X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/mpi-energy2.git/blobdiff_plain/054bc9f5e26028cd84aa816c5c06ed131959fdde..8635fccbf588d4a9dd25d07624be34eae4ab66ad:/Heter_paper.tex?ds=sidebyside

diff --git a/Heter_paper.tex b/Heter_paper.tex
index 9d743db..014ad60 100644
--- a/Heter_paper.tex
+++ b/Heter_paper.tex
@@ -8,7 +8,6 @@
 \usepackage{algorithm}
 \usepackage{subfig}
 \usepackage{amsmath}
-\usepackage{multirow}
 \usepackage{url}
 \DeclareUrlCommand\email{\urlstyle{same}}
 
@@ -63,8 +62,7 @@
     Arnaud Giersch
   } 
   \IEEEauthorblockA{%
-    FEMTO-ST Institute\\
-    University of Franche-ComtÃ©\\
+    FEMTO-ST Institute, University of Franche-Comte\\
     IUT de Belfort-MontbÃ©liard,
     19 avenue du MarÃ©chal Juin, BP 527, 90016 Belfort cedex, France\\
     % Telephone: \mbox{+33 3 84 58 77 86}, % RaphaÃ«l
@@ -82,16 +80,15 @@ platforms many techniques have been  used. Dynamic voltage and frequency scaling
 (DVFS) is  one of them. It  reduces the frequency of  a CPU to  lower its energy
 consumption.  However,  lowering the  frequency  of  a  CPU might  increase  the
 execution  time of  an application  running on  that processor.   Therefore, the
-frequency that  gives the best tradeoff  between the energy  consumption and the
-performance of an application must be selected.
-
+frequency that  gives the best trade-off  between the energy  consumption and the
+performance of an application must be selected.\\
 In this  paper, a new  online frequencies selecting algorithm  for heterogeneous
 platforms is presented.   It selects the frequency which tries  to give the best
-tradeoff  between  energy saving  and  performance  degradation,  for each  node
+trade-off  between  energy saving  and  performance  degradation,  for each  node
 computing the message  passing iterative application. The algorithm  has a small
 overhead and works without training or profiling. It uses a new energy model for
 message passing iterative applications  running on a heterogeneous platform. The
-proposed algorithm is  evaluated on the Simgrid simulator  while running the NAS
+proposed algorithm is  evaluated on the SimGrid simulator  while running the NAS
 parallel  benchmarks.  The  experiments   show  that  it  reduces  the  energy
 consumption by up to 35\% while  limiting the performance degradation as much as
 possible.   Finally,  the algorithm  is  compared  to  an existing  method,  the
@@ -121,13 +118,13 @@ This heterogeneous platform executes more than 5 GFLOPS per watt while consuming
 
 Besides platform  improvements, there are many software  and hardware techniques
 to lower  the energy consumption of  these platforms, such  as scheduling, DVFS,
-...   DVFS is  a  widely used  process to  reduce  the energy  consumption of  a
+\dots{}  DVFS is a widely used process to reduce the energy consumption of a
 processor            by             lowering            its            frequency
 \cite{Rizvandi_Some.Observations.on.Optimal.Frequency}. However, it also reduces
 the number of FLOPS executed by the processor which might increase the execution
 time of the application running over that processor.  Therefore, researchers use
 different optimization  strategies to select  the frequency that gives  the best
-tradeoff  between the  energy reduction  and performance  degradation  ratio. In
+trade-off  between the  energy reduction  and performance  degradation  ratio. In
 \cite{Our_first_paper}, a  frequency selecting algorithm was  proposed to reduce
 the energy  consumption of message  passing iterative applications  running over
 homogeneous platforms.  The results of  the experiments show  significant energy
@@ -154,24 +151,24 @@ Finally, in Section~\ref{sec.concl} the paper ends with a summary and some futur
 \section{Related works}
 \label{sec.relwork}
 DVFS is a technique used in modern processors to scale down both the voltage and
-the  frequency  of the  CPU  while  computing, in  order  to  reduce the  energy
-consumption of  the processor. DVFS is also  allowed in  GPUs  to achieve the
-same goal. Reducing the frequency of  a processor lowers its number of FLOPS and
-might  degrade the  performance of  the application  running on  that processor,
-especially if it is compute bound. Therefore selecting the appropriate frequency
-for a processor to satisfy some objectives while taking into account all the
-constraints,  is  not a  trivial  operation.   Many  researchers used  different
-strategies to  tackle this problem. Some  of them developed  online methods that
-compute   the  new   frequency  while   executing  the   application,   such  as
-~\cite{Hao_Learning.based.DVFS,Spiliopoulos_Green.governors.Adaptive.DVFS}. Others
-used  offline methods  that might  need to  run the  application and  profile it
-before       selecting       the        new       frequency,       such       as
-~\cite{Rountree_Bounding.energy.consumption.in.MPI,Cochran_Pack_and_Cap_Adaptive_DVFS}. The
-methods could  be heuristics, exact or  brute force methods  that satisfy varied
-objectives such as  energy reduction or performance. They  also could be adapted
-to  the  execution's  environment  and  the  type of  the  application  such  as
-sequential, parallel  or distributed architecture,  homogeneous or heterogeneous
-platform, synchronous or asynchronous application, ...
+the frequency of the CPU while computing, in order to reduce the energy
+consumption of the processor. DVFS is also allowed in GPUs to achieve the same
+goal. Reducing the frequency of a processor lowers its number of FLOPS and might
+degrade the performance of the application running on that processor, especially
+if it is compute bound. Therefore selecting the appropriate frequency for a
+processor to satisfy some objectives while taking into account all the
+constraints, is not a trivial operation.  Many researchers used different
+strategies to tackle this problem. Some of them developed online methods that
+compute the new frequency while executing the application, such
+as~\cite{Hao_Learning.based.DVFS,Spiliopoulos_Green.governors.Adaptive.DVFS}.
+Others used offline methods that might need to run the application and profile
+it before selecting the new frequency, such
+as~\cite{Rountree_Bounding.energy.consumption.in.MPI,Cochran_Pack_and_Cap_Adaptive_DVFS}.
+The methods could be heuristics, exact or brute force methods that satisfy
+varied objectives such as energy reduction or performance. They also could be
+adapted to the execution's environment and the type of the application such as
+sequential, parallel or distributed architecture, homogeneous or heterogeneous
+platform, synchronous or asynchronous application, \dots{}
 
 In this paper, we are interested in reducing energy for message passing iterative synchronous applications running over heterogeneous platforms.
 Some works have already been done for such platforms and they can be classified into two types of heterogeneous platforms: 
@@ -182,30 +179,44 @@ Some works have already been done for such platforms and they can be classified
 
 \end{itemize}
 
-For the first type of platform, the computing intensive parallel tasks are executed on the  GPUs and the rest are executed 
-on the CPUs.  Luley et al.
-~\cite{Luley_Energy.efficiency.evaluation.and.benchmarking}, proposed  a heterogeneous 
-cluster composed of Intel Xeon CPUs and NVIDIA GPUs. Their main goal was to maximize the 
-energy efficiency of the platform during computation by maximizing the number of FLOPS per watt generated. 
-In~\cite{KaiMa_Holistic.Approach.to.Energy.Efficiency.in.GPU-CPU}, Kai Ma et al. developed a scheduling 
-algorithm that distributes  workloads proportional to the computing power of the nodes which could be a GPU or a CPU. All the tasks must be completed at the same time.
-In~\cite{Rong_Effects.of.DVFS.on.K20.GPU}, Rong et al. showed that 
-a heterogeneous (GPUs and CPUs) cluster that enables DVFS gave better energy and performance 
-efficiency than other clusters only composed of  CPUs.
+For the first type of platform, the computing intensive parallel tasks are
+executed on the GPUs and the rest are executed on the CPUs.  Luley et
+al.~\cite{Luley_Energy.efficiency.evaluation.and.benchmarking}, proposed a
+heterogeneous cluster composed of Intel Xeon CPUs and NVIDIA GPUs. Their main
+goal was to maximize the energy efficiency of the platform during computation by
+maximizing the number of FLOPS per watt generated.
+In~\cite{KaiMa_Holistic.Approach.to.Energy.Efficiency.in.GPU-CPU}, Kai Ma et
+al. developed a scheduling algorithm that distributes workloads proportional to
+the computing power of the nodes which could be a GPU or a CPU. All the tasks
+must be completed at the same time.  In~\cite{Rong_Effects.of.DVFS.on.K20.GPU},
+Rong et al. showed that a heterogeneous (GPUs and CPUs) cluster that enables
+DVFS gave better energy and performance efficiency than other clusters only
+composed of CPUs.
  
-The work presented in this paper concerns the second type of platform, with heterogeneous CPUs.
-Many methods were conceived to reduce the energy consumption of this type of platform.  Naveen et al.~\cite{Naveen_Power.Efficient.Resource.Scaling}  
-developed a method that minimizes the value of $energy*delay^2$ (the delay is the sum of slack times that happen during synchronous communications) by dynamically assigning new frequencies to the CPUs of the heterogeneous cluster. Lizhe et al.~\cite{Lizhe_Energy.aware.parallel.task.scheduling} proposed
-an algorithm that divides the executed tasks into two types: the critical and 
-non critical tasks. The algorithm scales down the frequency of  non critical tasks proportionally to their  slack and communication times while limiting  the performance degradation percentage to less than 10\%. In~\cite{Joshi_Blackbox.prediction.of.impact.of.DVFS}, they developed 
-  a heterogeneous cluster composed of two  types 
-of Intel and AMD processors. They use a gradient method to predict the impact of DVFS operations on performance.
-In~\cite{Shelepov_Scheduling.on.Heterogeneous.Multicore} and \cite{Li_Minimizing.Energy.Consumption.for.Frame.Based.Tasks}, 
- the best frequencies for a specified heterogeneous cluster are selected offline using some 
-heuristic. Chen et al.~\cite{Chen_DVFS.under.quality.of.service.requirements} used a greedy dynamic programming approach to  
-minimize the power consumption of heterogeneous servers  while respecting given time constraints. This approach 
-had considerable overhead.
-In contrast to the above described papers, this paper presents the following contributions :
+The work presented in this paper concerns the second type of platform, with
+heterogeneous CPUs.  Many methods were conceived to reduce the energy
+consumption of this type of platform.  Naveen et
+al.~\cite{Naveen_Power.Efficient.Resource.Scaling} developed a method that
+minimizes the value of $energy\cdot delay^2$ (the delay is the sum of slack
+times that happen during synchronous communications) by dynamically assigning
+new frequencies to the CPUs of the heterogeneous cluster. Lizhe et
+al.~\cite{Lizhe_Energy.aware.parallel.task.scheduling} proposed an algorithm
+that divides the executed tasks into two types: the critical and non critical
+tasks. The algorithm scales down the frequency of non critical tasks
+proportionally to their slack and communication times while limiting the
+performance degradation percentage to less than
+10\%. In~\cite{Joshi_Blackbox.prediction.of.impact.of.DVFS}, they developed a
+heterogeneous cluster composed of two types of Intel and AMD processors. They
+use a gradient method to predict the impact of DVFS operations on performance.
+In~\cite{Shelepov_Scheduling.on.Heterogeneous.Multicore} and
+\cite{Li_Minimizing.Energy.Consumption.for.Frame.Based.Tasks}, the best
+frequencies for a specified heterogeneous cluster are selected offline using
+some heuristic. Chen et
+al.~\cite{Chen_DVFS.under.quality.of.service.requirements} used a greedy dynamic
+programming approach to minimize the power consumption of heterogeneous servers
+while respecting given time constraints. This approach had considerable
+overhead.  In contrast to the above described papers, this paper presents the
+following contributions :
 \begin{enumerate}
 \item  two new energy and performance models for message passing iterative synchronous applications running over 
        a heterogeneous platform. Both models take into account  communication and slack times. The models can predict the required energy and the execution time of the application.
@@ -231,16 +242,16 @@ network. Therefore, each node has different characteristics such as computing
 power (FLOPS), energy consumption, CPU's frequency range, \dots{} but they all
 have the same network bandwidth and latency.
 
-The overall execution time  of a distributed iterative synchronous application 
-over a heterogeneous platform  consists of the sum of the computation time and 
-the communication time for every iteration on a node. However, due to the 
-heterogeneous computation power of the computing nodes, slack times might occur 
-when fast nodes have to  wait, during synchronous communications, for  the slower 
-nodes to finish  their computations (see Figure~(\ref{fig:heter})). 
-Therefore,  the overall execution time  of the program is the execution time of the slowest
-task which has the highest computation time and no slack time.
+The overall execution time of a distributed iterative synchronous application
+over a heterogeneous platform consists of the sum of the computation time and
+the communication time for every iteration on a node. However, due to the
+heterogeneous computation power of the computing nodes, slack times might occur
+when fast nodes have to wait, during synchronous communications, for the slower
+nodes to finish their computations (see Figure~\ref{fig:heter}).  Therefore, the
+overall execution time of the program is the execution time of the slowest task
+which has the highest computation time and no slack time.
   
- \begin{figure}[t]
+ \begin{figure}[!t]
   \centering
    \includegraphics[scale=0.6]{fig/commtasks}
   \caption{Parallel tasks on a heterogeneous platform}
@@ -285,7 +296,7 @@ vector of scaling factors can be predicted using (\ref{eq:perf}).
  \textit  T_\textit{new} = 
  \max_{i=1,2,\dots,N} ({TcpOld_{i}} \cdot S_{i}) +  MinTcm 
 \end{equation}
-Where:\\
+Where:
 \begin{equation}
 \label{eq:perf2}
  MinTcm = \min_{i=1,2,\dots,N} (Tcm_i)
@@ -390,7 +401,7 @@ $Tcp_{i}$ might  be different and  different frequency scaling factors  might be
 computed in order to decrease  the overall energy consumption of the application
 and reduce slack  times.  The communication time of a processor  $i$ is noted as
 $Tcm_{i}$  and could  contain slack  times when  communicating  with slower
-nodes,  see figure(\ref{fig:heter}).  Therefore,  all nodes  do  not have  equal
+nodes,  see Figure~\ref{fig:heter}.  Therefore,  all nodes  do  not have  equal
 communication  times. While  the dynamic  energy  is computed  according to  the
 frequency  scaling   factor  and   the  dynamic  power   of  each  node   as  in
 (\ref{eq:Edyn}), the static energy is computed  as the sum of the execution time
@@ -486,7 +497,7 @@ normalized execution time is inverted which gives the normalized performance equ
 \end{multline}
 
 
-\begin{figure}
+\begin{figure}[!t]
   \centering
   \subfloat[Homogeneous platform]{%
     \includegraphics[width=.33\textwidth]{fig/homo}\label{fig:r1}}%
@@ -498,12 +509,12 @@ normalized execution time is inverted which gives the normalized performance equ
   \caption{The energy and performance relation}
 \end{figure}
 
-Then, the objective function can be modeled in order to find the maximum distance
-between the energy curve (\ref{eq:enorm}) and the  performance
-curve (\ref{eq:pnorm_inv}) over all available sets of scaling factors.  This
-represents the minimum energy consumption with minimum execution time (maximum 
-performance) at the same time, see figure~(\ref{fig:r1}) or figure~(\ref{fig:r2}). Then the objective
-function has the following form:
+Then, the objective function can be modeled in order to find the maximum
+distance between the energy curve (\ref{eq:enorm}) and the performance curve
+(\ref{eq:pnorm_inv}) over all available sets of scaling factors.  This
+represents the minimum energy consumption with minimum execution time (maximum
+performance) at the same time, see Figure~\ref{fig:r1} or
+Figure~\ref{fig:r2}. Then the objective function has the following form:
 \begin{equation}
   \label{eq:max}
   Max Dist = 
@@ -521,29 +532,33 @@ the energy curve has a convex form as shown in~\cite{Zhuo_Energy.efficient.Dynam
 \label{sec.optim}
 
 \subsection{The algorithm details}
-In this section, algorithm \ref{HSA} is presented. It selects the frequency scaling factors 
-vector that gives the best trade-off between minimizing the energy consumption  and maximizing 
-the performance of a message passing synchronous iterative application executed on a heterogeneous 
-platform. It works online during the execution time of the iterative message passing program.  
-It uses information gathered during the first iteration such as the computation time and the 
-communication time in one iteration for each node. The algorithm is executed  after the first 
-iteration and returns a vector of optimal frequency scaling factors   that satisfies the objective 
-function (\ref{eq:max}). The program applies DVFS operations to change the frequencies of the CPUs 
-according to the computed scaling factors.  This algorithm is called just once during the execution 
-of the program. Algorithm~(\ref{dvfs}) shows where and when the proposed scaling algorithm is called 
-in the iterative MPI program.
-
-The nodes in a heterogeneous platform have different computing powers, thus while executing message 
-passing iterative synchronous applications, fast nodes have to wait for the slower ones to finish their 
-computations before being able to synchronously communicate with them as in figure (\ref{fig:heter}). 
-These periods are called idle or slack times. 
-The algorithm takes into account this problem and tries to reduce these slack times when selecting the 
-frequency scaling factors vector. At first, it selects initial frequency scaling factors that increase 
-the execution times of fast nodes and  minimize the  differences between  the  computation times of 
-fast and slow nodes. The value of the initial frequency scaling factor  for each node is inversely 
-proportional to its computation time that was gathered from the first iteration. These initial frequency 
-scaling factors are computed as   a ratio between the computation time of the slowest node and the 
-computation time of the node $i$ as follows:
+In this section, Algorithm~\ref{HSA} is presented. It selects the frequency
+scaling factors vector that gives the best trade-off between minimizing the
+energy consumption and maximizing the performance of a message passing
+synchronous iterative application executed on a heterogeneous platform. It works
+online during the execution time of the iterative message passing program.  It
+uses information gathered during the first iteration such as the computation
+time and the communication time in one iteration for each node. The algorithm is
+executed after the first iteration and returns a vector of optimal frequency
+scaling factors that satisfies the objective function (\ref{eq:max}). The
+program applies DVFS operations to change the frequencies of the CPUs according
+to the computed scaling factors.  This algorithm is called just once during the
+execution of the program. Algorithm~\ref{dvfs} shows where and when the proposed
+scaling algorithm is called in the iterative MPI program.
+
+The nodes in a heterogeneous platform have different computing powers, thus
+while executing message passing iterative synchronous applications, fast nodes
+have to wait for the slower ones to finish their computations before being able
+to synchronously communicate with them as in Figure~\ref{fig:heter}.  These
+periods are called idle or slack times.  The algorithm takes into account this
+problem and tries to reduce these slack times when selecting the frequency
+scaling factors vector. At first, it selects initial frequency scaling factors
+that increase the execution times of fast nodes and minimize the differences
+between the computation times of fast and slow nodes. The value of the initial
+frequency scaling factor for each node is inversely proportional to its
+computation time that was gathered from the first iteration. These initial
+frequency scaling factors are computed as a ratio between the computation time
+of the slowest node and the computation time of the node $i$ as follows:
 \begin{equation}
   \label{eq:Scp}
  Scp_{i} = \frac{\max_{i=1,2,\dots,N}(Tcp_i)}{Tcp_i}
@@ -555,25 +570,25 @@ and the computation scaling factor $Scp_i$ as follows:
   \label{eq:Fint}
  F_{i} = \frac{Fmax_i}{Scp_i},~{i=1,2,\cdots,N}
 \end{equation}
-If the computed  initial frequency for a  node is not available in  the gears of
-that  node,  it  is replaced  by  the  nearest  available frequency.  In  figure
-(\ref{fig:st_freq}), the nodes are sorted by their computing power in ascending
-order and the  frequencies of the faster nodes are scaled  down according to the
-computed initial  frequency scaling factors.  The resulting new  frequencies are
-colored in  blue in figure (\ref{fig:st_freq}).  This set of  frequencies can be
-considered  as a higher  bound for  the search  space of  the optimal  vector of
-frequencies because  selecting frequency scaling factors higher  than the higher
-bound will not  improve the performance of the application  and it will increase
-its  overall  energy  consumption.  Therefore  the algorithm  that  selects  the
-frequency  scaling   factors  starts  the  search  method   from  these  initial
-frequencies and takes a downward  search direction toward lower frequencies. The
-algorithm  iterates on all  left frequencies,  from the  higher bound  until all
-nodes  reach  their  minimum   frequencies,  to  compute  their  overall  energy
-consumption and  performance, and select  the optimal frequency  scaling factors
-vector. At each iteration the algorithm determines the slowest node according to
-the equation (\ref{eq:perf}) and keeps  its frequency unchanged, while it lowers
-the  frequency  of  all  other  nodes  by one  gear.   The  new  overall  energy
-consumption  and  execution time  are  computed  according  to the  new  scaling
+If the computed initial frequency for a node is not available in the gears of
+that node, it is replaced by the nearest available frequency.  In
+Figure~\ref{fig:st_freq}, the nodes are sorted by their computing power in
+ascending order and the frequencies of the faster nodes are scaled down
+according to the computed initial frequency scaling factors.  The resulting new
+frequencies are colored in blue in Figure~\ref{fig:st_freq}.  This set of
+frequencies can be considered as a higher bound for the search space of the
+optimal vector of frequencies because selecting frequency scaling factors higher
+than the higher bound will not improve the performance of the application and it
+will increase its overall energy consumption.  Therefore the algorithm that
+selects the frequency scaling factors starts the search method from these
+initial frequencies and takes a downward search direction toward lower
+frequencies. The algorithm iterates on all left frequencies, from the higher
+bound until all nodes reach their minimum frequencies, to compute their overall
+energy consumption and performance, and select the optimal frequency scaling
+factors vector. At each iteration the algorithm determines the slowest node
+according to the equation (\ref{eq:perf}) and keeps its frequency unchanged,
+while it lowers the frequency of all other nodes by one gear.  The new overall
+energy consumption and execution time are computed according to the new scaling
 factors.  The optimal set of frequency scaling factors is the set that gives the
 highest distance according to the objective function (\ref{eq:max}).
 
@@ -590,7 +605,7 @@ words, until they reach the higher bound. It can also be noticed that the higher
 the difference between the faster nodes  and the slower nodes is, the bigger the
 maximum distance  between the  energy curve and  the performance curve  is while
  the scaling factors are varying which results in bigger energy savings.
-\begin{figure}[t]
+\begin{figure}[!t]
   \centering
     \includegraphics[scale=0.5]{fig/start_freq}
   \caption{Selecting the initial frequencies}
@@ -654,7 +669,7 @@ maximum distance  between the  energy curve and  the performance curve  is while
       \If {$(k=1)$}
         \State Gather all times of computation and\newline\hspace*{3em}%
                communication from each node.
-        \State Call algorithm \ref{HSA}.
+        \State Call Algorithm \ref{HSA}.
         \State Compute the new frequencies from the\newline\hspace*{3em}%
                returned optimal scaling factors.
         \State Set the new frequencies to nodes.
@@ -681,28 +696,33 @@ parallel benchmarks NPB v3.3  \cite{NAS.Parallel.Benchmarks}, running class B on
 very precise, the maximum  normalized difference between the predicted execution
 time and the real execution time is equal to 0.03 for all the NAS benchmarks.
 
-Since  the proposed algorithm is not an exact method it does not test all the possible solutions (vectors of scaling factors) 
-in the search space. To prove its efficiency, it was compared on small instances to a brute force search algorithm 
-that tests all the possible solutions. The brute force algorithm was applied to different NAS benchmarks classes with 
-different number of nodes. The solutions returned by the brute force algorithm and the proposed algorithm were identical 
-and the proposed algorithm was on average 10 times faster than the brute force algorithm. It has a small execution time: 
-for a heterogeneous cluster composed of four different types of nodes having the characteristics presented in 
-table~\ref{table:platform}, it takes on average \np[ms]{0.04}  for 4 nodes and \np[ms]{0.15} on average for 144 nodes 
-to compute the best scaling factors vector.  The algorithm complexity is $O(F\cdot (N \cdot4) )$, where $F$ is the number 
-of iterations and $N$ is the number of computing nodes. The algorithm needs  from 12 to 20 iterations to select the best 
-vector of frequency scaling factors that gives the results of the next sections.
+Since the proposed algorithm is not an exact method it does not test all the
+possible solutions (vectors of scaling factors) in the search space. To prove
+its efficiency, it was compared on small instances to a brute force search
+algorithm that tests all the possible solutions. The brute force algorithm was
+applied to different NAS benchmarks classes with different number of nodes. The
+solutions returned by the brute force algorithm and the proposed algorithm were
+identical and the proposed algorithm was on average 10 times faster than the
+brute force algorithm. It has a small execution time: for a heterogeneous
+cluster composed of four different types of nodes having the characteristics
+presented in Table~\ref{table:platform}, it takes on average \np[ms]{0.04} for 4
+nodes and \np[ms]{0.15} on average for 144 nodes to compute the best scaling
+factors vector.  The algorithm complexity is $O(F\cdot (N \cdot4) )$, where $F$
+is the number of iterations and $N$ is the number of computing nodes. The
+algorithm needs from 12 to 20 iterations to select the best vector of frequency
+scaling factors that gives the results of the next sections.
 
 \section{Experimental results}
 \label{sec.expe}
 To  evaluate the  efficiency and  the  overall energy  consumption reduction  of
-algorithm~\ref{HSA}, it was applied to the NAS parallel benchmarks NPB v3.3. The
+Algorithm~\ref{HSA}, it was applied to the NAS parallel benchmarks NPB v3.3. The
 experiments were executed on the  simulator SimGrid/SMPI which offers easy tools
 to create a heterogeneous platform and run message passing applications over it.
 The heterogeneous  platform that was used  in the experiments, had  one core per
 node because just one process was executed per node.  The heterogeneous platform
 was  composed  of  four  types  of  nodes. Each  type  of  nodes  had  different
 characteristics  such as  the maximum  CPU  frequency, the  number of  available
-frequencies  and the  computational power,  see Table  \ref{table:platform}. The
+frequencies  and the  computational power,  see Table~\ref{table:platform}. The
 characteristics  of  these  different  types  of nodes  are  inspired  from  the
 specifications of real  Intel processors.  The heterogeneous platform  had up to
 144 nodes and had nodes from the four types in equal proportions, for example if
@@ -714,10 +734,10 @@ highest frequency,  each node  consumed an amount  of power proportional  to its
 computing  power  (which  corresponds to  80\%  of  its  dynamic power  and  the
 remaining  20\%  to  the  static   power),  the  same  assumption  was  made  in
 \cite{Our_first_paper,Rauber_Analytical.Modeling.for.Energy}.    Finally,  These
-nodes were connected via an ethernet network with 1 Gbit/s bandwidth.
+nodes were connected via an Ethernet network with 1 Gbit/s bandwidth.
 
 
-\begin{table}[htb]
+\begin{table}[!t]
   \caption{Heterogeneous nodes characteristics}
   % title of Table
   \centering
@@ -727,16 +747,16 @@ nodes were connected via an ethernet network with 1 Gbit/s bandwidth.
     type          &GFLOPS     & Freq.    & Freq.        & Freq.          & power        & power \\
                   &           & GHz      & GHz          &GHz             &              &       \\
     \hline
-    1             &40         & 2.5      & 1.2          & 0.1            & 20~w         &4~w    \\
+    1             &40         & 2.5      & 1.2          & 0.1            & 20~W         &4~W    \\
          
     \hline
-    2             &50         & 2.66     & 1.6          & 0.133          & 25~w         &5~w    \\
+    2             &50         & 2.66     & 1.6          & 0.133          & 25~W         &5~W    \\
                   
     \hline
-    3             &60         & 2.9      & 1.2          & 0.1            & 30~w         &6~w    \\
+    3             &60         & 2.9      & 1.2          & 0.1            & 30~W         &6~W    \\
                   
     \hline
-    4             &70         & 3.4      & 1.6          & 0.133          & 35~w         &7~w    \\
+    4             &70         & 3.4      & 1.6          & 0.133          & 35~W         &7~W    \\
                   
     \hline
   \end{tabular}
@@ -762,12 +782,13 @@ be executed on $1, 4, 9, 16, 36, 64, 144$ nodes.
 
  
  
-\begin{table}[htb]
+\begin{table}[!t]
   \caption{Running NAS benchmarks on 4 nodes }
   % title of Table
   \centering
-  \begin{tabular}{|*{7}{l|}}
+  \begin{tabular}{|*{7}{r|}}
     \hline
+    \hspace{-2.2084pt}%
     Program     & Execution     & Energy         & Energy      & Performance        & Distance      \\
     name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
     \hline
@@ -787,14 +808,16 @@ be executed on $1, 4, 9, 16, 36, 64, 144$ nodes.
 \hline 
   \end{tabular}
   \label{table:res_4n}
-\end{table}
+% \end{table}
 
-\begin{table}[htb]
+\medskip
+% \begin{table}[!t]
   \caption{Running NAS benchmarks on 8 and 9 nodes }
   % title of Table
   \centering
-  \begin{tabular}{|*{7}{l|}}
+  \begin{tabular}{|*{7}{r|}}
     \hline
+     \hspace{-2.2084pt}%
     Program    & Execution     & Energy         & Energy      & Performance        & Distance      \\
     name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
     \hline
@@ -814,14 +837,16 @@ be executed on $1, 4, 9, 16, 36, 64, 144$ nodes.
 \hline 
   \end{tabular}
   \label{table:res_8n}
-\end{table}
+% \end{table}
 
-\begin{table}[htb]
+\medskip
+% \begin{table}[!t]
   \caption{Running NAS benchmarks on 16 nodes }
   % title of Table
   \centering
-  \begin{tabular}{|*{7}{l|}}
+  \begin{tabular}{|*{7}{r|}}
     \hline
+    \hspace{-2.2084pt}%
     Program     & Execution     & Energy         & Energy      & Performance        & Distance      \\
     name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
     \hline
@@ -841,14 +866,16 @@ be executed on $1, 4, 9, 16, 36, 64, 144$ nodes.
 \hline 
   \end{tabular}
   \label{table:res_16n}
-\end{table}
+% \end{table}
 
-\begin{table}[htb]
+\medskip
+% \begin{table}[!t]
   \caption{Running NAS benchmarks on 32 and 36 nodes }
   % title of Table
   \centering
-  \begin{tabular}{|*{7}{l|}}
+  \begin{tabular}{|*{7}{r|}}
     \hline
+    \hspace{-2.2084pt}%
     Program    & Execution     & Energy         & Energy      & Performance        & Distance      \\
     name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
     \hline
@@ -868,14 +895,16 @@ be executed on $1, 4, 9, 16, 36, 64, 144$ nodes.
 \hline 
   \end{tabular}
   \label{table:res_32n}
-\end{table}
+% \end{table}
 
-\begin{table}[htb]
+\medskip
+% \begin{table}[!t]
   \caption{Running NAS benchmarks on 64 nodes }
   % title of Table
   \centering
-  \begin{tabular}{|*{7}{l|}}
+  \begin{tabular}{|*{7}{r|}}
     \hline
+    \hspace{-2.2084pt}%
     Program    & Execution     & Energy         & Energy      & Performance        & Distance      \\
     name       & time/s        & consumption/J  & saving\%    & degradation\%      &               \\
     \hline
@@ -895,15 +924,16 @@ be executed on $1, 4, 9, 16, 36, 64, 144$ nodes.
 \hline 
   \end{tabular}
   \label{table:res_64n}
-\end{table}
-
+% \end{table}
 
-\begin{table}[htb]
+\medskip
+% \begin{table}[!t]
   \caption{Running NAS benchmarks on 128 and 144 nodes }
   % title of Table
   \centering
-  \begin{tabular}{|*{7}{l|}}
+  \begin{tabular}{|*{7}{r|}}
     \hline
+    \hspace{-2.2084pt}%
     Program    & Execution     & Energy         & Energy      & Performance        & Distance     \\
     name       & time/s        & consumption/J  & saving\%    & degradation\%      &              \\
     \hline
@@ -928,9 +958,9 @@ The overall energy  consumption was computed for each  instance according to the
 energy  consumption  model  (\ref{eq:energy}),  with and  without  applying  the
 algorithm. The execution time was also measured for all these experiments. Then,
 the energy saving and performance degradation percentages were computed for each
-instance.    The   results   are   presented  in   Tables   (\ref{table:res_4n},
+instance.    The   results   are   presented  in   Tables~\ref{table:res_4n},
 \ref{table:res_8n},           \ref{table:res_16n},          \ref{table:res_32n},
-\ref{table:res_64n} and \ref{table:res_128n}). All these results are the average
+\ref{table:res_64n} and \ref{table:res_128n}. All these results are the average
 values  from many experiments  for energy  savings and  performance degradation.
 The tables show the experimental results for running the NAS parallel benchmarks
 on  different  number  of  nodes.   The  experiments  show  that  the  algorithm
@@ -944,7 +974,7 @@ computation required  for each iteration is  divided by the  number of computing
 nodes.  On the other hand,  more communications are required when increasing the
 number  of  nodes so  the  static energy  increases  linearly  according to  the
 communication time and the dynamic power  is less relevant in the overall energy
-consumption.   Therefore, reducing the  frequency with  algorithm~(\ref{HSA}) is
+consumption.   Therefore, reducing the  frequency with  Algorithm~\ref{HSA} is
 less effective  in reducing the overall  energy savings. It can  also be noticed
 that for the benchmarks EP and  SP that contain little or no communications, the
 energy savings are  not significantly affected by the high  number of nodes.  No
@@ -959,7 +989,7 @@ small when compared to the communication times.
 
 
  
-\begin{figure}
+\begin{figure}[!t]
   \centering
   \subfloat[Energy saving]{%
     \includegraphics[width=.33\textwidth]{fig/energy}\label{fig:energy}}%
@@ -970,7 +1000,7 @@ small when compared to the communication times.
   \caption{The energy and performance for all NAS benchmarks running with a different number of nodes}
 \end{figure}
 
-Figures  \ref{fig:energy} and  \ref{fig:per_deg} present  the energy  saving and
+Figures~\ref{fig:energy} and  \ref{fig:per_deg} present  the energy  saving and
 performance  degradation respectively for  all the  benchmarks according  to the
 number of used nodes. As shown  in the first plot, the energy saving percentages
 of the benchmarks MG,  LU, BT and FT decrease linearly when  the number of nodes
@@ -1003,7 +1033,7 @@ two new power scenarios are the following:
 
 The NAS parallel benchmarks were  executed again over processors that follow the
 new power scenarios.   The class C of each  benchmark was run over 8  or 9 nodes
-and   the    results   are   presented   in    Tables   \ref{table:res_s1}   and
+and   the    results   are   presented   in    Tables~\ref{table:res_s1}   and
 \ref{table:res_s2}. These tables  show that the energy saving  percentage of the
 70\%-30\% scenario is  smaller for all benchmarks compared  to the energy saving
 of the 90\%-10\% scenario. Indeed, in  the latter more dynamic power is consumed
@@ -1014,13 +1044,13 @@ in the 70\%-30\% scenario compared to the 90\%-10\% scenario. This is due to the
 higher  static  power percentage  in  the first  scenario  which  makes it  more
 relevant in the  overall consumed energy.  Indeed, the  static energy is related
 to the execution time and if  the performance is degraded the amount of consumed
-static  energy directly  increas.  Therefore,  the proposed  algorithm  does not
+static  energy directly  increases.  Therefore,  the proposed  algorithm  does not
 really significantly  scale down much the  frequencies of the nodes  in order to
 limit the  increase of the  execution time and  thus limiting the effect  of the
 consumed static energy.
 
-Both   new  power   scenarios   are  compared   to   the  old   one  in   figure
-(\ref{fig:sen_comp}). It  shows the average of the  performance degradation, the
+Both   new  power   scenarios   are  compared   to   the  old   one  in
+Figure~\ref{fig:sen_comp}. It  shows the average of the  performance degradation, the
 energy saving and the  distances for all NAS benchmarks of class  C running on 8
 or 9 nodes.   The comparison shows that the energy  saving ratio is proportional
 to the dynamic power ratio: it is increased when applying the 90\%-10\% scenario
@@ -1034,17 +1064,17 @@ higher  ratio   for  static  power  (e.g.   70\%-30\%   scenario  and  80\%-20\%
 scenario). Since  the proposed algorithm  optimizes the energy  consumption when
 using a  higher ratio for dynamic  power the algorithm  selects bigger frequency
 scaling  factors that result  in more  energy saving  but less  performance, for
-example see  Figure (\ref{fig:scales_comp}). The  opposite happens when  using a
+example see  Figure~\ref{fig:scales_comp}. The  opposite happens when  using a
 higher  ratio for  static power,  the algorithm  proportionally  selects smaller
 scaling  values which result  in less  energy saving  but also  less performance
 degradation.
 
 
- \begin{table}[htb]
+ \begin{table}[!t]
   \caption{The results of the 70\%-30\% power scenario}
   % title of Table
   \centering
-  \begin{tabular}{|*{6}{l|}}
+  \begin{tabular}{|*{6}{r|}}
     \hline
     Program    & Energy          & Energy      & Performance        & Distance     \\
     name       & consumption/J   & saving\%    & degradation\%      &              \\
@@ -1069,11 +1099,11 @@ degradation.
 
 
 
-\begin{table}[htb]
+\begin{table}[!t]
   \caption{The results of the 90\%-10\% power scenario}
   % title of Table
   \centering
-  \begin{tabular}{|*{6}{l|}}
+  \begin{tabular}{|*{6}{r|}}
     \hline
     Program    & Energy          & Energy      & Performance        & Distance     \\
     name       & consumption/J   & saving\%    & degradation\%      &              \\
@@ -1097,7 +1127,7 @@ degradation.
 \end{table}
 
 
-\begin{figure}
+\begin{figure}[!t]
   \centering
   \subfloat[Comparison  between the results on 8 nodes]{%
     \includegraphics[width=.33\textwidth]{fig/sen_comp}\label{fig:sen_comp}}%
@@ -1115,36 +1145,42 @@ degradation.
 \label{sec.compare_EDP}
 In this section, the scaling  factors selection algorithm, called MaxDist,
 is compared to Spiliopoulos et al. algorithm \cite{Spiliopoulos_Green.governors.Adaptive.DVFS}, called EDP. 
-They developed a green governor that regularly applies an online frequency selecting algorithm to reduce the energy consumed by a multicore architecture without degrading much its performance. The algorithm selects the frequencies that minimize the energy and delay products, $EDP=Enegry*Delay$ using the predicted overall energy consumption and execution time delay for each frequency.
+They developed a green governor that regularly applies an online frequency selecting algorithm to reduce the energy consumed by a multicore architecture without degrading much its performance. The algorithm selects the frequencies that minimize the energy and delay products, $EDP=Energy\cdot Delay$ using the predicted overall energy consumption and execution time delay for each frequency.
 To fairly compare both algorithms, the same energy and execution time models, equations (\ref{eq:energy}) and  (\ref{eq:fnew}), were used for both algorithms to predict the energy consumption and the execution times. Also Spiliopoulos et al. algorithm was adapted to  start the search from the 
 initial frequencies computed using the equation (\ref{eq:Fint}). The resulting algorithm is an exhaustive search algorithm that minimizes the EDP and has the initial frequencies values as an upper bound.
 
-Both algorithms were applied to the parallel NAS benchmarks to compare their efficiency. Table \ref{table:compare_EDP}  presents the results of comparing the execution times and the energy consumptions for both versions of the NAS benchmarks while running the class C of each benchmark over 8 or 9 heterogeneous nodes. The results show that our algorithm provides better energy savings than Spiliopoulos et al. algorithm, 
-on average it results in 29.76\% energy saving while their algorithm returns just 25.75\%. The average of performance degradation percentage is approximately the same for both algorithms, about 4\%. 
+Both algorithms were applied to the parallel NAS benchmarks to compare their
+efficiency. Table~\ref{table:compare_EDP} presents the results of comparing the
+execution times and the energy consumption for both versions of the NAS
+benchmarks while running the class C of each benchmark over 8 or 9 heterogeneous
+nodes. The results show that our algorithm provides better energy savings than
+Spiliopoulos et al. algorithm, on average it results in 29.76\% energy saving
+while their algorithm returns just 25.75\%. The average of performance
+degradation percentage is approximately the same for both algorithms, about 4\%.
 
 
 For all benchmarks,  our algorithm outperforms Spiliopoulos et  al. algorithm in
-terms of  energy and  performance tradeoff, see  figure (\ref{fig:compare_EDP}),
+terms of  energy and  performance trade-off, see  Figure~\ref{fig:compare_EDP},
 because it maximizes the distance  between the energy saving and the performance
 degradation values while giving the same weight for both metrics.
 
 
 
 
-\begin{table}[h]
+\begin{table}[!t]
  \caption{Comparing the proposed algorithm}
  \centering
-\begin{tabular}{|l|l|l|l|l|l|l|l|}
+\begin{tabular}{|*{7}{r|}}
 \hline
-\multicolumn{2}{|l|}{\multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}Program \\ name\end{tabular}}} & \multicolumn{2}{l|}{Energy saving \%} & \multicolumn{2}{l|}{Perf.  degradation \%} & \multicolumn{2}{l|}{Distance} \\ \cline{3-8} 
-\multicolumn{2}{|l|}{}                                                                         & EDP             & MaxDist          & EDP            & MaxDist           & EDP          & MaxDist        \\ \hline
-\multicolumn{2}{|l|}{CG}                                                                       & 27.58           & 31.25            & 5.82           & 7.12              & 21.76        & 24.13          \\ \hline
-\multicolumn{2}{|l|}{MG}                                                                       & 29.49           & 33.78            & 3.74           & 6.41              & 25.75        & 27.37          \\ \hline
-\multicolumn{2}{|l|}{LU}                                                                       & 19.55           & 28.33            & 0.0            & 0.01              & 19.55        & 28.22          \\ \hline
-\multicolumn{2}{|l|}{EP}                                                                       & 28.40           & 27.04            & 4.29           & 0.49              & 24.11        & 26.55          \\ \hline
-\multicolumn{2}{|l|}{BT}                                                                       & 27.68           & 32.32            & 6.45           & 7.87              & 21.23        & 24.43          \\ \hline
-\multicolumn{2}{|l|}{SP}                                                                       & 20.52           & 24.73            & 5.21           & 2.78              & 15.31         & 21.95         \\ \hline
-\multicolumn{2}{|l|}{FT}                                                                       & 27.03           & 31.02            & 2.75           & 2.54              & 24.28        & 28.48           \\ \hline
+Program & \multicolumn{2}{c|}{Energy saving \%} & \multicolumn{2}{c|}{Perf.  degradation \%} & \multicolumn{2}{c|}{Distance} \\ \cline{2-7} 
+name    & EDP             & MaxDist          & EDP            & MaxDist           & EDP          & MaxDist        \\ \hline
+CG      & 27.58           & 31.25            & 5.82           & 7.12              & 21.76        & 24.13          \\ \hline
+MG      & 29.49           & 33.78            & 3.74           & 6.41              & 25.75        & 27.37          \\ \hline
+LU      & 19.55           & 28.33            & 0.0            & 0.01              & 19.55        & 28.22          \\ \hline
+EP      & 28.40           & 27.04            & 4.29           & 0.49              & 24.11        & 26.55          \\ \hline
+BT      & 27.68           & 32.32            & 6.45           & 7.87              & 21.23        & 24.43          \\ \hline
+SP      & 20.52           & 24.73            & 5.21           & 2.78              & 15.31         & 21.95         \\ \hline
+FT      & 27.03           & 31.02            & 2.75           & 2.54              & 24.28        & 28.48           \\ \hline
 
 \end{tabular}
 \label{table:compare_EDP}
@@ -1154,10 +1190,10 @@ degradation values while giving the same weight for both metrics.
 
 
 
-\begin{figure}[t]
+\begin{figure}[!t]
   \centering
    \includegraphics[scale=0.5]{fig/compare_EDP.pdf}
-  \caption{Tradeoff comparison for NAS benchmarks class C}
+  \caption{Trade-off comparison for NAS benchmarks class C}
   \label{fig:compare_EDP}
 \end{figure}
 
@@ -1166,19 +1202,19 @@ degradation values while giving the same weight for both metrics.
 \label{sec.concl} 
 In this paper, a new online frequency selecting algorithm has been presented. It
 selects the  best possible  vector of frequency  scaling factors that  gives the
-maximum  distance  (optimal  tradeoff)  between  the predicted  energy  and  the
+maximum  distance  (optimal  trade-off)  between  the predicted  energy  and  the
 predicted performance curves for a heterogeneous platform. This algorithm uses a
 new  energy  model  for  measuring  and predicting  the  energy  of  distributed
 iterative  applications running  over heterogeneous  platforms. To  evaluate the
 proposed method, it was applied on the NAS parallel benchmarks and executed over
-a heterogeneous  platform simulated by  Simgrid. The results of  the experiments
+a heterogeneous  platform simulated by  SimGrid. The results of  the experiments
 showed that the algorithm reduces up to 35\% the energy consumption of a message
 passing iterative method while limiting  the degradation of the performance. The
 algorithm also selects different scaling  factors according to the percentage of
 the computing and communication times, and according to the values of the static
 and  dynamic  powers  of the  CPUs.   Finally,  the  algorithm was  compared  to
 Spiliopoulos et al.  algorithm and  the results showed that it outperforms their
-algorithm in terms of energy-time tradeoff.
+algorithm in terms of energy-time trade-off.
 
 In the near future, this method  will be applied to real heterogeneous platforms
 to evaluate its  performance in a real study case. It  would also be interesting
@@ -1216,6 +1252,7 @@ Babylon (Iraq) for supporting his work.
 %%% End:
 
 % LocalWords:  Fanfakh Charr FIXME Tianhe DVFS HPC NAS NPB SMPI Rauber's Rauber
-% LocalWords:  CMOS EPSA Franche ComtÃ© Tflop RÃ¼nger IUT MarÃ©chal Juin cedex
+% LocalWords:  CMOS EPSA Franche ComtÃ© Tflop RÃ¼nger IUT MarÃ©chal Juin cedex GPU
 % LocalWords:  de badri muslim MPI TcpOld TcmOld dNew dOld cp Sopt Tcp Tcm Ps
-% LocalWords:  Scp Fmax Fdiff SimGrid GFlops Xeon EP BT
+% LocalWords:  Scp Fmax Fdiff SimGrid GFlops Xeon EP BT GPUs CPUs AMD
+%  LocalWords:  Spiliopoulos scalability
