X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/loba-papers.git/blobdiff_plain/69768dd70773e43e60a13d06dad73a65c00dfb85..3e0139b6d08c5073cf7665a95af6cb6ae750c923:/supercomp11/supercomp11.tex

diff --git a/supercomp11/supercomp11.tex b/supercomp11/supercomp11.tex
index 249b676..2a6c04b 100644
--- a/supercomp11/supercomp11.tex
+++ b/supercomp11/supercomp11.tex
@@ -2,9 +2,12 @@
 \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
@@ -39,14 +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
@@ -70,7 +73,7 @@ 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
@@ -80,7 +83,7 @@ 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,
@@ -94,17 +97,17 @@ 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
@@ -114,7 +117,7 @@ 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.
@@ -126,11 +129,11 @@ conclusion and some perspectives to this work.
 \label{BT algo}
 
 In  order  prove  the  convergence  of  asynchronous  iterative  load  balancing
-Bertesekas         and        Tsitsiklis         proposed         a        model
+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 differents processors. In this work, we
+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
@@ -141,7 +144,7 @@ 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 transfered to processor $j$ at time $t$ and let $r_{ij}(t)$ be the
+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}
@@ -151,18 +154,18 @@ x_i(t+1)=x_i(t)-\sum_{j\in V(i)} s_{ij}(t) + \sum_{j\in V(i)} r_{ji}(t)
 
 
 Some  conditions are  required to  ensure the  convergence. One  of them  can be
-called the \texttt{ping-pong} condition which specifies that:
+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 beeing
+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 wich 3 processors). Now consider we have the following values at time $t$:
+chain which 3 processors). Now consider we have the following values at time $t$:
 \begin{eqnarray*}
 x_1(t)=10   \\
 x_2(t)=100   \\
@@ -172,7 +175,7 @@ x_3(t)=99.99\\
 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 \texttt{ping-pong} condition is probably to
+$x_3^2(t)$.  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.
 
@@ -180,16 +183,60 @@ condition or with a weaker condition.
 \section{Best effort strategy}
 \label{Best-effort}
 
-\textbf{Ã traduire} Ordonne les voisins du moins chargÃ© au plus chargÃ©.
-Trouve ensuite, en les prenant dans ce ordre, le nombre maximal de
-voisins tels que tous ont une charge infÃ©rieure Ã  la moyenne des
-charges des voisins sÃ©lectionnÃ©s, et de soi-mÃªme.
-
-Les transferts de charge sont ensuite fait en visant cette moyenne pour
-tous les voisins sÃ©lectionnÃ©s.  On envoie une quantitÃ© de charge Ã©gale
-Ã  (moyenne - charge\_du\_voisin).
-
-~\\\textbf{Question} faut-il dÃ©crire les stratÃ©gies makhoul et simple ?
+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 dÃ©crire les stratÃ©gies makhoul et simple ?
 
 \paragraph{simple} Tentative de respecter simplement les conditions de Bertsekas.
 Parmi les voisins moins chargÃ©s que soi, on sÃ©lectionne :
@@ -332,5 +379,6 @@ Taille : 10 100 trÃ¨s gros
 %%% ispell-local-dictionary: "american"
 %%% End:
 
-% LocalWords:  RaphaÃ«l Couturier Arnaud Giersch Abderrahmane Sider
-% LocalWords:  Bertsekas Tsitsiklis SimGrid DASUD
+% LocalWords:  RaphaÃ«l Couturier Arnaud Giersch Abderrahmane Sider Franche ij
+% LocalWords:  Bertsekas Tsitsiklis SimGrid DASUD ComtÃ© BÃ©jaÃ¯a asynchronism ji
+% LocalWords:  ik
