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\begin{document}

\title{Best effort strategy and virtual load
  for asynchronous iterative load balancing}

\author{Raphaël Couturier \and
        Arnaud Giersch
}

\institute{R. Couturier \and A. Giersch \at
              FEMTO-ST, University of Franche-Comté, Belfort, France \\
              % Tel.: +123-45-678910\\
              % Fax: +123-45-678910\\
              \email{%
                raphael.couturier@femto-st.fr,
                arnaud.giersch@femto-st.fr}
}

\maketitle


\begin{abstract}

Most of the  time, asynchronous load balancing algorithms  have extensively been
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 a 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}.
\FIXME{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.  Former  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.  This strategy will be compared with other ones, presented in
Section~\ref{Other}.  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.
%
\FIXME{Develop: We have the feeling that such a weaker condition
  exists, because (it's not a proof, but) we have never seen any
  scenario that is not leading to convergence, even with LB-strategies
  that are not fulfilling these two conditions.}

\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}

\FIXME{describe parameter $k$}

\section{Other strategies}
\label{Other}

\FIXME{Réécrire en angliche.}

% \FIXME{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}

In this section,  we present the concept of \texttt{virtual  load}.  In order to
use this concept, load balancing messages must be sent using two different kinds
of  messages:  load information  messages  and  load  balancing messages.   More
precisely, a node  wanting to send a part  of its load to one  of its neighbors,
can first send  a load information message containing the load  it will send and
then it can send the load  balancing message containing data  to be transferred.
Load information  message are really  short, consequently they will  be received
very quickly.  In opposition, load  balancing messages are often bigger and thus
require more time to be transferred.

The  concept  of  \texttt{virtual load}  allows  a  node  that received  a  load
information message to integrate the load that it will receive later in its load
(virtually)  and consequently send  a (real)  part of  its load  to some  of its
neighbors. In fact,  a node that receives a load  information message knows that
later it  will receive the  corresponding load balancing message  containing the
corresponding data.  So  if this node detects it is too  loaded compared to some
of its neighbors  and if it has enough  load (real load), then it  can send more
load  to  some of  its  neighbors  without waiting  the  reception  of the  load
balancing message.

Doing  this, we  can  expect a  faster  convergence since  nodes  have a  faster
information of the load they will receive, so they can take in into account.

\FIXME{Est ce qu'on donne l'algo avec virtual load?}

\FIXME{describe integer mode}

\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}), and the experimental contexts are described in
section~\ref{Contexts}.  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.  Its behavior is sketched by Algorithm~\ref{algo.recv}.
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.

\begin{algorithm}
  \caption{Receiving thread}
  \label{algo.recv}
  \KwData{
    \begin{algodata}
      \VAR{ctrl\_chan}, \VAR{data\_chan}
      & communication channels (control and data) \\
      \VAR{ctrl\_fifo}, \VAR{data\_fifo}
      & buffers of received messages (control and data) \\
    \end{algodata}}
  \While{true}{%
    wait for a message to be available on either \VAR{ctrl\_chan},
    or \VAR{data\_chan}\;
    \If{a message is available on \VAR{ctrl\_chan}}{%
      get the message from \VAR{ctrl\_chan}, and push it into \VAR{ctrl\_fifo}\;
    }
    \If{a message is available on \VAR{data\_chan}}{%
      get the message from \VAR{data\_chan}, and push it into \VAR{data\_fifo}\;
    }
  }
\end{algorithm}

\paragraph{Computing thread} The computing thread is in charge of the
real load management.  As exposed in Algorithm~\ref{algo.comp}, 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 near zero).

\begin{algorithm}
  \caption{Computing thread}
  \label{algo.comp}
  \KwData{
    \begin{algodata}
      \VAR{data\_fifo} & buffer of received data messages \\
      \VAR{real\_load} & current load \\
    \end{algodata}}
  \While{true}{%
    \If{\VAR{data\_fifo} is empty and $\VAR{real\_load} = 0$}{%
      wait until a message is pushed into \VAR{data\_fifo}\;
    }
    \While{\VAR{data\_fifo} is not empty}{%
      pop a message from \VAR{data\_fifo}\;
      get the load embedded in the message, and add it to \VAR{real\_load}\;
    }
    \ForEach{neighbor $n$}{%
      \If{there is some amount of load $a$ to send to $n$}{%
        send $a$ units of load to $n$, and subtract it from \VAR{real\_load}\;
      }
    }
    \If{$\VAR{real\_load} > 0.0$}{
      simulate some computation, whose duration is function of \VAR{real\_load}\;
      ensure that the main loop does not iterate too fast\;
    }
  }
\end{algorithm}

\paragraph{Load-balancing thread} The load-balancing thread is in
charge of running the load-balancing algorithm, and exchange the
control messages.  As shown in Algorithm~\ref{algo.lb}, 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}

\begin{algorithm}
  \caption{Load-balancing}
  \label{algo.lb}
  \While{true}{%
    \While{\VAR{ctrl\_fifo} is not empty}{%
      pop a message from \VAR{ctrl\_fifo}\;
      identify the sender of the message,
      and update the current knowledge of its load\;
    }
    run the load-balancing algorithm to make the decision about load transfers\;
    \ForEach{neighbor $n$}{%
      send a control messages to $n$\;
    }
    ensure that the main loop does not iterate too fast\;
  }
\end{algorithm}

\paragraph{}
For the sake of simplicity, a few details were voluntary omitted from
these descriptions.  For an exhaustive presentation, we refer to the
actual source code that was used for the experiments%
\footnote{As mentioned before, our simulator relies on the SimGrid
  framework~\cite{casanova+legrand+quinson.2008.simgrid}.  For the
  experiments, we used a pre-release of SimGrid 3.7 (Git commit
  67d62fca5bdee96f590c942b50021cdde5ce0c07, available from
  \url{https://gforge.inria.fr/scm/?group_id=12})}, and which is
available at
\url{http://info.iut-bm.univ-fcomte.fr/staff/giersch/software/loba.tar.gz}.

\FIXME{ajouter des détails sur la gestion de la charge virtuelle ?}

\subsection{Experimental contexts}
\label{Contexts}

In order to assess the performances of our algorithms, we ran our
simulator with various parameters, and extracted several metrics, that
we will describe in this section.

\paragraph{Load balancing strategies}

Several load balancing strategies were compared.  We ran the experiments with
the \emph{Best effort}, and with the \emph{Makhoul} strategies.  \emph{Best
  effort} was tested with parameter $k = 1$, $k = 2$, and $k = 4$.  Secondly,
each strategy was run in its two variants: with, and without the management of
\emph{virtual load}.  Finally, we tested each configuration with \emph{real},
and with \emph{integer} load.

To summarize the different load balancing strategies, we have:
\begin{description}
\item[\textbf{strategies:}] \emph{Makhoul}, or \emph{Best effort} with $k\in
  \{1,2,4\}$
\item[\textbf{variants:}] with, or without virtual load
\item[\textbf{domain:}] real load, or integer load
\end{description}
%
This gives us as many as $4\times 2\times 2 = 16$ different strategies.


\paragraph{Configurations}

In order to show the behavior of the different strategies in different
settings, we simulated the executions on two sorts of platforms.  These two
sorts of platforms differ by their underlaid network topology.  On the one hand,
we have homogeneous platforms, modeled as a cluster.  On the other hand, we have
heterogeneous platforms, modeled as the interconnection of a number of clusters.
The heterogeneous platform descriptions were created by taking a subset of the
Grid'5000 infrastructure\footnote{Grid'5000 is a French large scale experimental
  Grid (see \url{https://www.grid5000.fr/}).}, as described in the platform file
\texttt{g5k.xml} distributed with SimGrid.  Note that the heterogeneity of the
platform only comes from the network topology.  The processor speeds, and
network bandwidths were normalized since our algorithms currently are not aware
of such heterogeneity.  We arbitrarily chose to fix the processor speed to
1~GFlop/s, and the network bandwidth to 125~MB/s, with a latency of 50~$\mu$s,
except for the links between geographically distant sites, where the network
bandwidth was fixed to 2.25~GB/s, with a latency of 500~$\mu$s.

Then we derived each sort of platform with four different number of computing
nodes: 16, 64, 256, and 1024 nodes.

The distributed processes of the application were then logically organized along
three possible topologies: a line, a torus or an hypercube.  We ran tests where
the total load was initially on an only node (at one end for the line topology),
and other tests where the load was initially randomly distributed across all
the participating nodes.

For each of the preceding configuration, we finally had to choose the
computation and communication costs of a load unit.  We chose them, such as to
have three different computation over communication cost ratios, and hence model
three different kinds of applications:
\begin{itemize}
\item mainly communicating, with a computation/communication cost ratio of $1/10$;
\item mainly computing, with a computation/communication cost ratio of $10/1$ ;
\item balanced, with a computation/communication cost ratio of $1/1$.
\end{itemize}

To summarize the various configurations, we have:
\begin{description}
\item[\textbf{platforms:}] homogeneous (cluster), or heterogeneous (subset of
  Grid'5000)
\item[\textbf{platform sizes:}] platforms with 16, 64, 256, or 1024 nodes
\item[\textbf{process topologies:}] line, torus, or hypercube
\item[\textbf{initial load distribution:}] initially on a only node, or
  initially randomly distributed over all nodes
\item[\textbf{computation/communication ratio:}] $10/1$, $1/1$, or $1/10$
\end{description}
%
This gives us as many as $2\times 4\times 3\times 2\times 3 = 144$ different
configurations.
%
Combined with the various load balancing strategies, we had $16\times 144 =
2304$ distinct settings to evaluate.  In fact, as it will be shown later, we
didn't run all the strategies, nor all the configurations for the bigger
platforms with 1024 nodes, since to simulations would have run for a too long
time.

Anyway, all these the experiments represent more than 240 hours of computing
time.

\paragraph{Metrics}

\begin{description}
\item[\textbf{average idle time}]
\item[\textbf{average convergence date}]
\item[\textbf{maximum convergence date}]
\item[\textbf{data transfer amount}] relative to the total data amount
\end{description}

\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}

\begin{acknowledgements}
  Computations have been performed on the supercomputer facilities of
  the Mésocentre de calcul de Franche-Comté.
\end{acknowledgements}

\bibliographystyle{spmpsci}
\bibliography{biblio}

\end{document}

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