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version 1.2, 2001/06/20 01:43:12 version 1.3, 2001/06/20 02:39:25
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 % $OpenXM: OpenXM/doc/ascm2001p/homogeneous-network.tex,v 1.1 2001/06/19 07:32:58 noro Exp $  % $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{Nesting of client-server communication}  \subsubsection{Competitive distributed computation by various strategies}
   
 Under OpenXM-RFC 100 an OpenXM server can be a client of other servers.  SINGULAR \cite{Singular} implements {\it MP} interface for distributed
 Figure \ref{tree} illustrates a tree-like structure of an OpenXM  computation and a competitive Gr\"obner basis computation is
 client-server communication.  illustrated as an example of distributed computation.
 \begin{figure}  Such a distributed computation is also possible on OpenXM as follows:
 \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}}  
   
 \put(90,70){\vector(-2,-1){43}}  The client creates two servers and it requests
 \put(90,70){\vector(0,-1){21}}  Gr\"obner basis comutations by the Buchberger algorithm the $F_4$ algorithm
 \put(90,70){\vector(2,-1){43}}  to the servers for the same input.
 \put(40,30){\vector(-2,-1){22}}  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(140,30){\vector(2,-1){22}}  server is reset.
 \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}
   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}
   %
   %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}(200,70)(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}}
   %
   %\put(90,70){\vector(-2,-1){43}}
   %\put(90,70){\vector(0,-1){21}}
   %\put(90,70){\vector(2,-1){43}}
   %\put(40,30){\vector(-2,-1){22}}
   %\put(40,30){\vector(2,-1){22}}
   %\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.
   %
 %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 90  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 */
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