From e98f1ccead5d206a6da91407b7298efbf0d828d7 Mon Sep 17 00:00:00 2001
From: Arnaud Giersch <arnaud.giersch@iut-bm.univ-fcomte.fr>
Date: Wed, 1 Jun 2011 15:11:03 +0200
Subject: [PATCH] Description best effort.

---
 supercomp11/supercomp11.tex | 67 +++++++++++++++++++++++++++++++------
 1 file changed, 57 insertions(+), 10 deletions(-)

diff --git a/supercomp11/supercomp11.tex b/supercomp11/supercomp11.tex
index edf5207..4cc971b 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
@@ -180,16 +183,60 @@ condition or with a weaker condition.
 \section{Best effort strategy}
 \label{Best-effort}
 
-\textbf{À traduire} Ordonne les voisins du moins chargé au plus chargé.
-Trouve ensuite, en les prenant dans ce ordre, le nombre maximal de
-voisins tels que tous ont une charge inférieure à la moyenne des
-charges des voisins sélectionnés, et de soi-même.
-
-Les transferts de charge sont ensuite fait en visant cette moyenne pour
-tous les voisins sélectionnés.  On envoie une quantité de charge égale
-à (moyenne - charge\_du\_voisin).
-
-~\\\textbf{Question} faut-il décrire les stratégies 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 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 :
-- 
2.39.5