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version 1.1, 2001/03/07 02:42:10 version 1.2, 2001/03/07 07:17:02
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 % $OpenXM$  % $OpenXM: OpenXM/doc/ascm2001/homogeneous-network.tex,v 1.1 2001/03/07 02:42:10 noro Exp $
   
 \subsection{Distributed computation with homogeneous servers}  \subsection{Distributed computation with homogeneous servers}
 \label{section:homog}  \label{section:homog}
Line 54  the computational cost and the communication cost for 
Line 54  the computational cost and the communication cost for 
 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 environment.  is not large, say, up to 10 under the above environment.
 If OpenXM provides operations for the broadcast and the reduction  If OpenXM provides collective operations for broadcast and reduction
 such as {\tt MPI\_Bcast} and {\tt MPI\_Reduce} respectively, the cost of  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 can expect better results in such a case.  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. The will be proposed as OpenXM-RFC-102 in future.
   We note that preliminary experiments shows the collective operations
   works well on OpenXM.
   
 \subsubsection{Competitive distributed computation by various strategies}  \subsubsection{Competitive distributed computation by various strategies}
   
Line 95  def dgr(G,V,O,P0,P1)
Line 99  def dgr(G,V,O,P0,P1)
   return [Win,R];    return [Win,R];
 }  }
 \end{verbatim}  \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
   partitioned into two sub-arrays and the algorithm is applied to each
   sub-array. In each level of recursion, two subtasks are generated
   and one can ask other OpenXM servers to execute them. Though
   this make little contribution to the efficiency, it is worth
   to show that such an attempt is very easy under OpenXM.
   Here is an Asir program.
   A predefined constant {\tt LevelMax} determines
   whether new servers are launched or whole subtasks are done on the server.
   
   \begin{verbatim}
   #define LevelMax 2
   extern Proc1, Proc2;
   Proc1 = -1$ Proc2 = -1$
   
   /* sort [A[P],...,A[Q]] by quicksort */
   def quickSort(A,P,Q,Level) {
     if (Q-P < 1) return A;
     Mp = idiv(P+Q,2); M = A[Mp]; B = P; E = Q;
     while (1) {
       while (A[B] < M) B++;
       while (A[E] > M && B <= E) E--;
       if (B >= E) break;
       else { T = A[B]; A[B] = A[E]; A[E] = T; E--; }
     }
     if (E < P) E = P;
     if (Level < LevelMax) {
      /* launch new servers if necessary */
      if (Proc1 == -1) Proc1 = ox_launch(0);
      if (Proc2 == -1) Proc2 = ox_launch(0);
      /* send the requests to the servers */
      ox_rpc(Proc1,"quickSort",A,P,E,Level+1);
      ox_rpc(Proc2,"quickSort",A,E+1,Q,Level+1);
      if (E-P < Q-E) {
        A1 = ox_pop_local(Proc1);
        A2 = ox_pop_local(Proc2);
      }else{
        A2 = ox_pop_local(Proc2);
        A1 = ox_pop_local(Proc1);
      }
      for (I=P; I<=E; I++) A[I] = A1[I];
      for (I=E+1; I<=Q; I++) A[I] = A2[I];
      return(A);
     }else{
      /* everything is done on this server */
      quickSort(A,P,E,Level+1);
      quickSort(A,E+1,Q,Level+1);
      return(A);
     }
   }
   \end{verbatim}
   
   Another example is a parallelization of the Cantor-Zassenhaus
   algorithm for polynomial factorization over finite fields. Its
   fundamental structure is similar to that of quicksort. By choosing a
   random polynomial, a polynomial is divided into two sub-factors with
   some probability. Then each subfactor is factorized recursively.  In
   the following program, one of the two sub-factors generated on a server
   is sent to another server and the other subfactor is factorized on the server
   itself.
   \begin{verbatim}
   /* factorization of F */
   /* E = degree of irreducible factors in F */
   def c_z(F,E,Level)
   {
     V = var(F); N = deg(F,V);
     if ( N == E ) return [F];
     M = field_order_ff(); K = idiv(N,E); L = [F];
     while ( 1 ) {
       W = monic_randpoly_ff(2*E,V);
       T = generic_pwrmod_ff(W,F,idiv(M^E-1,2));
       if ( !(W = T-1) ) continue;
       G = ugcd(F,W);
       if ( deg(G,V) && deg(G,V) < N ) {
         if ( Level >= LevelMax ) {
           /* everything is done on this server */
           L1 = c_z(G,E,Level+1);
           L2 = c_z(sdiv(F,G),E,Level+1);
         } else {
           /* launch a server if necessary */
           if ( Proc1 < 0 ) Proc1 = ox_launch();
           /* send a request with Level = Level+1 */
           ox_cmo_rpc(Proc1,"ox_c_z",lmptop(G),E,
               setmod_ff(),Level+1);
           /* the rest is done on this server */
           L2 = c_z(sdiv(F,G),E,Level+1);
           L1 = map(simp_ff,ox_pop_cmo(Proc1));
         }
         return append(L1,L2);
       }
     }
   }
   \end{verbatim}
   
   
   
   
   
   
   
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