Pseudo-random number generator for cluster environment

Question

How can I generate independent pseudo-random numbers on a cluster, for Monte Carlo simulation for example? I can have many compute nodes (e.g. 100), and I need to generate millions of numbers on each node. I need a warranty that a PRN sequence on one node will not overlap the PRN sequence on another node.

• I could generate all PRN on root node, then send them to other nodes. But it would be far too slow.
• I could jump to a know distance in the sequence, on each node. But is there such an algorithm for Mersenne-Twister or for any other good PRNG, which can be done with a reasonable amount of time and memory?
• I could use different generators on each node. But is it possible with good generators like Mersenne-Twister? How could it be done?
• Any other though?

Answer 1

You should never use potentially overlapping random streams obtained from the same original stream. If you have not tested the resulting interleaved stream, you have no idea of its statistic quality.

Fortunately, Mersenne Twister (MT) will help you in your distribution task. Using its dedicated algorithm, called Dynamic Creator (DC hereafter), you can create independent random number generators that will produce highly independent random streams.

Each stream will be created on the node that will be using it. Basically, think of DC as a constructor in object oriented paradigm that creates different instances of MT. Each different instance is designed to produce highly independent random sequences.

You can find DC here: http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/DC/dc.html
It's quite straightforward to use and you'll be able to fix different parameters such as the number of different MT instances you want to obtain or the period of these MTs. Depending on its input parameter, DC will runtime will change.

In addition of the README coming along with DC, take a look at the file example/new_example2.c in the DC archive. It shows example of calls to get independent sequences given a different input identifier, which is basically what you have to identify cluster jobs.

Finally, if you intend to learn more about how to use PRNGs in parallel or distributed environments, I suggest you read this scientific articles:

Practical distribution of random streams for stochastic High Performance Computing, David RC Hill, in International Conference on High Performance Computing and Simulation (HPCS), 2010

Answer 2

Okay, answer #2 ;-)

I am going to say ... keep it simple. Just use a "short" seed to prime the MT (imagine that this seed is 2³² bits for lack of better restriction). This assumes that the the short seed generates "sufficiently distributed" MT starting states (e.g. init_genrand in the code in my other answer, hopefully). This doesn't guarantee an equally distributed starting state but rather goes for "good enough", see below.

Each node will use it's own sequence of seeds which are pre-selected (either a list of random seeds which is transmitted or a formula like number_nodes * node_number * iteration). The important thing is that the initial "short" seed will never be re-used across nodes.

Each node will then use a MT PRNG initialized with this seed n times where n <<< MT_period / max_value_of_short_seed (TT800 is 2⁸⁰⁰-1 and MT19937 is 2¹⁹⁹³⁷-1, so n can still be a very large number). After n times, the node moves onto the next seed in the chosen list.

While I do not (nor can I) provide a "guarantee" that no node will ever have a duplicate sequence at the same time (or at all), here is what AMD says about Using Different Seends: (Obviously the initial seeding algorithm plays a role).

Of the four methods for creating multiple streams described here, this is the least satisfactory ... For example, sequences generated from different starting points may overlap if the initial values are not far enough apart. The potential for overlapping sequences is reduced if the period of the generator being used is large. Although there is no guarantee of the independence of the sequences, due to its extremely large period, using the Mersenne Twister with random starting values is unlikely to lead to problems, especially if the number of sequences required is small ...

Happy coding.

Answer 3

I could jump to a know distance in the sequence, on each node. But is there such an algorithm for Mersenne-Twister or for any other good PRNG, which can be done with a reasonable amount of time and memory?

Yes, see http://theo.phys.sci.hiroshima-u.ac.jp/~ishikawa/PRNG/mt_stream_en.html. This is a excellent solution to obtaining independent random number streams. By making jumps that are larger than the number of random numbers needed from each stream to create the starts of each stream, the streams won't overlap.

Answer 4

Disclaimer: I am not sure what guarantee MT has in terms of cycle overlap when starting from an arbitrary "uint" (or x, where x is a smaller arbitrary but unique value) seed, but that may be worth looking into, as if there is a guarantee then it may be sufficient to just start each node on a different "uint" seed and the rest of this post becomes largely moot. (The cycle length/period of MT is staggering and dividing out UINT_MAX still leaves an incomprehensible -- except on paper -- number.)

---

Well, here goes my comments to answer...

I like approach #2 with a pre-generated set of states; the MT in each node is then initialized with a given starting state.

Only the initial states must be preserved, of course, and once this is generated these states can

1. Be re-used indefinitely, if requirements are met, or;
2. The next states can generated forward on an external fast box why the simulation is running or;
3. The nodes can report back the end-state (if reliable messaging, and if sequence is used at same rate among nodes, and meets requirements, etc)

Considering that MT is fast to generate, I would not recommend #3 from above as it's just complicated and has a number of strings attached. Option #1 is simple, but might not be dynamic enough.

Option #2 seems like a very good possibility. The server (a "fast machine", not necessarily a node) only needs to transmit the starting state of the next "unused sequence block" (say, one billion cycles) -- the node would use the generator for one billion cycles before asking for a new block. This would make it a hybrid of #1 in the post with very infrequent messaging.

On my system, a Core2 Duo, I can generate one billion random numbers in 17 seconds using the code provided below (it runs in LINQPad). I am not sure what MT variant this is.

void Main()
{
    var mt = new MersenneTwister();
    var start = DateTime.UtcNow;
    var ct = 1000000000;
    int n = 0;
    for (var i = 0; i < ct; i++) {      
        n = mt.genrand_int32();
    }
    var end = DateTime.UtcNow;
    (end - start).TotalSeconds.Dump();
}

// From ... and modified (stripped) to work in LINQPad.
// http://mathnet-numerics.googlecode.com/svn-history/r190/trunk/src/Numerics/Random/MersenneTwister.cs
// See link for license and copyright information.
public class MersenneTwister
{
    private const uint _lower_mask = 0x7fffffff;
    private const int _m = 397;
    private const uint _matrix_a = 0x9908b0df;
    private const int _n = 624;
    private const double _reciprocal = 1.0/4294967295.0;
    private const uint _upper_mask = 0x80000000;
    private static readonly uint[] _mag01 = {0x0U, _matrix_a};
    private readonly uint[] _mt = new uint[624];
    private int mti = _n + 1;

    public MersenneTwister() : this((int) DateTime.Now.Ticks)
    {
    }       
    public MersenneTwister(int seed)
    {
                init_genrand((uint)seed);
    }       

    private void init_genrand(uint s)
    {
        _mt[0] = s & 0xffffffff;
        for (mti = 1; mti < _n; mti++)
        {
            _mt[mti] = (1812433253*(_mt[mti - 1] ^ (_mt[mti - 1] >> 30)) + (uint) mti);
            _mt[mti] &= 0xffffffff;
        }
    }

    public uint genrand_int32()
    {
        uint y;
        if (mti >= _n)
        {
            int kk;

            if (mti == _n + 1) /* if init_genrand() has not been called, */
                init_genrand(5489); /* a default initial seed is used */

            for (kk = 0; kk < _n - _m; kk++)
            {
                y = (_mt[kk] & _upper_mask) | (_mt[kk + 1] & _lower_mask);
                _mt[kk] = _mt[kk + _m] ^ (y >> 1) ^ _mag01[y & 0x1];
            }
            for (; kk < _n - 1; kk++)
            {
                y = (_mt[kk] & _upper_mask) | (_mt[kk + 1] & _lower_mask);
                _mt[kk] = _mt[kk + (_m - _n)] ^ (y >> 1) ^ _mag01[y & 0x1];
            }
            y = (_mt[_n - 1] & _upper_mask) | (_mt[0] & _lower_mask);
            _mt[_n - 1] = _mt[_m - 1] ^ (y >> 1) ^ _mag01[y & 0x1];

            mti = 0;
        }

        y = _mt[mti++];

        /* Tempering */
        y ^= (y >> 11);
        y ^= (y << 7) & 0x9d2c5680;
        y ^= (y << 15) & 0xefc60000;
        y ^= (y >> 18);

        return y;
    }
}

Happy coding.

Answer 5

TRNG is a random number generator built specifically with parallel cluster environments in mind (specifically it was built for the TINA super computer in Germany). Hence it is very eas to create independent random number streams and also generate non standard distributions. There is a tutorial on how to set it up here: http://www.lindonslog.com/programming/parallel-random-number-generation-trng/