X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/loba-papers.git/blobdiff_plain/e3080a36717b6b9710daaec8c4d1e529b19c3176..ab8360c25108171f3b2fdbda91fdf9747b5473ad:/supercomp11/supercomp11.tex?ds=inline

diff --git a/supercomp11/supercomp11.tex b/supercomp11/supercomp11.tex
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+++ b/supercomp11/supercomp11.tex
@@ -1,25 +1,34 @@
-
 \documentclass[smallextended]{svjour3}
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage{mathptmx}
+\usepackage{amsmath}
+\usepackage{courier}
 \usepackage{graphicx}
 
+\newcommand{\abs}[1]{\lvert#1\rvert} % \abs{x} -> |x|
+
 \begin{document}
 
-\title{Best effort strategy and virtual load for asynchronous iterative load balancing}
+\title{Best effort strategy and virtual load
+  for asynchronous iterative load balancing}
 
 \author{Raphaël Couturier \and
         Arnaud Giersch \and
         Abderrahmane Sider
 }
 
-\institute{F. Author \at
-              first address \\
-              Tel.: +123-45-678910\\
-              Fax: +123-45-678910\\
-              \email{fauthor@example.com}           %  \\
-%             \emph{Present address:} of F. Author  %  if needed
+\institute{R. Couturier \and A. Giersch \at
+              LIFC, University of Franche-Comté, Belfort, France \\
+              % Tel.: +123-45-678910\\
+              % Fax: +123-45-678910\\
+              \email{%
+                raphael.couturier@univ-fcomte.fr,
+                arnaud.giersch@univ-fcomte.fr}
            \and
-           S. Author \at
-              second address
+           A. Sider \at
+              University of Béjaïa, Béjaïa, Algeria \\
+              \email{ar.sider@univ-bejaia.dz}
 }
 
 \maketitle
@@ -28,32 +37,350 @@
 \begin{abstract}
 
 Most of the  time, asynchronous load balancing algorithms  have extensively been
-studied in  a theoretical point of  view. The Bertsekas'  algorithm 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 \texttt{best  effort} 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 \texttt{virtual
-  load}  which  allows a  node  that receives  an  load  information message  to
-integrate  the load  that it  will receive  latter 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.
+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 \emph{best effort} 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 an 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}
 
+\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}.
+{\bf Rajouter des choses ici}.
+
+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}
+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.  Formers  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{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. 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.
+
+
+
+
+\section{Bertsekas  and Tsitsiklis' asynchronous load balancing algorithm}
+\label{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.
+
+
+\section{Best effort strategy}
+\label{Best-effort}
+
+We will describe here a new load-balancing strategy that we called
+\emph{best effort}.  The general idea behind this strategy is, for a
+processor, to send some load to the most of its neighbors, doing its
+best to reach the equilibrium between those neighbors and himself.
+
+More precisely, when a processors $i$ 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}
+
+\section{Other strategies}
+\label{Other}
+
+\textbf{Question} faut-il décrire les stratégies makhoul et simple ?
+
+\paragraph{simple} Tentative de respecter simplement les conditions de Bertsekas.
+Parmi les voisins moins chargés que soi, on sélectionne :
+\begin{itemize}
+\item un des moins chargés (vmin) ;
+\item un des plus chargés (vmax),
+\end{itemize}
+puis on équilibre avec vmin en s'assurant que notre charge reste
+toujours supérieure à celle de vmin et à celle de vmax.
+
+On envoie donc (avec "self" pour soi-même) :
+\[
+    \min\left(\frac{load(self) - load(vmin)}{2}, load(self) - load(vmax)\right)
+\]
 
+\paragraph{makhoul} Ordonne les voisins du moins chargé au plus chargé
+puis calcule les différences de charge entre soi-même et chacun des
+voisins.
 
+Ensuite, pour chaque voisin, dans l'ordre, et tant qu'on reste plus
+chargé que le voisin en question, on lui envoie 1/(N+1) de la
+différence calculée au départ, avec N le nombre de voisins.
 
-qsdqsd
+C'est l'algorithme~2 dans~\cite{bahi+giersch+makhoul.2008.scalable}.
 
+\section{Virtual load}
+\label{Virtual load}
 
+\section{Simulations}
+\label{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{Sim
+  model}), then the results of the simulations are presented in
+section~\ref{Results}.
+
+\subsection{Simulation model}
+\label{Sim 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.
+
+\paragraph{Receiving thread} The receiving thread is in charge of
+waiting for messages to come, either on the control channel, or on the
+data channel.  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.
+
+\paragraph{Computing thread} The computing thread is in charge of the
+real load management.  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 zero).
+
+\paragraph{Load-balancing thread} The load-balancing thread is in
+charge of running the load-balancing algorithm, and exchange the
+control messages.  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}
+
+\subsection{Validation of our approaches}
+\label{Results}
+
+
+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}
+
+
+\bibliographystyle{spmpsci}
+\bibliography{biblio}
 
 \end{document}
+
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: t
+%%% ispell-local-dictionary: "american"
+%%% End:
+
+% LocalWords:  Raphaël Couturier Arnaud Giersch Abderrahmane Sider Franche ij
+% LocalWords:  Bertsekas Tsitsiklis SimGrid DASUD Comté Béjaïa asynchronism ji
+% LocalWords:  ik isend irecv