Remove useless \usepackage.

[mpi-energy.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index 2838f2db74b4ad49a12bba6ccf8179641a4700ec..949c10ca07224683d85186f5f19c9c820da0af5d 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -3,11 +3,9 @@
 \usepackage[T1]{fontenc}
 \usepackage[utf8]{inputenc}
 \usepackage[english]{babel}
 \usepackage[T1]{fontenc}
 \usepackage[utf8]{inputenc}
 \usepackage[english]{babel}

-\usepackage{algorithm,algorithmicx,algpseudocode}
+\usepackage{algpseudocode}

 \usepackage{graphicx,graphics}
 \usepackage{subfig}
 \usepackage{graphicx,graphics}
 \usepackage{subfig}

-\usepackage{listings}
-\usepackage{colortbl}

 \usepackage{amsmath}
 
 \usepackage{url}
 \usepackage{amsmath}
 
 \usepackage{url}


@@ -339,17 +337,19 @@ In the same way we can normalize the performance as follows:
 The second problem is that the optimization operation for both energy and
 performance is not in the same direction.  In other words, the normalized energy
 and the performance curves are not at the same direction see
 The second problem is that the optimization operation for both energy and
 performance is not in the same direction.  In other words, the normalized energy
 and the performance curves are not at the same direction see

-Figure~(\ref{fig:r2}).  While the main goal is to optimize the energy and
-performance in the same time.  According to the equations~(\ref{eq:enorm})
-and~(\ref{eq:pnorm}), the scaling factor $S$ reduce both the energy and the
-performance simultaneously.  But the main objective is to produce maximum energy
-reduction with minimum performance reduction.  Many researchers used different
-strategies to solve this nonlinear problem for example see~\cite{19,42}, their
-methods add big overheads to the algorithm to select the suitable frequency.
-In this paper we present a method to find the optimal scaling factor $S$ to optimize both energy and performance simultaneously without adding a big
-overhead.  Our solution for this problem is to make the optimization process
-for energy and performance follow the same direction.  Therefore, we inverse the equation of the normalized
-performance as follows:
+Figure~\ref{fig:rel}\subref{fig:r2}.  While the main goal is to optimize the
+energy and performance in the same time.  According to the
+equations~(\ref{eq:enorm}) and~(\ref{eq:pnorm}), the scaling factor $S$ reduce
+both the energy and the performance simultaneously.  But the main objective is
+to produce maximum energy reduction with minimum performance reduction.  Many
+researchers used different strategies to solve this nonlinear problem for
+example see~\cite{19,42}, their methods add big overheads to the algorithm to
+select the suitable frequency.  In this paper we present a method to find the
+optimal scaling factor $S$ to optimize both energy and performance
+simultaneously without adding a big overhead.  Our solution for this problem is
+to make the optimization process for energy and performance follow the same
+direction.  Therefore, we inverse the equation of the normalized performance as
+follows:

 \begin{equation}
   \label{eq:pnorm_en}
   P^{-1}_\textit{Norm} = \frac{ T_\textit{Old}}{ T_\textit{New}}
 \begin{equation}
   \label{eq:pnorm_en}
   P^{-1}_\textit{Norm} = \frac{ T_\textit{Old}}{ T_\textit{New}}


@@ -359,23 +359,22 @@ performance as follows:
 \end{equation}
 \begin{figure}
   \centering
 \end{equation}
 \begin{figure}
   \centering

-  \subfloat[Converted relation.]{%
-    \includegraphics[width=.24\textwidth]{fig/file}\label{fig:r1}}%
-%  \quad%

   \subfloat[Real relation.]{%
   \subfloat[Real relation.]{%

-    \includegraphics[width=.24\textwidth]{fig/file3}\label{fig:r2}}
-  \label{fig:rel}
+    \includegraphics[width=.5\linewidth]{fig/file3}\label{fig:r2}}%
+  \subfloat[Converted relation.]{%
+    \includegraphics[width=.5\linewidth]{fig/file}\label{fig:r1}}

   \caption{The energy and performance relation}
   \caption{The energy and performance relation}

+  \label{fig:rel}

 \end{figure}
 Then, we can model our objective function as finding the maximum distance
 between the energy curve EQ~(\ref{eq:enorm}) and the inverse of performance
 curve EQ~(\ref{eq:pnorm_en}) over all available scaling factors.  This
 represents the minimum energy consumption with minimum execution time (better
 \end{figure}
 Then, we can model our objective function as finding the maximum distance
 between the energy curve EQ~(\ref{eq:enorm}) and the inverse of performance
 curve EQ~(\ref{eq:pnorm_en}) over all available scaling factors.  This
 represents the minimum energy consumption with minimum execution time (better

-performance) at the same time, see Figure~(\ref{fig:r1}).  Then our objective
-function has the following form:
+performance) at the same time, see Figure~\ref{fig:rel}\subref{fig:r1}.  Then
+our objective function has the following form:

 \begin{equation}
   \label{eq:max}
 \begin{equation}
   \label{eq:max}

-  Max Dist = \max_{j=1,2,\dots,F}
+  \textit{Max Dist} = \max_{j=1,2,\dots,F}

       (\overbrace{P^{-1}_\textit{Norm}(S_j)}^{\text{Maximize}} -
        \overbrace{E_\textit{Norm}(S_j)}^{\text{Minimize}} )
 \end{equation}
       (\overbrace{P^{-1}_\textit{Norm}(S_j)}^{\text{Maximize}} -
        \overbrace{E_\textit{Norm}(S_j)}^{\text{Minimize}} )
 \end{equation}


@@ -388,12 +387,11 @@ form over the available frequency scaling factors as shown in~\cite{15,3,19}.
 \section{Optimal scaling factor for performance and energy}
 \label{sec.optim}
 
 \section{Optimal scaling factor for performance and energy}
 \label{sec.optim}
 

-Algorithm~\ref{EPSA} computes the optimal scaling factor according to the
-objective function described above.
-\begin{algorithm}[tp]
-  \caption{Scaling factor selection algorithm}
-  \label{EPSA}
+Algorithm on Figure~\ref{EPSA} computes the optimal scaling factor according to
+the objective function described above.
+\begin{figure}[tp]

   \begin{algorithmic}[1]
   \begin{algorithmic}[1]

+    % \footnotesize

     \State  Initialize the variable $Dist=0$
     \State Set dynamic and static power values.
     \State Set $P_{states}$ to the number of available frequencies.
     \State  Initialize the variable $Dist=0$
     \State Set dynamic and static power values.
     \State Set $P_{states}$ to the number of available frequencies.


@@ -421,19 +419,22 @@ objective function described above.
     \EndFor
     \State  Return $S_{opt}$
   \end{algorithmic}
     \EndFor
     \State  Return $S_{opt}$
   \end{algorithmic}

-\end{algorithm}  
+  \caption{Scaling factor selection algorithm}
+  \label{EPSA}
+\end{figure}

 
 The proposed algorithm works online during the execution time of the MPI
 program.  It selects the optimal scaling factor after gathering the computation
 and communication times from the program after one iteration.  Then the program
 changes the new frequencies of the CPUs according to the computed scaling
 
 The proposed algorithm works online during the execution time of the MPI
 program.  It selects the optimal scaling factor after gathering the computation
 and communication times from the program after one iteration.  Then the program
 changes the new frequencies of the CPUs according to the computed scaling

-factors. In our experiments over a  homogeneous cluster described in  section~\ref{sec.expe}, 
-this algorithm has a small execution time. It takes \np[$\mu$s]{1.52} on average for 4 nodes and
-\np[$\mu$s]{6.65} on average for 32 nodes.  The algorithm complexity is $O(F\cdot
-N)$, where $F$ is the number of available frequencies and $N$ is the number of
-computing nodes.  The algorithm is called just once during the execution of the
-program.  The DVFS algorithm~(\ref{dvfs}) shows where and when the algorithm is
-called in the MPI program.
+factors. In our experiments over a homogeneous cluster described in
+Section~\ref{sec.expe}, this algorithm has a small execution time. It takes
+\np[$\mu$s]{1.52} on average for 4 nodes and \np[$\mu$s]{6.65} on average for 32
+nodes.  The algorithm complexity is $O(F\cdot N)$, where $F$ is the number of
+available frequencies and $N$ is the number of computing nodes.  The algorithm
+is called just once during the execution of the program.  The DVFS algorithm on
+Figure~\ref{dvfs} shows where and when the algorithm is called in the MPI
+program.

 %\begin{table}[htb]
 %  \caption{Platform file parameters}
 %  % title of Table
 %\begin{table}[htb]
 %  \caption{Platform file parameters}
 %  % title of Table


@@ -450,24 +451,25 @@ called in the MPI program.
 %  \label{table:platform}
 %\end{table}
 
 %  \label{table:platform}
 %\end{table}
 

-\begin{algorithm}[tp]
-  \caption{DVFS}
-  \label{dvfs}
+\begin{figure}[tp]

   \begin{algorithmic}[1]
   \begin{algorithmic}[1]

+    % \footnotesize

     \For {$k:=1$ to \textit{some iterations}}
       \State Computations section.
       \State Communications section.
       \If {$(k=1)$}
         \State Gather all times of computation and\newline\hspace*{3em}%
                communication from each node.
     \For {$k:=1$ to \textit{some iterations}}
       \State Computations section.
       \State Communications section.
       \If {$(k=1)$}
         \State Gather all times of computation and\newline\hspace*{3em}%
                communication from each node.

-        \State Call algorithm~\ref{EPSA} with these times.
+        \State Call algorithm from Figure~\ref{EPSA} with these times.

         \State Compute the new frequency from the\newline\hspace*{3em}%
               returned optimal scaling factor.
         \State Set the new frequency to the CPU.
       \EndIf
     \EndFor
   \end{algorithmic}
         \State Compute the new frequency from the\newline\hspace*{3em}%
               returned optimal scaling factor.
         \State Set the new frequency to the CPU.
       \EndIf
     \EndFor
   \end{algorithmic}

-\end{algorithm}
+  \caption{DVFS algorithm}
+  \label{dvfs}
+\end{figure}

 After obtaining the optimal scaling factor, the program calculates the new
 frequency $F_i$ for each task proportionally to its time value $T_i$.  By
 substitution of EQ~(\ref{eq:s}) in EQ~(\ref{eq:si}), we can calculate the new
 After obtaining the optimal scaling factor, the program calculates the new
 frequency $F_i$ for each task proportionally to its time value $T_i$.  By
 substitution of EQ~(\ref{eq:s}) in EQ~(\ref{eq:si}), we can calculate the new


@@ -501,17 +503,17 @@ execution time values.  These scaling factors are computed by dividing the
 maximum frequency by the new one see EQ~(\ref{eq:s}).
 \begin{figure}
   \centering
 maximum frequency by the new one see EQ~(\ref{eq:s}).
 \begin{figure}
   \centering

-  \includegraphics[width=.24\textwidth]{fig/cg_per}\hfill%
- % \includegraphics[width=.328\textwidth]{fig/mg_pre}\hfill%
- % \includegraphics[width=.4\textwidth]{fig/bt_pre}\qquad%
-   \includegraphics[width=.24\textwidth]{fig/lu_pre}\hfill%
+  \includegraphics[width=.5\linewidth]{fig/cg_per}\hfill%
+ % \includegraphics[width=.5\linewidth]{fig/mg_pre}\hfill%
+ % \includegraphics[width=.5\linewidth]{fig/bt_pre}\qquad%
+   \includegraphics[width=.5\linewidth]{fig/lu_pre}\hfill%

   \caption{Comparing predicted to real execution times}
   \label{fig:pred}
 \end{figure}
 %see Figure~\ref{fig:pred}
 In our cluster there are 18 available frequency states for each processor.  This
 leads to 18 run states for each program.  We use seven MPI programs of the NAS
   \caption{Comparing predicted to real execution times}
   \label{fig:pred}
 \end{figure}
 %see Figure~\ref{fig:pred}
 In our cluster there are 18 available frequency states for each processor.  This
 leads to 18 run states for each program.  We use seven MPI programs of the NAS

-parallel benchmarks: CG, MG, EP, FT, BT, LU and SP.  Figure~(\ref{fig:pred})
+parallel benchmarks: CG, MG, EP, FT, BT, LU and SP.  Figure~\ref{fig:pred}

 presents plots of the real execution times and the simulated ones. The maximum
 normalized error between these two execution times varies between \np{0.0073} to
 \np{0.031} dependent on the executed benchmark.  The smallest prediction error
 presents plots of the real execution times and the simulated ones. The maximum
 normalized error between these two execution times varies between \np{0.0073} to
 \np{0.031} dependent on the executed benchmark.  The smallest prediction error


@@ -522,47 +524,48 @@ The proposed algorithm was applied to seven MPI programs of the NAS benchmarks
 (EP, CG, MG, FT, BT, LU and SP) which were run with three classes (A, B and C).
 For each instance the benchmarks were executed on a number of processors
 proportional to the size of the class.  Each class represents the problem size
 (EP, CG, MG, FT, BT, LU and SP) which were run with three classes (A, B and C).
 For each instance the benchmarks were executed on a number of processors
 proportional to the size of the class.  Each class represents the problem size

-ascending from  class A to C.  Additionally, depending on some speed up
-points for each class we run the classes A, B and C on 4, 8 or 9 and 16 nodes
+ascending from class A to C.  Additionally, depending on some speed up points
+for each class we run the classes A, B and C on 4, 8 or 9 and 16 nodes

 respectively.  Depending on EQ~(\ref{eq:energy}), we measure the energy
 respectively.  Depending on EQ~(\ref{eq:energy}), we measure the energy

-consumption for all the NAS MPI programs while assuming that the dynamic power with
-the highest frequency is equal to \np[W]{20} and the power static is equal to
-\np[W]{4} for all experiments.  These power values were also used by Rauber and
-Rünger in~\cite{3}.  The results showed that the algorithm selected different
-scaling factors for each program depending on the communication features of the
-program as in the plots~(\ref{fig:nas}).  These plots illustrate that there are
-different distances between the normalized energy and the normalized inverted
-performance curves, because there are different communication features for each
-benchmark.  When there are little or no communications, the inverted
-performance curve is very close to the energy curve.  Then the distance between
-the two curves is very small.  This leads to small energy savings.  The opposite
-happens when there are a lot of communication, the distance between the two
-curves is big.  This leads to more energy savings (e.g. CG and FT), see
-Table~(\ref{table:compareC}).  All discovered frequency scaling factors
-optimize both the energy and the performance simultaneously for all NAS
-benchmarks.  In Table~(\ref{table:compareC}), we record all optimal
-scaling factors results for each benchmark running class C.  These scaling
-factors give the maximum energy saving percentage and the minimum performance
-degradation percentage at the same time from all available scaling factors.
+consumption for all the NAS MPI programs while assuming that the dynamic power
+with the highest frequency is equal to \np[W]{20} and the power static is equal
+to \np[W]{4} for all experiments.  These power values were also used by Rauber
+and Rünger in~\cite{3}.  The results showed that the algorithm selected
+different scaling factors for each program depending on the communication
+features of the program as in the plots from Figure~\ref{fig:nas}.  These plots
+illustrate that there are different distances between the normalized energy and
+the normalized inverted performance curves, because there are different
+communication features for each benchmark.  When there are little or no
+communications, the inverted performance curve is very close to the energy
+curve.  Then the distance between the two curves is very small.  This leads to
+small energy savings.  The opposite happens when there are a lot of
+communication, the distance between the two curves is big.  This leads to more
+energy savings (e.g. CG and FT), see Table~\ref{table:compareC}.  All discovered
+frequency scaling factors optimize both the energy and the performance
+simultaneously for all NAS benchmarks.  In Table~\ref{table:compareC}, we record
+all optimal scaling factors results for each benchmark running class C.  These
+scaling factors give the maximum energy saving percentage and the minimum
+performance degradation percentage at the same time from all available scaling
+factors.

 \begin{figure*}[t]
   \centering
 \begin{figure*}[t]
   \centering

-  \includegraphics[width=.33\textwidth]{fig/ep}\hfill%
-  \includegraphics[width=.33\textwidth]{fig/cg}\hfill%
- % \includegraphics[width=.328\textwidth]{fig/sp}
- %  \includegraphics[width=.328\textwidth]{fig/lu}\hfill%
-  \includegraphics[width=.33\textwidth]{fig/bt}\hfill%
- % \includegraphics[width=.328\textwidth]{fig/ft}
+  \includegraphics[width=.33\linewidth]{fig/ep}\hfill%
+  \includegraphics[width=.33\linewidth]{fig/cg}\hfill%
+ % \includegraphics[width=.328\linewidth]{fig/sp}
+ %  \includegraphics[width=.328\linewidth]{fig/lu}\hfill%
+  \includegraphics[width=.33\linewidth]{fig/bt}
+ % \includegraphics[width=.328\linewidth]{fig/ft}

   \caption{Optimal scaling factors for the predicted energy and performance of NAS benchmarks}
   \label{fig:nas}
 \end{figure*}
 
   \caption{Optimal scaling factors for the predicted energy and performance of NAS benchmarks}
   \label{fig:nas}
 \end{figure*}
 

-As shown in Table~(\ref{table:compareC}), when the optimal scaling
-factor has a big value we can gain more energy savings  as in CG and
-FT benchmarks.  The opposite happens when the optimal scaling factor has a  
-small value as in BT and EP benchmarks.  Our algorithm selects a big scaling 
-factor value when the communication and the other slacks times are big and smaller 
-ones in opposite cases.  In EP there are no communication inside the iterations.  
-This leads our algorithm to select smaller scaling factor values (inducing smaller energy
+As shown in Table~\ref{table:compareC}, when the optimal scaling factor has a
+big value we can gain more energy savings as in CG and FT benchmarks.  The
+opposite happens when the optimal scaling factor has a small value as in BT and
+EP benchmarks.  Our algorithm selects a big scaling factor value when the
+communication and the other slacks times are big and smaller ones in opposite
+cases.  In EP there are no communication inside the iterations.  This leads our
+algorithm to select smaller scaling factor values (inducing smaller energy

 savings).
 
 \subsection{Results comparison}
 savings).
 
 \subsection{Results comparison}


@@ -575,7 +578,7 @@ is similar to the first except setting the slower task to the maximum frequency
 (when the scale $S=1$) to keep the performance from degradation as mush as
 possible.  We refer to this scenario as $R_{E-P}$.  While we refer to our
 algorithm as EPSA (Energy to Performance Scaling Algorithm).  The comparison is
 (when the scale $S=1$) to keep the performance from degradation as mush as
 possible.  We refer to this scenario as $R_{E-P}$.  While we refer to our
 algorithm as EPSA (Energy to Performance Scaling Algorithm).  The comparison is

-made in Table ~\ref{table:compareC}.  This table shows the results of our method and
+made in Table~\ref{table:compareC}.  This table shows the results of our method and

 Rauber and Rünger scenarios for all the NAS benchmarks programs for class C.
 
 \begin{table}
 Rauber and Rünger scenarios for all the NAS benchmarks programs for class C.
 
 \begin{table}


@@ -635,9 +638,9 @@ while Rauber and Rünger's method, ($R_{E-P}$), gives sometimes negative
 trade-offs such as in BT and EP.
 \begin{figure}[t]
   \centering
 trade-offs such as in BT and EP.
 \begin{figure}[t]
   \centering

-%  \includegraphics[width=.328\textwidth]{fig/compare_class_A}
-%  \includegraphics[width=.328\textwidth]{fig/compare_class_B}
-  \includegraphics[width=.49\textwidth]{fig/compare_class_C}
+%  \includegraphics[width=.328\linewidth]{fig/compare_class_A}
+%  \includegraphics[width=.328\linewidth]{fig/compare_class_B}
+  \includegraphics[width=\linewidth]{fig/compare_class_C}

   \caption{Comparing our method to Rauber and Rünger's methods}
   \label{fig:compare}
 \end{figure}
   \caption{Comparing our method to Rauber and Rünger's methods}
   \label{fig:compare}
 \end{figure}