X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/loba-papers.git/blobdiff_plain/a2a8d07a9e743174f5c7121b387e73dfa8d5eaaa..ccdcd0ac01b39dae92f6c456577fafdb046a36bd:/loba-besteffort/loba-besteffort.tex diff --git a/loba-besteffort/loba-besteffort.tex b/loba-besteffort/loba-besteffort.tex index d97c653..c387ad6 100644 --- a/loba-besteffort/loba-besteffort.tex +++ b/loba-besteffort/loba-besteffort.tex @@ -27,6 +27,9 @@ \newcommand{\VAR}[1]{\textit{#1}} +\newcommand{\besteffort}{\emph{best effort}} +\newcommand{\makhoul}{\emph{Makhoul}} + \begin{document} \begin{frontmatter} @@ -42,8 +45,13 @@ \author{Arnaud Giersch\corref{cor}} \ead{arnaud.giersch@femto-st.fr} -\address{FEMTO-ST, University of Franche-Comté\\ - 19 avenue du Maréchal Juin, BP 527, 90016 Belfort cedex, France} +\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.} @@ -54,7 +62,7 @@ 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 @@ -79,7 +87,8 @@ 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 +networks. They are iterative by nature.\FIXME{really?} +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 @@ -101,17 +110,17 @@ 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 \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, 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 +from data migration messages. Former ones allow 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 +quite often. For example, if a 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 +message to each neighbor at each iteration. Latter messages contain 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 @@ -121,14 +130,12 @@ 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 +So, in this work, we propose a new strategy to improve 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 that 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 @@ -151,8 +158,8 @@ 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 +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 $x_i(t)\geq 0$. Let $V(i)$ be the set of @@ -185,15 +192,15 @@ 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*} +\begin{align*} + x_1(t) &= 10 \\ + x_2(t) &= 100 \\ + x_3(t) &= 99.99 \\ + x_3^2(t) &= 99.99 \\ +\end{align*} 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 +\eqref{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. @@ -210,12 +217,12 @@ algorithm. \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. @@ -289,7 +296,7 @@ Section~\ref{sec.results}. The amount of data to send is then $s_{ij}(t) = 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. @@ -299,7 +306,7 @@ 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 +$1/(n+1)$ 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. @@ -310,8 +317,8 @@ neighbor. 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 +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 @@ -328,7 +335,7 @@ 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. +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?} @@ -339,7 +346,7 @@ information of the load they will receive, so they can take in into account. In order to test and validate our approaches, we wrote a simulator using the SimGrid -framework~\cite{casanova+legrand+quinson.2008.simgrid}. This +framework~\cite{simgrid.web,casanova+giersch+legrand+al.2014.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 @@ -375,7 +382,7 @@ 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 + framework~\cite{casanova+giersch+legrand+al.2014.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 @@ -501,7 +508,7 @@ we will describe in this section. \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}, @@ -509,7 +516,7 @@ and with \emph{integer} load. 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 @@ -553,7 +560,7 @@ 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 +Then we derived each kind of platform with four different numbers of computing nodes: 16, 64, 256, and 1024 nodes. \subsubsection{Configurations} @@ -645,7 +652,7 @@ With these constraints in mind, we defined the following metrics: \label{sec.results} In this section, the results for the different simulations will be presented, -and we'll try to explain our observations. +and we will try to explain our observations. \subsubsection{Cluster vs grid platforms} @@ -714,40 +721,123 @@ 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{donner les premières conclusions, annoncer le plan de la suite} -\FIXME{comparer be/makhoul -> be tient la route (parler du cas réel uniquement)} +\FIXME{annoncer le plan de la suite} -\subsubsection{With the virtual load extension} +\subsubsection{The \besteffort{} and \makhoul{} strategies without virtual load} -\FIXME{valider l'extension virtual load -> c'est 'achement bien} +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 Figures~\ref{fig.results1} and~\ref{fig.resultsN}, +these strategies are respectively labeled ``b'' and ``a''. -\subsubsection{The $k$ parameter} +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 faster than the \makhoul{} strategy. This can be explained by the +fact that the \besteffort{} strategy tries to distribute the load fairly between +all the nodes and with the line topology, it is easy to load balance the load +fairly. -\FIXME{proposer le -k -> ça peut aider dans certains cas} +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 equilibrium faster. -\subsubsection{With an initial random distribution, and larger platforms} +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 when the initial load is only on one node. The only case where the +\makhoul{} strategy is really faster than the \besteffort{} strategy is with the +random initial distribution when the communication are slow. -\FIXME{dire quoi ici ?} +Globally the number of interconnection is very important. The more +the interconnection links 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 integer load} -\FIXME{conclure avec la version entière -> on n'a pas l'effet d'escalier !} +\subsubsection{Virtual load} -\FIXME{what about the amount of data?} +The influence of virtual load is most of the time really significant compared to +the same configuration without it. Sometimes it has no effect but {\bf A + VERIFIER} it has never a negative effect on the load balancing we tested. -\FIXME{On constate quoi (vérifier avec les chiffres)? -\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 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 without it. This can be explained by the fact that, with virtual load, +processors will exchange all the load they need to exchange as soon as the +virtual load has been balanced between all the processors. So consequently they +cannot compute at the beginning. This is especially noticeable when the +communication are slow (on the left part of Figure ~\ref{fig.results1}. + +%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} + +%In +%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} + +As explained previously when the communication are slow the \besteffort{} +strategy is efficient. This is due to the fact that it tries to balance the load +fairly and consequently a significant amount of the load is transfered between +processors. In this situation, it is possible to reduce the convergence time by +using the leveler parameter (parameter $k$). The advantage of using this +solution is particularly efficient when the initial load is randomly distributed +on the nodes with torus and hypercube topology and slow communication. When +virtual load mechanism is used, the effect of this parameter is also visible +with the same condition. + + + +\subsubsection{With integer load} + +We also performed some experiments with integer load instead of load with real +value. In this case, the results have globally the same behavior. The most +intereting result, from our point of view, is that the virtual mode allows +processors in a line topology to converge to the uniform load balancing. Without +the virtual load, most of the time, processors converge to what we call the +``stairway effect'', that is to say that there is only a difference of one in +the load of each processor and its neighbors (for example with 10 processors, we +obtain 10 9 8 7 6 6 7 8 9 10 instead of 8 8 8 8 8 8 8 8 8 8). + +%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{ajouter une courbe avec l'équilibrage en entier} + +\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 ? : @@ -801,4 +891,6 @@ Mésocentre de calcul de Franche-Comté. % 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 +% LocalWords: biblio Institut UMR Université UFC Centre Scientifique CNRS des +% LocalWords: École Nationale Supérieure Mécanique Microtechniques ENSMM UTBM +% LocalWords: Technologie Bahi