+ε)) is arbitrarily superlinear (O(n^1) = O(n), and ε is some arbitrary small amount, making it superlinear by definition, and practically nothing else), it is a good choice when that is all you wish to say.

If you went with O(n log n), you'd get the same questions as with O(n^2): Why not O(...something else...): That's not the point! :-D

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

#57

Post by danjl »

bee_rider wrote: post #11

Huh. Not working in the field, I assumed startups went like sigmoids (everything is a sigmoid after all). Exponential at first as word of mouth spreads, then linear as your users start bumping into each other and word of mouth stops working, and then you eventually start leveling off near carrying capacity (you’ve hit your addressable market). I thought the game was to try to get bought by some massive company while…

If only it was exponential in the beginning, as word-of-mouth spreads. Sigh. The reality is that you need to claw and scrape your way to your first customers. The numbers vary depending on whether your product is b2b, consumer, or more niche, but the first customers are the hardest. You rarely get word-of-mouth in the beginning. Instead, it comes much later, typically after a long period of slow growth as you learn more about your customer's workflows and problems and adjust the product to get closer to PMF.

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

#58

Post by nis251413 »

thaumasiotes wrote: post #28
withinboredom wrote: post #15

Earlier quoted context omitted.

This is actually a better model and one that more closely reflects reality. You can see it on revenue as well, since even if growth is exponential, churn is a percentage of your total paying users. Thus, it produces a sigmoid curve unless you can get churn to 0% (pro-tip: you can’t). But, these are the two basic levers for a SaaS: growth and churn.

> You can see it on revenue as well, since even if growth is exponential, churn is a percentage of your total paying users. Thus, it produces a sigmoid curve unless you can get churn to 0% (pro-tip: you can’t). Exponential growth means that additional users are a percentage of your total users. It is trivial to see that adding a source of exponential decay will give you another exponential function. All churn (as you…

Sigmoid (or the logistic function specifically rather) is exponential until you get close to the "turning point" (or rather, its growth bounded from below by an exponential). It's as you approach that point that it becomes linear, and after that its growth decays.

However you are sort of right that "churn" does not necessarily have to do with it being sigmoid because it will be anyway. It may be bring it earlier if the churn rate surpasses the user growth, but that's probably not important here.

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

#59

Post by alangibson »

mgraczyk wrote: post #50
bee_rider wrote: post #11

Huh. Not working in the field, I assumed startups went like sigmoids (everything is a sigmoid after all). Exponential at first as word of mouth spreads, then linear as your users start bumping into each other and word of mouth stops working, and then you eventually start leveling off near carrying capacity (you’ve hit your addressable market). I thought the game was to try to get bought by some massive company while…

They are sigmoids, but for some the plateau is 30 years in the future with a 5T market cap. For example Facebook's revenue is still increasing at an increasing rate, 21 years later

This is an incredibly important thing to understand. Buffet himself said it's better to be an average business in a great market that to be a great business in an average market.

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

#60

Post by Snawoot »

xyst wrote: post #41

What ever happened to providing a good service? Why does everything have to be a "unicorn"? You greedy capitalists, and billionaires have sucked out the life of everything — tech, food, airlines. VC culture, private equity, and "hyper-growth" mentality has screwed over many good companies that once provided good services to the community. Good paying jobs with excellent benefits and providing upward mobility. Now the…

> What ever happened to providing a good service?

I'm getting an impression it's just not profitable enough. For many years I get a feeling that business is considered sound only if it is superprofitable (not exactly the right term, but still) in order to cover all losses.

Probably it's because of market competition required to be at least noticed. Some companies' spendings for marketing are greater than for R&D, production and operations combined. Maybe we got ourselves into a situation where everywhere competing for low-hanging fruits or trying to make customer believe it's the service they need while all of it doesn't really overlap with real society needs.

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