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Annotation of OpenXM/doc/issac2000/homogeneous-network.tex, Revision 1.6

1.6     ! takayama    1: % $OpenXM: OpenXM/doc/issac2000/homogeneous-network.tex,v 1.5 2000/01/15 00:20:45 takayama Exp $
1.2       takayama    2:
                      3: \section{Applications}
1.3       noro        4: \subsection{Distributed computation with homogeneous servers}
1.6     ! takayama    5: \label{section:homog}
1.2       takayama    6:
1.5       takayama    7: One of the aims of OpenXM is a parallel speedup by a distributed computation
1.3       noro        8: with homogeneous servers. As the current specification of OpenXM does
                      9: not include communication between servers, one cannot expect
                     10: the maximal parallel speedup. However it is possible to execute
                     11: several types of distributed computation as follows.
                     12:
                     13: \subsubsection{Product of univariate polynomials}
                     14:
                     15: Shoup \cite{Shoup} showed that the product of univariate polynomials
                     16: with large degrees and large coefficients can be computed efficiently
                     17: by FFT over small finite fields and Chinese remainder theorem.
                     18: It can be easily parallelized:
                     19:
                     20: \begin{tabbing}
                     21: Input :\= $f_1, f_2 \in Z[x]$\\
                     22: \> such that $deg(f_1), deg(f_2) < 2^M$\\
                     23: Output : $f = f_1f_2 \bmod p$\\
                     24: $P \leftarrow$ \= $\{m_1,\cdots,m_N\}$ where $m_i$ is a prime, \\
                     25: \> $2^{M+1}|m_i-1$ and $m=\prod m_i $ is sufficiently large. \\
                     26: Separate $P$ into disjoint subsets $P_1, \cdots, P_L$.\\
                     27: for \= $j=1$ to $L$ $M_j \leftarrow \prod_{m_i\in P_j} m_i$\\
                     28: Compute $F_j$ such that $F_j \equiv f_1f_2 \bmod M_j$\\
                     29: \> and $F_j \equiv 0 \bmod m/M_j$ in parallel.\\
                     30: \> ($f_1, f_2$ are regarded as integral.\\
                     31: \> The product is computed by FFT.)\\
                     32: return $\phi_m(\sum F_j)$\\
                     33: (For $a \in Z$, $\phi_m(a) \in (-m/2,m/2)$ and $\phi_m(a)\equiv a \bmod m$)
                     34: \end{tabbing}
                     35:
                     36: Figure \ref{speedup}
                     37: shows the speedup factor under the above distributed computation
                     38: on {\tt Risa/Asir}. For each $n$, two polynomials of degree $n$
                     39: with 3000bit coefficients are generated and the product is computed.
                     40: The machine is Fujitsu AP3000,
                     41: a cluster of Sun connected with a high speed network and MPI over the
                     42: network is used to implement OpenXM.
                     43: \begin{figure}[htbp]
                     44: \epsfxsize=8.5cm
                     45: \epsffile{speedup.ps}
                     46: \caption{Speedup factor}
                     47: \label{speedup}
                     48: \end{figure}
                     49:
                     50: The task of a client is the generation and partition of $P$, sending
                     51: and receiving of polynomials and the synthesis of the result. If the
1.5       takayama   52: number of servers is $L$ and the inputs are fixed, then the cost to
                     53: compute $F_j$ in parallel is $O(1/L)$, whereas the cost
                     54: to send and receive polynomials is $O(L)$
1.3       noro       55: because we don't have the broadcast and the reduce
                     56: operations. Therefore the speedup is limited and the upper bound of
1.4       noro       57: the speedup factor depends on the ratio of
                     58: the computational cost and the communication cost.
                     59: Figure \ref{speedup} shows that
1.5       takayama   60: the speedup is satisfactory if the degree is large and $L$
                     61: is not large, say, up to 10 under the above envionment.
                     62: If OpenXM provides the broadcast and the reduce operations, the cost of
                     63: sending $f_1$, $f_2$ and gathering $F_j$ may be reduced to $O(log_2L)$
                     64: and we will obtain better results in such a case.
1.3       noro       65:
1.5       takayama   66: \subsubsection{Competitive distributed computation by various strategies}
1.3       noro       67:
                     68: Singular \cite{Singular} implements {\tt MP} interface for distributed
                     69: computation and a competitive Gr\"obner basis computation is
1.5       takayama   70: illustrated as an example of distributed computation.
                     71: Such a distributed computation is also possible on OpenXM.
                     72: The following {\tt Risa/Asir} function computes a Gr\"obner basis by
1.3       noro       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}