Algorithm 806: SPRNG

Algorithm 806: SPRNG A scalable Library for pseudorandom number generation

About: Pseudo-random Libraries • Many Libraries • Parameterization – based libraries • Erroneous => Real world applications • New Library – SPRNG • Purpose of SPRNG ?

Introduction • Problems: • Erroneous => Monte Carlo Applications • High Performance Cycles • Parallelism – More Complicates • Solution: • SPRNG – Scalable and Distributed Architecture

PARALLEL Pseudorandom-number generation • Methods - Parameterization • Linear Congruential Generators: • Commonly used generator • Linear Recursion : • Periodic Sequence

Prime modulus • ‘m’ is prime • Determine family of a’s • Obtain Maximal period • ‘i ’ relatively prime to Produce Primitive element ‘j’: PROBLEM: Overall efficiency of PPRNG ? • Cost of Computing or • cost of modular multiplication

Exponential Sum: • Full period • Partially period • Upper Bounds: Equidistributed • Parallel: • Exponential Sum Correlation is :

Upper Bound: • Open Questions: • Choosing ‘m’ ? A. Fast modular multiplication or fast calculation of jth integer • Good intrastream correlation also ensure good intrastream independence to performance ? • Hence  Such techniques are less attractive for parallel Pseudo random number generation.

Lagged- Fibonacci Generator • Previous: Varying parameters - recursion • Now: Parameterized – Initial Value • ALFG Equation: • Popular : Serial and Scalable Parallel Machines • Reasons: • 1. Easy to implement • 2. Cheap to compute • It does well on statistical tests

Maximal Period : • Template :

O(K*K) Template : • Advantage: • Optimal Disadvantage : No proof and improvement of this analytical is open challenge

Advantage: Good for Floating point • However, care must be taken to maintain the uniqueness of the parallel streams

SPRNG • Random Number streams to parallel processes • Advantages: • 1. Several, Distinct, Well tested, Scalable PRNGs • 2. Initailization without Inter process communication • 3. Reproducibility • 4. Minimization of Interprocesscorrealtion • 5. Uniformity • 6. Integrated Test Suites • 7. Extensibility

Generators • Additive lagged Fibonacci • Multiplicative Lagged Fibonacci • Prime Modules Multiplicative Congruential • Power-of-two modules linear Congruential • Combined Multiplicative Recursive generator

Examples • Download: • http://sprng.cs.fsu.edu • Definition: init_sprng routine • int *init_sprng(int streamnum, int nstreams, int seed, int param)

#include <cstdio> • #define SIMPLE_SPRNG /* simple interface */ • #include "sprng.h" /* SPRNG header file */ • using namespace std; • int main() • { • double rn; • int i; • printf(" Printing 3 random numbers in [0,1):\n"); • for (i=0;i<3;i++) • { • rn = sprng(); /* generate double precision random number */ • printf("%f\n",rn); • } • return 0; • }

Execution • $> ./simple-simple • Printing 3 random numbers in [0,1): • 0.014267 • 0.749392 • 0.007316

SPRNG TEST SUITE • Intrastream Correlation • Interstream Correlation • Statistical Test

Example: • test.lib nstreamsncombine seed paramnblocks skip • Ex: eqi-distance.lcg 4 2 0 0 3 1 2 100 • mpirun -np 2 equidistnce.lcg 4 2 0 0 3 1 2 100 • Result: • mpirun -np 2 eqidist.lcg 4 2 0 0 3 1 2 1 00 • eqidistance.lcg 4 2 0 0 3 1 2 100 • KS value : 0.601252 • KS value prob : 17.50 • Hence, It is b/w 2.5% and 97.5% It is passed

There are different tests that are described by Knuth are described in SPRNG library such as • Collision tests • Coupons Collector tests • Equidistance tests • Gap test • Permutation tests • Poker tests • Serial test

Physically Based Tests: • Ising Model : • Random Walk Test:

CONCLUSION • Comprehensive Library • Recursions are simple • Reusable • Extensibility • Uniformity : C, C++, MPI and FORTRAN

REFERENCES • http://sprng.cs.fsu.edu • http://www.naadsm.org/opensource/sprng • http://www.cs.sunysb.edu/~algorith/implement/sprng/implement.shtml • https://computing.llnl.gov/LCdocs/math/index.jsp?show=s4.6 • http://www.berniepope.id.au/sprng.html

Algorithm 806: SPRNG