[contravariant]

Fail is an understatement, the ratio between the two volumes is basically (n/2)!(4/pi)^(n/2). Which is also the expected number of tries you'll need. The time doesn't merely grow exponentially it grows faster than exponential.

I don't actually know of any useful algorithms with worse asymptomatic behaviour.

[ianferrel]

Sure, but for the most-used real-world geometry, n = 2 or 3. So how fast it grows really doesn't matter.

Most maps are 2-dimensional and this is fine.

[throwaway81523]

When I saw "n" in the title I assumed they wanted something that worked reasonably for large n. Rejection is no good because of the notorious "curse of dimensionality". So my idea was to choose a suitable distribution on the radius, then draw from it and choose the angles at random (not sure what those angles are called). You might have to delete the point at the center for that to work.

[Certhas]

The article explicitly discusses both. Versions for low dimensions that have favorable properties, and those for high dimensions.

The method you propose is not used in high dimensions in practice as it would involve evaluating order n^2 trigonometric functions and is also far harder to implement than the methods discussed in the article.

[ls612]

Maybe n=4 if you have a timestamp as another dimension.

[alanbernstein]

Do your timestamps occupy a hypersphere?

[ls612]

A 1-dimensional hypersphere is a line segment so sure.

[bee_rider]

Quibble: I don’t think “fail” is an understatement, it is as strong a word as anything else for something that doesn’t work.

Actually, it is an overstatement I think, or something orthogonal; the algorithm is still correct just wildly inefficient. Fail should be reserved for algorithms that produce the wrong result, not ones that produce the right result very slowly, however slowly.

[fjkdlsjflkds]

> I don't actually know of any useful algorithms with worse asymptomatic behaviour.

Note: "asymptomatic" does not mean the same as "asymptotic"

[bee_rider]

Most of my code has bad asymptomatic bugs but they never show up in testing because the tests don’t have data sets big enough to show the symptoms!

[zeroonetwothree]

I doubt that the number of dimensions is the parameter that grows. It’s going to be constant for 99.99% of cases.