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\begin{document}

\begin{frontmatter}

\journal{Parallel Computing}

\title{Best effort strategy and virtual load for\\
  asynchronous iterative load balancing}

\author{Raphaël Couturier}
\ead{raphael.couturier@femto-st.fr}

\author{Arnaud Giersch\corref{cor}}
\ead{arnaud.giersch@femto-st.fr}

\address{%
  Institut FEMTO-ST (UMR 6174),
  Université de Franche-Comté (UFC),
  Centre National de la Recherche Scientifique (CNRS),
  École Nationale Supérieure de Mécanique et des Microtechniques (ENSMM),
  Université de Technologie de Belfort Montbéliard (UTBM)\\
  19 avenue du Maréchal Juin, BP 527, 90016 Belfort cedex, France}

\cortext[cor]{Corresponding author.}

\begin{abstract}
  Most of the time, asynchronous load balancing algorithms have extensively been
  studied in a theoretical point of view. The Bertsekas and Tsitsiklis'
  algorithm~\cite[section~7.4]{bertsekas+tsitsiklis.1997.parallel} is certainly
  the most well known algorithm for which the convergence proof is given. From a
  practical point of view, when a node wants to balance a part of its load to
  some of its neighbors, the strategy is not described.  In this paper, we
  propose a strategy called \besteffort{} which tries to balance the load
  of a node to all its less loaded neighbors while ensuring that all the nodes
  concerned by the load balancing phase have the same amount of load.  Moreover,
  asynchronous iterative algorithms in which an asynchronous load balancing
  algorithm is implemented most of the time can dissociate messages concerning
  load transfers and message concerning load information.  In order to increase
  the converge of a load balancing algorithm, we propose a simple heuristic
  called \emph{virtual load} which allows a node that receives a load
  information message to integrate the load that it will receive later in its
  load (virtually) and consequently sends a (real) part of its load to some of
  its neighbors.  In order to validate our approaches, we have defined a
  simulator based on SimGrid which allowed us to conduct many experiments.
\end{abstract}

% \begin{keywords}
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\end{frontmatter}

\section{Introduction}

Load  balancing algorithms  are  extensively used  in  parallel and  distributed
applications in  order to  reduce the  execution times. They  can be  applied in
different scientific  fields from high  performance computation to  micro sensor
networks.   They are  iterative by  nature.  In  literature many  kinds  of load
balancing  algorithms  have been  studied.   They  can  be classified  according
different  criteria:   centralized  or  decentralized,  in   static  or  dynamic
environment,  with  homogeneous  or  heterogeneous load,  using  synchronous  or
asynchronous iterations, with  a static topology or a  dynamic one which evolves
during time.  In  this work, we focus on  asynchronous load balancing algorithms
where computer nodes  are considered homogeneous and with  homogeneous load with
no external  load. In  this context, Bertsekas  and Tsitsiklis have  proposed an
algorithm which is definitively a reference  for many works. In their work, they
proved that under classical  hypotheses of asynchronous iterative algorithms and
a  special  constraint   avoiding  \emph{ping-pong}  effect,  an  asynchronous
iterative algorithm  converge to  the uniform load  distribution. This  work has
been extended by many authors. For example, Cortés et al., with
DASUD~\cite{cortes+ripoll+cedo+al.2002.asynchronous}, propose a
version working with integer load.  This work was later generalized by
the same authors in \cite{cedo+cortes+ripoll+al.2007.convergence}.
\FIXME{Rajouter des choses ici.  Lesquelles ?}

Although  the Bertsekas  and Tsitsiklis'  algorithm describes  the  condition to
ensure the convergence,  there is no indication or  strategy to really implement
the load distribution. In other word, a node  can send a part of its load to one
or   many  of   its  neighbors   while  all   the  convergence   conditions  are
followed. 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 concerned  by the load balancing phase have the same
amount of  load.  Moreover, when real asynchronous  applications are considered,
using  asynchronous   load  balancing   algorithms  can  reduce   the  execution
times. Most of the times, it is simpler to distinguish load information messages
from  data  migration  messages.  Former  ones  allows  a  node to  inform  its
neighbors of its  current load. These messages are very small,  they can be sent
quite often.  For example, if an  computing iteration takes  a significant times
(ranging from seconds to minutes), it is possible to send a new load information
message at each  neighbor at each iteration. Latter  messages contains data that
migrates from one node to another one. Depending on the application, it may have
sense or not  that nodes try to balance  a part of their load  at each computing
iteration. But the time to transfer a load message from a node to another one is
often much more longer that to  time to transfer a load information message. So,
when a node receives the information  that later it will receive a data message,
it can take this information into account  and it can consider that its new load
is larger.   Consequently, it can  send a part  of it real  load to some  of its
neighbors if required. We call this trick the \emph{virtual load} mechanism.



So, in  this work, we propose a  new strategy for improving  the distribution of
the  load  and  a  simple  but  efficient trick  that  also  improves  the  load
balancing. Moreover, we have conducted  many simulations with SimGrid in order to
validate our improvements are really efficient. Our simulations consider that in
order  to send a  message, a  latency delays  the sending  and according  to the
network  performance and  the message  size, the  time of  the reception  of the
message also varies.

In the following of this paper, Section~\ref{sec.bt-algo} describes the
Bertsekas and Tsitsiklis' asynchronous load balancing algorithm. Moreover, we
present a possible problem in the convergence conditions.
Section~\ref{sec.besteffort} presents the best effort strategy which provides an
efficient way to reduce the execution times.  This strategy will be compared
with other ones, presented in Section~\ref{sec.other}.  In
Section~\ref{sec.virtual-load}, the virtual load mechanism is proposed.
Simulations allowed to show that both our approaches are valid using a quite
realistic model detailed in Section~\ref{sec.simulations}.  Finally we give a
conclusion and some perspectives to this work.



\section{Bertsekas  and Tsitsiklis' asynchronous load balancing algorithm}
\label{sec.bt-algo}

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 $G=(N,V)$
where $V$ 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 $x_i(t)\geq  0$.   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 $j  \in V(i)$  represented by  $x_j^i(t)$.  According to
asynchronism and communication  delays, this estimate may be  outdated.  We also
consider that the load is described by a continuous variable.

When a processor  send a part of its  load to one or some of  its neighbors, the
transfer takes time to be completed.  Let $s_{ij}(t)$ be the amount of load that
processor $i$ has transferred to processor $j$ at time $t$ and let $r_{ij}(t)$ be the
amount of  load received by processor $j$  from processor $i$ at  time $t$. Then
the amount of load of processor $i$ at time $t+1$ is given by:
\begin{equation}
x_i(t+1)=x_i(t)-\sum_{j\in V(i)} s_{ij}(t) + \sum_{j\in V(i)} r_{ji}(t)
\label{eq.ping-pong}
\end{equation}


Some  conditions are  required to  ensure the  convergence. One  of them  can be
called the \emph{ping-pong} condition which specifies that:
\begin{equation}
x_i(t)-\sum _{k\in V(i)} s_{ik}(t) \geq x_j^i(t)+s_{ij}(t)
\end{equation}
for any  processor $i$ and any  $j \in V(i)$ such  that $x_i(t)>x_j^i(t)$.  This
condition aims  at avoiding a processor  to send a  part of its load  and being
less loaded after that.

Nevertheless,  we  think that  this  condition may  lead  to  deadlocks in  some
cases. For example, if we consider  only three processors and that processor $1$
is linked to processor $2$ which is  also linked to processor $3$ (i.e. a simple
chain which 3 processors). Now consider we have the following values at time $t$:
\begin{eqnarray*}
x_1(t)=10   \\
x_2(t)=100   \\
x_3(t)=99.99\\
 x_3^2(t)=99.99\\
\end{eqnarray*}
In this case, processor $2$ can either sends load to processor $1$ or processor
$3$.  If it sends load to processor $1$ it will not satisfy condition
(\ref{eq.ping-pong}) because after the sending it will be less loaded that
$x_3^2(t)$.  So we consider that the \emph{ping-pong} condition is probably to
strong. Currently, we did not try to make another convergence proof without this
condition or with a weaker condition.

Nevertheless, we conjecture that such a weaker condition exists.  In fact, we
have never seen any scenario that is not leading to convergence, even with
load-balancing strategies that are not exactly fulfilling these two conditions.

It may be the subject of future work to express weaker conditions, and to prove
that they are sufficient to ensure the convergence of the load-balancing
algorithm.

\section{Best effort strategy}
\label{sec.besteffort}

In this section we describe a new load-balancing strategy that we call
\besteffort{}.  First, we explain the general idea behind this strategy,
and then we describe some variants of this basic strategy.

\subsection{Basic strategy}

The general idea behind the \besteffort{} strategy is that each processor,
that detects it has more load than some of its neighbors, sends some load to the
most of its less loaded neighbors, doing its best to reach the equilibrium
between those neighbors and himself.

More precisely, when a processor $i$ is in its load-balancing phase,
he proceeds as following.
\begin{enumerate}
\item First, the neighbors are sorted in non-decreasing order of their
  known loads $x^i_j(t)$.

\item Then, this sorted list is traversed in order to find its largest
  prefix such as the load of each selected neighbor is lesser than:
  \begin{itemize}
  \item the processor's own load, and
  \item the mean of the loads of the selected neighbors and of the
    processor's load.
  \end{itemize}
  Let's call $S_i(t)$ the set of the selected neighbors, and
  $\bar{x}(t)$ the mean of the loads of the selected neighbors and of
  the processor load:
  \begin{equation*}
    \bar{x}(t) = \frac{1}{\abs{S_i(t)} + 1}
      \left( x_i(t) + \sum_{j\in S_i(t)} x^i_j(t) \right)
  \end{equation*}
  The following properties hold:
  \begin{equation*}
    \begin{cases}
      S_i(t) \subset V(i) \\
      x^i_j(t) < x_i(t) & \forall j \in S_i(t) \\
      x^i_j(t) < \bar{x} & \forall j \in S_i(t) \\
      x^i_j(t) \leq x^i_k(t) & \forall j \in S_i(t), \forall k \in V(i) \setminus S_i(t) \\
      \bar{x} \leq x_i(t)
    \end{cases}
  \end{equation*}

\item Once this selection is completed, processor $i$ sends to each of
  the selected neighbor $j\in S_i(t)$ an amount of load $s_{ij}(t) =
  \bar{x} - x^i_j(t)$.

  From the above equations, and notably from the definition of
  $\bar{x}$, it can easily be verified that:
  \begin{equation*}
    \begin{cases}
      x_i(t) - \sum_{j\in S_i(t)} s_{ij}(t) = \bar{x} \\
      x^i_j(t) + s_{ij}(t) = \bar{x} & \forall j \in S_i(t)
    \end{cases}
  \end{equation*}
\end{enumerate}

\subsection{Leveling the amount to send}

With the aforementioned basic strategy, each node does its best to reach the
equilibrium with its neighbors.  Since each node may be taking the same kind of
decision at the same moment, there is the risk that a node receives load from
several of its neighbors, and then is temporary going off the equilibrium state.
This is particularly true with strongly connected applications.

In order to reduce this effect, we add the ability to level the amount to send.
The idea, here, is to make smaller steps toward the equilibrium, such that a
potentially wrong decision has a lower impact.

Concretely, once $s_{ij}$ has been evaluated as before, it is simply divided by
some configurable factor.  That's what we named the ``parameter $k$'' in
Section~\ref{sec.results}.  The amount of data to send is then $s_{ij}(t) =
(\bar{x} - x^i_j(t))/k$.
\FIXME[check that it's still named $k$ in Sec.~\ref{sec.results}]{}

\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 Makhoul's.

Here is an outline of the Makhoul's 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 concept of \emph{virtual load}.  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 a part  of its load to one  of its neighbors,
can first send  a load information message containing the load  it will send and
then it can send the load  balancing message containing data  to be transferred.
Load information  message are really  short, consequently they will  be received
very quickly.  In opposition, load  balancing messages are often bigger and thus
require more time to be transferred.

The  concept  of  \emph{virtual load}  allows  a  node  that received  a  load
information message to integrate the load that it will receive later in its load
(virtually)  and consequently send  a (real)  part of  its load  to some  of its
neighbors. 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 in into account.

\FIXME{Est ce qu'on donne l'algo avec virtual load?}

\FIXME{describe integer mode}

\section{Simulations}
\label{sec.simulations}

In order to test and validate our approaches, we wrote a simulator
using the SimGrid
framework~\cite{casanova+legrand+quinson.2008.simgrid}.  This
simulator, which consists of about 2,700 lines of C++, allows to run
the different load-balancing strategies under various parameters, such
as the initial distribution of load, the interconnection topology, the
characteristics of the running platform, etc.  Then several metrics
are issued that permit to compare the strategies.

The simulation model is detailed in the next section (\ref{sec.model}), and the
experimental contexts are described in section~\ref{sec.exp-context}.  Then the
results of the simulations are presented in section~\ref{sec.results}.

\subsection{Simulation model}
\label{sec.model}

In the simulation model the processors exchange messages which are of
two kinds.  First, there are \emph{control messages} which only carry
information that is exchanged between the processors, such as the
current load, or the virtual load transfers if this option is
selected.  These messages are rather small, and their size is
constant.  Then, there are \emph{data messages} that carry the real
load transferred between the processors.  The size of a data message
is a function of the amount of load that it carries, and it can be
pretty large.  In order to receive the messages, each processor has
two receiving channels, one for each kind of messages.  Finally, when
a message is sent or received, this is done by using the non-blocking
primitives of SimGrid\footnote{That are \texttt{MSG\_task\_isend()},
  and \texttt{MSG\_task\_irecv()}.}.

During the simulation, each processor concurrently runs three threads:
a \emph{receiving thread}, a \emph{computing thread}, and a
\emph{load-balancing thread}, which we will briefly describe now.

For the sake of simplicity, a few details were voluntary omitted from
these descriptions.  For an exhaustive presentation, we refer to the
actual source code that was used for the experiments%
\footnote{As mentioned before, our simulator relies on the SimGrid
  framework~\cite{casanova+legrand+quinson.2008.simgrid}.  For the
  experiments, we used a pre-release of SimGrid 3.7 (Git commit
  67d62fca5bdee96f590c942b50021cdde5ce0c07, available from
  \url{https://gforge.inria.fr/scm/?group_id=12})}, and which is
available at
\url{http://info.iut-bm.univ-fcomte.fr/staff/giersch/software/loba.tar.gz}.

\subsubsection{Receiving thread}

The receiving thread is in charge of waiting for messages to come, either on the
control channel, or on the data channel.  Its behavior is sketched by
Algorithm~\ref{algo.recv}.  When a message is received, it is pushed in a buffer
of received message, to be later consumed by one of the other threads.  There
are two such buffers, one for the control messages, and one for the data
messages.  The buffers are implemented with a lock-free FIFO
\cite{sutter.2008.writing} to avoid contention between the threads.

\begin{algorithm}
  \caption{Receiving thread}
  \label{algo.recv}
  \KwData{
    \begin{algodata}
      \VAR{ctrl\_chan}, \VAR{data\_chan}
      & communication channels (control and data) \\
      \VAR{ctrl\_fifo}, \VAR{data\_fifo}
      & buffers of received messages (control and data) \\
    \end{algodata}}
  \While{true}{%
    wait for a message to be available on either \VAR{ctrl\_chan},
    or \VAR{data\_chan}\;
    \If{a message is available on \VAR{ctrl\_chan}}{%
      get the message from \VAR{ctrl\_chan}, and push it into \VAR{ctrl\_fifo}\;
    }
    \If{a message is available on \VAR{data\_chan}}{%
      get the message from \VAR{data\_chan}, and push it into \VAR{data\_fifo}\;
    }
  }
\end{algorithm}

\subsubsection{Computing thread}

The computing thread is in charge of the real load management.  As exposed in
Algorithm~\ref{algo.comp}, it iteratively runs the following operations:
\begin{itemize}
\item if some load was received from the neighbors, get it;
\item if there is some load to send to the neighbors, send it;
\item run some computation, whose duration is function of the current
  load of the processor.
\end{itemize}
Practically, after the computation, the computing thread waits for a
small amount of time if the iterations are looping too fast (for
example, when the current load is near zero).

\begin{algorithm}
  \caption{Computing thread}
  \label{algo.comp}
  \KwData{
    \begin{algodata}
      \VAR{data\_fifo} & buffer of received data messages \\
      \VAR{real\_load} & current load \\
    \end{algodata}}
  \While{true}{%
    \If{\VAR{data\_fifo} is empty and $\VAR{real\_load} = 0$}{%
      wait until a message is pushed into \VAR{data\_fifo}\;
    }
    \While{\VAR{data\_fifo} is not empty}{%
      pop a message from \VAR{data\_fifo}\;
      get the load embedded in the message, and add it to \VAR{real\_load}\;
    }
    \ForEach{neighbor $n$}{%
      \If{there is some amount of load $a$ to send to $n$}{%
        send $a$ units of load to $n$, and subtract it from \VAR{real\_load}\;
      }
    }
    \If{$\VAR{real\_load} > 0.0$}{
      simulate some computation, whose duration is function of \VAR{real\_load}\;
      ensure that the main loop does not iterate too fast\;
    }
  }
\end{algorithm}

\subsubsection{Load-balancing thread}

The load-balancing thread is in charge of running the load-balancing algorithm,
and exchange the control messages.  As shown in Algorithm~\ref{algo.lb}, it
iteratively runs the following operations:
\begin{itemize}
\item get the control messages that were received from the neighbors;
\item run the load-balancing algorithm;
\item send control messages to the neighbors, to inform them of the
  processor's current load, and possibly of virtual load transfers;
\item wait a minimum (configurable) amount of time, to avoid to
  iterate too fast.
\end{itemize}

\begin{algorithm}
  \caption{Load-balancing}
  \label{algo.lb}
  \While{true}{%
    \While{\VAR{ctrl\_fifo} is not empty}{%
      pop a message from \VAR{ctrl\_fifo}\;
      identify the sender of the message,
      and update the current knowledge of its load\;
    }
    run the load-balancing algorithm to make the decision about load transfers\;
    \ForEach{neighbor $n$}{%
      send a control messages to $n$\;
    }
    ensure that the main loop does not iterate too fast\;
  }
\end{algorithm}

\paragraph{}\FIXME{ajouter des détails sur la gestion de la charge virtuelle ?
  par ex, donner l'idée générale de l'implémentation.  l'idée générale est déja
  décrite en section~\ref{sec.virtual-load}}

\subsection{Experimental contexts}
\label{sec.exp-context}

In order to assess the performances of our algorithms, we ran our
simulator with various parameters, and extracted several metrics, that
we will describe in this section.

\subsubsection{Load balancing strategies}

Several load balancing strategies were compared.  We ran the experiments with
the \besteffort{}, and with the \makhoul{} strategies.  \emph{Best
  effort} was tested with parameter $k = 1$, $k = 2$, and $k = 4$.  Secondly,
each strategy was run in its two variants: with, and without the management of
\emph{virtual load}.  Finally, we tested each configuration with \emph{real},
and with \emph{integer} load.

To summarize the different load balancing strategies, we have:
\begin{description}
\item[\textbf{strategies:}] \makhoul{}, or \besteffort{} with $k\in
  \{1,2,4\}$
\item[\textbf{variants:}] with, or without virtual load
\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 the load was nearly balanced among the
participating nodes.  More precisely the simulation stops when each node holds
an amount of load at less than 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's obviously not realistic.  In a
real application we would have chosen a decentralized convergence detection
algorithm, like the one described by Bahi, Contassot-Vivier, Couturier, and
Vernier in \cite{10.1109/TPDS.2005.2}.

\subsubsection{Platforms}

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 underlaid 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 were 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.

The heterogeneous platform descriptions were created by taking a subset of the
Grid'5000 infrastructure\footnote{Grid'5000 is a French large scale experimental
  Grid (see \url{https://www.grid5000.fr/}).}, as described in the platform file
\texttt{g5k.xml} distributed with SimGrid.  Note that the heterogeneity of the
platform here only comes from the network topology.  Indeed, since our
algorithms currently do not handle heterogeneous computing resources, the
processor speeds were normalized, and we arbitrarily chose to fix them to
1~GFlop/s.

Then we derived each sort of platform with four different number of computing
nodes: 16, 64, 256, and 1024 nodes.

\subsubsection{Configurations}

The distributed processes of the application were then logically organized along
three possible topologies: a line, a torus or an hypercube.  We ran tests where
the total load was initially on an only node (at one end for the line topology),
and other tests where the load was initially randomly distributed across all the
participating nodes.  The total amount of load was fixed to a number of load
units equal to 1000 times the number of node.  The average load is then of 1000
load units.

For each of the preceding configuration, we finally had to choose the
computation and communication costs of a load unit.  We chose them, such as to
have three different computation over communication cost ratios, and hence model
three different kinds of applications:
\begin{itemize}
\item mainly communicating, with a computation/communication cost ratio of $1/10$;
\item mainly computing, with a computation/communication cost ratio of $10/1$ ;
\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, we had $16\times 144 =
2304$ distinct settings to evaluate.  In fact, as it will be shown later, we
didn't run all the strategies, nor all the configurations for the bigger
platforms with 1024 nodes, since to simulations would have run for a too long
time.

Anyway, all these the experiments represent more than 240 hours of computing
time.

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

In order to evaluate and compare the different load balancing strategies we had
to define several metrics.  Our goal, when choosing these metrics, was to have
something tending to a constant value, i.e. to have a measure which is not
changing anymore once the convergence state is reached.  Moreover, we wanted to
have some normalized value, in order to be able to compare them across different
settings.

With these constraints in mind, we defined the following metrics:
%
\begin{description}
\item[\textbf{average idle time:}] that's the total time spent, when the nodes
  don't 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.

\item[\textbf{average convergence date:}] that's the average of the dates when
  all nodes reached the convergence state.  The dates are measured as a number
  of (simulated) seconds since the beginning of the simulation.

\item[\textbf{maximum convergence date:}] that's the date when the last node
  reached the convergence state.

  These two dates give an idea of the time needed by the strategy to reach the
  equilibrium state.  A smaller value is better.

\item[\textbf{data transfer amount:}] that's 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}


\subsection{Experimental results}
\label{sec.results}

In this section, the results for the different simulations will be presented,
and we will try to explain our observations.

\subsubsection{Cluster vs grid platforms}

As mentioned earlier, we simulated the different algorithms on two kinds of
physical platforms: clusters and grids.  A first observation that we can make,
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 identical.

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'll only discuss the results for the grid
platforms.

\subsubsection{Main results}

\begin{figure*}[p]
  \centering
  \includegraphics[width=.5\linewidth]{data/graphs/R1-10:1-grid-line}%
  \includegraphics[width=.5\linewidth]{data/graphs/R1-1:10-grid-line}
  \includegraphics[width=.5\linewidth]{data/graphs/R1-10:1-grid-torus}%
  \includegraphics[width=.5\linewidth]{data/graphs/R1-1:10-grid-torus}
  \includegraphics[width=.5\linewidth]{data/graphs/R1-10:1-grid-hcube}%
  \includegraphics[width=.5\linewidth]{data/graphs/R1-1:10-grid-hcube}
  \caption{Real mode, initially on an only mode, comp/comm cost ratio = $10/1$ (left), or $1/10$ (right).}
  \label{fig.results1}
\end{figure*}

\begin{figure*}[p]
  \centering
  \includegraphics[width=.5\linewidth]{data/graphs/RN-10:1-grid-line}%
  \includegraphics[width=.5\linewidth]{data/graphs/RN-1:10-grid-line}
  \includegraphics[width=.5\linewidth]{data/graphs/RN-10:1-grid-torus}%
  \includegraphics[width=.5\linewidth]{data/graphs/RN-1:10-grid-torus}
  \includegraphics[width=.5\linewidth]{data/graphs/RN-10:1-grid-hcube}%
  \includegraphics[width=.5\linewidth]{data/graphs/RN-1:10-grid-hcube}
  \caption{Real mode, random initial distribution, comp/comm cost ratio = $10/1$ (left), or $1/10$ (right).}
  \label{fig.resultsN}
\end{figure*}

The main results for our simulations on grid platforms are presented on the
figures~\ref{fig.results1} and~\ref{fig.resultsN}.
%
The results on figure~\ref{fig.results1} are when the load to balance is
initially on an only node, while the results on figure~\ref{fig.resultsN} are
when the load to balance is initially randomly distributed over all nodes.

On both figures, the computation/communication cost ratio 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't give additional information, so we chose not to show them here.

On each of the figures~\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, on the graphs, the vertical bars show the measured times for each of
the algorithms.  These measured times are, from bottom to top, the average idle
time, the average convergence date, and the maximum convergence date (see
Section~\ref{sec.metrics}).  The measurements are repeated for the different
platform sizes.  Some bars are missing, specially for large platforms.  This is
either because the algorithm did not reach the convergence state in the
allocated time, or because we simply decided not to run it.

\FIXME{annoncer le plan de la suite}

\subsubsection{The \besteffort{} strategy with the load initially on only one
  node}

Before looking  at the different variations,  we will first show  that the plain
\besteffort{}  strategy  is valuable,  and  may be  as  good  as the  \makhoul{}
strategy.  On  the graphs  from the figure~\ref{fig.results1},  these strategies
(with virtual load feature) are respectively labeled ``b'' and ``a''.

We  can  see  that  the  relative  performance of  these  strategies  is  mainly
influenced by  the application topology.  It  is for the line  topology that the
difference is the  more important.  In this case,  the \besteffort{} strategy is
nearly twice as  fast as the \makhoul{} strategy.  This can  be explained by the
fact that the \besteffort{} strategy tries to distribute the load faitly between
all the nodes  and with the line topology,  it is easy to load  balance the load
fairly.

On the contrary, for the hypercube topology, the \besteffort{} strategy performs
worse than the \makhoul{} strategy. In this case, the \makhoul{} strategy which
tries to give more load to few neighbors reaches the equilibrum faster.

For the torus  topology, for which the  number of links is between  the line and
the hypercube, the \makhoul{} strategy  is slightly better but the difference is
more nuanced.

Globally   the  number  of   interconnection  is   very  important.    The  more
interconnection links there are, the  faster the \makhoul{} strategy is because
it distributes quickly significant amount of load even if this is unfair between
all the  neighbors.  In opposition,  the \besteffort{} strategy  distributes the
load fairly so this strategy is better for low connected strategy.


\subsubsection{With the virtual load extension with the load initially on only
  one node}

Dans ce cas légère amélioration de la cvg. max.  Temps moyen de cvg. amélioré,
mais plus de temps passé en idle, surtout quand les comms coutent cher.

\subsubsection{The \besteffort{} strategy with an initial random load
  distribution, and larger platforms}

Mêmes conclusions pour line et hcube.
Sur tore, BE se fait exploser quand les comms coutent cher.

\FIXME{virer les 1024 ?}

\subsubsection{With the virtual load extension with an initial random load
  distribution}

Soit c'est équivalent, soit on gagne -> surtout quand les comms coutent cher et
qu'il y a beaucoup de voisins.

\subsubsection{The $k$ parameter}
\label{results-k}

Dans le cas où les comms coutent cher et ou BE se fait avoir, on peut ameliorer
les perfs avec le param k.

\subsubsection{With integer load, 1 ou N}

Cas normal, ligne -> converge pas (effet d'escalier).
Avec vload, ça converge.

Dans les autres cas, résultats similaires au cas réel: redire que vload est
intéressant.

\FIXME{virer la metrique volume de comms}

\FIXME{ajouter une courbe ou on voit l'évolution de la charge en fonction du
  temps : avec et sans vload}

% \begin{itemize}
% \item cluster ou grid, entier ou réel, ne font pas de grosses différences
% \item bookkeeping? améliore souvent les choses, parfois au prix d'un retard au démarrage
% \item makhoul? se fait battre sur les grosses plateformes
% \item taille de plateforme?
% \item ratio comp/comm?
% \item option $k$? peut-être intéressant sur des plateformes fortement interconnectées (hypercube)
% \item volume de comm? souvent, besteffort/plain en fait plus. pourquoi?
% \item répartition initiale de la charge ?
% \item integer mode sur topo. line n'a jamais fini en plain? vérifier si ce n'est
%   pas à cause de l'effet d'escalier que bk est capable de gommer.
% \end{itemize}}

% On veut montrer quoi ? :

% 1) best plus rapide que les autres (simple, makhoul)
% 2) avantage virtual load

% Est ce qu'on peut trouver des contre exemple?
% Topologies variées


% Simulation avec temps définies assez long et on mesure la qualité avec : volume de calcul effectué, volume de données échangées
% Mais aussi simulation avec temps court qui montre que seul best converge

% Expés avec ratio calcul/comm rapide et lent

% Quelques expés avec charge initiale aléatoire plutot que sur le premier proc

% Cadre processeurs homogènes

% Topologies statiques

% On ne tient pas compte de la vitesse des liens donc on la considère homogène

% Prendre un réseau hétérogène et rendre processeur homogène

% Taille : 10 100 très gros

\section{Conclusion and perspectives}

\FIXME{conclude!}

\section*{Acknowledgments}

Computations have been performed on the supercomputer facilities of the
Mésocentre de calcul de Franche-Comté.

\bibliographystyle{elsarticle-num}
\bibliography{biblio}
\FIXME{find and add more references}

\end{document}

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% 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
