Probably looking for this: https://en.m.wikipedia.org/wiki/Fisher%E2%80%93Yates_shuffle

This seems to use an external random number generator.

I want the code to be complete and reproducible. So gimmeTheNumberAtPosition(x) will always return the same for the same x.

Also, FYShuffle is much more complex than I would like the algo to be.

Fisher-Yates is one of the simplest things out there IMO.

Isn't the Lehmer code the thing that you want? Then you can specify your permutation with an integer and extract the individual digits from it repeatedly.

    Fisher-Yates is one of the simplest
    things out there IMO

Just using "$n xor $something" would be simpler. I am just not sure yet, what a good "$something" is.

How does this guarantee the desired property of the output being a permutation?

If you seed your random number generator, then it is reproducible.

Are you saying you want a reproducible random number generator?

The equivalence classes modulo some n form a partition of the integers so you can use that to create a very efficient solution with very little code.

Here is a neat explanation:

https://preshing.com/20121224/how-to-generate-a-sequence-of-...

If you need even better properties (eg cryptographically secure) you can also look at PCG with k-dimensional equidistribution:

http://www.pcg-random.org/index.html

If size is a prime, repeated multiplication on a generator element (any element) will work if you compute `mod p`. But it's not going to have nice random properties for the most part. It's just a "permutation".

The reason it works is because the subgroup order generated by the element has to be divisible in the group order. There's only 1 and p. So unless it's trivial (i.e., the element 1), it's going to be p and thus it generates the whole group in p steps.

From memory you don't need size to be a prime, what you need is that it is coprime with your step.

Thus, the easiest solution is to take a step-size that is prime and different from you size (that works for any size). Given the index of the current element, the next index would be `(current_index + step) % size`.

Have you looked up LFSRs yet? https://en.wikipedia.org/wiki/Linear-feedback_shift_register

They are quite efficient for both HW and SW implementations.

If you’re using Go, you can use rand.Perm (https://golang.org/pkg/math/rand/#Perm), which uses a Fisher-Yates shuffle under the hood.

Are you talking about generating a random permutation?

I don't think so. Maybe a "random sequence" could be a name for it. But not sure.

What I mean is, in your example, do you need to have precisely the numbers 1, 2, 3, 4, and 5, but in any order? Or is the constraint something else?

Ah yes, you are right. In that sense, it is a random permutation of the numbers from 1 to max.

Yes, when initialized with "size=5" the numbers in the output must be precisely 1,2,3,4,5 but in random looking order.

And I don't want to store the whole sequence. I want to say gimmeTheNumberAtPosition(x) and it return just that number.

Generating random permutations is quite a different problem from creating random numbers. The standard algo for random permutations is the Fisher-Yates Shuffle.

https://en.wikipedia.org/wiki/Fisher–Yates_shuffle

ETA: As you don't want to store the permutation, you might want to pick a number randomly from 1 to n!, and then generate the permutation on the fly up to the desired element, using the techniques outlined here:

https://stackoverflow.com/questions/29236556/how-can-i-calcu...

Storing a number of the order of magnitude of n! will take the same amount of memory than storing a permutation of n elements, so what's the point ?

If one really wants to do it the slow and hard way, it's possible to compute a random permutation of n elements one element at a time with n bits of storage (less with fairly generic compression techniques): just remember the set of numbers that have been produced so far and sequentially pick one of the remaining ones. You'll waste some combination of additional memory (for arithmetic decoding state) and random bits (for rejected samples), but materializing a large randomly permuted vector should be slower.

Good point. Storing a number of magnitude n! will take n log n bits, and storing n numbers up to n will also take n log n bit.

In other words, good old Fisher Yates, storing the resulting permutation, and a simple lookup is probably the way to go.

I guess you could predictably enumerate the permutations, so that you could lookup with parameters size/n, position/i, seed/r - where r is the permutation.

For size=1, all parameters are 1, and only result is 1.

For size 2, r can be 1 or 2, naming the permutations [1,2];[2,1] - and eg: rand_at(n=2,r=2,i=2) would return 2, but i=1, would return 2.

I'm not sure how I'd implement this - I suppose it might be possible to generate a predictable "walk" based on n/r/i?

But for sizes less than, say, a million i would think that a shuffled array would be easier?

r would need to be in the range 1..n!

You can construct arbitrarily-sized permutations with log(n) time random access using Sometimes-Recurse Shuffle: https://eprint.iacr.org/2013/560.pdf.

What you want is a cipher from 1..N to 1..N. This problem is often approached as an encryption problem known as Format Preserving Encryption. The Sometimes Recurse Shuffle linked elsewhere in this thread is one solution but the most foundational and accessible paper on this is Black and Rogaway's "Ciphers with Arbitrary Domains" The Generalized Feistel Network (Method 3) is fast and easy to implement.

https://web.cs.ucdavis.edu/~rogaway/papers/subset.pdf

So you want a mapping of [1, n] to a permutation of [1, n] without having to generate the list [1, n] and shuffling it. An affine transformation modular n should work. Instead of [1, n], look at [0, n). Find a value `a` in [0, n) such that gcd(a, n) = 1 and pick a random integer b in [0, n). Then the `random` number at each position is `a * x + b mod n`.

This is simple and fast, but is not secure at all. You can solve for a, b by solving the linear congruence. It also does not generate every permutation of `n`. For n = 5, only 20 sequences can be found out of 5! = 120.

The value `a` has to fulfill some more properties: https://en.wikipedia.org/wiki/Linear_congruential_generator#...

Since the "randomness" of the permutation is not that great, I was looking for something better, but could not find anything. The closest I got was https://en.wikipedia.org/wiki/Xorshift#xoshiro_and_xoroshiro which only works for powers of two. A workaround would be to choose the next larger power of two and reject all random values which are smaller than `n`, but that introduces unpredictable latency and destroys the cool jump-ahead feature.

The LCG is the improved extension which causes you to have to compute values x_0 = seed to x_n in order to calculate x_{n+1}. What I am talking about is just a mapping of index x -> f(x) which is what the OP seemed to have wanted. In that case, `a` only needs to be relatively prime. This cannot account for every permutation of n since the number of values relatively prime to n, the totient, has an upper bound of n, which is the possible values for a. The number of values for b is also n, so at most n^2 possible sequences are generated, which is less than n! for n > 3.

For example, with n = 5: Let a = 3 and b = 2. x = [0, 1, 2, 3, 4], a * x = [0, 3, 6, 9, 12], a * x + b = [2, 5, 8, 11, 14], a * x + b mod n = [2, 0, 3, 1, 4]

If your value of size is a power of two you could just encrypt x with a block cipher where the block size is equal to size and the key is your seed.

for JS you can probably just do `[...Array(n).keys()].sort(()=>Math.random() - 0.5)` ?

Having an unstable comparator might wreak havoc and maybe not give a proper distributed space of possible results?

Edit: yes, here is an example (in the comments) of how biased this is: https://stackoverflow.com/a/18650169/923847

That is exactly the algorithm that Microsoft used to bias its purportedly random “browser choice” screen. Don’t use it.

https://www.robweir.com/blog/2010/02/microsoft-random-browse...

The Fisher–Yates shuffle is the right way to shuffle an array in an unbiased way.

https://en.wikipedia.org/wiki/Fisher%E2%80%93Yates_shuffle

Yes, xor can be used as part of this — see the paper[1] and code[2]. The idea is that you're generating a random shuffle of a sequence, but you can do it in a way that doesn't require storing the entire shuffled sequence. You can ask "what's in position 4" and it will return "item 1".

[1]: http://pcgworkshop.com/archive/mawhorter2019anarchy.pdf [2]: https://github.com/solsword/anarchy

Xoring won’t work:

- xoring will only produce permutations that have period two, and every element will be part of a 2-cycle.

- if your n isn’t a power of two, it may produce numbers larger than n

The set of 2-cycles will have a lot of structure, too (for example, if your magic constant is even, all cycles will have either two even numbers or two odd numbers; if it is odd, all cycles will have an even and an odd number), but the above should already be bad enough to drop that idea.

    if your n isn’t a power of two,
    it may produce numbers larger
    than n

This is the biggest problem.

The others are not such big problems as I don't need strong resemblence to real randomness.

I can’t look into your use case, but I’m not sure you realize how spectacularly bad xoring with a fixed value is as a way to generate a ‘random’ permutation.

For example, it will still alternate odd and even numbers.

Also, an adversary can derive the xor key from a single sample, and predict the entire sequence from it.

Alterning odd and even numbers is fine.

There is no adversary. The use case is not about security.

If you want a simple pseudo random sequence of N that just looks random (pseudocode):

  int start = getRandomInt(seed) % N;
  double k = N / 1.618;
  k = k % 2 == 0 ? k - 1 : k; // k and N must be coprime
  int nextElement = k * start % N;
  start++;
  

This will jump wildly across your sequence and visit all elements (as k and N are chosen coprime). No need to store a full array. Is it statistically random? Of course not, e.g. you will never see two values close to each other right after another.

Your algorithm fails to choose k and N coprime for about 19% of N, such as N = 6, 10, 14, 15, 25, 27, 35, 36, 45, 54, 55, 65, 66, 75, 84, 85, 90, 93, 95 ….

Look into linear feedback shift registers, they are very simple and can produce random-looking permutations.