# Independent Streams. Block-Splitting and Leapfrogging

One of the basic requirements for random number streams is their mutual independence and lack of intercorrelation. Even if you want random number samplings to be correlated, such correlation should be controllable.

You can get independent streams using various methods. This document discusses the following methods supported by VS:

- Using different parameter sets. For each stream, you may use the same type of generators (for example, linear congruential generators), but choose their parameters in such a way as to produce independent random number sequences. For example, the Mersenne Twister generator has 6024 parameter sets, which ensure that the resulting subsequences are independent (see [Matsum2000] for details). Another example is WH generator that can create up to 273 random number streams. The produced sequences are independent according to the spectral test (see [Knuth81] for the spectral test details).
- Block-splitting. Split the original sequence into
non-overlapping blocks, wherekis the number of independent streams. Each of the streams generates random numbers only from the corresponding block. This method is known as block-splitting or skipping-ahead.k - Leapfrogging. Split the original sequence into
disjoint subsequences, wherekis the number of independent streams, in such a way that the first stream would generate the random numberskx_{1},x_{k+1},x_{2k+1},x_{3k+1}, ..., the second stream would generate the random numbersx_{2},x_{k+2},x_{2k+2},x_{3k+2}, ..., and, finally, the k-th stream would generate the random numbersx_{k},x_{2k},x_{3k}, ... However, multidimensional uniformity properties of each subsequence deteriorate seriously asgrows. The method is useful ifkis fairly small.k

Karl Entacher presents data on inadequate subsequences produced by some commonly used linear congruential generators [Ent98].

VS permits you to use any of the above methods. Leapfrog and skip-ahead (block-split) methods are considered below in more detail.

Block-Splitting Method

VS implements block-splitting through

vslSkipAheadStream and vslSkipAheadStreamEx

functions:vslSkipAheadStream( stream, nskip )vslSkipAheadStreamEx( stream, n, nskip )

The functions change the current state of the stream

stream

so that with the further call of the generator the output subsequence begins with the element x

_{nskip}rather than with the current elementx

_{0}.Function

vslSkipAheadStream

supports number of skipped elements up to 2^{63}.Function

vslSkipAheadStreamEx

extends it by providing support for number of skipped elements greater than 2^{63}.Before calling this function, you should represent the number of skipped elements in the following format:

Number of skipped elements = nskip[0] + nskip[1] * 2

^{64}+nskip[2] * 2^{128}+ ... + nskip[n-1] * 2^{64*(n-1)},For example, if you want to skip 2

^{128}elements you should represent them as follows:Number of skipped elements = 0 * 2

^{0}* 2^{64}+ 1 * 2^{128}Then you should call

vslSkipAheadStreamEx

function.VSLStreamStatePtr stream;MKL_UINT64 nskip[3];nskip[0]=0;nskip[1]=0;nskip[2]=1;vslNewStream( &stream, brng, seed );vslSkipAheadStreamEx( stream, 3, nskip);

You can use either

vslSkipAheadStream

or vslSkipAheadStreamEx

to skip numbers of elements smaller than 2^{63}.Thus, if you wish to split the initial sequence into nstreams blocks of

nskip

size each, use the following sequence of operations:Option 1

VSLStreamStatePtr stream[nstreams];int k;for ( k=0; k<nstreams; k++ ){vslNewStream( &stream[k], brng, seed );vslSkipAheadStream( stream[k], nskip*k );}

Option 2

VSLStreamStatePtr stream[nstreams];int k;vslNewStream( &stream[0], brng, seed );for ( k=0; k<nstreams-1; k++ ){vslCopyStream( &stream[k+1], stream[k] );vslSkipAheadStream( stream[k+1], nskip );}

Leapfrog Method

VS implements the leapfrog method through function

vslLeapfrogStream

:vslLeapfrogStream( stream, k, nstreams )

The function changes the stream

stream

so that the further call of the generator generates the output subsequence x

_{k},x

_{k+nstreams},x

_{k+2nstreams}, ... rather than the output sequencex

_{0},x

_{1},x

_{2}, ... . Thus, if you wish to split the initial sequence into nstreams subsequences, the following sequence of operations should be implemented:VSLStreamStatePtr stream[nstreams];int k;for ( k=0; k<nstreams; k++ ){vslNewStream( &stream[k], brng, seed );vslLeapfrogStream( stream[k], k, nstreams );}

Block-splitting and leapfrog methods make programming with vector random number generators easier both in parallel applications and in sequential programs.

Not all VS BRNGs support both these methods of generating independent subsequences. The Leapfrog method is supported only when a BRNG provides a more efficient implementation than generation of the full sequence to pick out a required subsequence. The following table specifies the methods supported by different BRNGs:

BRNG | Leapfrog | Block-Splitting ( vslSkipAheadStream ) | Block-Splitting ( vslSkipAheadStreamEx ) |
---|---|---|---|

MCG31m1 | Supported | Supported | - |

R250 | - | - | - |

MRG32k3a | - | Supported | Supported |

MCG59 | Supported | Supported | - |

WH | Supported | Supported | - |

MT19937 | - | Supported | - |

SFMT19937 | - | Supported | - |

MT2203 | - | - | - |

SOBOL | Supported to pick out individual components of quasi-random vectors | Supported | - |

NIEDERREITER | Supported to pick out individual components of quasi-random vectors | Supported | - |

PHILOX4X32X10 | - | Supported | Supported |

ARS5 | - | Supported | Supported |

ABSTRACT | - | - | - |

NON-DETERMINISTIC | - | - | - |

To initialize

nstreams

independent streams for the MT2203 set of generators, you can use the following code sequence:...#define nstreams 6024...VSLStreamStatePtr stream[nstreams];int k;for ( k=0; k< nstreams; k++ ){vslNewStream( &stream[k], VSL_BRNG_MT2203+k, seed );}...