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-\usepackage{newtxtext}
-\usepackage[cmintegrals]{newtxmath}
-%\usepackage{mathptmx,helvet,courier}
+%\usepackage{newtxtext}
+%\usepackage[cmintegrals]{newtxmath}
+\usepackage{mathptmx,helvet,courier}
\usepackage{amsmath}
\usepackage{graphicx}
\newcommand{\VAR}[1]{\textit{#1}}
+\newcommand{\besteffort}{\emph{best effort}}
+\newcommand{\makhoul}{\emph{Makhoul}}
+
\begin{document}
\begin{frontmatter}
\author{Arnaud Giersch\corref{cor}}
\ead{arnaud.giersch@femto-st.fr}
-\address{FEMTO-ST, University of Franche-Comté\\
- 19 avenue de Maréchal Juin, BP 527, 90016 Belfort cedex , France\\
- % Tel.: +123-45-678910\\
- % Fax: +123-45-678910\\
-}
+\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.}
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 \emph{best effort} which tries to balance the load
+ 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
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 \emph{best effort}
+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,
network performance and the message size, the time of the reception of the
message also varies.
-In the following of this paper, Section~\ref{BT algo} describes the Bertsekas
-and Tsitsiklis' asynchronous load balancing algorithm. Moreover, we present a
-possible problem in the convergence conditions. Section~\ref{Best-effort}
-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{Other}. In Section~\ref{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{Simulations}. Finally we
-give a conclusion and some perspectives to this work.
+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{BT algo}
+\label{sec.bt-algo}
In order prove the convergence of asynchronous iterative load balancing
Bertsekas and Tsitsiklis proposed a model
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}
+\label{eq.ping-pong}
\end{equation}
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
+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.
algorithm.
\section{Best effort strategy}
-\label{Best-effort}
+\label{sec.besteffort}
In this section we describe a new load-balancing strategy that we call
-\emph{best effort}. First, we explain the general idea behind this strategy,
+\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 \emph{best effort} strategy is that each processor,
+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.
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{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{Results}]{}
+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{Other}
+\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 \emph{best effort}, we naturally chose to compare it to this anterior
+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.
\section{Virtual load}
-\label{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
\FIXME{describe integer mode}
\section{Simulations}
-\label{Simulations}
+\label{sec.simulations}
In order to test and validate our approaches, we wrote a simulator
using the SimGrid
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{Sim
- model}), and the experimental contexts are described in
-section~\ref{Contexts}. Then the results of the simulations are
-presented in section~\ref{Results}.
+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{Sim 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
\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{Virtual load}}
+ 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{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
\subsubsection{Load balancing strategies}
Several load balancing strategies were compared. We ran the experiments with
-the \emph{Best effort}, and with the \emph{Makhoul} strategies. \emph{Best
+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},
To summarize the different load balancing strategies, we have:
\begin{description}
-\item[\textbf{strategies:}] \emph{Makhoul}, or \emph{Best effort} with $k\in
+\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
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
\subsection{Experimental results}
-\label{Results}
+\label{sec.results}
In this section, the results for the different simulations will be presented,
and we'll try to explain our observations.
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 bandwith.
+distant sites are interconnected by links of smaller bandwidth.
Therefore, in the following, we'll only discuss the results for the grid
platforms.
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 computatio/communication cost
+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.
are given for the process topology being, from top to bottom, a line, a torus or
an hypercube.
-\FIXME{explain how to read the graphs}
+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.
-each bar -> times for an algorithm
-recall the different times
-no bar -> not run or did not converge in allocated time
+\FIXME{annoncer le plan de la suite}
-repeated for the different platform sizes.
+\subsubsection{The \besteffort{} strategy with the load initially on only one
+ node}
-\FIXME{donner les premières conclusions, annoncer le plan de la suite}
+Before looking at the different variations, we'll 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
+are respectively labeled ``b'' and ``a''.
-\subsubsection{With the virtual load extension}
+twice faster on lines
+almost equivalent on torus
+worse on hcubes
-\subsubsection{The $k$ parameter}
+-> interconnection
-\subsubsection{With an initial random repartition, and larger platforms}
+plus c'est connecté, moins bon est BE car à vouloir trop bien équilibrer
+localement, le processeurs se perturbent mutuellement. Du coup, makhoul qui
+équilibre moins bien localement est moins perturbé par ces interférences.
-\subsubsection{With integer load}
+\subsubsection{With the virtual load extension with the load initially on only
+ one node}
-\FIXME{what about the amount of data?}
+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.
-\begin{itshape}
-\FIXME{remove that part}
-Dans cet ordre:
-...
-- comparer be/makhoul -> be tient la route
- -> en réel uniquement
-- valider l'extension virtual load -> c'est 'achement bien
-- proposer le -k -> ça peut aider dans certains cas
-- conclure avec la version entière -> on n'a pas l'effet d'escalier !
-Q: comment inclure les types/tailles de platesformes ?
-Q: comment faire des moyennes ?
-Q: comment introduire les distrib 1/N ?
-...
+\subsubsection{The \besteffort{} strategy with an initial random load
+ distribution, and larger platforms}
-On constate quoi (vérifier avec les chiffres)?
-\begin{itemize}
-\item cluster ou grid, entier ou réel, ne font pas de grosses différences
+Mêmes conclusions pour line et hcube.
+Sur tore, BE se fait exploser quand les comms coutent cher.
-\item bookkeeping? améliore souvent les choses, parfois au prix d'un retard au démarrage
+\FIXME{virer les 1024 ?}
-\item makhoul? se fait battre sur les grosses plateformes
+\subsubsection{With the virtual load extension with an initial random load
+ distribution}
-\item taille de plateforme?
+Soit c'est équivalent, soit on gagne -> surtout quand les comms coutent cher et
+qu'il y a beaucoup de voisins.
-\item ratio comp/comm?
+\subsubsection{The $k$ parameter}
-\item option $k$? peut-être intéressant sur des plateformes fortement interconnectées (hypercube)
+Dans le cas où les comms coutent cher et ou BE se fait avoir, on peut ameliorer
+les perfs avec le param k.
-\item volume de comm? souvent, besteffort/plain en fait plus. pourquoi?
+\subsubsection{With integer load, 1 ou N}
-\item répartition initiale de la charge ?
+Cas normal, ligne -> converge pas (effet d'escalier).
+Avec vload, ça converge.
-\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.
+Dans les autres cas, résultats similaires au cas réel: redire que vload est
+intéressant.
-\end{itemize}
+\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 ? :
% Prendre un réseau hétérogène et rendre processeur homogène
% Taille : 10 100 très gros
-\end{itshape}
\section{Conclusion and perspectives}
\FIXME{conclude!}
-\section*{Acknowledgements}
+\section*{Acknowledgments}
Computations have been performed on the supercomputer facilities of the
Mésocentre de calcul de Franche-Comté.
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-% LocalWords: Raphaël Couturier Arnaud Giersch Abderrahmane Sider Franche ij
-% LocalWords: Bertsekas Tsitsiklis SimGrid DASUD Comté Béjaïa asynchronism ji
-% LocalWords: ik isend irecv Cortés et al chan ctrl fifo Makhoul GFlop xml pre
-% LocalWords: FEMTO Makhoul's fca bdee cdde Contassot Vivier underlaid
+% 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
