Fast Pseudo Random Number Generators for R

Ralf Stubner

2024-04-23

Source: vignettes/dqrng.Rmd

The dqrng package provides fast random number generators (RNG) with good statistical properties for usage with R. It combines these RNGs with fast distribution functions to sample from uniform, normal or exponential distributions. Both the RNGs and the distribution functions are distributed as C++ header-only library.

Supported Random Number Generators

Support for the following 64bit RNGs are currently included:

  • pcg64 The default 64 bit variant from the PCG family developed by Melissa O’Neill. See https://www.pcg-random.org/ for more details.
  • Xoroshiro128++ and Xoshiro256++
    RNGs developed by David Blackman and Sebastiano Vigna. See https://xoroshiro.di.unimi.it/ for more details. The older generators Xoroshiro128+ and Xoshiro256+ should be used only for backwards compatibility.
  • Threefry
    The 64 bit version of the 20 rounds Threefry engine (Salmon et al., 2011) as provided by the package ‘sitmo’.

Of these RNGs Xoroshiro128++ is used as default since it is fast, small and has good statistical properties.

Usage from R

Using these RNGs from R is deliberately similar to using R’s build-in RNGs:

Let’s look at the classical example of calculating \(\pi\) via simulation. The basic idea is to generate a large number of random points within the unit square. An approximation for \(\pi\) can then be calculated from the ratio of points within the unit circle to the total number of points. A vectorized implementation in R where we can switch the RNG might look like this:

N <- 1e7
piR <- function(n, rng = stats::runif) {
    x <- rng(n)
    y <- rng(n)
    4 * sum(sqrt(x^2 + y^2) < 1.0) / n
}
set.seed(42)
system.time(cat("pi ~= ", piR(N), "\n"))
#> pi ~=  3.140899
#>    user  system elapsed 
#>   0.385   0.020   0.404

Using dqrng is about three times faster:

library(dqrng)
dqset.seed(42)
system.time(cat("pi ~= ", piR(N, rng = dqrng::dqrunif), "\n"))
#> pi ~=  3.141457
#>    user  system elapsed 
#>   0.166   0.036   0.202

Since the calculations add a constant off-set, the speed-up for the RNGs alone has to be even greater:

system.time(stats::runif(N))
#>    user  system elapsed 
#>   0.083   0.008   0.091
system.time(dqrng::dqrunif(N))
#>    user  system elapsed 
#>   0.021   0.007   0.028

Similar for the exponential distribution:

system.time(stats::rexp(N))
#>    user  system elapsed 
#>   0.340   0.004   0.343
system.time(dqrng::dqrexp(N))
#>    user  system elapsed 
#>   0.038   0.008   0.046

And for the normal distribution:

system.time(stats::rnorm(N))
#>    user  system elapsed 
#>   0.354   0.012   0.367
system.time(dqrng::dqrnorm(N))
#>    user  system elapsed 
#>   0.059   0.012   0.071

As well as for sampling with and without replacement:

system.time(for (i in 1:100)   sample.int(N, N/100, replace = TRUE))
#>    user  system elapsed 
#>   0.556   0.020   0.576
system.time(for (i in 1:100) dqsample.int(N, N/100, replace = TRUE))
#>    user  system elapsed 
#>   0.030   0.008   0.039
system.time(for (i in 1:100)   sample.int(N, N/100))
#>    user  system elapsed 
#>   1.549   0.268   1.816
system.time(for (i in 1:100) dqsample.int(N, N/100))
#>    user  system elapsed 
#>   0.050   0.012   0.062

It is also possible to register the supplied generators as user-supplied RNGs. This way set.seed() and dqset.seed() influence both (dq)runif and (dq)rnorm in the same way. This is also true for other r<dist> functions, but note that rexp and dqrexp still give different results:

register_methods()
set.seed(4711);   stats::runif(5)
#> [1] 0.3143534 0.7835753 0.1443660 0.1109871 0.6433407
set.seed(4711);   dqrng::dqrunif(5)
#> [1] 0.3143534 0.7835753 0.1443660 0.1109871 0.6433407
dqset.seed(4711); stats::rnorm(5)
#> [1] -0.3618122  0.8199887 -0.4075635  0.2073972 -0.8038326
dqset.seed(4711); dqrng::dqrnorm(5)
#> [1] -0.3618122  0.8199887 -0.4075635  0.2073972 -0.8038326
set.seed(4711);   stats::rt(5, 10)
#> [1] -0.3196113 -0.4095873 -1.2928241  0.2399470 -0.1068945
dqset.seed(4711); stats::rt(5, 10)
#> [1] -0.3196113 -0.4095873 -1.2928241  0.2399470 -0.1068945
set.seed(4711);   stats::rexp(5, 10)
#> [1] 0.0950560698 0.0567150561 0.1541222748 0.2512966671 0.0002175758
set.seed(4711);   dqrng::dqrexp(5, 10)
#> [1] 0.03254731 0.06855303 0.06977124 0.02579004 0.07629535
restore_methods()

You can automatically register these methods when loading this package by setting the option dqrng.register_methods to TRUE, e.g. with options(dqrng.register_methods=TRUE).

For some workflows it is helpful to save and restore the RNG’s type and state, similar to how .Randome.seed can be saved and restored. The function pair dqrng_get_state() and dqrng_set_state() can be used for this task:

(state <- dqrng_get_state())
#> [1] "default"             "7442421893577288217" "2933090096537006399"
dqrunif(5)
#> [1] 0.850198175 0.184318214 0.003138956 0.071103977 0.430195275
# many other operations, that could even change the used RNG type
dqrng_set_state(state)
dqrunif(5)
#> [1] 0.850198175 0.184318214 0.003138956 0.071103977 0.430195275

Note that the state is represented by a character vector, since the unsigned 64 and 128 bit integers used by the supported RNGs cannot be represented in R otherwise. Generally this state should be treated as an implementation detail and not manipulated directly.

Usage from C++

The RNGs and distributions functions can also be used from C++ at various levels of abstraction. Technically there are three ways to make use of dqrng at the C++ level:

  • use // [[Rcpp::depends(dqrng)]] together with Rcpp::sourceCpp()
  • use Rcpp::cppFunction(depends = "dqrng", ...)
  • use an R package with LinkingTo: dqrng

Here only the first approach is shown.

Using the compiled library functions

The functions available in R directly call corresponding C++ functions. These functions are also available at the C++ level if you include dqrng.h. The full list of functions is available with vignette("cpp-api", package = "dqrng"). Revisiting the example of approximating \(\pi\) we can use:

// [[Rcpp::depends(dqrng)]]
#include <Rcpp.h>
#include <dqrng.h>

using Rcpp::IntegerVector;
using Rcpp::NumericVector;
using Rcpp::sqrt;
using Rcpp::sum;
using dqrng::dqrunif;

// [[Rcpp::export]]
double piCpp(const int n) {
  dqrng::dqset_seed(IntegerVector::create(42));
  NumericVector x = dqrunif(n);
  NumericVector y = dqrunif(n);
  NumericVector d = sqrt(x*x + y*y);
  return 4.0 * sum(d < 1.0) / n;
}
/*** R
system.time(cat("pi ~= ", piCpp(1e7), "\n"))
*/

Note that in C++ you have to use dqrng::dqset_seed(), whereas the analogue function in the R interface is called dqrng::dqset.seed(). For sampling with and without replacement dqrng::dqsample_int() and dqrng::dqsample_num() are the analogue of dqrng::dqsample.int() in the R interface:

// [[Rcpp::depends(dqrng)]]
#include <Rcpp.h>
#include <dqrng.h>

// [[Rcpp::export]]
void sampleCpp(const int n) {
  dqrng::dqset_seed(Rcpp::IntegerVector::create(42));
  Rcpp::IntegerVector sample = dqrng::dqsample_int(n, n/100, true);
  Rcpp::Rcout << sample << std::endl;
}
/*** R
sampleCpp(1000)
*/

Using the header only library

The RNG wrapper and distributions functions can be used from C++ by including dqrng_generator.h and dqrng_distribution.h. In order to use these header files, you have to use at least C++11. You have to link to the BH package as well to use dqrng’s distribution functions. For example, you can use the distribution functions from dqrng together with some foreign 64bit RNG. Here a minimal SplitMix generator is used together with dqrng::normal_distribution:

#include <Rcpp.h>
// [[Rcpp::depends(dqrng, BH)]]
#include <dqrng_distribution.h>

class SplitMix {
public:
  typedef uint64_t result_type;
  SplitMix (result_type seed) : state(seed) {};
  result_type operator() () {
    result_type z = (state += 0x9e3779b97f4a7c15ULL);
    z = (z ^ (z >> 30)) * 0xbf58476d1ce4e5b9ULL;
    z = (z ^ (z >> 27)) * 0x94d049bb133111ebULL;
    return z ^ (z >> 31);
  }
  void seed(result_type seed) {state = seed;}
  static constexpr result_type min() {return 0;};
  static constexpr result_type max() {return UINT64_MAX;};

private:
  result_type state;

public:  
  friend std::ostream& operator<<(std::ostream& ost, const SplitMix& e) {
    return ost << e.state;
  }
  friend std::istream& operator>>(std::istream& ist, SplitMix& e) {
    return ist >> e.state;
  }
};

// [[Rcpp::export]]
Rcpp::NumericVector splitmix_rnorm(const int n, const double mean = 0.0, const double sd = 1.0) {
  auto rng = dqrng::generator<SplitMix>(42);
  Rcpp::NumericVector result(n);
  rng->generate<dqrng::normal_distribution>(result, mean, sd);
  return result;
}
/*** R
splitmix_rnorm(10)
system.time(splitmix_rnorm(1e7))
*/

Since SplitMix is a very fast RNG, the speed of this function is comparable to dqrnorm. Generally speaking you can use any C++11 compliant RNG with 64 bit output size. For example, here the 64 bit Threefry engine with 13 rounds from package sitmo is used:

#include <Rcpp.h>
// [[Rcpp::depends(dqrng, BH, sitmo)]]
#include <dqrng_distribution.h>
#include <threefry.h>

// [[Rcpp::export]]
Rcpp::NumericVector threefry_rnorm(const int n, const double mean = 0.0, const double sd = 1.0) {
  auto rng = dqrng::generator<sitmo::threefry_13_64>(42);
  Rcpp::NumericVector result(n);
  rng->generate<dqrng::normal_distribution>(result, mean, sd);
  return result;
}

/*** R
threefry_rnorm(10)
system.time(threefry_rnorm(1e7))
*/

Note that for the (recommended) Threefry engine with 20 rounds some additional integration is provided in the dqrng_threefry.h header file.

Alternatively, you could combine the included RNGs together with dqrng’s tooling and some other distribution function. For example, this function generates random numbers according to the normal distribution using the standard library from C++11:

#include <random>
#include <Rcpp.h>
// [[Rcpp::depends(dqrng)]]
#include <dqrng_generator.h>
#include <xoshiro.h>

// [[Rcpp::export]]
Rcpp::NumericVector std_rnorm(const int n, const double mean = 0.0, const double sd = 1.0) {
  auto rng = dqrng::generator<dqrng::xoroshiro128plusplus>(42);
  Rcpp::NumericVector result(n);
  rng->generate<std::normal_distribution>(result, mean, sd);
  return result;
}

/*** R
std_rnorm(10)
system.time(std_rnorm(1e7))
*/

Typically this is not as fast as dqrnorm, but the technique is useful to support distributions not (yet) included in dqrng. Note however, that the algorithms used for the distributions from C++11 are implementation defined.

Finally you could directly use the base generators, which are provided as header-only libraries, without dqrng’s tooling. For example, the following function uses the 32 bit PCG variant together with Boost’s normal distribution function:

#include <Rcpp.h>
// [[Rcpp::depends(dqrng, BH)]]
#include <pcg_random.hpp>
#include <boost/random/normal_distribution.hpp>

// [[Rcpp::plugins(cpp11)]]   
// [[Rcpp::export]]
Rcpp::NumericVector boost_pcg_rnorm(const int n, const double mean = 0.0, const double sd = 1.0) {
  pcg32 rng(42);
  boost::random::normal_distribution<double> dist(mean, sd);
  Rcpp::NumericVector result(n);
  std::generate(result.begin(), result.end(), [&dist, &rng](){return dist(rng);});
  return result;
}
/*** R
boost_pcg_rnorm(10)
system.time(boost_pcg_rnorm(1e7))
*/

This is quite fast since boost::random::normal_distribution uses the fast Ziggurat algorithm. For some applications it is necessary to draw random numbers from multiple distributions with varying parameters. The following function uses a binomial distribution (from boost.random) as well as the normal distribution from dqrng. The parameters of the distributions are adjusted for every draw using <distribution>::param_type:

#include <Rcpp.h>
// [[Rcpp::depends(dqrng, BH)]]
#include <boost/random/binomial_distribution.hpp>
#include <dqrng_distribution.h>

// [[Rcpp::export]]
Rcpp::NumericMatrix multiple_distributions(int n) {
  auto rng = dqrng::generator<dqrng::xoshiro256plusplus>(42);
  Rcpp::NumericMatrix out(n, 3);
  double p = 0.0;
  for (int i = 0; i < n; ++i) {
    p = double(i) / double(n);
    out(i,0) = rng->variate<boost::random::binomial_distribution<int>>(1, p);
    out(i,1) = rng->variate<dqrng::normal_distribution>(p, 1.0);
    out(i,2) = rng->variate<dqrng::normal_distribution>(4.0, 3.0 - p);
  }
  Rcpp::colnames(out) = Rcpp::CharacterVector::create("Bernoulli", "Normal1", "Normal2");
  return out;
}

/*** R
multiple_distributions(5)
*/

Accessing the global RNG

You may use the class dqrng::random_64bit_accessor to use the seeded RNG engine of dqrng. Please note that the included RNG will be invalidated if dqRNGkind is called. You therefore should use this calls only within functions:

#include <Rcpp.h>
// [[Rcpp::depends(dqrng, BH)]]
#include <boost/random/binomial_distribution.hpp>
#include <dqrng.h>
#include <dqrng_distribution.h>

// [[Rcpp::export]]
Rcpp::NumericMatrix multiple_distributions(int n) {
  auto rng = dqrng::random_64bit_accessor{};
  Rcpp::NumericMatrix out(n, 3);
  double p = 0.0;
  for (int i = 0; i < n; ++i) {
    p = double(i) / double(n);
    out(i,0) = rng.variate<boost::random::binomial_distribution<int>>(1, p);
    out(i,1) = rng.variate<dqrng::normal_distribution>(p, 1.0);
    out(i,2) = rng.variate<dqrng::normal_distribution>(4.0, 3.0 - p);
  }
  Rcpp::colnames(out) = Rcpp::CharacterVector::create("Bernoulli", "Normal1", "Normal2");
  return out;
}

/*** R
dqRNGkind("Xoshiro256++")
dqset.seed(42)
multiple_distributions(5)
*/