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Annotation of OpenXM/doc/issac2000/homogeneous-network.tex, Revision 1.4

1.4     ! noro        1: % $OpenXM: OpenXM/doc/issac2000/homogeneous-network.tex,v 1.3 2000/01/07 06:27:55 noro Exp $
1.2       takayama    2:
                      3: \section{Applications}
1.3       noro        4: \subsection{Distributed computation with homogeneous servers}
1.2       takayama    5:
1.3       noro        6: OpenXM also aims at speedup by a distributed computation
                      7: with homogeneous servers. As the current specification of OpenXM does
                      8: not include communication between servers, one cannot expect
                      9: the maximal parallel speedup. However it is possible to execute
                     10: several types of distributed computation as follows.
                     11:
                     12: \subsubsection{Product of univariate polynomials}
                     13:
                     14: Shoup \cite{Shoup} showed that the product of univariate polynomials
                     15: with large degrees and large coefficients can be computed efficiently
                     16: by FFT over small finite fields and Chinese remainder theorem.
                     17: It can be easily parallelized:
                     18:
                     19: \begin{tabbing}
                     20: Input :\= $f_1, f_2 \in Z[x]$\\
                     21: \> such that $deg(f_1), deg(f_2) < 2^M$\\
                     22: Output : $f = f_1f_2 \bmod p$\\
                     23: $P \leftarrow$ \= $\{m_1,\cdots,m_N\}$ where $m_i$ is a prime, \\
                     24: \> $2^{M+1}|m_i-1$ and $m=\prod m_i $ is sufficiently large. \\
                     25: Separate $P$ into disjoint subsets $P_1, \cdots, P_L$.\\
                     26: for \= $j=1$ to $L$ $M_j \leftarrow \prod_{m_i\in P_j} m_i$\\
                     27: Compute $F_j$ such that $F_j \equiv f_1f_2 \bmod M_j$\\
                     28: \> and $F_j \equiv 0 \bmod m/M_j$ in parallel.\\
                     29: \> ($f_1, f_2$ are regarded as integral.\\
                     30: \> The product is computed by FFT.)\\
                     31: return $\phi_m(\sum F_j)$\\
                     32: (For $a \in Z$, $\phi_m(a) \in (-m/2,m/2)$ and $\phi_m(a)\equiv a \bmod m$)
                     33: \end{tabbing}
                     34:
                     35: Figure \ref{speedup}
                     36: shows the speedup factor under the above distributed computation
                     37: on {\tt Risa/Asir}. For each $n$, two polynomials of degree $n$
                     38: with 3000bit coefficients are generated and the product is computed.
                     39: The machine is Fujitsu AP3000,
                     40: a cluster of Sun connected with a high speed network and MPI over the
                     41: network is used to implement OpenXM.
                     42: \begin{figure}[htbp]
                     43: \epsfxsize=8.5cm
                     44: \epsffile{speedup.ps}
                     45: \caption{Speedup factor}
                     46: \label{speedup}
                     47: \end{figure}
                     48:
                     49: The task of a client is the generation and partition of $P$, sending
                     50: and receiving of polynomials and the synthesis of the result. If the
                     51: number of servers is $L$ and the inputs are fixed, then the time to
                     52: compute $F_j$ in parallel is proportional to $1/L$, whereas the time
                     53: for sending and receiving of polynomials is proportional to $L$
                     54: because we don't have the broadcast and the reduce
                     55: operations. Therefore the speedup is limited and the upper bound of
1.4     ! noro       56: the speedup factor depends on the ratio of
        !            57: the computational cost and the communication cost.
        !            58: Figure \ref{speedup} shows that
1.3       noro       59: the speedup is satisfactory if the degree is large and the number of
1.4     ! noro       60: servers is not large, say, up to 10 under the above envionment.
1.3       noro       61:
                     62: \subsubsection{Gr\"obner basis computation by various methods}
                     63:
                     64: Singular \cite{Singular} implements {\tt MP} interface for distributed
                     65: computation and a competitive Gr\"obner basis computation is
                     66: illustrated as an example of distributed computation.  However,
                     67: interruption has not implemented yet and the looser process have to be
                     68: killed explicitly. As stated in Section \ref{secsession} OpenXM
                     69: provides such a function and one can safely reset the server and
                     70: continue to use it.  Furthermore, if a client provides synchronous I/O
                     71: multiplexing by {\tt select()}, then a polling is not necessary.  The
                     72: following {\tt Risa/Asir} function computes a Gr\"obner basis by
                     73: starting the computations simultaneously from the homogenized input and
                     74: the input itself.  The client watches the streams by {\tt ox\_select()}
                     75: and The result which is returned first is taken. Then the remaining
                     76: server is reset.
                     77:
                     78: \begin{verbatim}
                     79: /* G:set of polys; V:list of variables */
                     80: /* O:type of order; P0,P1: id's of servers */
                     81: def dgr(G,V,O,P0,P1)
                     82: {
                     83:   P = [P0,P1]; /* server list */
                     84:   map(ox_reset,P); /* reset servers */
                     85:   /* P0 executes non-homogenized computation */
                     86:   ox_cmo_rpc(P0,"dp_gr_main",G,V,0,1,O);
                     87:   /* P1 executes homogenized computation */
                     88:   ox_cmo_rpc(P1,"dp_gr_main",G,V,1,1,O);
                     89:   map(ox_push_cmd,P,262); /* 262 = OX_popCMO */
                     90:   F = ox_select(P); /* wait for data */
                     91:   /* F[0] is a server's id which is ready */
                     92:   R = ox_get(F[0]);
                     93:   if ( F[0] == P0 ) {
                     94:     Win = "nonhomo"; Lose = P1;
                     95:   } else {
                     96:     Win = "homo"; Lose = P0;
                     97:   }
                     98:   ox_reset(Lose); /* reset the loser */
                     99:   return [Win,R];
                    100: }
                    101: \end{verbatim}