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Efficient rank and select operations on large bit-vectors based on the paper “Space-Efficient, High-Performance Rank & Select Structures on Uncompressed Bit Sequences”. This library does not yet implement the full cspoppy implementation. Notably, it still uses the sub-optimal “Straw man” design for “Combined Sampling” on page 10.

This library will use support for some BMI2 CPU instructions on some x86 based CPUs if compiled with the appropriate flags on ghc-8.4.1 or later.

Rank and select

This library provides the following functions on indexed bit-vectors:

  • rank1
  • rank0
  • select1
  • select0

The supported indexed bit-vector types are:

  • Poppy512
  • CsPoppy

Constructing and using an indexed bit-vector in the repl

The example below constructs an indexed bit-vector from a string and runs rank and select query operations on it. The bits in a string are in little-endian and can be of arbitrary length. The resulting bit-vector will be padded with 0-bits until the next 64-bit boundary.

$ stack repl --package hw-rankselect --flag bits-extra:bmi2 --flag hw-rankselect-base:bmi2 --flag hw-rankselect:bmi2
λ> import Data.Maybe
λ> import HaskellWorks.Data.Bits.BitRead
λ> import HaskellWorks.Data.RankSelect.Base.Rank1
λ> import HaskellWorks.Data.RankSelect.Base.Select1
λ> import HaskellWorks.Data.RankSelect.CsPoppy
λ> let bs = fromJust $ bitRead "10010010" :: CsPoppy
bs :: CsPoppy
λ> select1 bs 1
λ> select1 bs 2
λ> select1 bs 3
λ> rank1 bs 7
λ> rank1 bs 4
λ> rank1 bs 1

A valid bit string contains zero or more characters. Characters other than 1 and 0 are permitted, but are ignored. For example spaces can be used to group bits for clarity.

λ> let bs = fromJust $ bitRead "" :: CsPoppy
bs :: CsPoppy
λ> let bs = fromJust $ bitRead "10010010 10010010" :: CsPoppy
bs :: CsPoppy

Whilst the use of a bit string is convenient for the repl, for performance reasons, it is more typical to construct an indexed bit-vector from a 64-bit word vector:

> import qualified Data.Vector.Storable as DVS
λ> let bs = makeCsPoppy (DVS.fromList [23, 73, 55])
bs :: CsPoppy

Working with files

Bit strings are stored in files as a string of bits (little-endian, which is native for Intel platforms) padded to the nearest word8 (byte) without any additional structure.

Query such a structure directly is slow, so it is possible to load it into memory by way of memory mapping then constructing an additional Rank-Select-Bit-String index.

The following code shows how to query such bit vectors and run simple queries:

λ> import Data.Maybe
λ> import HaskellWorks.Data.Bits.BitRead
λ> import qualified HaskellWorks.Data.FromForeignRegion as IO
λ> v :: CsPoppy <- IO.mmapFromForeignRegion "data/sample-000.idx"
λ> rank1 v 100
λ> select1 v 8

Here the


It is sufficient to build, test and benchmark the library as follows for basic performance. The library will be compiled to use broadword implementation of rank & select, which has reasonable performance.

stack build
stack test
stack bench

To target the BMI2 instruction set, add the bmi2 flag:

stack build --flag bits-extra:bmi2 --flag hw-rankselect-base:bmi2 --flag hw-rankselect:bmi2
stack test  --flag bits-extra:bmi2 --flag hw-rankselect-base:bmi2 --flag hw-rankselect:bmi2
stack bench --flag bits-extra:bmi2 --flag hw-rankselect-base:bmi2 --flag hw-rankselect:bmi2

Benchmark results

The following benchmark shows the kinds of performance gain that can be expected from enabling the BMI2 instruction set for CPU targets that support them. Benchmarks were run on 2.9 GHz Intel Core i7, macOS High Sierra.

With BMI2 disabled:

benchmarking data/example.ib/CsPoppy Select1
time                 3.341 μs   (3.312 μs .. 3.385 μs)
                     0.999 R²   (0.998 R² .. 1.000 R²)
mean                 3.357 μs   (3.331 μs .. 3.403 μs)
std dev              109.2 ns   (81.13 ns .. 151.9 ns)
variance introduced by outliers: 42% (moderately inflated)

With BMI2 enabled:

benchmarking data/example.ib/CsPoppy Select1
time                 1.907 μs   (1.902 μs .. 1.911 μs)
                     1.000 R²   (1.000 R² .. 1.000 R²)
mean                 1.906 μs   (1.901 μs .. 1.911 μs)
std dev              17.94 ns   (14.44 ns .. 21.87 ns)