X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/loba-papers.git/blobdiff_plain/af32e62a6cd1228c1efbdad6d120bc81ea240282..f1aa5b9dfdd4ecd85ea65ea01584c8c452d8acec:/loba-besteffort/loba-besteffort.tex?ds=inline diff --git a/loba-besteffort/loba-besteffort.tex b/loba-besteffort/loba-besteffort.tex index 5570b35..2d37f5f 100644 --- a/loba-besteffort/loba-besteffort.tex +++ b/loba-besteffort/loba-besteffort.tex @@ -87,7 +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 @@ -115,11 +116,11 @@ 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 @@ -129,14 +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 @@ -159,8 +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 @@ -193,12 +192,12 @@ 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 @@ -307,7 +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. @@ -347,7 +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+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 @@ -653,7 +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} @@ -724,65 +723,103 @@ 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} +\subsubsection{The \besteffort{} and \makhoul{} strategies without virtual load} -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''. +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''. -We can see that the relative performance of these strategies is mainly -influenced by the application topology. It's 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. +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. On the contrary, for the hypercube topology, the \besteffort{} strategy performs -worse than the \makhoul{} strategy. +worse than the \makhoul{} strategy. In this case, the \makhoul{} strategy which +tries to give more load to few neighbors reaches the equilibrium faster. -Finally, the results are more nuanced for the torus topology. +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. -This can be explained by ... +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. --> interconnection -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{Virtual load} -\subsubsection{With the virtual load extension with the load initially on only - one node} +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. -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. +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}. -\subsubsection{The \besteffort{} strategy with an initial random load - distribution, and larger platforms} +%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. -Mêmes conclusions pour line et hcube. -Sur tore, BE se fait exploser quand les comms coutent cher. +%\subsubsection{The \besteffort{} strategy with an initial random load +% distribution, and larger platforms} -\FIXME{virer les 1024 ?} +%In +%Mêmes conclusions pour line et hcube. +%Sur tore, BE se fait exploser quand les comms coutent cher. -\subsubsection{With the virtual load extension with an initial random load - distribution} +%\FIXME{virer les 1024 ?} -Soit c'est équivalent, soit on gagne -> surtout quand les comms coutent cher et -qu'il y a beaucoup de voisins. +%\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. +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). -\subsubsection{With integer load, 1 ou N} +%Cas normal, ligne -> converge pas (effet d'escalier). +%Avec vload, ça converge. -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. -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}