X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/loba-papers.git/blobdiff_plain/d883af6fcab703af629fd6ca368b7076b5b3384a..bcf6cef9fa1df52f632849a68fc08e41f6163bb1:/supercomp11/supercomp11.tex?ds=inline

diff --git a/supercomp11/supercomp11.tex b/supercomp11/supercomp11.tex
index d1ff213..81ee704 100644
--- a/supercomp11/supercomp11.tex
+++ b/supercomp11/supercomp11.tex
@@ -2,27 +2,33 @@
 \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
@@ -36,14 +42,14 @@ 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 \texttt{best effort} which tries to balance
+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 \texttt{virtual load} which allows a node that receives an load
+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
@@ -67,17 +73,19 @@ 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  \texttt{ping-pong}  effect,  an  asynchronous
+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,
-DASUD~\cite{cortes+ripoll+cedo+al.2002.asynchronous} propose a version working
-with integer load. {\bf Rajouter des choses ici}.
+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  \texttt{best effort}
+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,
@@ -91,17 +99,17 @@ 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 nore longer that to  time to transfer a load information message. So,
+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 \texttt{virtual load} mecanism.
+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
-balacing. Moreover, we have conducted  many simulations with simgrid in order to
+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
@@ -111,7 +119,7 @@ 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  mecanism is
+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.
@@ -122,12 +130,140 @@ conclusion and some perspectives to this work.
 \section{Bertsekas  and Tsitsiklis' asynchronous load balancing algorithm}
 \label{BT algo}
 
-Comment on the problem in the convergence condition.
+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}
 
-
+In this section we  describe  a new load-balancing strategy that we call
+\emph{best effort}.  The general idea behind this 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}
+
+\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.
+
+C'est l'algorithme~2 dans~\cite{bahi+giersch+makhoul.2008.scalable}.
 
 \section{Virtual load}
 \label{Virtual load}
@@ -135,9 +271,74 @@ Comment on the problem in the convergence condition.
 \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 ? :
@@ -181,5 +382,6 @@ Taille : 10 100 très gros
 %%% ispell-local-dictionary: "american"
 %%% End:
 
-% LocalWords:  Raphaël Couturier Arnaud Giersch Abderrahmane Sider
-% LocalWords:  Bertsekas Tsitsiklis SimGrid DASUD
+% LocalWords:  Raphaël Couturier Arnaud Giersch Abderrahmane Sider Franche ij
+% LocalWords:  Bertsekas Tsitsiklis SimGrid DASUD Comté Béjaïa asynchronism ji
+% LocalWords:  ik isend irecv