X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/ThesisAhmed.git/blobdiff_plain/fc62606e9b5f178540e42e01dcc280b817c9fc70..399ba08408333d926285a5769d30aa91c8d5a120:/thesis-presentation/AhmedSlides.tex

diff --git a/thesis-presentation/AhmedSlides.tex b/thesis-presentation/AhmedSlides.tex
index a7b6a84..9bb4b6f 100644
--- a/thesis-presentation/AhmedSlides.tex
+++ b/thesis-presentation/AhmedSlides.tex
@@ -75,8 +75,8 @@
 %Iterations using CPU Frequency Scaling} 
 \vspace{2cm}
 
-\title{   \textbf{Energy Consumption Optimization of   Parallel Applications with Iterations   using CPU Frequency Scaling} \\ \vspace{0.2cm} \hspace{1.8cm}\textbf{\textcolor{cyan}{\small PhD Dissertation Defense}}}\vspace{-1cm}
-\author{ \textbf{Ahmed Badri Muslim Fanfakh} \\ \vspace{0.5cm}\small Under Supervision: \textcolor{cyan}{\small  Raphaël COUTURIER and Jean-Claude CHARR} \\\vspace{0.1cm} \textcolor{blue}{ University of Franche-Comté - FEMTO-ST - DISC Dept.  - AND Team} \\ ~~~~~~~~~~~~~~~~~~~~~ \textbf{\textcolor{blue}{ 17 October 2016 }}} 
+\title{   \textbf{Energy Consumption Optimization of   Parallel Applications with Iterations   using CPU Frequency Scaling} \\ \vspace{0.2cm} \hspace{1.8cm}\textbf{\textcolor{cyan}{\small PhD Dissertation Defense}}}\vspace{-0.5cm}
+\author{ \textbf{Ahmed Badri Muslim Fanfakh} \\ \vspace{0.5cm}\small Under the supervision of: \\ \textcolor{cyan}{\small  Raphaël COUTURIER and Jean-Claude CHARR} \\\vspace{0.1cm} \textcolor{blue}{ UBFC - FEMTO-ST - DISC Dept.  - AND Team} \\ ~~~~~~~~~~~~~~~~~~~~~ \textbf{\textcolor{blue}{ 17 October 2016 }}} 
 
 \date{}
 \vspace{-3cm}
@@ -115,9 +115,9 @@
 %%%%%%%%%%%%%%%%%%%% 
 \begin{frame}{Introduction and problem definition}
  \section{\small {Introduction and Problem definition}}
-   \bf \textcolor{blue}{Approaches to increase the computing power:}
+   \bf \textcolor{blue}{To get more computing power:}
      \begin{minipage}{0.5\textwidth} 
-      \textcolor{blue}{1)} \small  \bf \textcolor{black}{Increasing the frequency of a  processor}
+      \textcolor{blue}{1)} \small  \bf \textcolor{black}{Increase the frequency of a  processor.\\ (limited due to overheating)}
     \end{minipage}%
     \begin{minipage}{0.6\textwidth} 
     
@@ -128,7 +128,10 @@
     \end{minipage}%
     \vspace{0.2cm}
     \begin{minipage}{0.5\textwidth} 
-     \textcolor{blue}{2)} \small \bf \textcolor{black}{Increasing the number of nodes}        
+     \textcolor{blue}{2)} \small \bf \textcolor{black}{Use more nodes.}
+     
+ \textcolor{black}{The supercomputer Tianhe-2 has more than 3 million cores and consumes around 17.8 megawatts.}  
+          
     \end{minipage}%
     \begin{minipage}{0.6\textwidth} 
     \begin{figure}[h!]
@@ -143,14 +146,13 @@
  %%%%%%%%%%%%%%%%%%%
 %%    SLIDE 04   %%
 %%%%%%%%%%%%%%%%%%%% 
-\begin{frame}{Introduction and problem definition}
- \vspace{0.1cm}
- \bf \textcolor{blue}{Techniques for energy consumption reduction}
- 
+\begin{frame}{Techniques for energy consumption reduction}
+
      \textcolor{blue}{1)} \bf \textcolor{black}{Switch-off idle nodes method}  
     \vspace{-0.9cm}
     \begin{figure}
      \animategraphics[autopause,loop,controls,scale=0.25,buttonsize=0.2cm]{200}{on-off/a-}{0}{69}
+     %\includegraphics[width=0.6\textwidth]{on-off/a-69}
     \end{figure}
  \end{frame}
 
@@ -160,9 +162,10 @@
 \begin{frame}{Techniques for energy consumption reduction}
  
   \textcolor{blue}{2)} \bf \textcolor{black}{Dynamic voltage and frequency Scaling (DVFS)}
-     \vspace{-0.5cm}
+     \vspace{-0.9cm}
     \begin{figure}
-     \animategraphics[autopause,controls,scale=0.25,buttonsize=0.2cm]{10}{DVFS-meq/a-}{0}{109}
+    \animategraphics[autopause,controls,scale=0.25,buttonsize=0.2cm]{10}{DVFS-meq/a-}{0}{109}
+     %\includegraphics[width=0.6\textwidth]{DVFS-meq/a-109}
     \end{figure}
     \end{frame}
  
@@ -174,12 +177,12 @@
 \begin{frame}{Motivations}
 \vspace{0.05cm}
 \section{\small {Motivations}}
-\textcolor{blue}{Why we used DVFS method:}
+\textcolor{blue}{Why we used the DVFS method:}
 \vspace{-0.49cm}
 \begin{minipage}{0.5\textwidth} 
     \vspace{-0.49cm} 
       \begin{itemize} 
-       \item  \small \textcolor{black}{The biggest power consumption is consumed by a processor \textsuperscript{1}. }
+       \item  \small \textcolor{black}{ The CPU is the component that consumes the  highest amount of energy in a node \textsuperscript{1}. }
   		
  	 \end{itemize}
 
@@ -192,8 +195,9 @@
     \end{figure}
     \end{minipage}%
     
-  \begin{itemize} \item \small  \textcolor{black}{It used to reduce the energy consumption  while keeping all the node working, thus  it is more adapted to parallel computing.}
-  		\item \small \textcolor{black}{It has a very small overhead compared to switching-off the idle nodes method.}  \end{itemize} 
+  \begin{itemize} \item \small  \textcolor{black}{DVFS reduces the energy consumption while 
+   keeping all the nodes working.}
+  		\item \small \textcolor{black}{It has a very small overhead compared to switching-off the idle nodes.}  \end{itemize} 
     
 \vspace{-0.12cm}
 
@@ -202,7 +206,7 @@
   	\small	\textcolor{blue}{Challenge:} \textcolor{black}{DVFS is used to reduce the energy consumption, \textcolor{blue}{but} it degrades the performance simultaneously.}
   		
   		\vspace{0.1cm}
- \small	 \textcolor{blue}{Objective:} \textcolor{black}{Applying the DVFS to minimize the energy consumption while maintaining the performance of the parallel applications.}
+ \small	 \textcolor{blue}{Objective:} \textcolor{black}{Applying the DVFS to minimize the energy consumption while maintaining the performance of the parallel application.}
 \end{block}
  
  \tiny \textsuperscript{1} Fan, X., Weber, W., and Barroso, L. A. 2007.  Power provisioning
@@ -217,9 +221,12 @@ for a warehouse-sized computer.
 %%%%%%%%%%%%%%%%%%%% 
 
 
-\begin{frame}{Contribution}
+\begin{frame}{The first contribution}
+
+\section{\small {Energy optimization of a homogeneous platform}}
+%\vspace{-3cm}
+ % \includegraphics[width=0.6\textwidth]{white.pdf} 
 
-\section{\small {Energy optimization of homogeneous platform}}
 \begin{center}
 \bf  \Large \textcolor{blue}{Energy optimization of a parallel application with iterations running over a homogeneous platform}
 \end{center}
@@ -234,11 +241,10 @@ for a warehouse-sized computer.
 \begin{frame}{Objectives}
         \begin{femtoBlock}{} \vspace{-12 mm}
                 \begin{itemize} \small
-                   \item  Study the effect of the scaling factor $S$ on \textbf{energy consumption and performance } of parallel  applications with iterations such as NAS 
-                          Benchmarks. \includegraphics[width=.06\textwidth]{c1/nasa.pdf} \medskip
+                   \item  Study the effect of the scaling factor on the \textbf{energy consumption and performance } of parallel  applications with iterations. \medskip
                    
                    \item  Discovering the \textbf{energy-performance trade-off relation} when changing the frequency of the processor.\medskip
-                   \item  Proposing an algorithm for selecting the scaling factor $S$ producing \textbf {the optimal trade-off} between the energy consumption and the performance. \medskip
+                   \item  Proposing an algorithm for selecting the scaling factor that produces  \textbf {the optimal trade-off} between the energy consumption and the performance. \medskip
                    \item  Comparing the proposed algorithm to existing methods.
                    
                    
@@ -261,9 +267,9 @@ for a warehouse-sized computer.
 \vspace{-0.5 cm}
 \begin{figure}
   \centering
-  \subfloat[Sync. imbalanced communications]{%
+  \subfloat[Synchronous imbalanced communications]{%
     \includegraphics[scale=0.49]{c1/commtasks}\label{fig:h1}}
-  \subfloat[Sync. imbalanced computations]{%
+  \subfloat[Synchronous imbalanced computations]{%
     \includegraphics[scale=0.49]{c1/compt}\label{fig:h2}}
  % \caption{Parallel tasks on homogeneous platform}
   \label{fig:homo}
@@ -277,7 +283,7 @@ for a warehouse-sized computer.
 %%%%%%%%%%%%%%%%%%%%
 %%    SLIDE 11   %%
 %%%%%%%%%%%%%%%%%%%% 
-\begin{frame}{Energy model for homogeneous platform}    
+\begin{frame}{Energy model for a homogeneous platform}    
       The power consumed by a processor divided into two power metrics: the dynamic (\textcolor{red}{$P_d$}) and static   
        (\textcolor{red}{$P_s$}) power. 
     \begin{equation}
@@ -301,7 +307,7 @@ for a warehouse-sized computer.
 %%    SLIDE 12   %%
 %%%%%%%%%%%%%%%%%%%% 
 
-\begin{frame}{Energy model for homogeneous platform}
+\begin{frame}{Energy model for a homogeneous platform}
        
           The frequency scaling factor is the ratio between the maximum and the new frequency, \textcolor{blue}{$S = \frac{F_{max}}{F_{new}}$}.  \medskip     
               
@@ -312,12 +318,12 @@ for a warehouse-sized computer.
          \left( T_1 + \sum_{i=2}^{N} \frac{T_i^3}{T_1^2} \right) +
             P_{s} \cdot S_1  \cdot T_1 \cdot N$
         \end{block}     
-           \textcolor{blue}{$S_1$}: the max. scaling factor\\ 
-           \textcolor{blue}{$P_{d}$}: the dynamic power\\
-           \textcolor{blue}{$P_{s}$}: the static power\\
-           \textcolor{blue}{$T_I$}: the time of the slower task\\ 
-           \textcolor{blue}{$T_i$}: the time of the other tasks\\ 
-           \textcolor{blue}{$N$}:  the number of  nodes
+           \textcolor{blue}{$S_1$}: the maximum scaling factor.\\ 
+           \textcolor{blue}{$P_{d}$}: the dynamic power.\\
+           \textcolor{blue}{$P_{s}$}: the static power.\\
+           \textcolor{blue}{$T_I$}: the execution time of the slower task.\\ 
+           \textcolor{blue}{$T_i$}: the execution time of task i.\\ 
+           \textcolor{blue}{$N$}:  the number of  nodes.
        
 \end{frame}
   
@@ -394,7 +400,7 @@ for a warehouse-sized computer.
      
      \begin{figure}
   \animategraphics[autopause,controls,scale=0.28,buttonsize=0.2cm]{10}{dvfs-homo/a-}{0}{159}
-
+  %\includegraphics[width=0.6\textwidth]{dvfs-homo/a-159}
   \end{figure}
 \end{frame}
 
@@ -405,10 +411,10 @@ for a warehouse-sized computer.
       \begin{femtoBlock}{}      
         \begin{itemize}
          \small
-           \item The experiments are executed on the simulator SimGrid/SMPI v3.10.\medskip
-           \item The proposed algorithm is applied to the NAS parallel benchmarks.\medskip
+           \item The experiments were executed on the simulator SimGrid/SMPI v3.10.\medskip
+           \item The proposed algorithm was applied to the NAS parallel benchmarks.\medskip
            \item Each node in the cluster has 18 frequency values from \textbf{2.5$GHz$} to \textbf{800$MHz$}.\medskip
-           \item The proposed algorithm was evaluated over the A, B, C classes of the benchmarks using 4, 8 or 9 and 16 nodes respectively. \medskip
+           \item The proposed algorithm was evaluated over the A, B and C classes of the benchmarks using 4, 8 or 9 and 16 nodes respectively. \medskip
            \item $P_d=20W$,  $P_s=4W$.
                 \end{itemize}
         \end{femtoBlock}
@@ -438,13 +444,15 @@ for a warehouse-sized computer.
            $S_{opt} = \sqrt[3]{\frac{2}{N} \cdot \frac{P_{dyn}}{P_{static}} \cdot
             \left( 1 + \sum_{i=2}^{N} \frac{T_i^3}{T_1^3}\right) } $
         \end{block}   
+        
+        
     \centering {
          %\includegraphics[width=.33\textwidth]{c1/c1.pdf}
          %\qquad
          %\includegraphics[width=.33\textwidth]{c1/c2.pdf}}
            
          
-            \includegraphics[width=.55\textwidth]{c1/compare_c.pdf}}
+            \includegraphics[width=.55\textwidth]{c1/compare-c.pdf}}
         
 \end{frame}
 
@@ -456,6 +464,7 @@ for a warehouse-sized computer.
     \vspace{-0.75cm}     
   \begin{figure}
   \animategraphics[autopause,controls,scale=0.28,buttonsize=0.2cm]{10}{homo-model/a-}{0}{356}
+  %\includegraphics[width=0.6\textwidth]{homo-model/a-356}
   \end{figure}
 \end{frame}
 
@@ -463,7 +472,7 @@ for a warehouse-sized computer.
 %%%%%%%%%%%%%%%%%%%%
 %%    SLIDE 21   %%
 %%%%%%%%%%%%%%%%%%%% 
-\begin{frame}{Comparing the new model with Rauber model }
+\begin{frame}{\large Comparing the new model with Rauber's model }
  \vspace{0.1cm}    
  \centering
     \includegraphics[width=.45\textwidth]{c1/energy_con}
@@ -496,13 +505,13 @@ for a warehouse-sized computer.
 %%%%%%%%%%%%%%%%%%%% 
 
 
-\begin{frame}{Contribution}
+\begin{frame}{The second contribution}
 
-\section{\small {Energy optimization of heterogeneous platform}}
+\section{\small {Energy optimization of a heterogeneous platform}}
 \begin{center}
 
 
-\bf  \Large \textcolor{blue}{Energy optimization of a parallel application with iterations running over Heterogeneous platform}
+\bf  \Large \textcolor{blue}{Energy optimization of a parallel application with iterations running over a Heterogeneous platform}
 \end{center}
  \end{frame}
  
@@ -516,7 +525,7 @@ for a warehouse-sized computer.
         \begin{femtoBlock}{} \vspace{-12 mm}
                 \begin{itemize} \small
                   \item   Proposing  \textcolor{blue}{new energy and performance models} for message passing  applications with iterations running  
-                          over a heterogeneous platform (cluster and Grid). \medskip
+                          over a heterogeneous platform (cluster or Grid). \medskip
                    \item  Studying the effect of the scaling factor $S$ on both the \textcolor{blue}{energy consumption  and the performance} of
                           message passing iterative applications.    \medskip                      
                    
@@ -576,6 +585,7 @@ for a warehouse-sized computer.
   \vspace{-0.5cm}
  \begin{figure}
   \animategraphics[autopause,controls,scale=0.28,buttonsize=0.2cm]{10}{heter-model/a-}{0}{272}
+  %\includegraphics[width=0.6\textwidth]{heter-model/a-272}
   \end{figure}
  \end{frame}
  
@@ -626,6 +636,7 @@ for a warehouse-sized computer.
  
   \begin{figure}
   \animategraphics[autopause,controls,scale=0.28,buttonsize=0.2cm]{10}{dvfs-heter/a-}{0}{650}
+ % \includegraphics[width=0.6\textwidth]{dvfs-heter/a-650}
   \end{figure}
 \end{frame}
 
@@ -638,11 +649,11 @@ for a warehouse-sized computer.
 \begin{frame}{Experiments over a heterogeneous cluster  }   
         \begin{itemize}
          \small
-           \item The experiments executed on the simulator SimGrid/SMPI v3.10.\medskip
+           \item The experiments were executed on the simulator SimGrid/SMPI v3.10.\medskip
            \item The scaling algorithm was applied to the NAS parallel benchmarks class C.\medskip
            \item Four types of processors with different computing powers were used.\medskip
-           \item We ran the benchmarks on different number of nodes ranging from 4 to 144 nodes.\medskip
-           \item The total power consumption of the chosen CPUs  assumed to be composed of $80\%$ for the dynamic power and $20\%$ for the static power.
+           \item The benchmarks were executed with different number of nodes ranging from 4 to 144 nodes.\medskip
+           \item It was assumed that the total power consumption of the CPU consist of 80\% dynamic power and 20\% static power.
                   \medskip
          
         \end{itemize}
@@ -660,7 +671,7 @@ for a warehouse-sized computer.
     \includegraphics[width=0.8\textwidth]{c2/energy_saving.pdf}
     
     \textcolor{blue}{On average, it reduces the energy consumption by \textcolor{red}{29\%} 
-     for the class C of the NAS benchmarks executed over 8 nodes}
+     for the class C of the NAS Benchmarks executed over 8 nodes}
     
    \end{figure}
 \end{frame} 
@@ -678,7 +689,7 @@ for a warehouse-sized computer.
     \includegraphics[width=.8\textwidth]{c2/perf_degra.pdf}
    
    \textcolor{blue}{On average, it degrades  by \textcolor{red}{3.8\%} the performance
-     of NAS benchmarks class C executed over 8 nodes}
+     of NAS Benchmarks class C executed over 8 nodes}
      \end{figure}
 \end{frame} 
  
@@ -776,19 +787,21 @@ for a warehouse-sized computer.
 %%    SLIDE 37    %%
 %%%%%%%%%%%%%%%%%%%%
  \begin{frame}{Experiments over Grid'5000}
-  \centering
-
+ 
+   \textcolor{blue}{The experiments were conducted using three 
+          clusters distributed over one or two sites.}
+           \vspace{-7 mm}
+          \begin{center}
           \includegraphics[width=.5\textwidth]{c2/grid5000-2.pdf}
-          
-          \vspace{-3 mm}
-          \textcolor{blue}{Two experiments were conducted: over one site and two sites 
-          each one with three clusters }
-          
-              \vspace{1mm}
-
+          \end{center}          
+      \vspace{-10 mm}
+  \textcolor{blue}{Grid'5000 power measurement tools were used.} 
+        \vspace{-9 mm}
+  \begin{center}
           \includegraphics[width=.5\textwidth]{c2/power_consumption.pdf}
+          \end{center}
           
-        \textcolor{blue}{Grid'5000 power measurement tools were used} 
+      
 \end{frame}   
 
 
@@ -802,7 +815,7 @@ for a warehouse-sized computer.
    \begin{minipage}{0.4\textwidth}
        %\textcolor{blue}{Execution the NAS class D on 16 nodes saves the energy by  
         %\textcolor{red}{30\%}}
-        \textcolor{blue}{The energy saving =  \textcolor{red}{30\%}}
+     \small \textcolor{blue}{The average energy saving =  \textcolor{red}{30\%}}
    \end{minipage}  
      \begin{minipage}{0.55\textwidth}
         \begin{figure}[h!]
@@ -813,7 +826,7 @@ for a warehouse-sized computer.
          \begin{minipage}{0.4\textwidth}
            %\textcolor{blue}{Execution the NAS class D on 16 nodes degrades the 
                 %performance by \textcolor{red}{3.2\%}}
-              \textcolor{blue}{The performance degradation  =  \textcolor{red}{3.2\%}}  
+      \small  \textcolor{blue}{The average performance degradation  =  \textcolor{red}{3.2\%}}  
         \end{minipage}
        \begin{minipage}{0.55\textwidth}
          \begin{figure}[h!] 
@@ -864,10 +877,10 @@ for a warehouse-sized computer.
 %%%%%%%%%%%%%%%%%%%%
 %%    SLIDE 40    %%
 %%%%%%%%%%%%%%%%%%%%
-\begin{frame}{Contribution}
+\begin{frame}{The third contribution}
 \section{\small {Energy optimization of asynchronous applications}}
 \begin{center}
-\bf  \Large \textcolor{blue}{Energy optimization of asynchronous  message passing iterative applications}
+\bf  \Large \textcolor{blue}{Energy optimization of asynchronous iterative message passing  applications}
 \end{center}
  \end{frame}
 
@@ -880,7 +893,8 @@ for a warehouse-sized computer.
 \textcolor{blue}{The execution of a synchronous parallel iterative application over a grid }
 \vspace{-8 mm}
 \begin{figure}
-  \animategraphics[autopause,controls,scale=0.25,buttonsize=0.2cm]{10}{syn/a-}{0}{503}
+ \animategraphics[autopause,controls,scale=0.25,buttonsize=0.2cm]{10}{syn/a-}{0}{503}
+ %\includegraphics[width=0.6\textwidth]{syn/a-503}
   \end{figure}
 \end{frame}
 
@@ -893,7 +907,8 @@ for a warehouse-sized computer.
 \textcolor{blue}{The execution of an asynchronous parallel iterative application over a grid }
 \vspace{-8 mm}
 \begin{figure}
-  \animategraphics[autopause,controls,scale=0.25,buttonsize=0.2cm]{10}{asyn/a-}{0}{440}
+ \animategraphics[autopause,controls,scale=0.25,buttonsize=0.2cm]{10}{asyn/a-}{0}{440}
+ %\includegraphics[width=0.6\textwidth]{asyn/a-440}
   \end{figure}
 \end{frame}
 
@@ -907,6 +922,7 @@ for a warehouse-sized computer.
 \vspace{-8 mm}
 \begin{figure}
   \animategraphics[autopause,controls,scale=0.25,buttonsize=0.2cm]{10}{asyn+dvfs/a-}{0}{314}
+  %\includegraphics[width=0.6\textwidth]{asyn+dvfs/a-314}
   \end{figure}
 \end{frame}
 
@@ -1017,7 +1033,7 @@ for a warehouse-sized computer.
 \centering
     \includegraphics[scale=0.42]{c3/energy_saving.eps}
 
- \centering  The average of energy saving  = \textcolor{red}{22\%}
+ \centering  The average energy saving  = \textcolor{red}{22\%}
 \end{frame} 
 
 
@@ -1047,8 +1063,8 @@ for a warehouse-sized computer.
     \includegraphics[width=0.53\textwidth]{c3/perf-deg-compare.eps}
    \end{figure}
     \vspace{-5 mm}
-     \centering
-The energy saving = \textcolor{red}{26.93\%}, the average speed-up =  \textcolor{red}{21.48\%}
+     \centering \footnotesize
+The average energy saving = \textcolor{red}{26.93\%}, the average speed-up =  \textcolor{red}{21.48\%}
 \end{frame} 
 
 
@@ -1072,21 +1088,21 @@ The energy saving = \textcolor{red}{26.93\%}, the average speed-up =  \textcolor
 \section{Conclusions and Perspectives}
 \begin{itemize}
 
-\small  \barrow  Three \textcolor{blue}{ new energy consumption and performance} models were proposed for synchronous and asynchronous parallel applications with iterations running over 
-\textcolor{blue}{homogeneous and  heterogeneous clusters and grids}.  
+\small  \barrow  Three \textcolor{blue}{ new energy consumption and performance} models were proposed for synchronous or asynchronous parallel applications with iterations running over 
+\textcolor{blue}{homogeneous and  heterogeneous clusters or grids}.  
       
 
 
-\small \barrow \textcolor{blue}{A new objective function} was proposed to optimize both the energy consumption and the performance.
+\small \barrow \textcolor{blue}{A new objective function} to optimize both the energy consumption and the performance was proposed.
 
 \small \barrow \textcolor{blue}{New online frequency selecting algorithms} for clusters and grids were developed.
 
 \small \barrow The proposed algorithms were applied to the \textcolor{blue}{NAS parallel benchmarks} and \textcolor{blue}{the
 Multi-splitting} method.
 
-\small \barrow The proposed algorithms were evaluated over the \textcolor{blue}{SimGrid simulator} and over  \textcolor{blue}{Grid'5000 testbed}.
+\small \barrow The proposed algorithms were evaluated over the \textcolor{blue}{SimGrid simulator} and over  the \textcolor{blue}{Grid'5000 testbed}.
 
-\small  \barrow All the proposed methods were compared to either \textcolor{blue}{Rauber and Rünger's  method} or  \textcolor{blue}{the EDP objective function}.
+\small  \barrow All the proposed methods were compared to either \textcolor{blue}{Rauber and Rünger's  method} or to the \textcolor{blue}{EDP objective function}.
 
 
 \end{itemize}
