X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/prng_gpu.git/blobdiff_plain/3c253b8871482ed4e74d60ab23edaf08052e7c72..1d47039b10b11f257fe60db61313a82f5d688bd2:/prng_gpu.tex

diff --git a/prng_gpu.tex b/prng_gpu.tex
index fa71e5b..dccf50a 100644
--- a/prng_gpu.tex
+++ b/prng_gpu.tex
@@ -226,8 +226,7 @@ We have proposed in~\cite{bgw09:ip} a new family of generators that receives
 two PRNGs as inputs. These two generators are mixed with chaotic iterations, 
 leading thus to a new PRNG that improves the statistical properties of each
 generator taken alone. Furthermore, our generator 
-possesses various chaos properties
-that none of the generators used as input present.
+possesses various chaos properties that none of the generators used as input present.
 
 \begin{algorithm}[h!]
 %\begin{scriptsize}
@@ -287,7 +286,226 @@ We have proven in \cite{FCT11} that,
 
 
 
-\alert{Mettre encore un peu de blabla sur le PRNG, puis enchaÃ®ner en disant que, ok, on peut prÃ©server le chaos quand on passe sur machine, mais que le chaos dont il s'agit a Ã©tÃ© prouvÃ© pour une distance bizarroÃ¯de sur un espace non moins hÃ©moroÃ¯de, d'oÃ¹ ce qui suit}
+
+\section{Efficient prng based on chaotic iterations}
+
+In  order to  implement efficiently  a PRNG  based on  chaotic iterations  it is
+possible to improve  previous works [ref]. One solution  consists in considering
+that the  strategy used contains all the  bits for which the  negation is
+achieved out. Then in order to apply  the negation on these bits we can simply
+apply the  xor operator between  the current number  and the strategy. In
+order to obtain the strategy we also use a classical PRNG.
+
+Here  is an  example with  16-bits numbers  showing how  the bitwise  operations are
+applied.  Suppose  that $x$ and the  strategy $S^i$ are defined  in binary mode.
+Then the following table shows the result of $x$ xor $S^i$.
+$$
+\begin{array}{|cc|cccccccccccccccc|}
+\hline
+x      &=&1&0&1&1&1&0&1&0&1&0&0&1&0&0&1&0\\
+\hline
+S^i      &=&0&1&1&0&0&1&1&0&1&1&1&0&0&1&1&1\\
+\hline
+x \oplus S^i&=&1&1&0&1&1&1&0&0&0&1&1&1&0&1&0&1\\
+\hline
+
+\hline
+ \end{array}
+$$
+
+%% \begin{figure}[htbp]
+%% \begin{center}
+%% \fbox{
+%% \begin{minipage}{14cm}
+%% unsigned int CIprng() \{\\
+%%   static unsigned int x = 123123123;\\
+%%   unsigned long t1 = xorshift();\\
+%%   unsigned long t2 = xor128();\\
+%%   unsigned long t3 = xorwow();\\
+%%   x = x\textasciicircum (unsigned int)t1;\\
+%%   x = x\textasciicircum (unsigned int)(t2$>>$32);\\
+%%   x = x\textasciicircum (unsigned int)(t3$>>$32);\\
+%%   x = x\textasciicircum (unsigned int)t2;\\
+%%   x = x\textasciicircum (unsigned int)(t1$>>$32);\\
+%%   x = x\textasciicircum (unsigned int)t3;\\
+%%   return x;\\
+%% \}
+%% \end{minipage}
+%% }
+%% \end{center}
+%% \caption{sequential Chaotic Iteration PRNG}
+%% \label{algo:seqCIprng}
+%% \end{figure}
+
+
+
+\lstset{language=C,caption={C code of the sequential chaotic iterations based PRNG},label=algo:seqCIprng}
+\begin{lstlisting}
+unsigned int CIprng() {
+  static unsigned int x = 123123123;
+  unsigned long t1 = xorshift();
+  unsigned long t2 = xor128();
+  unsigned long t3 = xorwow();
+  x = x^(unsigned int)t1;
+  x = x^(unsigned int)(t2>>32);
+  x = x^(unsigned int)(t3>>32);
+  x = x^(unsigned int)t2;
+  x = x^(unsigned int)(t1>>32);
+  x = x^(unsigned int)t3;
+  return x;
+}
+\end{lstlisting}
+
+
+
+
+
+In listing~\ref{algo:seqCIprng}  a sequential version of  our chaotic iterations
+based   PRNG    is   presented.   The    xor   operator   is    represented   by
+\textasciicircum.  This   function  uses  three  classical   64-bits  PRNG:  the
+\texttt{xorshift},  the   \texttt{xor128}  and  the   \texttt{xorwow}.   In  the
+following,  we call  them  xor-like  PRNGSs.  These  three  PRNGs are  presented
+in~\cite{Marsaglia2003}.  As each  xor-like PRNG used works with  64-bits and as
+our PRNG works  with 32-bits, the use of \texttt{(unsigned  int)} selects the 32
+least significant bits whereas  \texttt{(unsigned int)(t3$>>$32)} selects the 32
+most  significants bits  of the  variable \texttt{t}.   So to  produce  a random
+number realizes  6 xor operations with  6 32-bits numbers produced  by 3 64-bits
+PRNG.  This version successes the  BigCrush of the TestU01 battery [P.  Lâecuyer
+  and R. Simard. Testu01].
+
+\section{Efficient prng based on chaotic iterations on GPU}
+
+In  order to benefit  from computing  power of  GPU, a  program needs  to define
+independent blocks of threads which  can be computed simultaneously. In general,
+the larger the number of threads is,  the more local memory is used and the less
+branching  instructions are  used (if,  while, ...),  the better  performance is
+obtained  on  GPU.  So  with  algorithm  \ref{algo:seqCIprng}  presented in  the
+previous section, it is possible to  build a similar program which computes PRNG
+on  GPU. In  the CUDA  [ref] environment,  threads have  a  local identificator,
+called \texttt{ThreadIdx} relative to the block containing them.
+
+
+\subsection{Naive version for GPU}
+
+From the CPU version, it is possible  to obtain a quite similar version for GPU.
+The principe consists in assigning the computation of a PRNG as in sequential to
+each thread  of the  GPU.  Of course,  it is  essential that the  three xor-like
+PRNGs  used for  our computation  have different  parameters. So  we  chose them
+randomly with  another PRNG. As the  initialisation is performed by  the CPU, we
+have chosen to use the ISAAC PRNG  [ref] to initalize all the parameters for the
+GPU version  of our  PRNG.  The  implementation of the  three xor-like  PRNGs is
+straightforward  as soon  as their  parameters have  been allocated  in  the GPU
+memory. Each xor-like  PRNGs used works with an internal  number $x$ which keeps
+the last generated random numbers. Other internal variables are also used by the
+xor-like PRNGs. More  precisely, the implementation of the  xor128, the xorshift
+and  the xorwow  respectively  require 4,  5  and 6  unsigned  long as  internal
+variables.
+
+\begin{algorithm}
+
+\KwIn{InternalVarXorLikeArray: array with internal variables of the 3 xor-like PRNGs in global memory\;
+NumThreads: Number of threads\;}
+\KwOut{NewNb: array containing random numbers in global memory}
+\If{threadIdx is concerned by the computation} {
+  retrieve data from InternalVarXorLikeArray[threadIdx] in local variables\;
+  \For{i=1 to n} {
+    compute a new PRNG as in Listing\ref{algo:seqCIprng}\;
+    store the new PRNG in NewNb[NumThreads*threadIdx+i]\;
+  }
+  store internal variables in InternalVarXorLikeArray[threadIdx]\;
+}
+
+\caption{main kernel for the chaotic iterations based PRNG GPU naive version}
+\label{algo:gpu_kernel}
+\end{algorithm}
+
+Algorithm~\ref{algo:gpu_kernel}  presents a naive  implementation of  PRNG using
+GPU.  According  to the available  memory in the  GPU and the number  of threads
+used simultenaously,  the number  of random numbers  that a thread  can generate
+inside   a    kernel   is   limited,   i.e.    the    variable   \texttt{n}   in
+algorithm~\ref{algo:gpu_kernel}. For example, if  $100,000$ threads are used and
+if $n=100$\footnote{in fact, we need to add the initial seed (a 32-bits number)}
+then   the  memory   required   to  store   internals   variables  of   xor-like
+PRNGs\footnote{we multiply this number by $2$ in order to count 32-bits numbers}
+and  random  number of  our  PRNG  is  equals to  $100,000\times  ((4+5+6)\times
+2+(1+100))=1,310,000$ 32-bits numbers, i.e. about $52$Mb.
+
+All the  tests performed  to pass the  BigCrush of TestU01  succeeded. Different
+number of threads, called \texttt{NumThreads} in our algorithm, have been tested
+upto $10$ millions.
+
+\begin{remark}
+Algorithm~\ref{algo:gpu_kernel}  has  the  advantage to  manipulate  independent
+PRNGs, so this version is easily usable on a cluster of computer. The only thing
+to ensure is to use a single ISAAC PRNG. For this, a simple solution consists in
+using a master node for the initialization which computes the initial parameters
+for all the differents nodes involves in the computation.
+\end{remark}
+
+\subsection{Improved version for GPU}
+
+As GPU cards using CUDA have shared memory between threads of the same block, it
+is possible  to use this  feature in order  to simplify the  previous algorithm,
+i.e. using less  than 3 xor-like PRNGs. The solution  consists in computing only
+one xor-like PRNG by thread, saving  it into shared memory and using the results
+of some  other threads in the  same block of  threads. In order to  define which
+thread uses the result of which other  one, we can use a permutation array which
+contains  the indexes  of  all threads  and  for which  a  permutation has  been
+performed.  In Algorithm~\ref{algo:gpu_kernel2}, 2 permutations arrays are used.
+The    variable   \texttt{offset}    is    computed   using    the   value    of
+\texttt{permutation\_size}.   Then we  can compute  \texttt{o1}  and \texttt{o2}
+which represent the indexes of the  other threads for which the results are used
+by the  current thread. In  the algorithm, we  consider that a  64-bits xor-like
+PRNG is used, that is why both 32-bits parts are used.
+
+This version also succeed to the BigCrush batteries of tests.
+
+\begin{algorithm}
+
+\KwIn{InternalVarXorLikeArray: array with internal variables of 1 xor-like PRNGs in global memory\;
+NumThreads: Number of threads\;
+tab1, tab2: Arrays containing permutations of size permutation\_size\;}
+
+\KwOut{NewNb: array containing random numbers in global memory}
+\If{threadId is concerned} {
+  retrieve data from InternalVarXorLikeArray[threadId] in local variables\;
+  offset = threadIdx\%permutation\_size\;
+  o1 = threadIdx-offset+tab1[offset]\;
+  o2 = threadIdx-offset+tab2[offset]\;
+  \For{i=1 to n} {
+    t=xor-like()\;
+    shared\_mem[threadId]=(unsigned int)t\;
+    x = x $\oplus$ (unsigned int) t\;
+    x = x $\oplus$ (unsigned int) (t>>32)\;
+    x = x $\oplus$ shared[o1]\;
+    x = x $\oplus$ shared[o2]\;
+
+    store the new PRNG in NewNb[NumThreads*threadId+i]\;
+  }
+  store internal variables in InternalVarXorLikeArray[threadId]\;
+}
+
+\caption{main kernel for the chaotic iterations based PRNG GPU efficient version}
+\label{algo:gpu_kernel2}
+\end{algorithm}
+
+
+
+\section{Experiments}
+
+Differents experiments have been performed in order to measure the generation speed.
+\begin{figure}[t]
+\begin{center}
+  \includegraphics[scale=.7]{curve_time_gpu.pdf}
+\end{center}
+\caption{Number of random numbers generated per second}
+\label{fig:time_naive_gpu}
+\end{figure}
+
+
+First of all we have compared the time to generate X random numbers with both the CPU version and the GPU version. 
+
+Faire une courbe du nombre de random en fonction du nombre de threads, Ã©ventuellement en fonction du nombres de threads par bloc.
 
 
 
@@ -665,212 +883,8 @@ Indeed this result is weaker than the theorem establishing the chaos for the fin
 
 
 
-\section{Efficient prng based on chaotic iterations}
-
-In  order to  implement efficiently  a PRNG  based on  chaotic iterations  it is
-possible to improve  previous works [ref]. One solution  consists in considering
-that the  strategy used contains all the  bits for which the  negation is
-achieved out. Then in order to apply  the negation on these bits we can simply
-apply the  xor operator between  the current number  and the strategy. In
-order to obtain the strategy we also use a classical PRNG.
-
-Here  is an  example with  16-bits numbers  showing how  the bit  operations are
-applied.  Suppose  that $x$ and the  strategy $S^i$ are defined  in binary mode.
-Then the following table shows the result of $x$ xor $S^i$.
-$$
-\begin{array}{|cc|cccccccccccccccc|}
-\hline
-x      &=&1&0&1&1&1&0&1&0&1&0&0&1&0&0&1&0\\
-\hline
-S^i      &=&0&1&1&0&0&1&1&0&1&1&1&0&0&1&1&1\\
-\hline
-x \oplus S^i&=&1&1&0&1&1&1&0&0&0&1&1&1&0&1&0&1\\
-\hline
 
-\hline
- \end{array}
-$$
 
-%% \begin{figure}[htbp]
-%% \begin{center}
-%% \fbox{
-%% \begin{minipage}{14cm}
-%% unsigned int CIprng() \{\\
-%%   static unsigned int x = 123123123;\\
-%%   unsigned long t1 = xorshift();\\
-%%   unsigned long t2 = xor128();\\
-%%   unsigned long t3 = xorwow();\\
-%%   x = x\textasciicircum (unsigned int)t1;\\
-%%   x = x\textasciicircum (unsigned int)(t2$>>$32);\\
-%%   x = x\textasciicircum (unsigned int)(t3$>>$32);\\
-%%   x = x\textasciicircum (unsigned int)t2;\\
-%%   x = x\textasciicircum (unsigned int)(t1$>>$32);\\
-%%   x = x\textasciicircum (unsigned int)t3;\\
-%%   return x;\\
-%% \}
-%% \end{minipage}
-%% }
-%% \end{center}
-%% \caption{sequential Chaotic Iteration PRNG}
-%% \label{algo:seqCIprng}
-%% \end{figure}
-
-
-
-\lstset{language=C,caption={C code of the sequential chaotic iterations based PRNG},label=algo:seqCIprng}
-\begin{lstlisting}
-unsigned int CIprng() {
-  static unsigned int x = 123123123;
-  unsigned long t1 = xorshift();
-  unsigned long t2 = xor128();
-  unsigned long t3 = xorwow();
-  x = x^(unsigned int)t1;
-  x = x^(unsigned int)(t2>>32);
-  x = x^(unsigned int)(t3>>32);
-  x = x^(unsigned int)t2;
-  x = x^(unsigned int)(t1>>32);
-  x = x^(unsigned int)t3;
-  return x;
-}
-\end{lstlisting}
-
-
-
-
-
-In listing~\ref{algo:seqCIprng}  a sequential  version of our  chaotic iterations
-based PRNG  is presented. The xor operator is represented by \textasciicircum. This function  uses three classical  64-bits PRNG: the
-\texttt{xorshift},   the  \texttt{xor128}  and   the  \texttt{xorwow}.   In  the
-following,  we  call them  xor-like  PRNGSs.   These  three PRNGs  are  presented
-in~\cite{Marsaglia2003}.  As each xor-like  PRNG used works with 64-bits  and as our PRNG
-works  with 32-bits, the  use of  \texttt{(unsigned int)}  selects the  32 least
-significant bits  whereas \texttt{(unsigned int)(t3$>>$32)} selects  the 32 most
-significants  bits of the  variable \texttt{t}.  So to  produce a  random number
-realizes 6  xor operations with  6 32-bits numbers  produced by 3  64-bits PRNG.
-This version  successes the  BigCrush of the  TestU01 battery [P.   Lâecuyer and
-  R. Simard. Testu01].
-
-\section{Efficient prng based on chaotic iterations on GPU}
-
-In  order to benefit  from computing  power of  GPU, a  program needs  to define
-independent blocks of threads which  can be computed simultaneously. In general,
-the larger the number of threads is,  the more local memory is used and the less
-branching  instructions are  used (if,  while, ...),  the better  performance is
-obtained  on  GPU.  So  with  algorithm  \ref{algo:seqCIprng}  presented in  the
-previous section, it is possible to  build a similar program which computes PRNG
-on  GPU. In  the CUDA  [ref] environment,  threads have  a  local identificator,
-called \texttt{ThreadIdx} relative to the block containing them.
-
-
-\subsection{Naive version for GPU}
-
-From the CPU version, it is possible  to obtain a quite similar version for GPU.
-The principe consists in assigning the computation of a PRNG as in sequential to
-each thread  of the  GPU.  Of course,  it is  essential that the  three xor-like
-PRNGs  used for  our computation  have different  parameters. So  we  chose them
-randomly with  another PRNG. As the  initialisation is performed by  the CPU, we
-have chosen to use the ISAAC PRNG  [ref] to initalize all the parameters for the
-GPU version  of our  PRNG.  The  implementation of the  three xor-like  PRNGs is
-straightforward  as soon  as their  parameters have  been allocated  in  the GPU
-memory. Each xor-like  PRNGs used works with an internal  number $x$ which keeps
-the last generated random numbers. Other internal variables are also used by the
-xor-like PRNGs. More  precisely, the implementation of the  xor128, the xorshift
-and  the xorwow  respectively  require 4,  5  and 6  unsigned  long as  internal
-variables.
-
-\begin{algorithm}
-
-\KwIn{InternalVarXorLikeArray: array with internal variables of the 3 xor-like PRNGs in global memory\;
-NumThreads: Number of threads\;}
-\KwOut{NewNb: array containing random numbers in global memory}
-\If{threadIdx is concerned by the computation} {
-  retrieve data from InternalVarXorLikeArray[threadIdx] in local variables\;
-  \For{i=1 to n} {
-    compute a new PRNG as in Listing\ref{algo:seqCIprng}\;
-    store the new PRNG in NewNb[NumThreads*threadIdx+i]\;
-  }
-  store internal variables in InternalVarXorLikeArray[threadIdx]\;
-}
-
-\caption{main kernel for the chaotic iterations based PRNG GPU naive version}
-\label{algo:gpu_kernel}
-\end{algorithm}
-
-Algorithm~\ref{algo:gpu_kernel}  presents a naive  implementation of  PRNG using
-GPU.  According  to the available  memory in the  GPU and the number  of threads
-used simultenaously,  the number  of random numbers  that a thread  can generate
-inside   a    kernel   is   limited,   i.e.    the    variable   \texttt{n}   in
-algorithm~\ref{algo:gpu_kernel}. For example, if  $100,000$ threads are used and
-if $n=100$\footnote{in fact, we need to add the initial seed (a 32-bits number)}
-then   the  memory   required   to  store   internals   variables  of   xor-like
-PRNGs\footnote{we multiply this number by $2$ in order to count 32-bits numbers}
-and  random  number of  our  PRNG  is  equals to  $100,000\times  ((4+5+6)\times
-2+(1+100))=1,310,000$ 32-bits numbers, i.e. about $52$Mb.
-
-All the  tests performed  to pass the  BigCrush of TestU01  succeeded. Different
-number of threads, called \texttt{NumThreads} in our algorithm, have been tested
-upto $10$ millions.
-
-\begin{remark}
-Algorithm~\ref{algo:gpu_kernel}  has  the  advantage to  manipulate  independent
-PRNGs, so this version is easily usable on a cluster of computer. The only thing
-to ensure is to use a single ISAAC PRNG. For this, a simple solution consists in
-using a master node for the initialization which computes the initial parameters
-for all the differents nodes involves in the computation.
-\end{remark}
-
-\subsection{Improved version for GPU}
-
-As GPU cards using CUDA have shared memory between threads of the same block, it
-is possible  to use this  feature in order  to simplify the  previous algorithm,
-i.e. using less  than 3 xor-like PRNGs. The solution  consists in computing only
-one xor-like PRNG by thread, saving  it into shared memory and using the results
-of some  other threads in the  same block of  threads. In order to  define which
-thread uses the result of which other  one, we can use a permutation array which
-contains  the indexes  of  all threads  and  for which  a  permutation has  been
-performed.  In Algorithm~\ref{algo:gpu_kernel2}, 2 permutations arrays are used.
-The    variable   \texttt{offset}    is    computed   using    the   value    of
-\texttt{permutation\_size}.   Then we  can compute  \texttt{o1}  and \texttt{o2}
-which represent the indexes of the  other threads for which the results are used
-by the  current thread. In  the algorithm, we  consider that a  64-bits xor-like
-PRNG is used, that is why both 32-bits parts are used.
-
-\begin{algorithm}
-
-\KwIn{InternalVarXorLikeArray: array with internal variables of 1 xor-like PRNGs in global memory\;
-NumThreads: Number of threads\;
-tab1, tab2: Arrays containing permutations of size permutation\_size\;}
-
-\KwOut{NewNb: array containing random numbers in global memory}
-\If{threadId is concerned} {
-  retrieve data from InternalVarXorLikeArray[threadId] in local variables\;
-  offset = threadIdx\%permutation\_size\;
-  o1 = threadIdx-offset+tab1[offset]\;
-  o2 = threadIdx-offset+tab2[offset]\;
-  \For{i=1 to n} {
-    t=xor-like()\;
-    shared\_mem[threadId]=(unsigned int)t\;
-    x = x $\oplus$ (unsigned int) t\;
-    x = x $\oplus$ (unsigned int) (t>>32)\;
-    x = x $\oplus$ shared[o1]\;
-    x = x $\oplus$ shared[o2]\;
-
-    store the new PRNG in NewNb[NumThreads*threadId+i]\;
-  }
-  store internal variables in InternalVarXorLikeArray[threadId]\;
-}
-
-\caption{main kernel for the chaotic iterations based PRNG GPU efficient version}
-\label{algo:gpu_kernel2}
-\end{algorithm}
-\section{Experiments}
-
-Differents experiments have been performed in order to measure the generation speed.
-
-
-First of all we have compared the time to generate X random numbers with both the CPU version and the GPU version. 
-
-Faire une courbe du nombre de random en fonction du nombre de threads, Ã©ventuellement en fonction du nombres de threads par bloc.
 
 
 \section{Conclusion}
