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version 1.7, 2000/01/15 06:11:17 version 1.11, 2000/01/17 07:15:52
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 % $OpenXM: OpenXM/doc/issac2000/homogeneous-network.tex,v 1.6 2000/01/15 02:24:18 takayama Exp $  % $OpenXM: OpenXM/doc/issac2000/homogeneous-network.tex,v 1.10 2000/01/17 07:06:53 noro Exp $
   
 \subsection{Distributed computation with homogeneous servers}  \subsection{Distributed computation with homogeneous servers}
 \label{section:homog}  \label{section:homog}
Line 17  by FFT over small finite fields and Chinese remainder 
Line 17  by FFT over small finite fields and Chinese remainder 
 It can be easily parallelized:  It can be easily parallelized:
   
 \begin{tabbing}  \begin{tabbing}
 Input :\= $f_1, f_2 \in Z[x]$\\  Input :\= $f_1, f_2 \in {\bf Z}[x]$ such that $deg(f_1), deg(f_2) < 2^M$\\
 \> such that $deg(f_1), deg(f_2) < 2^M$\\  Output : $f = f_1f_2$ \\
 Output : $f = f_1f_2 \bmod p$\\  $P \leftarrow$ \= $\{m_1,\cdots,m_N\}$ where $m_i$ is an odd prime, \\
 $P \leftarrow$ \= $\{m_1,\cdots,m_N\}$ where $m_i$ is a prime, \\  
 \> $2^{M+1}|m_i-1$ and $m=\prod m_i $ is sufficiently large. \\  \> $2^{M+1}|m_i-1$ and $m=\prod m_i $ is sufficiently large. \\
 Separate $P$ into disjoint subsets $P_1, \cdots, P_L$.\\  Separate $P$ into disjoint subsets $P_1, \cdots, P_L$.\\
 for \= $j=1$ to $L$ $M_j \leftarrow \prod_{m_i\in P_j} m_i$\\  for \= $j=1$ to $L$ $M_j \leftarrow \prod_{m_i\in P_j} m_i$\\
 Compute $F_j$ such that $F_j \equiv f_1f_2 \bmod M_j$\\  Compute $F_j$ such that $F_j \equiv f_1f_2 \bmod M_j$\\
 \> and $F_j \equiv 0 \bmod m/M_j$ in parallel.\\  \> and $F_j \equiv 0 \bmod m/M_j$ in parallel.\\
 \> ($f_1, f_2$ are regarded as integral.\\  \> (The product is computed by FFT.)\\
 \> The product is computed by FFT.)\\  
 return $\phi_m(\sum F_j)$\\  return $\phi_m(\sum F_j)$\\
 (For $a \in Z$, $\phi_m(a) \in (-m/2,m/2)$ and $\phi_m(a)\equiv a \bmod m$)  (For $a \in {\bf Z}$, $\phi_m(a) \in (-m/2,m/2)$ and $\phi_m(a)\equiv a \bmod m$)
 \end{tabbing}  \end{tabbing}
   
 Figure \ref{speedup}  Figure \ref{speedup}
 shows the speedup factor under the above distributed computation  shows the speedup factor under the above distributed computation
 on {\tt Risa/Asir}. For each $n$, two polynomials of degree $n$  on Risa/Asir. For each $n$, two polynomials of degree $n$
 with 3000bit coefficients are generated and the product is computed.  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 connected with a high speed network and MPI over the
Line 46  network is used to implement OpenXM.
Line 44  network is used to implement OpenXM.
 \label{speedup}  \label{speedup}
 \end{figure}  \end{figure}
   
 The task of a client is the generation and partition of $P$, sending  If the number of servers is $L$ and the inputs are fixed, then the cost to
 and receiving of polynomials and the synthesis of the result. If the  
 number of servers is $L$ and the inputs are fixed, then the cost to  
 compute $F_j$ in parallel is $O(1/L)$, whereas the cost  compute $F_j$ in parallel is $O(1/L)$, whereas the cost
 to send and receive polynomials is $O(L)$  to send and receive polynomials is $O(L)$ if {\tt ox\_push\_cmo()} and
 because we don't have the broadcast and the reduce  {\tt ox\_pop\_cmo()} are repeatedly applied on the client.
 operations. Therefore the speedup is limited and the upper bound of  Therefore the speedup is limited and the upper bound of
 the speedup factor depends on the ratio of  the speedup factor depends on the ratio of
 the computational cost and the communication cost.  the computational cost and the communication cost for each unit operation.
 Figure \ref{speedup} shows that  Figure \ref{speedup} shows that
 the speedup is satisfactory if the degree is large and $L$  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 envionment.
 If OpenXM provides the broadcast and the reduce operations, the cost of  If OpenXM provides operations for the broadcast and the reduction
   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 $O(log_2L)$  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.  and we can expect better results in such a case.
   
 \subsubsection{Competitive distributed computation by various strategies}  \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  computation and a competitive Gr\"obner basis computation is
 illustrated as an example of distributed computation.  illustrated as an example of distributed computation.
 Such a distributed computation is also possible on OpenXM.  Such a distributed computation is also possible on OpenXM.
 The following {\tt Risa/Asir} function computes a Gr\"obner basis by  The following Risa/Asir function computes a Gr\"obner basis by
 starting the computations simultaneously from the homogenized input and  starting the computations simultaneously from the homogenized input and
 the input itself.  The client watches the streams by {\tt ox\_select()}  the input itself.  The client watches the streams by {\tt ox\_select()}
 and The result which is returned first is taken. Then the remaining  and the result which is returned first is taken. Then the remaining
 server is reset.  server is reset.
   
 \begin{verbatim}  \begin{verbatim}
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