smallperm

Small library to generate permutations of a list of elements using pseudo-random permutations (PRP). Uses O(1) memory and O(1) time to generate the next element of the permutation.

>>> from smallperm import PseudoRandomPermutation
>>> list(PseudoRandomPermutation(42, 0xDEADBEEF))
[30, 11, 23, 21, 39, 9, 26, 5, 27, 38, 15, 37, 31, 35, 6, 13, 34, 10, 7, 0, 12, 22, 33, 17, 41, 29, 18, 20, 3, 40, 25, 4, 19, 24, 32, 16, 36, 14, 1, 28, 2, 8]

Motivation

In ML training, it is common to see things like

# Offline Shuffle
import numpy as np
sample_indices = np.arange(1_000_000)
np.random.shuffle(sample_indices)
for i in sample_indices:
    # do something with i
    ...

Or to do Fisher-Yates online

# Online Shuffle
import numpy as np
N = 1_000_000
sample_indices = np.arange(N)
for i in range(N):
    j = np.random.randint(i, N)
    sample_indices[i], sample_indices[j] = sample_indices[j], sample_indices[i]
    # do something with sample_indices[i]
    ...

The problem with either of these approaches is that they require O(n) memory to store the shuffled indices, and offline shuffle has a bad "time-to-first-sample" problem when we approach the scale of one billion data points. This library provides a way to generate a permutation of [0, n) using O(1) memory and O(1) time.

# Of course... first install us
pip install smallperm

import numpy as np
from smallperm import PseudoRandomPermutation as PRP
N = 1_000_000
prp = PRP(N, np.random.randint(0, np.iinfo(np.int64).max+1)) # O(1) time generates the permutation
print(prp[0], prp[50]) # We support O(1) random indexing, just like an array
assert 50 == prp.backward(prp[50]) # We support O(1) backward mapping

for ix in prp:
    # do something with ix
    ...

For most ML use cases this should be Pareto optimal: it is faster than Fisher-Yates, uses much less memory, and has a much better time-to-first-sample than offline shuffle. In other words, we used O(1) time and O(1) space to generate arr = np.arange(N); np.random.shuffle(arr), kind of magical, at the slight cost of some shuffling quality, but hey, in ML training when we constantly have > 1M data points it's not like our PRNG keys can represent the entire space of permutations anyway.

API

• Initialization: PseudoRandomPermutation(length: int, seed: int)

  • Generates a permutation of [0, length) using seed. We impose no restriction on length (except it fits under an unsigned 128-bit integer).

• Usage: Iterate over the instance to get the next element of the permutation.

  • Example: list(PseudoRandomPermutation(42, 0xDEADBEEF))

• O(1) forward/backward mapping:

  • forward(i: int) -> int: Returns the i-th element of the permutation (regardless of the current state of the iterator).
  • backward(el: int) -> int: Returns the index of el in the permutation.

Features

1. Hard-ware independent (i.e., reproducible across different machines, with the same seed) shuffling. This repo, barring major bugs, will not change the permutation generated by a given seed (in which case we will do major version bump).
2. Extremely fast. On my MBP iterating through the array is only 2x-3x slower than iterating throw a arange(N) array.

How

We use a (somewhat) weak albeit fast symmetric cipher to generate the permutation. The resulting shuffle quality is not as high as Fisher-Yates shuffle, but it is extremely efficient. Compared to Fisher-Yates, we use O(1) memory (as opposed to O(n), n the length of the shuffle); fix $\sigma$ a permutation (i.e., PseudoRandomPermutation(n, seed)) which maps ${0, 1, \ldots, n-1}$ to itself, we have $O(1)$ $\sigma(x)$ and $\sigma^{-1}(y)$, which can be very desirable properties in distributed ML training.

More examples

Shuffle non [0, n) sequences

PRP is a primitive yet powerful object, and can be composed.

from smallperm import PseudoRandomPermutation as PRP
from itertools import product

deck = list(product(range(4), range(13)))
prp = PRP(len(deck), 0xDEADBEEF)
shuffled_deck = [deck[i] for i in prp]

Replacing two random.shuffle

It is common to have two random.shuffle when there is a global shuffle and there is a "within-shard" shuffle.

from smallperm import PseudoRandomPermutation as PRP

seeds = (42, 0)
N = 1_000_000
local_indices = range(2, N, 8) # e.g., locally yield every 8th element
local_n = len(local_indices)

global_shuffle = PRP(N, seeds[0])
local_shuffle = PRP(len(local_indices), seeds[1])

for i in range(local_n):
  print(global_shuffle[local_indices[local_shuffle[i]]])
  # Compare and contrast with print(global_shuffle[local_indices[i]]),
  #   esp. with multiple epochs.

Infinite PRP with new shuffles per epoch

from itertools import chain, count
import numpy as np

N = 10_000_000
rng = np.random.default_rng(0xDEADBEEF)

uint32_upper_bound = 0x100000000

infinite_prp = chain.from_iterable(PRP(N, rng.integers(0, uint32_upper_bound)) for _ in count())

Acknowledgements

Gratefully modifies and reuses code from https://github.com/asimihsan/permutation-iterator-rs which does most of the heavy lifting. Because the heavy lifting is done in Rust, this library is very efficient.