[sharelatex-git-integration Best effort strategy and virtual load for asynchronous...

diff --git a/loba-besteffort/biblio.bib b/loba-besteffort/biblio.bib
index b800291cbe1964f5e3f6ae67606abc04a906895e..a0b06d08a060f4168c527c3ee2ec5f6aa0522a31 100644 (file)
--- a/loba-besteffort/biblio.bib
+++ b/loba-besteffort/biblio.bib
@@ -395,3 +395,27 @@ note = {},
   pages={238--253},
   year={2004}
 }
   pages={238--253},
   year={2004}
 }

+
+
+@book{Bharadwaj1996,
+ author = {Bharadwaj, Veeravalli and Robertazzi, Thomas G. and Ghose, Debasish},
+ title = {Scheduling Divisible Loads in Parallel and Distributed Systems},
+ year = {1996},
+ publisher = {IEEE Computer Society Press},
+ address = {Los Alamitos, CA, USA}
+} 
+
+@phdthesis{Drozdowski1998,
+    title    = {Selected Problems of Scheduling Tasks in Multiprocessor Computer Systems},
+    school   = {Poznan University of Technology, Poznan, Poland},
+    author   = {M. Drozdowski},
+    year     = {1998}
+}
+
+@book{Casanova2008,
+ author = {Casanova, Henri and Legrand, Arnaud and Robert, Yves},
+ title = {Parallel Algorithms},
+ year = {2008},
+ edition = {1st},
+ publisher = {Chapman \& Hall/CRC},
+}

diff --git a/loba-besteffort/loba-besteffort.tex b/loba-besteffort/loba-besteffort.tex
index b0885c1c00eec81def2e35872115f5ff3a34885f..ee788a5d85719fd08bb84528b60d6bc0c8fb7a52 100644 (file)
--- a/loba-besteffort/loba-besteffort.tex
+++ b/loba-besteffort/loba-besteffort.tex
@@ -91,7 +91,7 @@ polynomial time complexity.
 
 In this paper, we focus on asynchronous load balancing of non negative real numbers of {\it divisible loads}
 in homogeneous distributed systems. Loads can be divided in arbitrary {\it fine-grain} parallel parts size 
 
 In this paper, we focus on asynchronous load balancing of non negative real numbers of {\it divisible loads}
 in homogeneous distributed systems. Loads can be divided in arbitrary {\it fine-grain} parallel parts size 

-that can be processed independently of each other. This model of divisible loads arises in 
+that can be processed independently of each other~\cite{Bharadwaj1996, Drozdowski1998, Casanova2008}. This model of divisible loads arises in 

 a wide range of real-world applications. Common examples, among many, include signal processing, 
 feature extraction and edge detection in image processing, records search in huge databases, 
 average consensus in WSN, pattern search in Big data and so on. 
 a wide range of real-world applications. Common examples, among many, include signal processing, 
 feature extraction and edge detection in image processing, records search in huge databases, 
 average consensus in WSN, pattern search in Big data and so on. 


@@ -99,21 +99,14 @@ average consensus in WSN, pattern search in Big data and so on.
 
 In the literature, the problem of load balancing has been formulated and studied in various ways. The first pioneering work is due to Bertsekas  and Tsitsiklis~\cite{bertsekas+tsitsiklis.1997.parallel}. Under some specific hypothesis and {\it ping-pong} awareness conditions (see section~\ref{sec.bt-algo} for more details), an asymptotic convergence proof is derived. 
 
 
 In the literature, the problem of load balancing has been formulated and studied in various ways. The first pioneering work is due to Bertsekas  and Tsitsiklis~\cite{bertsekas+tsitsiklis.1997.parallel}. Under some specific hypothesis and {\it ping-pong} awareness conditions (see section~\ref{sec.bt-algo} for more details), an asymptotic convergence proof is derived. 
 

-%%RAPH Attention cette partie n'apparait plus
-\begin{comment}
-This algorithm has been borrowed and adapted in many works. For instance, in~\cite{CortesRCSL02} a static load balancing (called DASUD) for non negative integer number of divisible loads in arbitrary networks topologies is investigated. The term {\it "static"} stems from the fact that no loads are added or consumed during the load balancing process. The theoretical correctness proofs of the convergence property are given. Some generalizations of the same authors' own work for partially asynchronous discrete load balancing model are presented in~\cite{cedo+cortes+ripoll+al.2007.convergence}. The authors prove that the algorithm's convergence is finite and bounded by the straightforward network's diameter of the global equilibrium threshold in the network. In~\cite{bahi+giersch+makhoul.2008.scalable}, a fault tolerant communication version is addressed to deal with average consensus in wireless sensor networks. The objective is to have all nodes converged to the average of their initial measurements based only on nodes' local information. A slight adaptation is also considered  in~\cite{BahiCG10} for dynamic networks with bounded delays asynchronous diffusion. The dynamical aspect stands at the communication level as links between the network's resources may be intermittent.
-\end{comment}

 
 Although  Bertsekas  and Tsitsiklis  describe  the necessary conditions to
 ensure the algorithm's convergence,  there is no indication nor any strategy to really implement
 
 Although  Bertsekas  and Tsitsiklis  describe  the necessary conditions to
 ensure the algorithm's convergence,  there is no indication nor any strategy to really implement

-the load distribution. %In other word, a node  can send some amount of its load to one or   many  of   its  neighbors   while  all   the  convergence   conditions  are followed. 
+the load distribution. 

 Consequently,  we propose a  new strategy called  \besteffort{}
 that tries to balance the load of  a node to all its less loaded neighbors while
 ensuring that all the nodes involved in the load balancing phase have the same
 Consequently,  we propose a  new strategy called  \besteffort{}
 that tries to balance the load of  a node to all its less loaded neighbors while
 ensuring that all the nodes involved in the load balancing phase have the same

-amount of  load.  Moreover, %when real-world asynchronous  applications are considered,
-%using  asynchronous load  balancing  algorithms  can  reduce   the  execution
-%times. 
-most of the time, it is simpler to dissociate load information messages
+amount of  load.  Moreover, most of the time, it is simpler to dissociate load information messages

 from  data  migration  messages.  Former  ones  allow  a  node to  inform  its
 neighbors about its  current load. These messages are in fact very small and can often be sent 
 very quickly.  For example, if a computing iteration takes  a significant time
 from  data  migration  messages.  Former  ones  allow  a  node to  inform  its
 neighbors about its  current load. These messages are in fact very small and can often be sent 
 very quickly.  For example, if a computing iteration takes  a significant time


@@ -142,7 +135,7 @@ The main contributions and novelties of our work are summarized in the following
 \item Unlike earlier works, we use a new concept of virtual loads transfer which allows nodes to predict the future loads they will receive in the subsequent iterations. 
 This leads to a noticeable speedup of the global convergence time of the load balancing process.  
 
 \item Unlike earlier works, we use a new concept of virtual loads transfer which allows nodes to predict the future loads they will receive in the subsequent iterations. 
 This leads to a noticeable speedup of the global convergence time of the load balancing process.  
 

-\item We use SimGrid simulator which is known to be able to characterize and modelize realistic models of computation and communication in different types of platforms. We show that taking into account both loads transfers' costs and network contention is essential and has a real impact on the quality of the load balancing performances. 
+\item We use SimGrid simulator which is known to be able to characterize and model realistic models of computation and communication in different types of platforms. We show that taking into account both loads transfers' costs and network contention is essential and has a real impact on the quality of the load balancing performances. 

 
 \end{itemize}
 
 
 \end{itemize}
 


@@ -174,8 +167,7 @@ that characterize a set of negotiation rules for users' jobs. For instance, Izak
 
 Choi et al.~\cite{ChoiBH09} address the problem of robust task allocation in arbitrary networks. The proposed
 approaches combine a bidding approach for task selection and a consensus procedure scheme for
 
 Choi et al.~\cite{ChoiBH09} address the problem of robust task allocation in arbitrary networks. The proposed
 approaches combine a bidding approach for task selection and a consensus procedure scheme for

-decentralized conflict resolution. The developed algorithms are proven to converge to a conflict-free assignment in
-both single and multiple task assignment problems. An online stochastic dual gradient LB algorithm, which is called DGLB, is proposed in~\cite{chen2017dglb}. The authors deal with both workload and energy management for cloud networks consisting of multiple geo-distributed mapping nodes and data Centers. To enable online distributed implementation, tasks are decomposed both across time and space by leveraging a dual decomposition approach. Experimental results corroborate the merits of the proposed algorithm.
+decentralized conflict resolution. The developed algorithms are proven to converge to a conflict-free assignment in both single and multiple task assignment problem. An online stochastic dual gradient LB algorithm, which is called DGLB, is proposed in~\cite{chen2017dglb}. The authors deal with both workload and energy management for cloud networks consisting of multiple geo-distributed mapping nodes and data Centers. To enable online distributed implementation, tasks are decomposed both across time and space by leveraging a dual decomposition approach. Experimental results corroborate the merits of the proposed algorithm.

 
 
 In~\cite{tripathi2017non} a LB algorithm based on game theory is proposed for distributed data centers. The authors formulate the LB problem as a non-cooperative game among front-end proxy servers and characterize the structure of Nash equilibrium. Based on the obtained Nash equilibrium structure, they derive a LB algorithm to compute the Nash equilibrium. They show through simulations that the proposed algorithm ensures fairness among the users and a good average latency across all client regions. A hybrid task scheduling and load balancing dependent and independent tasks for master-slaves platforms are addressed in~In~\cite{liu2017dems}. To minimize the response time of the submitted jobs, the proposed algorithm which is called DeMS is split into three stages: i) communication overhead reduction between masters and slaves,  ii) task migration to keep the workload balanced iii) and precedence task graphs partitioning. 
 
 
 In~\cite{tripathi2017non} a LB algorithm based on game theory is proposed for distributed data centers. The authors formulate the LB problem as a non-cooperative game among front-end proxy servers and characterize the structure of Nash equilibrium. Based on the obtained Nash equilibrium structure, they derive a LB algorithm to compute the Nash equilibrium. They show through simulations that the proposed algorithm ensures fairness among the users and a good average latency across all client regions. A hybrid task scheduling and load balancing dependent and independent tasks for master-slaves platforms are addressed in~In~\cite{liu2017dems}. To minimize the response time of the submitted jobs, the proposed algorithm which is called DeMS is split into three stages: i) communication overhead reduction between masters and slaves,  ii) task migration to keep the workload balanced iii) and precedence task graphs partitioning. 


@@ -193,31 +185,14 @@ A network is modeled as a connected undirected graph $G=(N,A)$, where $N$ is a s
 of processors and $A$ is a set of communication links. The processors are 
 labeled $i = 1,...,n$, and a link between processors $i$ and
 $j$ is denoted by $(i, j)\in A$. The set of processor $i$'s neighbors is denoted by $V(i)$.
 of processors and $A$ is a set of communication links. The processors are 
 labeled $i = 1,...,n$, and a link between processors $i$ and
 $j$ is denoted by $(i, j)\in A$. The set of processor $i$'s neighbors is denoted by $V(i)$.

-%In this work, we consider that  
-%Processors are  considered to be homogeneous for the sake of simplicity. It is easily extendable to the case of heterogeneous platforms by scaling the processor's load by its computing power~\cite{ElsMonPre02}.
-%In  order  prove  the  convergence  of  asynchronous  iterative  load  balancing
-%Bertsekas         and        Tsitsiklis         proposed         a        model
-%in~\cite{bertsekas+tsitsiklis.1997.parallel}.   Here we  recall  some notations.
-%Consider  that  $N={1,...,n}$  processors   are  connected  through  a  network.
-%Communication links  are represented by  a connected undirected  graph $G=(N,A)$
-%where $A$ is the set of links connecting different processors. 
-%In this work, we
-%consider that  processors are  homogeneous for sake  of simplicity. It  is quite
-%easy to tackle the  heterogeneous case~\cite{ElsMonPre02}. 
+ 

 Load of processor $i$
 at  time $t$  is  represented  by $x_i(t)\geq  0$.   
 Load of processor $i$
 at  time $t$  is  represented  by $x_i(t)\geq  0$.   

-%Let $V(i)$  be  the set  of neighbors of processor  $i$.  

 Each processor $i$ has an estimate  of the load of
 each  of its  neighbors $j  \in V(i)$  denoted by  $x_j^i(t)$ and this estimate 
 may be outdated due to %.  According to
 asynchronism and communication  delays. 
 Each processor $i$ has an estimate  of the load of
 each  of its  neighbors $j  \in V(i)$  denoted by  $x_j^i(t)$ and this estimate 
 may be outdated due to %.  According to
 asynchronism and communication  delays. 

-%, this estimate may be  outdated.  
-%We also
-%consider that the load is described by a continuous variable.
-
-%Since we deal with large  {\it fine grain} parallelism of divisible loads, 
-%the processor's load is represented by a continuous variable for notational 
-%convenience.  
+ 

 \medskip
 When a processor  sends a part of its  load to one or to some of  its neighbors, the
 transfer takes time to be completed.  Let $s_{ij}(t)$ be the amount of load that
 \medskip
 When a processor  sends a part of its  load to one or to some of  its neighbors, the
 transfer takes time to be completed.  Let $s_{ij}(t)$ be the amount of load that


@@ -264,7 +239,7 @@ is linked to processor $2$ which is  also linked to processor $3$, but in which
 %{\bf RAPH, pourquoi il y a $x_3^2$?. Sinon il faudra reformuler la suite, c'est mal dit}
 
 Owing to the algorithm's specifications, processor $2$ can either send 
 %{\bf RAPH, pourquoi il y a $x_3^2$?. Sinon il faudra reformuler la suite, c'est mal dit}
 
 Owing to the algorithm's specifications, processor $2$ can either send 

-loads to processor $1$ or processor
+a load to processor $1$ or processor

 $3$.  If it sends loads to processor $1$, it will not satisfy condition
 \eqref{eq.ping-pong} because after that sending it will be less loaded than
 $x_3^2(t)$.  So we consider that the \emph{ping-pong} condition is probably too
 $3$.  If it sends loads to processor $1$, it will not satisfy condition
 \eqref{eq.ping-pong} because after that sending it will be less loaded than
 $x_3^2(t)$.  So we consider that the \emph{ping-pong} condition is probably too


@@ -391,36 +366,11 @@ Section~\ref{sec.results}.  The amount of data to send is then $s_{ij}(t) =
 
 
 
 
 
 

-%\section{Other strategies}
-%\label{sec.other}
-
-%Another load balancing strategy, working under the same conditions, was
-%previously developed by Bahi, Giersch, and Makhoul in
-%\cite{bahi+giersch+makhoul.2008.scalable}.  In order to assess the performances
-%of the new \besteffort{}, we naturally chose to compare it to this anterior
-%work.  More precisely, we will use the algorithm~2 from
-%\cite{bahi+giersch+makhoul.2008.scalable} and, in the following, we will
-%reference it under the name of naïve implementation of Bertsekas' load balancing algorithm.  {\bf : RAPH j'ai renommé MAKHOUL en naive, il faut valider !!!! LE SOUCI, il faudrait refaire les figures}
-
-%Here is an outline of the \makhoul{} algorithm.  When a given node needs to take
-%a load balancing decision, it starts by sorting its neighbors by increasing
-%order of their load.  Then, it computes the difference between its own load, and
-%the load of each of its neighbors.  Finally, taking the neighbors following the
-%order defined before, the amount of load to send $s_{ij}$ is computed as
-%$1/(n+1)$ of the load difference, with $n$ being the number of neighbors.  This
-%process continues as long as the node is more loaded than the considered
-%neighbor.
-
-

 \section{Virtual load}
 \label{sec.virtual-load}
 
 In this section,  we present the new concept of \emph{virtual load} which aims to improve the global convergence time. For this end, both load transfer messages and load information messages are dissociated. 
 \section{Virtual load}
 \label{sec.virtual-load}
 
 In this section,  we present the new concept of \emph{virtual load} which aims to improve the global convergence time. For this end, both load transfer messages and load information messages are dissociated. 

-%In order to
-%use this concept, load balancing messages must be sent using two different kinds
-%of  messages:  load information  messages  and  load  balancing messages.   
-More
-precisely, a node wanting to send some amount of its load to one (or more) of its neighbors
+More precisely, a node wanting to send some amount of its load to one (or more) of its neighbors

 can first send a load information message about the load it will send, and
 later it can send the load  message containing data to be transferred.
 Load information  messages are in fact short
 can first send a load information message about the load it will send, and
 later it can send the load  message containing data to be transferred.
 Load information  messages are in fact short


@@ -437,26 +387,8 @@ neighbors when needed. By and large, this allows a node on the one hand, to pred
 into account the information about the predictive loads not 
 received yet.
 
 into account the information about the predictive loads not 
 received yet.
 

-% repetition !
-%In fact,  a node that receives a load  information message knows that
-%later it  will receive the  corresponding load balancing message  containing the
-%corresponding data.  So, if this node detects it is too  loaded compared to some
-%of its neighbors  and if it has enough  load (real load), then it  can send more
-%load  to  some of  its  neighbors  without waiting  the  reception  of the  load
-%balancing message.
-
-%Doing  this, we  can  expect a  faster  convergence since  nodes  have a  faster
-%information of the load they will receive, so they can take it into account.
-
-%\FIXME{Est ce qu'on donne l'algo avec virtual load?}
-
-%With integer load, this algorithm has been adapted by rounding the load value. In fact, we consider that the total amount of load is big enough and that it can be split with integer numbers.
-
-

 
 
 
 

-%\FIXME{describe integer mode}
-

 \section{Implementation with SimGrid and simulations}
 \label{sec.simulations}
 
 \section{Implementation with SimGrid and simulations}
 \label{sec.simulations}
 


@@ -635,38 +567,16 @@ To summarize the different load balancing strategies, we have:
 \item[\textbf{variants:}] with, or without virtual loads
 %\item[\textbf{domain:}] real load, or integer load
 \end{description}
 \item[\textbf{variants:}] with, or without virtual loads
 %\item[\textbf{domain:}] real load, or integer load
 \end{description}

-%
-%This gives us as many as $4\times 2\times 2 = 16$ different strategies.

 
 \subsubsection{End of the simulation}
 
 The simulations were run until reaching the global equilibrium threshold. 
 
 \subsubsection{End of the simulation}
 
 The simulations were run until reaching the global equilibrium threshold. 

-%the load was nearly balanced among the participating nodes.  
+  

 More precisely, the simulation stops when each node holds
 an amount of load at least inferior to 1\% of the load average.
 More precisely, the simulation stops when each node holds
 an amount of load at least inferior to 1\% of the load average.

-%, during an arbitrary
-%number of computing iterations (2000 in our case).
-
-%Note that this convergence detection was implemented in a centralized manner.
-%This is easy to do within the simulator, but it is obviously not realistic.  In a
-%real application we would have chosen a decentralized convergence detection
-%algorithm, like the one described in \cite{ccl09:ij}.

 
 \subsubsection{Platform}
 
 
 \subsubsection{Platform}
 

-%In order to show the behavior of the different strategies 
-%in different
-%settings, we simulated the executions on two sorts of platforms.  These two
-%sorts of platforms differ by their network topology.  On the one hand,
-%we have homogeneous platforms, modeled as a cluster.  On the other hand, we have
-%heterogeneous platforms, modeled as the interconnection of a number of clusters.
-
-
-%The clusters are modeled by a fixed number of computing nodes interconnected
-%through a backbone link.  Each computing node has a computing power of
-%1~GFlop/s, and is connected to the backbone by a network link whose bandwidth is
-%of 125~MB/s, with a latency of 50~$\mu$s.  The backbone has a network bandwidth
-%of 2.25~GB/s, with a latency of 500~$\mu$s.

 
 In order to make our experiments, an heterogeneous grid platform description were created by taking a subset of the
 Grid'5000 infrastructure\footnote{Grid'5000 is a French large scale experimental
 
 In order to make our experiments, an heterogeneous grid platform description were created by taking a subset of the
 Grid'5000 infrastructure\footnote{Grid'5000 is a French large scale experimental


@@ -701,23 +611,6 @@ two different types of applications:
 %\item balanced, with a computation/communication cost ratio of $1/1$.
 \end{itemize}
 
 %\item balanced, with a computation/communication cost ratio of $1/1$.
 \end{itemize}
 

-% To summarize the various configurations, we have:
-% \begin{description}
-% \item[\textbf{platforms:}] homogeneous (cluster), or heterogeneous (subset of
-%   Grid'5000)
-% \item[\textbf{platform sizes:}] platforms with 16, 64, 256, or 1024 nodes
-% \item[\textbf{process topologies:}] line, torus, or hypercube
-% \item[\textbf{initial load distribution:}] initially on a only node, or
-%   initially randomly distributed over all nodes
-% \item[\textbf{computation/communication cost ratio:}] $10/1$, $1/1$, or $1/10$
-% \end{description}
-% %
-% This gives us as many as $2\times 4\times 3\times 2\times 3 = 144$ different
-% configurations.
-% %
-% Combined with the various load balancing strategies,  $16\times 144 =
-% 2,304$ distinct settings have been evaluated.  In fact, as it will be shown later, only configurations with a maximum number of 1,024 nodes are considered in order to limit the time of experiments.
-

 
 \subsubsection{Metrics}
 \label{sec.metrics}
 
 \subsubsection{Metrics}
 \label{sec.metrics}


@@ -731,30 +624,16 @@ settings. With these constraints in mind, the following metrics are defined:
 \begin{description}
 \item[\it{average idle time:}] that is the total time spent, when the nodes
   do not hold any share of load, and thus have nothing to compute. 
 \begin{description}
 \item[\it{average idle time:}] that is the total time spent, when the nodes
   do not hold any share of load, and thus have nothing to compute. 

-  %This total
-  %time is divided by the number of participating nodes, such as to have a number
-  %that can be compared between simulations of different sizes.
-  %This metric is expected to give an idea of the ability of the strategy to
-  %diffuse the load quickly.  
-  A smaller value is better.
+   A smaller value is better.

 
 \item[\it{average convergence time:}] that is the average of the times when
   all nodes reached the final balanced load distribution.  Times are measured as a number
   of (simulated) seconds from the beginning of the simulation.
 
 \item[\it{maximum convergence time:}] that is the time when the last node
 
 \item[\it{average convergence time:}] that is the average of the times when
   all nodes reached the final balanced load distribution.  Times are measured as a number
   of (simulated) seconds from the beginning of the simulation.
 
 \item[\it{maximum convergence time:}] that is the time when the last node

-  reached the final stable equilibrium.
-  %These two dates give an idea of the time needed by the strategy to reach the
-  %equilibrium state.  
-  A smaller value is better.
+  reached the final stable equilibrium.  A smaller value is better.

 
 

-% \item[\textbf{data transfer amount:}] that is the sum of the amount of all data
-%   transfers during the simulation.  This sum is then normalized by dividing it
-%   by the total amount of data present in the system.

 
 

-%   This metric is expected to give an idea of the efficiency of the strategy in
-%   terms of data movements, i.e. its ability to reach the equilibrium with fewer
-%   transfers.  Again, a smaller value is better.

 
 \end{description}
 
 
 \end{description}
 


@@ -765,22 +644,7 @@ settings. With these constraints in mind, the following metrics are defined:
 In this section, the results for the different simulations are presented,
 and our observations are explained.
 
 In this section, the results for the different simulations are presented,
 and our observations are explained.
 

-% \subsubsection{Cluster versus grid platforms}
-
-% As mentioned earlier, different algorithms have been simulated on two kinds of
-% physical platforms: clusters and grids.  A first observation,
-% is that the graphs we draw from the data have a similar aspect for the two kinds
-% of platforms.  The only noticeable difference is that the algorithms need a bit
-% more time to achieve the convergence on the grid platforms, than on clusters.
-% Nevertheless their relative performances remain generally similar.
-
-% This suggests that the relative performances of the different strategies are not
-% influenced by the characteristics of the physical platform.  The differences in
-% the convergence times can be explained by the fact that on the grid platforms,
-% distant sites are interconnected by links of smaller bandwidth.

 
 

-% Therefore, in the following, we only discuss the results for the grid
-% platforms.

 
 \subsubsection{Main results}
 
 
 \subsubsection{Main results}
 


@@ -815,16 +679,12 @@ The results in Figure~\ref{fig.results1} are when the load to balance is
 initially on only one node, while the results in Figure~\ref{fig.resultsN} are
 when the load to balance is initially randomly distributed over all nodes. 
 On both figures, the CCR is $10/1$ on the left
 initially on only one node, while the results in Figure~\ref{fig.resultsN} are
 when the load to balance is initially randomly distributed over all nodes. 
 On both figures, the CCR is $10/1$ on the left

-column, and $1/10$ on the right column.  %With a computation/communication cost
-%ratio of $1/1$ the results are just between these two extrema, and definitely
-%don not give additional information, so we chose not to show them here.
+column, and $1/10$ on the right column.  

 On each Figure, ~\ref{fig.results1} and~\ref{fig.resultsN}, the results
 are given for the process topology being, from top to bottom, a line, a torus or
 an hypercube.
 
 On each Figure, ~\ref{fig.results1} and~\ref{fig.resultsN}, the results
 are given for the process topology being, from top to bottom, a line, a torus or
 an hypercube.
 

-Finally, the vertical bars show the measured times for the evaluated metrics
-%each of the algorithms
-. These measured times are, starting at $t=0$ and from bottom to top, the average idle
+Finally, the vertical bars show the measured times for the evaluated metrics. These measured times are, starting at $t=0$ and from bottom to top, the average idle

 time, the average convergence time, and the maximum convergence time (see
 Section~\ref{sec.metrics}).  The measurements are repeated for the different
 platform sizes.  Some bars are missing, especially for large platforms.  This is
 time, the average convergence time, and the maximum convergence time (see
 Section~\ref{sec.metrics}).  The measurements are repeated for the different
 platform sizes.  Some bars are missing, especially for large platforms.  This is


@@ -832,7 +692,6 @@ because the algorithm did not reach the convergence state in the
 allocated time.
 
 
 allocated time.
 
 

-%\FIXME{annoncer le plan de la suite}

 
 \subsubsection{The \besteffort{} and  \makhoul{} strategies without virtual load}
 
 
 \subsubsection{The \besteffort{} and  \makhoul{} strategies without virtual load}
 


@@ -877,7 +736,7 @@ load fairly when needed and is better for sparse connected applications.
 \subsubsection{With virtual load}
 
 The impact of virtual load scheme is most of the time really significant compared to
 \subsubsection{With virtual load}
 
 The impact of virtual load scheme is most of the time really significant compared to

-the simple version of the algorithm with the same configuration. %Sometimes  it  has no  effect  but, based on our observations,  it has never a negative effect on the load balancing we tested.
+the simple version of the algorithm with the same configuration. 

 For instance, as can be seen from Figure~\ref{fig.results1}, when the load is  initially on one node, it can be
 noticed that the  average idle times are generally longer  with the virtual load
 than the simple version. This  can be explained  by the  fact that, with  virtual load,
 For instance, as can be seen from Figure~\ref{fig.results1}, when the load is  initially on one node, it can be
 noticed that the  average idle times are generally longer  with the virtual load
 than the simple version. This  can be explained  by the  fact that, with  virtual load,


@@ -958,21 +817,8 @@ ANR-11-LABX-01-01).  We also thank the supercomputer facilities of the Mésocent
  
 \bibliographystyle{elsarticle-num}
 \bibliography{biblio}
  
 \bibliographystyle{elsarticle-num}
 \bibliography{biblio}

-%\FIXME{find and add more references}
+

 
 \end{document}
 
 
 \end{document}
 

-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: t
-%%% fill-column: 80
-%%% ispell-local-dictionary: "american"
-%%% End:
-
-% LocalWords:  Raphaël Couturier Arnaud Giersch Franche ij Bertsekas Tsitsiklis
-% LocalWords:  SimGrid DASUD Comté asynchronism ji ik isend irecv Cortés et al
-% LocalWords:  chan ctrl fifo Makhoul GFlop xml pre FEMTO Makhoul's fca bdee
-% LocalWords:  cdde Contassot Vivier underlaid du de Maréchal Juin cedex calcul
-% LocalWords:  biblio Institut UMR Université UFC Centre Scientifique CNRS des
-% LocalWords:  École Nationale Supérieure Mécanique Microtechniques ENSMM UTBM
-% LocalWords:  Technologie Bahi
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