\section{Efficient prng based on chaotic iterations}
-On parle du séquentiel avec des nombres 64 bits\\
-
-
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 instead of applying the negation on these bits we can simply
+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{
-In listing~\ref{algo:seqCIprng} a sequential version of our chaotic iterations
-based PRNG is presented. This version 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$>>$32)} selects the 32 most
-significants bits of the variable \texttt{t}. So to produce a random number
-realizes 6 xor operations with 6 32-bits numbers produced by 3 64-bits PRNG.
-This version successes the BigCrush of the TestU01 battery [P. L’ecuyer and
- R. Simard. Testu01].
+In listing~\ref{algo:seqCIprng} a sequential version of our chaotic iterations
+based PRNG is presented. The xor operator is represented by
+\textasciicircum. This function uses three classical 64-bits PRNG: the
+\texttt{xorshift}, the \texttt{xor128} and the \texttt{xorwow}. In the
+following, we call them xor-like PRNGSs. These three PRNGs are presented
+in~\cite{Marsaglia2003}. As each xor-like PRNG used works with 64-bits and as
+our PRNG works with 32-bits, the use of \texttt{(unsigned int)} selects the 32
+least significant bits whereas \texttt{(unsigned int)(t3$>>$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. L’ecuyer
+ and R. Simard. Testu01].
\section{Efficient prng based on chaotic iterations on GPU}
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. 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 to initalize all the parameters for the GPU
-version of our PRNG. The implementation of the three xor-like PRNGs is
+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
\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{threadId is concerned} {
- retrieve data from InternalVarXorLikeArray[threadId] in local variables\;
+\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*threadId+i]\;
+ store the new PRNG in NewNb[NumThreads*threadIdx+i]\;
}
- store internal variables in InternalVarXorLikeArray[threadId]\;
+ store internal variables in InternalVarXorLikeArray[threadIdx]\;
}
-\caption{main kernel for the chaotic iterations based PRNG GPU version}
+\caption{main kernel for the chaotic iterations based PRNG GPU naive version}
\label{algo:gpu_kernel}
\end{algorithm}
-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} 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
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}
-On passe le BigCrush\\
-On donne des temps de générations sur GPU/CPU\\
-On donne des temps de générations de nombre sur GPU puis on rappatrie sur CPU / CPU ? bof bof, on verra
+Differents experiments have been performed in order to measure the generation speed.
+\begin{figure}[t]
+\begin{center}
+ \includegraphics[scale=.5]{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, éventuellement en fonction du nombres de threads par bloc.
\section{Conclusion}
