A newtype over Bool with a better Vector instance: 8x less memory, up to 3500x faster.

The vector package represents unboxed arrays of Bools spending 1 byte (8 bits) per boolean. This library provides a newtype wrapper Bit and a custom instance of an unboxed Vector, which packs bits densely, achieving an 8x smaller memory footprint. The performance stays mostly the same; the most significant degradation happens for random writes (up to 10% slower). On the other hand, for certain bulk bit operations Vector Bit is up to 3500x faster than Vector Bool.

• Data.Bit is faster, but writes and flips are not thread-safe. This is because naive updates are not atomic: they read the whole word from memory, then modify a bit, then write the whole word back. Concurrently modifying non-intersecting slices of the same underlying array may also lead to unexpected results, since they can share a word in memory.
• Data.Bit.ThreadSafe is slower (usually 10-20%), but writes and flips are thread-safe. Additionally, concurrently modifying non-intersecting slices of the same underlying array works as expected. However, operations that affect multiple elements are not guaranteed to be atomic.

Consider the following (very naive) implementation of the sieve of Eratosthenes. It returns a vector with True at prime indices and False at composite indices.

import Control.Monad
import Control.Monad.ST
import qualified Data.Vector.Unboxed as U
import qualified Data.Vector.Unboxed.Mutable as MU

eratosthenes :: U.Vector Bool
eratosthenes = runST $ do
  let len = 100
  sieve <- MU.replicate len True
  MU.write sieve 0 False
  MU.write sieve 1 False
  forM_ [2 .. floor (sqrt (fromIntegral len))] $ \p -> do
    isPrime <- MU.read sieve p
    when isPrime $
      forM_ [2 * p, 3 * p .. len - 1] $ \i ->
        MU.write sieve i False
  U.unsafeFreeze sieve

We can switch from Bool to Bit just by adding newtype constructors:

import Data.Bit

import Control.Monad
import Control.Monad.ST
import qualified Data.Vector.Unboxed as U
import qualified Data.Vector.Unboxed.Mutable as MU

eratosthenes :: U.Vector Bit
eratosthenes = runST $ do
  let len = 100
  sieve <- MU.replicate len (Bit True)
  MU.write sieve 0 (Bit False)
  MU.write sieve 1 (Bit False)
  forM_ [2 .. floor (sqrt (fromIntegral len))] $ \p -> do
    Bit isPrime <- MU.read sieve p
    when isPrime $
      forM_ [2 * p, 3 * p .. len - 1] $ \i ->
        MU.write sieve i (Bit False)
  U.unsafeFreeze sieve

The Bit-based implementation requires 8x less memory to store the vector. For large sizes it allows to crunch more data in RAM without swapping. For smaller arrays it helps to fit into CPU caches.

> listBits eratosthenes
[2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97]

There are several high-level helpers, digesting bits in bulk, which makes them up to 64x faster than the respective counterparts for Vector Bool. One can query the population count (popcount) of a vector (giving us the prime-counting function):

> countBits eratosthenes
25

And vice versa, query an address of the n-th set bit (which corresponds to the n-th prime number here):

> nthBitIndex (Bit True) 10 eratosthenes
Just 29

One may notice that the order of the inner traversal by i does not matter and get tempted to run it in several parallel threads. In this case it is vital to switch from Data.Bit to Data.Bit.ThreadSafe, because the former is not thread-safe with regards to writes. There is a moderate performance penalty (usually 10-20%) for using the thread-safe interface.

Bit vectors can be used as a blazingly fast representation of sets, as long as their elements are Enumeratable and sufficiently dense, leaving IntSet far behind.

For example, consider three possible representations of a set of Word16:

• As an IntSet with a readily available union function.
• As a 64k-long unboxed Vector Bool, implementing union as zipWith (||).
• As a 64k-long unboxed Vector Bit, implementing union as zipBits (.|.).

When the simd flag is enabled, according to our benchmarks (see bench folder), the union of Vector Bit evaluates magnitudes faster than the union of not-too-sparse IntSets and stunningly outperforms Vector Bool. Here are benchmarks on MacBook M2:

union
  16384
    Vector Bit:
      61.2 ns ± 3.2 ns
    Vector Bool:
      96.1 μs ± 4.5 μs, 1570.84x
    IntSet:
      2.15 μs ± 211 ns, 35.06x
  32768
    Vector Bit:
      143  ns ± 7.4 ns
    Vector Bool:
      225  μs ±  16 μs, 1578.60x
    IntSet:
      4.34 μs ± 429 ns, 30.39x
  65536
    Vector Bit:
      249  ns ±  18 ns
    Vector Bool:
      483  μs ±  28 μs, 1936.42x
    IntSet:
      8.77 μs ± 835 ns, 35.18x
  131072
    Vector Bit:
      322  ns ±  30 ns
    Vector Bool:
      988  μs ±  53 μs, 3071.83x
    IntSet:
      17.6 μs ± 1.6 μs, 54.79x
  262144
    Vector Bit:
      563  ns ±  27 ns
    Vector Bool:
      2.00 ms ± 112 μs, 3555.36x
    IntSet:
      36.8 μs ± 3.3 μs, 65.40x

Binary polynomials are polynomials with coefficients modulo 2. Their applications include coding theory and cryptography. While one can successfully implement them with the poly package, operating on UPoly Bit, this package provides even faster arithmetic routines exposed via the F2Poly data type and its instances.

> :set -XBinaryLiterals
> -- (1 + x) * (1 + x + x^2) = 1 + x^3 (mod 2)
> 0b11 * 0b111 :: F2Poly
F2Poly {unF2Poly = [1,0,0,1]}

Use fromInteger / toInteger to convert binary polynomials from Integer to F2Poly and back.

• Flag simd, enabled by default.

  Use a C SIMD implementation for the ultimate performance of zipBits, invertBits and countBits.

• bv and bv-little do not offer mutable vectors.

• array is memory-efficient for Bool, but lacks a handy Vector interface and is not thread-safe.

• Bit vectors without compromises, Haskell Love, 31.07.2020: slides, video.