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Comment by dustingetz | Hacker News Reader

> However, I still haven’t found any studies which show what this relationship is like. Does the number of bugs grow linearly with code size? Sub-linearly? Super-linearly? My gut feeling still says “sub-linear”.

interesting, my gut says exponential, which is why cost and likelihood of project cancellation shoot up in the largest projects.

edit: i have been corrected below, i concur with quadratic.

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Are you sure you mean exponential? I think quadratic is a more reasonable guess.

He might be using "exponential" to mean "superlinear", which seems to be the sense a lot of my students try to use it in (as well as non-technical people).

Now that you mention it, I guess non-technical people do tend to use that definition. I had never thought about it before. It's unfortunate, considering there's a big difference between, say, quadratic and exponential growth. In fact, it tends to be a more impactful difference than between linear and quadratic, especially regarding algorithms.

I am not aware of the difference. Is it that exponential means something like c^n, superlinear is n^c, and linear cn?

Superlinear is anything above linear. This includes pseudolinear (e.g., O(nlog n)). Exponential is O(m^n). Linear is O(n).

Edit: And polynomial is O(n^m).

What Locke1689 said. If you're not familiar with big O notation, linear is like f(x) = 2x, quadratic is like f(x) = x^2, and exponential is like f(x) = 2^x. There's a huge difference between quadratic and exponential: exponential grows significantly faster as x increases. For computing, the difference is significant. Cobham's thesis says that polynomial algorithms (which includes quadratic) are reasonable to perform, while exponential algorithms aren't.

without straining my brain, to me its reasonable to measure complexity by number of interacting components, which is a combination, which is geometric, which is a discrete exponential, right?

The number of edges in a complete graph is n*(n - 1)/2, so by that metric it would be quadratic.

Yes but the number of nodes is n, and an "interaction" among m nodes might not be decomposable to a chain of pairwise interactions, which raises the ceiling back to exponetial (actually factorial, which is worse).

What corresponds to interactions that aren't decomposable as pairwise interactions? Race conditions? Resource utilization? Real bugs (and the nastier ones), but probably a minority. So factorial with a relatively small constant in front of it.

But obviously the real answer is to write a program that will correctly verify all other programs.

It can be quadratic. Think about drawing all possible lines between points (possible components in software):

Make a table with columns being # points and the # lines you can draw between them.

1 0 2 1 3 3 (triangle) 4 6 (box with a criss cross) 5 10 (5 point star, with everything connected) ...

This relation is n * (n - 1) / 2

Yes, combination, which is factorial, which is yet larger than exponential.

Not sure what you mean by "geometric" in this context.

The Art of Unix Programming by esr covers just this question[1]. Nowadays module/file size is one of the most important factors in how I design and write my code.

[1]: http://catb.org/~esr/writings/taoup/html/ch04s01.html

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