Analytic Combinatorics – A Worked Example

I'm not a researcher, so start with the default assumption that I don't know what I'm talking about, but I had the same impression when studying those topics, asymptotic analysis is kind of broken.

For example, we assume that arbitrary arithmetic takes constant time. Good. Except that if you take that seriously, then P=PSPACE. So we might say that bit operations on fixed registers are constant time. But nobody does that because then arithmetic is O(lg n) and it's too much trouble to keep track of everything (and if you do, then hashtables aren't O(1) any longer!) If we are more sophisticated we might take a transidchotomous model, but that's really weird and we have results like sorting in less than n lg n time. Not to mention that the mathematical foundations get a bit shaky with more than one variable.

I think the hard problem underneath is that it's hard to define a computational model that (a) can be reasoned about and isn't extremely weird, (b) has solid mathematical foundations, (c) is reasonably predictive of real world performance, and (d) is not linked to a specific architecture. Asymptotic analysis is kind of a compromise. It can be formulated in very elementary terms, it makes mathematical sense and if you plot the performance of a sorting algorithm on a chart it kind of looks like the graph of n log n if you squint a bit.

Knuth does something similar to what you're suggesting IIRC, albeit without the analytic combinatorics machinery. For some programs he derives a precise expression as a function of the size of input that describes how many operations of each kind the underlying machine will perform. In numerical analysis there is a similar concept, e.g. evaluating a polynomial with Horner has the same asymptotic complexity as the naive method, but we say it's better because it takes exactly the optimal amount of floating point operations (and, in practice, because it uses fused multiply-adds, but that's another story...)

A research program along those lines would be certainly fascinating, but the combination of "needs to be reasonably predictive" and "needs to be portable" is a tough nut to crack.

Yep, I mentioned this elsewhere in the threads—Knuth does do this quite a bit in TAOCP. I think in a lot of ways we've ended up completely full circle, in that we are now counting FLOPs. This time though I think we have very little hope for something as nice and pretty as asymptotics. At least ones that are useful.

> For example, we assume that arbitrary arithmetic takes constant time.

Arithmetic on fixed-width number types is constant. It's very much not constant on arbitrary precision integers, something that underpins much of cryptography. Indeed, if you try to e.g. use Gaussian elimination (which uses O(n^3) arithmetic operations) to compute the determinant of a large integer matrix, you'll run into trouble, which is why there is another algorithm for this use case: https://en.m.wikipedia.org/wiki/Bareiss_algorithm

(Disclaimer: Also not a researcher)

My sentence there is a bit unclear -- I meant it as an implication.

Arithmetic on fixed-width registers in O(1) is equivalent to saying that bit operations are O(1), which intuitively makes the most sense, after all adding two uint64 is not the same as adding two arbitrary precision integers. But that's not always the model we pick, often we just assume that arithmetic is O(1) and sweep length considerations under the rug.

Also, what does fixed mean? Is it like the word size in an actual computer architecture, where registers are 64 bits and that's all you'll ever get, or it depends on the size of input? Because the latter is the transdichotomous model, yet another option. Unsurprisingly you can get different results with different computational models.

This is not to say that asymptotic analysis is wrong, obviously, but it's a bit of a mess and it's important to remember what computational model the analysis was performed under.

> But that's not always the model we pick, often we just assume that arithmetic is O(1) and sweep length considerations under the rug.

So, I only have rudimentary undergraduate knowledge about complexity theory, but my materials (e.g. in theoretical CS, cryptography and computational algebra) were always incredibly clear about the fact that arbitrary integer arithmetic isn't constant - this is why complexity classes such as weakly vs. strongly NP-complete exist. Do you have any examples of where people would treat arbitrary length integer arithmetic as constant? (I think most people just never have to use arbitrary precision arithmetic, so that's why they'd not talk about it explicitly.)

You're right that computational complexity depends on the model and that this makes it nontrivial to use in practice. The theoreticians would sweep this under the rug with "these models are all polynomially equivalent to Turing machines" (which lets us formulate a conjecture such as P=NP precisely), but of course that doesn't in practice help you distinguish between a constant, linear or quadratic algorithm.