hw-rankselect-base

Rank and select operations.

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 various types:

• rank1
• rank0
• select1
• select0

Type class instances are provided for the following primitive types:

• Bool
• Word8
• Word16
• Word32
• Word64

Moreover additional type class instances are provided for [], Vector from both Data.Vector, and Data.Vector.Storable of these primitive types.

Examples

Check the convenience imports in the project’s .ghci file.

Run the repl in convenience script (uses stack).

$ ./run-stack.sh repl

Then create a rank-select bit-string of the desired type:

λ> let bs = fromJust $ bitRead "0001001001100001000001000110101000101000" :: Word64
"00010010 01100001 00000100 01101010 00101000 00000000 00000000 00000000"

Call the rank-select operations on the bit-string

λ> rank1 bs 20
1
λ> select1 bs 4
11

Vector indexing conventions

This library follows standard 1-based counting conventions typically found in Computer Science literature where select1 10 2 = 4 as illustrated here:

  8 7 6 5  [4]3 2 1
  0 0 0 0   1 0 1 0

The standard convention for the bmi2 implementation, comes at a small cost.

An internal function select1Word64Bmi2Base0 demonstrates 0-based counting that is slightly faster when implemented with the bmi2 instruction set where select1 10 1 = 3 as illustrated here:

  7 6 5 4  [3]2 1 0
  0 0 0 0   1 0 1 0

Performance notes

The word-vector-based type classes instances are not intended to be used in high-performance code because where random-access on large bit-vectors are needed because they have poor performance due to having to do a linear scan.

For smaller bit-vectors that fit on one page of memory, they do quite well. In fact, the hw-dsv library uses them for small vectors.

Bit-vectors larger than say 4096-bits need indexing to achieve reasonable random-access performance.

An indexed bit-vector implementation can found in the hw-rankselect package.

Architecture notes

This library has only been tested on little-endian CPU architectures.

Anyone wishing to use this on big-endian CPU architectures will need to confirm that this works properly.

Compilation

It is sufficient to build, test and benchmark the library as follows for emulated behaviour:

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

benchmarking 64-bit/Once: Select1 Broadword
time                 14.75 ns   (14.63 ns .. 14.90 ns)
                     0.996 R²   (0.987 R² .. 0.999 R²)
mean                 15.35 ns   (14.92 ns .. 16.70 ns)
std dev              2.355 ns   (607.2 ps .. 4.849 ns)
variance introduced by outliers: 96% (severely inflated)

benchmarking 64-bit/Once: Select1 Bmi2
time                 6.026 ns   (5.933 ns .. 6.134 ns)
                     0.999 R²   (0.998 R² .. 0.999 R²)
mean                 6.024 ns   (5.966 ns .. 6.096 ns)
std dev              224.4 ps   (176.9 ps .. 318.6 ps)
variance introduced by outliers: 62% (severely inflated)

benchmarking 32-bit/Once: Select1 Broadword
time                 26.09 ns   (25.84 ns .. 26.40 ns)
                     0.999 R²   (0.998 R² .. 0.999 R²)
mean                 26.67 ns   (26.37 ns .. 27.01 ns)
std dev              1.017 ns   (848.4 ps .. 1.291 ns)
variance introduced by outliers: 61% (severely inflated)

benchmarking 32-bit/Once: Select1 Bmi2
time                 8.613 ns   (8.543 ns .. 8.687 ns)
                     0.999 R²   (0.999 R² .. 1.000 R²)
mean                 8.592 ns   (8.515 ns .. 8.671 ns)
std dev              248.3 ps   (216.2 ps .. 294.8 ps)
variance introduced by outliers: 48% (moderately inflated)