Alias sampling is a really beautiful algorithm, but implementing it correctly and efficiently isn't as easy as you seem to imply.

I would also very much like to know what a generalization of this algorithm to continuous distribution would look like.

You make a list of below-average bars and a list of above-average bars. Iteratively pop the top of both, donate from the larger to the smaller until the smaller is average, then put the larger back into the appropriate list based on its new value. Stop when there's no too-large bars left. Compared to writing a red black tree it's trivial.

A correct implementation wrt probabilities being exact given a frequency table (the "average" can be a very large rational number), as well as efficient when constructing the data structure to be sampled from during sampling phase is IMO not as trivial as you seem to imply.

I have asked people to implement this algorithm during engineering interviews on a regular basis for the better part of five years, and I can promise you, strictly none of the candidate who were subjected to it would describe their experience as "I was asked a trivial question".

But then everyone has their own bar for the word "trivial". I know for example that I have my own bar for the word "arrogance".

Oh wow, how do you prompt the interviewees? Do you tell them about the average value threshold or the guarantee you only need 2N buckets? If not, I’d guess this is probably way too hard of an interview question, even though a decent implementation really isn’t that difficult. Is this like a bonus difficult question for someone who aces all your other questions?

The implementation isn’t hard once you’ve read the Vose paper or the Keith Schwarz link I posted. There’s only one tricky part about handling rounding that is pretty easy to fix.

I would never expect anyone who hasn’t heard of the idea to be able to implement it on the spot without being more or less given a recipe for the math and how to use it. If you ever find someone who can, hire them immediately!

This is one of those problems that only feels somewhat easy in retrospect, but almost nobody could derive any of it on their own. The alias method is extremely clever, and not at all trivial.

> Oh wow, how do you prompt the interviewees?

Back when I was still doing this, I used to start the interview like this:

        - given: a non-uniform, finite, frequency table with actual observations of some real-world phenomenon (and if that's too generic, just tell the candidate you've thrown a dice a million time an recorded the outcomes).
        - given: a uniform pseudo random number generator
        - please design a pseudo random number generator that will, if sampled for long enough, produce the same distribution (non generic version: RNG that statistically behave like the dice).

Unfortunately, a non-negligible proportion of candidates fail to produce any solution at all :(

Decent candidates quickly produce the standard: "build the CDF once and then at sample time, draw a uniform number in the correct range and walk the CDF until we're above that random number, return the bucket where that occurred"

I then ask for time and space complexity for the sampling phase.

Slightly better than average candidate notice that the CDF is naturally sorted and can be binary searched.

I then ask those if they can improve what we have to a O(1) solution (independent of the size of the histogram).

The top 20% candidates usually manage to put their intuition at work: from the fact that O(1) almost always implies a simple table lookup, they come up with the: "build a very large array, where the index of each bucket B of the histogram is replicated histo[B] times and then sample uniformly from it".

We then discuss when this algorithm is/isn't practical.

The candidates that get to this point within ~15mn can then be gently walked towards redesigning the alias method using some hints.

The starting hint goes like this: what if you were to completely ignore the fact that the histogram is not uniform, and just stubbornly sample from it, can you quantify how much of a mistake you would make ?

The second hint is: for some buckets, picking them will be an overshoot, and for some we will undershoot. Explain when that is.

The third hint: say we picked a bucket that's chosen too often, can we decide to only keep it some of the time (as in: with some carefully chosen probability)

The fourth hint: and if we decide not keep it, why waste the dice throw, why not pick another, carefully chosen bucket

etc ...

The nice property of this question is: it's very gradual, starting with a problem most people with a CS degree should be able to solve, then going to a solution that's less standard and requires more than base regurgitation of CS lessons, to finally get to what I consider to be (at the risk of repeating myself) the non-trivial state of the art solution to the problem.

The point where the candidate start to struggle gives you a nice reading on their strength.

And there's the rare few new grads that would vaguely recall having seen the alias method and could rebuild it cleanly on the white board from scratch and without help. These rare occurrences were very enjoyable.

I agree that the discretizing step is hard to get right. I agree that you don't have enough time in an engineering interview to get Vose's algorithm right (nevermind think of it in the first place). I was definitely imagining that you had a day to implement it and test it, not an hour and a whiteboard.

The closest continuous distribution analogy would be inverse transform sampling - analytically determine the CDF and invert it. Then you have an O(1) function to map a uniform sample into your continuous PDF. Unfortunately, it’s not always possible, lots of continuous functions can’t be integrated & inverted.

There's actually an interesting experiment to run, which is as follows:

       - take some non-uniform a continuous distribution with finite support.
       - discretize it into a N-bucket histogram
       - apply the alias algorithm
       - display the result
       - increase N
       - rinse and repeat

The resulting animation is likely going to be interesting.

I suspect it won't be stable at all and might veer into the fractal at some point.

Should be easy to try, maybe I’ll do it if I understand correctly what you’re thinking about...

It sounds like you’re thinking of a feedback loop, where I pick, say 1000 uniform samples, then run them through the alias map, and then run those results back through the alias map, etc., while each time increasing the number of buckets for the alias map?

I would assume the output distribution range is either [0,1] or we re-normalize the samples every time through the alias map?

Is that what you’re thinking of, or are you suggesting to visualize the map directly, and not a bunch of samples?

For a feedback loop on samples, I guess I might expect the samples to eventually converge with all samples at the function’s maximum. But if you mean to track and visualize all the little intervals in the alias map through itself several times, yeah, that might be a bit wild to see.