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version 1.1, 2001/06/19 07:32:58 version 1.3, 2001/06/20 02:39:25
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 % $OpenXM$  % $OpenXM: OpenXM/doc/ascm2001p/homogeneous-network.tex,v 1.2 2001/06/20 01:43:12 noro Exp $
   
 \subsection{Distributed computation with homogeneous servers}  \subsection{Distributed computation with homogeneous servers}
 \label{section:homog}  \label{section:homog}
Line 9  not include communication between servers, one cannot 
Line 9  not include communication between servers, one cannot 
 the maximal parallel speedup. However it is possible to execute  the maximal parallel speedup. However it is possible to execute
 several types of distributed computation as follows.  several types of distributed computation as follows.
   
 \subsubsection{Product of univariate polynomials}  \subsubsection{Competitive distributed computation by various strategies}
   
 Shoup \cite{Shoup} showed that the product of univariate polynomials  SINGULAR \cite{Singular} implements {\it MP} interface for distributed
 with large degrees and large coefficients can be computed efficiently  computation and a competitive Gr\"obner basis computation is
 by FFT over small finite fields and Chinese remainder theorem,  illustrated as an example of distributed computation.
 which can be easily parallelized.  Such a distributed computation is also possible on OpenXM as follows:
 %  
 %\begin{tabbing}  
 %Input :\= $f_1, f_2 \in {\bf Z}[x]$ such that $deg(f_1), deg(f_2) < 2^M$\\  
 %Output : $f = f_1f_2$ \\  
 %$P \leftarrow$ \= $\{m_1,\cdots,m_N\}$ where $m_i$ is an odd prime, \\  
 %\> $2^{M+1}|m_i-1$ and $m=\prod m_i $ is sufficiently large. \\  
 %Separate $P$ into disjoint subsets $P_1, \cdots, P_L$.\\  
 %for \= $j=1$ to $L$ $M_j \leftarrow \prod_{m_i\in P_j} m_i$\\  
 %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.\\  
 %\> (The product is computed by FFT.)\\  
 %return $\phi_m(\sum F_j)$\\  
 %(For $a \in {\bf Z}$, $\phi_m(a) \in (-m/2,m/2)$ and $\phi_m(a)\equiv a \bmod m$)  
 %\end{tabbing}  
 %  
 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,  
 a cluster of Sun workstations connected with a high speed network  
 and MPI over the network is used to implement OpenXM.  
 \begin{figure}[htbp]  
 \epsfxsize=10cm  
 \epsffile{speedup.ps}  
 \caption{Speedup factor}  
 \label{speedup}  
 \end{figure}  
 If the number of servers is $L$ and the inputs are fixed, then the cost to  
 compute the products modulo some integers in parallel is $O(1/L)$,  
 whereas the cost  
 to send and receive polynomials is $O(L)$ if {\tt ox\_push\_cmo()} and  
 {\tt ox\_pop\_cmo()} are repeatedly applied on the client.  
 Therefore the speedup is limited and the upper bound of  
 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 environment.  
 If OpenXM provides collective operations for broadcast and reduction  
 such as {\tt MPI\_Bcast} and {\tt MPI\_Reduce} respectively, the cost of  
 broadcasting the inputs and gathering the results on the servers  
 may be reduced to $O(\log_2L)$  
 and we can expect better results in such a case. In order to implement  
 such operations we need new specifications for inter-sever communication  
 and the session management, which will be proposed as OpenXM-RFC 102.  
 We note that preliminary experiments show the collective operations  
 work well on OpenXM.  
   
 %\subsubsection{Competitive distributed computation by various strategies}  The client creates two servers and it requests
   Gr\"obner basis comutations by the Buchberger algorithm the $F_4$ algorithm
   to the servers for the same input.
   The client watches the streams by {\tt ox\_select()}
   and the result which is returned first is taken. Then the remaining
   server is reset.
   
   \begin{verbatim}
   extern Proc1,Proc2$
   Proc1 = -1$ Proc2 = -1$
   /* G:set of polys; V:list of variables */
   /* Mod: the Ground field GF(Mod); O:type of order */
   def dgr(G,V,Mod,O)
   {
     /* invoke servers if necessary */
     if ( Proc1 == -1 ) Proc1 = ox_launch();
     if ( Proc2 == -1 ) Proc2 = ox_launch();
     P = [Proc1,Proc2];
     map(ox_reset,P); /* reset servers */
     /* P0 executes Buchberger algorithm over GF(Mod) */
     ox_cmo_rpc(P[0],"dp_gr_mod_main",G,V,0,Mod,O);
     /* P1 executes F4 algorithm over GF(Mod) */
     ox_cmo_rpc(P[1],"dp_f4_mod_main",G,V,Mod,O);
     map(ox_push_cmd,P,262); /* 262 = OX_popCMO */
     F = ox_select(P); /* wait for data */
     /* F[0] is a server's id which is ready */
     R = ox_get(F[0]);
     if ( F[0] == P[0] ) { Win = "Buchberger"; Lose = P[1]; }
     else { Win = "F4"; Lose = P[0]; }
     ox_reset(Lose); /* reset the loser */
     return [Win,R];
   }
   \end{verbatim}
   
   %\subsubsection{Nesting of client-server communication}
 %  %
 %SINGULAR \cite{Singular} implements {\it MP} interface for distributed  %Under OpenXM-RFC 100 an OpenXM server can be a client of other servers.
 %computation and a competitive Gr\"obner basis computation is  %Figure \ref{tree} illustrates a tree-like structure of an OpenXM
 %illustrated as an example of distributed computation.  %client-server communication.
 %Such a distributed computation is also possible on OpenXM as follows:  %\begin{figure}
   %\label{tree}
   %\begin{center}
   %\begin{picture}(200,70)(0,0)
   %\put(70,70){\framebox(40,15){client}}
   %\put(20,30){\framebox(40,15){server}}
   %\put(70,30){\framebox(40,15){server}}
   %\put(120,30){\framebox(40,15){server}}
   %\put(0,0){\framebox(40,15){server}}
   %\put(50,0){\framebox(40,15){server}}
   %\put(150,0){\framebox(40,15){server}}
 %  %
 %The client creates two servers and it requests  %\put(90,70){\vector(-2,-1){43}}
 %Gr\"obner basis comutations from the homogenized input and the input itself  %\put(90,70){\vector(0,-1){21}}
 %to the servers.  %\put(90,70){\vector(2,-1){43}}
 %The client watches the streams by {\tt ox\_select()}  %\put(40,30){\vector(-2,-1){22}}
 %and the result which is returned first is taken. Then the remaining  %\put(40,30){\vector(2,-1){22}}
 %server is reset.  %\put(140,30){\vector(2,-1){22}}
   %\end{picture}
   %\caption{Tree-like structure of client-server communication}
   %\end{center}
   %\end{figure}
   %Such a computational model is useful for parallel implementation of
   %algorithms whose task can be divided into subtasks recursively.
 %  %
 %\begin{verbatim}  
 %/* G:set of polys; V:list of variables */  
 %/* O:type of order; P0,P1: id's of servers */  
 %def dgr(G,V,O,P0,P1)  
 %{  
 %  P = [P0,P1]; /* server list */  
 %  map(ox_reset,P); /* reset servers */  
 %  /* P0 executes non-homogenized computation */  
 %  ox_cmo_rpc(P0,"dp_gr_main",G,V,0,1,O);  
 %  /* P1 executes homogenized computation */  
 %  ox_cmo_rpc(P1,"dp_gr_main",G,V,1,1,O);  
 %  map(ox_push_cmd,P,262); /* 262 = OX_popCMO */  
 %  F = ox_select(P); /* wait for data */  
 %  /* F[0] is a server's id which is ready */  
 %  R = ox_get(F[0]);  
 %  if ( F[0] == P0 ) {  
 %    Win = "nonhomo"; Lose = P1;  
 %  } else {  
 %    Win = "homo"; Lose = P0;  
 %  }  
 %  ox_reset(Lose); /* reset the loser */  
 %  return [Win,R];  
 %}  
 %\end{verbatim}  
   
 \subsubsection{Nesting of client-server communication}  
   
 Under OpenXM-RFC 100 an OpenXM server can be a client of other servers.  
 Figure \ref{tree} illustrates a tree-like structure of an OpenXM  
 client-server communication.  
   
 \begin{figure}  
 \label{tree}  
 \begin{center}  
 \begin{picture}(200,140)(0,0)  
 \put(70,120){\framebox(40,15){client}}  
 \put(20,60){\framebox(40,15){server}}  
 \put(70,60){\framebox(40,15){server}}  
 \put(120,60){\framebox(40,15){server}}  
 \put(0,0){\framebox(40,15){server}}  
 \put(50,0){\framebox(40,15){server}}  
 \put(135,0){\framebox(40,15){server}}  
   
 \put(90,120){\vector(-1,-1){43}}  
 \put(90,120){\vector(0,-1){43}}  
 \put(90,120){\vector(1,-1){43}}  
 \put(40,60){\vector(-1,-2){22}}  
 \put(40,60){\vector(1,-2){22}}  
 \put(140,60){\vector(1,-3){14}}  
 \end{picture}  
 \caption{Tree-like structure of client-server communication}  
 \end{center}  
 \end{figure}  
   
 Such a computational model is useful for parallel implementation of  
 algorithms whose task can be divided into subtasks recursively.  
   
 %A typical example is {\it quicksort}, where an array to be sorted is  %A typical example is {\it quicksort}, where an array to be sorted is
 %partitioned into two sub-arrays and the algorithm is applied to each  %partitioned into two sub-arrays and the algorithm is applied to each
 %sub-array. In each level of recursion, two subtasks are generated  %sub-array. In each level of recursion, two subtasks are generated
Line 186  algorithms whose task can be divided into subtasks rec
Line 131  algorithms whose task can be divided into subtasks rec
 %  }  %  }
 %}  %}
 %\end{verbatim}  %\end{verbatim}
   %
 A typical example is a parallelization of the Cantor-Zassenhaus  %A typical example is a parallelization of the Cantor-Zassenhaus
 algorithm for polynomial factorization over finite fields.  %algorithm for polynomial factorization over finite fields.
 which is a recursive algorithm.  %which is a recursive algorithm.
 At each level of the recursion, a given polynomial can be  %At each level of the recursion, a given polynomial can be
 divided into two non-trivial factors with some probability by using  %divided into two non-trivial factors with some probability by using
 a randomly generated polynomial as a {\it separator}.  %a randomly generated polynomial as a {\it separator}.
 We can apply the following simple parallelization:  %We can apply the following simple parallelization:
 When two non-trivial factors are generated on a server,  %When two non-trivial factors are generated on a server,
 one is sent to another server and the other factor is factorized on the server  %one is sent to another server and the other factor is factorized on the server
 itself.  %itself.
 %\begin{verbatim}  %\begin{verbatim}
 %/* factorization of F */  %/* factorization of F */
 %/* E = degree of irreducible factors in F */  %/* E = degree of irreducible factors in F */
Line 242  itself. 
Line 187  itself. 
 %  %
 %  %
 %  %
   
   \subsubsection{Product of univariate polynomials}
   
   Shoup \cite{Shoup} showed that the product of univariate polynomials
   with large degrees and large coefficients can be computed efficiently
   by FFT over small finite fields and Chinese remainder theorem.
   It can be easily parallelized:
   
   \begin{tabbing}
   Input :\= $f_1, f_2 \in {\bf Z}[x]$ such that $deg(f_1), deg(f_2) < 2^M$\\
   Output : $f = f_1f_2$ \\
   $P \leftarrow$ \= $\{m_1,\cdots,m_N\}$ where $m_i$ is an odd prime, \\
   \> $2^{M+1}|m_i-1$ and $m=\prod m_i $ is sufficiently large. \\
   Separate $P$ into disjoint subsets $P_1, \cdots, P_L$.\\
   for \= $j=1$ to $L$ $M_j \leftarrow \prod_{m_i\in P_j} m_i$\\
   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.\\
   \> (The product is computed by FFT.)\\
   return $\phi_m(\sum F_j)$\\
   (For $a \in {\bf Z}$, $\phi_m(a) \in (-m/2,m/2)$ and $\phi_m(a)\equiv a \bmod m$)
   \end{tabbing}
   
   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,
   a cluster of Sun workstations connected with a high speed network
   and MPI over the network is used to implement OpenXM.
   \begin{figure}[htbp]
   \epsfxsize=8.5cm
   \epsffile{speedup.ps}
   \caption{Speedup factor}
   \label{speedup}
   \end{figure}
   
   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
   to send and receive polynomials is $O(L)$ if {\tt ox\_push\_cmo()} and
   {\tt ox\_pop\_cmo()} are repeatedly applied on the client.
   Therefore the speedup is limited and the upper bound of
   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 environment.
   If OpenXM provides collective operations for broadcast and 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 $O(\log_2L)$
   and we can expect better results in such a case. In order to implement
   such operations we need new specifications for inter-sever communication
   and the session management, which will be proposed as OpenXM-RFC 102.
   We note that preliminary experiments show the collective operations
   work well on OpenXM.
   
   %\subsubsection{Competitive distributed computation by various strategies}
   %
   %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 as follows:
   %
   %The client creates two servers and it requests
   %Gr\"obner basis comutations from the homogenized input and the input itself
   %to the servers.
   %The client watches the streams by {\tt ox\_select()}
   %and the result which is returned first is taken. Then the remaining
   %server is reset.
   %
   %\begin{verbatim}
   %/* G:set of polys; V:list of variables */
   %/* O:type of order; P0,P1: id's of servers */
   %def dgr(G,V,O,P0,P1)
   %{
   %  P = [P0,P1]; /* server list */
   %  map(ox_reset,P); /* reset servers */
   %  /* P0 executes non-homogenized computation */
   %  ox_cmo_rpc(P0,"dp_gr_main",G,V,0,1,O);
   %  /* P1 executes homogenized computation */
   %  ox_cmo_rpc(P1,"dp_gr_main",G,V,1,1,O);
   %  map(ox_push_cmd,P,262); /* 262 = OX_popCMO */
   %  F = ox_select(P); /* wait for data */
   %  /* F[0] is a server's id which is ready */
   %  R = ox_get(F[0]);
   %  if ( F[0] == P0 ) {
   %    Win = "nonhomo"; Lose = P1;
   %  } else {
   %    Win = "homo"; Lose = P0;
   %  }
   %  ox_reset(Lose); /* reset the loser */
   %  return [Win,R];
   %}
   %\end{verbatim}
   
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