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% $OpenXM: OpenXM/doc/issac2000/homogeneous-network.tex,v 1.8 2000/01/16 03:15:49 noro Exp $

% $OpenXM: OpenXM/doc/issac2000/homogeneous-network.tex,v 1.12 2000/01/17 08:06:15 noro Exp $

\subsection{Distributed computation with homogeneous servers}

\label{section:homog}

Line 34 Figure \ref{speedup}

shows the speedup factor under the above distributed computation

on Risa/Asir. For each $n$, two polynomials of degree $n$

with 3000bit coefficients are generated and the product is computed.

The machine is Fujitsu AP3000,

The machine is FUJITSU AP3000,

a cluster of Sun connected with a high speed network and MPI over the

a cluster of Sun workstations connected with a high speed network

network is used to implement OpenXM.

and MPI over the network is used to implement OpenXM.

\begin{figure}[htbp]

\epsfxsize=8.5cm

\epsffile{speedup.ps}

Line 53 the speedup factor depends on the ratio of

the computational cost and the communication cost for each unit operation.

Figure \ref{speedup} shows that

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

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

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

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

If OpenXM provides operations for the broadcast and the reduction

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

such as {\tt MPI\_Bcast} and {\tt MPI\_Reduce} respectively, the cost of

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

and we can expect better results in such a case.

\subsubsection{Competitive distributed computation by various strategies}

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

SINGULAR \cite{Singular} implements {\it MP} interface for distributed

computation and a competitive Gr\"obner basis computation is

illustrated as an example of distributed computation.

Such a distributed computation is also possible on OpenXM.