\newcommand{\MI}{\mathit{MaxIter}}
+\usepackage{array}
+\usepackage{color, colortbl}
+\newcolumntype{M}[1]{>{\centering\arraybackslash}m{#1}}
+\newcolumntype{Z}[1]{>{\raggedleft}m{#1}}
\begin{document}
As a first step, the algorithm was run on a network consisting of two clusters
containing fifty hosts each, totaling one hundred hosts. Various combinations of
-the above factors have providing the results shown in Table~\ref{tab.cluster.2x50} with a matrix size
+the above factors have providing the results shown in Table I with a matrix size
ranging from $N_x = N_y = N_z = 62 \text{ to } 171$ elements or from $62^{3} = \np{238328}$ to
$171^{3} = \np{5211000}$ entries.
+\begin{table}[h!]
+ \centering
+
+ \tiny
+
+\begin{tabular}{|Z{0.55cm}|Z{0.25cm}|Z{0.25cm}|M{0.25cm}|Z{0.25cm}|M{0.25cm}|M{0.25cm}|M{0.25cm}|M{0.25cm}|M{0.25cm}|M{0.25cm}|M{0.25cm}|M{0.25cm}|M{0.25cm}|}
+ \hline
+ \bf bw & 5 &5 & 5 & 5 & 5 & 50 & 50 & 50 & 50 & 50 & 10 & 10\\
+ \hline
+ \bf lat & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.03 & 0.01\\
+ \hline
+ \bf power & 1 & 1 & 1 & 1.5 & 1.5 & 1.5 & 1.5 & 1.5 & 1.5 & 1.5 & 1 & 1.5\\ \hline \bf size & 62 & 62 & 62 & 100 & 100 & 110 & 120& 130 & 140 & 150 & 171 & 171\\ \hline
+ \bf Prec/Eprec & 10$^{-5}$ & 10$^{-8}$ & 10$^{-9}$ & 10$^{-11}$ & 10$^{-11}$ & 10$^{-11}$ & 10$^{-11}$ & 10$^{-11}$ & 10$^{-11}$ & 10$^{-11}$ & 10$^{-5}$ & 10$^{-5}$\\ \hline
+ \bf speedup & 0.396 & 0.392 & 0.396 & 0.391 & 0.393 & 0.395 & 0.398 & 0.388 & 0.393 & 0.394 & 0.63 & 0.778\\ \hline
+ \end{tabular}
+ \smallskip
+ \caption{2 Clusters x 50 nodes each} \label{tab1}
+\end{table}
+
Then we have changed the network configuration using three clusters containing
respectively 33, 33 and 34 hosts, or again by on hundred hosts for all the
clusters. In the same way as above, a judicious choice of key parameters has
-permitted to get the results in Table~\ref{tab.cluster.3x33} which shows the speedups less than 1 with
+permitted to get the results in Table II which shows the speedups less than 1 with
a matrix size from 62 to 100 elements.
+\begin{table}[h!]
+ \centering
+
+ \tiny
+
+\begin{tabular}{|Z{0.55cm}|Z{0.25cm}|Z{0.25cm}|M{0.25cm}|Z{0.25cm}|M{0.25cm}|M{0.25cm}|}
+ \hline
+ \bf bw & 10 &5 & 4 & 3 & 2 & 6\\ \hline
+ \bf lat & 0.01 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02\\
+ \hline
+ \bf power & 1 & 1 & 1 & 1 & 1 & 1\\ \hline
+ \bf size & 62 & 100 & 100 & 100 & 100 & 171\\ \hline
+ \bf Prec/Eprec & 10$^{-5}$ & 10$^{-5}$ & 10$^{-5}$ & 10$^{-5}$ & 10$^{-5}$ & 10$^{-5}$\\ \hline
+ \bf speedup & 0.997 & 0.99 & 0.93 & 0.84 & 0.78 & 0.99\\
+ \hline
+ \end{tabular}
+ \smallskip
+ \caption{3 Clusters x 33 nodes each} \label{tab2}
+\end{table}
+
+
In a final step, results of an execution attempt to scale up the three clustered
-configuration but increasing by two hundreds hosts has been recorded in Table~\ref{tab.cluster.3x67}.
+configuration but increasing by two hundreds hosts has been recorded in Table III.
+
+\begin{table}[h!]
+ \centering
+ \tiny
+\begin{tabular}{|M{0.55cm}|M{0.25cm}|}
+ \hline
+ \bf bw & 1\\ \hline
+ \bf lat & 0.02\\
+ \hline
+ \bf power & 1\\
+ \hline
+ \bf size & 62\\
+ \hline
+ \bf Prec/Eprec & 10$^{-5}$\\
+ \hline
+ \bf speedup & 0.9\\
+ \hline
+ \end{tabular}
+ \smallskip
+ \caption{3 Clusters x 66 nodes each} \label{tab3}
+\end{table}
Note that the program was run with the following parameters:
\item Execution Mode: synchronous or asynchronous.
\end{itemize}
-\begin{table}
- \centering
- \caption{2 clusters X 50 nodes}
- \label{tab.cluster.2x50}
- \AG{Ces tableaux (\ref{tab.cluster.2x50}, \ref{tab.cluster.3x33} et
- \ref{tab.cluster.3x67}) sont affreux. Utiliser un format vectoriel (eps ou
- pdf) ou, mieux, les réécrire en \LaTeX{}. Réécrire les légendes proprement
- également (\texttt{\textbackslash{}times} au lieu de \texttt{X} par ex.)}
- \includegraphics[width=209pt]{img1.jpg}
-\end{table}
-
-\begin{table}
- \centering
- \caption{3 clusters X 33 nodes}
- \label{tab.cluster.3x33}
- \AG{Refaire le tableau.}
- \includegraphics[width=209pt]{img2.jpg}
-\end{table}
-
-\begin{table}
- \centering
- \caption{3 clusters X 67 nodes}
- \label{tab.cluster.3x67}
- \AG{Refaire le tableau.}
-% \includegraphics[width=160pt]{img3.jpg}
- \includegraphics[scale=0.5]{img3.jpg}
-\end{table}
\paragraph*{Interpretations and comments}
After analyzing the outputs, generally, for the configuration with two or three
-clusters including one hundred hosts (Tables~\ref{tab.cluster.2x50} and~\ref{tab.cluster.3x33}), some combinations of the
+clusters including one hundred hosts (Table I and II), some combinations of the
used parameters affecting the results have given a speedup less than 1, showing
the effectiveness of the asynchronous performance compared to the synchronous
mode.
-In the case of a two clusters configuration, Table~\ref{tab.cluster.2x50} shows that with a
+In the case of a two clusters configuration, Table I shows that with a
deterioration of inter cluster network set with \np[Mbits/s]{5} of bandwidth, a latency
in order of a hundredth of a millisecond and a system power of one GFlops, an
efficiency of about \np[\%]{40} in asynchronous mode is obtained for a matrix size of 62
obtained with high external precision of \np{E-11} for a matrix size from 110 to 150
side elements.
-For the 3 clusters architecture including a total of 100 hosts, Table~\ref{tab.cluster.3x33} shows
+For the 3 clusters architecture including a total of 100 hosts, Table II shows
that it was difficult to have a combination which gives an efficiency of
asynchronous below \np[\%]{80}. Indeed, for a matrix size of 62 elements, equality
between the performance of the two modes (synchronous and asynchronous) is
A last attempt was made for a configuration of three clusters but more powerful
with 200 nodes in total. The convergence with a speedup of \np[\%]{90} was obtained
-with a bandwidth of \np[Mbits/s]{1} as shown in Table~\ref{tab.cluster.3x67}.
+with a bandwidth of \np[Mbits/s]{1} as shown in Table III.
\section{Conclusion}
The experimental results on executing a parallel iterative algorithm in
