Shufflish

Shufflish is the answer whenever you need to kind of shuffle ranges of many integers. Think Billions, Trillions, ... of integers, where you have to question whether they all fit into memory.

The key advantages of shufflish are virtually no setup time, a permutation occupies just 80 bytes, and yet it can be randomly accessed like an array. When shuffling 100M integers, it is 25 times faster than random.shuffle(), three times faster than numpy.random.shuffle(), and even ten times faster than random.randrange(), even though that obviously does not produce a permutation.

With shufflish, you can effortlessly run massively parallel tasks on large datasets with some degree of randomness by simply sharing a seed and reading different parts of the permutation.

How does this even work?

We use an affine cipher to generate different permutations of a domain. It maps an index i to (i * prime + offset) % domain, where domain is the size of the range of integers. If we select prime to be coprime with domain, then this function is bijective, i.e., for every output in the desired range, there exists exactly one input from the same range that maps to it.

This means we can directly calculate any index or slice of a permutation. It also means that the result does not have the same quality as a true shuffle, hence shuffl-ish. It will also only ever generate a small fraction of all possible permutations. And while the generated permutations look random at first glance, they do not fool proper randomness tests like PractRand. As a workaround, we added the local_shuffle() function, which reads small chunks from some iterable and performs a true shuffle on them. This mostly fools PractRand for chunk sizes as low as 16k.

Basic usage

To obtain a permutation for some domain, simply call the permutation() function. It determines suitable parameters and returns an AffineCipher instance. The most important parameters are the domain that sets the range of integers, and an optional seed value. If no seed is provided, a random value is chosen instead. Based on the seed, num_primes (default 3) values are chosen from a list of primes (default are the 100 largest primes less than 2^64).

The returned object is iterable, sliceable, and indexable, exactly like a list or array. For example, with domain=10 and seed=42:

from shufflish import permutation
p = permutation(10, 42)

for i in p:
    print(i)

print(list(p))
print(list(p[3:8]))
print(p[3])

Also note the strategic use of list. Where multiple values can be returned, iterators are used to conserve memory.

Advanced usage

Affine ciphers are invertible (they would be bad ciphers if they were not). You can use AffineCipher.invert to obtain the inverse chipher.

from shufflish import permutation
p = permutation(10, 42)
ip = p.invert()

for i in range(10):
    assert ip[p[i]] == i

Note that, while you can invert a slice of a chipher, this effectively swaps the meaning of index and value, i.e., if p[x]=v then ip[v]=x. Since slice start/stop/step lose their meaning after inversion, AffineCipher.invert ignores them and thus p[:10].invert() produces the same result as p.invert().

The extended Euclidean algorithm is used to obtain the multiplicative inverse, which has a complexity of O(log(N)). In practice this takes anywhere from 4 to 10 times as long as getting one value from the cipher when first called. The inverse is then cached inside the AffineCipher instance, so subsequent calls will be very fast. Even the first call is still considerably faster than random.randrange(), so it is probably not worth worrying about.

Creating many permutations

One performance caveat is that the permutation() function needs to determine the correct coprime value for the seed. By default, it uses a combination of num_primes=3 primes and skips repetitions mod domain. As you can imagine, this can take a little while. If you need to create many permutations for the same domain, consider using the Permutations class instead. It computes and stores all valid coprimes upon initialization, which makes getting permutations effectively instantaneous. Note that the coprimes array can use up to 1.3 MiB of memory with the default settings, though it will be shared between instances with identical parameters.

Once you have your instance, using it is straightforward:

from shufflish import Permutations
perms = Permutations(10)
p = perms.get(seed=42)

for i in p:
    print(i)

print(list(p))
print(list(p[3:8]))
print(p[3])

Alternatively, you can set allow_repetition=True to skip detection of repetitions. The permutation() function can then determine the correct combination of primes much faster (using combinatorial unraking), with the caveat that there is now a small chance that permutations are repeated early. Empirically, we find that repetitions occur at the earliest after domain seeds.

Project status

Shufflish is currently in alpha. You can expect permutations to be correct and complete, but updates may change which permutation is generated for a given set of parameters. For instance, the algorithm that determines the affine cipher's parameters based on the seed may change, e.g., to reduce collisions. Though unlikely, the API may also change if it proves annoying to use. Once the project reaches a stable state, we will guarantee API stability and that a set of parameters always produces the same permutation.

Acknowledgements

Shufflish is supported by the Albatross and SustainML projects.