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% $OpenXM: OpenXM/doc/issac2000/homogeneous-network.tex,v 1.2 2000/01/02 07:32:12 takayama Exp $

% $OpenXM: OpenXM/doc/issac2000/homogeneous-network.tex,v 1.5 2000/01/15 00:20:45 takayama Exp $

\section{Applications}

\subsection{Distributed computation with homogeneous servers}

\label{section:homog}

OpenXM also aims at speedup by a distributed computation

One of the aims of OpenXM is a parallel speedup by a distributed computation

with homogeneous servers. As the current specification of OpenXM does

not include communication between servers, one cannot expect

the maximal parallel speedup. However it is possible to execute

Line 48 network is used to implement OpenXM.

Line 49 network is used to implement OpenXM.

The task of a client is the generation and partition of $P$, sending

and receiving of polynomials and the synthesis of the result. If the

number of servers is $L$ and the inputs are fixed, then the time to

number of servers is $L$ and the inputs are fixed, then the cost to

compute $F_j$ in parallel is proportional to $1/L$, whereas the time

compute $F_j$ in parallel is $O(1/L)$, whereas the cost

for sending and receiving of polynomials is proportional to $L$

to send and receive polynomials is $O(L)$

because we don't have the broadcast and the reduce

operations. Therefore the speedup is limited and the upper bound of

the speedup factor depends on the communication cost and the degree

the speedup factor depends on the ratio of

of inputs. Figure \ref{speedup} shows that

the computational cost and the communication cost.

the speedup is satisfactory if the degree is large and the number of

Figure \ref{speedup} shows that

servers is not large, say, up to 10.

the speedup is satisfactory if the degree is large and $L$

is not large, say, up to 10 under the above envionment.

If OpenXM provides the broadcast and the reduce operations, the cost of

sending $f_1$, $f_2$ and gathering $F_j$ may be reduced to $O(log_2L)$

and we will obtain better results in such a case.

\subsubsection{Order counting of an elliptic curve}

\subsubsection{Competitive distributed computation by various strategies}

\subsubsection{Gr\"obner basis computation by various methods}

Singular \cite{Singular} implements {\tt MP} interface for distributed

computation and a competitive Gr\"obner basis computation is

illustrated as an example of distributed computation. However,

illustrated as an example of distributed computation.

interruption has not implemented yet and the looser process have to be

Such a distributed computation is also possible on OpenXM.

killed explicitly. As stated in Section \ref{secsession} OpenXM

The following {\tt Risa/Asir} function computes a Gr\"obner basis by

provides such a function and one can safely reset the server and

continue to use it. Furthermore, if a client provides synchronous I/O

multiplexing by {\tt select()}, then a polling is not necessary. The

following {\tt Risa/Asir} function computes a Gr\"obner basis by

starting the computations simultaneously from the homogenized input and

the input itself. The client watches the streams by {\tt ox\_select()}

and The result which is returned first is taken. Then the remaining

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